**Meet the editor**

Dr. Ayse Irmak is a faculty member of the School of Natural Resources and Department of Civil Engineering at the University of Nebraska-Lincoln. Her M.S. and Ph.D. are from the University of Florida. Her research and teaching areas include hydrological information systems, geographical information systems (GIS) for water resources, remote sensing-based evapotranspiration and

other surface energy fluxes, remote sensing in agricultural and natural resources systems, ımpacts of land use/land cover on climate change, and simulation of crop production, soil water processes, and interactions with climate. She has received six journal paper awards and a number of presentation awards. Dr. Irmak's research has been funded by NASA, USDA, USGS and Nebraska agencies. She has published 44 articles in refereed journals and four book chapters. She teaches courses in GIS, remote sensing, data analysis, and surface water hydrology. Dr. Irmak is a member of the American Society of Agricultural and Biological Engineers, American Society of Civil Engineers-EWRI, United States Committee on Irrigation and Drainage, American Society of Agronomy, Soil Science Society of America, Soil and Water Conservation Society, and American Water Works Association.

## Contents

## **Preface XIII**


Tadanobu Nakayama


Chapter 20 **Possibilities of Deriving Crop Evapotranspiration from Satellite Data with the Integration with Other Sources of Information 437**  Gheorghe Stancalie and Argentina Nertan

VI Contents

Chapter 8 **Estimation of Evapotranspiration Using** 

Zoubeida Kebaili Bargaoui

**Shortage Conditions 197** 

Daniel Szejba

Georgeta Bandoc

Sungwon Kim

Yann Chemin

**Soil Water Balance Modelling 147** 

Chapter 9 **Evapotranspiration of Grasslands and Pastures in North-Eastern Part of Poland 179**

Chapter 10 **The Role of Evapotranspiration in the Framework of** 

Giuseppe Mendicino and Alfonso Senatore

**Water Resource Management and Planning Under** 

Chapter 11 **Guidelines for Remote Sensing of Evapotranspiration 227**  Christiaan van der Tol and Gabriel Norberto Parodi

Chapter 12 **Estimation of the Annual and Interannual Variation of** 

Chapter 13 **Evapotranspiration of Partially Vegetated Surfaces 273**

Chapter 15 **Critical Review of Methods for the Estimation of Actual Evapotranspiration in Hydrological Models 329** 

Chapter 16 **Development of Hybrid Method for the Modeling of Evaporation and Evapotranspiration 351** 

Chapter 17 **Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments 377** 

Giacomo Bertoldi, Riccardo Rigon and Ulrike Tappeiner

L.O. Lagos, G. Merino, D. Martin, S. Verma and A. Suyker

Martina Eiseltová, Jan Pokorný, Petra Hesslerová and Wilhelm Ripl

**Potential Evapotranspiration 251**

Chapter 14 **Evapotranspiration – A Driving Force in Landscape Sustainability 305** 

Nebo Jovanovic and Sumaya Israel

Chapter 18 **Stomatal Conductance Modeling to Estimate the Evapotranspiration of Natural and Agricultural Ecosystems 403** Giacomo Gerosa, Simone Mereu, Angelo Finco and Riccardo Marzuoli

Chapter 19 **A Distributed Benchmarking Framework for Actual ET Models 421** 


## Preface

The transfer of liquid water from soil to vapor in the atmosphere (Evapotranspiration) is one of the most profound and consequential processes on Earth. Evapotranspiration (ET), along with evaporation from open water, supplies vapor to the atmosphere to replace that condensed as precipitation. The flux of water through plants via transpiration transports minerals and nutrients required for plant growth and creates a beneficial cooling process to plant canopies in many climates. At the global scale, ET measures nearly one hundred trillion cubic meters per year and is the largest component of the hydrologic cycle, following precipitation. The large spatial variability in water consumption from land surfaces is strongly related to vegetation type, vegetation amount, soil water holding characteristics, and of course, precipitation or irrigation amount. There are very strong feedbacks from all of these factors and consequent ET rates. In this book, Evapotranspiration is defined as the aggregate sum of evaporation (E) direct from the soil surface and the surfaces of plant canopies and transpiration (T), where T is the evaporation of water from the plant system via the plant leaf, stem and root-soil system.

In addition to consuming enormous amounts of water, ET substantially modifies the Earth's energy balance through its consumption of enormous amounts of energy during conversion of liquid water to vapor. Each cubic meter of water evaporated requires 2.45 billion Joules of energy. That consumption of energy cools the evaporating surface and reduces the heating of air by the surface. On a global basis, the cooling effect to the land surface is measured in trillions of GigaJoules per day. Much of that 'latent' energy absorbed by ET later reenters the surface energy balance when the vapor recondenses as precipitation.

Even though the magnitude of ET is enormous over the Earth's surface, and even though ET has such high bearing on vegetation growth and health, its spatial distribution and magnitudes are poorly understood and poorly quantified. Although man has been able to estimate general magnitudes of ET via its strong correlation with precipitation for centuries, it has only been during the past thirty years, with the advent of satellites and remote sensing technologies, along with sophisticated modeling approaches, that we have been able to view and quantify the complex and variable geospatial structure of ET. The combination of thermally-equipped satellites, such as Landsat, AVHRR, MODIS and ASTER, and the improved ability to simulate the energy balance at the Earth's surface has enabled a substantial revolution in 'mapping' of ET over large, variable landscapes.

This edition of *Evapotranspiration* contains 23 chapters, covering a broad range of topics related to the modeling and simulation of ET, as well as to the remote sensing of ET. Both of these areas are at the forefront of technologies required to quantify the highly spatial ET from the Earth's surface. The chapters cover mechanics of ET simulation, including ET from partially vegetated surfaces and the modeling of stomatal conductance for natural and agricultural ecosystems, ET estimation using soil water balance, weather data and vegetation cover, ET estimation based on the Complementary Relationship, and adaptability of woody plants in conditions of soil aridity. Modeling descriptions include chapters focusing on distributed benchmarking frameworks for ET models, Hargreaves and other temperature-radiation based methods, Fuzzy-Probabilistic calculations, a hybrid-method for modeling evaporation and ET, and estimation of ET using water balance modeling. One chapter provides a critical review of methods for estimation of actual ET in hydrological models. In addition to that, six chapters describe modeling applications for determining ET patterns in alpine catchments, ET assessment and water resource management planning under shortage conditions, estimation of the annual and interannual variation of potential ET, impacts of irrigation on hydrologic change in a highly cultivated basin, ET of grasslands and pastures in north-eastern part of Poland, and climatological aspects of water balance components for Croatia.

Remote sensing based approaches are described in five chapters that include deriving crop ET from satellite data, integration with other information sources and an assessment of ET using MODIS products with energy balance algorithms. Importantly, the book includes two chapters describing an overview of recommended guidelines for operational remote sensing of ET, and a review of operational remote sensingbased energy balance models including SEBAL and METRIC, and specific challenges and insights for their application.

These 23 chapters represent the current state of the art in ET modeling and remote sensing applications, and provide valuable insights and experiences of developers and appliers of the technologies that have been gained over decades of development work, experimentation and modeling. This text provides valuable background information and theory for university students and courses on ET, as well as guidance and ideas for those that apply these modern methods. I wish to express my thanks to the authors of all chapters for making these timely and very useful contributions available, and to all anonymous reviewers of chapters. I also wish to thank Mr Baburao Kamble, University of Nebraska, for assistance in the handling of chapter manuscripts during reviews and for providing technical assistance.

**Dr. Ayse Irmak**

School of Natural Resources and Civil Engineering, Center for Advanced Land Management Information Technologies (CALMIT), University of Nebraska-Lincoln, USA

## **Assessment of Evapotranspiration in North Fluminense Region, Brazil, Using Modis Products and Sebal Algorithm**

José Carlos Mendonça1, Elias Fernandes de Sousa2, Romísio Geraldo Bouhid André3, Bernardo Barbosa da Silva4 and Nelson de Jesus Ferreira5 *1Laboratório de Meteorologia (LAMET/UENF). Rod. Amaral Peixoto, Av. Brennand s/n Imboassica, Macaé, RJ 2Laboratorio de Engenharia Agrícola (LEAG/UENF); Avenida Alberto Lamego, CCTA, sl 209, Parque Califórnia, Campos dos Goytacazes, RJ 3Instituto Nacional de Meteorologia (INMET/MAPA); Eixo Monumental, Via S1 – Sudoeste, Brasília, DF 4Departamento de Ciências Atmosféricas (DCA/UFCG); Avenida Aprígio Veloso, Bodocongó, Campina Grande, PB 5Centro de Previsão de Tempo e Estudos Climáticos (CPTEC/INPE); Av. dos Astronautas, Jardim da Granja, São José dos Campos, SP Brazil* 

## **1. Introduction**

North Fluminense Region, Rio de Janeiro State, Brazil (Fig. 1) is known as a sugar cane producer. The production during harvest season 2007/08 were 4 million tons of sugar cane, that were transformed into 4.8 million sacks of sugar, 36,786 liters anhydrous alcohol (ethanol) and 91,008 liters of hydrated alcohol. Economically generated 250 million U. S. dollars (Morgado, 2009). However, this activity is declining in the region due to different factors, including hidric deficit and the use of irrigation techniques may reverse this situation(Azevedo et al., 2002). Some authors (Ide e Oliveira, 1986; Magalhães, 1987) define temperature as a factor of greater importance for sugar cane physiology maturation (ripening) because more the affecting nutrients and water absorption through transpiration flux is a non-controllable condition. Soil humidity is another preponderant factor to sugar cane physiology and varies in function of the cultivation cycle, development stage, climactic conditions and others factors, such as spare water in the soil. The soil moisture content varies during the growth that corresponds to the main cause of production variation. However, the precipitation distribution along the year and spare soil water for the plant disposition are more important in the vegetative cycle of the sugar cane that total precipitation. (Magalhães, 1987).

The physical properties of energy exchange between the plant community and environment such as momentum, latent heat, sensible heat and others are evidenced by the influence they exert on physiological processes of plants and the occurrence of pests and diseases, which affect the productive potential of plants species exploited economically (Frota, 1978). The radiation components measurements of energy balance in field conditions have direct applicability in agricultural practices, especially in irrigation rational planning, appropriate use of land in regional agricultural zoning, weather variations impact on agricultural crops, protecting plants, among others. The knowledge advance in micro-scale weather, as well as the instrumental monitoring technology evolution has allowed a research increase in this area. Energy balance studies on a natural surface based on energy conservation principle. By accounting means for components that make up this balance, can be evaluate the net radiation plots used for the flow of sensible and latent heat.

The analysis of data collected by artificial satellites orbiting planet earth, allows the determination of various physical properties of planet, consequently, spatial and temporal modifications of different ecosystems are able to be identified.

According Moran et al. (1989), estimative of evepotranspiration – ET, based in data collected in meteorological stations have the limitation of representing punctual values that are capable of satisfactory representing local conditions but, if the objective is to obtain analysis of a regional variation of ET using a method with interpolation and extrapolation from micro-meteorological parameters of an specific area, these punctual data may increase the uncertainty of the analysis.

Trying to reduce such uncertainty degree, different algorithms were developed during the last decades to estimate surface energy flux based in the use of remote sensing techniques.

Bastiaanssen (1995) developed the 'Surface Energy Balance Algorithm for Land - SEBAL', with its validation performed in experimental campaigns in Spain and Egypt (arid climate) using Landsat 5 –TM images. This model involves the spatial variability of the most agrometeorological variables and can be applied to various ecosystems and requires spatial distributed visible, near-infrared and thermal infrared data together with routine weather data. The algorithm computes net radiation flux – Rn, sensible heat flux - H and soil heat flux - G for every pixel of a satellite image and latent heat flux - LE is acquired as a residual in energy balance equation (Equation 01). This is accomplished by firt computing the surface radiation balance, flowed by the surface energy balance. Althoygh SEBAL has been designed to calculate the energy partition at the regional scale with minimum ground data (Teixeira, 2008).

Roerink et al. (1997) also used Landsat 5 –TM images to evaluate irrigation's performance in Argentina and AVHRR/NOAA sensor images in Pakistan. Combination of Landsat 5 – TM and NOAA/AVHRR images were used by Timmermans and Meijerink (1999) in Africa. Latter, Hafeez et al. (2002) used the SEBAL algorithm with the ASTER sensor installed onboard 'Terra' satellite while studying Pumpanga river region in Philippines. These authors concluded that the combination of the high spatial resolution of ETM+ and ASTER sensors, together with the high temporal resolution from AVHRR and MODIS, provided high precision results of water balance and water use studies on regional scale.

In Brazil, several research center are conducting research using the SEBAL algorithm specially 'Federal University of Campina Grande, PB - UFCG', 'National Institute of Space Research - INPE' and others.

Sebal was developed and validated in arid locations and one of its peculiarities is the use of two anchors pixels (hot pixel – LE = 0 and cold pixel – H =0) with the determination or selection of hot pixel easier in dry climates. In humid and sub-humid climates is not easy determine a hot pixel, where the latent heat flux is zero or null.

The objectives of the research described in this work are (i) to evaluate two propositions to estimate the sensible heat flux (H) and (ii) to evaluate two methods for conversion of ETinst values to ET24h on the daily evepotranspiration to estimate evepotranspiration in regional scale using SEBAL algorithm, MODIS images, the two propositions to estimate H and meteorological data of the four surface meteorological stations.

## **2. Materials and methods**

## **2.1 Study area**

2 Evapotranspiration – Remote Sensing and Modeling

exert on physiological processes of plants and the occurrence of pests and diseases, which affect the productive potential of plants species exploited economically (Frota, 1978). The radiation components measurements of energy balance in field conditions have direct applicability in agricultural practices, especially in irrigation rational planning, appropriate use of land in regional agricultural zoning, weather variations impact on agricultural crops, protecting plants, among others. The knowledge advance in micro-scale weather, as well as the instrumental monitoring technology evolution has allowed a research increase in this area. Energy balance studies on a natural surface based on energy conservation principle. By accounting means for components that make up this balance, can be evaluate the net

The analysis of data collected by artificial satellites orbiting planet earth, allows the determination of various physical properties of planet, consequently, spatial and temporal

According Moran et al. (1989), estimative of evepotranspiration – ET, based in data collected in meteorological stations have the limitation of representing punctual values that are capable of satisfactory representing local conditions but, if the objective is to obtain analysis of a regional variation of ET using a method with interpolation and extrapolation from micro-meteorological parameters of an specific area, these punctual data may increase the

Trying to reduce such uncertainty degree, different algorithms were developed during the last decades to estimate surface energy flux based in the use of remote sensing techniques. Bastiaanssen (1995) developed the 'Surface Energy Balance Algorithm for Land - SEBAL', with its validation performed in experimental campaigns in Spain and Egypt (arid climate) using Landsat 5 –TM images. This model involves the spatial variability of the most agrometeorological variables and can be applied to various ecosystems and requires spatial distributed visible, near-infrared and thermal infrared data together with routine weather data. The algorithm computes net radiation flux – Rn, sensible heat flux - H and soil heat flux - G for every pixel of a satellite image and latent heat flux - LE is acquired as a residual in energy balance equation (Equation 01). This is accomplished by firt computing the surface radiation balance, flowed by the surface energy balance. Althoygh SEBAL has been designed to calculate the energy partition at the regional scale with minimum ground data

Roerink et al. (1997) also used Landsat 5 –TM images to evaluate irrigation's performance in Argentina and AVHRR/NOAA sensor images in Pakistan. Combination of Landsat 5 – TM and NOAA/AVHRR images were used by Timmermans and Meijerink (1999) in Africa. Latter, Hafeez et al. (2002) used the SEBAL algorithm with the ASTER sensor installed onboard 'Terra' satellite while studying Pumpanga river region in Philippines. These authors concluded that the combination of the high spatial resolution of ETM+ and ASTER sensors, together with the high temporal resolution from AVHRR and MODIS, provided

In Brazil, several research center are conducting research using the SEBAL algorithm specially 'Federal University of Campina Grande, PB - UFCG', 'National Institute of Space

Sebal was developed and validated in arid locations and one of its peculiarities is the use of two anchors pixels (hot pixel – LE = 0 and cold pixel – H =0) with the determination or

high precision results of water balance and water use studies on regional scale.

radiation plots used for the flow of sensible and latent heat.

modifications of different ecosystems are able to be identified.

uncertainty of the analysis.

(Teixeira, 2008).

Research - INPE' and others.

The Norte Fluminense region in Rio de Janeiro State, Brazil, has an area of 9.755,1 km2, corresponding to 22% of the state's total area. Among its agricultural production, sugar cane plantations are predominant as well as cattle production. In the last years irrigation technologies for fruit production are being promoted and implemented by the government. Nowadays, passion fruit, guava, coconut and pineapple plantations extend for more than 4.000 ha (SEAAPI, 2006).

According Koppen, this region's clime is classified as Aw, that is, tropical humid with rainy summers, dry winters and temperatures average above 18 oC during the coolest months. The annual mean temperatures are of 24oC, with a little thermal amplitude and mean rain precipitation values of 1.023 mm (Gomes, 1999).

The area under study is showed in Figure 1, comparing the area of the Norte Fluminense region within the Rio de Janeiro state and the RJ state within Brazil.

Fig. 1. Study area localization.

## **2.2 Digital orbital images – MODIS images**

Daily MOD09 and MYD09 data (Surface Reflectance – GHK / 500 m and GQK / 250 m) and MOD11A1 and MYD11A1 data (Surface Temperature - LST) were used in this research, totalizing 24 scenes over the 'tile' h14/v11 corresponding to Julian Day 218th, 227th, 230th, 241st, 255th, 285th, 320th and 339th in 2005 and 15th, 36th, 63rd , 102nd, 116th, 139th, 166th, 186th, 189th, 190th, 191st, 200th, 201st, 205th, 208th and 221st in 2006. These days were selected because no cloud covering was registered over the study area during the satellite's course over the area were obtained from the Land Processes Distributed Active Archive Center (LP-DAAC), of the National Aeronautics and Space Administration (NASA), at http://edcimswww.cr.usgs.gov/pub/imswelcome/.

The GHK – 500 m (Blue, Green, Red, Nir, Mir, Fir, Xir) reflectance band were resampled fron 500 m to 250 m. The Red and Nir bands were excluded and GQK (250 m) bands included. This operation aimed to input the value of the red and nir bands in the algorithm. The LST bands were also resampled from 1000 m to 250 m.

The software Erdas Image – Pro, version 8.7 was used for the piles, compositions, clippings and algebra. The Model Maker tool was used to application of the algorithm and the thematic maps were produced using the software ArcGis 9.0.

## **2.3 Meteorological data**

Surface data were collected in two micro-meteorological stations from the Universidade Estadual do Norte Fluminense – UENF, installed over agricultural areas cultivated with sugar cane (geographical coordinates: 21º 43' 21,8" S and 41º 24' 26,1" W), and 'dwarf green' coconut irrigated (geographical coordinates: 21º 48' 31,2" S and 41º 10' 46,2" W).

The micrometeorological stations installed in both areas (sugar cane and coconut) were equipped with the following sensor: 1 Net radiometer NR Lite (Kipp and Zonen), 2 Piranometer LI 200 (Li-Cor), 2 Probe HMP45C-L (Vaissala), 2 Met One Anemometer (RN Yong) and 3 HFP01SC\_L Soil Healt Flux Plat (Hukseflux). All data from were collected every minute and average values extracted and stores every 15 min in a datalogger CR21X (Sugar cane) and CR 1000 (coconut). Both dataloggers are Campbell Scientific's (USA). The horizontal bars were placed 0.50 m above crop canopy (first level) and 2.0 m between the first and second bars. This standard was maintained all crop cycle and bars relocated where necessary (sugar cane station). In coconut station the relocated was not necessary.

These stations were installed in the center of an area of 5,000 hectare (sugar cane – Santa Cruz Agroindustry) 256 hectare (coconut – Agriculture Taí).

Fig. 2. Localization of the surface micro-meteorological and meteorological stations installed in the study area.

The GHK – 500 m (Blue, Green, Red, Nir, Mir, Fir, Xir) reflectance band were resampled fron 500 m to 250 m. The Red and Nir bands were excluded and GQK (250 m) bands included. This operation aimed to input the value of the red and nir bands in the algorithm. The LST

The software Erdas Image – Pro, version 8.7 was used for the piles, compositions, clippings and algebra. The Model Maker tool was used to application of the algorithm and the

Surface data were collected in two micro-meteorological stations from the Universidade Estadual do Norte Fluminense – UENF, installed over agricultural areas cultivated with sugar cane (geographical coordinates: 21º 43' 21,8" S and 41º 24' 26,1" W), and 'dwarf green'

The micrometeorological stations installed in both areas (sugar cane and coconut) were equipped with the following sensor: 1 Net radiometer NR Lite (Kipp and Zonen), 2 Piranometer LI 200 (Li-Cor), 2 Probe HMP45C-L (Vaissala), 2 Met One Anemometer (RN Yong) and 3 HFP01SC\_L Soil Healt Flux Plat (Hukseflux). All data from were collected every minute and average values extracted and stores every 15 min in a datalogger CR21X (Sugar cane) and CR 1000 (coconut). Both dataloggers are Campbell Scientific's (USA). The horizontal bars were placed 0.50 m above crop canopy (first level) and 2.0 m between the first and second bars. This standard was maintained all crop cycle and bars relocated where

These stations were installed in the center of an area of 5,000 hectare (sugar cane – Santa

Fig. 2. Localization of the surface micro-meteorological and meteorological stations installed

coconut irrigated (geographical coordinates: 21º 48' 31,2" S and 41º 10' 46,2" W).

necessary (sugar cane station). In coconut station the relocated was not necessary.

bands were also resampled from 1000 m to 250 m.

**2.3 Meteorological data** 

in the study area.

thematic maps were produced using the software ArcGis 9.0.

Cruz Agroindustry) 256 hectare (coconut – Agriculture Taí).

The meteorological stations, both installed on grass (*Paspalum Notatum L.*) are property of research center. The Thies Clima model (Germany) installed at the UENF's Evapotranspiration Station – Pesagro Research Center, (geographical coordinates: 21º 24' 48" S and 41º 44' 48" W) is an automatic station. Is equipped with 1 Anemometer, 1 Barometer, 1 Termohygrometer, 1 Piranometer and 1 Pluviometer. All sensor are connected to a datalogger model DL 12 – V. 2.00 – Thies Clima, recording values every minute and stored an average every 10 minutes.

The Agrosystem model install at the Meteorological Station of the Experimental Campus 'Dr. Leonel Miranda' – UFRRJ, (geographical coordinates: 21º 17' 36" S and 41º 48' 09" W) contains 1 Anemometer, 1 Barometer, 1 Termohygrometer, 1 Piranometer and 1 Pluviometer and recording values every minute and stored an average every 10 minutes.

All geographical coordinates are related to Datum WGS 84 – zone 24, with average altitude of 11 m. The localization of the surface stations, where meteorological data used in this study were collected are showed in Figure 2.

## **2.4 Real evapotranspiration estimation with SEBAL**

To calculate surface radiation balance was used the Model Maker tool from the software Erdas Image 8.6. The estimations of the incident solar radiation and the long wave radiation emitted by the atmosphere to the surface were performed in electronic sheet.

To better understand the different phases of the Sebal algorithm using Modis products, a general diagram of the computational routines are shown in Figure 3.

Fig. 3. Diagram of the computational routines for determination of the Surface Energy Balance using SEBAL, form MODIS products. (Modified from Trezza (2002).

A schematic diagram for the estimation of the surface radiation balance (Rn), adapted to MODIS images is showed in Figure 4.

Fig. 4. Diagram showing the process steps of the surface radiation balance adapted for MODIS images.

Detailed processes, as well as the equations for the SEBAL algorithm development, may be obtained in Bastiaanssen et al. (1998). In the present work two propositions were assumed to select the anchor pixels, the first was similar to the one used by Bastiaanssen (1995), with the selection of two pixels with external temperatures (hot pixel/LE = 0 and cool pixel/H = 0). The hot pixel always comprising an area of exposed soil with little vegetation and the cool pixel localized in the interior of a great extension water body. The first proposition was called as 'H\_Classic'.

With the hypothesis that the linear relation dT = a + d.Ts would be better represented with the selection of a hot pixel with its energy balance components previously known, specially the sensible heat flux (H) and in regions of humid and sub-humid climate be difficult identifying de hot pixels, which can hardly meet the condition of being dry, or have LE = 0, the second hypothesis was formulated. The criterion used for the selection of the cool pixel was the same as in the first hypothesis, that is, to be localized inside a water body of a great extension, but the selection of the hot pixel, where determination of the H values estimate as residue of the Penman-Monteih FAO56 equation using meteorological data from installed at the UENF's Evapotranspiration Station – Pesagro Research Center. This second hypothesis was called 'H\_Pesagro'.

#### **2.5 Latent heat flux (** *L*E **)**

Latent heat flux (vapor transference to the atmosphere trough the process of vegetal transpiration and soil water evaporation) was computed by the simple difference between the radiation balance cards, soil heat flux and sensible heat flux:

$$LE = \mathbf{R}\mathbf{n} - \mathbf{G} - \mathbf{H} \tag{1}$$

Fig. 4. Diagram showing the process steps of the surface radiation balance adapted for

Detailed processes, as well as the equations for the SEBAL algorithm development, may be obtained in Bastiaanssen et al. (1998). In the present work two propositions were assumed to select the anchor pixels, the first was similar to the one used by Bastiaanssen (1995), with the selection of two pixels with external temperatures (hot pixel/LE = 0 and cool pixel/H = 0). The hot pixel always comprising an area of exposed soil with little vegetation and the cool pixel localized in the interior of a great extension water body. The first proposition was

With the hypothesis that the linear relation dT = a + d.Ts would be better represented with the selection of a hot pixel with its energy balance components previously known, specially the sensible heat flux (H) and in regions of humid and sub-humid climate be difficult identifying de hot pixels, which can hardly meet the condition of being dry, or have LE = 0, the second hypothesis was formulated. The criterion used for the selection of the cool pixel was the same as in the first hypothesis, that is, to be localized inside a water body of a great extension, but the selection of the hot pixel, where determination of the H values estimate as residue of the Penman-Monteih FAO56 equation using meteorological data from installed at the UENF's Evapotranspiration Station – Pesagro Research Center. This second hypothesis

Latent heat flux (vapor transference to the atmosphere trough the process of vegetal transpiration and soil water evaporation) was computed by the simple difference between

E Rn G H *L* (1)

the radiation balance cards, soil heat flux and sensible heat flux:

MODIS images.

called as 'H\_Classic'.

was called 'H\_Pesagro'.

**2.5 Latent heat flux (** *L*E **)** 

where: *L*E represents the latent heat flux, Rn is the radiation balance and G is the soil heat flux, all expressed in W m-2 and obtained during the course of the satellite over the study area.

The value of the instantaneously latent heat flux ( *L*Einst ), integrated at the time (hour) of the satellites passage (mm h-1) is:

$$\mathrm{L\,E}\_{inst} = \text{3600} \frac{\mathrm{L\,E}}{\lambda} \tag{2}$$

where: *L*E *inst* is the value of instantaneously ET, expressed in mm h-1; *L*E is the latent heat flux at the moment of the sensor's course and λ is the water vaporization latent heat, expressed by the equation:

$$\text{Ca} = 2,501-0,00236 \text{ (Ts}-273,16) \text{\*} 10^6 \tag{3}$$

where: Ts is the surface temperature chart (oC) obtained by the product MOD11A1 (K). With the radiation balance, soil heat flux and latent heat flux charts, the evaporative fraction was obtained and expressed by the equation:

$$\Lambda = \frac{L \,\text{ET}}{\text{R }\text{n} - \text{G}} \tag{4}$$

The evaporative fraction has an important characteristic, it regularity and constancy in clear sky days. In this sense, we can admit that its instantaneously character represents its diurnal mean value satisfactorily, enabling the estimation of daily evapotranspiration by the equation:

$$ET\_{24h} = \frac{86400 \text{ \AA} \text{Ru}\_{24h}}{\text{\AA}}\tag{5}$$

where: Rn24h, is the mean radiation balance occurred during a period of 24 h, expressed in W.m-2, obtained by the equation:

$$\mathrm{Rn}\_{24h} = (1 - \alpha) \,\mathrm{Rs} \, 24h - 110 \,\tau\_{sw} \, 24h \tag{6}$$

where: α, is the surface albedo; Rs24h, is the daily mean radiation of short incident wave expressed in W m-2 and 24 *sw h* , is the mean daily atmospheric transmissivity.

To determine Rs24h values, an approximation similar to the method proposed by Lagouarde and Brunet (1983) for the estimation of diurnal cycles of Rn and Rs↓ in clear sky days, was used. With the values of Rn24h, Rs24h and the surface albedo, extracted from the PESAGRO pixel, a linear regression between these values was performed to obtain a regression equation, its coefficients a1 and b1 and then to calculate the Rn24h chart as a function of the short wave balance. To determine the linear regression the following equation was used:

$$R n \, 24h = a\_1 (1 - \alpha) \, ^\ast R \, 24h + b\_1 \, \tag{7}$$

Allen et al. (2002) defined the evaporative fraction of reference (ETrF) as the relation between the ETinst chart and the ETo integrated at the same moment and computed with data obtained from a meteorological station, that is:

$$ETrF = \frac{ET\_{inst}}{ET\_{FAO56}}\tag{8}$$

This procedure generates a type of hourly-cultive coefficient (kc\_h), admitting that this relation represents the daily relation expressed by the equation:

$$\text{Kc\\_h} = \frac{\text{ETinst}}{\text{ETolv}} = \frac{\text{ET24}}{\text{ETo24}} \tag{9}$$

Admitting the relation represented in equation 09 it is possible to obtain the ET24h expressed in mm day-1 from the equation:

$$ET\_{24h} = ETrF \, ^\ast ETo\_{24} \tag{10}$$

In the present work, four values of ET24hSEBAL were estimated for the same day, applying equations 5 and 10 to the 'H\_Classic' and H\_Pesagro' propositions.

#### **3. Results and discusion**

#### **3.1 Daily evapotranspiration (ET24h)**

#### **3.1.1 Determination of Rn24h values**

To determine Rn24h charts, an adaptation proposed by Ataide (2006) for the sinusoidal model estimator of the cycle of radiation balance for clear sky days, based in an approximation similar to the Lagourade and Brunet (1983) method, was adopted.

Looking forward for reliability and applicability in the generation of the Rn24h charts form values of Rs↓24h, a linear regression between the short wave balance and the daily radiation balance was performed, where the regression equation coefficients were determined as *a* = 0,9111 and *b =* -23,918.

The coefficients obtained (*a* and *b*) are next to the values found by Alados et al. (2003), whit values of *a* = 0,709 and *b* = -25,4 where values of global solar radiation (Rg) and not short wave balance (BOC) were used in the linear regression, thus excluding the effect of the surface albedo in the calculation. Considering that values of Rg were determined in a standard meteorological station, installed on a grass field, with values of albedo varying between 20 and 25 %, the coefficients determined by the linear regression between values of BOC and Rn24h tent to be in agreement with the values mentioned by Alados et al. (2003).

Thus, the radiation balance for the daily period (Rn24h) was ultimately determined for each pixel of the study scene by the equation:

$$\text{Rp}\_{24h} = 0.9111^{\circ} \text{ (1 - chart of albedo)}^{\circ} \text{Rs} |24h \text{ -} 23.918 \tag{11}$$

#### **3.1.2 Determination of the ET24h values**

Based on charts of Rn, G, H, LE, Ts and α and values of ETo24h and EToinst, estimated from data observed at Pesagro's meteorological station, four values of ET24h were estimated for each scene studied: ET24h\_'Classic' w/ETrF; ET24h\_'Classic' w/Rn24h; ET24h\_'H\_Pesagro' w/ETrf and ET24h\_'H\_Pesagro' w/Rn24h.

Mean, maximum and minimum values obtained in charts of daily evapotranspiration (ET24h) estimated with the 'H\_Classic' proposition and expressed in mm day-1, are showed in Table 1.

*ET ETrF*

relation represents the daily relation expressed by the equation:

equations 5 and 10 to the 'H\_Classic' and H\_Pesagro' propositions.

in mm day-1 from the equation:

**3. Results and discusion** 

0,9111 and *b =* -23,918.

in Table 1.

**3.1 Daily evapotranspiration (ET24h) 3.1.1 Determination of Rn24h values** 

pixel of the study scene by the equation:

**3.1.2 Determination of the ET24h values** 

w/ETrf and ET24h\_'H\_Pesagro' w/Rn24h.

 *inst FAO*

This procedure generates a type of hourly-cultive coefficient (kc\_h), admitting that this

Admitting the relation represented in equation 09 it is possible to obtain the ET24h expressed

In the present work, four values of ET24hSEBAL were estimated for the same day, applying

To determine Rn24h charts, an adaptation proposed by Ataide (2006) for the sinusoidal model estimator of the cycle of radiation balance for clear sky days, based in an

Looking forward for reliability and applicability in the generation of the Rn24h charts form values of Rs↓24h, a linear regression between the short wave balance and the daily radiation balance was performed, where the regression equation coefficients were determined as *a* =

The coefficients obtained (*a* and *b*) are next to the values found by Alados et al. (2003), whit values of *a* = 0,709 and *b* = -25,4 where values of global solar radiation (Rg) and not short wave balance (BOC) were used in the linear regression, thus excluding the effect of the surface albedo in the calculation. Considering that values of Rg were determined in a standard meteorological station, installed on a grass field, with values of albedo varying between 20 and 25 %, the coefficients determined by the linear regression between values of BOC and Rn24h tent to be in agreement with the values mentioned by Alados et al. (2003). Thus, the radiation balance for the daily period (Rn24h) was ultimately determined for each

Rn24h = 0,9111\* (1 – chart of albedo) \* Rs↓24h -23,918 (11)

Based on charts of Rn, G, H, LE, Ts and α and values of ETo24h and EToinst, estimated from data observed at Pesagro's meteorological station, four values of ET24h were estimated for each scene studied: ET24h\_'Classic' w/ETrF; ET24h\_'Classic' w/Rn24h; ET24h\_'H\_Pesagro'

Mean, maximum and minimum values obtained in charts of daily evapotranspiration (ET24h) estimated with the 'H\_Classic' proposition and expressed in mm day-1, are showed

approximation similar to the Lagourade and Brunet (1983) method, was adopted.

56

*ET* (8)

*ETinst ET24 Kc\_h = = EToh ETo24* (9)

*ET ETrF \* ETo* <sup>24</sup>*<sup>h</sup>* 24 (10)


Table 1. Statistical data of daily evapotranspiration charts (ET24h) of the study area using the 'H\_Classic' proposition w/ Rn24h and w/ ETr\_F, in mm day-1.

Average mean data showed in Table 1 are similar, with a slight superiority for the values estimated by the method using Rn24h for the ET estimative. Minimum values for ETr\_F have negative values. Tasumi et al. (2003), using SEBAL in Idaho, U.S.A., also observed negative values for ET and attributed such results to systematic errors caused by diverse parameterizations used during the process of energy balance estimation.

Average mean, maximum and minimum values obtained in charts of daily evapotranspiration (ET24h) estimated with the "H\_Pesagro' proposition, expressed in mm day-1, are showed in Table 2.


Table 2. Statistical data of daily evapotranspiration charts (ET 24h) of the study area using the 'H\_Pesagro' proposition w/ Rn 24hs and w/ ETr\_F, in mm day-1.

Average mean values of the same magnitude order and with a slight superiority to values estimated using Rn24h are obse4rved in Table 2. In a general way, by the use of the 'Classic' proposal as well as by 'Pesagro' proposal, a higher amplitude of the estimated values is observed when using the method of ETr\_F.

Values of ET 24h\_SEBAL, observed in pixels where the micro-meteorological and meteorological stations were located (pixels from Pesagro, UFFRJ, Sugar-cane and Coconut), were correlated with values of ETo estimated by the equation of Penman-Monteith\_FAO (ETo PM\_FAO56) with data observed in Pesagro Station. Figures 5, 6, 7 and 8 show graphical representations of the regression analysis, the adjustment equation and the correlation coefficient (R2), obtained among the values estimated by SEBAL for all four methods used.

Table 2. Statistical data of daily evapotranspiration charts (ET 24h) of the study area using

Average mean values of the same magnitude order and with a slight superiority to values estimated using Rn24h are obse4rved in Table 2. In a general way, by the use of the 'Classic' proposal as well as by 'Pesagro' proposal, a higher amplitude of the estimated values is

Values of ET 24h\_SEBAL, observed in pixels where the micro-meteorological and meteorological stations were located (pixels from Pesagro, UFFRJ, Sugar-cane and Coconut), were correlated with values of ETo estimated by the equation of Penman-Monteith\_FAO (ETo PM\_FAO56) with data observed in Pesagro Station. Figures 5, 6, 7 and 8 show graphical representations of the regression analysis, the adjustment equation and the correlation coefficient (R2), obtained among the values estimated by SEBAL for all four

the 'H\_Pesagro' proposition w/ Rn 24hs and w/ ETr\_F, in mm day-1.

observed when using the method of ETr\_F.

methods used.

Fig. 5. Correlation between values of ET24h estimated with the method FAO (PM\_FAO56) with data collected at PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel from Pesagro, expressed in mm day-1.

Fig. 6. Correlation between values of ET24h estimated with the method FAO (PM\_FAO56) with data collected in PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel pixel from UFRRJ, expressed in mm day-1.

Fig. 6. Correlation between values of ET24h estimated with the method FAO (PM\_FAO56) with data collected in PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel pixel from UFRRJ, expressed in mm

day-1.

Fig. 7. Correlation between values of ET24h estimated by the method FAO (PM\_FAO56) with data collected from PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel from Sugar-cane (SANTA CRUZ **AGROINDUSTRY**), expressed in mm day-1.

Fig. 8. Correlation between values of ET24h estimated by the method FAO (PM\_FAO56) with data collected from PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel from Coconut (**AGRICULTURE TAÍ**) expressed in mm day-1.

Observing Figures 5, 6, 7, and 8, it is possible to conclude that the proposition 'H\_Classic' under estimated values projected by PM\_FAO56 method, showing better results for values estimated using Rn24h.

Proposition 'H\_Pesagro', although in a slight way, super estimated values of the ETo estimated with data from the meteorological station Pesagro, in all four control points, showing higher correction coefficients than the others with emphasis for the method using Rn24h.

Hafeez et al. (2002) applied SEBAL using MODIS images in Philippines and observed that the ET\_SEBAL super estimated in 13,5 % the values of ETo estimated by PM\_FAO56, justifying such behavior due to the spatial resolution of 1.000 m of the surface temperature chart (MOD11A1).

Fig. 8. Correlation between values of ET24h estimated by the method FAO (PM\_FAO56) with data collected from PESAGRO station and values of ET24h estimated by SEBAL with propositions "H\_Classic" w/Rn24h (A), "H\_Classic" w/ETr\_F (B), "H\_Pesagro" w/Rn24h (C) and "H\_Pesagro" w/ETr\_F (D) observed in pixel from Coconut (**AGRICULTURE TAÍ**)

Observing Figures 5, 6, 7, and 8, it is possible to conclude that the proposition 'H\_Classic' under estimated values projected by PM\_FAO56 method, showing better results for values

Proposition 'H\_Pesagro', although in a slight way, super estimated values of the ETo estimated with data from the meteorological station Pesagro, in all four control points, showing higher correction coefficients than the others with emphasis for the method using

Hafeez et al. (2002) applied SEBAL using MODIS images in Philippines and observed that the ET\_SEBAL super estimated in 13,5 % the values of ETo estimated by PM\_FAO56, justifying such behavior due to the spatial resolution of 1.000 m of the surface temperature

expressed in mm day-1.

estimated using Rn24h.

chart (MOD11A1).

Rn24h.

Fig. 9. Images of the daily evapotranspiration for the dry period in the Fluminense North Region, Rio de Janeiro State. DJ 2005218.

Fig. 10. Images of the daily evapotranspiration for the humid period in the Fluminense North Region, Rio de Janeiro State. DJ 2006015.

Allen et al. (2001), using images of LANDSAT in the basin of river Bear, North-East region of the U.S.A., observed that SEBAL showed a good precision for the estimation of ET, compared with weighing lysimeters, super estimating monthly mean values in 16% and 4 % for seasonal values.

Images of the daily evapotranspiration for the dry and humid periods in the Fluminense North Region, Rio de Janeiro State is showed in Figures 9 (DJ 2005218 ) and 10. (DJ 2006015).

## **4. Conclusion**

In accordance with the proposed objectives in this work, it is possible to conclude that in conditions de sub-humid climate: For the estimative of sensible heath flux, the use of proposition 'H\_Pesagro' resulted more efficient than 'H\_Classic'; The method that uses values of mean radiation balance integrated in 24 hours (Rn24h) is more consistent than the method that uses the reference evaporative fraction (ETr\_F) for the conversion of instantaneous evapotranspiration values (ETinst) in daily values (ET24h).

## **5. Acknowledgements**

The authors are grateful for the National Counsel for Scientific and Technological Development – CNPq and the Coordenação *de Aperfeiçoamento de Pessoal de Nível Superior – CAPES*, for the financial support and logistics that made this study possible.

## **6. References**


Allen et al. (2001), using images of LANDSAT in the basin of river Bear, North-East region of the U.S.A., observed that SEBAL showed a good precision for the estimation of ET, compared with weighing lysimeters, super estimating monthly mean values in 16% and 4 %

Images of the daily evapotranspiration for the dry and humid periods in the Fluminense North Region, Rio de Janeiro State is showed in Figures 9 (DJ 2005218 ) and 10. (DJ 2006015).

In accordance with the proposed objectives in this work, it is possible to conclude that in conditions de sub-humid climate: For the estimative of sensible heath flux, the use of proposition 'H\_Pesagro' resulted more efficient than 'H\_Classic'; The method that uses values of mean radiation balance integrated in 24 hours (Rn24h) is more consistent than the method that uses the reference evaporative fraction (ETr\_F) for the conversion of

The authors are grateful for the National Counsel for Scientific and Technological Development – CNPq and the Coordenação *de Aperfeiçoamento de Pessoal de Nível Superior –* 

Alados, C.L.; Pueyo, Y.; Giner, M.L.; Navarro, T.; Escos, J.; Barroso, F.; Cabezudo, B.; Emlen,

Allen, R.G.; Tasumi, M.; Trezza, R.; Bastiaanssen, W.G.M., 2002. SEBAL - Surface Energy

Ataíde, K.R.P., 2006. Determinação do saldo de radiação e radiação solar global com

Bastiaanssen, W.G.M., 1995. Regionalization of surface flux densities and moisture

Bastiaanssen, W.G.M.; Pelgrum, H.; Wang, J.; Ma, Y.; Moreno, J.; Roerink, G. J.; van der Val,

fractal analysis of plant spatial patterns. Ecological Modell. v.163, p.1-17. Allen, R. G.; Pereira, L. S.; Raes, D.; Smith, M., 1998. Crop evapotranspiration – Guidelines

G.M., 2003. Quantitative characterization of the regressive ecological succession by

for computing crop water requeriments. FAO Irrigation and Drainage Paper 56,

Balance Algorithms for Land. Advanced training and users manual, Version *1.0*.

produtos do sensor MODIS Terra e Aqua. Tese (Mestrado em Meteorologia) - Campina Grande, PB - Universidade Federal de Campina Grande – UFCG, 88p. Azevedo, H.J.; Silva Neto, R.; Carvalho, A. M.; Viana, J.L.; Mansur, A.F.U., 2002. Uma

análise da cadeia produtiva da cana-de-açúcar na Região Norte Fluminense. Observatório sócio-econômico da Região Norte Fluminense – Boletim Técnico nº 6,

indicators in composite terrain. Ph,D Thesis, Wageningen Agricultural University,

T., 1998. A remote sensing surface energy balance algorithm for land (SEBAL):Part

instantaneous evapotranspiration values (ETinst) in daily values (ET24h).

*CAPES*, for the financial support and logistics that made this study possible.

for seasonal values.

**4. Conclusion** 

**5. Acknowledgements** 

Rome, Italy, 318 p.

University of Idaho, EUA. 97 p.

Wageningen, The Netherlands. 273p.

2 validation, Journal of Hidrology, v, 212-213: 213-229.

**6. References** 

51p.


http://www.infoagro.ucam-campos.br/agro\_in\_rio.htm. (Accessed on 25 June/2010).


## **Evapotranspiration Estimation Based on the Complementary Relationships**

Virginia Venturini1, Carlos Krepper1,2 and Leticia Rodriguez1 *1Centro de Estudios Hidro-Ambientales-Facultad de Ingeniería y Ciencias Hídricas Universidad Nacional del Litoral 2Consejo Nacional de Investigaciones Científicas y Técnicas Argentina* 

## **1. Introduction**

18 Evapotranspiration – Remote Sensing and Modeling

Timmermans, W.J.; Meijerink, A.M.J., 1999. Remotely sensed actual evapotranspiration:

Trezza, R., 2002. Evapotranspiration using a satellite-based energy balance with sandarized ground control. PhD Dissertation. Utah State University. Logan. USA. 247p.

Geohydrology. 1:222-233.

implications for groundwater management in Botzwana. Journal of Applied

Many hydrologic modeling and agricultural management applications require accurate estimates of the actual evapotranspiration (ET), the relative evaporation (F) and the evaporative fraction (EF). In this chapter, we define ET as the actual amount of water that is removed from a surface due to the processes of evaporation-transpiration whilst the potential evapotranspiration (Epot) is any other evaporation concept. There are as many potential concepts as developed mathematical formulations. In this chapter, F represents the ratio between ET and Epot, as it was introduced by Granger & Gray (1989). Meanwhile, EF is the ratio of latent flux over available energy.

It is worthy to note that, in general, the available evapotranspiration concepts and models involve three sets of variables, i.e. available net radiation (Rn), atmospheric water vapor content or temperature and the surface humidity. Hence, different Epot formulations were derived with one or two of those sets of variables. For instance, Penman (1948) established an equation by using the Rn and the air water vapor pressure. Priestley & Taylor (1972) derived their formulations with only the available Rn.

In the last three decades, several models have been developed to estimate ET for a wide range of spatial and temporal scales provided by remote sensing data. The methods could be categorized as proposed by Courault et al. (2005).

*Empirical and semi-empirical methods*: These methods use site specific or semi-empirical relationships between two o more variables. The models proposed by Priestley & Taylor (1972), hereafter referred to as P-T, Jackson et al. (1977*);* Seguin et al. (1989*);* Granger & Gray (1989)*;* Holwill & Stewart (1992); Carlson et al. (1995); Jiang & Islam (2001) and Rivas & Caselles (2004), lie within this category.

*Residual methods*: This type of models commonly calculates the energy budged, then ET is estimated as the residual of the energy balance. The following models are examples of residual methods: The Surface Energy Balance Algorithm for Land (SEBAL) *(*Bastiaanssen et al., 1998; Bastiaanssen, 2000), the Surface Energy Balance System (SEBS) model (Su, 2002) and the two-source model proposed by Norman et al. (1995*)*, among others.

*Indirect methods*: These physically based methods involve Soil-Vegetation-Atmosphere Transfer (SVAT) models, presenting different levels of complexity often reflected in the number of parameters. For example, the ISBA (Interactions between Soil, Biosphere, and Atmosphere) model by Noilhan & Planton (1989), developed to be included within large scale meteorological models, parameterizes the land surface processes. The ISBA Ags model (Calvet et al., 1998) improved the canopy stomatal conductance and CO2 concentration with respect to the ISBA original model.

Among the first category (Empirical and semi-empirical methods), only few methodologies to calculate ET have taken advantage of the complementary relationship (CR)*.*

It is worth mentioning that there are only two CR approaches known so far, one attributed to Bouchet (1963) and the other to Granger & Gray (1989). Even though various ET models derived from these two fundamental approaches are referenced to throughout the chapter, it is not the intention of the authors to review them in detail.

Bouchet (1963) proposed the first complementary model based on an experimental design. He postulated that, for a large homogeneous surface and in absence of advection of heat and moisture, regional ET could be estimated as a complementary function of Epot and the wet environment evapotranspiration (Ew) for a wide range of available energy. Ew is the ET of a surface with unlimited moisture. Thus, if Epot is defined as the evaporation that would occur over a saturated surface, while the energy and atmospheric conditions remain unchanged, it seems reasonable to anticipate that Epot would decrease as ET increases. The underlying argument is that ET incorporates humidity to the surface sub-layer reducing the possibility for the atmosphere to transport that humidity away from the surface. Bouchet´s idea that Epot and ET have this complementary relationship has been the subject of many studies and discussions, mainly due to its empirical background (Brutsaert & Parlange, 1998; Ramírez et al., 2005). Examples of successful models based on Bouchet's heuristic relationship include those developed by Brutsaert & Stricker (1979); Morton (1983) and Hobbins et al. (2001). These models have been widely applied to a broad range of surface and atmospheric conditions (Brutsaert & Parlange, 1998; Sugita et al., 2001; Kahler & Brutsaert, 2006; Ozdogan et al., 2006; Lhomme & Guilioni, 2006; Szilagyi, 2007; Szilagyi & Jozsa, 2008).

Granger (1989a) developed a physically based complementary relationship after a meticulous analysis of potential evaporation concepts. He remarked that "*Bouchet corrected the misconception that a larger potential evaporation necessarily signified a larger actual evaporation*". The author used the term "potential evaporation" for the Epot and Ew concepts, and clearly presented the complementary behavior of common potential evaporation theories. This author suggested that Ew is the value of the potential evaporation when the actual evaporation rate is equal to the potential rate. The use of two potential parameters, i.e. Epot and Ew, seems to generate a universal relationship, and therefore, universal ET models. Conversely, attempting to estimate ET from only one potential formulation may need site-specific calibration or auxiliary relationships (Granger, 1989b). In addition, the relative evaporation coefficient introduced by Granger & Gray (1989) enhances the complementary relationship with a dimensionless coefficient that yields a simpler complementary model.

The foundation of the complementary relationship is the basis for operational estimates of areal ET by Morton (1983), who formulated the Complementary Relationship Areal Evapotranspiration (CRAE) model. The reliability of the independent operational estimates of areal evapotranspiration was tested with comparable, long-term water budget estimates for 143 river basins in North America, Africa, Ireland, Australia and New Zealand.

A procedure to calculate ET requiring only common meteorological data was presented by Brutsaert & Stricker (1979). Their Advection-Aridity approach (AA) is based on a conceptual

Atmosphere) model by Noilhan & Planton (1989), developed to be included within large scale meteorological models, parameterizes the land surface processes. The ISBA Ags model (Calvet et al., 1998) improved the canopy stomatal conductance and CO2 concentration with

Among the first category (Empirical and semi-empirical methods), only few methodologies

It is worth mentioning that there are only two CR approaches known so far, one attributed to Bouchet (1963) and the other to Granger & Gray (1989). Even though various ET models derived from these two fundamental approaches are referenced to throughout the chapter, it

Bouchet (1963) proposed the first complementary model based on an experimental design. He postulated that, for a large homogeneous surface and in absence of advection of heat and moisture, regional ET could be estimated as a complementary function of Epot and the wet environment evapotranspiration (Ew) for a wide range of available energy. Ew is the ET of a surface with unlimited moisture. Thus, if Epot is defined as the evaporation that would occur over a saturated surface, while the energy and atmospheric conditions remain unchanged, it seems reasonable to anticipate that Epot would decrease as ET increases. The underlying argument is that ET incorporates humidity to the surface sub-layer reducing the possibility for the atmosphere to transport that humidity away from the surface. Bouchet´s idea that Epot and ET have this complementary relationship has been the subject of many studies and discussions, mainly due to its empirical background (Brutsaert & Parlange, 1998; Ramírez et al., 2005). Examples of successful models based on Bouchet's heuristic relationship include those developed by Brutsaert & Stricker (1979); Morton (1983) and Hobbins et al. (2001). These models have been widely applied to a broad range of surface and atmospheric conditions (Brutsaert & Parlange, 1998; Sugita et al., 2001; Kahler & Brutsaert, 2006; Ozdogan et al., 2006;

Granger (1989a) developed a physically based complementary relationship after a meticulous analysis of potential evaporation concepts. He remarked that "*Bouchet corrected the misconception that a larger potential evaporation necessarily signified a larger actual evaporation*". The author used the term "potential evaporation" for the Epot and Ew concepts, and clearly presented the complementary behavior of common potential evaporation theories. This author suggested that Ew is the value of the potential evaporation when the actual evaporation rate is equal to the potential rate. The use of two potential parameters, i.e. Epot and Ew, seems to generate a universal relationship, and therefore, universal ET models. Conversely, attempting to estimate ET from only one potential formulation may need site-specific calibration or auxiliary relationships (Granger, 1989b). In addition, the relative evaporation coefficient introduced by Granger & Gray (1989) enhances the complementary relationship with a dimensionless coefficient that yields a simpler

The foundation of the complementary relationship is the basis for operational estimates of areal ET by Morton (1983), who formulated the Complementary Relationship Areal Evapotranspiration (CRAE) model. The reliability of the independent operational estimates of areal evapotranspiration was tested with comparable, long-term water budget estimates

A procedure to calculate ET requiring only common meteorological data was presented by Brutsaert & Stricker (1979). Their Advection-Aridity approach (AA) is based on a conceptual

for 143 river basins in North America, Africa, Ireland, Australia and New Zealand.

to calculate ET have taken advantage of the complementary relationship (CR)*.*

is not the intention of the authors to review them in detail.

Lhomme & Guilioni, 2006; Szilagyi, 2007; Szilagyi & Jozsa, 2008).

respect to the ISBA original model.

complementary model.

model involving the effect of the regional advection on potential evaporation and Bouchet's complementary model. Thus, the aridity of the region is deduced from the regional advection of the drying power of the air. The authors validated their model in a rural watershed finding a good agreement between estimated daily ET and ET obtained with the energy budget method.

Morton's CRAE model was tested by Granger & Gray (1990) for field-size land units under a specific land use, for short intervals of time such as 1 to 10 days. They examined the CRAE model with respect to the algorithms used to describe different terms and its applicability to reduced spatial and temporal scales. The assumption in CRAE that the vapor transfer coefficient is independent of wind speed may lead to appreciable errors in computing ET. Comparisons of ET estimates and measurements demonstrated that the assumptions that the soil heat flux and storage terms are negligible, lead to large overestimation by the model during periods of soil thaw.

Hobbins et al. (2001) and Hobbins & Ramírez (2001) evaluated the implementations of the complementary relationship hypothesis for regional evapotranspiration using CRAE and AA models. Both models were assessed against independent estimates of regional evapotranspiration derived from long-term, large-scale water balances for 120 minimally impacted basins in the conterminous United States. The results suggested that CRAE model overestimates annual evapotranspiration by 2.5% of mean annual precipitation, whereas the AA model underestimates annual evapotranspiration by 10.6% of mean annual precipitation. Generally, increasing humidity leads to decreasing absolute errors for both models. On the contrary, increasing aridity leads to increasing overestimation by the CRAE model and underestimation by the AA model, except at high aridity basins, where the AA model overestimates evapotranspiration.

Three evapotranspiration models using the complementary relationship approach for estimating areal ET were evaluated by Xu & Singh (2005*).* The tested models were the CRAE model, the AA model, and the model proposed by Granger & Gray (1989) (GG), using the concept of relative evaporation. The ET estimates were compared in three study regions representing a wide geographic and climatic diversity: the NOPEX region in Central Sweden (typifying a cool temperate humid region), the Baixi catchment in Eastern China (typifying a subtropical, humid region), and the Potamos tou Pyrgou River catchment in Northwestern Cyprus (typifying a semiarid to arid region). The calculation was made on a daily basis whilst comparisons were made on monthly and annual bases. The results showed that using the original parameter values, all three complementary relationship models worked reasonably well for the temperate humid region, while their predictive power decreased as soil moisture exerts increasing control over the region, i.e. increased aridity. In such regions, the parameters need to be calibrated.

Ramírez et al. (2005) provided direct observational evidence of the complementary relationship in regional evapotranspiration hypothesized by Bouchet in 1963. They used independent observations of ET and Epot at a wide range of spatial scales. This work is the first to assemble a data set of direct observations demonstrating the complementary relationship between regional ET and Epot. These results provided strong evidence for the complementary relationship hypothesis, raising its status above that of a mere conjecture.

A drawback among the aforementioned complementary ET models is the use of Penman or Penman-Monteith equation (Monteith & Unsworth, 1990) to estimate Epot. Specifically, the Morton's CRAE model *(*Morton, 1983*)* uses Penman equation to calculate Epot, and a modified P-T equation to approximate Ew. Brutsaert & Stricker (1979) developed their AA model using Penman for Epot and the P-T equilibrium evaporation to model Ew. At the time those models were developed, networks of meteorological stations constituted the main source of atmospheric data, while the surface temperature (Ts) or the soil temperature were available only at some locations around the World. The advent of satellite technology provided routinely observations of the surface temperature, but the source of atmospheric data was still ancillary. Thus, many of the current remote sensing approaches were developed to estimate ET with little amount of atmospheric data (Price, 1990; Jiang & Islam, 2001).

The recent introduction of the Atmospheric Profiles Product derived from Moderate Resolution Imaging Spectroradiometer (MODIS) sensors onboard of EOS-Terra and EOS-Aqua satellites meant a significant advance for the scientific community. The MODIS Atmospheric profile product provides atmospheric and dew point temperature profiles on a daily basis at 20 vertical atmospheric pressure levels and at 5x5km of spatial resolution (Menzel et al., 2002*).* When combined with readily available Ts maps obtained from different sensors, this new remote source of atmospheric data provides a new opportunity to revise the complementary relationship concepts that relate ET and Epot (Crago & Crowley, 2005; Ramírez et al., 2005).

A new method to derive spatially distributed EF and ET maps from remotely sensed data without using auxiliary relationships such as those relating a vegetation index (VI) with the land surface temperature (Ts) or site-specific relationships, was proposed by Venturini et al. (2008*)*. Their method for computing ET is based on Granger's complementary relationship, the P-T equation and a new parameter introduced to calculate the relative evaporation (F=ET/Epot). The ratio F can be expressed in terms of Tu, which is the temperature of the surface if it is brought to saturation without changing the actual surface vapor pressure. The concept of Tu proposed by these authors is analogous to the dew point temperature (Td) definition.

Szilagyi & Jozsa (2008) presented a long term ET calculation using the AA model. In their work the authors presented a novel method to calculate the equilibrium temperature of Ew and P-T equation that yields better long-term ET estimates. The relationship between ET and Epot was studied at daily and monthly scales with data from 210 stations distributed all across the USA. They reported that only the original Rome wind function of Penman yields a truly symmetric CR between ET and Epot which makes Epot estimates true potential evaporation values. In this case, the long-term mean value of evaporation from the modified AA model becomes similar to CRAE model, especially in arid environments with possible strong convection. An R2 of approximately 0.95 was obtained for the 210 stations and all wind functions used. Likewise, Szilagyi & Jozsa in (2009) investigated the environmental conditions required for the complementary ET and Epot relationship to occur. In their work, the coupled turbulent diffusion equations of heat and vapor transport were solved under specific atmospheric, energy and surface conditions. Their results showed that, under nearneutral atmospheric conditions and a constant energy term at the evaporating surface, the analytical solution across a moisture discontinuity of the surface yields a symmetrical complementary relationship assuming a smooth wet area.

Recently, Crago et al. (2010) presented a modified AA model in which the specific humidity at the minimum daily temperature is assumed equal to the daily average specific humidity. The authors also modified the drying power calculation in Penman equation using Monin-Obukhov theory (Monin & Obukhov, 1954). They found promising results with these modifications. Han et al. (2011*)* proposed and verified a new evaporation model based on

model using Penman for Epot and the P-T equilibrium evaporation to model Ew. At the time those models were developed, networks of meteorological stations constituted the main source of atmospheric data, while the surface temperature (Ts) or the soil temperature were available only at some locations around the World. The advent of satellite technology provided routinely observations of the surface temperature, but the source of atmospheric data was still ancillary. Thus, many of the current remote sensing approaches were developed to estimate ET with little amount of atmospheric data (Price,

The recent introduction of the Atmospheric Profiles Product derived from Moderate Resolution Imaging Spectroradiometer (MODIS) sensors onboard of EOS-Terra and EOS-Aqua satellites meant a significant advance for the scientific community. The MODIS Atmospheric profile product provides atmospheric and dew point temperature profiles on a daily basis at 20 vertical atmospheric pressure levels and at 5x5km of spatial resolution (Menzel et al., 2002*).* When combined with readily available Ts maps obtained from different sensors, this new remote source of atmospheric data provides a new opportunity to revise the complementary relationship concepts that relate ET and Epot (Crago &

A new method to derive spatially distributed EF and ET maps from remotely sensed data without using auxiliary relationships such as those relating a vegetation index (VI) with the land surface temperature (Ts) or site-specific relationships, was proposed by Venturini et al. (2008*)*. Their method for computing ET is based on Granger's complementary relationship, the P-T equation and a new parameter introduced to calculate the relative evaporation (F=ET/Epot). The ratio F can be expressed in terms of Tu, which is the temperature of the surface if it is brought to saturation without changing the actual surface vapor pressure. The concept of Tu proposed by these authors is analogous to the

Szilagyi & Jozsa (2008) presented a long term ET calculation using the AA model. In their work the authors presented a novel method to calculate the equilibrium temperature of Ew and P-T equation that yields better long-term ET estimates. The relationship between ET and Epot was studied at daily and monthly scales with data from 210 stations distributed all across the USA. They reported that only the original Rome wind function of Penman yields a truly symmetric CR between ET and Epot which makes Epot estimates true potential evaporation values. In this case, the long-term mean value of evaporation from the modified AA model becomes similar to CRAE model, especially in arid environments with possible strong convection. An R2 of approximately 0.95 was obtained for the 210 stations and all wind functions used. Likewise, Szilagyi & Jozsa in (2009) investigated the environmental conditions required for the complementary ET and Epot relationship to occur. In their work, the coupled turbulent diffusion equations of heat and vapor transport were solved under specific atmospheric, energy and surface conditions. Their results showed that, under nearneutral atmospheric conditions and a constant energy term at the evaporating surface, the analytical solution across a moisture discontinuity of the surface yields a symmetrical

Recently, Crago et al. (2010) presented a modified AA model in which the specific humidity at the minimum daily temperature is assumed equal to the daily average specific humidity. The authors also modified the drying power calculation in Penman equation using Monin-Obukhov theory (Monin & Obukhov, 1954). They found promising results with these modifications. Han et al. (2011*)* proposed and verified a new evaporation model based on

1990; Jiang & Islam, 2001).

Crowley, 2005; Ramírez et al., 2005).

dew point temperature (Td) definition.

complementary relationship assuming a smooth wet area.

the AA model and the Granger's CR model (Granger, 1989b). This newly proposed model transformed Granger´s and AA models into similar, dimensionless forms by normalizing the equations with Penman potential model. The evaporation ratio (i.e. the ratio of ET to Penman potential evaporation) was expressed as a function of dimensionless variables based on radiation and atmospheric conditions. From the validation with ground observations, the authors concluded that the new model is an enhanced Granger`s model, with better evaporation predictions. In addition, the model somewhat approximates the AA model under neither too-wet nor too-dry conditions. As the reader can conclude, the complementary approach is nowadays the subject of many ongoing researches.

## **2. A review of Bouchet's and Granger's models**

Bouchet (1963) set an experiment over a large homogeneous surface without advective effects. Initially, the surface was saturated and evaporated at potential rate. With time, the region dried, but a small parcel was kept saturated (see Figure 1), evaporating at potential rate. The region and the parcel scales were such that the atmosphere could be considered stable. Bouchet described his experiment, dimension and scales as follows1,


As mentioned, initially the surface was saturated and evaporated at its potential rate, i.e. at the so-called reference evapotranspiration (or Ew). In this initial condition, Epot = Ew = ET. When ET is lower than Ew due to limited water availability, a certain excess of energy would become available. This remaining energy not used for evaporation may, in tern, warm the lower layer of the atmosphere. The resulting increase in air temperature due to the heating, and the decrease in humidity caused by the reduction of ET, would lead to a new value of Epot larger than Ew by the amount of energy left over.

<sup>1</sup> The following text was translated by the authors of this chapter from Bouchet's original paper (in French).

Fig. 1. Reproduction of Bouchet´s schematic representation of the Oasis Effect experiment.

Thus, Bouchet's complementary relationship was obtained from the balance of these evaporation rates,

$$ET + Epot = 2\, Ew \tag{1}$$

Bouchet postulated that in such a system, under a constant energy input and away from sharp discontinuities, there exists a complementary feedback mechanism between ET and Epot, that causes changes in each to be complementary, that is, a positive change in ET causes a negative change in Epot (Ozdogan et al., 2006), as sketched in Figure 2. Later, Morton (1969) utilized Bouchet's experiment to derive the potential evaporation as a manifestation of regional evapotranspiration, i.e. the evapotranspiration of an area so large that the heat and water vapor transfer from the surface controls the evaporative capacity of the lower atmosphere.

Fig. 2. Sketch of Bouchet´s complementary ET and Epot relationship

The hypothesis asserts that when ET falls below Ew as a result of limited moisture availability, a large quantity of energy becomes available for sensible heat flux that warms and dries the atmospheric boundary layer thereby causing Epot to increase, and *vise versa*.

Fig. 1. Reproduction of Bouchet´s schematic representation of the Oasis Effect experiment. Thus, Bouchet's complementary relationship was obtained from the balance of these

Bouchet postulated that in such a system, under a constant energy input and away from sharp discontinuities, there exists a complementary feedback mechanism between ET and Epot, that causes changes in each to be complementary, that is, a positive change in ET causes a negative change in Epot (Ozdogan et al., 2006), as sketched in Figure 2. Later, Morton (1969) utilized Bouchet's experiment to derive the potential evaporation as a manifestation of regional evapotranspiration, i.e. the evapotranspiration of an area so large that the heat and water vapor transfer from the surface controls the evaporative capacity of

Fig. 2. Sketch of Bouchet´s complementary ET and Epot relationship

The hypothesis asserts that when ET falls below Ew as a result of limited moisture availability, a large quantity of energy becomes available for sensible heat flux that warms and dries the atmospheric boundary layer thereby causing Epot to increase, and *vise versa*.

*ET Epot 2Ew* (1)

evaporation rates,

the lower atmosphere.

Equation (1) holds true if the energy budget remains unchanged and all the excess energy goes into sensible heat (Ramírez et al., 2005). It should be noted that Bouchet´s experimental system is the so-called advection-free-surface in P-T formulation.

This relationship assumes that as ET increases, Epot decreases by the same amount, i.e. δET = -δEpot, where the symbol δ means small variations. Bouchet's equation has been widely used in conjunction with Penman (1948) and Priestley-Taylor (1972) (Brutsaert & Stricker, 1979; Morton, 1983; Hobbins el al., 2001*).*

Granger (1989b*)* argued that the above relationship lacked a theoretical background, mainly due to Bouchet's symmetry assumption (δET=-δEpot). Nonetheless, the author recognized that Bouchet´s CR set the basis for the complementary behavior between two potential concepts of evaporation and ET. One of the benefits of using two potential evaporation concepts rather than a single one is that the resulting CR would be universal, without the need of tuning parameters from local data.

Granger (1989a) revised the diversity of potential evaporation concepts available at that moment and expertly established an inequity among them. The resulting comparison yielded that Penman (1948) and Priestley & Taylor (1972) concepts are Ew concepts, and that the true potential evaporation would be that proposed by van Bavel (1966). Thus, these parameterizations would result in the following inequity, Epot Ew ET, where Epot would be van Bavel´s concept, Ew could be obtained with either Penman or P-T, knowing that ET-Penman is larger than ET-Priestley-Taylor (Granger, 1989a). Hence, the author postulated that the above inequity comprises Bouchet´s equity (δET = -δEpot) but it is based on a new CR. Granger (1989b*)* then proposed the following CR formulation,

$$ET + Epot\frac{\gamma}{\Delta} = Ew\left(\frac{\Delta + \gamma}{\Delta}\right) \tag{2}$$

where is the psychrometric constant and is the slope of the saturation vapor pressure (SVP) curve.

Equation (2) shows that for constant available energy and atmospheric conditions, -/ is equal to the ratio δET/δEpot. In addition, this CR is not symmetric with respect to Ew. It can be easily verified that equation (2) is equivalent to equation (1) when The condition that the slope of the SVP curve equals the psychrometric constant is only true when the temperature is near 6 °C (Granger, 1989b). This has been widely tested (Granger & Gray, 1989*;* Crago & Crowley, 2005*;* Crago et al., 2005*;* Xu & Singh, 2005*;* Venturini et al., 2008; Venturini et al., 2011).

#### **3. Bouchet`s versus Granger`s complementary models**

A review of the two complementary models widely used for ET calculations was presented. Both methods are not only conceptually different, but also differ in their derivations. Mathematically speaking, Bouchet's complementary relationship (equation 1) results a simplification of Granger's complementary equation (equation 2) for the case =. Equations (1) and (2) can also be written, respectively, as follows,

$$\frac{1}{2}ET + \frac{1}{2}Epot = Ew\tag{3}$$

$$\left(\frac{\Delta}{\Delta + \gamma}\right) ET + \left(\frac{\gamma}{\Delta + \gamma}\right)Epot = Ew \tag{4}$$

The re-written Bouchet´s complementary model, equation (3), clearly expresses Ew as the middle point between the ET and the Epot processes. In contrast, the re-written Granger's complementary relationship, equation (4), shows how both, ET and Epot contribute to Ew with different coefficients, the coefficients varying with the slope of the SVP curve at the air temperature Ta, since is commonly assumed constant. For clarity, Table 1 summarizes all symbols and definitions used in this Chapter.

Recently, Ramírez et al., (2005) discussed Bouchet's coefficient "2" with monthly average ground measurements. In their application, Epot was calculated with the Penman-Monteith equation and Ew with the P-T model. They concluded that the appropriate coefficient should be slightly lower than 2.

Venturini et al. (2008) and Venturini et al. (2011) introduced the concept of the relative evaporation, F= ET/Epot, proposed earlier by Granger & Gray (1989), along with P-T equation in both CR models. Thus, Epot is replaced by ET/F and Ew is equated to P-T equation. Hence, replacing Epot in equation (3),

$$ET + \frac{ET}{F} = k \ Ew \tag{5}$$

where k is Bouchet´s coefficient, originally assumed k=2 Then, when Ew is replaced in (5) by the P-T equation, results

$$ET\left(1+\frac{1}{F}\right) = ka \quad \left(\text{Rn}-\text{G}\right)\frac{\Delta}{\Delta+\chi} \tag{6}$$

where α is the P-T's coefficient, and the rest of the variables are defined in Table 1. Finally, Bouchet's CR is obtained by rearranging the terms in equation (6),

$$ET = ka \left(\frac{F}{F+1}\right) \left(\frac{\Delta}{\Delta + \gamma}\right) \left(\text{Rn} - G\right) \tag{7}$$

Following the same procedure with equation (4), the equivalent equation for Granger´s CR model is,

$$ET = a \left(\frac{F\Lambda}{F\Lambda + \mathcal{Y}}\right) \left(\text{Rn} - G\right) \tag{8}$$

It should be noted that the underlying assumptions of equation (7) are the same as those behind equation (8), plus the condition that is approximately equal to .

Both, equations (7) and (8), require calculating the F parameter, otherwise the equations would have only theoretical advantages and would not be operative models. Venturini et al. (2008) developed an equation for F that can be estimated using MODIS products. Their F method is briefly presented here.

Consider the relative evaporation expression proposed by Granger & Gray (1989),

$$\frac{\text{ET}}{\text{Epot}} = \frac{f\_{\text{u}} \left(\mathbf{e}\_{\text{s}} - \mathbf{e}\_{\text{a}}\right)}{f\_{\text{u}} \left(\mathbf{e}\_{\text{s}}^{\text{\textbullet}} - \mathbf{e}\_{\text{a}}\right)}\tag{9}$$

where fu is a function of the wind speed and vegetation height, es is the surface actual water vapor pressure, ea is the air actual water vapor pressure, e\* s is the surface saturation water vapor pressure.


Table 1. Symbols and units

26 Evapotranspiration – Remote Sensing and Modeling

The re-written Bouchet´s complementary model, equation (3), clearly expresses Ew as the middle point between the ET and the Epot processes. In contrast, the re-written Granger's complementary relationship, equation (4), shows how both, ET and Epot contribute to Ew with different coefficients, the coefficients varying with the slope of the SVP curve at the air temperature Ta, since is commonly assumed constant. For clarity, Table 1 summarizes all

Recently, Ramírez et al., (2005) discussed Bouchet's coefficient "2" with monthly average ground measurements. In their application, Epot was calculated with the Penman-Monteith equation and Ew with the P-T model. They concluded that the appropriate coefficient

Venturini et al. (2008) and Venturini et al. (2011) introduced the concept of the relative evaporation, F= ET/Epot, proposed earlier by Granger & Gray (1989), along with P-T equation in both CR models. Thus, Epot is replaced by ET/F and Ew is equated to P-T

*ET ET + = k Ew*

 <sup>1</sup> *ET ka G* 1 Rn *F*

 

where α is the P-T's coefficient, and the rest of the variables are defined in Table 1. Finally,

*<sup>F</sup> ET k<sup>α</sup> <sup>G</sup>*

 

Following the same procedure with equation (4), the equivalent equation for Granger´s CR

 Rn *<sup>F</sup> ET <sup>G</sup> F* 

It should be noted that the underlying assumptions of equation (7) are the same as those

Both, equations (7) and (8), require calculating the F parameter, otherwise the equations would have only theoretical advantages and would not be operative models. Venturini et al. (2008) developed an equation for F that can be estimated using MODIS products. Their F

> )e(e )ee

a \* s as *u u f*

*F*

behind equation (8), plus the condition that is approximately equal to .

Consider the relative evaporation expression proposed by Granger & Gray (1989),

Epot ET

Rn <sup>1</sup>

  (4)

*F* (5)

(7)

(8)

*( f* (9)

(6)

 *ET Epot Ew* 

symbols and definitions used in this Chapter.

equation. Hence, replacing Epot in equation (3),

where k is Bouchet´s coefficient, originally assumed k=2 Then, when Ew is replaced in (5) by the P-T equation, results

Bouchet's CR is obtained by rearranging the terms in equation (6),

should be slightly lower than 2.

model is,

method is briefly presented here.

This form of the relative evaporation equation needs readily available meteorological data. A key difficulty in applying equation (9) lies on the estimation of (es-ea), since there is no simple way to relate es to any readily available surface temperature. Thus, a new temperature should be defined. Many studies have used temperature as a surrogate for vapor pressure (Monteith & Unsworth, 1990; Nishida et al., 2003). Although the relationship between vapor pressure and temperature is not linear, it is commonly linearized for small temperature differences. Hence, es and es \* should be related to soil+vegetation at a temperature that would account for water vapor pressure. Figure 3 shows the relationship between es, e\* s and ea and their corresponding temperatures; where eu\* is the SVP at an unknown surface temperature Tu.

An analogy to the dew point temperature concept (Td) suggests that Tu would be the temperature of the surface if the surface is brought to saturation without changing the surface actual water vapor pressure. Accordingly, Tu must be lower than Ts if the surface is not saturated and close to Ts if the surface is saturated. Consequently, es could be derived from the temperature Tu. Although Tu may not possibly be observed in the same way as Td, it can be derived, for instance, from the slope of the exponential SVP curve as a function of Ts and Td. This calculation is further discussed later in this chapter.

Assuming that the surface saturation vapor pressure at Tu would be the actual soil vapor pressure and that the SVP can be linearized, (es -ea) can be approximated by 1(Tu-Td) and (e\* s -ea) by 2(Ts-Td), respectively. Figure 3 shows a schematic of these concepts.

Fig. 3. Schematic of the linearized saturation vapor pressure curve and the relationship between (es -ea) and 1(Tu-Td), and (e\* s -ea) and 2(Ts-Td).

Therefore, ET/Epot (see equation 9) can be rewritten as follows,

$$\mathbf{F} = \frac{\mathbf{ET}}{\mathbf{Epot}} = \frac{(\mathbf{Tu} - \mathbf{Td})}{(\mathbf{Ts} - \mathbf{Td})} \left(\frac{\Delta\_1}{\Delta\_2}\right) \tag{10}$$

The wind function, fu, depends on the vegetation height and the wind speed, but it is independent of surface moisture. In other words, it is reasonable to expect that the wind function will affect ET and Epot in a similar fashion (Granger, 1989b), so its effect on ET and Epot cancels out. The slopes of the SVP curve, 1 and 2, can be computed from the SVP first derivative at Td and Ts without adding further complexity to this method. However, 1 and 2 will be assumed approximately equal from now on, as they will be estimated as the first derivative of the SVP at Ta.

The relationship between Ts and Tu can be examined throughout the definition of Tu, which represents the saturation temperature of the surface. For a saturated surface, Tu is expected to be very close or equal to Ts. In contrast, for a dry surface, Ts would be much larger than Tu. Since Epot is larger than or equal to ET, F ranges from 0 to 1. For a dry surface, with Ts >> Tu, (Ts-Td) would be larger than (Tu-Td) and ET/Epot would tend to 0. In the case of a saturated surface with es close to es \* and Ts close to Tu, (Ts-Td) would be similar to (Tu-Td) and ET/Epot would tend to 1.

The calculation of Tu proposed by Venturini et al. (2008) is presented in the next section, where results from MODIS data are shown. However, it is emphasized that the definition of Tu is not linked to any data source; therefore it can be estimated with different approaches.

## **4. Complementary models application using remotely sensed data**

In order to show the potential of the complementary relationships, equations (7) and (8) were applied to the Southern Great Plains of the USA region and the results compared and analyzed.

## **4.1 Study area**

28 Evapotranspiration – Remote Sensing and Modeling

Assuming that the surface saturation vapor pressure at Tu would be the actual soil vapor pressure and that the SVP can be linearized, (es -ea) can be approximated by 1(Tu-Td) and

s -ea) by 2(Ts-Td), respectively. Figure 3 shows a schematic of these concepts.

Fig. 3. Schematic of the linearized saturation vapor pressure curve and the relationship

ET (Tu Td) <sup>Δ</sup> <sup>F</sup> Epot (Ts Td) Δ 

The wind function, fu, depends on the vegetation height and the wind speed, but it is independent of surface moisture. In other words, it is reasonable to expect that the wind function will affect ET and Epot in a similar fashion (Granger, 1989b), so its effect on ET and Epot cancels out. The slopes of the SVP curve, 1 and 2, can be computed from the SVP first derivative at Td and Ts without adding further complexity to this method. However, 1 and 2 will be assumed approximately equal from now on, as they will be estimated as the first

The relationship between Ts and Tu can be examined throughout the definition of Tu, which represents the saturation temperature of the surface. For a saturated surface, Tu is expected to be very close or equal to Ts. In contrast, for a dry surface, Ts would be much larger than Tu. Since Epot is larger than or equal to ET, F ranges from 0 to 1. For a dry surface, with Ts >> Tu, (Ts-Td) would be larger than (Tu-Td) and ET/Epot would tend to 0. In the case of a

The calculation of Tu proposed by Venturini et al. (2008) is presented in the next section, where results from MODIS data are shown. However, it is emphasized that the definition of Tu is not linked to any data source; therefore it can be estimated with different

Therefore, ET/Epot (see equation 9) can be rewritten as follows,

s -ea) and 2(Ts-Td).

1 2

\* and Ts close to Tu, (Ts-Td) would be similar to (Tu-Td)

(10)

between (es -ea) and 1(Tu-Td), and (e\*

derivative of the SVP at Ta.

saturated surface with es close to es

and ET/Epot would tend to 1.

approaches.

(e\*

The Southern Great Plains (SGP) region in the United States of America extends over the State of Oklahoma and southern parts of Kansas. The area broadens in longitude from 95.3º W to 99.5º W and in latitude from 34.5º N to 38.5º N (Figure 4). This region was the first field measurement site established by the Atmospheric Radiation Measurement (ARM) Program. At present, the ARM program has three experimental sites. Scientists from all over the World are using the information obtained from these sites to improve the performance of atmospheric general circulation models used for climate change research. The SGP was chosen as the first ARM field measurement site for several reasons, among them, its relatively homogeneous geography, easy accessibility, wide variability of climate cloud types, surface flux properties, and large seasonal variations in temperature and specific humidity (http://www.arm.gov/sites/sgp).

Most of this region is characterized by irregular plains. Altitudes range from approximately 500 m to 90 m, increasing gradually from East to West. In southwestern Oklahoma, the highest Wichita Mountains rise as much as 800 m above the surrounding landscape (Heilman & Brittin, 1989; Venturini et al., 2008). The climate is semiarid-subtropical. Although the maximum rainfall occurs in summer, high temperatures make summer relatively dry. Average annual temperatures range from 14°C to 18°C. Winters are cold and dry, and summers are warm to hot. The frost-free season stretches from 185 to 230 days. Precipitation ranges from 490 to 740 mm, with most of it falling as rain.

Grass is the dominant prairie vegetation. Most of it is moderately tall and usually grows in bunches. The most prevalent type of grassland is the bluestem prairie (*Andropogon gerardii and Andropogon hallii)*, along with many species of wildflowers and legumes. In many places where grazing and fire are controlled, deciduous forest is encroaching on the prairies.

Fig. 4. Study area map

Due to generally favorable conditions of climate and soil, most of the area is cultivated, and little of the original vegetation remains intact. Oak savanna occurs along the eastern border of the region and along some of the major river valleys.

## **4.2 Ground data availability**

The latent heat data was obtained from the ARM program Web site (http://www.arm.gov). The ARM instruments and measurement applications are well established and have been used for validation purposes in many studies (Halldin & Lindroth, 1992; Fritschen & Simpson, 1989). The site and name, elevation, geographic coordinates (latitude and longitude) and surface cover of the stations used in this work are shown in Table 2.


Table 2. Site name and station name, elevation, latitude, longitude and surface type

The first instrumentation installation to the SGP site took place in 1992, with data processing capabilities incrementally added in the succeeding years. This region has relatively extensive and well-distributed coverage of surface fluxes and meteorological observation stations. In this study, Energy Balance Bowen Ratio stations (EBBR), maintained by the ARM program were used for the validation of surface fluxes. The EBBR system produces 30 minute estimates of the vertical fluxes of sensible and latent heat at the local points. The EBBR fluxes estimates are calculated from observations of net radiation, soil surface heat flux, the vertical gradients of temperature and relative humidity.

## **4.3 MODIS products**

The method proposed here was physically derived from universal relationships. Moreover, data sources do not represent a limitation for the applicability of equations (6) and (8), nonetheless remotely sensed data such as that provided by MODIS scientific team would empower the potential applications of the methods. Hence, the equations applicability using MODIS products was explored. The sensor's bands specifications can be obtained from http://modis.gsfc.nasa.gov/about/specifications.php.

Due to generally favorable conditions of climate and soil, most of the area is cultivated, and little of the original vegetation remains intact. Oak savanna occurs along the eastern border

The latent heat data was obtained from the ARM program Web site (http://www.arm.gov). The ARM instruments and measurement applications are well established and have been used for validation purposes in many studies (Halldin & Lindroth, 1992; Fritschen & Simpson, 1989). The site and name, elevation, geographic coordinates (latitude and

**(m a.m.s.l.) Lat/Lon Vegetation Type** 

longitude) and surface cover of the stations used in this work are shown in Table 2.

Ashton, Kansas E-9 386 37.133 N/97.266 W Pasture Coldwater, Kansas E-8 664 37.333 N/99.309 W Rangeland (grazed) Cordell, Oklahoma: E-22 465 35.354 N/98.977 W Rangeland (grazed) Cyril, Oklahoma: E-24 409 34.883 N/98.205 W Wheat (gypsum hill) Earlsboro, Oklahoma: E-27 300 35.269 N/96.740 W Pasture Elk Falls, Kansas E-7 283 37.383 N/96.180 W Pasture El Reno, Oklahoma: E-19 421 35.557 N/98.017 W Pasture (ungrazed) Hillsboro, Kansas E-2 447 38.305 N/97.301 W Grass Lamont, Oklahoma: E-13 318 36.605 N/97.485 W Pasture and wheat Meeker, Oklahoma: E-20 309 35.564 N/96.988 W Pasture Morris, Oklahoma: E-18 217 35.687 N/95.856 W Pasture (ungrazed) Pawhuska, Oklahoma: E-12 331 36.841 N/96.427 W Native prairie Plevna, Kansas E-4 513 37.953 N/98.329 W Rangeland (ungrazed) Ringwood, Oklahoma: E-15 418 36.431 N/98.284 W Pasture Table 2. Site name and station name, elevation, latitude, longitude and surface type

The first instrumentation installation to the SGP site took place in 1992, with data processing capabilities incrementally added in the succeeding years. This region has relatively extensive and well-distributed coverage of surface fluxes and meteorological observation stations. In this study, Energy Balance Bowen Ratio stations (EBBR), maintained by the ARM program were used for the validation of surface fluxes. The EBBR system produces 30 minute estimates of the vertical fluxes of sensible and latent heat at the local points. The EBBR fluxes estimates are calculated from observations of net radiation, soil surface heat

The method proposed here was physically derived from universal relationships. Moreover, data sources do not represent a limitation for the applicability of equations (6) and (8), nonetheless remotely sensed data such as that provided by MODIS scientific team would empower the potential applications of the methods. Hence, the equations applicability using MODIS products was explored. The sensor's bands specifications can be obtained from

flux, the vertical gradients of temperature and relative humidity.

http://modis.gsfc.nasa.gov/about/specifications.php.

**4.3 MODIS products** 

of the region and along some of the major river valleys.

**Site Elevation**

**4.2 Ground data availability** 

Daytime images for seven days in year 2003 with at least 80% of the study area free of clouds were selected. Table 3 summarizes the images information including date, day of the year, satellite overpass time and image quality.

Geolocation is the process by which scientists specify where a specific radiance signal was detected on the Earth's surface. The MODIS geolocation dataset, called MOD03, includes eight Earth location data fields, e.g. geodetic latitude and longitude, height above the Earth ellipsoid, satellite zenith angle, satellite azimuth, range to the satellite, solar zenith angle, and solar azimuth. Similarly Earth location algorithms are widely used in modeling and geometrically correct image data from the Land Remote Sensing Satellite (Landsat) Multispectral Scanner (MSS), Landsat Thematic Mapper (TM), System pour l'Observation de la Terre (SPOT), and Advanced Very High Resolution Radiometer (AVHRR) missions.


Table 3. Date, Day of the Year, overpass time and image quality of the seven study days.

MOD11 is the Land Surface Temperature (LST), and emissivity product, providing per-pixel temperature and emissivity values. Average temperatures are extracted in Kelvin with a day/night LST algorithm applied to a pair of MODIS daytime and nighttime observations. This method yields 1 K accuracy for materials with known emissivities, and the view angle information is included in each LST product. The LST algorithms use other MODIS data as input, including geolocation, radiance, cloud masking, atmospheric temperature, water vapor, snow, and land cover. These products are validated, meaning that product uncertainties are well defined over a range of representative conditions. The theories behind this product can be found in Wan (1999), available at http://modis.gsfc.nasa.gov/ data/atbd/atbd\_mod11.pdf.

In particular, MODIS Atmospheric Profile product consists on several parameters: total ozone burden, atmospheric stability, temperature and moisture profiles, and atmospheric water vapor. All of these parameters are produced day and night at 5×5 km pixel resolution. There are two MODIS Atmosphere Profile data product files: MOD07\_L2, containing data collected from the Terra platform and MYD07\_L2 collecting data from Aqua platform. The MODIS temperature and moisture profiles are defined at 20 vertical levels. A simultaneous direct physical solution to the infrared radiative-transfer equation in a cloudless sky is used. The profiles are also utilized to correct for atmospheric effects for some of the MODIS products (e.g., sea-surface temperature and LST, ocean aerosol properties, etc) as well as to characterize the atmosphere for global greenhouse studies. Temperature and moisture profile retrieval algorithms are adapted from the International TIROS Operational Vertical Sounder (TOVS) Processing Package (ITPP), taking into account MODIS' lack of stratospheric channels and far higher horizontal resolution. The profile retrieval algorithm requires calibrated, navigated, and co-registered 1-km field of the view (FOV) radiances from MODIS channels 20, 22-25, 27-29, and 30-36. The atmospheric water vapor is most directly obtained by integrating the moisture profile through the atmospheric column. Data validation was conducted by comparing results from the Aqua platform with *in situ* data (Menzel et al., 2002). In the present study, air temperature and dew point temperature at 1000 hPa level are used to calculate the vapor pressure deficit. Also the temperatures are assumed to be homogenous over the 5x5 km grid.

#### **5. Results**

In this section, the results are divided in two parts. The results of variables and parameters needed to apply the CR models are presented in first place, followed by a comparison of results between equations (7) and (8).

#### **5.1 Variables calculation**

In order to apply Bouchet´s and Granger´s CR, Rn, G and F for each pixel of every image of the study area must be computed. The other parameters, and , can be assumed constant for the entire region. Alternatively, they can be estimated with spatially distributed information of Ta over the region. The constants α and k are assumed equal to 1.26 and 2, respectively.

The Rn maps were estimated with the methodology published by Bisht et al. (2005), which provides a spatially consistent and distributed Rn map over a large domain for clear sky days. With this method, Rn can be evaluated in terms of its components of downward and upward short wave radiation fluxes, and downward and upward long wave radiation fluxes. Several MODIS data products are utilized to estimate every component. Details of these calculations for the study days presented in this work can be found in Bisht et al. (2005), from where we took the Rn maps.

Soil heat fluxes G were calculated according to Moran et al. (1989) with the daily Normalized Difference Vegetation Index (NDVI) maps (Kogan et al., 2003), calculated with MOD021KM products. The equations used are

$$\mathbf{G} = \mathbf{0}.583 \text{ Rn } \mathbf{e}^{\{2.13^\ast N \text{NDVI}\}} \qquad \text{for } \mathbf{NIDVI} \ge \mathbf{0} \tag{11}$$

$$G = 0.583 \text{ Rn} \tag{12}$$

$$\text{for NDVI} \le 0 \tag{12}$$

The slope of the SVP curve, , was calculated at Ta using Buck's equation *(Buck, 1981*) and the MODIS Ta product.

In order to determine F, a methodology to estimate Tu is needed. By definition, different types of soils and water content would render different Tu values. Here, it is proposed to estimate the variable Tu from the SVP curve. It can be assumed that es is larger or equal to ea and lower or equal to e\* s, thus Tu must lie between Ts and Td.

The first derivative of the SVP curve at Ts and at Td represents the slope of the curve between those points. It can also be computed from the linearized SVP curve between the intervals [Tu,Ts] and [Td,Tu], which are symbolized as 1 and 2, respectively. Thus, an expression for Tu is derived from a simple system of two equations with two unknowns, as follows,

$$T\_u = \frac{\left(e\_s^\ast - e\_a\right) - \Delta\_1 Ts + \Delta\_2 Td}{\Delta\_2 - \Delta\_1} \tag{13}$$

There are many published SVP equations that can be used to obtain the derivative of e as function of the temperature. Here, Buck's formulation (Buck, 1981) was chosen for its simple form (equation 14),

$$e = 6.1121 \exp\left(\frac{17.502 \text{ T}}{240.97 + T}\right) \tag{14}$$

where "e" is water vapor pressure [hPa] and T is temperature [°C]. Thus, the first derivative of equation 14 is computed at Td and Ts to estimate 1 and 2 in equation (13).

$$\frac{dc}{dT} = \left[\frac{4217.45694}{\left(240.97 + \mathrm{T}\right)^2}\right] \text{\*} 6.1121 \exp\left(\frac{17.502 \,\mathrm{T}}{240.97 + \mathrm{T}}\right) \tag{15}$$

The estimation of Tu could be improved by introducing another surface variable, such as soil moisture or any other surface variable that accounts for the surface wetness. However, in order to demonstrate the strength of the CR models, the Tu calculation is kept simple, with minimum data requirements. It is recognized, however, that this calculation simplifies the physical process and may introduce errors and uncertainties to the F ratio.

Figure 5 shows Rn maps obtained for April 1st, 2003 as an example of what can be expected in terms of spatial resolution with Bisht et al. methodology. Figure 6 displays Tu map for the same date obtained with the MOD07 spatial resolution (5x5 km).

#### **5.2 Comparison of the CR models**

32 Evapotranspiration – Remote Sensing and Modeling

requires calibrated, navigated, and co-registered 1-km field of the view (FOV) radiances from MODIS channels 20, 22-25, 27-29, and 30-36. The atmospheric water vapor is most directly obtained by integrating the moisture profile through the atmospheric column. Data validation was conducted by comparing results from the Aqua platform with *in situ* data (Menzel et al., 2002). In the present study, air temperature and dew point temperature at 1000 hPa level are used to calculate the vapor pressure deficit. Also the temperatures are

In this section, the results are divided in two parts. The results of variables and parameters needed to apply the CR models are presented in first place, followed by a comparison of

In order to apply Bouchet´s and Granger´s CR, Rn, G and F for each pixel of every image of the study area must be computed. The other parameters, and , can be assumed constant for the entire region. Alternatively, they can be estimated with spatially distributed information of Ta over the region. The constants α and k are assumed equal to 1.26 and 2,

The Rn maps were estimated with the methodology published by Bisht et al. (2005), which provides a spatially consistent and distributed Rn map over a large domain for clear sky days. With this method, Rn can be evaluated in terms of its components of downward and upward short wave radiation fluxes, and downward and upward long wave radiation fluxes. Several MODIS data products are utilized to estimate every component. Details of these calculations for the study days presented in this work can be found in Bisht et al.

Soil heat fluxes G were calculated according to Moran et al. (1989) with the daily Normalized Difference Vegetation Index (NDVI) maps (Kogan et al., 2003), calculated with

The slope of the SVP curve, , was calculated at Ta using Buck's equation *(Buck, 1981*) and

In order to determine F, a methodology to estimate Tu is needed. By definition, different types of soils and water content would render different Tu values. Here, it is proposed to estimate the variable Tu from the SVP curve. It can be assumed that es is larger or equal to ea

s, thus Tu must lie between Ts and Td. The first derivative of the SVP curve at Ts and at Td represents the slope of the curve between those points. It can also be computed from the linearized SVP curve between the intervals [Tu,Ts] and [Td,Tu], which are symbolized as 1 and 2, respectively. Thus, an expression for

Tu is derived from a simple system of two equations with two unknowns, as follows,

\*

*u*

1 2

2 1 *s a*

*e e Ts Td*

2.13\* 0.583 Rn e *NDVI G* for NDVI > 0 (11)

*G* 0.583 Rn for NDVI 0 (12)

*T* (13)

assumed to be homogenous over the 5x5 km grid.

results between equations (7) and (8).

(2005), from where we took the Rn maps.

MOD021KM products. The equations used are

**5.1 Variables calculation** 

the MODIS Ta product.

and lower or equal to e\*

**5. Results** 

respectively.

The results obtained from equations (7) and (8) are compared to demonstrate the strength of the complementary relationship. The contrasted results were computed assuming k=2, α=1.26, =0.67 hPa/C, was obtained with Ta maps, estimating F as proposed in Venturini et al. (2008). The resulting ET estimates are shown in Table 4, where average root mean square errors (RMSEs) and biases are about 25 Wm-2, indicating that equation (7), obtained with Bouchet`s complementary model, would lead to larger ET estimates. However, only the "ground truth" would tell which equation is more precise. In this case, the ground truth is considered to be the ground measurements of ET described in section 4.2. Then, observed ET values were compared with the results obtained using equations (7) and (8), (see Figure 7). The overall RMSE is about 52.29 and the bias (Observed-Bouchet) is –37.90 Wm-2. For Granger`s CR, the overall RMSE and bias (Observed-Granger) are 33.89 and -10.96 Wm-2 respectively, with an R2 of about 0.79.


Table 4. ET(Wm-2) comparison between Bouchet´s and Granger´s CR.

From Table 4 it can be concluded that Bouchet's simplification results in larger ET estimates, with biases up to approximately 32 Wm-2, than those obtained with Granger`s CR. From Figure 7 it can be seen that Bouchet`s CR overestimates ground observations as well.

Ramírez et al. (2005) derived the value of Bouchet´s k parameter from ground data. The authors presented evidences of the complementary relationship from independent measurements of ET and Epot. Then, k values were calculated for different hypothesis. These authors reported a mean k of about 2.21 and a k variance equal to 0.07 using uncorrected pan evaporation data as a surrogate of Epot.

In this chapter, equations (7) and (8) are equated and k calculated for instantaneous ET values. Thus,

Fig. 5. Net radiation map of the SGP for April 1st, 2003

$$\frac{F\Delta}{F\Delta+\gamma} = \frac{kF\Delta}{(F+1)(\Delta+\gamma)}\tag{16}$$

$$k = \frac{(F+1)(\Lambda + \gamma)}{F\Lambda + \gamma} \tag{17}$$

Bouchet's coefficient k was calculated for each pixel in every day. The overall mean k value is 2.341, with an overall minimum of 1.784 and a maximum of 2.710, standard deviations varying from 0.025 to 0.078. These results are close to those reported by Ramírez et al. (2005).

From Table 4 it can be concluded that Bouchet's simplification results in larger ET estimates, with biases up to approximately 32 Wm-2, than those obtained with Granger`s CR. From

Ramírez et al. (2005) derived the value of Bouchet´s k parameter from ground data. The authors presented evidences of the complementary relationship from independent measurements of ET and Epot. Then, k values were calculated for different hypothesis. These authors reported a mean k of about 2.21 and a k variance equal to 0.07 using

In this chapter, equations (7) and (8) are equated and k calculated for instantaneous ET

( 1)( ) *F kF*

( 1)( )

Bouchet's coefficient k was calculated for each pixel in every day. The overall mean k value is 2.341, with an overall minimum of 1.784 and a maximum of 2.710, standard deviations varying from 0.025 to 0.078. These results are close to those reported by

*<sup>F</sup> <sup>k</sup>*

*F F* (16)

*<sup>F</sup>* (17)

Figure 7 it can be seen that Bouchet`s CR overestimates ground observations as well.

uncorrected pan evaporation data as a surrogate of Epot.

Fig. 5. Net radiation map of the SGP for April 1st, 2003

Ramírez et al. (2005).

values. Thus,

Fig. 6. Tu map of the SGP for April 1st, 2003

Fig. 7. Comparison between Bouchet's and Granger`s complementary models against ground measurements

Both complementary models yield similar ET estimates, however Granger´s model lead to more accurate results than Bouchet's method. The slope of the SVP curve at the air temperature sets a k value slightly different from 2.

## **6. Spatial and temporal scales considerations**

The complementary theory assumes a surface without advection influences and so does the regional evapotranspiration concept (Penman, 1948; Priestly & Taylor, 1972; Brutsaert & Stricker 1979). In fact, in his original work, Bouchet (1963) described five scales implicated in the oasis effect (see Table 5). Therefore for each scale of heterogeneity (*s*), we can define the oasis effects that give the lateral energy exchange of Q1, Q2, Q3, Q4, Q5. In the development of his theory he assumed that only Q3 is variable with ET while Q4 and Q5 are not affected by changes of ET and Epot associated with water availability. For the other two scales, *s*1 and *s*2, *Q*<sup>1</sup> and *Q*2 are not involved in the complementary relationship. Bouchet´s experiment established an energy balance over 24 hours, avoiding taking into account the phenomena of accumulation and restoration of heat during the day and night phases. These particular assumptions left smaller time and space scales out of the CR, therefore a review of the scales of applicability of the CR might be interesting.

The "evaporation paradox" mentioned by Brutsaert & Parlange (1998) refers to the seemingly opposing trends observed between pan evaporation and actual evaporation. The authors suggested that the paradox is solved in the CR framework.

The usefulness of the CR for understanding global scale in climate studies have been analized by Brutsaert & Parlange (1998)*,* Szilagyi (2001) and Hobbins et al. (2001), among others. Szilagy & Josza, (2009), coupled Bouchet´s CR with a long-term water-energy balance based on considerations of the precipitation time series and the soil water balance. The authors show that important ecosystem characteristics, such as the maximum soil water storage, can be derived from this "long-term" application of the CR. The scales shown in Table 5 seem to be compatible with those used in the aforementioned works. Nonetheless, the applicability of the CR at small scales is not evident from Bouchet´s publication.

Crago & Crowley (2005*)* evaluated the complementary relationship at relatively small temporal scales (10 to 30 min) using data from meteorological stations in different grassland sites. The authors demonstrated that the CR holds true also at small scales. Kahler and Brutsaert (2006) used properly scaled data of daily ET and daily pan evaporation observed at two experimental sites to demonstrate the validity of the CR. The CR at daily scales was confirmed by this research. The authors argue that for unscaled daily data of pan evaporation the CR may not be noticeable.


Table 5. Translation of Table 1 published by Bouchet in 1963

In a more practical way, the method proposed by Venturini et al. (2008) corrects the ET from a saturated surface with the local surface-atmosphere conditions at the pixel scale. The absence of regional assumptions makes the method applicable to a wide range of spatial scales even though the background of their method is Granger´s CR. Venturini´s method has been applied with instantaneous data, i.e. remotely sensed data with MODIS. The comparison between observed and estimated ET values yields errors of about 15% of observed instantaneous ET( Venturini et al., 2011).

### **7. References**

36 Evapotranspiration – Remote Sensing and Modeling

Both complementary models yield similar ET estimates, however Granger´s model lead to more accurate results than Bouchet's method. The slope of the SVP curve at the air

The complementary theory assumes a surface without advection influences and so does the regional evapotranspiration concept (Penman, 1948; Priestly & Taylor, 1972; Brutsaert & Stricker 1979). In fact, in his original work, Bouchet (1963) described five scales implicated in the oasis effect (see Table 5). Therefore for each scale of heterogeneity (*s*), we can define the oasis effects that give the lateral energy exchange of Q1, Q2, Q3, Q4, Q5. In the development of his theory he assumed that only Q3 is variable with ET while Q4 and Q5 are not affected by changes of ET and Epot associated with water availability. For the other two scales, *s*1 and *s*2, *Q*<sup>1</sup> and *Q*2 are not involved in the complementary relationship. Bouchet´s experiment established an energy balance over 24 hours, avoiding taking into account the phenomena of accumulation and restoration of heat during the day and night phases. These particular assumptions left smaller time and space scales out of the CR, therefore a review of the scales

The "evaporation paradox" mentioned by Brutsaert & Parlange (1998) refers to the seemingly opposing trends observed between pan evaporation and actual evaporation. The

The usefulness of the CR for understanding global scale in climate studies have been analized by Brutsaert & Parlange (1998)*,* Szilagyi (2001) and Hobbins et al. (2001), among others. Szilagy & Josza, (2009), coupled Bouchet´s CR with a long-term water-energy balance based on considerations of the precipitation time series and the soil water balance. The authors show that important ecosystem characteristics, such as the maximum soil water storage, can be derived from this "long-term" application of the CR. The scales shown in Table 5 seem to be compatible with those used in the aforementioned works. Nonetheless,

Crago & Crowley (2005*)* evaluated the complementary relationship at relatively small temporal scales (10 to 30 min) using data from meteorological stations in different grassland sites. The authors demonstrated that the CR holds true also at small scales. Kahler and Brutsaert (2006) used properly scaled data of daily ET and daily pan evaporation observed at two experimental sites to demonstrate the validity of the CR. The CR at daily scales was confirmed by this research. The authors argue that for unscaled daily data of pan

(symbol in the text) Timescales Spatial Scales Effects of oasis

minutes few hundred meters *Q2*

hours few kilometers *Q3*

Molecular - s*<sup>1</sup>* 10'9 second few hundred meters *Q1*

Cyclonic - s*<sup>4</sup>* 3 to 4 hours 1000 a 2000 kilometers *Q4* Global - s*<sup>5</sup>* 10 to 30 hours 5000 to 10000 km *Q5*

1 second to some

10 minutes to a few

Table 5. Translation of Table 1 published by Bouchet in 1963

corresponding

the applicability of the CR at small scales is not evident from Bouchet´s publication.

temperature sets a k value slightly different from 2.

of applicability of the CR might be interesting.

evaporation the CR may not be noticeable.

Scale

Turbulent - *s2*

Convection and related movements - s*<sup>3</sup>*

authors suggested that the paradox is solved in the CR framework.

**6. Spatial and temporal scales considerations** 


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## **Evapotranspiration Estimation Using Soil Water Balance, Weather and Crop Data**

Ketema Tilahun Zeleke and Leonard John Wade *School of Agricultural and Wine Sciences, EH Graham Centre for Agricultural Innovation, Charles Sturt University Australia* 

## **1. Introduction**

40 Evapotranspiration – Remote Sensing and Modeling

Su, B. (2002). The surface energy balance system (SEBS) for estimation of turbulent heat fluxes. *Hydrology and Earth System Sciences,* 6, pp. 85-99, ISSN 1027-5606. Sugita, M., Usui, J., Tamagawa, I. & Kaihotsu, I. (2001).Complementary relationship with a

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Szilagyi J., & Jozsa J. (2009). Analytical solution of the coupled 2-D turbulent heat and

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Venturini, V., Islam, S., & Rodríguez, L., (2008). Estimation of evaporative fraction and

Venturini, V., Rodriguez L., & Bisht G. (2011). A comparison among different modified

Xu, C.Y., & Singh, V.P. (2005). Evaluation of three complementary relationship

Wan, Z. (1999). *MODIS Land-Surface Temperature Algorithm Basis Document (LST ATBD*),

version 3.3, NASA, www.icess.ucsb.edu/modis/atbd-mod-11.pdf.

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vapor transport equations and the complementary relationship of evaporation .

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evapotranspiration from MODIS products using a complementary based model.

Priestley and Taylor´s equation to calculate actual evapotranspiration".

evapotranspiration models by water balance approach to estimate actual regional evapotranspiration in different climatic regions. *Journal of Hydrology*, 308, pp. 105The rise in water demand for agriculture, industry, domestic, and environmental needs requires sagacious use of this limited resource. Since agriculture (mainly irrigation) is the major user of water, improving agricultural water management is essential. Efficient agricultural water management requires reliable estimation of crop water requirement (evapotranspiration). Evapotranspiration (ET) is the transfer of water from the soil surface (evaporation) and plants (transpiration) to the atmosphere. ET is a critical component of water balance at plot, field, farm, catchment, basin or global level. From an agricultural point of view, ET determines the amount of water to be applied through artificial means (irrigation). Reliable estimation of ET is important in that it determines the size of canals, pumps, and dams. The use of the terms 'reference evapotranspiration', 'potential evapotranspiration', 'crop evapotranspiration', 'actual evapotranspiration' in this chapter is based on FAO-56 (FAO Irrigation and Drainage publication No 56) (Allen et al., 1998).

There are different methods of determining evapotranspiration: direct measurement, indirect methods from weather data and soil water balance. These methods can be generally classified as empirical methods (eg. Thornthwaite, 1948; Blaney and Criddle, 1950) and physical based methods (eg. Penman, 1948; Montheith, 1981 and FAO Penman Montheith (Allen et al., (1998)). They vary in terms of data requirement and accuracy. At present, the FAO Penman Montheith approach is considered as a standard method for ET estimation in agriculture (Allen et al., 1998). A case study from a semiarid region of Australia will be used to demonstrate ET estimation for a canola (*Brassica napus* L*.*) crop using soil water balance and crop coefficient approaches. Daily rainfall data, soil moisture measurement data using neutron probe, and AquaCrop (Steduto et al., 2009) -estimated deep percolation below the crop root zone will be used to determine actual evapotranspiration of the crop using soil water balance. Reference evapotranspiration ETo will be determined using FAO *ETo calculator* (Raes, 2009). Crop canopy cover measured using a handheld *GreenSeekerTM* and expressed as normalized difference vegetation index (NDVI) will be used to interpret evolution of evapotranspiration during the growing season (life cycle) of the canola crop.

## **2. Field experiment**

#### **2.1 Description of study area and field experiment**

The study area is in Wagga Wagga, New South Wales (Australia). Wagga Wagga, referred to as 'the capital of Riverina', is located in the Riverina region of NSW. The Riverina extends from the foot hills of the Great Dividing Range in the east to the flat and dry inland plains in the west. Agriculture in the Riverina is significantly diversified with dry land farming of winter cereals and irrigation in Murrumbidgee and Colleambally irrigation areas. It has a Mediterranean type climate with a mixed farming system of winter cereal crops, summer crops, and pastures grazing lands. In addition to the major grain crops of rice, canola, wheat, and maize, the area also produces a quarter of NSW fruit and vegetable production (RDA, 2011). The Riverina region is characterized by the semiarid climate, with hot summers and cool winters (Stern et al., 2000). Seasonal temperature varies little across the region. More consistent rainfall occurs in winter months. Mean annual temperature is 15-18oC. January is the hottest month of the year while July is the coolest. Mean annual rainfall varies from 238 mm in the west to 617 mm in the east. Long term and 2010 mean monthly rainfall, reference evapotranspiration, and temperature are presented in Fig. 1. Rainfall in 2010 was much higher than the long term average while evapotranspiration in 2010 was lower than the long term average.

Fig. 1. (a) Rain and reference evapotranspiration ETo (long term average and in 2010) (b) Monthly average temperature (long term average and in 2010) at Wagga Wagga, NSW (Australia).

A field experiment was carried out during the growing season of 2010 at canola field experimental site of Wagga Wagga Agricultural Research Institute located at Wagga Wagga (35o03'N; 147o21'E; 235 m asl), NSW (Australia). There was enough rainfall (930 mm) in contrast to long term average of 522 mm in 2010 to provode ideal growing conditions. A popular variety of canola (Hyola50) was sown on 30 April 2010. The experiment was conducted on a 24 m x 24 m area. There were 24 plots, 12 experimental plots and 12 buffer plots. The plots were 6 m long with 1 meter buffer on either end. Plot width was 1.8 m with a 0.5 m walking strip between plots for data collection.

About a month before the experimental season, neutron probe access tubes were installed to a depth of 1.5 m for soil moisture measurement. Two access tubes were installed at 2 m from either end of the plot and 2 m from each other. Soil moisture content was measured at 15, 30, 45, 60, 90, and 120 cm depths every two weeks. The probe was calibrated using gravimetric soil moisture measurements done when access tubes were installed on site.

## **2.2 Weather data**

42 Evapotranspiration – Remote Sensing and Modeling

The study area is in Wagga Wagga, New South Wales (Australia). Wagga Wagga, referred to as 'the capital of Riverina', is located in the Riverina region of NSW. The Riverina extends from the foot hills of the Great Dividing Range in the east to the flat and dry inland plains in the west. Agriculture in the Riverina is significantly diversified with dry land farming of winter cereals and irrigation in Murrumbidgee and Colleambally irrigation areas. It has a Mediterranean type climate with a mixed farming system of winter cereal crops, summer crops, and pastures grazing lands. In addition to the major grain crops of rice, canola, wheat, and maize, the area also produces a quarter of NSW fruit and vegetable production (RDA, 2011). The Riverina region is characterized by the semiarid climate, with hot summers and cool winters (Stern et al., 2000). Seasonal temperature varies little across the region. More consistent rainfall occurs in winter months. Mean annual temperature is 15-18oC. January is the hottest month of the year while July is the coolest. Mean annual rainfall varies from 238 mm in the west to 617 mm in the east. Long term and 2010 mean monthly rainfall, reference evapotranspiration, and temperature are presented in Fig. 1. Rainfall in 2010 was much higher than the long term average while evapotranspiration in 2010 was lower than the long

0

Fig. 1. (a) Rain and reference evapotranspiration ETo (long term average and in 2010) (b) Monthly average temperature (long term average and in 2010) at Wagga Wagga, NSW

A field experiment was carried out during the growing season of 2010 at canola field experimental site of Wagga Wagga Agricultural Research Institute located at Wagga Wagga (35o03'N; 147o21'E; 235 m asl), NSW (Australia). There was enough rainfall (930 mm) in contrast to long term average of 522 mm in 2010 to provode ideal growing conditions. A popular variety of canola (Hyola50) was sown on 30 April 2010. The experiment was conducted on a 24 m x 24 m area. There were 24 plots, 12 experimental plots and 12 buffer plots. The plots were 6 m long with 1 meter buffer on either end. Plot width was 1.8 m with

About a month before the experimental season, neutron probe access tubes were installed to a depth of 1.5 m for soil moisture measurement. Two access tubes were installed at 2 m from

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2010 Mean

Months

5

10

15

Temperature (oC)

20

25

30

**2. Field experiment** 

term average.

0

(Australia).

50

100

150

Rain and Evapotranspiration (mm)

200

250

**2.1 Description of study area and field experiment** 

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Rain 2010 Rain Mean ETo 2010 ETo Mean

**a b**

Months

a 0.5 m walking strip between plots for data collection.

Daily weather data (rainfall, minimum and maximum temperature, solar radiation, relative humidity, and wind speed) were collected from the meteorological station of the Wagga Wagga Agricultural Institute located adjacent to the experimental site. Out of the total annual rainfall of 930 mm, the amount or proportion (in percentage) during the canola growing season (May to November) was 514 mm (53%) while the long term average was 333 mm (64% of the long term average of 522 mm). Monthly average maximum and minimum temperature was 26oC and 3oC respectively. Reference evapotranspiration ETo was calculated using the procedure described in the FAO Irrigation and Drainage Paper 56 (Allen et al., 1998) with the help of the program FAO *ETo Calculator* (Raes, 2009).

## **2.3 Soil hydraulic characteristics**

A 1.5m x 1.5m x 1.5m soil trench was dug for soil texture, field capacity (*θFC*), and wilting point (*θWP*) determination. Soil samples were retrieved from 0-30, 30-60, 60-90, and 90-120 cm depths for soil texture, *θFC*, and *θWP* determination using standard laboratory procedures hydrometer and pressure plate apparatus apparatus.

## **2.4 Crop parameters**

The following crop phenological stages were recorded during the growing season: planting date, 90% emergence, beginning and end of flowering, senescence and maturity. The canopy cover was measured using *GreenSeekerTM*, an Optical Sensor Unit (NTech Industries, Inc., USA). G*reenSeekerTM*, is a handheld tool that determines Normalized Difference Vegetative Index (NDVI), is an integrated optical sensing and application system that measures green crop canopy cover.

## **3. Soil water balance method**

Rain or irrigation reaching a unit area of soil surface, may infiltrate into the soil, or leave the area as surface runoff. The infiltrated water may (a) evaporate directly from the soil surface, (b) taken up by plants for growth or transpiration, (c) drain downward beyond the root zone as deep percolation, or (d) accumulate within the root zone. The water balance method is based on the conservation of mass which states that change in soil water content ∆S of a root zone of a crop is equal to the difference between the amount of water added to the root zone, Qi, and the amount of water withdrawn from it, Qo (Hillel, 1998) in a given time interval expressed as in Eq. (1).

$$
\Delta S = Q\_i - Q\_o \tag{1}
$$

Eq. (1) can be used to determine evapotranspiration of a given crop as follows

$$ET = P + I + \mathcal{U} - R - D - \Delta S \tag{2}$$

where ∆S = change in root zone soil moisture storage, P = Precipitation, I = Irrigation, U = upward capillary rise into the root zone, R = Runoff, D = Deep percolation beyond the root zone, ET = evapotranspiration. All quantities are expressed as volume of water per unit land area (depth units).

In order to use Eq. (2) to determine evapotranspiration (ET), other parameters must be measured or estimated. It is relatively easy to measure the amount of water added to the field by rain and irrigation. In agricultural fields, the amount of runoff is generally small so is often considered negligible. When the groundwater table is deep, capillary rise U is negligible. The most difficult parameter to measure is deep percolation D. If soil water potential and moisture content are monitored, D can be estimated using Darcy's Principle. In this study, deep percolation estimated using AquaCrop (Raes et al., 2009), was adopted. Runoff R was also estimated using AquaCrop following USDA curve number approach (Hawkins et al., 1985). The change in soil water storage ∆S is measured using specialized instruments such as neutron probe and time-domain reflectrometer.

#### **4. Crop coefficient method**

#### **4.1 Introduction**

The crop coefficient approach relates evapotranspiration from a reference crop surface (ETo) to evapotranspiration from a given crop (ETc) through a coefficient. Estimation of crop water requirement from weather and crop data is a simpler and cost effective method compared to other methods such as soil water balance method. In this method, potential evapotranspiration of a crop is presumed to be determined by the evaporative demand of the atmosphere and crop characteristics. Evaporative demand of the air is determined as the evapotranspiration from a reference crop. The reference crop is a hypothetical crop (grass or alfalfa) with specific characteristics such as crop height of 0.12 m and albedo of 0.23 (Allen et al., 1998). Penman (1956) defined reference evapotranspiration as "the amount of water transpired in unit time by a shorter green crop, completely shading the ground, of uniform height and never short of water." It is a useful standard of reference for the comparison of different regions and of different measured evapotranspiration values within a given region. As such, ETo is a climatic parameter expressing the evaporation power of the atmosphere independent of crop type, crop development and management practices (Allen et al., 1998). FAO Penman Montheith approach is considered as the standard method. In this method, reference evapotranspiration ETo is estimated from weather data as given in Eq. (3).

$$ET\_o = \frac{0.408\Delta \left(R\_n - G\right) + \gamma \frac{900}{T + 273} \mu\_2 \left(e\_s - e\_a\right)}{\Delta + \gamma \left(1 + 0.34\mu\_2\right)}\tag{3}$$

where ETo = reference evapotranspiration (mm/day); Rn = net radiation at the crop surface (MJ/m² day); G = soil heat flux density (MG/m² day); T = air temperature at 2 m height (°C); u2 = wind speed at 2 m height (m/s); es= saturation vapor pressure (kPa); ea = actual vapor pressure (kPa); es-ea = saturation vapor pressure deficit (kPa); = slope vapor pressure curve (kPa/°C); = psychrometric constant (kPa/°C).

Reference evapotranspiration ETo can be calculated using a spreadsheet or computer programs which are designed for various level of data availability eg. *CROPWAT* (Smith, 1992) and *ETo Calculator* (Raes, 2009). In this study, the latter program was used. It is important to make clear distinction between reference evapotranspiration ETo and potential crop evapotranspiration ETc. The latter is also called maximum crop evapotranspiration. Evapotranspiration from a given crop grown and managed under standard conditions is called potential crop evapotranspiration ETc. Standard condition is a disease-free, wellfertilized crops, grown in large fields, under optimum soil water conditions, and achieving full production under the given climatic conditions. ETo depends evapotranspiration (ETc) represents the climatic "demand" for water by a given crop. Potential crop depends primarily on the evaporative demand of the air.

#### **4.2 Single crop coefficient method**

44 Evapotranspiration – Remote Sensing and Modeling

zone, ET = evapotranspiration. All quantities are expressed as volume of water per unit land

In order to use Eq. (2) to determine evapotranspiration (ET), other parameters must be measured or estimated. It is relatively easy to measure the amount of water added to the field by rain and irrigation. In agricultural fields, the amount of runoff is generally small so is often considered negligible. When the groundwater table is deep, capillary rise U is negligible. The most difficult parameter to measure is deep percolation D. If soil water potential and moisture content are monitored, D can be estimated using Darcy's Principle. In this study, deep percolation estimated using AquaCrop (Raes et al., 2009), was adopted. Runoff R was also estimated using AquaCrop following USDA curve number approach (Hawkins et al., 1985). The change in soil water storage ∆S is measured using specialized

The crop coefficient approach relates evapotranspiration from a reference crop surface (ETo) to evapotranspiration from a given crop (ETc) through a coefficient. Estimation of crop water requirement from weather and crop data is a simpler and cost effective method compared to other methods such as soil water balance method. In this method, potential evapotranspiration of a crop is presumed to be determined by the evaporative demand of the atmosphere and crop characteristics. Evaporative demand of the air is determined as the evapotranspiration from a reference crop. The reference crop is a hypothetical crop (grass or alfalfa) with specific characteristics such as crop height of 0.12 m and albedo of 0.23 (Allen et al., 1998). Penman (1956) defined reference evapotranspiration as "the amount of water transpired in unit time by a shorter green crop, completely shading the ground, of uniform height and never short of water." It is a useful standard of reference for the comparison of different regions and of different measured evapotranspiration values within a given region. As such, ETo is a climatic parameter expressing the evaporation power of the atmosphere independent of crop type, crop development and management practices (Allen et al., 1998). FAO Penman Montheith approach is considered as the standard method. In this method,

reference evapotranspiration ETo is estimated from weather data as given in Eq. (3).

<sup>900</sup> 0.408

where ETo = reference evapotranspiration (mm/day); Rn = net radiation at the crop surface (MJ/m² day); G = soil heat flux density (MG/m² day); T = air temperature at 2 m height (°C); u2 = wind speed at 2 m height (m/s); es= saturation vapor pressure (kPa); ea = actual vapor pressure (kPa); es-ea = saturation vapor pressure deficit (kPa); = slope vapor

Reference evapotranspiration ETo can be calculated using a spreadsheet or computer programs which are designed for various level of data availability eg. *CROPWAT* (Smith, 1992) and *ETo Calculator* (Raes, 2009). In this study, the latter program was used. It is important to make clear distinction between reference evapotranspiration ETo and potential crop evapotranspiration ETc. The latter is also called maximum crop evapotranspiration.

*T ET*

*o*

pressure curve (kPa/°C); = psychrometric constant (kPa/°C).

 

*R G ue e*

1 0.34 *n s a*

2

(3)

2

273

*u*

instruments such as neutron probe and time-domain reflectrometer.

area (depth units).

**4. Crop coefficient method** 

**4.1 Introduction** 

The single crop coefficient (Kc) method is used to determine soil evaporation and transpiration lumped over a number of days or weeks. The single "time-averaged" Kc curve incorporates averaged transpiration and soil wetting effects into a single Kc factor. The FAO-56 publication divides the crop growth stages into four phenological stages. Initial stage is from planting to 10% ground cover. Development stage is from 10% groundcover to maximum cover. Midseason stage is from the beginning of full cover to the start of senescence. The late season stage is from the start of senescence to full senescence or harvest. The evolution of crop coefficients during these stages is tabulated in FAO-56 for a number of crops including canola. Three coefficients are given for the initial, midseason, and end of season stages as Kc ini, Kc mid, and Kc end respectively. Kc ini is assumed to be constant and relatively small (<0.4). The Kc begins to increase during the crop development stage and reaches a maximum value Kc mid which is relatively constant for most growing and cultural conditions. During the late season period, as leaves begin to age and senesce, the Kc begins to decrease until it reaches a lower value at the end of the growing period equal to Kc end. The Kc during the development is estimated using linear interpolation between Kc ini and Kc mid. Similarly, Kc during the late season stage is determined using linear interpolation between Kc mid and Kc end. The value of Kc ini and Kc end can vary considerably on a daily basis, depending on the frequency of wetting by irrigation and rainfall. The single crop coefficient method can be used for irrigation planning and design. It is accurate enough for systems with large interval such as surface and set sprinkler irrigation. It is also used for catchment level hydrologic water balance studies (Allen et al., 1998).

In the single crop coefficient method, potential crop evapotranspiration ETc is estimated from a single crop coefficient (Kc) and reference evapotranspirations ETo as in Eq. (4).

$$ET\_c = ET\_o \mathbb{K}\_c \tag{4}$$

Eq. (4) gives the potential (maximum) evapotranspiration of the crop when the soil moisture is not limiting. Since localized Kc values are not always available in many parts of the world, the values of Kc as suggested by FAO (Allen et al., 1998) are being widely used to estimate evapotranspiration.

When rainfall amount and irrigation are not sufficient to keep the soil moisture high enough, the soil moisture content in the root zone is reduced to levels too low to sustain the potential crop evapotranspiration ETc. This results in an evapotranspiration less than the potential, and the plants are said to be under water stress. This evapotranspiration is called actual evapotranspiration (ETa). In general, the actual evapotranspiration ETa from various crops will not be equal to the potential value ETc. Actual evapotranspiration ETa is generally a fraction of ETc depending on soil moisture availability. Actual evapotranspiration ETa from a well-watered crop might generally approach ETc during the active growing stage, but may fall below during the early growth stage, prior to full canopy coverage, and again toward the end of the growing season as the matured plant starts to dry out (Hillel, 1997). The actual evapotranspiration ETa is calculated by combining the effects of Kc and soil water stress coefficient (Ks) as shown in Eq. (5).

$$ET\_a = ET\_o \mathcal{K}\_c \mathcal{K}\_s \tag{5}$$

The stress reduction coefficient Ks [0-1] reduces Kc when the average soil water content of the root zone is not high enough to sustain full crop transpiration. The stress coefficient Ks is determined by the amount of moisture the crop depleted from the rootzone of a crop. The amount of water depleted from the rootzone is expressed by root zone depletion Dr, i.e. water storage relative to field capacity. Stress is presumed to initiate when Dr exceeds the readily available water (RAW), Fig. 2. When more than RAW is extracted from the rootzone (Dr >RAW), Ks is expressed (Allen et al., 1998) as

$$K\_s = \frac{TAMV - D\_r}{TAMV - RAW} = \frac{TAMV - D\_r}{(1 - p)TAMV} \tag{6}$$

Where TAW = total plant available soil water in the root zone (mm), and p = fraction of TAW that a crop can extract from the root zone without suffering water stress. When Dr ≤ RAW, Ks =1 indicating no water stress. The total available water in the root zone (TAW, mm) is estimated as the difference between the water content at the field capacity and wilting point

$$TAV = 1000 \left(\theta\_{\rm FC} - \theta\_{\rm WP}\right) Z\_r \tag{7}$$

Where Zr = effective rooting depth (m); θFC is soil moisture content at field capacity (m3 m-3); θWP is soil moisture content at permanent wilting point (m3 m-3).

Fig. 2. Schematic of moisture stress coefficient (adapted from Allen et al., 1998).

Readily available water (RAW) is the amount of water which the crop can extract without experiencing stress. It is expressed as

$$\text{RAW} = \text{pTAW} \tag{8}$$

Soil moisture depletion fraction (p) is the fraction of soil water in the root zone that can be depleted before stress occurred. It varies from crop to crop and also varies at different growth stages of a given crop. Shallow rooted and sensitive crops such as vegetables have low p value while deep rooted and stress tolerant crops have a higher p value.

Canola crop coefficient values given in FAO 56 (Allen et al., 1998) are Kc ini = 0.35, Kc mid = 1.0-1.15, Kc end = 0.35. These values represent Kc for a sub humid climate with RHmin = 45% and wind speed of 2 m/s. To take account for impacts of differences in aerodynamic roughness between crops and the grass reference with changing climate, the Kc mid and Kc end values larger than 0.45 must be adjusted using the following equation:

$$K\_c = K\_{c\text{ (tab)}} + \left[0.04\left(\text{u}\_2 - 2\right) - 0.004\left(RH\_{\text{min}} - 45\right)\right] \left(\frac{h}{3}\right)^{0.3} \tag{9}$$

Where Kc (tab) is the value of Kc taken from Table 12 of Allen et al. (1998); h is the mean plant height during the mid or late season stage (m); RHmin the mean value for daily minimum relative humidity during the mid or late season growth stages (%) for 20%≤RHmin≤ 80%; u2 is the mean value for daily wind speed at 2 m during the mid season or late season stages (m/s) for 1m/s ≤ u2 ≤ 6 m/s. In this study, Kc ini = 0.35, Kc mid = 1.10, and Kc end = 0.35 were used. Accordingly, Kc mid value was adjusted to 1.08 for RHmin = 48%, u2 = 1.91 m/s, and plant height of 1.0 m during this stage. Since Kc end was less than 0.4, it was not necessary to adjust it. Once the Kcb values for the initial stage, mid season stage, and end-of-season stage were determined, Kcb values for development and late season stages were determined using linear interpolation.

#### **4.3 Dual crop coefficient method**

46 Evapotranspiration – Remote Sensing and Modeling

toward the end of the growing season as the matured plant starts to dry out (Hillel, 1997). The actual evapotranspiration ETa is calculated by combining the effects of Kc and soil water

The stress reduction coefficient Ks [0-1] reduces Kc when the average soil water content of the root zone is not high enough to sustain full crop transpiration. The stress coefficient Ks is determined by the amount of moisture the crop depleted from the rootzone of a crop. The amount of water depleted from the rootzone is expressed by root zone depletion Dr, i.e. water storage relative to field capacity. Stress is presumed to initiate when Dr exceeds the readily available water (RAW), Fig. 2. When more than RAW is extracted from the rootzone

> *r r <sup>s</sup> TAW D TAW D <sup>K</sup> TAW RAW p TAW*

Where TAW = total plant available soil water in the root zone (mm), and p = fraction of TAW that a crop can extract from the root zone without suffering water stress. When Dr ≤ RAW, Ks =1 indicating no water stress. The total available water in the root zone (TAW, mm) is estimated as the difference between the water content at the field capacity and

> 

Where Zr = effective rooting depth (m); θFC is soil moisture content at field capacity (m3 m-3);

*TAW* 1000

Fig. 2. Schematic of moisture stress coefficient (adapted from Allen et al., 1998).

experiencing stress. It is expressed as

Readily available water (RAW) is the amount of water which the crop can extract without

*ET ET K K a ocs* (5)

(6)

*FC WP r Z* (7)

stress coefficient (Ks) as shown in Eq. (5).

(Dr >RAW), Ks is expressed (Allen et al., 1998) as

wilting point

<sup>1</sup>

θWP is soil moisture content at permanent wilting point (m3 m-3).

The single coefficient method does not separate evaporation and transpiration components of evapotranspiration. The dual crop coefficient approach calculates the actual increase in Kc for each day as a function of plant development and the wetness of the soil surface. It is best for high frequency irrigation such as microirrigation, centre pivots, and linear move systems (Suleiman et al., 2007). The effects of crop transpiration and soil evaporation are determined separately using two coefficients: the basal crop coefficient (Kcb) to describe plant transpiration and the soil water evaporation coefficient (Ke) to describe evaporation from the soil surface, Eq (10). AquaCrop determines crop transpiration (Tr) and soil evaporation (E) by multiplying ETo with their specific coefficients Kcb and Ke (Eq. 11) (Steduto et al., 2009).

$$\mathbf{K}\_{\mathbf{c}} = \mathbf{K}\_{\mathbf{c}b} + \mathbf{K}\_{\mathbf{c}\mathbf{y}} \text{ and } \tag{10}$$

$$\rm ET\_c = (K\_{cb} + K\_c) \, ET\_o \tag{11}$$

The range of Kcb and Ke is [0-1.4]. When soil moisture is limiting, Kcb is multiplied by a coefficient Ks which is equal to 1 when Dr≤RAW and declines linearly to zero when all the available water in the rooting zone has been used. Evapotranspiration under such a condition is calculated using Eq. (12).

$$\rm ET\_a = (K\_s K\_{cb} + K\_e) \, ET\_o \tag{12}$$

Because the water stress coefficient impacts only crop transpiration, rather than evaporation from the soil, the application using Eq. (12) is generally more valid than is application using Eq. (5) in the single crop coefficient approach. Allen et al. (1998) reported that in situations where evaporation from soil is not a large component of ETc, use of Eq. (5) will provide reasonable results. The dual coefficient approach can be summarized into the following three steps: Calculate reference evapotranspiration (ETo) from climatic data using Eq. (3), calculate individual crops potential evapotranspiration ETc using Eq. (11), and when the soil moisture content is limited, Kcb coefficient is multiplied by stress factors Ks to calculate actual evapotranspiration ETa using Eq. (12).

#### **4.3.1 Basal crop coefficient**

The basal crop coefficient Kcb is defined as the ratio of ETc to ETo when the soil surface layer is dry but where the average soil water content of the rootzone is adequate to sustain full plant transpiration (Bonder et al., 2007). The dual crop coefficient approach uses daily time step and is readily adapted to spreadsheet program. Some models such as AquaCrop (Steduto et al., 2009) determine crop water productivity from the "productive" component of evapotranspiration i.e. transpiration. AquaCrop requires regression of daily values of biomass and crop transpiration to determine crop water productivity. Therefore, transpiration should be measured or estimated.

FAO-56 has tabulated Kcb values for a number of crops, including canola, at the initial, mid season, and end of season stages. Since localized Kcb values were not available for the study area, the values of Kcb suggested by FAO-56 (Allen et al., 1998) were used. For canola these value were Kcb ini = 0.15, Kcb mid = 0.95-1.10, and Kcb end = 0.25. In this study, Kcb of 0.15, 1, and 0.25, respectively, for the initial, mid-season, and end of season stages were selected. The growing season of canola vary from 5 months to 7 months in Australia i.e. 150 -210 days depending on the planting date and the weather conditions (rainfall and temperature) during the season. Initial, development, mid-season, and late season stage lengths for canola grown during the 2010 winter season in Wagga Wagga (Australia) were 10, 64, 84, 48 days respectively.

The values for Kcb in the FAO-56 table represent values for a sub humid climate with RHmin = 45% and wind speed of 2 m/s. To take account for impacts of differences in aerodynamic roughness between crops and the grass reference, the Kcb mid and Kcb end values larger than 0.45 must be adjusted using the following equation:

$$K\_{cb} = K\_{cb \text{ (tab)}} + \left[0.04 \left(\text{u}\_2 - 2\right) - 0.004 \left(RH\_{\text{min}} - 45\right)\right] \left(\frac{h}{3}\right)^3 \tag{13}$$

Where Kcb (tab) is the value of Kcb mid taken from Table 17 of Allen et al. (1998). The other parameters are as defined in Eq. (9). The Kcb values for the mid-season stage was adjusted using Eq. (13) to 0.98 for for RHmin = 48%, u2 = 1.91 m/s, and plant height of 1.0 m. Once the Kcb values for the initial stage, mid season stage, and end-of-season stage were determined, Kcb values for development and late season stages were determined using linear interpolation.

The Kcb coefficient for any period (day) of the growing season can be derived by considering that during the initial and mid-season stages Kcb is constant and equal to the Kcb value of the growth stage under consideration. During the crop development and late season stage, Kcb varies linearly between the Kcb at the end of the initial stage (Kc ini) and the Kcb at the beginning of the midseason stage (Kcb mid). During the mid season stage Kcb is constant as Kcb mid. During late season stage, Kcb varies linearly between Kcb mid and Kcb

end. In the case of canola the end of season Kcb does not need adjustment since it is 0.25 which is less than 0.45.

#### **4.3.2 Soil evaporation coefficient**

48 Evapotranspiration – Remote Sensing and Modeling

Eq. (5) in the single crop coefficient approach. Allen et al. (1998) reported that in situations where evaporation from soil is not a large component of ETc, use of Eq. (5) will provide reasonable results. The dual coefficient approach can be summarized into the following three steps: Calculate reference evapotranspiration (ETo) from climatic data using Eq. (3), calculate individual crops potential evapotranspiration ETc using Eq. (11), and when the soil moisture content is limited, Kcb coefficient is multiplied by stress factors Ks to calculate

The basal crop coefficient Kcb is defined as the ratio of ETc to ETo when the soil surface layer is dry but where the average soil water content of the rootzone is adequate to sustain full plant transpiration (Bonder et al., 2007). The dual crop coefficient approach uses daily time step and is readily adapted to spreadsheet program. Some models such as AquaCrop (Steduto et al., 2009) determine crop water productivity from the "productive" component of evapotranspiration i.e. transpiration. AquaCrop requires regression of daily values of biomass and crop transpiration to determine crop water productivity. Therefore,

FAO-56 has tabulated Kcb values for a number of crops, including canola, at the initial, mid season, and end of season stages. Since localized Kcb values were not available for the study area, the values of Kcb suggested by FAO-56 (Allen et al., 1998) were used. For canola these value were Kcb ini = 0.15, Kcb mid = 0.95-1.10, and Kcb end = 0.25. In this study, Kcb of 0.15, 1, and 0.25, respectively, for the initial, mid-season, and end of season stages were selected. The growing season of canola vary from 5 months to 7 months in Australia i.e. 150 -210 days depending on the planting date and the weather conditions (rainfall and temperature) during the season. Initial, development, mid-season, and late season stage lengths for canola grown during the 2010 winter season in Wagga Wagga (Australia) were 10, 64, 84, 48 days

The values for Kcb in the FAO-56 table represent values for a sub humid climate with RHmin = 45% and wind speed of 2 m/s. To take account for impacts of differences in aerodynamic roughness between crops and the grass reference, the Kcb mid and Kcb end values larger than

> (tab) <sup>2</sup> min 0.04 u 2 0.004 45 3 *cb cb <sup>h</sup> K K RH*

Where Kcb (tab) is the value of Kcb mid taken from Table 17 of Allen et al. (1998). The other parameters are as defined in Eq. (9). The Kcb values for the mid-season stage was adjusted using Eq. (13) to 0.98 for for RHmin = 48%, u2 = 1.91 m/s, and plant height of 1.0 m. Once the Kcb values for the initial stage, mid season stage, and end-of-season stage were determined, Kcb values for development and late season stages were determined using linear

The Kcb coefficient for any period (day) of the growing season can be derived by considering that during the initial and mid-season stages Kcb is constant and equal to the Kcb value of the growth stage under consideration. During the crop development and late season stage, Kcb varies linearly between the Kcb at the end of the initial stage (Kc ini) and the Kcb at the beginning of the midseason stage (Kcb mid). During the mid season stage Kcb is constant as Kcb mid. During late season stage, Kcb varies linearly between Kcb mid and Kcb

3

(13)

actual evapotranspiration ETa using Eq. (12).

transpiration should be measured or estimated.

0.45 must be adjusted using the following equation:

**4.3.1 Basal crop coefficient** 

respectively.

interpolation.

Similar to Kcb, soil evaporation coefficient Ke needs to be calculated on a daily basis. Ke is a function of soil water characteristics, exposed and wetted soil fraction, and top layer soil water balance (Allen et al., 2005). In the initial stage of crop growth, the fraction of soil surface covered by the crop is small, and thus, soil evaporation losses are considerable. Following rain or irrigation, Ke can be as high as 1. When the soil surface is dry, Ke is small and even zero. Ke is determined using Eq. (14).

$$K\_c = \min\{ [K\_r \left( \text{Kc } \max \text{ - K}\_{\text{cb}} \right)]\_l [f\_{ew} \text{ Kc } \max] \} \tag{14}$$

Where Kc max = maximum value of crop coefficient Kc following rain or irrigation; Kr = evaporation reduction coefficient which depends on the cumulative depth of water depleted; and few = fraction of the soil that is both wetted and exposed to solar radiation. Kc max represents an upper limit on evaporation and transpiration from the cropped surface. Kc max ranges [1.05-1.30] (Allen et al., 2005). Its value is calculated for initial, development, mid-season, or late season using Eq. 15.

$$K\_{c\text{ max}} = \max\left( \left\{ 1.2 + \left[ 0.04 \left( u\_2 - 2 \right) - 0.004 \left( RH\_{\text{min}} - 45 \right) \right] \left( \frac{h}{3} \right)^{0.3} \right\} , \left\{ K\_{cb} + 0.05 \right\} \right) \tag{15}$$

Evaporation occurs predominantly from the exposed soil fraction. Hence, evaporation is restricted at any moment by the energy available at the exposed soil fraction, i.e. Ke cannot exceed few x Kc max. The calculation of Ke consists in determining Kc max, Kr, and few. Kc max for initial, development, midseason, and late season stages were calculated to be 1.196, 1.181, 1.187, and 1.195 respectively.

#### **4.3.3 Evaporation reduction coefficient**

The estimation of evaporation reduction coefficient Kr requires a daily water balance computation for the surface soil layer. Evaporation from exposed soil takes place in two stages: an energy limiting stage (Stage 1) and a falling rate stage (Stage 2) (Ritchie 1972) as indicated in Fig. 3. During stage 1, evaporation occurs at the maximum rate limited only by energy availability at the soil surface and therefore, Kr = 1. As the soil surface dries, the evaporation rate decreases below the potential evaporation rate (Kc max – Kcb). Kr becomes zero when no water is left for evaporation in the evaporation layer. Stage 1 holds until the cumulative depth of evaporation De is depleted which depends on the hydraulic properties of the upper soil. At the end of Stage 1 drying, De is equal to readily evaporable water (REW). REW ranges from 5 to 12 mm and highest for medium and fine textured soils (Table 1 of Allen et al., 2005). The evolution of Kr is presented in Fig. 3.

The second stage begins when De exceeds REW. Evaporation from the soil decreases in proportion to the amount of water remaining at the surface layer. Therefore reduction in evaporation during stage 2 is proportional to the cumulative evaporation from the surface soil layer as expressed in Eq. (16).

$$K\_r = \frac{\text{TE}\mathcal{W} - D\_{e,j-1}}{\text{TE}\mathcal{W} - \text{RE}\mathcal{W}}\text{ for }\mathcal{D}\_{ij} \text{ > REW} \tag{16}$$

where De, j-1 = cumulative depletion from the soil surface layer at the end of previous day (mm); The TEW and REW are in mm. The amount of water that can be removed by evaporation during a complete drying cycle is estimated as in Eq. (17).

$$\text{TE} \text{W} = 1000 \left( \theta\_{\text{FC}} - 0.5 \theta\_{\text{WP}} \right) Z\_e \tag{17}$$

Where TEW =maximum depth of water that can be evaporated from the surface soil layer when the layer has been initially completely wetted (mm). θFC and θwp are in (m3 m-3) and Ze (m) = depth of the surface soil subject to evaporation. FAO-56 recommended values for Ze of 0.10-0.15m, with 0.10 m for coarse soils and 0.15 m for fine textured soils.

Fig. 3. Soil evaporation reduction coefficient Kr (adapted from Allen et al., 2005). REW stands for readily extractable water and TEW stands for total extractable water.

Calculation of Ke requires a daily water balance for the wetted and exposed fraction of the surface soil layer (few). Eq. (18) is used to determine cumulative evaporation from the top soil layer (Allen et al., 2005).

$$D\_{e,j} = D\_{e,j-1} - \left(P\_j - R\_j\right) - \frac{I\_j}{f\_w} + \frac{E\_j}{f\_{ew}} + T\_{ci,j} + D\_{ci,j} \tag{18}$$

where De,j-1 and De,j = cumulative depletion at the ends of days j-1 and j (mm); Pj and Rj = precipitation and runoff from the soil surface on day j (mm); Ij = irrigation on day j (mm); Ej = evaporation on day j (i.e., Ej = Ke x ETo) (mm); Tei,j = depth of transpiration from exposed and wetted fraction of the soil surface layer (few) on day j (mm); and Dei,j = deep percolation from the soil surface layer on day j (mm) if soil water content exceeds field capacity (mm). Assuming that the surface layer is at field capacity following heavy rain or irrigation, the minimum value of De,j is zero and limits imposed are 0≤De,j≤TEW. Tei can be ignored except for shallow rooted crops (0.5-0.6m).

Evaporation is greater between plants exposed to sunlight and with air ventilation. The fraction of the soil surface from which most evaporation occurs is few = 1-fc.

$$\mathbf{f\_{ew}} = \min(\mathbf{1} \mathbf{-f\_{cr}} \ \mathbf{f\_{w}}) \tag{19}$$

where De, j-1 = cumulative depletion from the soil surface layer at the end of previous day (mm); The TEW and REW are in mm. The amount of water that can be removed by

Where TEW =maximum depth of water that can be evaporated from the surface soil layer when the layer has been initially completely wetted (mm). θFC and θwp are in (m3 m-3) and Ze (m) = depth of the surface soil subject to evaporation. FAO-56 recommended values for

 *FC WP e* 

for De,j-1 > REW (16)

*Z* (17)

*<sup>e</sup>*, 1 *<sup>j</sup>*

*r*

*K*

*TEW* 1000 0.5

soil layer (Allen et al., 2005).

for shallow rooted crops (0.5-0.6m).

*TEW D*

evaporation during a complete drying cycle is estimated as in Eq. (17).

*TEW REW*

Ze of 0.10-0.15m, with 0.10 m for coarse soils and 0.15 m for fine textured soils.

Fig. 3. Soil evaporation reduction coefficient Kr (adapted from Allen et al., 2005). REW stands for readily extractable water and TEW stands for total extractable water.

, ,1 , ,

fraction of the soil surface from which most evaporation occurs is few = 1-fc.

Calculation of Ke requires a daily water balance for the wetted and exposed fraction of the surface soil layer (few). Eq. (18) is used to determine cumulative evaporation from the top

*e j e j jj ei j ei j*

where De,j-1 and De,j = cumulative depletion at the ends of days j-1 and j (mm); Pj and Rj = precipitation and runoff from the soil surface on day j (mm); Ij = irrigation on day j (mm); Ej = evaporation on day j (i.e., Ej = Ke x ETo) (mm); Tei,j = depth of transpiration from exposed and wetted fraction of the soil surface layer (few) on day j (mm); and Dei,j = deep percolation from the soil surface layer on day j (mm) if soil water content exceeds field capacity (mm). Assuming that the surface layer is at field capacity following heavy rain or irrigation, the minimum value of De,j is zero and limits imposed are 0≤De,j≤TEW. Tei can be ignored except

Evaporation is greater between plants exposed to sunlight and with air ventilation. The

*D D PR T D*

*j j*

(18)

few = min(1-fc, fw) (19)

*w ew I E*

*f f*

Where 1-fc = 1-CC; fw is fraction of soil surface wetted by irrigation or rainfall; fw is 1 for rainfall (Table 20 of Allen et al., 1998); fc is fraction of soil surface covered by vegetation. In this study fc is the canopy cover measured using *GreenSeekerTM*. Values of parameters used in the dual coefficient approach are presented in Table 1.


Table 1. The parameters of the soil used in the determination of Ks, Ke, and Kr in the FAO dual coefficient method.

The top soil layer (0-0.15 m) of the soil in this study is sandy clay loam. Readily extractable water (REW) is 9 mm for this soil texture (Table 1 of Allen et al., 2005). Field capacity and wilting point of this soil were determined as part of soil hydraulic properties characterization. Canola effective rooting depth was determined as part of National Brasicca Germaplasm Improvement Program (David Luckett, personal communication). Soil moisture content was monitored using on-site calibrated neutron probe. Soil moisture depletion fraction (p) of 0.6 m was taken from FAO-56 publication (Allen et al., 1998). Since the only source of water was rainfall, wetting fraction fw of 1 was used.

#### **4.4 AquaCrop approach of determining dual evapotranspiration coefficients**

Eq. (11) gives evapotranspiration when the soil water is not limiting. When the soil evaporation and transpiration drops below their respective maximum rates, AquaCrop simulates ETa by multiplying the crop transpiration coefficient with the water stress coefficient for stomatal closure (Kssto), and the soil water evaporation coefficient with a reduction Kr [0-1] (Steduto et al., 2009) as

$$\rm ET\_a = \left( K s\_{\rm sto} K\_{\rm cb} + K\_{\rm r} K\_{\rm c} \right) \,\rm ET\_o \tag{20}$$

AquaCrop calculates basal crop coefficient at any stage as a product of basal crop coefficient at mid-season stage Kcb(x) and green canopy cover (CC). For canola Kcb(x) = 0.95 was used.

$$\mathbf{K}\_{\text{cb}} = \mathbf{K}\_{\text{cb}(\text{x})} \times \mathbf{CC} \tag{21}$$

$$\mathbf{K\_{e}} = \mathbf{K\_{e(x)}} \propto \text{(1-CC)}\tag{22}$$

Evaporation from a fully wet soil surface is inversely proportional to the effective canopy cover. The proportional factor is the soil evaporation coefficient for fully wet and unshaded soil surface (Ke(x)) which is a program parameter with a default value of Ke(x) = 1.1 (Raes et al., 2009).

During the energy limiting (non-water limiting) stage of evaporation, maximum evaporation (Ex) is given by

$$\mathbf{E}\_{\alpha} = \mathbf{K}\_{\alpha} \mathbf{E} \mathbf{T}\_{\alpha} \quad \text{=} \begin{bmatrix} (\text{1-CC})\mathbf{K}\_{\text{ex}} \end{bmatrix} \mathbf{E} \mathbf{T}\_{\alpha} \tag{23}$$

Where CC is green canopy cover; Kex is soil evaporation coefficient for fully wet and non shaded soil surface (Steduto et al., 2009). In AquaCrop, Kex is a program parameter with a default value of 1.10 (Allen et al., 1998). When the soil water is limiting, actual evaporation rate is given by

$$\mathbf{E\_a = K\_r E\_x} \tag{24}$$

Maximum crop transpiration (Trx) for a well-watered crop is calculated as

$$\mathbf{T\_{rx}} = \mathbf{K\_{cb}} \, \text{ET}\_{o-} \, [\mathbf{CC} \, \mathbf{K\_{cb}}] \text{ET}\_{o} \tag{25}$$

Kcbx is the basal crop coefficient for well-watered soil and complete canopy cover.

#### **5. Results and discussion**

#### **5.1 Soil water balance**

The actual evapotranspiration determined using soil water balance method is presented in Table 2. Evapotranspiration was determined using Eq. (2) from measurement of 12 neutron probes several times during the season. Deep percolation and runoff were not measured. Therefore, values estimated by AquaCrop (Steduto et al., 2009; Raes et al., 2009) during the canola water productivity simulation were adopted.


Table 2. Evapotranspiration determined using soil water balance method for canola planted on 30 April 2010 at Wagga Wagga (Australia).

The runoff estimated using AquaCrop was low, supporting the consensus that runoff from agricultural land is low. However, deep percolation past the 1.2 m was significant. The actual annual crop evapotranspiration estimated using this method was 313 mm. It can be observed that evapotranspiration was higher during the mid season and highly evaporative months.

### **5.2 Evapotranspiration coefficient**

52 Evapotranspiration – Remote Sensing and Modeling

soil surface (Ke(x)) which is a program parameter with a default value of Ke(x) = 1.1 (Raes et

During the energy limiting (non-water limiting) stage of evaporation, maximum

 Ex = Ke ETo = [(1-CC)Kex]ETo (23) Where CC is green canopy cover; Kex is soil evaporation coefficient for fully wet and non shaded soil surface (Steduto et al., 2009). In AquaCrop, Kex is a program parameter with a default value of 1.10 (Allen et al., 1998). When the soil water is limiting, actual evaporation

Ea = KrEx (24)

Trx = Kcb ETo = [CC Kcbx]ETo (25)

The actual evapotranspiration determined using soil water balance method is presented in Table 2. Evapotranspiration was determined using Eq. (2) from measurement of 12 neutron probes several times during the season. Deep percolation and runoff were not measured. Therefore, values estimated by AquaCrop (Steduto et al., 2009; Raes et al., 2009) during the

> Runoff (mm)

Table 2. Evapotranspiration determined using soil water balance method for canola planted

0-13 6.5 0 0 -2.1 8.6 14-21 0 0 0 -1.8 1.8 22-28 36.9 4.6 0.5 13.4 18.4 29-35 23.4 24.6 1.4 -10 7.4 36-42 1.8 1.8 0 -3.1 3.1 43-49 6 2.2 0 -1.1 4.9 50-63 21.8 6.7 0 4.6 10.5 64-77 60 20.2 4.1 17.7 18 78-94 3.2 18.9 0 -25.6 9.9 95-118 58.7 21.2 1.6 6.7 29.2 119-143 81 34.3 3.8 -20.8 63.7 144-159 0 1.5 0 -39.6 38.1 160-173 103.9 8.6 14 30.3 51 174-196 31.6 3.8 0 -20.7 48.5 \*DAP stands for days after planting Seasonal 313

Change in storage (mm)

Evapotranspiration ETa using water balance (mm)

Maximum crop transpiration (Trx) for a well-watered crop is calculated as

Kcbx is the basal crop coefficient for well-watered soil and complete canopy cover.

al., 2009).

rate is given by

evaporation (Ex) is given by

**5. Results and discussion** 

DAP\* Rainfall

(mm)

on 30 April 2010 at Wagga Wagga (Australia).

canola water productivity simulation were adopted.

Deep percolation (mm)

**5.1 Soil water balance** 

Single and dual evapotranspiration coefficients and crop canopy cover data are presented in Fig. 4. The Kc and Kcb values adopted from FAO-56 publication and adjusted for the local condition are shown in the Figure. The Kc and Kcb curves follow similar trend as the measured canopy cover curve. The canopy cover values were higher than the Kc and Kcb curves towards the end of the season. This is due to the fact that as an indeterminate crop, canola still had green canopy due to the ample rainfall during this late season stage of the crop. The soil evaporation coefficient Ke was correctly simulated using the top-layer soil water balance model. It can be seen that Ke is high during the initial and late season stages. It remained low and steady during the midseason stage. The higher number of Ke spikes are

Fig. 4. Single crop coefficient (Kc), basal coefficient (Kcb), soil evaporation coefficient (Ke), crop canopy cover (CC) curves for canola having growth stage lengths of 10, 64, 84, and 48 days during initial, development, midseason, and late season stages. Indicated on curve are also single and basal crop coefficient (Kc and Kcb) at initial, midseason, and end of season stages. Day of planting is 30 April 2010.

due to frequent rainfall during the season. The Ke value estimated using AquaCrop followed similar trend to the manually calculated using Eq. (14). However, AquaCrop did not simulate response to individual rainfall events.

In the development stage, the soil surface covered by the crop gradually increases and the Ke value decreases. In the midseason stage, the soil surface covered by the crop reaches maximum and water loss is mainly by crop transpiration and Ke is as low as 0.05. In the late season stage, the Ke values are greater than that in the mid-season stage because of the senescence.

Evaporation and transpiration estimated using the dual coefficient approach (Fig. 5) are correctly simulated, with high evaporation during the initial and late stages, and low during the developmental and mid season stages. The fluctuation in the evaporation component is high at these stages and low and steady during the mid season stage except minor spikes after rainfall events. Evaporation during the late stage (late spring months) was high compared with the initial stage which is a winter period. The transpiration component was steady increasing during the crop development stage before reaching a maximum in late mid season stage and declined during the late season stage due to senescence. The trends in evaporation and transpiration were in perfect phase with the weather and crop phenology.

Fig. 5. Daily soil evaporation and transpiration estimated using dual coefficient method for canola planted on 30 April 2010 at Wagga Wagga, NSW (Australia).

Evapotranspiration varies during the growing period of a crop due to variation in crop canopy and climatic conditions (Allen et al., 1998). Variation in crop canopy changes the

due to frequent rainfall during the season. The Ke value estimated using AquaCrop followed similar trend to the manually calculated using Eq. (14). However, AquaCrop did not

In the development stage, the soil surface covered by the crop gradually increases and the Ke value decreases. In the midseason stage, the soil surface covered by the crop reaches maximum and water loss is mainly by crop transpiration and Ke is as low as 0.05. In the late season stage, the Ke values are greater than that in the mid-season stage because of the

Evaporation and transpiration estimated using the dual coefficient approach (Fig. 5) are correctly simulated, with high evaporation during the initial and late stages, and low during the developmental and mid season stages. The fluctuation in the evaporation component is high at these stages and low and steady during the mid season stage except minor spikes after rainfall events. Evaporation during the late stage (late spring months) was high compared with the initial stage which is a winter period. The transpiration component was steady increasing during the crop development stage before reaching a maximum in late mid season stage and declined during the late season stage due to senescence. The trends in evaporation and transpiration were in perfect phase with the weather and crop phenology.

> Evaporation Transpiration

0 30 60 90 120 150 180 210 240

Days after planting

Fig. 5. Daily soil evaporation and transpiration estimated using dual coefficient method for

Evapotranspiration varies during the growing period of a crop due to variation in crop canopy and climatic conditions (Allen et al., 1998). Variation in crop canopy changes the

canola planted on 30 April 2010 at Wagga Wagga, NSW (Australia).

simulate response to individual rainfall events.

senescence.

0

1

2

3

Evaporation and transpiration (mm)

4

5

6

proportion of evaporation and transpiration components of evapotranspiration. The spikes in basal crop coefficient were high during the initial and crop development phases and decreases as the soil dries (Fig. 4). The spikes decrease as the canopy closes and much of ET is by transpiration. During the late season stage, there were fewer spikes because soil evaporation was low and almost constant. The largest difference between Kc and Kcb is found in the initial growth stage where evapotranspiration is predominantly in the form of soil evaporation and crop transpiration. Because crop canopies are near or at full ground cover during the mid-season stage, soil evaporation beneath the canopy has less effect on crop transpiration and the value of Kcb in the mid season stage is very close to Kc. Depending on the ground cover, the basal crop coefficient during the mid season stage may be only 0.05-0.10 lower than the Kc value. In this study Kcb mid is 0.10 lower than Kc mid.

Some studies, carried out in different regions of the world, have compared the results obtained using the approach described by Allen et al. (1998) with those resulting from other methodologies. From this comparison, some limitations should be expected in the application of the dual crop coefficient FAO-56 approach. Dragoni et al. (2004), which measured actual transpiration in an apple orchard in cool, humid climate (New York, USA), showed a significant overestimation (over 15%) of basal crop coefficients by the FAO 56 method compared to measurements (sap flow). This suggests that dual crop coefficient method is more appropriate if there is substantial evaporation during the season and for incomplete cover and drip irrigation.

Fig. 6. Crop evapotranspiration determined using single and dual coefficient approaches of FAO 56 for a canola planted on 30 April 2010 at Wagga Wagga, NSW (Australia). ETc estimated using AquaCrop (dual coefficient) is also presented.

Crop evapotranspiration estimated using single and double coefficients is presented in Fig. 6. ETc estimated using AquaCrop is also presented in the Figure. It can be observed that ETc estimated using the three approaches is similar except in the initial and late season stages. During the initial stage, the ETc estimated using Eq. (14) and AquaCrop (Eqs. 21 and 22) are very close. However, the single coefficient method underestimated ETc at this stage. During the initial stage when most of the soil is bare, evaporation is high especially if the soil is wet due to irrigation or rainfall. The single crop coefficient approach does not sufficiently take this into account. A similar pattern was observed during the late season stage. However, AquaCrop overestimated ETc during this stage compared to the other two methods. The annual evapotranspiration estimated using different approaches was as follows: soil water balance (ETa = 313 mm), single crop coefficient (ETc = 332 mm), dual coefficient approach (ETc = 366 mm with E of 79 mm and T of 288 mm), AquaCrop (ETc = 382 mm with E of 139 mm and T of 243 mm). The evapotranspiration determined using soil water balance method is the "actual" evapotranspiration while the other methods measure potential evapotranspiration ETc. Soil water depletion (Dr) in Eq. (6) was determined using soil moisture content measured during the season and it was found that Dr<RAW throughout the season indicating that there was no soil moisture stress (Ks = 1). That might be why the ETc estimated using single coefficient method is close to the ETc determined using soil water balance method. Approaches using dual coefficient (Eq. 14) and Eqs. (21 and 22) resulted in higher ETc values. This might be due to the fact that in these approaches, the evaporation during the initial and late season stages was well simulated.

## **6. Conclusion**

Two approaches of estimating crop evapotranspiration were demonstrated using a field crop grown in a semiarid environment of Australia. These approaches were the rootzone soil water balance and the crop coefficient methods. The components of rootzone water balance, except evapotranspiration, were measured/estimated. Evapotranspiration was calculated as an independent parameter in the soil water balance equation. Single crop coefficient and dual coefficient approaches were based on adjustment of the FAO 56 coefficients for local condition. AquaCrop was also used to estimate crop evapotranspiration using the dual coefficient approach. It was found that the dual coefficients, basal or transpiration coefficient Kcb and evaporation coefficient Ke, correctly depict the actual process. The effects of weather (rainfall and radiation) and crop phenology were correctly simulated in this method. However, single coefficient does not show the high evaporation component during the initial and late season stages. Generally, there is a strong agreement among different estimation methods except that the dual coefficient approach had better estimate during the initial and late season stages. The evapotranspiration estimated using different approaches was as follows: soil water balance (ETa = 313 mm), single crop coefficient (ETc = 332 mm), dual coefficient approach (ETc = 366 mm with E of 79 mm and T of 288 mm), AquaCrop (ETc = 382 mm with E of 139 mm and T of 243 mm). Evapotranspiration estimated using soil water balance method is actual evapotranspiration ETa, while other methods estimate potential (maximum) evapotranspiration. Accordingly, ET estimated using rootzone water balance is lower than the ET estimated using the other methods. The single coefficient approach resulted in the lowest ETc as it is not taking into account the evaporation spikes after rainfall during the initial and late season stages.

## **7. Acknowledgments**

The senior author was research fellow at EH Graham Centre for Agricultural Innovation during this study. We also would like to thank David Luckett, Raymond Cowley, Peter Heffernan, David Roberts, and Peter Deane for professional and technical assistance.

## **8. References**

56 Evapotranspiration – Remote Sensing and Modeling

Crop evapotranspiration estimated using single and double coefficients is presented in Fig. 6. ETc estimated using AquaCrop is also presented in the Figure. It can be observed that ETc estimated using the three approaches is similar except in the initial and late season stages. During the initial stage, the ETc estimated using Eq. (14) and AquaCrop (Eqs. 21 and 22) are very close. However, the single coefficient method underestimated ETc at this stage. During the initial stage when most of the soil is bare, evaporation is high especially if the soil is wet due to irrigation or rainfall. The single crop coefficient approach does not sufficiently take this into account. A similar pattern was observed during the late season stage. However, AquaCrop overestimated ETc during this stage compared to the other two methods. The annual evapotranspiration estimated using different approaches was as follows: soil water balance (ETa = 313 mm), single crop coefficient (ETc = 332 mm), dual coefficient approach (ETc = 366 mm with E of 79 mm and T of 288 mm), AquaCrop (ETc = 382 mm with E of 139 mm and T of 243 mm). The evapotranspiration determined using soil water balance method is the "actual" evapotranspiration while the other methods measure potential evapotranspiration ETc. Soil water depletion (Dr) in Eq. (6) was determined using soil moisture content measured during the season and it was found that Dr<RAW throughout the season indicating that there was no soil moisture stress (Ks = 1). That might be why the ETc estimated using single coefficient method is close to the ETc determined using soil water balance method. Approaches using dual coefficient (Eq. 14) and Eqs. (21 and 22) resulted in higher ETc values. This might be due to the fact that in these approaches, the evaporation

Two approaches of estimating crop evapotranspiration were demonstrated using a field crop grown in a semiarid environment of Australia. These approaches were the rootzone soil water balance and the crop coefficient methods. The components of rootzone water balance, except evapotranspiration, were measured/estimated. Evapotranspiration was calculated as an independent parameter in the soil water balance equation. Single crop coefficient and dual coefficient approaches were based on adjustment of the FAO 56 coefficients for local condition. AquaCrop was also used to estimate crop evapotranspiration using the dual coefficient approach. It was found that the dual coefficients, basal or transpiration coefficient Kcb and evaporation coefficient Ke, correctly depict the actual process. The effects of weather (rainfall and radiation) and crop phenology were correctly simulated in this method. However, single coefficient does not show the high evaporation component during the initial and late season stages. Generally, there is a strong agreement among different estimation methods except that the dual coefficient approach had better estimate during the initial and late season stages. The evapotranspiration estimated using different approaches was as follows: soil water balance (ETa = 313 mm), single crop coefficient (ETc = 332 mm), dual coefficient approach (ETc = 366 mm with E of 79 mm and T of 288 mm), AquaCrop (ETc = 382 mm with E of 139 mm and T of 243 mm). Evapotranspiration estimated using soil water balance method is actual evapotranspiration ETa, while other methods estimate potential (maximum) evapotranspiration. Accordingly, ET estimated using rootzone water balance is lower than the ET estimated using the other methods. The single coefficient approach resulted in the lowest ETc as it is not taking into

account the evaporation spikes after rainfall during the initial and late season stages.

during the initial and late season stages was well simulated.

**6. Conclusion** 


## **Hargreaves and Other Reduced-Set Methods for Calculating Evapotranspiration**

Shakib Shahidian1, Ricardo Serralheiro1, João Serrano1,

José Teixeira2, Naim Haie3 and Francisco Santos1 *1University of Évora/ICAAM 2Instituto Superior de Agronomia 3Universidade do Minho Portugal* 

## **1. Introduction**

58 Evapotranspiration – Remote Sensing and Modeling

Suleiman A.A., Tojo Soler, C.M., Hoogenboom, G. 2007. Evaluation of FAO-56 crop

Thornthwaite, C.W. 1948. An approach toward a rational classification of climate. Geograph.

climate. Agric. Water Maneg., 91:33-42.

Rev., 38:55-94.

coefficient procedures for deficit irrigation management of cotton in a humid

Globally, irrigation is the main user of fresh water, and with the growing scarcity of this essential natural resource, it is becoming increasingly important to maximize efficiency of water usage. This implies proper management of irrigation and control of application depths in order to apply water effectively according to crop needs. Daily calculation of the Reference Potential Evapotranspiration (ETo) is an important tool in determining the water needs of different crops. The United Nations Food and Agriculture Organization (FAO) has adopted the Penman-Monteith method as a global standard for estimating ETo from four meteorological data (temperature, wind speed, radiation and relative humidity), with details presented in the Irrigation and Drainage Paper no. 56 (Allen et al., 1998), referred to hereafter as PM:

$$ET\_o = \frac{0.408\Delta (R\_n - G) + \gamma \frac{900}{T + 273} \mu\_2 (e\_s - e\_a)}{\Delta + \gamma (1 + 0.34\mu\_2)}\tag{1}$$

where:

*Rn* – net radiation at crop surface [MJ m-2 day-1],

*G* – soil heat flux density [MJ m-2 day-1],

*T* – air temperature at 2 m height [ºC],

*u2* – wind speed at 2 m height [m s-1],

*es* – saturation vapor pressure [kPa],

*ea* – actual vapor pressure [kPa],

*es-ea* – saturation vapor pressure deficit [kPa],

*∆* – slope vapor pressure curve [kPa ºC-1],

*γ* – psychrometric constant [kPa ºC-1],

The PM model uses a hypothetical green grass reference surface that is actively growing and is adequately watered with an assumed height of 0.12m, with a surface resistance of 70s m-1 and an albedo of 0.23 (Allen et al., 1998) which closely resemble evapotranspiration from an extensive surface of green grass cover of uniform height, completely shading the ground and with no water shortage. This methodology is generally considered as the most reliable, in a wide range of climates and locations, because it is based on physical principles and considers the main climatic factors, which affect evapotranspiration.

#### *Need for reduced-set methods*

The main limitation to generalized application of this methodology in irrigation practice is the time and cost involved in daily acquisition and processing of the necessary meteorological data. Additionally, the number of meteorological stations where all these parameters are observed is limited, in many areas of the globe. The number of stations where *reliable* data for these parameters exist is an even smaller subset.

There are also concerns about the accuracy of the observed meteorological parameters (Droogers and Allen, 2002), since the actual instruments, specifically pyranometers (solar radiation) and hygrometers (relative humidity), are often subject to stability errors. It is common to see a drift, of as much as 10 percent, in pyranometers (Samani, 2000, 1998). Henggeler et al. (1996) have observed that hygrometers loose about 1 percent in accuracy per installed month. There are also issues related to the proper irrigation and maintenance of the reference grass, at the weather stations. Jensen et al. (1997) observed that many weather stations are often not irrigated or inadequately irrigated, during the summer months, and thus the use of relative humidity and air temperature from these stations could introduce a bias in the computed values for *ETo*. Additionally, they observed that the measured values of solar radiation, *Rs,* are not always reliable or available and that wind data are quite site specific, unavailable, or of questionable reliability. Thus, they recommend the use of *ETo* equations that require fewer variables. These authors compared various methods, including FAO Penman Monteith, PM, and Hargreaves and Samani, HS, with lysimeter data and noted *r2* values of 0.94-0.97, with monthly SEE values of 0.30-0.34mm. Based on these data they concluded that the differences in *ETo* values, calculated by the different methods, are minor when compared with the uncertainties in estimating actual crop evapotranspiration from ETo. Additionally, these equations can be more easily used in adaptive or smart irrigation controllers that adjust the application depth according to the daily *ETo* demand (Shahidian et al., 2009).

This has created interest and has encouraged development of practical methods, based on a single or a reduced number of weather parameters for computing *ETo*. These models are usually classified according to the weather parameters that play the dominant role in the model. Generally these classifications include the *temperature-based models* such as Thornthwaite (1948); Blaney-Criddle (1950) and Hargreaves and Samani (1982); The *radiation models* which are based on solar radiation, such as Priestly-Taylor (1972) and Makkink (1957); and the *combination models* which are based on the energy balance and mass transfer principles and include the Penman (1948), modified Penman (Doorenbos and Pruitt, 1977) and FAO PM (Allen et al., 1998).

#### *Objectives and methods*

The objective of this chapter is to review the underlying principles and the genesis of these methodologies and provide some insight into their applicability in various climates and regions. To obtain a global view of the applicability of the reduced-set equations, each equation is presented together with a review of the published studies on its regional calibration as well as its application under different climates.

The main approach for evaluation and calibration of the reduced-set equations has been to use the PM methodology or lysimeter measurements as the benchmark for assessing their performance. Usually a linear regression equation, established with PM *ETo* values or lysimeter readings plotted as the dependent variable and values from the reduced-set equation plotted as the independent variable. The intercept, *a*, and calibration slope, *b*, of the best fit regression line, are then used as regional calibration coefficients:

$$ET\_oPM = a + b(ET\_oEquation) \tag{2}$$

The quality of the fit between the two methodologies is usually presented in terms of the coefficient of determination, *r2*, which is the ratio of the explained variance to the total variance or through the Root Mean Square Error, *RMSE*:

$$RMSE = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left( ETo\_{yi} - ETo\_{PM} \right)^2} \tag{3}$$

and the mean Bias error:

60 Evapotranspiration – Remote Sensing and Modeling

and with no water shortage. This methodology is generally considered as the most reliable, in a wide range of climates and locations, because it is based on physical principles and

The main limitation to generalized application of this methodology in irrigation practice is the time and cost involved in daily acquisition and processing of the necessary meteorological data. Additionally, the number of meteorological stations where all these parameters are observed is limited, in many areas of the globe. The number of stations

There are also concerns about the accuracy of the observed meteorological parameters (Droogers and Allen, 2002), since the actual instruments, specifically pyranometers (solar radiation) and hygrometers (relative humidity), are often subject to stability errors. It is common to see a drift, of as much as 10 percent, in pyranometers (Samani, 2000, 1998). Henggeler et al. (1996) have observed that hygrometers loose about 1 percent in accuracy per installed month. There are also issues related to the proper irrigation and maintenance of the reference grass, at the weather stations. Jensen et al. (1997) observed that many weather stations are often not irrigated or inadequately irrigated, during the summer months, and thus the use of relative humidity and air temperature from these stations could introduce a bias in the computed values for *ETo*. Additionally, they observed that the measured values of solar radiation, *Rs,* are not always reliable or available and that wind data are quite site specific, unavailable, or of questionable reliability. Thus, they recommend the use of *ETo* equations that require fewer variables. These authors compared various methods, including FAO Penman Monteith, PM, and Hargreaves and Samani, HS, with lysimeter data and noted *r2* values of 0.94-0.97, with monthly SEE values of 0.30-0.34mm. Based on these data they concluded that the differences in *ETo* values, calculated by the different methods, are minor when compared with the uncertainties in estimating actual crop evapotranspiration from ETo. Additionally, these equations can be more easily used in adaptive or smart irrigation controllers that adjust the application depth according to the

This has created interest and has encouraged development of practical methods, based on a single or a reduced number of weather parameters for computing *ETo*. These models are usually classified according to the weather parameters that play the dominant role in the model. Generally these classifications include the *temperature-based models* such as Thornthwaite (1948); Blaney-Criddle (1950) and Hargreaves and Samani (1982); The *radiation models* which are based on solar radiation, such as Priestly-Taylor (1972) and Makkink (1957); and the *combination models* which are based on the energy balance and mass transfer principles and include the Penman (1948), modified Penman (Doorenbos and Pruitt, 1977)

The objective of this chapter is to review the underlying principles and the genesis of these methodologies and provide some insight into their applicability in various climates and regions. To obtain a global view of the applicability of the reduced-set equations, each equation is presented together with a review of the published studies on its regional

considers the main climatic factors, which affect evapotranspiration.

where *reliable* data for these parameters exist is an even smaller subset.

*Need for reduced-set methods* 

daily *ETo* demand (Shahidian et al., 2009).

and FAO PM (Allen et al., 1998).

calibration as well as its application under different climates.

*Objectives and methods* 

$$MBE = \frac{1}{n} \sum\_{i=1}^{n} \left( ETo\_{yi} - ETo\_{PM} \right) \tag{4}$$

where *n* is the number of estimates and *ETo yi* is the estimated values from the reduced-set equation.

## **2. Temperature based equations**

Temperature is probably the easiest, most widely available and most reliable climate parameter. The assumption that temperature is an indicator of the evaporative power of the atmosphere is the basis for temperature-based methods, such as the Hargreaves-Samani. These methods are useful when there are no data on the other meteorological parameters. However, some authors (McKenny and Rosenberg, 1993, Jabloun and Sahli, 2007) consider that the obtained estimates are generally less reliable than those which also take into account other climatic factors.

Mohan and Araumugam (1995) and Nandagiri and Kovoor (2006) carried out a multivariate analysis of the importance of various meteorological parameters in evapotranspiration. They concluded that temperature related variables are the most crucial required inputs for obtaining ETo estimates, comparable to those from the PM method across all types of climates. However, while wind speed is considered to be an important variable in arid climate, the number of sunshine hours is considered to be the more dominant variable in sub-humid and humid climates.

#### **2.1 The Hargreaves- Samani methodology**

Hargreaves, using grass evapotranspiration data from a precision lysimeter and weather data from Davis, California, over a period of eight years, observed, through regressions, that for five-day time steps, 94% of the variance in measured *ET* can be explained through average temperature and global solar radiation, *Rs*. As a result, in 1975, he published an equation for predicting *ETo* based only on these two parameters:

$$ET\_o = 0.0135 \, R\_s (T + 17.8) \tag{5}$$

where *Rs* is in units of water evaporation, in mm day-1, and *T* in ºC. Subsequent attempts to use wind velocity, *U2,* and relative humidity, *RH,* to improve the results were not encouraging so these parameters have been left out (Hargreaves and Allen, 2003).

The clearness index, or the fraction of the extraterrestrial radiation that actually passes through the clouds and reaches the earth's surface, is the main energy source for evapotranspiration, and later studies by Hargreaves and Samani (1982) show that it can be estimated by the difference between the maximum, *Tmax*, and the minimum, T*min* daily temperatures. Under clear skies the atmosphere is transparent to incoming solar radiation so the *Tmax* is high, while night temperatures are low due to the outgoing longwave radiation (Allen et al., 1998). On the other hand, under cloudy conditions, *Tmax* is lower, since part of the incoming solar radiation never reaches the earth, while night temperatures are relatively higher, as the clouds limit heat loss by outgoing longwave radiation. Based on this principle, Hargreaves and Samani (1982) recommended a simple equation to estimate solar radiation using the temperature difference, *T*:

$$\begin{aligned} \left\langle \begin{array}{c} \text{R}\_{\text{s}} \\ \text{R}\_{a} \end{array} \right\rangle\_{\text{H}} &= \text{K}\_{T} \left( T\_{\text{max}} - T\_{\text{min}} \right)^{0.5} \end{aligned} \tag{6}$$

where *Ra* is the extraterrestial radiation in mm day-1, and can be obtained from tables (Samani, 2000) or calculated (Allen et al., 1998). The empirical coefficient, *KT* was initially fixed at 0.17 for Salt Lake City and other semi-arid regions, and later Hargreaves (1994) recommended the use of 0.162 for interior regions where land mass dominates, and 0.190 for coastal regions, where air masses are influenced by a nearby water body. It can be assumed that this equation accounts for the effect of cloudiness and humidity on the solar radiation at a location (Samani, 2000). The clearness index (*Rs/Ra*) ranges from 0.75 on a clear day to 0.25 on a day with dense clouds.

Based on equations (5) and (6), Hargreaves and Samani (1985) developed a simplified equation requiring only temperature, day of year and latitude for calculating *ETo*:

$$ET\_o = 0.0135 \, K\_T \, (T + 17.78) (T\_{max} - T\_{min}) \, ^{0.5}R\_a \tag{7}$$

Since *KT* usually assumes the value of 0.17, sometimes the 0.0135 *KT* coefficient is replaced by 0.0023. The equation can also be used with *Ra* in MJ m-2 day-1, by multiplying the right hand side by 0.408.

This method (designated as HS in this chapter) has produced good results, because at least 80 percent of *ETo* can be explained by temperature and solar radiation (Jensen, 1985) and *T* is related to humidity and cloudiness (Samani and Pessarakli, 1986). Thus, although this equation only needs a daily measurement of maximum and minimum temperatures, and is presented here as a temperature-based method, it effectively incorporates measurement of radiation, albeit indirectly. As will be seen later, the ability of the methodology to account for both temperature and radiation provides it with great resilience in diverse climates around the world.

Sepashkhah and Razzaghi (2009) used lysimeters to compare the Thornthwaithe and the HS in semi-arid regions of Iran and concluded that a calibrated HS method was the most accurate method. Jensen et al.(1997) compared this and other *ETo* calculation methods and concluded that the differences in *ETo* values computed by the different methods are not larger than those introduced as a result of measuring and recording weather variables or the uncertainties

where *Rs* is in units of water evaporation, in mm day-1, and *T* in ºC. Subsequent attempts to use wind velocity, *U2,* and relative humidity, *RH,* to improve the results were not

The clearness index, or the fraction of the extraterrestrial radiation that actually passes through the clouds and reaches the earth's surface, is the main energy source for evapotranspiration, and later studies by Hargreaves and Samani (1982) show that it can be estimated by the difference between the maximum, *Tmax*, and the minimum, T*min* daily temperatures. Under clear skies the atmosphere is transparent to incoming solar radiation so the *Tmax* is high, while night temperatures are low due to the outgoing longwave radiation (Allen et al., 1998). On the other hand, under cloudy conditions, *Tmax* is lower, since part of the incoming solar radiation never reaches the earth, while night temperatures are relatively higher, as the clouds limit heat loss by outgoing longwave radiation. Based on this principle, Hargreaves and Samani (1982) recommended a simple equation to estimate solar radiation

max min ( ) *<sup>s</sup> <sup>T</sup> <sup>a</sup>*

where *Ra* is the extraterrestial radiation in mm day-1, and can be obtained from tables (Samani, 2000) or calculated (Allen et al., 1998). The empirical coefficient, *KT* was initially fixed at 0.17 for Salt Lake City and other semi-arid regions, and later Hargreaves (1994) recommended the use of 0.162 for interior regions where land mass dominates, and 0.190 for coastal regions, where air masses are influenced by a nearby water body. It can be assumed that this equation accounts for the effect of cloudiness and humidity on the solar radiation at a location (Samani, 2000). The clearness index (*Rs/Ra*) ranges from 0.75 on a clear day to 0.25

Based on equations (5) and (6), Hargreaves and Samani (1985) developed a simplified

Since *KT* usually assumes the value of 0.17, sometimes the 0.0135 *KT* coefficient is replaced by 0.0023. The equation can also be used with *Ra* in MJ m-2 day-1, by multiplying the right

This method (designated as HS in this chapter) has produced good results, because at least 80 percent of *ETo* can be explained by temperature and solar radiation (Jensen, 1985) and

is related to humidity and cloudiness (Samani and Pessarakli, 1986). Thus, although this equation only needs a daily measurement of maximum and minimum temperatures, and is presented here as a temperature-based method, it effectively incorporates measurement of radiation, albeit indirectly. As will be seen later, the ability of the methodology to account for both temperature and radiation provides it with great resilience in diverse climates

Sepashkhah and Razzaghi (2009) used lysimeters to compare the Thornthwaithe and the HS in semi-arid regions of Iran and concluded that a calibrated HS method was the most accurate method. Jensen et al.(1997) compared this and other *ETo* calculation methods and concluded that the differences in *ETo* values computed by the different methods are not larger than those introduced as a result of measuring and recording weather variables or the uncertainties

equation requiring only temperature, day of year and latitude for calculating *ETo*:

encouraging so these parameters have been left out (Hargreaves and Allen, 2003).

*T*:

using the temperature difference,

on a day with dense clouds.

hand side by 0.408.

around the world.

0.0135 ( 17.8) *ET R T o s* (5)

0.5

*<sup>R</sup> KT T <sup>R</sup>* (6)

0.5

*T*

min 0.0135 ( 17.78)( ) *ET K T T T R oT m zx a* (7)

associated with estimating crop evapotranspiration from *ETo*. López-Urrea et al. (2006) compared seven *ETo* equations in arid southern Spain with Lysimeter data, and observed daily RMSE values between 0.67 for FAO PM and 2.39 for FAO Blaney-Criddle. They also observed that the Hargreaves equation was the second best after PM, with an RMSE of only 0.88.

Since the HS method was originally calibrated for the semi-arid conditions of California, and does not explicitly account for relative humidity, it has been observed that it can overestimate *ETo* in humid regions such as Southeastern US (Lu et al. 2005), North Carolina (Amatya et al. 1995), or Serbia (Trajkovic, 2007).

In Brasil, Reis et al. (2007) studied three regions of the Espírito Santo State: The north with a moderately humid climate, the south with a sub-humid climate, and the mountains with a humid climate (Table 1). The HS equation overestimated *ETo* in all three regions by as much as 32%, but the performance of the HS equation improved progressively as the climate became drier. Only further south, at a latitude of 24º S, and in a warm temperate climate did HS provide good agreement with PM, though still with a small overestimation. Borges and Mendiondo (2007) obtained an r2 of 0.997 for HS when compared to PM, when using a calibrated of 0.0022 (Sept-April) and 0.0020 for the rest of the year.

On the other hand, in dry regions such as Mahshad, Iran and Jodhpur, India, the HS equation tends to underestimate *ETo* by as much as 24% (Rahimkoob, 2008; Nandagiri and Kovoor, 2006). Rahimkoob (2008) studied the *ETo* estimates obtained from the HS equation in the very dry south of Iran. His data indicate that the HS equation fails to calculate *ETo* values above 9 mm day-1, even when the PM reaches values of more than 13 mm day-1 (Fig. 1).

Wind removes saturated air from the boundary layer and thus increases evapotranspiration (Brutsaert, 1991). Since most of the reduced-set equations do not explicitly account for wind speed, it is natural for the calibration slope to be influenced by this parameter. Itensifu et al. (2003) carried out a major study using weather data from 49 diverse sites in the United States. They obtained ratios ranging from 0.805 to 1.242 between HS and PM and concluded that the HS equation has difficulty in accounting for the effects of high winds and high vapor pressure deficits, typical of the Great Plains region. They also observed that the HS equation tends to overestimate *ETo* when mean daily *ETo* is relatively low, as in most sites in the eastern region of the US, and to underestimate when *ETo* is relatively high, as in the lower Midwest of the US. As will be seen later, this seems to be a common issue with most of the reduced set evapotranspiration equations (see section 4.3, Fig. 7).

For the Mkoji sub-catchment of the Great Ruaha River in Tanzania, Igbadun et al. (2006) calculated the monthly *ETo* values of three very distinct areas of the catchment: the humid Upper Mkoji with an altitude of 1700m, the middle Mkoji with an average altitude of 1100 m, and the semi-arid lower Mkoji with an altitude of 900m. Their data indicate a strong relation between the monthly average wind speed and the performance of the HS equation as measured by the slope of the calibration equation (PM/HS ratio). Although the three areas have distinct climates, the HS equation clearly underestimated ETo for wind speed values below 2-2.3 ms-1, and overestimated it for higher wind speed values (Fig. 2).

Trajkovic, et al. (2005) studied the HS equation in seven locations in continental Europe with different altitudes (42-433m) with RH ranging from 55 to 71%, representative of the distinct climates of Serbia. Their data show that despite the different altitudes and climatic conditions, wind speed was the major determinant for the calibration of the HS equation (Fig. 3). The results from these works indicate that wind is the main factor affecting the calibration of the HS equation and that the equation should be calibrated in areas with very high or low wind speeds.

Fig. 1. Relation between *ETo* calculated with the HS equation and the PM for the dry conditions of Abadan, Iran. The Hargreaves Samani equation fails to calculate *ETo* values above 9 mm day-1 (data kindly provided by Rahimkoob)

Fig. 2. Correlation between average wind speed and the calibration slope in distinct climates of the Great Ruana River in Tanzania (based on the original data from Igbadun et al. 2006).

Jabloun and Sahli (2008) studied eight stations in the semi-arid Tunisia and concluded that in inland stations, HS tends to overestimate *ETo* due to high *T* values. In the coastal station of Tunis, HS underestimated *ETo* values, which they attributed to an underestimation of *Rs*. Various attempts have been made to improve the accuracy of the HS equation through incorporation of additional measured parameters, such as rainfall (Droogers and Allen, 2002) and altitude (Allen, 1995). These methodologies have had limited global application, probably because *ETo* is influenced by a combination of different parameters, and although in a certain region there appears to be a good correlation between the calibration slope and a certain parameter, this might not be so in a different climate.

The alternative is to use regional calibration, in which, based on the climatic characteristics of the region, the *ETo* calculated by the HS equation is adjusted to account for the combined

*0 2 4 6 8 10 12 14 Hargreaves Samani ETo, mm day -1*

01234 *Average monthly wind speed, m s -1*

Fig. 2. Correlation between average wind speed and the calibration slope in distinct climates of the Great Ruana River in Tanzania (based on the original data from Igbadun et al. 2006). Jabloun and Sahli (2008) studied eight stations in the semi-arid Tunisia and concluded that

of Tunis, HS underestimated *ETo* values, which they attributed to an underestimation of *Rs*. Various attempts have been made to improve the accuracy of the HS equation through incorporation of additional measured parameters, such as rainfall (Droogers and Allen, 2002) and altitude (Allen, 1995). These methodologies have had limited global application, probably because *ETo* is influenced by a combination of different parameters, and although in a certain region there appears to be a good correlation between the calibration slope and a

The alternative is to use regional calibration, in which, based on the climatic characteristics of the region, the *ETo* calculated by the HS equation is adjusted to account for the combined

y = 0,0947x + 0,7636 R2 = 0,7598

*T* values. In the coastal station

Fig. 1. Relation between *ETo* calculated with the HS equation and the PM for the dry conditions of Abadan, Iran. The Hargreaves Samani equation fails to calculate *ETo* values

*0*

0

0,2

0,4

0,6

*Ratio of PM to HS*

0,8

1

1,2

above 9 mm day-1 (data kindly provided by Rahimkoob)

in inland stations, HS tends to overestimate *ETo* due to high

certain parameter, this might not be so in a different climate.

*2*

*4*

*6*

*FAO Penman Monteith, ETo, mm day*

*8*

*10*

*12*

 *-1* *14*

effect of the dominant climate parameters, and thus accuracy of the equations is improved (Teixeira et al., 2008). Table 1 presents a compilation of most of the published studies on the regional calibration of the HS equation. This compilation contains 33 published works covering 21 countries with all types of climatic conditions according to the Koppen classification. Whenever various stations from a similar climate were studied, only parameters from one representative station are presented. In some studies, HS and PM were calibrated against a third methodology (such as Pan A) and thus no direct calibration parameters for the PM/HS regression were provided. In these cases, a linear regression was obtained by plotting the PM calibration equation as the dependent variable and the HS calibration equation as the independent variable. The parameters of the resulting regression equation are then presented as the PM-HS calibration parameters.

In order to contextualize the information and allow for extension of the results to other regions with a similar climate, the locations are grouped according to Koppen climate classification. These calibration coefficients can be used in the area where they were obtained or can be extrapolated for areas with similar conditions where no actual calibration has been carried out yet.

Fig. 3. Correlation between wind speed and the calibration slope for seven different locations in Serbia, representing the diverse local climates (original data from Trajkovic, 2005).

#### **2.2 The Thornthwaite method**

Thornthwaite (1948) devised a methodology to estimate *ETo* for short vegetation with an adequate water supply in certain parts of the USA. The procedure uses the mean air temperature and number of hours of daylight, and is thus classified as a temperature based method. Monthly *ETo* can be estimated according to Thornthwaite (1948) by the following equation:

$$\mathbf{E}t\_0 = \mathbf{E}T\_0 \mathbf{s} \mathbf{c} \begin{pmatrix} \mathbf{N} \bigvee\_{12} \big( \mathbf{d}m \bigvee\_{30} \big) \end{pmatrix} \tag{8}$$



Table 1. Regional calibration for the Hargreaves Samani equation compiled from published works

Country

**Arid** *Desert* China, NW

 Minle Aquila Jodhpur Heydarabad

Tel Hadya

Shiraz Shiraz Progreso (Yucatán)

China, NW

US *Steppe* India

India Syria

Iran Iran México *Dry summer*

Spain Spain Bolivia

Spain Spain Tanzania

**Mesothermal**

*Mediterranean*

Spain Spain Spain Portugal, S

US Portugal

Spain Spain Greece

USA Spain *Dry winter* Tanzania

 Middle Mkoji Douradas, Mato G. Sul

S. Mantiqueira, MG

Brasil Brasil *fully humid* Netherlands

 Haarweg Louisiana, inland Lousisana, coastal

North Carolina, Plymouth

Palotina, Paraná Jacupiranga river, SP

Values in grey are annual averages obtained from Climwat data base.

US US US Brasil Brasil

Athens Prosser, WA

Lleida

 Evora Davis Elvas Niebla (Andalucia)

Vejer Frontera (Andalucia)

Malaga (Andalucia) Coast

Sevilla (Andalucia) interior

La Mojonera, coast

 36º40' N

 37º125' N

37º45' N

38º55' N

38º32' N

38º60' N

37º21' N

 36º 17' N 38º23' N

46º15' N

41º42' N

8º30'

 22º16'S

 452 1500

51º58' N

31º N low land 29º N low land

 35º52' 24º18'S

24º29'S

 310

 52

 Cfa

 Cfa

 1700 73.8 1.74 1879 91.5 0.97 When calibration parameters of the HS vsFAO PM were not directly provided, linear regression equations were established with FAO-56 PM daily ET0 estimates as the dependent variable and daily ET0 values estimated by HS as an independent variable. The parameters of the regression equation were then presented as the calibration parameters.

 6

 Cfa

 1299 80.2 4.9

 9

 Cfb

 Cfa

 Cfa

 1500 88.7 0.6

 1500 92 0.82

 -0.28

 -0.17 0.03



 1.05

 0.87

 0.83

 1

 1.042 778 87.3 2.41

 Cwa

 Cwb

 2150

 1603 73.8 1.74

 1070

 Cwa

 800


 1.73 0.153

 0.67

 1.16 1.02

 0.91

Stockle, 2004 Fontenot, 2004 Fontenot, 2004

1.23

Amatya et al. 1995

Syperreck, 2006 Borges and Mendiondo, 2007

 0.7

 0.955

 100

 380

 221

 Csb

 Csb

 Csa

 202

 52

 24

 Csa

 571 69.4 2.9 371 61.8 1.87

994 601 68.8 0.97

69.7 1.62

 0.264

 0.781 1.02

1.1

 0.98

 0.95

 Csa

 702 65.3 1.3

 Csa

 18.3

 Csa

 458 63.3 2.62 508 58.2 1.97

 -0.844 -0.08

 1.245

 1.04 1.035

1.404

 0.93

 142

 246

 Csa

 Csa

 272 62.3 1.9 627 63.3 4.3

 31

 Csa

 7

 Csa

531 68.1 1.9 473 67.8 0.93

0.962 1.165 1.27 0.866

Vanderlinden et al., 2004

Vanderlinden et al., 2004

Gavilán et al, 2008

Santos and Maia, 2007

Alexandris, 2006 Teixeira et al. 2008

Gavilán et al., 2008

Gavilán et al., 2008

Alexandris, 2006

Stockle, 2004 Stockle, 2004 Igbadun et al, 2006

Fietz, 2004 Pereira et al. 2009

 Lower Mkoji

Daroca (NE Spain)

Zaragoza (NE Spain)

Spain Cordoba, inland

Patacamaya and Oruro

Albacete Cordoba, inland

41º07' N

41º43' N

37º52' N

 17º15'S 39º14' N

37º51' N

7º80'

 695

 110

 900

 Bsh

 520

 Bsk

 696 63.3 1.6

 Bsk

 283

 3749

 779

 225

 117

 Bsk

 Bsk

 375

57.4 1.2 68.7 1.08

0.8622

0.34\*

 -1.49 -0.0027

 0.9092

 0.6422

 1.14\*

 1.3

 696 63.3 1.6

 Bsk

 353 73.7 2.43

 Bsk

 364 66.5 1.08

 -0.203

 -0.012

 0.93

 0.99 1.06

Shandan Heihe R.

38º90' N

38º80' N

33º56' N

26º18' N

17º32' N

36º01' N

30º07'N

30º07' N

21º17' N

 2

 BSh

 511

 1650

 BSh

 1650

 BSh

 306 305.6 36.4 2.49

36.4 2.49

0.41


 1.012

 0.78

 0.82 1.13

 224

 545

 293

 BSh

 BSh

 BSh

402 38.9 2.1 820 65.6 2.8 231 57.4 2.82



 1.1924

 1.48 1.04

 0.91

 655

 BWh

 2271

 BWk

 1483

 BWk

 250

 100 195 35.3 3.2

40 1.98

0.5431


0.0378

 1.148

 1.065

 1.3155

Zhao et al. 2005 Zhao et al. 2005 Alexandris, 2006 Nandagiri and Kovoor, 2006

Nandagiri and Kovoor, 2006

Stockle, 2004 Razzaghi and Sepahskah, 2009

Sepashkah and Razzaghi, 2009

Bautista et al 2009

Mártinez-Cob andTejero-Juste, 2004

Mártinez-Cob andTejero-Juste, 2004

Gavilán et al, 2008

Garcia et al 2004 Lopéz-Urra et al 2005

Berengena and Gavilan, 2005

Igbadun et al

Station

latitude

m

 m classification

Altitude

 Koppen

 Rainfall RH U2

 mm %

ms-1

intercept

a

 slope

 b

R2 RMSE Source

Regression adjustment

estimated by HS as an independent variable. The parameters of the regression equation were then presented as the calibration parameters.

Where *N* is the maximum number of sunny hours as a function of the month and latitude and *dm* is the number of days per month. *ETosc* is the gross evapotranspiration (without corrections) and can be calculated as:

$$E t\_0 s c = 16 \binom{10T\_a}{\searrow 1} a \tag{9}$$

where *Ta* is the mean daily temperature (°C), *a* is an exponent as a function of the annual index: *a* = 0.49239 + 1792 × 10-5 I - 771 × 10 -7 I2 + 675 × 10-9 I3; and *I* is the annual heat index obtained form the monthly heat indecies:

$$I = \sum\_{m=1}^{12} \binom{T\_m}{\heartsuit} \text{1.514} \tag{10}$$

Bautista et al. (2009) found that the precision of the Thorntwaite methodology improved during the winter months in Mexico. Garcia et al. (2004) observed that under the dry and arid conditions of the Bolivian highlands the Thornthwaite equation strongly underestimates *ETo* because the equation does not consider the saturation deficit of the air (Stanhill, 1961; Pruitt, 1964; Pruitt and Doorenbos, 1977). Additionally, at high altitudes, the Thornthwaite equation also underestimates the effect of radiation, because the equation is calibrated for temperate low altitude climates. Studies in Brazil have shown that the underestimation of *ETo* produced by temperature-based equations under arid conditions, may be reduced by using the daily thermal amplitude instead of the mean temperature (Paes de Camargo, 2000) as in the case of the Hargreaves–Samani equation.

Gonzalez et al. (2009) studied the Thorthwaite method in the Bolivian Amazon. They observed that the Thornthwaite method underestimates evapotranspiration at all the three stations studied. This is expected, considering that normally this method leads to underestimations in humid areas (Jensen et al., 1990).

#### **2.3 Blaney-Criddle method**

The FAO Temperature Methodology recommended by Doorenbos and Pruitt (1977) is based on the Blaney-Criddle method (Blaney and Criddle, 1950), introducing a correction factor based on estimates of humidity, sunshine and wind.

$$ET\_o = \alpha + \beta \left[ p \left( 0.46T + 8.13 \right) \right] \tag{11}$$

where and *β* are calibration parameters and *p* is the mean annual percentage of daytime hours. Values for can be calculated using the daily *RHmin* and *n/N* as follows:

$$a = 0.043RH\_{\text{min}} - \left(\frac{n}{N}\right) - 1.41\tag{12}$$

$$\frac{n}{N} = 2\left(\text{Rs} \;/\; Ra\right) - 0.5\tag{13}$$

For windy South Nebraska, Irmak et al. (2008) compared 12 different ET methodologies and found that the Blaney–Criddle method was the best temperature method and it had an RMSE value (0.64 mm d−1) which was similar to some of the combination methods. The

Where *N* is the maximum number of sunny hours as a function of the month and latitude and *dm* is the number of days per month. *ETosc* is the gross evapotranspiration (without

> <sup>10</sup> <sup>16</sup> *Et sc Ta <sup>a</sup> <sup>I</sup>*

where *Ta* is the mean daily temperature (°C), *a* is an exponent as a function of the annual index: *a* = 0.49239 + 1792 × 10-5 I - 771 × 10 -7 I2 + 675 × 10-9 I3; and *I* is the annual heat index

Bautista et al. (2009) found that the precision of the Thorntwaite methodology improved during the winter months in Mexico. Garcia et al. (2004) observed that under the dry and arid conditions of the Bolivian highlands the Thornthwaite equation strongly underestimates *ETo* because the equation does not consider the saturation deficit of the air (Stanhill, 1961; Pruitt, 1964; Pruitt and Doorenbos, 1977). Additionally, at high altitudes, the Thornthwaite equation also underestimates the effect of radiation, because the equation is calibrated for temperate low altitude climates. Studies in Brazil have shown that the underestimation of *ETo* produced by temperature-based equations under arid conditions, may be reduced by using the daily thermal amplitude instead of the mean temperature

Gonzalez et al. (2009) studied the Thorthwaite method in the Bolivian Amazon. They observed that the Thornthwaite method underestimates evapotranspiration at all the three stations studied. This is expected, considering that normally this method leads to

The FAO Temperature Methodology recommended by Doorenbos and Pruitt (1977) is based on the Blaney-Criddle method (Blaney and Criddle, 1950), introducing a correction factor

and *β* are calibration parameters and *p* is the mean annual percentage of daytime

can be calculated using the daily *RHmin* and *n/N* as follows:

min 0.043 1.41 *<sup>n</sup> RH N*

2 / 0.5 *<sup>n</sup> Rs Ra*

For windy South Nebraska, Irmak et al. (2008) compared 12 different ET methodologies and found that the Blaney–Criddle method was the best temperature method and it had an RMSE value (0.64 mm d−1) which was similar to some of the combination methods. The

*ET p T <sup>o</sup>* 

1.514 <sup>5</sup> *<sup>m</sup>*

(9)

(10)

0.46 8.13 (11)

*<sup>N</sup>* (13)

(12)

0

12

*m <sup>T</sup> <sup>I</sup>* 

(Paes de Camargo, 2000) as in the case of the Hargreaves–Samani equation.

underestimations in humid areas (Jensen et al., 1990).

based on estimates of humidity, sunshine and wind.

**2.3 Blaney-Criddle method** 

where 

hours. Values for

1

corrections) and can be calculated as:

obtained form the monthly heat indecies:

obtained estimates were good and were within 3% of the ASCE-PM *ETo* with a high r2 of 0.94. The estimates were consistent with no large under or over estimations for the majority of the dataset. They attributed this to the fact that, unlike most of the other temperature methods, this method takes into account humidity and wind speed in addition to air temperature.

Lee et al. (2004) compared various *ETo* calculation methods in the West Coast of Malaysia and concluded that the Blaney-Criddle method was the best, among the reduced-set equations, for estimating ET in the region. They also observed that HS gave the highest estimates followed by the Priestly-Taylor equation. Similarly, in the humid Goiânia region of Brazil, Oliveira et al. (2005) observed that the Blaney-Criddle method produced the best results, next to the full PM equation.

Various studies indicate that the Blaney-Criddle equation might show some bias under arid conditions. For semi-arid conditions of Iran, Dehghani Sanij et al. (2004) found the Blaney-Criddle and the Makkink method to overestimate ETo during the growing season. Lopéz-Urrea et al. (2006) compared seven different methods for calculating ETo in the semiarid regions of Spain and observed that the Blaney-Criddle method significantly over-estimated average daily ETo.

For arid conditions of Iran, Fard et al. (2009) compared nine different methodologies with lysimeter data and observed that the Turc and the Blaney-Criddle methods showed very close agreement with the lysimeter data, while PM showed moderate agreement with the lysimeter data. The other methods showed bias, systematically over estimating the lysimeter data (Fig. 4).

Although recognizing the historical value of the Blaney-Criddle method and its validity, the FAO Expert Commission on Revision of FAO Methodologies for Crop Water Requirements (Smith et al. 1992) did not recommend the method further, in view of difficulties in estimating humidity, sunshine and wind parameters in remote areas. Nevertheless, they emphasized the value of the method for areas having only the mean daily temperature, and where appropriate correction factors can be found.

Fig. 4. Comparision of six ET methods with lysimeter data for Isfahan (adapted from Fard et al., 2009).

#### **2.4 Reduced-set PM**

The PM methodology has provisions for application in data-short situations (Allen et al. 1998), including the use of temperature data alone. The reduced-set PM equation requiring only the measured maximum and minimum temperatures uses estimates of solar radiation, relative humidity, and wind speed. Solar radiation, *Rs*, MJ m−2 d−1 can be estimated using equation 3 (Hargreaves and Samani, 1985) or using averages from nearby stations. For island locations *Rs* can be estimated as (Allen et al. 1998):

$$R\_s = 0.7R\_a - b \tag{14}$$

where *b* is an empirical constant with a value of 4 MJ m−2 d-1 . Relative humidity can be estimated by assuming that the dewpoint temperature is approximately equal to *Tmin* (Allen 1996; Allen et al. 1998) which is usually experienced at sunrise. In this case, *ea* can be calculated as:

$$e\_d = e^o \left( T\_{\rm min} \right) = 0.611 \exp \left[ \frac{17.27 T\_{\rm min}}{T\_{\rm min} + 237.3} \right] \tag{15}$$

where *eo(Tmin)* is the vapour pressure at the minimum temperature, expressed in mbar. For wind speed, Allen et al. (1998) recommend using average wind speed data from nearby locations or using a wind speed of 2 m s−1, since, they consider, the impact of wind speed on the *ETo* results is relatively small, except in arid and windy areas. The soil heat flux density, *G*, for monthly periods can be estimated as:

$$G\_i = 0.07(T\_{i+1} - T\_{i-1})\tag{16}$$

where *Gi* is the soil heat flux density in month *I* in MJ m−2 d−1; and *Ti*+1 and *Ti*−1 are the mean air temperatures in the previous and following months, respectively.

Allen (1995) evaluated the reduced-set PM (using only *Tmax* and *Tmin*) and HS using the mean annual monthly data from the 3,000 stations in the FAO CLIMWAT data base, with the full PM serving as the comparative basis. He found little difference in the mean monthly *ETo* between the two methods. Wright et al. (2000) found similar results in Kimberly, and 75 years of data from California (Hargreaves and Allen, 2003). Other data generally indicate that the reduced-set PM performs better in humid areas (Popova, 2005, Pereira et al., 2003), while HS performs better in dry climates (Temesgen et al. 2005, Jabloun et al. 2008).

Trajkovic (2005) compared the reduced-set PM, Hargreaves, and Thornthwaite temperaturebased methods with the full PM in Serbia and found that the reduced-set PM estimates were better than those produced from the Hargreaves and Thornthwaite equations. Popova et al. (2006) found the reduced-set PM to provide more accurate results compared to the Hargreaves equation, which tended to overestimate reference evapotranspiration in the Trace plain in south Bulgaria. Jabloun and Sahli (2008) also found the Hargreaves equation to overestimate reference evapotranspiration in Tunisia and found the reduced-set PM equation to provide better estimates. Nevertheless, the reduced-set PM can produce poor results in areas where wind speed is significantly different from 2 ms-1 (Trajkovic, 2005).

#### **3. Radiation based methods**

It is known that water loss from a crop is related to the incident solar energy, and thus it is possible to develop a simple model that relates solar radiation to evapotranspiration. Various models have been developed, over the years, for relating the measured net global radiation to the estimated reference evapotranspiration; such as the Priestley-Taylor method (1972), the Makkink method (1957), the Turc radiation method (1961), and the Jensen and Haise method (1965).

Irmak et al. (2008) compared 11 ET models and studied the relevance of their complexity for direct prediction of hourly, daily and seasonal scales. They concluded that radiation is the dominant driver of evaporative losses, over seasonal time scales, and that other meteorological variables, such as temperature and wind speed, gained importance in daily and hourly calculations.

#### **3.1 The Priestley-Taylor method**

70 Evapotranspiration – Remote Sensing and Modeling

The PM methodology has provisions for application in data-short situations (Allen et al. 1998), including the use of temperature data alone. The reduced-set PM equation requiring only the measured maximum and minimum temperatures uses estimates of solar radiation, relative humidity, and wind speed. Solar radiation, *Rs*, MJ m−2 d−1 can be estimated using equation 3 (Hargreaves and Samani, 1985) or using averages from nearby stations. For

where *b* is an empirical constant with a value of 4 MJ m−2 d-1 . Relative humidity can be estimated by assuming that the dewpoint temperature is approximately equal to *Tmin* (Allen 1996; Allen et al. 1998) which is usually experienced at sunrise. In this case, *ea* can be

min

where *eo(Tmin)* is the vapour pressure at the minimum temperature, expressed in mbar. For wind speed, Allen et al. (1998) recommend using average wind speed data from nearby locations or using a wind speed of 2 m s−1, since, they consider, the impact of wind speed on the *ETo* results is relatively small, except in arid and windy areas. The soil heat flux density,

where *Gi* is the soil heat flux density in month *I* in MJ m−2 d−1; and *Ti*+1 and *Ti*−1 are the mean

Allen (1995) evaluated the reduced-set PM (using only *Tmax* and *Tmin*) and HS using the mean annual monthly data from the 3,000 stations in the FAO CLIMWAT data base, with the full PM serving as the comparative basis. He found little difference in the mean monthly *ETo* between the two methods. Wright et al. (2000) found similar results in Kimberly, and 75 years of data from California (Hargreaves and Allen, 2003). Other data generally indicate that the reduced-set PM performs better in humid areas (Popova, 2005, Pereira et al., 2003),

Trajkovic (2005) compared the reduced-set PM, Hargreaves, and Thornthwaite temperaturebased methods with the full PM in Serbia and found that the reduced-set PM estimates were better than those produced from the Hargreaves and Thornthwaite equations. Popova et al. (2006) found the reduced-set PM to provide more accurate results compared to the Hargreaves equation, which tended to overestimate reference evapotranspiration in the Trace plain in south Bulgaria. Jabloun and Sahli (2008) also found the Hargreaves equation to overestimate reference evapotranspiration in Tunisia and found the reduced-set PM equation to provide better estimates. Nevertheless, the reduced-set PM can produce poor results in areas where wind speed is significantly different from 2 ms-1 (Trajkovic, 2005).

It is known that water loss from a crop is related to the incident solar energy, and thus it is possible to develop a simple model that relates solar radiation to evapotranspiration.

while HS performs better in dry climates (Temesgen et al. 2005, Jabloun et al. 2008).

*<sup>T</sup> e eT*

min 17.27 0.611exp 237.3

*T*

min

*o a*

air temperatures in the previous and following months, respectively.

*G*, for monthly periods can be estimated as:

**3. Radiation based methods** 

0.7 *R Rb s a* (14)

1 1 0.07( ) *G TT i ii* (16)

(15)

island locations *Rs* can be estimated as (Allen et al. 1998):

**2.4 Reduced-set PM** 

calculated as:

The Priestley-Taylor method (Priestley and Taylor, 1972; De Bruin, 1983) is a simplified form of the Penman equation, that only needs net radiation and temperature to calculate *ETo*. This simplification is based on the fact that *ETo* is more dependant on radiation than on relative humidity and wind. The Priestly-Taylor method is basically the radiation driven part of the Penman Equation, multiplied by a coefficient, and can be expressed as:

$$ET\_o = \alpha \frac{\Delta \left(R\_n - G\right)}{\Delta + \gamma} + \beta \tag{17}$$

where and are calibration factors, assuming values of 1.26 and 0, respectively. This model was calibrated for Switzerland (Xu and Singh, 1998) and values of 0.98 and 0.94 were obtained for and , respectively. In the Priestley-Taylor equation, evapotranspiration is proportional to net radiation, while in the Makkink equation (section 3.2), it is proportional to short-wave radiation.

Van Kraalingen and Stol (1997) found that application of the Priestly-Taylor equation during the Dutch winter months was not possible because it is based on net radiation. Since net radiation is often negative in the winter, it predicts dew formation, whereas the actual ET is positive. The situation would be different for a humid climate such as the Philippines, or in a semi-arid climate such as Israel, where the equation should compare well with PM.

Irmak et al. (2003) calibrated the Priestly-Taylor method against the FAO PM method using 15 years of climate data (1980–1994) in humid Florida, United States. The monthly values of the calibration coefficient (Fig. 5) show a considerable seasonal variation, aside from the natural difference in annual values. In general, the calibration coefficients are lower in winter months indicating that the Priestley and Taylor method underestimates *ETo*, and they are higher than 1.0 during the summer months, indicating that the method overestimates during the summer months. The long-term average lowest calibration values were obtained in January and December (0.70) and the highest values in July (1.10). These results indicate the importance of developing monthly calibration coefficients for regional use based on historic records. For the semi-arid conditions of southern Portugal, the authors also observed that the Priestley-Taylor method over-estimates daily ETo during the summer months (Shahidian et al., 2007).

Shuttleworth and Calder (1979) showed that Priestley-Taylor significantly underestimates wet forest evaporation, but also overestimates dry forest transpiration by as much as 20%. Berengena and Gavilán (2005) found that the Priestley–Taylor equation shows a considerable tendency to underestimate ETo, on average 23%, under convective conditions. They concluded that the Priestly-Taylor equation is very sensitive to advection, and local calibration does not ensure an acceptable level of accuracy.

Fig. 5. Average monthly calibration coefficient for the Priestly-Taylor equation against PM for humid southern United States (based on data from Irmak et al. 2003).

#### **3.2 The Makkink method**

The Makkink method can be seen as a simplified form of the Priestley-Taylor method and was developed for grass lands in Holland. The difference is that the Makkink method uses incoming short-wave radiation *Rs* and temperature, instead of using net radiation, *Rn*, and temperature. This is possible, because on average, there is a constant ratio of 50% between net radiation and short wave radiation. The equation can be expressed as:

$$E t\_o = a \frac{\Delta}{\Delta + \chi} \frac{R\_s}{2\text{\textdegree\textdegree R}} + \beta$$

where is usually 0.61, and is -0.012. Doorenbos and Pruitt (1975) proposed the FAO Radiation method based on the Makkink equation (1957), introducing a correction factor based on estimates for wind and humidity conditions to compensate for advective conditions. This radiation method has been proven valid, in particular under humid conditions, but can differ systematically from the PM reference method under special conditions, such as during dry months (Bruin and Lablands, 1998).

It has also been observed that it is difficult to use this radiation based method during winter months: Van Kraalingen and Stol (1997) found that application of the Makkink equation in Dutch winter months was not possible, though the Makkink equation did not produce negative values for ET, as was the case with the Priestley-Taylor method. Bruin and Lablans (1998) also concluded that there is no relationship between Makkink and PM in the winter months, December and January, since Makkink's method has no physical meaning, in this period.

It is reasonable to expect the Makkink and the Priestley-Taylor equations to compare well with the Penman's method, since in all these approaches the radiation terms are dominant and radiation is the main driving force for evaporation in short vegetation.

ET models tend to perform best in climates in which they were designed. A study by Amayta et al. (1995) showed that while the Makkink model generally performed well in North Carolina, the model underestimated *ETo* in the peak months of summer. Yet, the Makkink model shows excellent results in Western Europe where it was designed, both in comparison to PM as well as to the measured ETo data (Bruin and Lablans 1998, Xu and Singh 2000, Bruin and Stricker 2000, Barnett et al., 1998).

#### **3.3 The Turc method**

72 Evapotranspiration – Remote Sensing and Modeling

They concluded that the Priestly-Taylor equation is very sensitive to advection, and local

*Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month of the year*

Fig. 5. Average monthly calibration coefficient for the Priestly-Taylor equation against PM

The Makkink method can be seen as a simplified form of the Priestley-Taylor method and was developed for grass lands in Holland. The difference is that the Makkink method uses incoming short-wave radiation *Rs* and temperature, instead of using net radiation, *Rn*, and temperature. This is possible, because on average, there is a constant ratio of 50% between

> *<sup>s</sup> <sup>o</sup> <sup>R</sup> Et*

Radiation method based on the Makkink equation (1957), introducing a correction factor based on estimates for wind and humidity conditions to compensate for advective conditions. This radiation method has been proven valid, in particular under humid conditions, but can differ systematically from the PM reference method under special

It has also been observed that it is difficult to use this radiation based method during winter months: Van Kraalingen and Stol (1997) found that application of the Makkink equation in Dutch winter months was not possible, though the Makkink equation did not produce negative values for ET, as was the case with the Priestley-Taylor method. Bruin and Lablans (1998) also concluded that there is no relationship between Makkink and PM in the winter months, December and January, since Makkink's method has no physical meaning, in this

It is reasonable to expect the Makkink and the Priestley-Taylor equations to compare well with the Penman's method, since in all these approaches the radiation terms are dominant

2,45

(18)

is -0.012. Doorenbos and Pruitt (1975) proposed the FAO

for humid southern United States (based on data from Irmak et al. 2003).

net radiation and short wave radiation. The equation can be expressed as:

conditions, such as during dry months (Bruin and Lablands, 1998).

and radiation is the main driving force for evaporation in short vegetation.

calibration does not ensure an acceptable level of accuracy.

*0*

is usually 0.61, and

*0.2*

*0.4*

*0.6*

*0.8*

*Priestly Taylor/Pm calibration coefficient*

**3.2 The Makkink method** 

where 

period.

*1*

*1.2*

*1.4*

Also known as the Turc-Radiation equation, this method was presented by Turc in 1961, using data from the humid climate of Western Europe (France). This method only uses two parameters, average daily radiation and temperature and for RH>50% can be expressed as:

$$ET\_p = a \left( \left( 23, 9001R\_s \right) + 50 \right) \left( \frac{T}{T + 15} \right) \tag{19}$$

And for RH < 50% as:

$$ET\_p = \alpha \left( (23, 9001R\_s) + 50 \right) \left( \frac{T}{T + 15} \right) \left( 1 + \left( \frac{50 - RH}{70} \right) \right) \tag{20}$$

Where is 0.01333 and *Rs* is expressed in MJ m-2 day-1.

Yoder et al. (2005) compared six different ET equations in humid southeast United States, and found the Turc equation to be second best only to the full PM. Jensen et al. (1990) analyzed the properties of twenty different methods against carefully selected lysimeter data from eleven stations, located worldwide in different climates. They observed that the Turc method compared very favorably with combination methods at the humid lysimeter locations. The Turc method was ranked second when only humid locations were considered, with only the Penman-Monteith method performing better. Trajkovic and Stojnic (2007) compared the Turc method with full PM in 52 European sites and found a SEE (Standard Error of Estimate) of between 0.10 and 0.37 mm d-1. They also found that the reliability of the Turc method depends on the wind speed (Fig. 6). The Turc method overestimated PM ETo in windless locations and generally underestimated ETo in windy locations.

Amatya et al. (1995) compared 5 different ETo methodologies in North Carolina and concluded that the Turc and the Priestley-Taylor methods were generally the best in estimating ETo. They observed that all other radiation methods and the temperature based Thorntwaite method underestimated the annual ET by as much as 16%.

Kashyap and Panda (2001) compared 10 different methods with lysimeter data in the sub humid Kharagupur region of India and observed that the Turc method had a deviation of only 2.72% from lysimeter values, followed by Blaney-Criddle with a 3.16% and Priestly Taylor with a 6.28% deviation (Fig. 7). The Kashyap and Panda data are also important because they show that under sub humid conditions, most of the equations, including the PM, tend to overestimate when evapotranspiration is low, and underestimate when it is high.

Fig. 6. Effect of wind on the ratio of evapotranspiration calculated with the FAO PM and the Turc methods (based on data from Trajkovic and Stojnic (2007), using average annual values).

Fig. 7. Comparison of various ETo methods with Lysimeter readings in the sub-humid region of Kharagpur, India (adapted from Kashyap and Panda, 2001).

For Florida, Martinez and Thepadia (2010) compared the reduced-set PM equation with various temperature and radiation based equations and concluded that in the absence of regionally calibrated methods, the Turc equation has the least error and bias when using measured maximum and minimum temperatures. They also observed that the reduced-set PM and Hargreaves equations overestimate ET.

Fontenote (2004) studied the accuracy of seven evapotranspiraiton models for estimating grass reference ET in Louisiana. He observed that, statewide and in the coastal region, the Turc model was the most accurate daily model with a MAE of 0.26mm day-1. Inland, the Blaney-Criddle performed best with a MAE of 0.31mm day-1 (Fig. 8).

Hence, it can be safely concluded that the Turc model can be expected to perform well in warm, humid climates such as those found in North Carolina (Amatya et al., 1995), India (George et al., 2002), and Florida (Irmak et al., 2003; Martinez and Thepadia, 2010).

Fig. 8. Comparison of five ET methods with PM in two different regions of Louisiana (Adapted from Fontenote, 1999).

#### **3.4 The Jensen and Haise method**

This method was derived for the drier parts of the United States and is based on 3,000 observations of ET. Jensen and Haise used 35 years of measured evapotranspiration and solar radiation to derive the equation, based on the assumption that net radiation is more closely related to ET than other variables such as air temperature and humidity (Jensen and Haise, 1965). The equation can be expressed as:

$$ET = C\_t \left( T - T\_x \right) R\_s \tag{21}$$

The original study of Jensen and Haise provides a calculation procedure to obtain *Rs* from the cloudiness, *Cl*, and the solar and sky radiation flux on cloudless days. The temperature Constant, *Ct*, and the intercept of the temperature exis, *Tx*, can be calculated as follows:

$$C\_t = \frac{1}{\left[ \left( 45 - \frac{h}{137} \right) + \left( \frac{365}{e^0 \left( T\_{\text{max}} \right) - e^0 \left( T\_{\text{min}} \right)} \right) \right]} \tag{22}$$

and

74 Evapotranspiration – Remote Sensing and Modeling

0 0.5 1 1.5 2 2.5 3 3.5 *Wind speed, m s -1*

*0 2 4 6 810 ET Lysimeter, mm*

Fig. 7. Comparison of various ETo methods with Lysimeter readings in the sub-humid

For Florida, Martinez and Thepadia (2010) compared the reduced-set PM equation with various temperature and radiation based equations and concluded that in the absence of regionally calibrated methods, the Turc equation has the least error and bias when using measured maximum and minimum temperatures. They also observed that the reduced-set

Fontenote (2004) studied the accuracy of seven evapotranspiraiton models for estimating grass reference ET in Louisiana. He observed that, statewide and in the coastal region, the Turc model was the most accurate daily model with a MAE of 0.26mm day-1. Inland, the

Hence, it can be safely concluded that the Turc model can be expected to perform well in warm, humid climates such as those found in North Carolina (Amatya et al., 1995), India

(George et al., 2002), and Florida (Irmak et al., 2003; Martinez and Thepadia, 2010).

region of Kharagpur, India (adapted from Kashyap and Panda, 2001).

Blaney-Criddle performed best with a MAE of 0.31mm day-1 (Fig. 8).

*1:1 Blaney Criddle Hargreaves Samani Priestley Taylor PM Turc*

Fig. 6. Effect of wind on the ratio of evapotranspiration calculated with the FAO PM and the Turc methods (based on data from Trajkovic and Stojnic (2007), using average annual

y = 0.9838x0.0935 R2 = 0.6747

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

*0*

PM and Hargreaves equations overestimate ET.

*2*

*4*

*ET different methods, mm*

*6*

*8*

*10*

values).

*Ratio of FAO-PM /Turc* 

$$T\_x = -2.5 - 0.14 \left( e^0 \left( T\_{\text{max}} \right) - e^0 \left( T\_{\text{min}} \right) \right) - \bigvee\_{500} \tag{23}$$

where *h* is the altitude of the location in m, *Rs* is solar radiation (MJ m-2 d-1); *eoTmax* and *eoTmin* are vapour pressures of the month with the mean maximum temperature and the month with the mean minimum temperature, respectively, expressed in mbar.

For the humid and rainy Rio Grande watershed in Brazil, Pereira et al. (2009) compared 10 different equations and concluded that the methods based on solar radiation are more accurate than those based only on air temperature, with the Jensen and Haise method presenting the smallest MBE, and thus being the method most recommended for this region.

## **4. Conclusions**

Both temperature and radiation can be used successfully to calculate daily *ETo* values with relative accuracy. All the equations can be used for areas that have a climate that is similar to the one for which the equations were originally developed; while most of the equations can be used with some confidence for areas with moderate conditions of humidity and wind speed.

Regional calibration, especially if including monthly calibration coefficients, is important in decreasing the bias of the ETo estimates. Wind speed can greatly influence the results obtained with reduced-set equations, since wind removes the boundary layer from the leaf surface and can significantly increase evapotranspiration. Relative Humidity is another important factor that can affect the results.

Globally, it is observed that the Turc equation is highly recommended for humid or semihumid areas, where it can produce very good results even without calibration, while the Thornthwaite equation tends to underestimate *ETo*.

The Priestley-Taylor and the Makkinik equations should not be used in the winter months in locations with high latitude, such as northern Europe.

Both the Hargreaves and the reduced-set Panman-Monteith can be effectively used with only temperature measurements, although the results can be improved if wind speed is taken into consideration.

The use of the reduced-set equations can be very important in actual irrigation management, since the error involved in using these equations can be much smaller than that resulting from using data from a weather station located many miles away.

## **5. References**


Both temperature and radiation can be used successfully to calculate daily *ETo* values with relative accuracy. All the equations can be used for areas that have a climate that is similar to the one for which the equations were originally developed; while most of the equations can be used with some confidence for areas with moderate conditions of humidity and wind

Regional calibration, especially if including monthly calibration coefficients, is important in decreasing the bias of the ETo estimates. Wind speed can greatly influence the results obtained with reduced-set equations, since wind removes the boundary layer from the leaf surface and can significantly increase evapotranspiration. Relative Humidity is another

Globally, it is observed that the Turc equation is highly recommended for humid or semihumid areas, where it can produce very good results even without calibration, while the

The Priestley-Taylor and the Makkinik equations should not be used in the winter months

Both the Hargreaves and the reduced-set Panman-Monteith can be effectively used with only temperature measurements, although the results can be improved if wind speed is

The use of the reduced-set equations can be very important in actual irrigation management, since the error involved in using these equations can be much smaller than that resulting

Allen RG (1993) Evaluation of a temperature difference method for computing grass

Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspiration: Guidelines for

Amatya DM, Skaggs RW, Gregory JD(1995) Comparison of Methods for Estimating REF-ET.

Bautista F, Bautista D, Delgado-Carranza (2009) Calibration of the equations of Hargreaves

Borges AC, Mendiondo EM (2007) Comparação entre equações empíricas para estimativa da

Blaney, HF, Criddle, WD (1950). Determining water requirements in irrigated áreas from climatological and irrigation data. In ISDA Soil Conserv. Serv., SCS-TP-96,

and Man. Serv., Land and Water Develop. Div., FAO, Rome. 49 p.

Journal of Irrigation and Drainage Engineering 121:427-435.

semiarid environment. J. Irrig. Drain. Eng. ASCE 131 (2):147–163.

Engenharia Agrícola e Ambiental 11(3): 293–300.

reference evapotranspiration. Report submitted to the Water Resources Develop.

computing crop requirements. Irrigation and Drainage Paper No. 56, FAO, Rome,

and Thornthwaite to estimate the potential evapotranspiration in semi-arid and sub-humid tropical climates for regional applications. Atmósfera 22(4): 331-348 Berengena J, Gavilán P (2005) Reference evapotranspiration estimation in a highly advective

evapotranspiração de referência na Bacia do Rio Jacupiranga. Revista Brasileira de

**4. Conclusions** 

important factor that can affect the results.

taken into consideration.

**5. References** 

Italy.

Thornthwaite equation tends to underestimate *ETo*.

in locations with high latitude, such as northern Europe.

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## **Fuzzy-Probabilistic Calculations of Evapotranspiration**

Boris Faybishenko

*Lawrence Berkeley National Laboratory Berkeley, CA USA* 

### **1. Introduction**

80 Evapotranspiration – Remote Sensing and Modeling

Xu, CY, Singh, VP (2000) Evaluation and Generalization of Radiation-based Methods for

Xu CY, Singh VP (2002) Cross Comparison of Empirical Equations for Calculating Potential

evapotranspiration with data from Switzerland. Water Resources Management 16:

Calculating Evaporation, *Hydrolog. Processes* 14: 339–349.

197-219.

Evaluation of evapotranspiration uncertainty is needed for proper decision-making in the fields of water resources and climatic predictions (Buttafuoco et al., 2010; Or and Hanks, 1992; Zhu et al., 2007). However, in spite of the recent progress in soil-water and climatic uncertainty quantification, using stochastic simulations, the estimates of potential (reference) evapotranspiration (*E*o) and actual evapotranspiration (*ET*) using different methods/models, with input parameters presented as PDFs or fuzzy numbers, is a somewhat overlooked aspect of water-balance uncertainty evaluation (Kingston et al., 2009). One of the reasons for using a combination of different methods/models and presenting the final results as fuzzy numbers is that the selection of the model is often based on vague, inconsistent, incomplete, or subjective information. Such information would be insufficient for constructing a single reliable model with probability distributions, which, in turn, would limit the application of conventional stochastic methods.

Several alternative approaches for modeling complex systems with uncertain models and parameters have been developed over the past ~50 years, based on fuzzy set theory and possibility theory (Zadeh, 1978; 1986; Dubois & Prade, 1994; Yager & Kelman, 1996). Some of these approaches include the blending of fuzzy-interval analysis with probabilistic methods (Ferson & Ginzburg, 1995; Ferson, 2002; Ferson et al., 2003). This type of analysis has recently been applied to hydrological research, risk assessment, and sustainable waterresource management under uncertainty (Chang, 2005), as well as to calculations of *E*o, ET, and infiltration (Faybishenko, 2010).

The objectives of this chapter are to illustrate the application of a combination of probability and possibility conceptual-mathematical approaches—using fuzzy-probabilistic models for predictions of potential evapotranspiration (*E*o) and actual evapotranspiration (*ET*) and their uncertainties, and to compare the results of calculations with field evapotranspiration measurements.

As a case study, statistics based on monthly and annual climatic data from the Hanford site, Washington, USA, are used as input parameters into calculations of potential evapotranspiration, using the Bair-Robertson, Blaney-Criddle, Caprio, Hargreaves, Hamon, Jensen-Haise, Linacre, Makkink, Penman, Penman-Monteith, Priestly-Taylor, Thornthwaite, and Turc equations. These results are then used for calculations of evapotranspiration based on the modified Budyko (1974) model. Probabilistic calculations are performed using Monte Carlo and p-box approaches, and fuzzy-probabilistic and fuzzy simulations are conducted using the RAMAS Risk Calc code. Note that this work is a further extension of this author's recently published work (Faybishenko, 2007, 2010).

The structure of this chapter is as follows: Section 2 includes a review of semi-empirical equations describing potential evapotranspiration, and a modified Budyko's model for evaluating evapotranspiration. Section 3 includes a discussion of two types of uncertainties—epistemic and aleatory uncertainties—involved in assessing evapotranspiration, and a general approach to fuzzy-probabilistic simulations by means of combining possibility and probability approaches. Section 4 presents a summary of input parameters and the results of *E*o and *ET* calculations for the Hanford site, and Section 5 provides conclusions.

## **2. Calculating potential evapotranspiration and evapotranspiration**

### **2.1 Equations for calculations of potential evapotranspiration**

The potential (reference) evapotranspiration *E*o is defined as evapotranspiration from a hypothetical 12 cm grass reference crop under well-watered conditions, with a fixed surface resistance of 70 s m-1 and an albedo of 0.23 (Allen et al., 1998). Note that this subsection includes a general description of equations used for calculations of potential evapotranspiration; it does not provide an analysis of the various advantages and disadvantages in applying these equations, which are given in other publications (for example, Allen et al., 1998; Allen & Pruitt, 1986; Batchelor, 1984; Maulé et al., 2006; Sumner & Jacobs, 2005; Walter et al., 2002).

The two forms of Baier-Robertson equations (Baier, 1971; Baier & Robertson, 1965) are given by:

$$E\_0 = 0.157 T\_{\text{max}} + 0.158 \left( T\_{\text{max}} \cdot T\_{\text{min}} \right) + 0.109 R\_u \cdot 5.39 \tag{1}$$

$$E\_o = -0.0039T\_{\text{max}} + 0.1844(T\_{\text{max}} \cdot T\_{\text{min}}) + 0.1136 \ R\_a + 2.811(e\_s - e\_a) - 4.0\tag{2}$$

where *E*o= daily evapotranspiration (mm day-1); *T*max = the maximum daily air temperature, oC; *T*min= minimum temperature, oC; *Ra* = extraterrestrial radiation (MJ m-2 day-1) (ASCE 2005), *es* = saturation vapor pressure (kPa), and *ea* = mean actual vapor pressure (kPa). Equation (1) takes into account the effect of temperature, and Equation (2) takes into account the effects of temperature and relative humidity.

The Blaney-Criddle equation (Allen & Pruitt, 1986) is used to calculate evapotranspiration for a reference crop, which is assumed to be actively growing green grass of 8–15 cm height:

$$E\_v = p \left( 0.46 \cdot T\_{\text{mean}} + 8 \right) \tag{3}$$

where *Eo* is the reference (monthly averaged) evapotranspiration (mm day−1), *T*mean is the mean daily temperature (°C) given as *Tmean =* (*Tmax + T*min)/2, and *p* is the mean daily percentage of annual daytime hours.

The Caprio (1974) equation for calculating the potential evapotranspiration is given by

$$E\_o = 6.1 \cdot 10^{\circ} \, R\_s \left[ \begin{pmatrix} 1.8 \ \cdot \, T\_{\text{mean}} \end{pmatrix} + 1.0 \right] \tag{4}$$

where *E*o = mean daily potential evapotranspiration (mm day-1); *R*<sup>s</sup> = daily global (total) solar radiation (kJ m-2 day-1); and *T*mean = mean daily air temperature (°C).

The Hansen (1984) equation is given by:

82 Evapotranspiration – Remote Sensing and Modeling

Carlo and p-box approaches, and fuzzy-probabilistic and fuzzy simulations are conducted using the RAMAS Risk Calc code. Note that this work is a further extension of this author's

The structure of this chapter is as follows: Section 2 includes a review of semi-empirical equations describing potential evapotranspiration, and a modified Budyko's model for evaluating evapotranspiration. Section 3 includes a discussion of two types of uncertainties—epistemic and aleatory uncertainties—involved in assessing evapotranspiration, and a general approach to fuzzy-probabilistic simulations by means of combining possibility and probability approaches. Section 4 presents a summary of input parameters and the results of *E*o and *ET* calculations for the Hanford site, and Section 5

The potential (reference) evapotranspiration *E*o is defined as evapotranspiration from a hypothetical 12 cm grass reference crop under well-watered conditions, with a fixed surface resistance of 70 s m-1 and an albedo of 0.23 (Allen et al., 1998). Note that this subsection includes a general description of equations used for calculations of potential evapotranspiration; it does not provide an analysis of the various advantages and disadvantages in applying these equations, which are given in other publications (for example, Allen et al., 1998; Allen & Pruitt, 1986; Batchelor, 1984; Maulé et al., 2006; Sumner

The two forms of Baier-Robertson equations (Baier, 1971; Baier & Robertson, 1965) are given

 *E*o= 0.157*T*max + 0.158 (*T*max - *T*min) + 0.109*R*a - 5.39 (1)

 *E*o= -0.0039*T*max + 0.1844(*T*max - *T*min) + 0.1136 *R*a + 2.811(*es* − *ea* ) − 4.0 (2) where *E*o= daily evapotranspiration (mm day-1); *T*max = the maximum daily air temperature, oC; *T*min= minimum temperature, oC; *Ra* = extraterrestrial radiation (MJ m-2 day-1) (ASCE 2005), *es* = saturation vapor pressure (kPa), and *ea* = mean actual vapor pressure (kPa). Equation (1) takes into account the effect of temperature, and Equation (2) takes into account

The Blaney-Criddle equation (Allen & Pruitt, 1986) is used to calculate evapotranspiration for a reference crop, which is assumed to be actively growing green grass of 8–15 cm height:

 *Eo* = *p* (0.46·*T*mean + 8) (3) where *Eo* is the reference (monthly averaged) evapotranspiration (mm day−1), *T*mean is the mean daily temperature (°C) given as *Tmean =* (*Tmax + T*min)/2, and *p* is the mean daily

 *E*o = 6.1·10-6 *R*s [(1.8 ·*T*mean) + 1.0] (4)

= daily global (total)

The Caprio (1974) equation for calculating the potential evapotranspiration is given by

where *E*o = mean daily potential evapotranspiration (mm day-1); *R*<sup>s</sup>

solar radiation (kJ m-2 day-1); and *T*mean = mean daily air temperature (°C).

**2. Calculating potential evapotranspiration and evapotranspiration** 

**2.1 Equations for calculations of potential evapotranspiration** 

recently published work (Faybishenko, 2007, 2010).

provides conclusions.

& Jacobs, 2005; Walter et al., 2002).

the effects of temperature and relative humidity.

percentage of annual daytime hours.

by:

$$E\_o = 0.7 \,\text{\AA/} \,\text{ (}\Delta + \eta\text{)} \cdot R\_\text{/} \lambda \,\text{\AA} \tag{5}$$

where = slope of the saturation vapor pressure vs. temperature curve, = psychrometric constant, *R*i = global radiation, and = latent heat of water vaporization. The Hargreaves equation (Hargreaves & Samani, 1985) is given by

$$E\_o = 0.0023(T\_{\text{mean}} + 17.8)(T\_{\text{max}} - T\_{\text{min}})^{0.5}R\_\text{a} \tag{6}$$

where both *E*o and *R*a (extraterrestrial radiation) are in millimeters per day-1 (mm day-1). The Jensen and Haise (1963) equation is given by

$$E\_o = R\_s / 2450 \left[ (0.025 \ T\_{\text{mean}}) + 0.08 \right] \tag{7}$$

where *E*o = monthly mean of daily potential evapotranspiration (mm day-1); *R*<sup>s</sup> = monthly mean of daily global (total) solar radiation (kJ m-2 day-1); and *T*mean = monthly mean temperature.

The Linacre (1977) equation is given by:

$$E\_o = \left[500T\_m \text{ / (100-L)} + 15(T \text{-Td})\right] \text{ / (80-T)}\tag{8}$$

where *E*o is in mm day-1, *T*m = temperature adjusted for elevation, *T*m = *T* + 0.006h (°C), *h* = elevation (m), *T*d = dew point temperature (°C), and *L* = latitude (°). The Makkink (1957) model is given by

 *E*o= 0.61 / ( + ) *R*s/2.45 – 0.12 (9)

where *R*s = solar radiation (MJ m-2 day-1), and and are the parameters defined above. The Penman (1963) equation is given by

$$E\_o = mR\_n + \gamma \left\{ 6.43(1 + 0.536 \text{ } \mu\_2) \text{ è } \begin{array}{c} \lambda\_v \text{ (} m + \gamma \text{)} \end{array} \right. \tag{10}$$

where = slope of the saturation vapor pressure curve (kPa K-1), *R*n = net irradiance (MJ m-2 day-1), *ρ*a = density of air (kg m-3), *c*p = heat capacity of air (J kg-1 K-1), *e* = vapor pressure deficit (Pa), *v* = latent heat of vaporization (J kg-1), = psychrometric constant (Pa K-1), and *E*o is in units of kg/(m²s).

The general form of the Penman-Monteith equation (Allen et al., 1998) is given by

$$E\_o = \left[ 0.408 \,\Lambda \, \left( \text{R}\_n - \text{G} \right) + \, \text{C}\_n \, \gamma \, \left/ \left( \text{T} + 27 \text{S} \right) \, \mu\_2 \left( e\_5 \, e\_4 \right) \right] / \left[ \Delta + \gamma \, \left( \text{1} + \text{C}\_4 \, \mu\_2 \right) \right] \tag{11}$$

where *E*o is the standardized reference crop evapotranspiration (in mm day-1) for a short (0.12 m, with values *C*n=900 and *C*d=0.34) reference crop or a tall (0.5 m, with values *C*n=1600 and *C*d=0.38) reference crop, *R*n = net radiation at the crop surface (MJ m-2 day-1), *G* = soil heat flux density (MJ m-2 day-1), *T* = air temperature at 2 m height (°C), *u*2 = wind speed at 2 m height (m s-1), *e*s = saturation vapor pressure (kPa), *e*a = actual vapor pressure (kPa), (*e*s *e*a) = saturation vapor pressure deficit (kPa), = slope of the vapor pressure curve (kPa °C-1), and = psychrometric constant (kPa °C-1).

The Priestley–Taylor (1972) equation is given by

$$E\_o = \alpha \, 1/\lambda \, \Delta \, (\mathrm{R}\_n - \mathrm{G}) \, / \, (\Delta + \gamma) \tag{12}$$

where = latent heat of vaporization (MJ kg-1), *R*n = net radiation (MJ m-2 day-1), G = soil heat flux (MJ m-2 day-1), = slope of the saturation vapor pressure-temperature relationship (kPa °C-1), = psychrometric constant (kPa °C-1), and = 1.26. Eichinger et al. (1996) showed that is practically constant for all typically observed atmospheric conditions and relatively insensitive to small changes in atmospheric parameters. (On the other hand, Sumner and Jacobs [2005] showed that is a function of the green-leaf area index [LAI] and solar radiation.)

The Thornthwaite (1948) equation is given by

$$E\_o = 1.6 \text{ (L/12) (N/30) (10 } T\_{\text{mean} \text{(i)}} / l)^{a} \tag{13}$$

where *E*<sup>o</sup> is the estimated potential evapotranspiration (cm/month), *T*mean (i) = average monthly (*i*) temperature (oC); if *T*mean (*<sup>i</sup>*) < 0, *E*o = 0 of the month (*i*) being calculated, *N* = number of days in the month, *L* = average day length (hours) of the month being calculated, and *I* = heat index given by

$$I = \sum\_{i=1}^{12} \left(\frac{T\_{\text{mean}(i)}}{5}\right)^{1.514}$$

and = (6.75·10-7) *I*3 – (7.71·10-5) *I*2 + (1.792·10-2)*I* + 0.49239 The Turc (1963) equation is given by

$$E\_o = \left(0.0239 \cdot R\_s + 50\right) \left[0.4/30 \cdot T\_{\text{mean}} \Big/ \left(T\_{\text{mean}} + 15.0\right)\right] \tag{14}$$

where *Eo* = mean daily potential evapotranspiration (mm/day); *Rs* = daily global (total) solar radiation (kJ/m2/day); *T*mean = mean daily air temperature (°C).

#### **2.2 Modified Budyko's equation for evaluating evapotranspiration**

For regional-scale, long-term water-balance calculations within arid and semi-arid areas, we can reasonably assume that (1) soil water storage does not change, (2) lateral water motion within the shallow subsurface is negligible, (3) the surface-water runoff and runon for regional-scale calculations simply cancel each other out, and (4) *ET* is determined as a function of the aridity index, *ET*=*f*(where  *E*o/*P*, which is the ratio of potential evapotranspiration, *E*o, to precipitation, *P* (Arora 2002).

Budyko's (1974) empirical formula for the relationship between the ratio of *ET/P* and the aridity index was developed using the data from a number of catchments around the world, and is given by:

$$ET/\!P = \{ \phi \text{ tanh } (1/\phi) \left[ 1 - \exp \left( -\phi \right) \right] \}^{0.5} \tag{15}$$

Equation (1) can also be given as a simple exponential expression (Faybishenko, 2010):

$$ET/\mathbb{P} \equiv a \left[ 1 \cdot \exp \left( \cdot b \,\phi \right) \right] \tag{16}$$

with coefficients *a* =0.9946 and *b* =1.1493. The correlation coefficient between the calculations using (15) and (16) is *R*=0.999. Application of the modified Budyko's equation, given by an exponential function (2) with the value in single term, will simplify further calculations of *ET*.

where = latent heat of vaporization (MJ kg-1), *R*n = net radiation (MJ m-2 day-1), G = soil heat flux (MJ m-2 day-1), = slope of the saturation vapor pressure-temperature relationship (kPa °C-1), = psychrometric constant (kPa °C-1), and = 1.26. Eichinger et al. (1996) showed that is practically constant for all typically observed atmospheric conditions and relatively insensitive to small changes in atmospheric parameters. (On the other hand, Sumner and Jacobs [2005] showed that is a function of the green-leaf area index [LAI] and

where *E*<sup>o</sup> is the estimated potential evapotranspiration (cm/month), *T*mean (i) = average monthly (*i*) temperature (oC); if *T*mean (*<sup>i</sup>*) < 0, *E*o = 0 of the month (*i*) being calculated, *N* = number of days in the month, *L* = average day length (hours) of the month being calculated,

1.514 <sup>12</sup> mean( )

*i*

<sup>1</sup> 5

 *E*o = (0.0239 · *R*s + 50) [0.4/30 · *T*mean / (*T*mean + 15.0)] (14) where *Eo* = mean daily potential evapotranspiration (mm/day); *Rs* = daily global (total) solar

For regional-scale, long-term water-balance calculations within arid and semi-arid areas, we can reasonably assume that (1) soil water storage does not change, (2) lateral water motion within the shallow subsurface is negligible, (3) the surface-water runoff and runon for regional-scale calculations simply cancel each other out, and (4) *ET* is determined as a function of the aridity index, *ET*=*f*(where  *E*o/*P*, which is the ratio of potential

Budyko's (1974) empirical formula for the relationship between the ratio of *ET/P* and the aridity index was developed using the data from a number of catchments around the world,

with coefficients *a* =0.9946 and *b* =1.1493. The correlation coefficient between the calculations using (15) and (16) is *R*=0.999. Application of the modified Budyko's equation, given by an

Equation (1) can also be given as a simple exponential expression (Faybishenko, 2010):

*ET/P* = { tanh (1/exp (-)]}0.5 (15)

value in single term, will simplify further calculations of

)] (16)

*T*

 

*i*

*I*  (13)

solar radiation.)

and is given by:

*ET*.

and *I* = heat index given by

The Turc (1963) equation is given by

The Thornthwaite (1948) equation is given by

 *E*o= 1.6 (*L*/12) (*N*/30) (10 *T*mean (*<sup>i</sup>*) /*I*)

and = (6.75·10-7) *I*3 – (7.71·10-5) *I*2 + (1.792·10-2)*I* + 0.49239

radiation (kJ/m2/day); *T*mean = mean daily air temperature (°C).

evapotranspiration, *E*o, to precipitation, *P* (Arora 2002).

 *ET/P*=*a*[1-exp(-*b*

exponential function (2) with the

**2.2 Modified Budyko's equation for evaluating evapotranspiration** 

### **3. Types of uncertainties in calculating evapotranspiration and simulation approaches**

#### **3.1 Epistemic and aleatory uncertainties**

The uncertainties involved in predictions of evapotranspiration, as a component of soilwater balance, can generally be categorized into two groups—*aleatory* and *epistemic* uncertainties. Aleatory uncertainty arises because of the natural, inherent variability of soil and meteorological parameters, caused by the subsurface heterogeneity and variability of meteorological parameters. If sufficient information is available, probability density functions (PDFs) of input parameters can be used for stochastic simulations to assess aleatory evapotranspiration uncertainty. In the event of a lack of reliable experimental data, fuzzy numbers can be used for fuzzy or fuzzy-probabilistic calculations of the aleatory evapotranspiration uncertainty (Faybishenko 2010).

Epistemic uncertainty arises because of a lack of knowledge or poor understanding, ambiguous, conflicting, or insufficient experimental data needed to characterize coupledphysics phenomena and processes, as well as to select or derive appropriate conceptualmathematical models and their parameters. This type of uncertainty is also referred to as subjective or reducible uncertainty, because it can be reduced as new information becomes available, and by using various models for uncertainty evaluation. Generally, variability, imprecise measurements, and errors are distinct features of uncertainty; however, they are very difficult, if not impossible, to distinguish (Ferson & Ginzburg, 1995).

In this chapter the author will consider the effect of aleatory uncertainty on evapotranspiration calculations by assigning the probability distributions of input meteorological parameters, and the effect of epistemic uncertainty is considered by using different evapotranspiration models.

### **3.2 Simulation approaches**

#### **3.2.1 Probability approach**

A common approach for assessing uncertainty is based on Monte Carlo simulations, using PDFs describing model parameters. Another probability-based approach to the specification of uncertain parameters is based on the application of probability boxes (Ferson, 2002; Ferson et al., 2003). The probability box (p-box) approach is used to impose bounds on a cumulative distribution function (CDF), expressing different sources of uncertainty. This method provides an envelope of distribution functions that bounds all possible dependencies. An uncertain variable *x* expressed with a probability distribution, as shown in Figure 1a, can be represented as a variable that is bounded by a p-box [ *F* , *F* ], with the right curve *F* (*x*) bounding the higher values of *x* and the lower probability of *x*, and the left curve *F* (*x*) bounding the lower values and the higher probability of *x*. With better or sufficiently abundant empirical information, the p-box bounds are usually narrower, and the results of predictions come close to a PDF from traditional probability theory.

#### **3.2.2 Possibility approach**

In the event of imprecise, vague, inconsistent, incomplete, or subjective information about models and input parameters, the uncertainty is captured using *fuzzy modeling theory,* or *possibility theory*, introduced by Zadeh (1978). For the past 50 years or so, possibility theory has successfully been applied to describe such systems as complex, large-scale engineering systems, social and economic systems, management systems, medical diagnostic processes, human perception, and others. The term *fuzziness* is, in general, used in possibility theory to

Fig. 1. Graphical illustration of uncertain numbers: (a) Cumulative normal distribution function (dashed line), with mean=10 and standard deviation =1, and a p-box—left bound with mean=9.5 and =0.9, and right bound with mean=10.5 and =1.1; and (b) Fuzzy trapezoidal (solid line) number, plotted using Eq. (17) with a=6, b=9, c=11, and d=14. Interval [b,c]=[9, 11] corresponds to FMF=1. Triangular (short dashes) and Gaussian (long dashes) fuzzy numbers are also shown. Figure (b) also shows an -cut=0.5 (thick horizontal line) through the trapezoidal fuzzy number (Faybishenko 2010).

describe objects or processes that cannot be given precise definition or precisely measured. *Fuzziness* identifies a class (set) of objects with nonsharp (i.e., fuzzy) boundaries, which may result from imprecision in the meaning of a concept, model, or measurements used to characterize and model the system. Fuzzification implies replacing a set of crisp (i.e., precise) numbers with a set of fuzzy numbers, using fuzzy membership functions based on the results of measurements and perception-based information (Zadeh 1978). A fuzzy number is a quantity whose value is imprecise, rather than exact (as is the case of a singlevalued number). Any fuzzy number can be thought of as a function whose domain is a specified set of real numbers. Each numerical value in the domain is assigned a specific "grade of membership," with 0 representing the smallest possible grade (full nonmembership), and 1 representing the largest possible grade (full membership). The grade of membership is also called the degree of possibility and is expressed using fuzzy membership functions (FMFs). In other words, a fuzzy number is a fuzzy subset of the domain of real numbers, which is an alternative approach to expressing uncertainty.

Several types of FMFs are commonly used to define fuzzy numbers: triangular, trapezoidal, Gaussian, sigmoid, bell-curve, Pi-, *S*-, and *Z*-shaped curves. As an illustration, Figure 1b shows a trapezoidal fuzzy number given by

$$f(\mathbf{x}) = \begin{cases} 0, \mathbf{x} \le a \\ \frac{\mathbf{x} - a}{b - a}, a \le \mathbf{x} \le b \\ 1, b \le \mathbf{x} \le c \\ \frac{d - \mathbf{x}}{d - c}, c \le \mathbf{x} \le d \\ 0, d \le \mathbf{x} \end{cases} \tag{17}$$

**(b)**

**FMF/Possibility**

Fig. 1. Graphical illustration of uncertain numbers: (a) Cumulative normal distribution function (dashed line), with mean=10 and standard deviation =1, and a p-box—left bound with mean=9.5 and =0.9, and right bound with mean=10.5 and =1.1; and (b) Fuzzy trapezoidal (solid line) number, plotted using Eq. (17) with a=6, b=9, c=11, and d=14. Interval [b,c]=[9, 11] corresponds to FMF=1. Triangular (short dashes) and Gaussian (long dashes) fuzzy numbers are also shown. Figure (b) also shows an -cut=0.5 (thick horizontal

describe objects or processes that cannot be given precise definition or precisely measured. *Fuzziness* identifies a class (set) of objects with nonsharp (i.e., fuzzy) boundaries, which may result from imprecision in the meaning of a concept, model, or measurements used to characterize and model the system. Fuzzification implies replacing a set of crisp (i.e., precise) numbers with a set of fuzzy numbers, using fuzzy membership functions based on the results of measurements and perception-based information (Zadeh 1978). A fuzzy number is a quantity whose value is imprecise, rather than exact (as is the case of a singlevalued number). Any fuzzy number can be thought of as a function whose domain is a specified set of real numbers. Each numerical value in the domain is assigned a specific "grade of membership," with 0 representing the smallest possible grade (full nonmembership), and 1 representing the largest possible grade (full membership). The grade of membership is also called the degree of possibility and is expressed using fuzzy membership functions (FMFs). In other words, a fuzzy number is a fuzzy subset of the

domain of real numbers, which is an alternative approach to expressing uncertainty.

0,

*b a fx b x c*

> *d c d x*

0,

( ) 1,

Several types of FMFs are commonly used to define fuzzy numbers: triangular, trapezoidal, Gaussian, sigmoid, bell-curve, Pi-, *S*-, and *Z*-shaped curves. As an illustration, Figure 1b

,

*x a axb*

*x a*

,

*d x cxd*

line) through the trapezoidal fuzzy number (Faybishenko 2010).

shows a trapezoidal fuzzy number given by

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1**

> **6 8 10 12 14 16** *x*

, (17)

where coefficients *a*, *b*, *c* , and *d* are used to define the shape of the trapezoidal FMF. When *a= b*, the trapezoidal number becomes a triangular fuzzy number.

Figure 1b also illustrates one of the most important attributes of fuzzy numbers, which is the notion of an -cut. The -cut interval is a crisp interval, limited by a pair of real numbers. An -cut of 0 of the fuzzy variable represents the widest range of uncertainty of the variable, and an -cut value of 1 represents the narrowest range of uncertainty of the variable.

Possibility theory is generally applicable for evaluating all kinds of uncertainty, regardless of its source or nature. It is based on the application of both hard data and the subjective (perception-based) interpretation of data. Fuzzy approaches provide a distribution characterizing the results of all possible magnitudes, rather than just specifying upper or lower bounds. Fuzzy methods can be combined with calculations of PDFs, interval numbers, or p-boxes, using the RAMAS Risk Calc code (Ferson 2002). In this paper, the RAMAS Risk Calc code is used to assess the following characteristic parameters of the fuzzy numbers and p-boxes:


When fuzzy measures serve as upper bounds on probability measures, one could expect to obtain a conservative (bounding) prediction of system behavior. Therefore, fuzzy calculations may overestimate uncertainty. For example, the application of fuzzy methods is not optimal (i.e., it overestimates uncertainty) when sufficient data are available to construct reliable PDFs needed to perform a Monte Carlo analysis.

In a recent paper (Faybishenko 2010), this author demonstrated the application of the fuzzyprobabilistic method using a hybrid approach, with direct calculations, when some quantities can be represented by fuzzy numbers and other quantities by probability distributions and interval numbers (Kaufmann and Gupta 1985; Ferson 2002; Guyonnet et al. 2003; Cooper et al. 2006). In this paper, the author combines (aggregates) the results of Monte Carlo calculations with multiple *E*o models by means of fuzzy numbers and p-boxes, using the RAMAS Risk Calc software (Ferson 2002).

## **4. Hanford case study**

#### **4.1 Input parameters and modeling scenarios for the Hanford Site**

The Hanford Site in Southeastern Washington State is one of the largest environmental cleanup sites in the USA, comprising 1,450 km2 of semiarid desert. Located north of Richland, Washington, the Hanford Site is bordered on the east by the Columbia River and on the south by the Yakima River, which joins the Columbia River near Richland, in the Pasco Basin, one of the structural and topographic basins of the Columbia Plateau. The areal topography is gently rolling and covered with unconsolidated materials, which are sufficiently thick to mask the surface irregularities of the underlying material. Areas adjacent to the Hanford Site are primarily agricultural lands.

Meteorological parameters used to assign model input parameters were taken from the Hanford Meteorological Station (HMS—see http://hms.pnl.gov/), located at the center of the Hanford Site just outside the northeast corner of the 200 West Area, as well as from publications (DOE, 1996; Hoitink et al., 2002; Neitzel, 1996.) At the Hanford Site, the *E*o is estimated to be from 1,400 to 1,611 mm/yr (Ward et al. 2005), and the *ET* is estimated to be 160 mm/yr (Figure 2). A comparison of field estimates with the results of calculations performed in this paper is shown in Section 4.2. Calculations are performed using the temperature and precipitation time-series data representing a period of active soil-water balance (i.e., with no freezing) from March through October for the years 1990–2007. A set of meteorological parameters is summarized in Table 1, which are then used to develop the input PDFs and fuzzy numbers shown in Figure 3.

Several modeling scenarios were developed (Table 2) to assess how the application of different models for input parameters affects the uncertainty of *E*o and *ET* calculations. For the sake of simulation simplicity, the input parameters are assumed to be independent variables. Scenarios 0 to 8, described in detail in Faybishenko (2010), are based on the application of a single Penman model for *E*o calculations, with annual average values of input parameters. Scenario 0 was modeled using input PDFs by means of Monte Carlo simulations, using RiskAMP Monte Carlo Add-In Library version 2.10 for Excel. Scenarios 1 through 8 were simulated by means of the RAMAS Risk Calc code. Scenario 1 was simulated using input PDFs, and the results are given as p-box numbers. Scenarios 2 through 6 were simulated applying both PDFs and fuzzy number inputs, corresponding to -cuts from 0 to 1). Scenarios 7 and 8 were simulated using only fuzzy numbers. The calculation results of Scenarios 0 through 8 are compared in this chapter with newly calculated Scenarios 9 and 10, which are based on Monte Carlo calculations by means of all *E*o models, described in Section 2, and then bounding the resulting PDFs by a trapezoidal fuzzy number (Scenario 9) and the p-box (Scenario 10).


Table 1. Meteorological parameters from the Hanford Meteorological Station used for *E*<sup>o</sup> calculations for all scenarios (the data sources are given in the text).

Meteorological parameters used to assign model input parameters were taken from the Hanford Meteorological Station (HMS—see http://hms.pnl.gov/), located at the center of the Hanford Site just outside the northeast corner of the 200 West Area, as well as from publications (DOE, 1996; Hoitink et al., 2002; Neitzel, 1996.) At the Hanford Site, the *E*o is estimated to be from 1,400 to 1,611 mm/yr (Ward et al. 2005), and the *ET* is estimated to be 160 mm/yr (Figure 2). A comparison of field estimates with the results of calculations performed in this paper is shown in Section 4.2. Calculations are performed using the temperature and precipitation time-series data representing a period of active soil-water balance (i.e., with no freezing) from March through October for the years 1990–2007. A set of meteorological parameters is summarized in Table 1, which are then used to develop the

Several modeling scenarios were developed (Table 2) to assess how the application of different models for input parameters affects the uncertainty of *E*o and *ET* calculations. For the sake of simulation simplicity, the input parameters are assumed to be independent variables. Scenarios 0 to 8, described in detail in Faybishenko (2010), are based on the application of a single Penman model for *E*o calculations, with annual average values of input parameters. Scenario 0 was modeled using input PDFs by means of Monte Carlo simulations, using RiskAMP Monte Carlo Add-In Library version 2.10 for Excel. Scenarios 1 through 8 were simulated by means of the RAMAS Risk Calc code. Scenario 1 was simulated using input PDFs, and the results are given as p-box numbers. Scenarios 2 through 6 were simulated applying both PDFs and fuzzy number inputs, corresponding to -cuts from 0 to 1). Scenarios 7 and 8 were simulated using only fuzzy numbers. The calculation results of Scenarios 0 through 8 are compared in this chapter with newly calculated Scenarios 9 and 10, which are based on Monte Carlo calculations by means of all *E*o models, described in Section 2, and then bounding the resulting PDFs by a trapezoidal

input PDFs and fuzzy numbers shown in Figure 3.

fuzzy number (Scenario 9) and the p-box (Scenario 10).

**speed (km/hr)**

**Relative humidity (%)** 

**PDFs Mean** 15.07 80.2 33.3 0.21 332.55 185 33.41 2.87

Table 1. Meteorological parameters from the Hanford Meteorological Station used for *E*<sup>o</sup>

calculations for all scenarios (the data sources are given in the text).

**= 0 Min** 12.31 68.17 28.29 0.15 282.66 46.0 30.17 0.0

**=1 Min** 14.61 78.2 32.47 0.22 324.24 157.2 32.87 2.32

**Max** 17.84 92.23 38.31 0.27 382.44 324.1 36.65 6.17

**Max** 15.53 82.2 34.14 0.27 382.44 212.8 33.95 3.42

**Albedo Solar** 

0.92 4.01 1.66 0.021 16.63 55.62 1.08 1.11

**radiation (Ly/day)**

**Max Min Max Min** 

**Annual precipitation (mm/yr)**

**Temperature (oC)** 

**Parameters Wind** 

**Standard Deviation** 

**Type of data** 

**Trapezoidal FMFs** 

Fig. 2. Estimated water balance *ET* and recharge/infiltration at the Hanford site (Gee et al, 2007).


Notes:

1) In Scenario 7, all FMFs are trapezoidal.

2) In Scenario 8, all FMFs are triangular: the mean values of parameters, which are given in Table 1, are used for =1; and the minimum and maximum values of parameters, given in Table 1 for trapezoidal FMFs (Scenario 7), are also used for =0 of triangular FMFs in Scenario 8. 3) In Scenarios 9 and 10, input parameters are monthly averaged.

Table 2. Scenarios of input and output parameters used for water-balance calculations (Scenarios 0, and 1-8 are from Faybishenko, 2010).

#### **4.2 Results and comparison with field data 4.2.1 Potential evapotranspiration (***Eo***)**

Figure 4a shows cumulative distributions of *E*o from different models, along with an aggregated p-box, and Figure 4b shows the corresponding FMFs (calculated as normalized PDFs) of *E*o from different models, along with an aggregated trapezoidal fuzzy *E*o. These figures illustrate that the Baier-Robertson (Eq. 1), Blaney-Criddle (Eq. 3), Hargreaves (Eq. 6), Penman (Eq. 10), Penman-Monteith (Eq. 11) (for tall plants), and Priestly-Taylor (Eq. 12) models provide the best match with field data, while the Makkink (Eq. 9) and Thornthwaite (Eq. 13) models significantly underestimate the *E*o, and the Linacre (Eq. 8) and Baier-Robertson (Eq. 2) models greatly overestimate *E*o.

Fig. 3. Input PDFs (solid lines) and fuzzy numbers (dashed lines) used for calculations (Faybishenko, 2010).

Figure 4a shows cumulative distributions of *E*o from different models, along with an aggregated p-box, and Figure 4b shows the corresponding FMFs (calculated as normalized PDFs) of *E*o from different models, along with an aggregated trapezoidal fuzzy *E*o. These figures illustrate that the Baier-Robertson (Eq. 1), Blaney-Criddle (Eq. 3), Hargreaves (Eq. 6), Penman (Eq. 10), Penman-Monteith (Eq. 11) (for tall plants), and Priestly-Taylor (Eq. 12) models provide the best match with field data, while the Makkink (Eq. 9) and Thornthwaite (Eq. 13) models significantly underestimate the *E*o, and the Linacre (Eq. 8) and Baier-

0

0

Fig. 3. Input PDFs (solid lines) and fuzzy numbers (dashed lines) used for calculations

0.2

0.4

0.6

Probability/FMF

0.8

1

0

0.2

0.4

0.6

Probability/FMF

0.8

1

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> Temperature (oC)

0.1 0.15 0.2 0.25 0.3 Albedo

10 12 14 16 18 20 Wind (km/d)

min max

0.2

0.4

0.6

Probability/FMF

0.8

1

**4.2 Results and comparison with field data 4.2.1 Potential evapotranspiration (***Eo***)** 

Robertson (Eq. 2) models greatly overestimate *E*o.

0 100 200 300 400 Precipitatin (mm/day)

20 40 60 80 100 Humidity (%)

min max

250 300 350 400 Solar radiation (Ly/d)

0

0

(Faybishenko, 2010).

0.2

0.4

0.6

Probability/FMF

0.8

1

0

0.2

0.4

0.6

Probability/FMF

0.8

1

0.2

0.4

0.6

Probability/FMF

0.8

1

Figure 5a demonstrates that the *E*o mean from Monte Carlo simulations is within the mean ranges from the p-box (Scenario 1) and fuzzy-probabilistic scenarios (Scenarios 2-6). It also corresponds to a midcore of the fuzzy scenario with trapezoidal FMFs (Scenario 7), the core of the fuzzy scenario with triangular FMFs (Scenario 8), and the centroid values of the fuzzy *E*o of Scenario 9, as well as a p-box of Scenario 10.

Fig. 4. (a) Cumulative probability of potential evapotranspiration calculated using different *E*o formulae; an aggregated p-box, which is shown by a black line with solid squares: normal distribution with the left/minimum curve—mean=933, var=1070, and the right /max curve—mean=1763, var=35755; and (b) corresponding fuzzy numbers (calculated from normalized PDFs); an aggregated trapezoidal fuzzy number is shown by a black line—Eq. (17) with *a*=772, *b*=933, *c*=1763, and *d*=2222. (all numbers of *E*o are in mm/yr)

The range of means from the p-box and fuzzy-probabilistic calculations for =1 is practically the same, indicating that including fuzziness within the input parameters does not change the range of most possible *E*o values. Figure 5a shows that the core uncertainty of the trapezoidal FMFs (Scenario 7) is the same as the uncertainty of means for fuzzy-probabilistic calculations for =1. Obviously, the output uncertainty decreases for the input triangular FMFs (Scenario 8), because these FMFs resemble more tightly the PDFs used in other scenarios. Figure 5a also illustrates that a relatively narrow range of field estimates of *E*o from 1,400 to 1,611 mm/yr for the Hanford site (Ward 2005)—is well within the calculated uncertainty of *E*o values. Note from Figure 5a that the uncertainty ranges from p-box, hybrid, and fuzzy calculations significantly exceed those from Monte Carlo simulations for a single Penman model, but are practically the same as those from calculations using multiple *E*o models.

Characteristic parameters (Figures 5a) and the breadth of uncertainty (Figure 6a) of *E*<sup>o</sup> calculated from multiple models—Scenarios 9 and 10—are in a good agreement with field measurements and other calculation scenarios.

#### **4.2.2 Evapotranspiration (***ET***)**

Figure 5b shows that the mean *ET* of ~184 mm/yr from Monte Carlo simulations (Scenario 0) is practically the same as the *ET* means for Scenarios 1 through 5 and the core value for Scenario 8. The greater *ET* uncertainty for Scenario 6 (precipitation is simulated using a fuzzy number) can be explained by the relatively large precipitation range for =0—from 46 to 324 mm/yr. At the same time, the means of *ET* values for =1 range within relatively narrow limits, as the precipitation for =1 changes from 157.2 to 212.8 mm/yr (see Table 1).

The breadth of uncertainty of *ET* (Figure 6b) is practically the same for Scenarios 1 through 5, increase for Scenarios 6, 7, and 8 in the account of calculations using a fuzzy precipitation, and then decrease for Scenarios 9 and 10 using multiple *E*o models. A smaller range of *ET* uncertainty calculated using multiple *E*o models can be explained by the fact that the Budyko curve asymptotically reaches the limit of *ET*/*P*=1 for high values of the aridity index, which are typical for the semi-arid climatic conditions of the Hanford site.

Fig. 5. Results of calculations of *E*o (a) and *ET* (b) and comparison with field measurements. Red vertical lines are the mean intervals (Scenarios 1-6, and 10) and core intervals (Scenarios 7, 8, and 9), the blue diamonds indicate the interquartile ranges with endpoints at the 25th and 75th percentiles of the underlying distribution. Red open diamonds for Scenarios 2-6 indicate the mean intervals for the hybrid level=10 (Faybishenko 2010), and red solid diamonds for Scenarios 7-10 indicate centroid values. The height of a shaded area in figure *a* indicates the range of *E*o from field measurements. (Results of calculations of Scenarios 0-8 are from Faybishenko, 2010.)

(a)

within relatively narrow limits, as the precipitation for =1 changes from 157.2 to 212.8

The breadth of uncertainty of *ET* (Figure 6b) is practically the same for Scenarios 1 through 5, increase for Scenarios 6, 7, and 8 in the account of calculations using a fuzzy precipitation, and then decrease for Scenarios 9 and 10 using multiple *E*o models. A smaller range of *ET* uncertainty calculated using multiple *E*o models can be explained by the fact that the Budyko curve asymptotically reaches the limit of *ET*/*P*=1 for high values of the aridity

index, which are typical for the semi-arid climatic conditions of the Hanford site.

1543

184.5 180.1

185.2 179.4

184.6 180

184.6 179.8

Fig. 5. Results of calculations of *E*o (a) and *ET* (b) and comparison with field measurements. Red vertical lines are the mean intervals (Scenarios 1-6, and 10) and core intervals (Scenarios 7, 8, and 9), the blue diamonds indicate the interquartile ranges with endpoints at the 25th and 75th percentiles of the underlying distribution. Red open diamonds for Scenarios 2-6 indicate the mean intervals for the hybrid level=10 (Faybishenko 2010), and red solid diamonds for Scenarios 7-10 indicate centroid values. The height of a shaded area in figure *a* indicates the range of *E*o from field measurements. (Results of calculations of Scenarios 0-8

184.6 179.4

0 1 2 3 4 5 6 7 8 9 10 Scenario

1241 1235

1549

1229

MC p-box Fuzzy-probabilistic Fuzzy

1215

0 1 2 3 4 5 6 7 8 9 10 Scenario

1215

322.4

43.1

156.1

184

211.7

1576

Penman model Multiple

<sup>1458</sup> <sup>1447</sup> <sup>1423</sup> <sup>1369</sup>

Fuzzy-

Prob p-box

<sup>163</sup> <sup>163</sup> 163.2

163.2

Field

Field

models

1576

1557

mm/yr (see Table 1).

400

184

40

are from Faybishenko, 2010.)

90

140

190

*ET* (mm/yr)

240

290

340

800

1200

*E*

(b)

o (mm.yr)

1600

2000

2400

(a)

The calculated means for Scenarios 0, 1–5, and 8 exceed the field estimates of *ET* of 160 mm/yr (Gee et al., 1992; 2007) by 22 to 24 mm/yr. This difference can be explained by Gee et al. using a lower value of annual precipitation (160 mm/yr for the period prior to 1990) in their calculations, while our calculations are based on using a greater mean annual precipitation (185 mm/yr), averaged for the years from 1990 to 2007. The field-based data are within the *ET* uncertainty range for Scenarios 6 and 7, since the precipitation range is wider for these scenarios. Calculations using multiple *E*o models generated the *ET* values (Scenarios 9 and 10), which are practically the same as those from field measurements.

Fig. 6. Breadth of uncertainty of *E*o and *ET*. For Scenarios 2-6, grey and white bars indicate the maximum and minimum uncertainty, correspondingly. (Results of calculations of Scenarios 0-8 are from Faybishenko, 2010.)

## **5. Conclusions**

The objectives of this chapter are to illustrate the application of a fuzzy-probabilistic approach for predictions of *E*o and *ET*, and to compare the results of calculations with those from field measurements at the Hanford site. Using historical monthly averaged data from the Hanford Meteorological Station, this author employed Monte-Carlo simulations to assess the frequency distribution and statistics of input parameters for these models, which are then used as input into probabilistic simulations. The effect of aleatory uncertainty on calculations of evapotranspiration is assessed by assigning the probability distributions of input meteorological parameters, and the combined effect of aleatory and epistemic (model) uncertainty is then expressed by means of aggregating the results of calculations using a pbox and fuzzy numbers. To illustrate the application of these approaches, the potential evapotranspiration is calculated using the Bair-Robertson, Blaney-Criddle, Caprio, Hargreaves-Samani, Hamon, Jensen-Haise, Linacre, Makkink, Priestly-Taylor, Penman, Penman-Monteith, Thornthwaite, and Turc models, and evapotranspiration is then determined based on the modified Budyko (1974) model. Probabilistic and fuzzyprobabilistic calculations using multiple *E*o models generate the *E*o and *ET* results, which are well within the range of field measurements and the application of a single Penman model. The Baier-Robertson, Blaney-Criddle, Hargreaves, Penman, Penman-Monteith, and Priestly-Taylor models provide the best match with field data.

## **6. Acknowledgment**

This work was partially supported by the Director, Office of Science, Office of Biological and Environmental Remediation Sciences of the U.S. Department of Energy, and the DOE EM-32 Office of Soil and Groundwater Remediation (ASCEM project) under Contract No. DE-AC02-05CH11231 to Lawrence Berkeley National Laboratory.

## **7. References**


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compatibility, partial aggregation, and reinforcement. *International Journal of* 

Water Discharge by Evapotranspiration for the BARCAS Study Area, DHS

In humid regions such as west-central Florida, evapotranspiration (ET) is estimated to be 70% of precipitation on an average annual basis (Bidlake et al. 1993; Knowles 1996; Sumner 2001). ET is traditionally inferred from values of potential ET (PET) or reference ET (Doorenabos and Pruitt 1977). PET data are more readily available and can be computed from either pan evaporation or from energy budget methods (Penman 1948; Thornthwaite 1948; Monteith 1965; Priestly and Taylor 1972, etc.). The above methodology though simple, suffer from the fact that meteorological data collected in the field for PET are mostly under non-potential conditions, rendering ET estimates as erroneous (Brutsaert 1982; Sumner 2006). Lysimeters can be used to determine ET from mass balance, however, for shallow water table environments, they are found to give erroneous readings due to air entrapment (Fayer and Hillel 1986), as well as fluctuating water table (Yang et al. 2000). Remote sensing techniques such as, satellite-derived feedback model and Surface Energy Balance Algorithm (SEBAL) as reviewed by Kite and Droogers (2000) and remotely sensed Normalized Difference Vegetation Index (NDVI) as used by Mo et al. (2004) are especially useful for large scale studies. However, in the case of highly heterogeneous landscapes , the resolution of ET may become problematic owing to the coarse resolution of the data (Nachabe et al. 2005). The energy budget or eddy correlation methodologies are also limited to computing net ET and cannot resolve ET contribution from different sources. For shallow water table environments, continuous soil moisture measurements and water table estimation have been found to accurately determine ET (Nachabe et al. 2005; Fares and Alva 2000). Past studies, e.g., Robock et al. (2000), Mahmood and Hubbard (2003), and Nachabe et al. (2005), have clearly shown that soil moisture monitoring can be successfully used to determine ET from a hydrologic balance. The approach used herein involves use of soil moisture and water table data measurements. Using point measurement of soil moisture and water table observations from an individual monitoring well ET values can be accurately determined. Additionally, if similar measurements of soil moisture content and water table are available from a set of wells along a flow transect , other components of water budgets and attempts to comprehensively resolve other components of the water budget at the study site.

The following section describes a particular configuration of the instruments, development of a methodology, and an example case study where the authors have successfully applied measurement of soil moisture and water table in the past to estimate and model ET at the study site. The authors also used the soil moisture dataset to compute actual root water uptake for two different land-covers (grassed and forested). The new methodology of estimating ET is based on an eco-hydrological framework that includes plant physiological characteristics. The new methodology is shown to provide a much better representation of the ET process with varying antecedent conditions for a given land-cover as compared to traditional hydrological models.

## **2. Study site**

The study site for gathering field data and using it for ET estimation and vadose zone process modeling was located in the sub basin of Long Flat Creek, a tributary of the Alafia River, adjacent to the Tampa bay regional reservoir, in Lithia, Florida. **Figure 1** shows the regional and aerial view of the site location. Two sets of monitoring well transects were installed on the west side of Long Flat Creek. One set of wells designated as PS-39, PS-40, PS-41, PS-42, and PS-43 ran from east to west while the other set consisting of two wells was roughly parallel to the stream (Long Flat Creek), running in the North South direction. The wells were designated as USF-1 and USF-3.

Fig. 1. Location of the study site in Hillsborough County, Florida

The topography of the area slopes towards the stream with PS-43 being located at roughly the highest point for both transects. The vegetation varied from un-grazed Bahia pasture grass in the upland areas (in proximity of PS-43, USF-1, and USF-3), to alluvial wetland forest comprised of slash pine and hardwood trees near the stream. The area close to PS-42 is characterized as a mixed (grassed and forested) zone. Horizontal distance between the wells is approximately 16, 22, 96, 153 m from PS-39 to PS-43, with PS-39 being approximately 6 m from the creek. The horizontal distance between USF-1 and USF-3 was 33 m. All wells were surveyed and land surface elevations were determined with respect to National Geodetic Vertical Datum 1929 (NGVD).

The data captured from this configuration was used both for point estimation as well as transect modeling, however , for this particular chapter, only point estimation of ET and and point data set will be used to develop conceptualizations of vadose zone processes will be discussed. For details regarding transect modeling to generate water budget estimates refer to Shah (2007).

## **3. Instrumentation**

98 Evapotranspiration – Remote Sensing and Modeling

measurement of soil moisture and water table in the past to estimate and model ET at the study site. The authors also used the soil moisture dataset to compute actual root water uptake for two different land-covers (grassed and forested). The new methodology of estimating ET is based on an eco-hydrological framework that includes plant physiological characteristics. The new methodology is shown to provide a much better representation of the ET process with varying antecedent conditions for a given land-cover as compared to

The study site for gathering field data and using it for ET estimation and vadose zone process modeling was located in the sub basin of Long Flat Creek, a tributary of the Alafia River, adjacent to the Tampa bay regional reservoir, in Lithia, Florida. **Figure 1** shows the regional and aerial view of the site location. Two sets of monitoring well transects were installed on the west side of Long Flat Creek. One set of wells designated as PS-39, PS-40, PS-41, PS-42, and PS-43 ran from east to west while the other set consisting of two wells was roughly parallel to the stream (Long Flat Creek), running in the North South direction. The

traditional hydrological models.

wells were designated as USF-1 and USF-3.

Fig. 1. Location of the study site in Hillsborough County, Florida

National Geodetic Vertical Datum 1929 (NGVD).

The topography of the area slopes towards the stream with PS-43 being located at roughly the highest point for both transects. The vegetation varied from un-grazed Bahia pasture grass in the upland areas (in proximity of PS-43, USF-1, and USF-3), to alluvial wetland forest comprised of slash pine and hardwood trees near the stream. The area close to PS-42 is characterized as a mixed (grassed and forested) zone. Horizontal distance between the wells is approximately 16, 22, 96, 153 m from PS-39 to PS-43, with PS-39 being approximately 6 m from the creek. The horizontal distance between USF-1 and USF-3 was 33 m. All wells were surveyed and land surface elevations were determined with respect to

**2. Study site** 

For measurement of water table at a particular location a monitoring well instrumented with submersible pressure transducer (manufactured by Instrumentation Northwest, Kirkland, WA) 0-34 kPa (0-5 psi), accurate to 0.034 kPa (0.005 psi) was installed. Adjacent to each well, an EnviroSMART® soil moisture probe (Sentek Pty. Ltd., Adelaide, Australia) carrying eight sensors was installed (see **Figure 2**). The soil moisture sensors allowed measurement of volumetric moisture content along a vertical profile at different depths from land surface. The sensors were deployed at 10, 20, 30, 50, 70, 90, 110, 150 cm from the land surface. The sensors work on the principle of frequency domain reflectometery (FDR) to convert electrical capacitance shift to volumetric water content ranging from oven dryness to saturation with a resolution of 0.1% (Buss 1993). Default factory calibration equations were used for calibrating these sensors. Fares and Alva (2000) and Morgan et al. (1999) found no significant difference in the values of observed recorded water content from the sensors when compared with the manually measured values. Two tipping bucket and two manual rain gages were also installed to record the amount of precipitation.

Fig. 2. Soil moisture probe on the left showing the mounted sensors along with schematics on the right showing sample stratiagraphy at different depths.

#### **4. Point estimation of evapotranspiration using soil moisture data**

At any given well location variation in total soil moisture on non-rainy days can be due to (a) subsurface flow from or to the one dimensional soil column (0 – 155 cm below land surface) over which soil moisture is measured, and (b) evapotranspiration from this soil column. Mathematically

$$\frac{\partial \text{TSM}}{\partial t} = Q - ET \tag{1}$$

where t is time [T], *Q* is subsurface flow rate [LT-1], and *ET* is evapotranspiration rate [LT-1]. *TSM* is total soil moisture, determined as below

$$TSM = \bigcap\_{\xi} \theta \, dz \tag{2}$$

 where *θ*[L3L-3] is the measured water content, *z* [L] is the depth below land surface *ζ*[L] is the depth of monitored soil column (155 cm). The values in the square brackets (for all the variables) represent the dimensions (instead of units) e.g. L is length, T is time.

The negative sign in front of ET in **Equation 1** indicates that ET depletes the TSM in the column. The subsurface flow rate can be either positive or negative. In a groundwater discharge area, the subsurface flow rate, *Q*, is positive because it acts to replenish the TSM in the soil column (Freeze and Cherry, 1979). Thus, this flow rate is negative in a groundwater recharge area. **Figure 3** illustrates the role of subsurface flow in replenishing or depleting total soil moisture in the column. An inherent assumption in this approach is that the deepest sensor is below the water table which allows accounting for all the soil moisture in the vadose zone. Hence, monitoring of water table is critical to make sure that the water table is shallower than the bottom most sensor. To estimate both ET and Q in **Equation 1**, it was important to decouple these fluxes. In this model the subsurface flow rate was estimated from the diurnal fluctuation in TSM. Assuming ET is effectively zero between midnight and 0400 h, Q can be easily calculated from **Equation 3** using:

$$Q = \frac{\text{TSM}\_{0400h} - \text{TSM}\_{midnight}}{4} \tag{3}$$

where TSM0400h and TSMmidnight are total soil moisture measured at 0400 h and midnight, respectively. The denominator in **Equation 3** is 4 h, corresponding to the time difference between the two TSM measurements. The assumption of negligible ET between midnight and 0400h is not new, but was adopted in the early works of White (1932) and Meyboom (1967) in analyzing diurnal water table fluctuation. It is a reasonable assumption to make at night when sunlight is absent.

Taking Q as constant for a 24h period (White 1932; Meyboom, 1967), the ET consumption in any single day was calculated from the following equation

$$ET = TSM\_j - TSM\_{j+1} + 24 \times Q \tag{4}$$

where TSMj is the total soil moisture at midnight on day *j*, and TSM j+1 is the total soil moisture 24h later (midnight the following day). Q is multiplied by 24 as the **Equation 4**

At any given well location variation in total soil moisture on non-rainy days can be due to (a) subsurface flow from or to the one dimensional soil column (0 – 155 cm below land surface) over which soil moisture is measured, and (b) evapotranspiration from this soil

*TSM Q ET <sup>t</sup>*

where t is time [T], *Q* is subsurface flow rate [LT-1], and *ET* is evapotranspiration rate [LT-1].

*TSM dz* 

 where *θ*[L3L-3] is the measured water content, *z* [L] is the depth below land surface *ζ*[L] is the depth of monitored soil column (155 cm). The values in the square brackets (for all the

The negative sign in front of ET in **Equation 1** indicates that ET depletes the TSM in the column. The subsurface flow rate can be either positive or negative. In a groundwater discharge area, the subsurface flow rate, *Q*, is positive because it acts to replenish the TSM in the soil column (Freeze and Cherry, 1979). Thus, this flow rate is negative in a groundwater recharge area. **Figure 3** illustrates the role of subsurface flow in replenishing or depleting total soil moisture in the column. An inherent assumption in this approach is that the deepest sensor is below the water table which allows accounting for all the soil moisture in the vadose zone. Hence, monitoring of water table is critical to make sure that the water table is shallower than the bottom most sensor. To estimate both ET and Q in **Equation 1**, it was important to decouple these fluxes. In this model the subsurface flow rate was estimated from the diurnal fluctuation in TSM. Assuming ET is effectively zero between

variables) represent the dimensions (instead of units) e.g. L is length, T is time.

midnight and 0400 h, Q can be easily calculated from **Equation 3** using:

any single day was calculated from the following equation

0400

4

where TSM0400h and TSMmidnight are total soil moisture measured at 0400 h and midnight, respectively. The denominator in **Equation 3** is 4 h, corresponding to the time difference between the two TSM measurements. The assumption of negligible ET between midnight and 0400h is not new, but was adopted in the early works of White (1932) and Meyboom (1967) in analyzing diurnal water table fluctuation. It is a reasonable assumption to make at

Taking Q as constant for a 24h period (White 1932; Meyboom, 1967), the ET consumption in

where TSMj is the total soil moisture at midnight on day *j*, and TSM j+1 is the total soil moisture 24h later (midnight the following day). Q is multiplied by 24 as the **Equation 4**

*TSM TSM h midnight <sup>Q</sup>* (3)

<sup>1</sup> 24 *ET TSM TSM Q j j* (4)

 

(1)

(2)

**4. Point estimation of evapotranspiration using soil moisture data** 

column. Mathematically

night when sunlight is absent.

*TSM* is total soil moisture, determined as below

provides daily ET values. **Figure 4** show a sample observations for 5 day period showing the evolution of TSM in a groundwater discharge and recharge area respectively. Also marked on the graphs are different quantities calculated to determine *ET* from the observations.

Fig. 3. Total soil moisture is estimated in two soil columns. The first is in a groundwater recharge area (pasture), and the second is in a groundwater discharge area (forested). In a groundwater discharge area, subsurface flow acts to replenish the total soil.

**Equation 1** applies for dry periods only, because it does not account for the contribution of interception storage to ET on rainy days. Also, the changes in soil moisture on rainy days can occur due to other processes like infiltration, upstream runoff infiltration (as will be discussed later) etc. The results obtained from the above model were averaged based on the land cover of each well and are presented as ET values for grass or forested land cover. The values for the grassed land cover were also compared against ET values derived from pan evaporation measurements.

The ET estimates from the data collected at the study site using the above methodology are shown in **Figure 5**. **Figure 5** shows variability in the values of *ET* for a period of about a year and half. It can be seen from **Figure 5** that the method was successful in capturing spatial variability in the ET rates based on the changes in the land cover, as the ET rate of forested (alluvial wetland forest) land cover was found to be always higher than that of the grassland (in this case un-grazed Bahia grass). In addition to spatial variability, the method seemed to capture well the temporal variability in ET. The temporal variability for this particular analysis existed at two time scales, a short-term daily variation associated with daily changes in atmospheric conditions (e.g. local cloud cover, wind speed etc.) and a long-term, seasonal, climatic variation. The short-term variation tends to be less systematic and is demonstrated in **Figure 5** by the range marks. The seasonal variation is more systematic and pronounced and is clearly captured by the method.

Fig. 4. Total soil moisture versus time in the (a) groundwater discharge area and (b) ground water recharge area. The subsurface flux is the positive slope of the line between midnight and 4 AM.

Fig. 4. Total soil moisture versus time in the (a) groundwater discharge area and (b) ground water recharge area. The subsurface flux is the positive slope of the line between midnight

and 4 AM.

A

B

Fig. 5. Monthly average of evapotranspiration (ET) daily values in forested (diamonds) and pasture (triangles) areas. The gap in the graph represents a period of missing data. Standard deviations of daily values are also shown in the range limits.

To assess the reasonableness of the methodology, the estimated ET values for pasture were compared with ET estimated from the evaporation pan. The measured pan evaporation was multiplied by a pan coefficient for pasture to estimate ET for this vegetation cover. A monthly variable crop coefficient was adopted (Doorenbos and Pruitt, 1977) to account for changes associated with seasonal plant phenology (see **Table 1**). The consumptive water use or the crop evapotranspiration is calculated as:

$$\rm{ET}\_{\rm C} = \rm{E}\_{\rm P} \times \rm{K}\_{\rm C} \tag{5}$$

where *EP* is the measured pan evaporation, *KC* is a pan coefficient for pastureland, and *ETC* is the estimated evapotranspiration [LT-1] (mm/d) by the pan evaporation method. **Figure 6** compares the ET estimated by both the evaporation pan and moisture sensors for pasture. Although the two methods are fundamentally different, on average, estimated *ET* agreed well with an R2 coefficient of 0.78. This supported the validity of the soil moisture methodology, which further captured the daily variability of ET ranging from a low of 0.3 mm/d to a maximum of 4.9 mm/d. The differences between the two methods can be attributed to fundamental discrepancies. The pan results are based on atmospheric potential with crude average monthly coefficients while the TSM approach inherently incorporates plant physiology and actual moisture limitations. Indeed, both methods suffer from limitations. The pan coefficient is generic and does not account for regional variation in vegetation phenology or other local influences such as soil texture and fertility. Similarly, the accuracy of the soil moisture method proposed in this study depends on the number of sensors used in monitoring total moisture in the soil column.


Table 1. Pan coefficients used to obtain pasture evapotranspiration for different months.

Fig. 6. Evapotranspiration estimates for pasture by the pan and point scale model. Data points represent the daily values of ET from both techniques.

## **5. Development of root water uptake model**

The preceding sections described a novel data collection approach that can be used to measure ET (and other water budget components). The measured values can be subsequently used to develop modeling parameters or validate modeling results for areas which are similar to the study site in terms of climatic and land-cover conditions. The next step is the development of a generic modeling framework to accurately determine ET.

Month Coefficient

Table 1. Pan coefficients used to obtain pasture evapotranspiration for different months.

Fig. 6. Evapotranspiration estimates for pasture by the pan and point scale model. Data

The preceding sections described a novel data collection approach that can be used to measure ET (and other water budget components). The measured values can be subsequently used to develop modeling parameters or validate modeling results for areas which are similar to the study site in terms of climatic and land-cover conditions. The next step is the development of a generic modeling framework to accurately determine ET.

points represent the daily values of ET from both techniques.

**5. Development of root water uptake model** 

January 0.4 February 0.45 March 0.55 April 0.64 May 0.7 June 0.7 July 0.7 August 0.7 September 0.7 October 0.6 November 0.5 December 0.5

Transpiration by its very nature is a process that is primarily based on plant physiology and the better one can determine root water uptake the more accurate will be the estimation of actual transpiration and, therefore ET. Traditionally used models and concepts, however, make over simplifying assumptions about plants (Shah et al. 2007), hence casting doubt on the model results. What needs to be done is to try and combine land cover characteristics in the root water uptake models to produce more reliable results. With this intent in mind, recently, a new branch of study called "Eco-Hydrology" has been initiated. The aim of ecohydrology is to encourage the interdisciplinary work on ecology and hydrology with an objective of improving hydrological modeling capabilities.

Soil moisture datasets (as described in Section 2) can be used to provide insight into the process of root water uptake which can then be combined with plant characteristics to develop a more physically based ET model. The next sections describe how the soil moisture dataset has been used by the authors to estimate vertical distribution of root-water uptake for two land-cover classes (shallow rooted and deep rooted) and how the results were then used to develop a land-cover based modeling framework.

## **5.1 Traditional root water uptake models**

The governing equation for soil moisture dynamics in the unsaturated soil zone is the Richards's equation (Richards 1931). Richards's equation is derived from Darcy's law and the continuity equation. What follows is a brief description of Richards's equation and how can it incorporates root water uptake. For more detailed information about formulation of Richards's equation, including its derivation in three dimensions, the readers are directed to any text book on soil physics e.g. Hillel (1998).

Due to ease of measurement and conceptualization, energy of water (E) is represented in terms of height of liquid column and is called the hydraulic head (h). It is defined as the total energy of water per unit weight. Mathematically hydraulic head, h, can be represented as

$$h = \frac{E}{\rho\_W \mathcal{g}}\tag{6}$$

where ρW is the density of water and g is the acceleration due to gravity. The flow of water always occurs along decreasing head. In soil physics the fundamental equation used to model the flow of water along a head gradient is known as Darcy Law (Hillel 1998). Mathematically the equation can be written as

$$q = K \frac{\Delta h}{l} \tag{7}$$

where *q* [L3L-2T-1] is known as the specific discharge and is defined as the flow per unit cross-sectional area, *K*[LT-1] is termed as the hydraulic conductivity, which indicates ease of flow, *∆h* [L] is the head difference between the points of interest and *l*[L] is the distance between them. Darcy's Law is analogous to Ohm's law with head gradient being analogous to the potential difference and current being analogous to specific discharge and hydraulic conductivity being similar to the conductance of a wire.

The second component of Richards's equation is the continuity equation. Continuity equation is based on the law of mass conservation, and for any given volume it states that net increase in storage in the given volume is inflow minus sum of outflow and any sink present in the volume of soil. Mathematically, it is this sink term that allows the modeling of water extracted from the given volume of soil.

In one dimension for flow occurring in the vertical direction (z axis is positive downwards) Richards's equation can be written as

$$\mathbf{E}\left(\frac{\partial \mathcal{O}}{\partial t}\right) = \left(\frac{\partial}{\partial z}\mathcal{K}\left(\frac{\partial \mathcal{H}}{\partial z} + \mathbf{1}\right)\right) - \mathbf{S} \tag{8}$$

where *θ* is the water content, defined as the ratio of volume of water present and total volume of the soil element , *t* is time, *S* represents the sink term while other terms are as defined before.

If flow in X and Y directions is also considered , Richards's equation in three dimensions can be derived. Solution of Equation 8 can theoretically provide the spatial and temporal variability of moisture in the soil. However, due to high degree of non linearity of the equation no analytical solution exists for Richards's equation and numerical techniques are used to solve it. For a numerical solution of Richards's equation two essential properties that need to be defined a-priori are (a) relationship between soil water content and hydraulic head, also known as, soil moisture retention curves, and (b) a model that relate hydraulic head to root water uptake. Details about the soil moisture retention curves and numerical techniques used to solve Richards's equation can be found in Simunek et al. (2005). While much literature and field data exist describing the soil moisture retention curves, relatively less information exists about root water uptake models. The root water uptake models generally used, especially, on a watershed scale, are mostly empirical and lack any field verification.

The most common approach used to model root water uptake is to define a sink term S as a function of hydraulic head using the following equation

$$S(h) = \alpha(h)S\_p \tag{9}$$

where S(h)[L3L-3T-1] is the actual root water uptake (RWU) from roots subjected to hydraulic or capillary pressure head '*h*'. On the right hand side of the equation *Sp* [L3L-3T-1] is the maximum (also known as potential) uptake of water by the roots. The *α(h)* is a root water uptake stress response function, with its values varying between 0 and 1.

The idea behind conceptualization of **Equation 9** is based on three basic assumptions. The first assumption being , as the soil becomes dryer the amount of water that can be extracted will decrease proportionally. Secondly, the amount of water extracted by the roots is affected by the ambient climatic conditions. Drier and hotter conditions result in more water loss from surface of leaves, hence, initiating more water extraction from the soil. The third and final assumption is that the uptake of water from a particular section of a root is directly proportional to the amount of roots present in that section.

The root water stress response function () is a result of the first assumption. Two models commonly used to define are the Feddes model (Feddes et al. 1978) and the van Genuchten model (van Genuchten 1987). **Figure 7** (a and b, respectively) show the variation of with decreasing hydraulic head which is same as decreasing water content or increasing soil dryness. Both models for α are empirical and do not involve any plant physiology to define the thresholds for the water stress response function. An interesting

net increase in storage in the given volume is inflow minus sum of outflow and any sink present in the volume of soil. Mathematically, it is this sink term that allows the modeling of

In one dimension for flow occurring in the vertical direction (z axis is positive downwards)

 

where *θ* is the water content, defined as the ratio of volume of water present and total volume of the soil element , *t* is time, *S* represents the sink term while other terms are as

If flow in X and Y directions is also considered , Richards's equation in three dimensions can be derived. Solution of Equation 8 can theoretically provide the spatial and temporal variability of moisture in the soil. However, due to high degree of non linearity of the equation no analytical solution exists for Richards's equation and numerical techniques are used to solve it. For a numerical solution of Richards's equation two essential properties that need to be defined a-priori are (a) relationship between soil water content and hydraulic head, also known as, soil moisture retention curves, and (b) a model that relate hydraulic head to root water uptake. Details about the soil moisture retention curves and numerical techniques used to solve Richards's equation can be found in Simunek et al. (2005). While much literature and field data exist describing the soil moisture retention curves, relatively less information exists about root water uptake models. The root water uptake models generally used, especially, on a watershed scale, are mostly empirical and lack any field

The most common approach used to model root water uptake is to define a sink term S as a

() () *Sh hS* 

where S(h)[L3L-3T-1] is the actual root water uptake (RWU) from roots subjected to hydraulic or capillary pressure head '*h*'. On the right hand side of the equation *Sp* [L3L-3T-1] is the maximum (also known as potential) uptake of water by the roots. The *α(h)* is a root water

The idea behind conceptualization of **Equation 9** is based on three basic assumptions. The first assumption being , as the soil becomes dryer the amount of water that can be extracted will decrease proportionally. Secondly, the amount of water extracted by the roots is affected by the ambient climatic conditions. Drier and hotter conditions result in more water loss from surface of leaves, hence, initiating more water extraction from the soil. The third and final assumption is that the uptake of water from a particular section of a root is directly

The root water stress response function () is a result of the first assumption. Two models commonly used to define are the Feddes model (Feddes et al. 1978) and the van Genuchten model (van Genuchten 1987). **Figure 7** (a and b, respectively) show the

or increasing soil dryness. Both models for α are empirical and do not involve any plant physiology to define the thresholds for the water stress response function. An interesting

with decreasing hydraulic head which is same as decreasing water content

uptake stress response function, with its values varying between 0 and 1.

*t zz* 

<sup>1</sup> *<sup>h</sup> K S*

(8)

*<sup>p</sup>* (9)

water extracted from the given volume of soil.

function of hydraulic head using the following equation

proportional to the amount of roots present in that section.

Richards's equation can be written as

defined before.

verification.

variation of

contrast, due to empiricism that is clearly evident is the value of α during saturated conditions. While the Feddes model predict the value of *α* to decrease to zero van Genuchten model predicts totally opposite with α rising to become unity under saturated conditions.

Fig. 7. Water stress response function as given conceptualized by (a) Feddes et al. 1978 and (b) van Genuchten (1980) [Adapted from Simunek et al. 2005].

Recently couple of different models (Li et al. 2001, Li et al. 2006) have been presented to overcome the empiricism in *.* However these models are more a result of observation fitting and fail to bring in the plant physiology, which is what causes the changes in the water uptake rate due variation in soil moisture conditions.

Combining the second and the third assumptions in **Equation 9** results in the definition of *Sp*. *Sp* for any section of roots is defined as the product of root fraction in that section and the maximum possible water loss by the plant which is also known as the potential evapotranspiration. Potential evapotranspiration is a function of ambient atmospheric conditions and standard models like Penman-Monteith (Allen et al. 1998) are used to calculate the potential evapotranspiration rate. The problem with this definition of Sp is that for any given value of potential evapotranspiration, limiting the value of root water uptake by the root-fraction restrict the amount of water that can be extracted from a particular section. In other words, the amount of water extracted by a particular section of root is directly proportional to the amount of roots present and ignores the amount of ambient soil moisture present. This as will discussed later using field data is a significant limitation especially during dry period when the top soil with maximum roots get dry while the deep soil layer with lesser root mass still have soil moisture available for extraction.

#### **5.2 Use of soil moisture data to estimate root water uptake**

For the current analysis, the soil moisture data as described in Section 2 are used. Soil moisture and water-table data from well locations PS-43 and PS-40 were used to determine root water uptake from forested versus grassed land cover. The well PS-43 is referred to as Site A while PS-40 will be called Site B. Hourly averaged data at four hour time step were used for the analysis.

Extensive soil investigations including in-situ and laboratory analysis were performed for the study site. The soil in the study area is primarily sandy marine sediments with high permeability in the surface and subsurface layers. Detailed information about soil and site characteristics can be found in Said et al. (2005), and Trout and Ross (2004). Data for period of record January 2003 to December 2003 were used in this analysis.

van Genuchten (1980) proposed a model relating the water content and hydraulic conductivity with the suction head (soil suction pressure) represented by the following equations

$$S\_{\rm c} = \frac{\theta - \theta\_r}{\theta\_s - \theta\_r} \tag{10}$$

$$\mathbf{h}(\theta) = \frac{\frac{1}{\left(\mathbf{S}\_e^{\mathrm{m}} - 1\right)^{\mathrm{n}}}}{\phi} \tag{11}$$

$$K(h) = \begin{cases} K\_S S\_e^l [1 - (1 - S\_e^{1/m})^m] \ h < 0 \\ K\_S & h \ge 0 \end{cases} \tag{12}$$

where m = 1 – 1/n for n > 1, *Se* [-] is the normalized water content, varying between 0 and 1. *θ* is the observed water content, while *θ<sup>r</sup>* and *θs* are the residual and saturated water content values respectively KS [LT-1] is the hydraulic conductivity when the soil matrix is saturated, *l*[-] is the pore connectivity parameter assumed to be 0.5 as an average for most soils (Mualem, 1976), and [L-1], n[-] and m[-] are the van Genuchten empirical parameters. Negative values of hydraulic head (suction head) indicate the water content in the soil matrix is less than saturated while the positive value indicate saturated conditions. From the **Equations 11** and **12**, it is clear that for each type of soil five parameters, namely, *KS*, *n*,*, θ<sup>r</sup>* and *θs* have to be determined to uniquely define relationship of hydraulic conductivity and water content with soil suction head.

**Figure 8** shows the schematics of the vertical soil column which is monitored using eight soil moisture sensors and a pressure transducer measuring the water table elevation, at each of the two locations. Shown also in **Figure 8** is the zone of influence of each sensor along with the elevation of water table and arrows showing possible flow directions. For the purpose of defining moisture retention and hydraulic conductivity curves, each section is treated as a different soil layer and independently parameterized. Hence, for each of the two locations for this particular study eight soil cores from depths corresponding to the zone of influence of each sensor were taken and analyzed (see Shah, 2007 for more details). **Table 2(a)** and **(b)** shows the parameters values that were obtained following the all the soil tests.

For the current analysis, the soil moisture data as described in Section 2 are used. Soil moisture and water-table data from well locations PS-43 and PS-40 were used to determine root water uptake from forested versus grassed land cover. The well PS-43 is referred to as Site A while PS-40 will be called Site B. Hourly averaged data at four hour time step were

Extensive soil investigations including in-situ and laboratory analysis were performed for the study site. The soil in the study area is primarily sandy marine sediments with high permeability in the surface and subsurface layers. Detailed information about soil and site characteristics can be found in Said et al. (2005), and Trout and Ross (2004). Data for period

van Genuchten (1980) proposed a model relating the water content and hydraulic conductivity with the suction head (soil suction pressure) represented by the following

> *<sup>r</sup> <sup>e</sup> s r*

> > 1 1 m n

(10)

(11)

(12)

*, θ<sup>r</sup>*

 

<sup>e</sup> (S 1) h( ) 

1/ [1 (1 ) 0 ( ) <sup>0</sup> *l m m S e e*

where m = 1 – 1/n for n > 1, *Se* [-] is the normalized water content, varying between 0 and 1. *θ* is the observed water content, while *θ<sup>r</sup>* and *θs* are the residual and saturated water content values respectively KS [LT-1] is the hydraulic conductivity when the soil matrix is saturated, *l*[-] is the pore connectivity parameter assumed to be 0.5 as an average for most soils

Negative values of hydraulic head (suction head) indicate the water content in the soil matrix is less than saturated while the positive value indicate saturated conditions. From the **Equations 11** and **12**, it is clear that for each type of soil five parameters, namely, *KS*, *n*,

and *θs* have to be determined to uniquely define relationship of hydraulic conductivity and

**Figure 8** shows the schematics of the vertical soil column which is monitored using eight soil moisture sensors and a pressure transducer measuring the water table elevation, at each of the two locations. Shown also in **Figure 8** is the zone of influence of each sensor along with the elevation of water table and arrows showing possible flow directions. For the purpose of defining moisture retention and hydraulic conductivity curves, each section is treated as a different soil layer and independently parameterized. Hence, for each of the two locations for this particular study eight soil cores from depths corresponding to the zone of influence of each sensor were taken and analyzed (see Shah, 2007 for more details). **Table 2(a)** and **(b)** shows the parameters values that were obtained

*K h* 

[L-1], n[-] and m[-] are the van Genuchten empirical parameters.

*S* 

*S KS S h K h*

**5.2 Use of soil moisture data to estimate root water uptake** 

of record January 2003 to December 2003 were used in this analysis.

used for the analysis.

(Mualem, 1976), and

water content with soil suction head.

following the all the soil tests.

equations


(b)

Table 2. Soil parameters for study locations in (a) Grassland and (b) Forested area.

Once the soil parameterization is complete root water uptake from each section can be calculated. For any given soil layer in the vertical soil column (**Figure 8**), above the observed water table, observed water content and **Equation 11** can be used to calculate the hydraulic head. For soil layers below the water table hydraulic head is same as the depth of soil layer below the water table due to assumption of hydrostatic pressure. Similarly using **Equation 12** hydraulic conductivity can be calculated. Hence, at any instant in time hydraulic head in each of the eight soil layers can be calculated. To determine total head, gravity head, which is the height of the soil layer above a common datum, has to be added to the hydraulic head.

Fig. 8. Schematics of the vertical soil column with location of the soil moisture sensors and water table.

To quantify flow across each soil layer, Darcy's Law (**Equation 7**) is used. Average head values between two consecutive time steps are used to determine the head difference. Also, flow across different soil layers is assumed to be occurring between the midpoints of one layer to another, hence, to determine the head gradient (*∆h/l*) the distance between the midpoints of each soil layer is used. The last component needed to solve Darcy's Law is the value of hydraulic conductivity. For flow occurring between layers of different hydraulic conductivities equivalent hydraulic conductivity is calculated by taking harmonic means of

below the water table due to assumption of hydrostatic pressure. Similarly using **Equation 12** hydraulic conductivity can be calculated. Hence, at any instant in time hydraulic head in each of the eight soil layers can be calculated. To determine total head, gravity head, which is the height of the soil layer above a common datum, has to be added to the hydraulic head.

> **Sensor @ 10 cm Sensor @ 20 cm Sensor @ 30 cm**

**Sensor @ 50 cm**

**Sensor @ 70 cm**

**Sensor @ 90 cm**

**Sensor @ 110 cm**

W T

water table.

**Sensor @ 150 cm**

Fig. 8. Schematics of the vertical soil column with location of the soil moisture sensors and

To quantify flow across each soil layer, Darcy's Law (**Equation 7**) is used. Average head values between two consecutive time steps are used to determine the head difference. Also, flow across different soil layers is assumed to be occurring between the midpoints of one layer to another, hence, to determine the head gradient (*∆h/l*) the distance between the midpoints of each soil layer is used. The last component needed to solve Darcy's Law is the value of hydraulic conductivity. For flow occurring between layers of different hydraulic conductivities equivalent hydraulic conductivity is calculated by taking harmonic means of the hydraulic conductivities of both the layers (Freeze and Cherry 1979). Hence for each time step harmonically averaged hydraulic conductivity values (**Equation 13**) were used to calculate the flow across soil layers.

$$K\_{eq} = \frac{2K\_1K\_2}{K\_1 + K\_2} \tag{13a}$$

where *K1* [LT-1]and *K2* [LT-1]are the two hydraulic conductivity values for any two adjacent soil layers and *Ke*q [LT-1]is the equivalent hydraulic conductivity for flow occurring between those two layers.

**Figure 9** shows a typical flow layer with inflow and outflow marked. Now using **simple mass balance** changes in water content at two consecutive time steps can be attributed to net inflow minus the root water uptake (assuming no other sink is present). Equation 6.9 can hence be used to determine root water uptake from any given soil layer

$$\text{RVML} = (\theta^t - \theta^{t+1}) - (q\_{out} - q\_{in}) \tag{13b}$$

Using the described methodology one can determine the root water uptake from each soil layer at both study locations (site A and site B).Time step for calculation of the root water uptake was set as four hours and the root water uptake values obtained were summed up to get a daily value for each soil layer.

Fig. 9. Schematics of a section of vertical soil column showing fluxes and change in storage.

Using the above methodology root water uptake was calculated from each section of roots for tree and grass land cover from January to December 2003 at a daily time step. **Figure 10** (a and b) shows the variation of root water uptake for a representative period from May 1st to May 15th 2003, This particular period was selected as the conditions were dry and their was no rainfall. Graphs in **Figure 10** (a and b) show the root water uptake variation from

section corresponding to each section. Also plotted on the graphs is the normalized water content, which also gives an indication, of water lost from the section.

Fig. 10. Root water uptake from sections of soil corresponding to each sensor on the soil moisture instrument for (a, c) Grass land and (b, d) Forest land cover

**Figure 10(a)** shows the root water uptake from grassed site while panel of graphs in **Figure 10(b)** plots RWU from the forested area. From **Figure 10** (a and b) it can be seen that in both the cases of grass and forest the root water uptake varies with water content and as the top layers starts to get dry, the water uptake from the lower layer increases so as to keep the root water uptake constant clearly indicating that the compensation do take place and hence the models need to account for it. Another important point to note is that in **Figure 10(a)** root water uptake from top three sensors is accounts for the almost all the water uptake while in **Figure 10(b)** the contribution from fourth and fifth sensor is also significant. Also, as will be shown later, in case of forested land cover, root water uptake is observed from the sections that are even deeper than 70 cm below land surface. This is expected owing to the differences in the root system of both land cover types. While grasses have shallow roots, forest trees tend to put their roots deeper into the soil to meet their high water consumptive use.

**Figure 10**(**c** and **d**) show the values of PET plotted along with the observed values of root water uptake. On comparing the grass versus forested graphs it is evident while the grass is

section corresponding to each section. Also plotted on the graphs is the normalized water

Fig. 10. Root water uptake from sections of soil corresponding to each sensor on the soil

**Figure 10(a)** shows the root water uptake from grassed site while panel of graphs in **Figure 10(b)** plots RWU from the forested area. From **Figure 10** (a and b) it can be seen that in both the cases of grass and forest the root water uptake varies with water content and as the top layers starts to get dry, the water uptake from the lower layer increases so as to keep the root water uptake constant clearly indicating that the compensation do take place and hence the models need to account for it. Another important point to note is that in **Figure 10(a)** root water uptake from top three sensors is accounts for the almost all the water uptake while in **Figure 10(b)** the contribution from fourth and fifth sensor is also significant. Also, as will be shown later, in case of forested land cover, root water uptake is observed from the sections that are even deeper than 70 cm below land surface. This is expected owing to the differences in the root system of both land cover types. While grasses have shallow roots, forest trees tend to put their roots deeper into the soil to meet

**Figure 10**(**c** and **d**) show the values of PET plotted along with the observed values of root water uptake. On comparing the grass versus forested graphs it is evident while the grass is

moisture instrument for (a, c) Grass land and (b, d) Forest land cover

their high water consumptive use.

content, which also gives an indication, of water lost from the section.

still evapotranspiring at values close to PET root water uptake from forested land covers is occurring at less than potential. This behavior can be explained by the fact that water content in the grassed region (as shown by the normalized water content graph, Se) is greater than that of the forest and even though the 70 cm sensor shows significant contribution the uptake is still not sufficient to meet the potential demand.

**Figure 11** shows an interesting scenario when a rainfall event occurs right after a long dry stretch that caused the upper soil layers to dry out. **Figure 11(a)** shows the root water uptake profile on 5/18/2003 for forested land cover with maximum water being taken from section of soil profile corresponding to 70 cm below the land surface. A rainfall event of 1inch took place on 5/19/2003. As can be clearly seen in **Figure 11(b)** the maximum water uptake shifts right back up to 10 cm below the land surface, clearly showing that the ambient water content directly and quickly affects the root water uptake distribution. **Figure 11(c)** which shows the snapshot on 5/20/2003 a day after the rainfall where the root water uptake starts redistributing and shifting toward deeper wetter layers. In fact this behavior was observed for all the data analyzed for the period of record for both the grass and forested land covers. With roots taking water from deeper wetter layers and as soon as the shallower layer becomes wet the uptakes shifts to the top layers. **Figure 12** (a and b) show a long duration of record spanning 2 months (starting October to end November), with the whiter shade indicating higher root water uptake. From both the figures it is evident that water uptake significantly shifts in lieu of drier soil layers especially in case of forest land cover (**Figure 12(b)**), while in case of grass uptake is primarily concentrated in the top layers.

As a quick summary the results indicate that


Hence, traditionally used models are not adequate, to model this behavior. Changes in regard to the modeling techniques as well as conceptualizations, hence, need to occur. Plant physiology is one area that needs to be looked into to see what plant properties affect the water uptake and how can they be modeled mathematically. The next section discusses a modeling framework based on plant root characteristics which can be employed to model the aforesaid observations.

## **5.3 Incorporation of plant physiology in modeling root water uptake**

Any framework to model root water uptake dynamically, will have to explicitly account for all the four points listed above. The dynamic model should be able to adjust the uptake pattern based on root density as well as available water across the root zone. The model should use physically based parameters so as to remove empiricism from the formulation of the equations. For a given distribution of water content along the root zone (observed or modeled) knowledge of root distribution as well as hydraulic characteristics of roots is hence essential to develop a physically based root water uptake model. The following two sections will describe how root distributions can be modeled as well as how do roots need to be characterized to model uptake from root's perspective.

Fig. 11. Root water uptake variation due to a one inch rainfall even on 5/19/2003.

Fig. 11. Root water uptake variation due to a one inch rainfall even on 5/19/2003.

Fig. 12. Daily root water uptake variation for two October and November 2003 for (a) grass land cover and (b) forested land cover.

## **5.3.1 Root distribution**

Schenk and Jackson (2002) expanded an earlier work of Jackson et al. (1996) to develop a global root database having 475 observed root profiles from different geographic regions of the world. It was found that by varying parameter values the root distribution model given by Gale and Grigal (1987) can be used with sufficient accuracy to describe the observed root distributions. **Equation 14** describes the root distribution model.

$$Y = \mathbf{1} \cdot \gamma^{\mathbf{d}} \tag{14}$$

where Y is the cumulative fraction of roots from the surface to depth d, and is a numerical index of rooting distribution which depends on vegetation type. **Figure 13** shows the observed distribution (shown by data points) versus the fitted distribution using **Equation 14** for different vegetation types. The figure clearly indicates the goodness of fit of the above model. Hence, for a given type of vegetation a suitable can be used to describe the root distribution.

Fig. 13. Observed and Fitted Root Distribution for different type of land covers. [Adapted from Jackson et al. 1996]

#### **5.3.2 Hydraulic characterization of roots**

Hydraulically, soil and xylem are similar as they both show a decrease in hydraulic conductivity with reduction in soil moisture (increase in soil suction). For xylem the

Schenk and Jackson (2002) expanded an earlier work of Jackson et al. (1996) to develop a global root database having 475 observed root profiles from different geographic regions of the world. It was found that by varying parameter values the root distribution model given by Gale and Grigal (1987) can be used with sufficient accuracy to describe the observed root

where Y is the cumulative fraction of roots from the surface to depth d, and is a numerical index of rooting distribution which depends on vegetation type. **Figure 13** shows the observed distribution (shown by data points) versus the fitted distribution using **Equation 14** for different vegetation types. The figure clearly indicates the goodness of fit of the above model. Hence, for a given type of vegetation a suitable can be used to describe the root

Fig. 13. Observed and Fitted Root Distribution for different type of land covers. [Adapted

Hydraulically, soil and xylem are similar as they both show a decrease in hydraulic conductivity with reduction in soil moisture (increase in soil suction). For xylem the

Y = 1 - d (14)

distributions. **Equation 14** describes the root distribution model.

**5.3.1 Root distribution** 

distribution.

from Jackson et al. 1996]

**5.3.2 Hydraulic characterization of roots** 

relationship between hydraulic conductivity and soil suction pressure is called 'vulnerability curve' (Sperry et al. 2003) (see **Figure 14**). The curves are drawn as a percentage loss in conductivity rather than absolute value of conductivity due to the ease of determination of former. Tyree et al (1994) and Hacke et al (2000) have described methods for determination of vulnerability curves for different types of vegetation.

Commonly, the stems and/or root segments are spun to generate negative xylem pressure (as a result of centrifugal force) which results in loss of hydraulic conductivity due to air seeding into the xylem vessels (Pammenter and Willigen 1998). This loss of hydraulic conductivity is plotted against the xylem pressure to get the desired vulnerability curve.

Fig. 14. Vulnerability curves for various species. [Adapted from Tyree, 1999]

For different plant species the vulnerability curve follows an S-Shape function, see **Figure 14** (Tyree 1999). In **Figure 14**, y-axis is percentage loss of hydraulic conductivity induced by the xylem pressure potential Px, shown on the x-axis. C= Ceanothus megacarpus, J = Juniperus virginiana, R = Rhizphora mangel, A = Acer saccharum, T= Thuja occidentalis, P = Populus deltoids.

Pammenter and Willigen (1998) derived an equation to model the vulnerability curve by parametrizing the equation for different plant species. **Equation 15** describes the model mathematically.

$$PLC = \frac{100}{1 + e^{a.(P - P\_{50PIC})}} \tag{15}$$

where PLC denotes the percentage loss of conductivity P50PLC denotes the negative pressure causing 50% loss in the hydraulic conductivity of xylems, P represents the negative pressure and a is a plant based parameter. **Figure 15** shows the model plotted against the data points for different plants. Oliveras et al. (2003) and references cited therein have parameterize the model for different type of pine and oak trees and found the model to be successful in modeling the vulnerability characteristics of xylem.

Fig. 15. Observed values and fitted vulnerability curve for roots and stem sections of different Eucylaptus trees. [Adapted from Pammenter and Willigen, 1998].

The knowledge of hydraulic conductivity loss can be used analogous to the water stress response function *α* (**Equation 9**) by scaling PLC from 0 to 1 and converting the suction pressure to water head. The advantage of using vulnerability curves instead of Feddes or van Genuchten model is that vulnerability curves are based on xylem hydraulics and hence can be physically characterized for each plant species.

where PLC denotes the percentage loss of conductivity P50PLC denotes the negative pressure causing 50% loss in the hydraulic conductivity of xylems, P represents the negative pressure and a is a plant based parameter. **Figure 15** shows the model plotted against the data points for different plants. Oliveras et al. (2003) and references cited therein have parameterize the model for different type of pine and oak trees and found the model to be successful in

Fig. 15. Observed values and fitted vulnerability curve for roots and stem sections of

The knowledge of hydraulic conductivity loss can be used analogous to the water stress response function *α* (**Equation 9**) by scaling PLC from 0 to 1 and converting the suction pressure to water head. The advantage of using vulnerability curves instead of Feddes or van Genuchten model is that vulnerability curves are based on xylem hydraulics and hence

different Eucylaptus trees. [Adapted from Pammenter and Willigen, 1998].

can be physically characterized for each plant species.

modeling the vulnerability characteristics of xylem.

#### **5.3.3 Development of a physically based root water uptake model**

The current model development is based on model conceptualization proposed by Jarvis (1989) however the parameters for the current model are physically defined and include plant physiological characteristics.

For a given land cover type **Equation 14** and **15** can be parameterize to determine the root fraction for any given segment in root zone and percentage loss of conductivity for a given soil suction pressure. For consistency of representation percentage loss of conductivity will be hence forth represented by α (scaled between 0 and 1 similar to **Equation 9**) and will be called stress index.

For any section of root zone, for example *ith* section, root fraction can be written as *Ri* and stress index, determined from vulnerability curve and ambient soil moisture condition, can be written as *αi*. Average stress level over the root zone can be defined as the

$$
\bar{\alpha} = \sum\_{i=1}^{n} R\_i \alpha\_i \tag{16}
$$

where *n* represents the number of soil layers and the other symbols are as previously defined. Thus, as can be seen from **Equation 16** the average stress level combines the effect of both the root distribution and the available water content (via vulnerability curve).

As shown in **Figure 12(b)** if there is available moisture in the root zone, plant can transpire at potential by increasing the uptake from the lower wetter section of the roots. In terms of modeling it can be conceptualized that above a certain critical average stress level ( *<sup>C</sup>* ) plants can transpire at potential and below *<sup>C</sup>* the value of total evapotranspiration decreases. The decrease in the ET value can be modeled linearly as shown by Li et al (2001). The graph of average stress level versus ET (expresses as a ratio with potential ET rate) can hence be plotted as shown in **Figure 16**. In **Figure 16**, ETa is the actual ET out of the soil column while *ETp* is the potential value of ET. **Figure 16** can be used to determine the value of actual ET for any given average stress level.

Once the actual ET value is known, the contribution from individual sections can be modeled depending on the weighted stress index using the relationship defined by

$$S\_i = \left(\frac{E\_a}{\Delta Z\_i}\right) \left(\frac{R\_i \alpha\_i}{\overline{a}}\right) \tag{17}$$

where *Si* defined as the water uptake from the ith *section, ∆Zi* is the depth of ith section and other symbols are as previously defined

Jarvis (1989) used empirical values to simulate the behavior of the above function and **Figure 17** shows the result of root water uptake obtained from his simulation. The values next to each curve in **Figure 17** represent the day after the start of simulation and actual ET rate as expressed in mm/day. On comparison with **Figure 12**, the model successfully reproduced the shift in root water uptake pattern with the uptake being close to potential value (*ETP* = 5.0 mm/d) for about a month from the start of simulation. The decline in ET rate occurred long after the start of the simulation in accordance with the observed values. The model was successful not only in simulating peak but also in the observed magnitude of the root water uptake.

From the above analysis it can be concluded that the root water uptake is just not directly proportional to the distribution of the roots but also depends on the ambient water content. Under dry conditions roots can easily take water from deeper wetter soil layers.

Fig. 16. Variation of ratio of actual to potential ET with location of the critical stress level.

Fig. 17. Variation in the vertical distribution of root water uptake, at different times. [Adapted from Jarvis (1989)]

The methodology described here involves initial laboratory analyses to determine the hydraulic characteristics of the plant. However, once a particular plant specie is characterized then the parameters can be use for that specie elsewhere under similar conditions. The approach shows that eco-hydrological framework has great potential for improving predictive hydrological modeling.

## **6. Conclusion**

120 Evapotranspiration – Remote Sensing and Modeling

From the above analysis it can be concluded that the root water uptake is just not directly proportional to the distribution of the roots but also depends on the ambient water content.

Fig. 16. Variation of ratio of actual to potential ET with location of the critical stress level.

Fig. 17. Variation in the vertical distribution of root water uptake, at different times.

[Adapted from Jarvis (1989)]

Under dry conditions roots can easily take water from deeper wetter soil layers.

The chapter described a method of data collection for soil moisture and water table that can be used for estimation of evapotranspiration. Also described in the chapter is the use of vertical soil moisture measurements to compute the root water uptake in the vadose zone and use that uptake to validate a root water uptake model based on plant physiology based root water uptake model. As evaporation takes place primarily from the first few centimeters (under normal conditions) of the soil profile and the biggest component of the ET is the root water uptake. Hence to improve our estimates of ET, which constitutes ~70% of the rainfall, the estimation and modeling of root water uptake needs to be improved. Ecohydrology provides one such avenue where plant physiology can be incorporated to better represent the water loss. Also, hydrological model incorporating plant physiology can be modified easily in future to be used to predict land-cover changes due to changes in rainfall pattern or other climatic variables.

## **7. References**


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Kite, G.W., and P. Droogers. 2000. Comparing evapotranspiration estimates from satellites, hydrological models and field data. Journal of Hydrology 229:3–18. Knowles, L., Jr. 1996. Estimation of evapotranspiration in the Rainbow Springs and Silver

Li, K.Y., R.De jong, and J.B. Boisvert. 2001. An exponential root-water-uptake model with

Li,K.Y., R.De Jong, and M.T.Coe. 2006. Root water uptake based upon a new water stress

Mahmood, R. and K.G. Hubbard. 2003. Simulating sensitivity of soil moisture and

Meyboom, P. 1967. Ground water studies in the Assiniboine river drainage basin: II.

Monteith, J. L. 1965. Evaporation and environment. *In* G.E.Fogg (ed). The state and

Mo, X., S. Liu, Z. Lin, and W. Zhao. 2004. Simulating temporal and spatial variation of evapotranspiration over the Lushi basin. Journal of Hydrology 285:125–142. Mualem, Y. 1976. A new model predicting the hydraulic conductivity of unsaturated porous

Nachabe, M., N.Shah, M.Ross, and J.Vomacka. 2005. Evapotranspiration of two vegetation

Oliveras,I., J.Martinez-Vilalta, T.Jimenez-Ortiz, M.J Lledo, A.Escarre, and J.Pinol. 2003.

Pammenter.N.W. and C.V.Willigen. 1998. A Mathematical and Statistical Analysis of the

Priestley, C.H.B., and Taylor, R.J. 1972. On the assessment of surface heat flux and

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Biology: San Diego, California, Academic Press, New York, p.205-234 Morgan,K.T., L.R.Parsona, T.A. Wheaton, D.J.Pitts and T.A.Oberza. 1999. Field calibration of

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## **Impact of Irrigation on Hydrologic Change in Highly Cultivated Basin**

## Tadanobu Nakayama1,2

*1National Institute for Environmental Studies (NIES) 16-2 Onogawa, Tsukuba, Ibaraki 2Centre for Ecology & Hydrology (CEH) Crowmarsh Gifford, Wallingford, Oxfordshire 1Japan 2United Kingdom* 

## **1. Introduction**

124 Evapotranspiration – Remote Sensing and Modeling

White,W.N. 1932.A method of estimating ground-water supplies based on discharge by

Water-Supply Paper 659-A.

plants and evaporation from soil: Results of investigation in Escalante Valley, Utah.

With the development of regional economies, the water use environment in the Yellow River Basin, China, has changed greatly (Fig. 1). The river is well known for its high sediment content, frequent floods, unique channel characteristics in the downstream (where the river bed lies above the surrounding land), and limited water resources. This region is heavily irrigated, and combinations of increased food demand and declining water availability are creating substantial pressures. Some research emphasized human activities such as irrigation water withdrawals dominate annual streamflow changes in the downstream in addition to climate change (Tang et al., 2008a). The North China Plain (NCP), located in the downstream area of the Yellow River, is one of the most important grain cropping areas in China, where water resources are also the key to agricultural development, and the demand for groundwater has been increasing. Groundwater has declined dramatically over the previous half century due to over-pumping and drought, and the area of saline-alkaline land has expanded (Brown and Halweil, 1998; Shimada, 2000; Chen et al., 2003b; Nakayama et al., 2006).

Since the completion of a large-scale irrigation project in 1969, noticeable cessation of flow has been observed in the Yellow River (Yang et al., 1998; Fu et al., 2004) resulting from intense competition between water supply and demand, which has occurred increasingly often. The ratio of irrigation water use (defined as the ratio of the annual gross use for irrigation relative to the annual natural runoff) having increased continuously from 21% to 68% during the last 50 years, indicating that the current water shortage is closely related to irrigation development (Yang et al., 2004a). This shortage also reduces the water renewal time (Liu et al., 2003) and renewability of water resources (Xia et al., 2004). This has been accompanied by a decrease in precipitation in most parts of the basin (Tang et al., 2008b). To ensure sustainable water resource use, it is also important to understand the contributions of human intervention to climate change in this basin (Xu et al., 2002), in addition to clarifying the rather complex and diverse water system in the highly cultivated region.

The objective of this research is to clarify the impact of irrigation on the hydrologic change in the Yellow River Basin, an arid to semi-arid environment with intensive cultivation. Combination of the National Integrated Catchment-based Eco-hydrology (NICE) model (Nakayama, 2008a, 2008b, 2009, 2010, 2011a, 2011b; Nakayama and Fujita, 2010; Nakayama and Hashimoto, 2011; Nakayama and Watanabe, 2004, 2006, 2008a, 2008b, 2008c; Nakayama et al., 2006, 2007, 2010, 2011) with complex components such as irrigation, urban water use, and dam/canal systems has led to the improvement in the model, which simulates the balance of both water budget and energy in the entire basin with a resolution of 10 km. The simulated results also evaluates the complex hydrological processes of river dry-up, agricultural/urban water use, groundwater pumping, and dam/canal effects, and to reveal the impact of irrigation on both surface water and groundwater in the basin. This approach will help to clarify how the substantial pressures of combinations of increased food demand and declining water availability can be overcome, and how effective decisions can be made regarding sustainable development under sound socio-economic conditions in the basin.

## **2. Material and methods**

#### **2.1 Coupling of process-based model with complex irrigation procedures**

Previously, the author developed the process-based NICE model, which includes surfaceunsaturated-saturated water processes and assimilates land-surface processes describing the variations of LAI (leaf area index) and FPAR (fraction of photosynthetically active radiation) from satellite data (Fig. 2) (Nakayama, 2008a, 2008b, 2009, 2010, 2011a, 2011b; Nakayama and Fujita, 2010; Nakayama and Hashimoto, 2011; Nakayama and Watanabe, 2004, 2006, 2008a, 2008b, 2008c; Nakayama et al., 2006, 2007, 2010, 2011). The unsaturated layer divides canopy into two layers, and soil into three layers in the vertical dimension in the SiB2 (Simple Biosphere model 2) (Sellers et al., 1996). About the saturated layer, the NICE solves three-dimensional groundwater flow for both unconfined and confined aquifers. The hillslope hydrology can be expressed by the two-layer surface runoff model including freezing/thawing processes. The NICE connects each sub-model by considering water/heat fluxes: gradient of hydraulic potentials between the deepest unsaturated layer and the groundwater, effective precipitation, and seepage between river and groundwater.

In an agricultural field, NICE is coupled with DSSAT (Decision Support Systems for Agrotechnology Transfer) (Ritchie et al., 1998), in which automatic irrigation mode supplies crop water requirement, assuming that average available water in the top layer falls below soil moisture at field capacity for cultivated fields (Nakayama et al., 2006). The model includes different functions of representative crops (wheat, maize, soybean, and rice) and simulates automatically dynamic growth processes. Potential evaporation is calculated on Priestley and Taylor equation (Priestley and Taylor, 1972), and plant growth is based on biomass formulation, which is limited by various reduction factors like light, temperature, water, and nutrient, et al. (Nakayama et al., 2006; Nakayama and Watanabe, 2008b; Nakayama, 2011a).

In this study, the NICE was coupled with complex sub-systems in irrigation and dam/canal in order to develop coupled human and natural systems and to analyze impact of irrigation on hydrologic change in highly cultivated basin. The return flow was evaluated from surface drainage and from groundwater, whereas previous studies had considered only surface drainage (Liu et al., 2003; Xia et al., 2004; Yang et al., 2004a). The gross loss of river water to irrigation includes losses via canals and leakage into groundwater in the field, and can be estimated as the difference between intake from, and return to the river.

Combination of the National Integrated Catchment-based Eco-hydrology (NICE) model (Nakayama, 2008a, 2008b, 2009, 2010, 2011a, 2011b; Nakayama and Fujita, 2010; Nakayama and Hashimoto, 2011; Nakayama and Watanabe, 2004, 2006, 2008a, 2008b, 2008c; Nakayama et al., 2006, 2007, 2010, 2011) with complex components such as irrigation, urban water use, and dam/canal systems has led to the improvement in the model, which simulates the balance of both water budget and energy in the entire basin with a resolution of 10 km. The simulated results also evaluates the complex hydrological processes of river dry-up, agricultural/urban water use, groundwater pumping, and dam/canal effects, and to reveal the impact of irrigation on both surface water and groundwater in the basin. This approach will help to clarify how the substantial pressures of combinations of increased food demand and declining water availability can be overcome, and how effective decisions can be made regarding sustainable development under sound socio-economic conditions in the basin.

**2.1 Coupling of process-based model with complex irrigation procedures** 

groundwater, effective precipitation, and seepage between river and groundwater.

In an agricultural field, NICE is coupled with DSSAT (Decision Support Systems for Agrotechnology Transfer) (Ritchie et al., 1998), in which automatic irrigation mode supplies crop water requirement, assuming that average available water in the top layer falls below soil moisture at field capacity for cultivated fields (Nakayama et al., 2006). The model includes different functions of representative crops (wheat, maize, soybean, and rice) and simulates automatically dynamic growth processes. Potential evaporation is calculated on Priestley and Taylor equation (Priestley and Taylor, 1972), and plant growth is based on biomass formulation, which is limited by various reduction factors like light, temperature, water, and nutrient, et al. (Nakayama et al., 2006; Nakayama and Watanabe, 2008b; Nakayama,

In this study, the NICE was coupled with complex sub-systems in irrigation and dam/canal in order to develop coupled human and natural systems and to analyze impact of irrigation on hydrologic change in highly cultivated basin. The return flow was evaluated from surface drainage and from groundwater, whereas previous studies had considered only surface drainage (Liu et al., 2003; Xia et al., 2004; Yang et al., 2004a). The gross loss of river water to irrigation includes losses via canals and leakage into groundwater in the field, and

can be estimated as the difference between intake from, and return to the river.

Previously, the author developed the process-based NICE model, which includes surfaceunsaturated-saturated water processes and assimilates land-surface processes describing the variations of LAI (leaf area index) and FPAR (fraction of photosynthetically active radiation) from satellite data (Fig. 2) (Nakayama, 2008a, 2008b, 2009, 2010, 2011a, 2011b; Nakayama and Fujita, 2010; Nakayama and Hashimoto, 2011; Nakayama and Watanabe, 2004, 2006, 2008a, 2008b, 2008c; Nakayama et al., 2006, 2007, 2010, 2011). The unsaturated layer divides canopy into two layers, and soil into three layers in the vertical dimension in the SiB2 (Simple Biosphere model 2) (Sellers et al., 1996). About the saturated layer, the NICE solves three-dimensional groundwater flow for both unconfined and confined aquifers. The hillslope hydrology can be expressed by the two-layer surface runoff model including freezing/thawing processes. The NICE connects each sub-model by considering water/heat fluxes: gradient of hydraulic potentials between the deepest unsaturated layer and the

**2. Material and methods** 

2011a).

Fig. 1. Land cover in the study area of the Yellow River Basin in China. Bold black line shows the boundary of the basin. Black dotted line is the border of the North China Plain (NCP), which includes the downstream of the Yellow River Basin. Verification data are also plotted in this figure: river discharge (open red circle), soil moisture (open brown triangle), and groundwater level (green dot).

Irrigation withdrawals in the basin account for about 90% of total surface abstraction and 60% of groundwater withdrawal (Chen et al., 2003a). The model was improved for application to irrigated fields where water is withdrawn from both groundwater and river, and therefore the river dry-up process can be reproduced well. As the initial conditions, the ratios of river to aquifer irrigation were set at constant values. In the calibration procedure, these values were changed from initial conditions in order to reproduce the observation data as closely as possible after repeated trial and error (Oreskes et al., 1994). A validation procedure was then conducted in order to confirm the simulation under the same set of parameters, which resulted into reproducing reasonably the observed values. Spring/winter wheat, summer maize, and summer rice were automatically simulated in sequence analysis mode in succession by inputting previous point data for each crop type (Wang et al., 2001; Liu et al., 2002; Tao et al., 2006) and spatial distribution data (Chinese Academy of Sciences, 1988; Fang et al., 2006). The deficit water in the irrigated fields was automatically withdrawn and supplied from the river or the aquifer in the model in order to satisfy the observed hydrologic variables like soil moisture, river discharge, groundwater level, LAI, evapotranspiration, and crop coefficient. So, NICE simulates drought impact and includes the effect of water stress implicitly. Details are given in the previous researches (Nakayama et al., 2006; Nakayama and Watanabe, 2008b; Nakayama, 2011a, 2011b).

Fig. 2. National Integrated Catchment-based Eco-hydrology (NICE) model.

Another important characteristics of the study area is that there are many dams and canals to meet the huge demand for agricultural, industrial, and domestic water use (Ren et al., 2002) (Fig. 1), and exist six large dams on the main river (Yang et al., 2004a). Because there are few available data on discharge control at most of these dams, the model uses a constant ratio of dam inflow to outflow, which is a simpler approach than that of the storage-runoff function model (Sato et al., 2008). There are also many complex canals in the three large irrigation zones (Qingtongxia in Ningxia Hui, Hetao in Inner Mongolia, and Weisan in Shandong Province), in addition to the NCP, making it very difficult to evaluate the flow dynamics there. Because it is impossible to obtain the observed discharge and data related to the control of the weir/gate at every canal, it is effective to estimate the flow dynamics only in main canals as the first approximation when attempting to evaluate the hydrologic cycle in the entire basin in the same way as (Nakayama et al., 2006; Nakayama, 2011a). Therefore, NICE simulates the discharge only in a main canal assuming that this is defined as the difference in hydraulic potentials at both junctions similar to the stream junction model (Nakayama and Watanabe, 2008b). The dynamic wave effect is also important for the simulation of meandering rivers and smaller slopes, because the backwater effect is predominant (Nakayama and Watanabe, 2004). When a river flow is very low and almost

the effect of water stress implicitly. Details are given in the previous researches (Nakayama

et al., 2006; Nakayama and Watanabe, 2008b; Nakayama, 2011a, 2011b).

Fig. 2. National Integrated Catchment-based Eco-hydrology (NICE) model.

Another important characteristics of the study area is that there are many dams and canals to meet the huge demand for agricultural, industrial, and domestic water use (Ren et al., 2002) (Fig. 1), and exist six large dams on the main river (Yang et al., 2004a). Because there are few available data on discharge control at most of these dams, the model uses a constant ratio of dam inflow to outflow, which is a simpler approach than that of the storage-runoff function model (Sato et al., 2008). There are also many complex canals in the three large irrigation zones (Qingtongxia in Ningxia Hui, Hetao in Inner Mongolia, and Weisan in Shandong Province), in addition to the NCP, making it very difficult to evaluate the flow dynamics there. Because it is impossible to obtain the observed discharge and data related to the control of the weir/gate at every canal, it is effective to estimate the flow dynamics only in main canals as the first approximation when attempting to evaluate the hydrologic cycle in the entire basin in the same way as (Nakayama et al., 2006; Nakayama, 2011a). Therefore, NICE simulates the discharge only in a main canal assuming that this is defined as the difference in hydraulic potentials at both junctions similar to the stream junction model (Nakayama and Watanabe, 2008b). The dynamic wave effect is also important for the simulation of meandering rivers and smaller slopes, because the backwater effect is predominant (Nakayama and Watanabe, 2004). When a river flow is very low and almost zero at some point in the simulation, the dynamic wave theory requires a lot more computation time and sometimes becomes unstable. Therefore, the model applies a threshold water level of 1 mm to ensure simulation stability and to include the dry-up process. The model also includes the seepage process which is decided by some parameters such as hydraulic conductivity of the river bed, cross-sectional area of the groundwater section, and river bed thickness. Details are described in Nakayama (2011b).

#### **2.2 Model input data and running the simulation**

Six-hour reanalysed data for downward radiation, precipitation, atmospheric pressure, air temperature, air humidity, wind speed at a reference level, FPAR, and LAI were input into the model after interpolation of ISLSCP (International Satellite Land Surface Climatology Project) data with a resolution of 1° x 1° (Sellers et al., 1996) in inverse proportion to the distance back-calculated in each grid. Because the ISLSCP precipitation data had the least reliability and underestimated the observed values at peak times, rain gauge daily precipitation data collected at 3,352 meteorological stations throughout the study area were used to correct the ISLSCP precipitation data. Mean elevation of each 10-km grid cell was calculated from the spatial average of a global digital elevation model (DEM; GTOPO30) with a horizontal grid spacing of 30 arc–seconds (~1 km) (USGS, 1996). Digital land cover data produced by the Chinese Academy of Sciences (CAS) based on Landsat TM data from the early 1990s (Liu, 1996) were categorized for the simulation (Fig. 1). Vegetation class and soil texture were categorized and digitized into 1-km mesh data by using 1:4,000,000 and 1:1,000,000 vegetation and soil maps of China (Chinese Academy of Sciences, 1988, 2003). The author's previous research showed that these finer-resolution products are highly correlated with the ISLSCP (Nakayama, 2011b). The geological structure was divided into four types on the basis of hydraulic conductivity, the specific storage of porous material, and specific yield by scanning and digitizing the geological material (Geological Atlas of China, 2002) and core–sampling data at some points (Zhu, 1992).

The irrigation area was calculated from the GIS data based on Landsat TM data from the early 1990s (Liu, 1996), and the calculated value agrees well with the previous results from that period (Yang et al., 2004a) (Table 1), as described in Nakayama (2011b). Most of the irrigated fields are distributed in the middle and lower regions of the Yellow River mainstream and in the NCP (Fig. 3). The agricultural areas in the upper regions and Erdos Plateau are dominated by dryland fields. Spring/winter wheat was predominant in the upper and middle of the arid and semi-arid regions, and double cropping of winter wheat and summer maize was usually practised in the middle and downstream and in the NCP's relatively warm and humid environment (Wang et al., 2001; Liu et al., 2002; Fang et al., 2006; Nakayama et al., 2006; Tao et al., 2006). The averaged water use during 1987-1988 at the main cities in the Yellow River Basin and the NCP (Hebei Department of Water Conservancy, 1987-1988; Yellow River Conservancy Commission, 2002) was directly input to the model. In the 1990s, return flow was as much as 35% of withdrawal in the upper and 25% in the middle, but close to 0% in the downstream (Chen et al., 2003a; Cai and Rosegrant, 2004). The return flows at Qingtongxia and Hetao irrigation zones are 59% and 25% of withdrawal, whereas that at Weisan irrigation zone is close to 0%, because the river bed is above the level of the plain (Chen et al., 2003a; Cai and Rosegrant, 2004). This information was also input into the model.

At the upstream boundaries, a reflecting condition on the hydraulic head was used assuming that there is no inflow from the mountains in the opposite direction (Nakayama and Watanabe, 2004). At the eastern sea boundary, a constant head was set at 0 m. The hydraulic head values parallel to the observed ground level were input as initial conditions for the groundwater sub-model. As initial conditions, the ratios of river and aquifer irrigation were set at the same constant values as in the above section. In river grids decided by digital river network from 1:50,000 and 1:100,000 topographic maps (CAS, 1982), inflows or outflows from the riverbeds were simulated at each time step depending on the difference in the hydraulic heads of groundwater and river. The simulation area covered 3,000 km by 1,000 km with a grid spacing of 10 km, covering the entire Yellow River Basin and the NCP. The vertical layer was discretized in thickness with depth, with each layer increased in thickness by a factor of 1.1 (Nakayama, 2011b; Nakayama and Watanabe, 2008b; Nakayama et al., 2006). The upper layer was set at 2 m depth, and the 20th layer was defined as an elevation of –500 m from the sea surface. Simulations were performed with a time step of 6 h for two years during 1987–1988 after 6 months of warm-up period until equilibrium. The author first calibrated the simulated values including irrigation water use in 1987 against previous results, and then validated them in 1988. Previously observed data about river discharge (9 points; Yellow River Conservancy Commission, 1987-1988), soil moisture (7 points of the Global Soil Moisture Data Bank; Entin et al., 2000; Robock et al., 2000), and groundwater level (26 points; China Institute for Geo-Environmental Monitoring, 2003) were also used for the verification of the model (Fig. 1 and Table 2) in addition to values published in the literature (Clapp and Hornberger, 1978; Rawls et al., 1982). Details are described in Nakayama (2011b).


aAbbreviation in the following; LZ, Lanzhou (R-1);

TDG, Toudaoguai (R-4); LM, Longmen; SMX, Sanmenxia;

HYK, Huayuankou (R-6).

Table 1. Comparison of irrigation area in the simulated condition with that in the previous research.

At the upstream boundaries, a reflecting condition on the hydraulic head was used assuming that there is no inflow from the mountains in the opposite direction (Nakayama and Watanabe, 2004). At the eastern sea boundary, a constant head was set at 0 m. The hydraulic head values parallel to the observed ground level were input as initial conditions for the groundwater sub-model. As initial conditions, the ratios of river and aquifer irrigation were set at the same constant values as in the above section. In river grids decided by digital river network from 1:50,000 and 1:100,000 topographic maps (CAS, 1982), inflows or outflows from the riverbeds were simulated at each time step depending on the difference in the hydraulic heads of groundwater and river. The simulation area covered 3,000 km by 1,000 km with a grid spacing of 10 km, covering the entire Yellow River Basin and the NCP. The vertical layer was discretized in thickness with depth, with each layer increased in thickness by a factor of 1.1 (Nakayama, 2011b; Nakayama and Watanabe, 2008b; Nakayama et al., 2006). The upper layer was set at 2 m depth, and the 20th layer was defined as an elevation of –500 m from the sea surface. Simulations were performed with a time step of 6 h for two years during 1987–1988 after 6 months of warm-up period until equilibrium. The author first calibrated the simulated values including irrigation water use in 1987 against previous results, and then validated them in 1988. Previously observed data about river discharge (9 points; Yellow River Conservancy Commission, 1987-1988), soil moisture (7 points of the Global Soil Moisture Data Bank; Entin et al., 2000; Robock et al., 2000), and groundwater level (26 points; China Institute for Geo-Environmental Monitoring, 2003) were also used for the verification of the model (Fig. 1 and Table 2) in addition to values published in the literature (Clapp and Hornberger, 1978; Rawls et al., 1982). Details

Reachesa Irrigation area (x 104 ha)

GIS database (Liu 1996)

Above LZ 46.8 39.5 LZ – TDG 342.1 344.1 TDG – LM 43.0 53.4 LM – SMX 295.6 281.3 SMX – HYK 59.2 60.6 Below HYK 160.7 155.0

Sum 947.3 933.9

Table 1. Comparison of irrigation area in the simulated condition with that in the previous

Previous research (Yang et al. 2004a)

are described in Nakayama (2011b).

aAbbreviation in the following; LZ, Lanzhou (R-1); TDG, Toudaoguai (R-4); LM, Longmen; SMX, Sanmenxia;

HYK, Huayuankou (R-6).

research.


Table 2. Lists of observation stations for calibration and validation shown in Fig. 1.

Fig. 3. Crop types in the agricultural areas. Irrigation areas are also overshaded. Irrigated fields cover most of the NCP for double cropping of winter wheat and summer maize in addition to three large irrigation zones.

## **3. Result and discussion**

## **3.1 Verification of hydrologic cycle in the basin**

The irrigation water use simulated in 1987 was firstly calibrated against previous results (Cai and Rosegrant, 2004; Liu and Xia, 2004; Yang et al., 2004a; Cai, 2006), showing a close agreement with the results of Cai and Rosegrant (2004). Then, the simulated value in 1988 was validated with the previous researches (Table 3), which indicates that there was reasonable agreement with each other and that irrigation water use is higher in large irrigation zones (LZ-TDG), along the Wei and Fen rivers (LM-SMX) in the middle, and in the downstream (below HYK). The results also show a high correlation between irrigation area and water use: *r*2 = 0.986 (Chen et al., 2003a) and *r*2 = 0.826 (Liu and Xia, 2004). Details of calibration and validation procedures are described in Nakayama (2011b).

The actual ET simulated by NICE reproduces reasonably the general trend estimated by integrated AVHRR NDVI data (Sun et al., 2004), which may give a good support on the predictive skill of the model (Fig. 4a-b). Although there are some discrepancies particularly for the lowest ET area (EP < 200 mm/year) mainly because of the banded colour figures, the simulated result reproduces the characteristics that the value is lowest in the downstream area of middle and on the Erdos Plateau—less than 200-300 mm per year (except in the irrigated area)—where vegetation is dominated by desert and soil is dominated by sand, and increases gradually towards the south-east. The simulated result also indicates that this spatial heterogeneity is related to human interventions and the resultant water stress by spring/winter cultivation in the upper/middle areas (Chen et al., 2003a; Tao et al., 2006), and winter wheat and summer maize cultivations in the middle/downstream (including the

Fig. 3. Crop types in the agricultural areas. Irrigation areas are also overshaded. Irrigated fields cover most of the NCP for double cropping of winter wheat and summer maize in

The irrigation water use simulated in 1987 was firstly calibrated against previous results (Cai and Rosegrant, 2004; Liu and Xia, 2004; Yang et al., 2004a; Cai, 2006), showing a close agreement with the results of Cai and Rosegrant (2004). Then, the simulated value in 1988 was validated with the previous researches (Table 3), which indicates that there was reasonable agreement with each other and that irrigation water use is higher in large irrigation zones (LZ-TDG), along the Wei and Fen rivers (LM-SMX) in the middle, and in the downstream (below HYK). The results also show a high correlation between irrigation area and water use: *r*2 = 0.986 (Chen et al., 2003a) and *r*2 = 0.826 (Liu and Xia, 2004). Details of

The actual ET simulated by NICE reproduces reasonably the general trend estimated by integrated AVHRR NDVI data (Sun et al., 2004), which may give a good support on the predictive skill of the model (Fig. 4a-b). Although there are some discrepancies particularly for the lowest ET area (EP < 200 mm/year) mainly because of the banded colour figures, the simulated result reproduces the characteristics that the value is lowest in the downstream area of middle and on the Erdos Plateau—less than 200-300 mm per year (except in the irrigated area)—where vegetation is dominated by desert and soil is dominated by sand, and increases gradually towards the south-east. The simulated result also indicates that this spatial heterogeneity is related to human interventions and the resultant water stress by spring/winter cultivation in the upper/middle areas (Chen et al., 2003a; Tao et al., 2006), and winter wheat and summer maize cultivations in the middle/downstream (including the

calibration and validation procedures are described in Nakayama (2011b).

addition to three large irrigation zones.

**3.1 Verification of hydrologic cycle in the basin** 

**3. Result and discussion** 

Wei and Fen tributaries) and the NCP (Wang et al., 2001; Liu et al., 2002; Nakayama et al., 2006). Although the satellite-derived data are effective for grasping the spatial distribution of actual ET, there are some inefficiencies with regard to underestimation in sparsely vegetated regions (Inner Mongolia and Shaanxi Province) and overestimation in densely vegetated or irrigated regions (source area and Henan Province), as suggested by previous research (Sun et al., 2004; Zhou et al., 2007), which the simulation overcomes and improves mainly due to the inclusion of drought impact in the model. Details are described in Nakayama (2011b).

The model also simulated effect of irrigation on evapotranspiration at rotation between winter wheat and summer maize in the downstream of Yellow River (Fig. 4c). Because more water is withdrawn during winter-wheat period due to small rainfall in the north, the irrigation in this period affects greatly the increase in evapotranspiration. The simulated result indicates that the evapotranspiration increases predominantly during the seasons of grain filling and harvest of winter wheat with the effect of irrigation. In particular, most of the irrigation is withdrawn from aquifer in the NCP because surface water is seriously limited there (Nakayama, 2011b; Nakayama et al., 2006). This over-irrigation also affects the hydrologic change such as river discharge, soil moisture, and groundwater level in addition to evapotranspiration, as described in the following.


aAbbreviation in the following; LZ, Lanzhou (R-1); TDG, Toudaoguai (R-4); LM, Longmen; SMX, Sanmenxia; HYK, Huayuankou (R-6).

bValue in parenthesis shows the target year in the simulation and the literatures.

Table 3. Validation of irrigation water use simulated by the model with that in the previous research.

The model could simulate reasonably the spatial distribution of irrigation water use after the comparison with a previous study based on the Penman-Monteith method and the crop coefficient (Fang et al., 2006) not only in reach level but also in the spatial distribution, as described in Nakayama (2011b). In particular, simulated ratios of river to total irrigation

Fig. 4. Annual-averaged spatial distribution of evapotranspiration in 1987; (a) previous research; (b) simulated result; and (c) simulated value about impact of irrigation on evapotranspiration at rotation between winter wheat and summer maize. In Fig. 4c, right axis (dotted line) shows a period of each crop (WH; wheat, and MZ; maize).

(=river + aquifer) showed great variation and spatial heterogeneity in the basin. Fig. 5 shows the effect of over-irrigation on the decrease in river discharge on the downstream. The model reproduces reasonably the observed discharge for a low flow, and sometimes dry-up in the downstream (Yellow River Conservancy Commission, 1987-1988) with relatively high correlation *r*2 and Nash-Sutcliffe criterion (NS; Nash and Sutcliffe, 1970) because the model includes the irrigation procedure and dynamic wave effect (Nakayama and Watanabe, 2004) in the model (Fig. 5b). The discharge decreases seriously in the downstream area mainly because of the water withdrawal for agriculture, which is more than 90% of the total withdrawal (Cai, 2006). At the downstream point at Lijin (R-9 in Table 2), the river discharge dries out during the spring mainly because most of the water is used for the irrigation of winter wheat in correspondence with the great increase in evapotranspiration shown in Fig. 4c. The model also indicated that the effect of groundwater irrigation is predominant in the downstream (data not shown), mainly on account of intensified water-use conflicts between upstream and downstream, and between various sectors like agriculture, municipality, and industry (Brown and Halweil, 1998; Nakayama, 2011a, 2011b; Nakayama et al., 2006). The smaller change in groundwater level in the upper was largely attributable to its unsuitability for crop production and the higher dependence of irrigation on surface water, as described previously (Yellow River Conservancy Commission, 2002).

Fig. 4. Annual-averaged spatial distribution of evapotranspiration in 1987; (a) previous research; (b) simulated result; and (c) simulated value about impact of irrigation on evapotranspiration at rotation between winter wheat and summer maize. In Fig. 4c, right

(=river + aquifer) showed great variation and spatial heterogeneity in the basin. Fig. 5 shows the effect of over-irrigation on the decrease in river discharge on the downstream. The model reproduces reasonably the observed discharge for a low flow, and sometimes dry-up in the downstream (Yellow River Conservancy Commission, 1987-1988) with relatively high correlation *r*2 and Nash-Sutcliffe criterion (NS; Nash and Sutcliffe, 1970) because the model includes the irrigation procedure and dynamic wave effect (Nakayama and Watanabe, 2004) in the model (Fig. 5b). The discharge decreases seriously in the downstream area mainly because of the water withdrawal for agriculture, which is more than 90% of the total withdrawal (Cai, 2006). At the downstream point at Lijin (R-9 in Table 2), the river discharge dries out during the spring mainly because most of the water is used for the irrigation of winter wheat in correspondence with the great increase in evapotranspiration shown in Fig. 4c. The model also indicated that the effect of groundwater irrigation is predominant in the downstream (data not shown), mainly on account of intensified water-use conflicts between upstream and downstream, and between various sectors like agriculture, municipality, and industry (Brown and Halweil, 1998; Nakayama, 2011a, 2011b; Nakayama et al., 2006). The smaller change in groundwater level in the upper was largely attributable to its unsuitability for crop production and the higher dependence of irrigation on surface water, as described

axis (dotted line) shows a period of each crop (WH; wheat, and MZ; maize).

previously (Yellow River Conservancy Commission, 2002).

Fig. 5. Decrease in discharge caused by over-irrigation in the downstream region; (a) simulated result of river irrigation in 1987; (b) river discharge at the downstream. In Fig. 5b, solid line is the simulated result with irrigation, and circle is the observed value.

The simulated groundwater levels and soil moisture contents were calibrated and validated against observed data (Entin et al., 2000; Robock et al., 2000; China Institute for Geo-Environmental Monitoring, 2003) shown in Fig. 1 and Table 2 (data not shown). Although the correlation of groundwater level relative to the surface was not as good (*r*2 = 0.401) as that of the absolute groundwater level (*r*2 = 0.983) and the simulated value showed a tendency to overestimate the observed value in the calibration procedure for 1987, the simulation reproduced well the general distribution (BIAS = -21.2%, RMSE = 5.6 m, RRMSE = -0.468, MSSS = 0.356) (Nakayama, 2011b). This disagreement was due to the difference in surface elevation on the point-scale and mesh-scale (scale dependence), and the resolution of the groundwater flow model (changes in elevation from 0 m to 3000–4000 m in the basin). Because the simulated level is the hydraulic head in an aquifer, it might take a larger value than the land surface, particularly for a grid cell near or on the river. Another reason is that the irrigation water use simulated by the model might be underestimated because automatic irrigation supplied the water requirement for crops in order to satisfy the observed soil moisture, river discharge, groundwater level, LAI, evapotranspiration, and crop coefficient, which was theoretically pumped up from the river or the aquifer in the model. In reality, it has a possibility that farmers might use more irrigation water than the theoretical water requirement for crops if possible though there were not enough statistical or observed data to support it. The simulated water level decreases rapidly around the source area, indicating that there are many springs in this region. It is very low in the downstream (below sea level in some regions) because of the low elevation and overexploitation, as is the case in the NCP (Nakayama, 2011a, 2011b; Nakayama et al., 2006). The soil moisture is higher in the source area and in the paddy-dominated Qingtongxia Irrigation Zone (data not shown), corresponding closely with the distribution of the groundwater level. Details are described in Nakayama (2011b).

#### **3.2 Impact of irrigation on hydrologic changes**

Scenario analysis of conversion from unirrigated to irrigated run predicted the hydrologic changes (Fig. 6). The predicted result without irrigation generally overestimates the observed river discharge (Fig. 6a) and this effect is more prominent in the middle and downstream, as supported by reports that the difference between natural and observed runoff is larger downstream (Ren et al., 2002; Fu et al., 2004; Liu and Zheng, 2004). The difference between simulations considering and not considering irrigation strongly supports previous studies from the point of view that the influence of human interventions on river runoff has increased downstream over the last five decades (Chen et al., 2003a; Liu and Xia, 2004; Yang et al., 2004a; Cai, 2006; Tang et al., 2007) (Table 3), as also represented by the decline of water renewal times (Liu et al., 2003) and water resource renewability (Xia et al., 2004). This difference is greatly affected by complex irrigation procedures of various crops, which are roughly represented by spring/winter wheat in the upper-middle, and double cropping of winter wheat and summer maize in the middle-downstream regions (Wang et al., 2001; Liu et al., 2002; Fang et al., 2006; Nakayama et al., 2006; Tao et al., 2006).

Because there is some time lag between periods of increase in irrigation and decrease in runoff, the river discharge does not necessarily decrease in the winter and sometimes decreases in the summer. Further, the discharge sometimes increases slightly in the flood season, which indicates that the precipitation in irrigated fields sometimes responds quickly to flood drainage. Although both *r*2 and NS have relatively low values across the basin (max: *r*2 = 0.447, NS = 0.452), the simulated results with irrigation reproduce these characteristics better, and the statistics for MV (mean value), SD (standard deviation), and CV (coefficient of variation; CV = SD/MV) generally agree better with the observed values, as also supported by the better reproduction of other components of the hydrologic cycle, such as annual ET (Fig. 4) (data not shown in the case without irrigation) and irrigation water use (Fig. 5a, Table 3). The simulated result considering irrigation also reproduces the observed data for a low flow, and sometimes dry-up in the same way as Fig. 5b (Zhang et al., 1990; Yang et al., 1998; Ren et al., 2002), being attributable to inclusion of the dynamic wave effect in NICE, which other previous NICE series were unable to reproduce. Furthermore, the model improves the reproduction of river discharge in the basin in comparison with previous research (Yang and Musiake, 2003), where the ratio of absolute error to the mean was more than 60% at Huayuankou hydrological station, one of the worst such cases on a major river in Asia. The major reason for this disagreement is artificial water regulation such as reservoirs, water intakes, and diversions, which the model generally includes in addition to the extreme annual variation in flood seasons (Nakayama and Watanabe, 2008b).

Scenario analysis also predicts the groundwater level change and indicates that the effect of groundwater over-irrigation is predominant in the middle and downstream (Fig. 6b), where surface water is seriously limited, as shown in Fig. 5b and described in section 2.1 (Yellow River Conservancy Commission, 2002). The predicted result indicates a serious situation of water shortage in the downstream region and the NCP where groundwater level degrades over a wide area (Brown and Halweil). The result also implies that the model accounts for

in some regions) because of the low elevation and overexploitation, as is the case in the NCP (Nakayama, 2011a, 2011b; Nakayama et al., 2006). The soil moisture is higher in the source area and in the paddy-dominated Qingtongxia Irrigation Zone (data not shown), corresponding closely with the distribution of the groundwater level. Details are described

Scenario analysis of conversion from unirrigated to irrigated run predicted the hydrologic changes (Fig. 6). The predicted result without irrigation generally overestimates the observed river discharge (Fig. 6a) and this effect is more prominent in the middle and downstream, as supported by reports that the difference between natural and observed runoff is larger downstream (Ren et al., 2002; Fu et al., 2004; Liu and Zheng, 2004). The difference between simulations considering and not considering irrigation strongly supports previous studies from the point of view that the influence of human interventions on river runoff has increased downstream over the last five decades (Chen et al., 2003a; Liu and Xia, 2004; Yang et al., 2004a; Cai, 2006; Tang et al., 2007) (Table 3), as also represented by the decline of water renewal times (Liu et al., 2003) and water resource renewability (Xia et al., 2004). This difference is greatly affected by complex irrigation procedures of various crops, which are roughly represented by spring/winter wheat in the upper-middle, and double cropping of winter wheat and summer maize in the middle-downstream regions (Wang et

al., 2001; Liu et al., 2002; Fang et al., 2006; Nakayama et al., 2006; Tao et al., 2006).

to the extreme annual variation in flood seasons (Nakayama and Watanabe, 2008b).

Scenario analysis also predicts the groundwater level change and indicates that the effect of groundwater over-irrigation is predominant in the middle and downstream (Fig. 6b), where surface water is seriously limited, as shown in Fig. 5b and described in section 2.1 (Yellow River Conservancy Commission, 2002). The predicted result indicates a serious situation of water shortage in the downstream region and the NCP where groundwater level degrades over a wide area (Brown and Halweil). The result also implies that the model accounts for

Because there is some time lag between periods of increase in irrigation and decrease in runoff, the river discharge does not necessarily decrease in the winter and sometimes decreases in the summer. Further, the discharge sometimes increases slightly in the flood season, which indicates that the precipitation in irrigated fields sometimes responds quickly to flood drainage. Although both *r*2 and NS have relatively low values across the basin (max: *r*2 = 0.447, NS = 0.452), the simulated results with irrigation reproduce these characteristics better, and the statistics for MV (mean value), SD (standard deviation), and CV (coefficient of variation; CV = SD/MV) generally agree better with the observed values, as also supported by the better reproduction of other components of the hydrologic cycle, such as annual ET (Fig. 4) (data not shown in the case without irrigation) and irrigation water use (Fig. 5a, Table 3). The simulated result considering irrigation also reproduces the observed data for a low flow, and sometimes dry-up in the same way as Fig. 5b (Zhang et al., 1990; Yang et al., 1998; Ren et al., 2002), being attributable to inclusion of the dynamic wave effect in NICE, which other previous NICE series were unable to reproduce. Furthermore, the model improves the reproduction of river discharge in the basin in comparison with previous research (Yang and Musiake, 2003), where the ratio of absolute error to the mean was more than 60% at Huayuankou hydrological station, one of the worst such cases on a major river in Asia. The major reason for this disagreement is artificial water regulation such as reservoirs, water intakes, and diversions, which the model generally includes in addition

in Nakayama (2011b).

**3.2 Impact of irrigation on hydrologic changes** 

intensified water-use conflicts between upstream and downstream areas, and between agriculture, municipal, and industrial sectors (Brown and Halweil, 1998; Shimada, 2000; Chen et al., 2003b; Nakayama et al., 2006). These analyses of the impact of human intervention on hydrologic changes present strong indicatives of the seriousness of the situation, and imply the need for further correct estimation and appropriate measures against such irrigation loss and the low irrigation efficiency described previously (Wang et al., 2001). Details are described in Nakayama (2011b).

Fig. 6. Scenario analysis of conversion from unirrigated to irrigated run; (a) prediction of river discharge at the upper-middle (R–3; Qingtongxia) and the lower (R–6; Huayuankou) in Fig. 1 and Table 2; (b) groundwater level change in the middle-downstream regions. In Fig. 6a, circles show observation data; solid line is the simulated result without irrigation effect; bold line is the simulated result with irrigation. Right axis (dotted line) shows a period of each crop (WH; wheat, and MZ; maize) in the same way as Fig. 4c.

## **3.3 Discussion**

Water scarcity and resource depletion in the downstream and the NCP, referred to as the 'bread basket' of China, is becoming more severe every year against increased crop production based on irrigation water, in addition to the expansion of municipal and industrial usage (Nakayama, 2011a; Nakayama et al., 2006). The simulated result shows the discharge was affected greatly by the rapid development of cities and industries and the increase in farmland irrigation (Fig. 6), which is closely related to severe groundwater degradation owing to the high clay content of the surface soil (Nakayama et al., 2006; Nakayama, 2011a, 2011b). Because the dry-up of river reaches and groundwater exhaustion have been very severe so far (Chen et al., 2003b; Xia et al., 2004; Yang et al., 2004a), it is urgently necessary to perform effective control of water diversions (Liu and Xia, 2004; Liu and Zheng, 2004); the results simulated by NICE can be taken as strong indicatives of the seriousness of the situation. There are some reasons for the gap between irrigation water use (Fig. 5a) and groundwater level distribution (Fig. 6b). Firstly, the simulated levels have a temporally averaged distribution, and it takes some time for water levels to reach equilibrium after the boundary conditions have changed. This in turn affects the replenishment of groundwater from adjacent regions in addition to the heterogeneity of three-dimensional groundwater flow. Secondly, irrigation water is drawn not only from groundwater but also from river, and the ratio of river to total irrigation changes spatiotemporally in the basin (Fig. 5b); more river irrigation is drawn in the upper, and most of the irrigation depends on groundwater in the downstream, particularly in the NCP. This effect is clearly evident in comparison with the simulated results and the degradation value in the downstream is smaller that that in Fig. 6b.

Though the simulation reproduced reasonably hydrologic cycle such as evapotranspiration (Fig. 4), irrigation water use (Fig. 5a and Table 3), groundwater level, and river discharge (Fig. 5b and Fig. 6a), there were some discrepancies due to very complex and inaccurate nature of water withdrawal in the basin. In particular, the model achieved a relatively reasonable agreement though the model tried to calibrate and validate irrigation water use during only two years against other studies focusing on irrigation during long period (Fig. 5a and Table 3), which might lead to a substantial bias on model parameters. Because the objective of this study is primarily to evaluate the complex hydrological processes and reveal the impact of irrigation on hydrologic cycle in the basin through the verification during a fixed period, it is a future work to run model for the long period in the next step. At the same time, it will be of importance to derive better estimates of water demand in agricultural and urban areas during the long period by using more detailed statistical data, GIS data, and satellite data in longer period. Although the geological structure in the model included the general characteristics of several aquifers by reference to previous literature (Geological Atlas of China, 2002), the detailed structure of each aquifer layer was simplified as much as possible (Nakayama and Watanabe, 2008b; Nakayama et al., 2006). It will be necessary to obtain more precise data for the complex channel geometry of both natural and artificial rivers, soil properties, and geological structure. The spatial and temporal resolution used in the simulation also requires further improvement in order to overcome the problem of scale dependence and to improve verification and future reliability (Nakayama, 2011b).

Simulated results about the impact of irrigation on evapotranspiration change show a heterogeneous distribution (Fig. 4a-b). In particular, the irrigation of winter wheat increases greatly evapotranspiration, which is supplied by the limited water resources of river discharge and groundwater there (Fig. 5). This implies that energy supply is abundant relative to the water supply and the hydrological process is more sensitive to precipitation in the north, whereas the water supply is abundant relative to the energy supply and sun duration has a more significant impact in the south (Cong et al., 2010). The NICE is effective to provide better evaluation of hydrological trends in longer period including 'evaporation paradox' (Roderick and Farquhar, 2002; Cong et al., 2010) together with observation networks because the model does not need the crop coefficient (depending on a growing stage and a kind of crop) for the calculation of actual evaporation and simulates it directly without detailed site-specific information or empirical relation to calculate effective

Nakayama, 2011a, 2011b). Because the dry-up of river reaches and groundwater exhaustion have been very severe so far (Chen et al., 2003b; Xia et al., 2004; Yang et al., 2004a), it is urgently necessary to perform effective control of water diversions (Liu and Xia, 2004; Liu and Zheng, 2004); the results simulated by NICE can be taken as strong indicatives of the seriousness of the situation. There are some reasons for the gap between irrigation water use (Fig. 5a) and groundwater level distribution (Fig. 6b). Firstly, the simulated levels have a temporally averaged distribution, and it takes some time for water levels to reach equilibrium after the boundary conditions have changed. This in turn affects the replenishment of groundwater from adjacent regions in addition to the heterogeneity of three-dimensional groundwater flow. Secondly, irrigation water is drawn not only from groundwater but also from river, and the ratio of river to total irrigation changes spatiotemporally in the basin (Fig. 5b); more river irrigation is drawn in the upper, and most of the irrigation depends on groundwater in the downstream, particularly in the NCP. This effect is clearly evident in comparison with the simulated results and the degradation value in the

Though the simulation reproduced reasonably hydrologic cycle such as evapotranspiration (Fig. 4), irrigation water use (Fig. 5a and Table 3), groundwater level, and river discharge (Fig. 5b and Fig. 6a), there were some discrepancies due to very complex and inaccurate nature of water withdrawal in the basin. In particular, the model achieved a relatively reasonable agreement though the model tried to calibrate and validate irrigation water use during only two years against other studies focusing on irrigation during long period (Fig. 5a and Table 3), which might lead to a substantial bias on model parameters. Because the objective of this study is primarily to evaluate the complex hydrological processes and reveal the impact of irrigation on hydrologic cycle in the basin through the verification during a fixed period, it is a future work to run model for the long period in the next step. At the same time, it will be of importance to derive better estimates of water demand in agricultural and urban areas during the long period by using more detailed statistical data, GIS data, and satellite data in longer period. Although the geological structure in the model included the general characteristics of several aquifers by reference to previous literature (Geological Atlas of China, 2002), the detailed structure of each aquifer layer was simplified as much as possible (Nakayama and Watanabe, 2008b; Nakayama et al., 2006). It will be necessary to obtain more precise data for the complex channel geometry of both natural and artificial rivers, soil properties, and geological structure. The spatial and temporal resolution used in the simulation also requires further improvement in order to overcome the problem of scale dependence and to improve verification and future reliability (Nakayama, 2011b). Simulated results about the impact of irrigation on evapotranspiration change show a heterogeneous distribution (Fig. 4a-b). In particular, the irrigation of winter wheat increases greatly evapotranspiration, which is supplied by the limited water resources of river discharge and groundwater there (Fig. 5). This implies that energy supply is abundant relative to the water supply and the hydrological process is more sensitive to precipitation in the north, whereas the water supply is abundant relative to the energy supply and sun duration has a more significant impact in the south (Cong et al., 2010). The NICE is effective to provide better evaluation of hydrological trends in longer period including 'evaporation paradox' (Roderick and Farquhar, 2002; Cong et al., 2010) together with observation networks because the model does not need the crop coefficient (depending on a growing stage and a kind of crop) for the calculation of actual evaporation and simulates it directly without detailed site-specific information or empirical relation to calculate effective

downstream is smaller that that in Fig. 6b.

precipitation (Nakayama, 2011a; Nakayama et al., 2006). It is further necessary to clarify feedback and inter-relationship between micro, regional, and global scales; Linkage with global-scale dynamic vegetation model including two-way interactions between seasonal crop growth and atmospheric variability (Bondeau et al., 2007; Oleson et al., 2008); From stochastic to deterministic processes towards relationship between seedling establishment, mortality, and regeneration, and growth process based on carbon balance (Bugmann et al., 1996); From CERES-DSSAT to generic (hybrid) crop model by combinations of growthdevelopment functions and mechanistic formulation of photosynthesis and respiration (Yang et al., 2004b); Improvement of nutrient fixation in seedlings, growth rate parameter, and stress factor, etc. for longer time-scale (Hendrickson et al., 1990). These future works might make a great contribution to the construction of powerful strategy for climate change problems in global scale.

Importance is that authority for water management in the basin is delineated by water source (surface water or groundwater) in addition to topographic boundaries (basin) and integrated water management concepts. In China, surface water and groundwater are managed by different authorities; the Ministry of Water Resources is responsible for surface water, while groundwater is considered a mineral resource and is administered by the Ministry of Minerals. In order to manage water resources effectively, any change in water accounting procedures may need to be negotiated through agreements brokered at relatively high levels of government, because surface water and groundwater are physically closely related to each other. Furthermore, the future development of irrigated and unirrigated fields and the associated crop production would affect greatly hydrologic change and usable irrigation water from river and aquifer, and vice versa (Nakayama, 2011b). The changes seen in this water resource are also related to climate change because groundwater storage moderates basin responses and climate feedback through evapotranspiration (Maxwell and Kollet, 2008). This is also related to a necessity of further evaluation about the evaporation paradox as described in the above. Although the groundwater level has decreased rapidly mainly due to overexploitation in the middle and downstream (Nakayama et al., 2006; Nakayama, 2011a, 2011b), regions where the land surface energy budget is very sensitive to groundwater storage are dominated by a critical water level (Kollet and Maxwell, 2008). The predicted hydrologic change indicates heterogeneous vulnerability of water resources and implies the associated impact on climate change (Fig. 6).

Basin responses will also be accelerated by an ambitious project to divert water from the Changjiang to the Yellow River, so-called, the South-to-North Water Transfer Project (SNWTP) (Rich, 1983; Yang and Zehnder, 2001). It can be estimated that the degradation of crop productivity may become severe, because most of the irrigation is dependent on vulnerable water resources (McVicar et al., 2002). Further research is necessary to examine the optimum amount of water that can be transferred, the effective management of the Three Gorges Dam (TGD) in the Changjiang River, the overall economic and social consequences of both projects, and their environmental assessment. It will be further necessary to obtain more observed and statistical data relating to water level, soil and water temperatures, water quality, and various phenological characteristics and crop productivity of spring/winter wheat and summer maize, in addition to satellite data of higher spatiotemporal resolution describing the seasonal and spatial vegetation phenology more accurately. The linear relationship between evapotranspiration and biomass production, which is very conservative and physiologically determined, is also valuable for further evaluation of the relationship between changes in water use and crop production by coupling with the numerical simulation and the satellite data analysis. Furthermore, it is powerful to develop a more realistic mechanism for sub-models, and to predict future hydrologic cycle and associated climate change using the model in order to achieve sustainable development under sound socio-economic conditions.

## **4. Conclusion**

This study coupled National Integrated Catchment-based Eco-hydrology (NICE) model series with complex sub-models involving various factors, and clarified the importance of and diverse water system in the highly cultivated Yellow River Basin, including hydrological processes such as river dry-up, groundwater deterioration, agricultural water use, et al. The model includes different functions of representative crops (wheat, maize, soybean, and rice) and simulates automatically dynamic growth processes and biomass formulation. The model reproduced reasonably evapotranspiration, irrigation water use, groundwater level, and river discharge during spring/winter wheat and summer maize cultivations. Scenario analysis predicted the impact of irrigation on both surface water and groundwater, which had previously been difficult to evaluate. The simulated discharge with irrigation was improved in terms of mean value, standard deviation, and coefficient of variation. Because this region has experienced substantial river dry-up and groundwater degradation at the end of the 20th century, this approach would help to overcome substantial pressures of increasing food demand and declining water availability, and to decide on appropriate measures for whole water resources management to achieve sustainable development under sound socio-economic conditions.

## **5. Acknowledgment**

The author thanks Dr. Y. Yang, Shijiazhuang Institute of Agricultural Modernization of the Chinese Academy of Sciences (CAS), China, and Dr. M. Watanabe, Keio University, Japan, for valuable comments about the study area. Some of the simulations in this study were run on an NEC SX–6 supercomputer at the Center for Global Environmental Research (CGER), NIES. The support of the Asia Pacific Environmental Innovation Strategy (APEIS) Project and the Environmental Technology Development Fund from the Japanese Ministry of Environment is also acknowledged.

## **6. References**


which is very conservative and physiologically determined, is also valuable for further evaluation of the relationship between changes in water use and crop production by coupling with the numerical simulation and the satellite data analysis. Furthermore, it is powerful to develop a more realistic mechanism for sub-models, and to predict future hydrologic cycle and associated climate change using the model in order to achieve

This study coupled National Integrated Catchment-based Eco-hydrology (NICE) model series with complex sub-models involving various factors, and clarified the importance of and diverse water system in the highly cultivated Yellow River Basin, including hydrological processes such as river dry-up, groundwater deterioration, agricultural water use, et al. The model includes different functions of representative crops (wheat, maize, soybean, and rice) and simulates automatically dynamic growth processes and biomass formulation. The model reproduced reasonably evapotranspiration, irrigation water use, groundwater level, and river discharge during spring/winter wheat and summer maize cultivations. Scenario analysis predicted the impact of irrigation on both surface water and groundwater, which had previously been difficult to evaluate. The simulated discharge with irrigation was improved in terms of mean value, standard deviation, and coefficient of variation. Because this region has experienced substantial river dry-up and groundwater degradation at the end of the 20th century, this approach would help to overcome substantial pressures of increasing food demand and declining water availability, and to decide on appropriate measures for whole water resources management to achieve

The author thanks Dr. Y. Yang, Shijiazhuang Institute of Agricultural Modernization of the Chinese Academy of Sciences (CAS), China, and Dr. M. Watanabe, Keio University, Japan, for valuable comments about the study area. Some of the simulations in this study were run on an NEC SX–6 supercomputer at the Center for Global Environmental Research (CGER), NIES. The support of the Asia Pacific Environmental Innovation Strategy (APEIS) Project and the Environmental Technology Development Fund from the Japanese Ministry of

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## **Estimation of Evapotranspiration Using Soil Water Balance Modelling**

Zoubeida Kebaili Bargaoui *Tunis El Manar University Tunisia* 

## **1. Introduction**

146 Evapotranspiration – Remote Sensing and Modeling

Zhu, Y. (1992). *Comprehensive hydro-geological evaluation of the Huang-Huai-Hai Plain*,

Assessing evapotranspiration is a key issue for natural vegetation and crop survey. It is a very important step to achieve the soil water budget and for deriving drought awareness indices. It is also a basis for calculating soil-atmosphere Carbon flux. Hence, models of evapotranspiration, as part of land surface models, are assumed as key parts of hydrological and atmospheric general circulation models (Johnson et al., 1993). Under particular climate (represented by energy limiting evapotranspiration rate corresponding to potential evapotranspiration) and soil vegetation complex, evapotranspiration is controlled by soil moisture dynamics. Although radiative balance approaches are worth noting for evapotranspiration evaluation, according to Hofius (2008), the soil water balance seems the best method for determining evapotranspiration from land over limited periods of time. This chapter aims to discuss methods of computing and updating evapotranspiration rates using soil water balance representations.

At large scale, Budyko (1974) proposed calculating annual evapotranspiration from data of meteorological stations using one single parameter w0 representing a critical soil water storage. Using a statistical description of the sequences of wet and dry days, Eagleson (1978 a) developed an average annual water balance equation in terms of 23 variables including soil, climate and vegetation parameters with the assumption of a homogeneous soilatmosphere column using Richards (1931) equation. On the other hand, the daily bucket with bottom hole model (BBH) proposed by Kobayashi et al. (2001) was introduced based on Manabe model (1969) involving one single layer bucket but including gravity drainage (leakage) as well as capillary rise. Vrugt et al. (2004) concluded that the daily Bucket model and the 3-D model (MODHMS) based on Richards equation have similar results. Also, Kalma & Boulet (1998) compared simulation results of the rainfall runoff hydrological model VIC which assumes a bucket representation including spatial variability of soil parameters to the one dimensional physically based model SiSPAT (Braud et al. , 1995). Using soil moisture profile data for calibration, they conclude that catchment's scale wetness index for very dry and very wet periods are misrepresented by SiSPAT while captured by VIC. Analyzing VIC parameter identifiability using streamflow data, DeMaria et al. (2007) concluded that soil parameters sensitivity was more strongly dictated by climatic gradients than by changes in soil properties especially for dry environments. Also, studying the measurements of soil moisture of sandy soils under semi-arid conditions, Ceballos et al. (2002) outlined the dependence of soil moisture time series on intra annual rainfall variability. Kobayachi et al. (2001) adjusted soil humidity profiles measurements for model calibration while Vrugt et al. (2004) suggested that effective soil hydraulic properties are poorly identifiable using drainage discharge data.

The aim of the chapter is to provide a review of evapotranspiration soil water balance models. A large variety of models is available. It is worth noting that they do differ with respect to their structure involving empirical as well as conceptual and physically based models. Also, they generally refer to soil properties as important drivers. Thus, the chapter will first focus on the description of the water balance equation for a column of soilatmosphere (one dimensional vertical equation) (section 2). Also, the unsaturated hydrodynamic properties of soils as well as some analytical solutions of the water balance equation are reviewed in section 2. In section 3, key parameterizations generally adopted to compute actual evapotranspiration will be reported. Hence, several soil water balance models developed for large spatial and time scales assuming the piecewise linear form are outlined. In section 4, it is focused on rainfall-runoff models running on smaller space scales with emphasizing on their evapotranspiration components and on calibration methods. Three case studies are also presented and discussed in section 4. Finally, the conclusions are drawn in section 5.

### **2. The one dimensional vertical soil water balance equation**

As pointed out by Rodriguez-Iturbe (2000) the soil moisture balance equation (mass conservation equation) is "likely to be the fundamental equation in hydrology". Considering large spatial scales, Sutcliffe (2004) might agree with this assumption. In section 2.1 we first focus on the presentation of the equation relating relative soil moisture content to the water balance components: infiltration into the soil, evapotranspiration and leakage. Then water loss through vegetation is addressed. Finally, infiltration models are discussed in section 2.2.

#### **2.1 Water balance**

For a control volume composed by a vertical soil column, the land surface, and the corresponding atmospheric column, and under solar radiation and precipitation as forcing variables, this equation relates relative soil moisture content *s* to infiltration into the soil I(s,t), evapotranspiration E(s,t) and leakage L(s,t).

$$\mathbf{nZ\_u} \circledast \mathbf{s} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{t} \mathbf{$$

Where t is time, n is soil effective porosity (the ratio of volume of voids to the total soil matrix volume); and Za is the active depth of soil.

Soil moisture exchanges as well as surface heat exchanges depend on physical soil properties and vegetation (through albedo , soil emissivity, canopy conductance) as well as atmosphere properties (turbulent temperature and water vapour transfer coefficients, aerodynamic conductance in presence of vegetation) and weather conditions (solar radiation, air temperature, air humidity, cloud cover, wind speed). Soil moisture measurements require sampling soil moisture content by digging or soil augering and determining soil moisture by drying samples in ovens and measuring weight losses; also, in situ use of tensiometry, neutron scattering, gamma ray attenuation, soil electrical conductivity analysis, are of common practice (Gardner et al. (2001) ; Sutcliffe, 2004; Jeffrey et al. (2004) ).

variability. Kobayachi et al. (2001) adjusted soil humidity profiles measurements for model calibration while Vrugt et al. (2004) suggested that effective soil hydraulic properties are

The aim of the chapter is to provide a review of evapotranspiration soil water balance models. A large variety of models is available. It is worth noting that they do differ with respect to their structure involving empirical as well as conceptual and physically based models. Also, they generally refer to soil properties as important drivers. Thus, the chapter will first focus on the description of the water balance equation for a column of soilatmosphere (one dimensional vertical equation) (section 2). Also, the unsaturated hydrodynamic properties of soils as well as some analytical solutions of the water balance equation are reviewed in section 2. In section 3, key parameterizations generally adopted to compute actual evapotranspiration will be reported. Hence, several soil water balance models developed for large spatial and time scales assuming the piecewise linear form are outlined. In section 4, it is focused on rainfall-runoff models running on smaller space scales with emphasizing on their evapotranspiration components and on calibration methods. Three case studies are also presented and discussed in section 4. Finally, the conclusions are

As pointed out by Rodriguez-Iturbe (2000) the soil moisture balance equation (mass conservation equation) is "likely to be the fundamental equation in hydrology". Considering large spatial scales, Sutcliffe (2004) might agree with this assumption. In section 2.1 we first focus on the presentation of the equation relating relative soil moisture content to the water balance components: infiltration into the soil, evapotranspiration and leakage. Then water loss through vegetation is addressed. Finally, infiltration models are discussed in section 2.2.

For a control volume composed by a vertical soil column, the land surface, and the corresponding atmospheric column, and under solar radiation and precipitation as forcing variables, this equation relates relative soil moisture content *s* to infiltration into the soil

 nZa st= I(s,t) – E(s,t) – L(s,t) (1a) Where t is time, n is soil effective porosity (the ratio of volume of voids to the total soil

Soil moisture exchanges as well as surface heat exchanges depend on physical soil properties and vegetation (through albedo , soil emissivity, canopy conductance) as well as atmosphere properties (turbulent temperature and water vapour transfer coefficients, aerodynamic conductance in presence of vegetation) and weather conditions (solar radiation, air temperature, air humidity, cloud cover, wind speed). Soil moisture measurements require sampling soil moisture content by digging or soil augering and determining soil moisture by drying samples in ovens and measuring weight losses; also, in situ use of tensiometry, neutron scattering, gamma ray attenuation, soil electrical conductivity analysis, are of common practice (Gardner et al. (2001) ; Sutcliffe, 2004; Jeffrey

**2. The one dimensional vertical soil water balance equation** 

poorly identifiable using drainage discharge data.

I(s,t), evapotranspiration E(s,t) and leakage L(s,t).

matrix volume); and Za is the active depth of soil.

drawn in section 5.

**2.1 Water balance** 

et al. (2004) ).

The basis of soil water movement has been experimentally proposed by Darcy in 1856 and expresses the average flow velocity in a porous media in steady-state flow conditions of groundwater. Darcy introduced the notion of hydraulic conductivity. Boussinesq in 1904 introduced the notion of specific yield so as to represent the drainage from the unsaturated zone to the flow in the water table. The specific yield is the flux per unit area draining for a unit fall in water table height. Richards (1931) proposed a theory of water movement in the unsaturated homogeneous bare soil represented by a semi infinite homogeneous column:

$$\left[\partial\theta/\partial t = \partial/\partial\mathbf{z}\left[\mathbf{K}\,\left\{\mathbf{K}\,\left\{\theta\psi/\partial\mathbf{z} - \mathbf{K}(\theta)\right\}\right\}\right]\right] \tag{1b}$$

Where t is time; is volumetric water content (which is the ratio between soil moisture volume and the total soil matrix volume cm3cm-3); z is the vertical coordinate (z>0 downward from surface); K is hydraulic conductivity (cms-1); is the soil water matrix potential. Both K and are function of the volumetric water content. Richards equation assumes that the effect of air on water flow is negligible. If accounting for the slope surface, it comes:

$$\text{R\ } \\$\theta\text{/\\$t=\\$/\\$z} \text{ [K } \\$\psi\text{/\\$}\theta\text{/\\$}\theta\text{/\\$z}\text{]} - \\$\text{ K/\\$}\theta\text{ } \\$\theta\text{/\\$z}\text{]}\cos\text{\\$}\tag{2}$$

Where is surface slope angle and cos is the cosinus function. We notice that the term [K /z – K()] represents the vertical moisture flux. In particular, as reported by Youngs (1988) the soil-water diffusivity parameter D has been proposed by Childs and Collis-George (1950) as key soil-water property controlling the water movement.

$$\mathbf{D}(\theta) = \,\,\mathbf{K}(\theta)\,\,\delta\psi\,\delta\theta\,\,\tag{3}$$

Thus, the Richards equation is often written as following:

tzDz–z

Eq. (4) is generally completed by source and sink terms to take into account the occurrence of precipitation infiltrating into the soil Inf(,z0) where z0 is the vertical coordinate at the surface and vegetation uptake of soil moisture gr(,z),. Vegetation uptake (transpiration) depends on vegetation characteristics (species, roots, leaf area, and transfer coefficients) and on the potential rate of evapotranspiration E0 which characterizes the climate. Consequently, Eq. (4) becomes:

$$\delta\Theta\theta\text{\%} = \delta\Diamond\text{z}\left[\text{D}(\theta)\text{\\$}\theta\text{\%}\text{z} - \text{K}(\theta)\right] - \text{g}\_{\text{z}}(\theta, \text{z}) + \text{I}\_{\text{nf}}(\theta, \text{z}\mathfrak{d})\tag{5}$$

Youngs (1988) noticed that near the soil surface where temperature gradients are important Richards equation may be inadequate. We find in Raats (2001) an important review of evapotranspiration models and analytical and numerical solutions of Richards equation. However, it should be noticed that after Feddes et al. (2001) "in case of catchments with complex sloping terrain and groundwater tables, a vertical domain model has to be coupled with either a process or a statistically based scheme that incorporates lateral water transfer". So, a key task in the soil water balance model evaluation is the estimation of Inf(,z0) and gr(,z). Both depend on the distribution of soil moisture. We focus here on vegetation uptake (or transpiration) gr(,z) which is regulated by stomata and is driven by atmospheric demand. Based on an Ohm's law analogy which was primary proposed by Honert in 1948 as outlined by Eagleson (1978 b), the conceptual model of local transpiration uptake u(z,t)= gr(,z) as volume of water per area per time is expressed as (Guswa, 2005)

$$\mathbf{u}(\mathbf{z}, \mathbf{t}) \equiv \Delta \mathbf{z} \left(\psi(\mathbf{z}, \mathbf{t}) \cdot \boldsymbol{\upmu}\_{\mathbb{P}}\right) / \left[\mathcal{R}\_{\mathbb{I}}(\boldsymbol{\theta} \text{ (}\mathbf{z}, \mathbf{t})) \mathbf{+} \mathbb{R}\_{\mathbb{Z}}\right] \tag{6}$$

soil moisture potential (bars), p leaf moisture potential (bars); R1 (s cm-1) a resistance to moisture flow in soil; it depends on soil and root characteristics and is function of the volumetric water content; R2 (s cm-1) is vegetation resistance to moisture flow; z is soil depth. It is worth noting that p > where is the wilting point potential; In Ceballos et al. (2002) the wilting point is taken as the soil-moisture content at a soil-water potential of - 1500 kPa.

Estimations of air and canopy resistances R1 and R2 often use semi-empirical models based on meteorological data such as wind speed as explanatory variables (Monteith (1965); Villalobos et al., 2000). Jackson et al. (2000) pointed out the role of the Hydraulic Lift process which is the movement of water through roots from wetter, deeper soil layers into drier, shallower layers along a gradient in . On the basis of such redistribution at depth, Guswa (2005) introduced a parameter to represent the minimum fraction of roots that must be wetted to the field capacity in order to meet the potential rate of transpiration. The field capacity is defined as the saturation for which gravity drainage becomes negligible relative to potential transpiration (Guswa, 2005). The potential matrix at field capacity is assumed equal to 330 hPa (330 cm) (Nachabe, 1998). The resulting u(z,t) function is strongly non linear versus the average root moisture with a relative insensitivity to changes in moisture when moisture is high and sensitivity to changes in moisture when the moisture is near the wilting point conditions. We also emphasize the Perrochet model (Perrochet, 1987) which links transpiration to potential evapotranspiration E0 through:

$$\mathbf{g}\_{\mathbf{f}}(\theta, \mathbf{z}, \mathbf{t}) \equiv \alpha(\theta) \mathbf{r}(\mathbf{z}) \to \mathfrak{z}(\mathbf{t}) \tag{7}$$

Where r(z) (cm-1) is a root density function which depends both on vegetation type and climatic conditions, (is the root efficiency function. Both r(z) and (represent macroscopic properties of the root soil system; they depend on layer thickness and root distribution . Lai and Katul (2000) and Laio (2006) reported some models assigned to r(z) which are linear or non linear. As out pointed by Laio (2006), models generally assume that vegetation uptake at a certain depth depends only on the local soil moisture. It is noticeable that in Feddes et al. (2001), a decrease of uptake is assumed when the soil moisture exceeds a certain limit and transpiration ceases for soil moisture values above a limit related to oxygen deficiency.

#### **2.2 Review of models for hydrodynamic properties of soils**

Many functional forms are proposed to describe soil properties evolution as a function of the volumetric water content (Clapp et al. , 1978). They are called retention curves or pedo transfer functions. We first present the main functional forms adopted to describe hydraulic parameters (section 2.2.1). Then, we report some solutions of Richards equation (section 2.2.2).

#### **2.2.1 Functional forms of soil properties**

According to Raats (2001), four classes of models are distinguishable for representing soil hydraulic parameters. Among them the linear form with D as constant and K linear with and the function Delta type as proposed by Green Ampt D= ½ s² (1 - 0)-1 (1 - 0) where s is the degree of saturation (which is the ratio between soil moisture volume and voids

 u(z,t)=z (z,t) -p) /[ R1( (z,t))+R2] (6) soil moisture potential (bars), p leaf moisture potential (bars); R1 (s cm-1) a resistance to moisture flow in soil; it depends on soil and root characteristics and is function of the volumetric water content; R2 (s cm-1) is vegetation resistance to moisture flow; z is soil depth. It is worth noting that p > where is the wilting point potential; In Ceballos et al. (2002) the wilting point is taken as the soil-moisture content at a soil-water potential of -

Estimations of air and canopy resistances R1 and R2 often use semi-empirical models based on meteorological data such as wind speed as explanatory variables (Monteith (1965); Villalobos et al., 2000). Jackson et al. (2000) pointed out the role of the Hydraulic Lift process which is the movement of water through roots from wetter, deeper soil layers into drier, shallower layers along a gradient in . On the basis of such redistribution at depth, Guswa (2005) introduced a parameter to represent the minimum fraction of roots that must be wetted to the field capacity in order to meet the potential rate of transpiration. The field capacity is defined as the saturation for which gravity drainage becomes negligible relative to potential transpiration (Guswa, 2005). The potential matrix at field capacity is assumed equal to 330 hPa (330 cm) (Nachabe, 1998). The resulting u(z,t) function is strongly non linear versus the average root moisture with a relative insensitivity to changes in moisture when moisture is high and sensitivity to changes in moisture when the moisture is near the wilting point conditions. We also emphasize the Perrochet model (Perrochet, 1987) which

 gr(,z,t) = (r(z) E0(t) (7) Where r(z) (cm-1) is a root density function which depends both on vegetation type and climatic conditions, (is the root efficiency function. Both r(z) and (represent macroscopic properties of the root soil system; they depend on layer thickness and root distribution . Lai and Katul (2000) and Laio (2006) reported some models assigned to r(z) which are linear or non linear. As out pointed by Laio (2006), models generally assume that vegetation uptake at a certain depth depends only on the local soil moisture. It is noticeable that in Feddes et al. (2001), a decrease of uptake is assumed when the soil moisture exceeds a certain limit and transpiration ceases for soil moisture values above a limit related to

Many functional forms are proposed to describe soil properties evolution as a function of the volumetric water content (Clapp et al. , 1978). They are called retention curves or pedo transfer functions. We first present the main functional forms adopted to describe hydraulic parameters (section 2.2.1). Then, we report some solutions of Richards equation (section

According to Raats (2001), four classes of models are distinguishable for representing soil hydraulic parameters. Among them the linear form with D as constant and K linear with and the function Delta type as proposed by Green Ampt D= ½ s² (1 - 0)-1 (1 - 0) where s is the degree of saturation (which is the ratio between soil moisture volume and voids

links transpiration to potential evapotranspiration E0 through:

**2.2 Review of models for hydrodynamic properties of soils** 

**2.2.1 Functional forms of soil properties** 

1500 kPa.

oxygen deficiency.

2.2.2).

volume; s=1 in case of saturation) and 1; 0 parameters. Also power law functions for and K) are proposed by Brooks and Corey (1964) on the basis of experimental observations while Gardner (1958) assumes exponential functions. The power type model proposed by Brooks & Corey (1964) are the most often adopted forms in rainfall-runoff transformation models. The Brooks and Corey model for K and is written as:

$$\mathbf{K(s) = K(1) s \mathbf{s}^\circ ; \psi \text{ (s) = }} \psi \text{(1) s \text{-1/m} \tag{8}$$

where m is a pore size index and c' a pore disconnectedness index (Eagleson 1978 a,b); After Eagleson (1978a, b), c' is linked to m with c'=(2+3m)/m. In Eq. (8), K(1) is hydraulic conductivity at saturation (for s=1); (1) is the bubbling pressure head which represents matrix potential at saturation. During dewatering of a sample, it corresponds to the suction at which gas is first drawn from the sample; As a result, Brooks and Corey (*BC*) model for diffusivity is derived as:

$$\mathbf{D}(\theta) = \mathrm{s}^d \,\mathrm{\upmu(l)}\,\,\,\mathbf{K}(1) \,\,\mathrm{\upleft(rm\right)}\,\,\mathrm{\upleft(rm\right)}\tag{9}$$

where n is effective soil porosity; and d=(c'-1- (1/m)). Let's consider the intrinsic permeability k which is a soil property. (K and k are related by K= kw where dynamic viscosity of water; <sup>w</sup> specific weight of pore water). After Eagleson (1978 a, b), three parameters involved in pedo transfer functions may be considered as independent parameters: n, c' and k(1) where k(1) is intrinsic permeability at saturation.

On the other hand, Gardner (1958) model assumed the exponential form for the hydraulic conductivity parameter (Eq. 10):

$$\mathbf{K}(\psi) = \mathbf{K}\_{\mathbb{S}} \text{ e }^{\cdot \cdot \Psi} \tag{10}$$

Where KS saturated hydraulic conductivity at soil surface; *a'* pore size distribution parameter. Also, in Gardner (1958) model, the degree of saturation and the soil moisture potential are linked according to Eq. (11). The power function introduces a parameter *l* which is a factor linked to soil matrix tortuosity (*l*= 0.5 is recommended for different types of soils);.

$$\mathbf{s}(\boldsymbol{\psi}) = [\mathbf{e}^{\ \ 0.5 \ u' \ \mathbf{V}} \ (1 + 0.5 \ a' \ \boldsymbol{\psi} \ )]^{2/(0 \ \mathbf{V})} \tag{11}$$

Van Genutchen model (1980) is another kind of power law model but it is highly non linear

$$\mathbf{K}(\boldsymbol{\psi}) = \mathbf{K}\_{\mathbb{S}} \mathbf{s}^{\hat{\lambda}+1} \left[ \mathbf{1} \text{ - (1-s}^{(\hat{\lambda}+1)\hat{\lambda}} \right]^{\hat{\lambda}(\hat{\lambda}+1)} \right]^2 \tag{12}$$

$$\mathbf{s}(\boldsymbol{\psi}) = \left[1 + \left(\boldsymbol{\psi}(1)\,\boldsymbol{\psi}\right)\right]^{-(\lambda+1)} \mathbf{l}^{\lambda\,(\lambda+1)} \text{ for } \boldsymbol{\psi} \lessapprox \boldsymbol{\psi}(1);$$

$$\mathbf{s} = \mathbf{1} \tag{13}$$

In Eq. (12) and (13) is a parameter to be calibrated. Calibration is generally performed on the basis of the comparison of computed and observed retention curves. In order to determine KS one way is to adopt Cosby et al. (1984) model (Eq. 14).

$$\text{Log(Ks)} = -0.6 + (0.0126 \,\text{S}\_{\%} - 0.0064 \,\text{C}\_{\%}) \tag{14}$$

Where S% and C% stand for soil percents of sand and clay. Also, we may find tabulated values of KS (in m/day) according to soil texture and structure properties in FAO (1980). On the other hand, soil field capacity SFC plays a key role in many soil water budget models. In Ceballos et al. (2002) the field capacity was considered as "the content in humidity corresponding to the inflection point of the retention curve before it reached a trend parallel to the soil water potential axis". In Guswa (2005), it is defined as the saturation for which gravity drainage becomes negligible relative to potential transpiration. As pointed out by Liao (2006) who agreed with Nachabe (1998), there is an "intrinsic subjectivity in the definition of field capacity". Nevertheless, many semi-empirical models are offered in the literature for SFC estimation as a function of soil properties (Nachabe, 1988). In Cosby (1984), SFC expressed as a degree of saturation is assumed s:

$$\mathbf{S\_{FC}} = \mathbf{50.1} + \text{(-0.142 S\_{\%} - 0.037 C\_{\%})} \tag{15}$$

On the other hand, according to Cosby (1984) and Saxton et al. (1986) SFC may be derived as:

$$\mathbf{S\_{FC}} = (20/\text{A}')^{1/\text{B}'} \tag{16}$$

where

A'=100\*exp(a1+a2C%+a3S%2+a4S%2C%); B'=a5+a6C%2+a7S%2+a8S%2C%; a1= - 4,396; a2 = - 0,0715; a3 = - 0,000488; a4 = -0,00004285; a5 = -0,00222; a6 = -0,00222 ; a7 = -0,00003484; a8 = -0,00003484 Recently, this model was adopted by Zhan et al. (2008) to estimate actual evapotranspiration in eastern China using soil texture information. Also, soil characteristics such as SFC may be obtained from Rawls & Brakensiek (1989) according to soil classification (Soil Survey Division Staff, 1998). Nasta et al. (2009) proposed a method taking advantage of the similarity between shapes of the particle-size distribution and the soil water retention function and adopted a log-Normal Probability Density Function to represent the matrix pressure head function retention curve.

#### **2.2.2 Review of analytical solutions of the movement equation**

Two well-known solutions of Richards equation are reported here (Green &Ampt model (1911), Philip model (1957)) as well as a more recent solution proposed by Zhao and Liu (1995). These solutions are widely adopted in rainfall-runoff models to derive infiltration.

In the Green &Ampt method (1911), it is assumed that infiltration capacity f from a ponded surface is:

$$\mathbf{f} = \mathbf{K}\_{\text{av}} \left( \mathbf{1} + \Lambda \boldsymbol{\mu} \,\,\Delta \boldsymbol{\theta} \,\, \mathbf{F}^{-1} \right) \tag{17}$$

av average saturated hydraulic conductivity ; difference in average matrix potential before and after wetting; difference in average soil water content before and after wetting; F the cumulative infiltration for a rainfall event (with f = dF/dt).

In the Philip (1957) solution, it is assumed that the gravity term is negligible so that K()/z]≈0. A time series development considers the soil water profile of the form:

$$\mathbf{z}(\theta, \mathbf{t}) = \mathbf{f}\_1 \text{ (\(\theta\)} \text{ t}^{1/2} + \mathbf{f}\_2 \text{ (\(\theta\)} \text{ t} + \mathbf{f}\_3 \text{ (\(\theta\)} \text{ t}^{3/2} + \dots \text{ } \tag{18}$$

Where f1, f2, … are functions of . Hence, the cumulative infiltration f (t) is:

$$\mathbf{Q}\_{\mathbf{i}}\mathbf{Q}\_{\mathbf{i}}\mathbf{(t)} = \mathbf{S}\_{\mathbf{i}}\mathbf{t}^{1/2} + (\mathbf{A}\_{\mathbf{2}} + \mathbf{K}\_{\mathbf{S}})\mathbf{t} + \mathbf{A}\_{\mathbf{3}}\mathbf{t}^{3/2} + \dots \tag{19}$$

Where S soil sorptivity, KS is saturated hydraulic conductivity of the soil and A1, A2, … are parameters. Philip suggested adopting a truncation that results in:

the other hand, soil field capacity SFC plays a key role in many soil water budget models. In Ceballos et al. (2002) the field capacity was considered as "the content in humidity corresponding to the inflection point of the retention curve before it reached a trend parallel to the soil water potential axis". In Guswa (2005), it is defined as the saturation for which gravity drainage becomes negligible relative to potential transpiration. As pointed out by Liao (2006) who agreed with Nachabe (1998), there is an "intrinsic subjectivity in the definition of field capacity". Nevertheless, many semi-empirical models are offered in the literature for SFC estimation as a function of soil properties (Nachabe, 1988). In Cosby (1984),

 SFC = 50.1 + (-0.142 S% - 0.037 C%) (15) On the other hand, according to Cosby (1984) and Saxton et al. (1986) SFC may be derived as:

SFC= (20/A')1/B' (16)

A'=100\*exp(a1+a2C%+a3S%2+a4S%2C%); B'=a5+a6C%2+a7S%2+a8S%2C%; a1= - 4,396; a2 = - 0,0715; a3 = - 0,000488; a4 = -0,00004285; a5 = -0,00222; a6 = -0,00222 ; a7 = -0,00003484; a8 = -0,00003484 Recently, this model was adopted by Zhan et al. (2008) to estimate actual evapotranspiration in eastern China using soil texture information. Also, soil characteristics such as SFC may be obtained from Rawls & Brakensiek (1989) according to soil classification (Soil Survey Division Staff, 1998). Nasta et al. (2009) proposed a method taking advantage of the similarity between shapes of the particle-size distribution and the soil water retention function and adopted a log-Normal Probability Density Function to represent the matrix

Two well-known solutions of Richards equation are reported here (Green &Ampt model (1911), Philip model (1957)) as well as a more recent solution proposed by Zhao and Liu (1995). These solutions are widely adopted in rainfall-runoff models to derive

In the Green &Ampt method (1911), it is assumed that infiltration capacity f from a ponded

av average saturated hydraulic conductivity ; difference in average matrix potential before and after wetting; difference in average soil water content before and after

In the Philip (1957) solution, it is assumed that the gravity term is negligible so that

z(,t) = f1 () t 1/2+ f2 () t + f3 () t 3/2 +… (18)

Where S soil sorptivity, KS is saturated hydraulic conductivity of the soil and A1, A2, … are

K()/z]≈0. A time series development considers the soil water profile of the form:

f av ( 1 + F) (17)

f (t)= S t1/2 + (A 2 +KS) t + A 3 t 3/2 + … (19)

SFC expressed as a degree of saturation is assumed s:

pressure head function retention curve.

**2.2.2 Review of analytical solutions of the movement equation** 

wetting; F the cumulative infiltration for a rainfall event (with f = dF/dt).

Where f1, f2, … are functions of . Hence, the cumulative infiltration f (t) is:

parameters. Philip suggested adopting a truncation that results in:

where

infiltration.

surface is:

$$
\Omega\_{\rm f} \text{(t)} = \mathbf{S} \text{ t}^{1/2} \,\, + \, \mathbf{K}\_{\rm S} / n' \,\, \text{t} \tag{20}
$$

Where *n'* is a factor 0.3 < *n'* < 0.7. It is worth noting that the soil sorptivity S depends on initial water content. So it has to be adjusted for each rainfall event. This is usually performed by comparing observed and simulated cumulative infiltration. For further discussion of Philip model, the reader may profitably refer to Youngs (1988).

Another model of infiltration is worth noting. It is the model of Zhao and Liu (1995) which introduced the fraction of area under the infiltration capacity:

$$\mathbf{i(t)} = \mathbf{i\_{max}} \left[ \mathbf{1-(1-A(t))^{1/b'}} \right] \tag{21}$$

Where i(t) is infiltration capacity at time t. Its maximum value is imax. A(t) is the fraction of area for which the infiltration capacity is less than i(t) and b'' is the infiltration shape parameter. As out pointed by DeMaria et al. (2007), the parameter b'' plays a key role. Effectively, an increase in b'' results in a decrease in infiltration.

### **3. Review of various parameterizations of actual evapotranspiration**

Many early works on radiative balance combination methods for estimating latent heat using Penman – Monteith method (Monteith, 1965) were coupled with empirical models for representing the conductance of the soil-plant system (the conductance is the inverse function of the resistance). Based on observational evidence, these works have assumed a linear piecewise relation between volumetric soil moisture and actual evapotranspiration. Thus, several water balance models have been developed for large spatial and time scales assuming this piecewise linear form beginning from the work of Budyko in 1956 as pointed out by Manabe (1969)), Budyko (1974), Eagleson (1978 a, b), Entekhabi & Eagleson (1989) and Milly (1993). In fact, soil water models for computing actual evapotranspiration differ according to the time and space scales and the number of soil layers adopted as well as the degree of schematization of the water and energy balances. Moreover, specific canopy interception schemes, pedo transfer sub-models and runoff sub-models often distinguish between actual evapotranspiration schemes. Also, models differ by the consideration of mixed bare soil and vegetation surface conditions or by differencing between vegetation and soil cover. In the former, there is a separation between bare soil evapotranspiration and vegetation transpiration as distinct terms in the computation of evapotranspiration. In the following, we first present a brief review of land surface models which fully couple energy and mass transfers (section 3.1). Then, we make a general presentation of soil water balance models based on the actualisation of soil water storage in the upper soil zone assuming homogeneous soil (section 3.2).Further, it is focused on the estimation of long term actual evapotranspiration using approximation of the solution of the water balance model (section 3.3). In section 3.4, large scale soil water balance models (bucket schematization) are outlined with much more details. Finally a discussion is performed in section 3.5.

#### **3.1 Review of land surface models**

In Soil-Vegetation-Atmosphere-Transfer (SVAT) models or land surface models, energy and mass transfers are fully coupled solving both the energy balance (net radiation equation, soil heat fluxes, sensible heat fluxes, and latent heat fluxes) in addition to water movement equations. Usually this is achieved using small time scales (as for example one hour time increment). The specificity of SVAT models is to describe properly the role of vegetation in the evolution of water and energy budgets. This is achieved by assigning land type and soil information to each model grid square and by considering the physiology of plant uptake. Many SVAT models have been developed in the last 25 years. We may find in Dickinson and al. (1986) perhaps one of the first comprehensive SVAT models which was addressed to be used for General circulation modelling and climate modelling. It was called BATS (Biosphere-Atmosphere Transfer Scheme). It was able to compute surface temperature in response to solar radiation, water budget terms (soil moisture, evapotranspiration and, runoff), plant water budget (interception and transpiration) and foliage temperature. ISBA model (Noilhan et Mahfouf, 1996) was further developed in France and belongs to "simple models with mono layer energy balance combined with a bulk soil description" (after Olioso et al. (2002)). An example of using ISBA scheme is presented in Olioso et al. (2002). The following variables are considered: surface temperature, mean surface temperature, soil volumetric moisture at the ground surface, total soil moisture, canopy interception reservoir. The soil volumetric moisture at the ground surface is adopted to compute the soil evaporation while the total soil moisture is used to compute transpiration. The total latent heat is assumed as a weighted average between soil evaporation and transpiration using a weight coefficient depending on the degree of canopy cover. Canopy albedo and emissivity, vegetation Leaf area index LAI, stomatal resistance, turbulent heat and transfer coefficients are parameters of the energy balance equations. It is worth noting that soil parameters in temperature and moisture are computed using soil classification databases. Without loss of generality we briefly present the two layers water movement model adopted by Montaldo et al. (2001)

$$\delta\Theta\_{\mathbb{B}}/\delta\mathfrak{t} \equiv \mathrm{C}\_{1}/\left(\mathfrak{p}\_{\text{w}}\mathrm{d}\_{1}\right)\left[\mathrm{P}\_{\mathbb{B}}\cdot\mathrm{E}\_{\mathbb{B}}\right]\mathrm{-C}\_{2}/\mathfrak{r}\left[\Theta\_{\mathbb{B}}\cdot\Theta\_{\mathbb{B}^{\text{eq}}}\right] \qquad \qquad 0\leq\theta\_{\mathbb{B}}\leq\Theta\_{\mathbb{B}}\tag{22}$$

$$\begin{array}{cccc}\hline \end{array} \delta \Theta\_2 / \delta \mathbf{t} \equiv \mathbf{C}\_1 / \begin{pmatrix} \mathbf{p}\_{\text{w}} \mathbf{d}\_2 \end{pmatrix} \begin{bmatrix} \mathbf{P}\_{\text{g}} \ -\mathbf{E}\_{\text{g}} \ -\mathbf{E}\_{\text{tr}} - \mathbf{q}\_2 \end{bmatrix} \tag{23}$$

d1 and d2 depth of near surface and root zone soil layers; w density of the water; <sup>g</sup> and <sup>2</sup> volumetric water contents of near surface and root zone soil layers; geq equilibrium surface volumetric soil moisture content ideally describing a reference soil moisture for which gravity balances capillary forces such that no flow crosses the bottom of the near surface zone of depth d1; Pg precipitation infiltrating into the soil; Eg bare soil evaporation rate at the surface; Etr transpiration rate from the root zone of depth d2; q2 rate of drainage out of the bottom of the root zone; It is assumed to be equal to the hydraulic conductivity of the root zone at =2 ; C1 and C2 are parameters. In this model, the rescaling of the root zone soil moisture 2 seems to be highly recommended in order to achieve adequate prediction of <sup>g</sup> in comparison to observations (Montaldo et al. (2001)). Using an assimilation procedure, Montaldo et al. (2001) achieved overcoming misspecification of KS of two orders magnitude in the simulation of 2.

According to Franks et al. (1997), the calibration of SVAT schemes requires a large number of parameters. Also, field experimentations needed to calibrate these parameters are rather important. Moreover up scaling procedures are to be implemented. Boulet and al. (2000) argued that "detailed SVAT models especially when they exhibit small time and space steps are difficult to use for the investigation of the spatial and temporal variability of land surface fluxes".

increment). The specificity of SVAT models is to describe properly the role of vegetation in the evolution of water and energy budgets. This is achieved by assigning land type and soil information to each model grid square and by considering the physiology of plant uptake. Many SVAT models have been developed in the last 25 years. We may find in Dickinson and al. (1986) perhaps one of the first comprehensive SVAT models which was addressed to be used for General circulation modelling and climate modelling. It was called BATS (Biosphere-Atmosphere Transfer Scheme). It was able to compute surface temperature in response to solar radiation, water budget terms (soil moisture, evapotranspiration and, runoff), plant water budget (interception and transpiration) and foliage temperature. ISBA model (Noilhan et Mahfouf, 1996) was further developed in France and belongs to "simple models with mono layer energy balance combined with a bulk soil description" (after Olioso et al. (2002)). An example of using ISBA scheme is presented in Olioso et al. (2002). The following variables are considered: surface temperature, mean surface temperature, soil volumetric moisture at the ground surface, total soil moisture, canopy interception reservoir. The soil volumetric moisture at the ground surface is adopted to compute the soil evaporation while the total soil moisture is used to compute transpiration. The total latent heat is assumed as a weighted average between soil evaporation and transpiration using a weight coefficient depending on the degree of canopy cover. Canopy albedo and emissivity, vegetation Leaf area index LAI, stomatal resistance, turbulent heat and transfer coefficients are parameters of the energy balance equations. It is worth noting that soil parameters in temperature and moisture are computed using soil classification databases. Without loss of generality we briefly present the two layers water movement model adopted by Montaldo

gt= C1/ (wd1) [ Pg -Eg] –C2/ [g - geq] 0≤<sup>g</sup> ≤s (22)

2t= C1/ (wd2) [ Pg -Eg –Etr – q2] 0≤<sup>2</sup> ≤s (23)

d1 and d2 depth of near surface and root zone soil layers; w density of the water; <sup>g</sup> and <sup>2</sup> volumetric water contents of near surface and root zone soil layers; geq equilibrium surface volumetric soil moisture content ideally describing a reference soil moisture for which gravity balances capillary forces such that no flow crosses the bottom of the near surface zone of depth d1; Pg precipitation infiltrating into the soil; Eg bare soil evaporation rate at the surface; Etr transpiration rate from the root zone of depth d2; q2 rate of drainage out of the bottom of the root zone; It is assumed to be equal to the hydraulic conductivity of the root zone at =2 ; C1 and C2 are parameters. In this model, the rescaling of the root zone soil moisture 2 seems to be highly recommended in order to achieve adequate prediction of <sup>g</sup> in comparison to observations (Montaldo et al. (2001)). Using an assimilation procedure, Montaldo et al. (2001) achieved overcoming misspecification of KS of two orders magnitude

According to Franks et al. (1997), the calibration of SVAT schemes requires a large number of parameters. Also, field experimentations needed to calibrate these parameters are rather important. Moreover up scaling procedures are to be implemented. Boulet and al. (2000) argued that "detailed SVAT models especially when they exhibit small time and space steps are difficult to use for the investigation of the spatial and temporal variability of land

et al. (2001)

in the simulation of 2.

surface fluxes".

#### **3.2 Review of average long term evapotranspiration or "regional" evapotranspiration models**

Considering the soil water balance at monthly time scale, Budyko (1974) introduced one single parameter which is a critical soil water storage w0 corresponding to 1 m homogeneous soil depth. According to Budyko (1974), w0 is a regional parameter seasonally constant and essentially depending on the climate-vegetation complex. The main assumption is that monthly actual evapotranspiration starts from zero and is a piecewise linear function of the degree of saturation expressed as the ratio w/w0 where w is the actual soil water storage. Either, for w≥ w0 actual evapotranspiration is assumed at potential value E0.

Average annual water balance equation is also developed in Eagleson (1978 a) in terms of 23 variables (six for soil, six for climate and one for vegetation) with the assumption of a homogeneous soil-atmosphere column using Richards equation. Further, the behaviour of soil moisture in the upper soil zone (1 m deep or root zone) is expressed in terms of the following three independent soil parameters: effective porosity n, pore disconnectedness index c' and saturated hydraulic conductivity at soil surface KS while storm and inter storm net soil moisture flux are coupled to storm and inter storm Probability Density Functions. The average annual evapotranspiration Em is finally expressed as :

$$\mathbf{E\_m = f(E\_{ev}M\_{v}, k\_{v}) \ (E\_{pu} \cdot E\_{ra})} \tag{24}$$

J(.) evapotranspiration function; Epa average annual potential evapotranspiration; Era average annual surface retention; Ee exfiltration parameter as function of initial degree of saturation s0; kv plant coefficient. It is approximately equal to effective transpiring leaf surface per unit of vegetated land surface; Mv vegetation fraction of surface.

Further, Milly (1993) developed similar probabilistic approach for soil water storage dynamics based on Manabe model (Manabe, 1969). A key assumption is that the soil is of high infiltration capacity. The model adopts the so-called water holding capacity W0, which is a storage capacity parameter allowing the definition of the state "reservoir is full". For well developed vegetation, W0 is interpreted as the difference between the volumetric moisture contents θf of the soil at field capacity and the wilting point θw (W0=θf-θw). Furthermore, Milly (1994) adopted seasonally Poisson and exponential Probability Density Functions, together with seasonality of evapotranspiration forcing. To take into account horizontal large length scales, the spatial variability of water holding capacity W0 was introduced, adopting a Gamma Probability Density Function with mean Wm0. In total, the model involved only seven parameters: a dryness index EDI = P / ETP, the mean holding capacity of soil Wm0 and a shape parameter of the Gamma distribution,, mean storm arrival rate, and one measure of seasonality for respectively annual precipitation, potential evapotranspiration and storm arrival rate. Performing a comparison with observed annual runoff in US, it was found that the geographical distribution of calculated runoff shares at least qualitatively the large scale features of observed maps. In effect, 88% of the variance of grid runoff and 85% of the variance of grid evapotranspiration is reproduced by this model. However, it is outlined that the model presents failures within areas with elevation. Average annual precipitation and runoff over 73 large basins worldwide were also studied by (Milly and Dunne, 2002). Using precipitation and net radiation as independent variables, they compared observed mean runoff amounts to those computed by Turc-Pike and Budyko models. In northern Europe, they found a tendency for underestimation of observed evapotranspiration.

#### **3.3 Empirical model for estimating regional evapotranspiration**

Combining the water balance to the radiative balance at monthly scale, Budyko proposed an asymptotic solution in which Rn stands for average annual net radiation (which is the net energy exchange with the atmosphere equal to net radiation – sensible heat flux – latent heat flux), P average annual precipitation, Em average (long term) annual evapotranspiration, a function expressed in Eq. (26).

$$\mathbf{E\_m} \,/\mathbf{P} = \boldsymbol{\phi} \,\mathrm{(R\_n/P)}\tag{25}$$

$$\phi\_{\mathbf{(x)}} = \left[ \mathbf{x} \left( \tanh(\mathbf{x}^1) \right) \left( \mathbf{1} \cdot \cosh(\mathbf{x}) + \sinh(\mathbf{x}) \right) \right] \mathbf{l}^{1/2} \tag{26}$$

Where tanh(.) stands for hyperbolic tangent, cosh(.) hyperbolic cosines, sinh(.) hyperbolic sinus

According to Shiklomavov (1989) and Budyko (1974), Ol'dekop was the first to propose in 1911 an empirical formulation of the relationship between climate characteristics and water balance terms (rainfall and runoff) assuming the concept of « maximum probable evaporation» Emax and using the ratio P / Emax. According to Milly (1994), works of Budyko in 1948 resulted, on the basis of dimensional analysis, to propose the ratio Rn/P as radiative index of aridity. Conversely, the function (Eq. 26) was empirical and was derived assuming that in arid climate Em approaches P while it approaches Rn under humid climate.Budyko model was validated using 1200 watersheds world wild computing Em as the difference between average long term annual observed rainfall and annual observed runoff. Model accuracy is reflected by the fact that the ratio Em /P is simulated within a relative error of 10% (Budyko, 1974). However, larger discrepancy values are found for basins with important orography. Choudhury (1999) proposed to adopt Eq. (27) to derive :

$$\phi\_{\mathbf{(x)}} = (\mathbf{1} + \mathbf{x}^{\cdots})^{\mathbf{1}/\mathbf{v}} \tag{27}$$

where is a parameter depending of the basin characteristics. Milly et Dunne (2002) reported that =2.1 closely approximates Budyko model, while =2 corresponds to Turc-Pike model. According to Choudhury (1999), the more the basin area is large, the more is small and smaller is Em. =2.6 is recommended for micro-basins while =1.8 for large basins. According to Milly et Dunne (2002), it was found that for a large interval of watershed areas, =1.5 to 2.6.

Another approximation of Budyko model is the Hsuen Chun (1988) model (H.C.) introducing the ratio IDetp =E0/P and an empirical parameter k'.

$$\mathbf{E\_m = E\_0} \left[ \mathbf{ID\_{epp}} \,\mathrm{k}\,\,\,\right] \left( \mathbf{1} + \mathbf{ID\_{ep}} \,\mathrm{k}\,\right) \left[ \mathbf{1}/\mathrm{k}\,\,\right] \tag{28}$$

After Hsuen Chun (1988) the value *k'*=2.2 reproduces Budyko model results. According to Pinol et al. (1991), the adjusted values of k' are in the interval 1.03 <k'< 2.40. Also, they noticed that k' depends on the type of vegetation cover. After Donohue et al. (2007), Eq. (28) may be adopted for basins with area < 1000 Km² and series of at least 5 year length.

#### **3.4 Modeling of actual evapotranspiration for long time series and large scale applications**

Simple soil water balance models based on bucket schematization have been developed to fulfil the need to simulate long time series of water balance outputs allowing the calculation of actual evapotranspiration. We focus the review on the Manabe model (1969), the Rodriguez-Iturbe et al. (1999) model and the Bottom hole bucket model of Kobayachi et al. (2001).

## **3.4.1 Manabe bucket model**

In fact, the single layer single bucket model of Manabe (1969) takes a central place in large scale water budget modelling. It was proposed as part of the climate and ocean circulation model. This conceptual model runs at the monthly scale and adopts the field capacity SFC as key parameter. Also, it assumes an effective parameter Wk representing a fraction of the field capacity (Wk = 0.75\* SFC). Here we notice that the field capacity SFC is now expressed as a water content. The climatic forcing is represented by the potential evapotranspiration E0. Let w be the actual soil water content. The actual evapotranspiration Ea is expressed as a linear piecewise function:

For w≥Wk Ea= E0

156 Evapotranspiration – Remote Sensing and Modeling

Combining the water balance to the radiative balance at monthly scale, Budyko proposed an asymptotic solution in which Rn stands for average annual net radiation (which is the net energy exchange with the atmosphere equal to net radiation – sensible heat flux – latent heat flux), P average annual precipitation, Em average (long term) annual evapotranspiration, a

Em /P = (Rn/P) (25)

 (x) = [x (tanh(x-1)) (1 - cosh(x) + sinh(x)) ]1/2 (26) Where tanh(.) stands for hyperbolic tangent, cosh(.) hyperbolic cosines, sinh(.) hyperbolic

According to Shiklomavov (1989) and Budyko (1974), Ol'dekop was the first to propose in 1911 an empirical formulation of the relationship between climate characteristics and water balance terms (rainfall and runoff) assuming the concept of « maximum probable evaporation» Emax and using the ratio P / Emax. According to Milly (1994), works of Budyko in 1948 resulted, on the basis of dimensional analysis, to propose the ratio Rn/P as radiative index of aridity. Conversely, the function (Eq. 26) was empirical and was derived assuming that in arid climate Em approaches P while it approaches Rn under humid climate.Budyko model was validated using 1200 watersheds world wild computing Em as the difference between average long term annual observed rainfall and annual observed runoff. Model accuracy is reflected by the fact that the ratio Em /P is simulated within a relative error of 10% (Budyko, 1974). However, larger discrepancy values are found for basins with important orography. Choudhury (1999) proposed to adopt Eq. (27) to derive :

where is a parameter depending of the basin characteristics. Milly et Dunne (2002) reported that =2.1 closely approximates Budyko model, while =2 corresponds to Turc-Pike model. According to Choudhury (1999), the more the basin area is large, the more is small and smaller is Em. =2.6 is recommended for micro-basins while =1.8 for large basins. According to Milly et Dunne (2002), it was found that for a large interval of watershed areas,

Another approximation of Budyko model is the Hsuen Chun (1988) model (H.C.)

 Em=E0 [IDetp k' / (1+ IDetpk')]1/k' (28) After Hsuen Chun (1988) the value *k'*=2.2 reproduces Budyko model results. According to Pinol et al. (1991), the adjusted values of k' are in the interval 1.03 <k'< 2.40. Also, they noticed that k' depends on the type of vegetation cover. After Donohue et al. (2007), Eq. (28)

Simple soil water balance models based on bucket schematization have been developed to fulfil the need to simulate long time series of water balance outputs allowing the calculation of actual evapotranspiration. We focus the review on the Manabe model (1969), the

may be adopted for basins with area < 1000 Km² and series of at least 5 year length.

**3.4 Modeling of actual evapotranspiration for long time series and large scale** 

)-1/

**3.3 Empirical model for estimating regional evapotranspiration** 

(x) = (1+x –

introducing the ratio IDetp =E0/P and an empirical parameter k'.

function expressed in Eq. (26).

sinus

=1.5 to 2.6.

**applications** 

For w<Wk Ea= E0\*(w/Wk)

On the other hand, the surface runoff Rs component in Manabe model depends on the actual soil moisture content in comparison to the field capacity as well as on the precipitation forcing compared to the potential evapotranspiration uptake. Let ∆w the change in soil water content. Thus, surface runoff is assumed as following:

For w= SFC and P> E0; ∆w=0 and Rs= P- E0

For w< SFC ; ∆w=P-Ea; Rs=0

Another well-known model is FAO-56 model (Allen et al. (1998)). In fact, it is based on Manabe soil water budget. However, it takes into account the water stress through an empirical coefficient K's. First of all, in FAO-56 model, it is important to outline that the potential evapotranspiration is replaced by a reference evapotranspiration Er computed using Penman-Montheith model with respect to a reference grass corresponding to an albedo value equal 0.23. Then, a seasonal crop coefficient Kc is introduced. The parameter Kc depends on both the crop type and the vegetative stage. Default Kc values are reported in (Allen et al. (1998)) for various crop types. This crop coefficient corresponds to ideal soil moisture conditions related to no water stress conditions and to good biological conditions. In real conditions, Kc is corrected by a correction coefficient K's (0<K's <1) such that the product Kc K's includes the vegetation type as well as the water stress conditions. So actual evapotranspiration is written as:

$$\mathbf{E}\_a = \mathbf{K}\_c \, \mathbf{K}'\_s \, \mathbf{E}\_r \tag{29}$$

According to Biggs et al. (2008) mild stress conditions would correspond to K's of 0.8 and moderate stress conditions to K's of 0.6. Based on the findings that default Kc values underestimate lysimeter experiments Kc values, Biggs et al. (2008) built a non linear regression relationships between the product (Kc K's) and the ratio of seasonal precipitation to potential evapotranspiration for various crop types. To that purpose they fitted a Beta Probability Density Function to the correction factor K's. They adopted lysimeter observations to fit this modified FAO-56 model.. The model explained (49–90%) of the variance in actual evapotranspiration, depending on the crop type.

### **3.4.2 Rodriguez-Iturbe model**

In Rodriguez-Iturbe et al. (1999), the point of departure is infiltration into the soil which is expressed as function of the existing soil moisture which is reported in terms of saturation (corresponding to s= w/nZa where Za is effective depth of soil and n soil effective porosity). Soil drainage varies according to a power law although it is approximated by two linear segments. Consequently, it is assumed that soil drainage occurs for s exceeding a threshold value s1, going from zero for s=s1 to KS for saturated condition (s=1) where KS is the saturated hydraulic conductivity of the soil. Moreover, a saturation threshold s\* is assumed to reduce evapotranspiration in case of water stress. Its value depends on the type of vegetation. Thus, for s≤s\*, the evapotranspiration is computed as the potential rate scaled by the ratio s/s\* while the evapotranspiration is at potential value for s> s\*.

$$\mathbf{E}\_{\mathbf{a}}(\mathbf{s}) = \mathbf{E}\_{\mathbf{0}} \mathbf{s} / \mathbf{s}^\* \qquad \text{For } \mathbf{s} \mathbf{s} \mathbf{s}^\* \tag{30}$$

$$\mathbf{E\_{a}(s)=E\_{0}}\qquad\text{For }s\succ s^{\*}\tag{31}$$

Milly (2001) model corresponds to the case s\* → 0 and KS → infinity. According to Milly (2001), the introduction of the threshold parameter s\* is much recommended especially under arid conditions. In the case where no distinction is made between forested and bare soil areas, Rodriguez-Iturbe et al. (1999) pointed out that s\* is considerably lower than the field capacity SFC conversely to Manabe model which corresponds to s\* = 0.75 SFC. Laio (2006) adopted a generalized form of Rodriguez-Iturbe et al. (1999) model by accounting for the reduction of evapotranspiration in case of water stress by introducing the soil moisture at wilting point sw. He represented s\* as a soil moisture level above which plant stomata are completely opened (Eq. 32 and Eq. 33).

$$\mathbf{E\_{a}(s) = E\_{0} \ (s \text{-} s\_{\text{w}}) / (s^{\ast} \cdot s\_{\text{w}})} \qquad \text{For } s \lesssim s^{\ast} \tag{32}$$

$$\mathbf{E\_a(s)=E\_0} \qquad \text{For } \mathbf{S\_{FC}} \rhd \mathbf{s} \rhd \mathbf{s}^\* \tag{33}$$

On the other hand, Rodriguez-Iturbe et al. (1999) model the leakage component is represented by the exponential decay Gardner model. This model was also adopted by Guswa et al. (2002). Leakage component is assumed as exponential decay function of the effective degree of soil saturation, as well as soil characteristics (saturated hydraulic conductivity, drainage curve parameter and field capacity).

#### **3.4.3 Bottom hole bucket model**

The daily bucket with bottom hole model (BBH) proposed by Kobayashi et al. (2001) is also based on Manabe model involving one layer bucket but including gravity drainage (leakage) as well as capillary rise. Kobayashi et al. (2001) outlined that the soil moisture dynamics is better simulated by BBH than by Bucket (Manabe) model. Kobayashi et al. (2007) developed a new version of BBH named BBH-B including a second soil layer in order to take into account for the variability of the soil profile when the root zone is rather deep (1 m or more).

In the following, we focus on BBH model where forcing variables are precipitation P and potential evapotranspiration E0. The actual evapotranspiration is assumed as:

$$\begin{array}{ll} \mathbf{E\_{a} = M \; \prime \to 0} & \text{For s\; \mathbf{s} \; \approx \mathbf{s}\*}\\\\ \mathbf{E\_{a} = E\_{0}} & \text{For s\; \mathbf{s}\*} \mathbf{s}\* \end{array} \tag{34}$$

Where M' is a water stress factor updated at each time step and expressed as:

$$\mathbf{M}' \equiv \text{Min}\left(1, \mathbf{w} / (\sigma \mathbf{W}\_{\text{max}})\right) \qquad \text{For s\!\!/} \mathbf{s}\*\tag{35}$$

parameter representing the resistance of vegetation to evapotranspiration; Wmax=nZa where Wmax: total water-holding capacity (mm); Za: thickness of active soil layer (mm); n: effective soil porosity.

Percolation and capillary rise term Gd(t) is assumed according to exponential function.

$$\text{Gd(t)=exp}\left(\left(\mathbf{w(t)-a}\right)/\mathbf{b}\right)\text{-c}\tag{36}$$

Where a: parameter related to the field capacity (mm); b: parameter representing the decay of soil moisture (mm); c: parameter representing the daily maximal capillary rise (mm). On the other hand, daily surface runoff Rs(t) is expressed as:

$$\text{Rs(t)} \equiv \text{Max } \left[ \text{P(t)-} (\text{W}\_{\text{BC}} \text{-} \text{W(t)}) \text{-E}\_{\text{a}} (\text{t)-} \text{Gd(t)}, 0 \right] \tag{37}$$

Where WBC= η Wmax; η : parameter representing the moisture retaining capacity (0< η <1). According to Kobayachi and al. (2001) the parameter a (which corresponds here to a/Wmax) is "nearly equal to or somewhat smaller than the field capacity". After Teshima et al. (2006), parameter b is a measure of soil moisture recession that depends on hydraulic conductivity and thickness of active soil layer Za. In Iwanaga et al. (2005), a sensitivity analysis of BBH model applied to an irrigated area in semi-arid region suggests that error soil moisture is most sensitive to and c.

#### **3.5 Discussion**

158 Evapotranspiration – Remote Sensing and Modeling

(corresponding to s= w/nZa where Za is effective depth of soil and n soil effective porosity). Soil drainage varies according to a power law although it is approximated by two linear segments. Consequently, it is assumed that soil drainage occurs for s exceeding a threshold value s1, going from zero for s=s1 to KS for saturated condition (s=1) where KS is the saturated hydraulic conductivity of the soil. Moreover, a saturation threshold s\* is assumed to reduce evapotranspiration in case of water stress. Its value depends on the type of vegetation. Thus, for s≤s\*, the evapotranspiration is computed as the potential rate scaled by

Ea(s)=E0 s/s\* For s≤s\* (30)

 Ea(s)=E0 For s>s\* (31) Milly (2001) model corresponds to the case s\* → 0 and KS → infinity. According to Milly (2001), the introduction of the threshold parameter s\* is much recommended especially under arid conditions. In the case where no distinction is made between forested and bare soil areas, Rodriguez-Iturbe et al. (1999) pointed out that s\* is considerably lower than the field capacity SFC conversely to Manabe model which corresponds to s\* = 0.75 SFC. Laio (2006) adopted a generalized form of Rodriguez-Iturbe et al. (1999) model by accounting for the reduction of evapotranspiration in case of water stress by introducing the soil moisture at wilting point sw. He represented s\* as a soil moisture level above which plant stomata are

Ea(s)=E0 (s-sw)/(s\*-sw) For s≤s\* (32)

 Ea (s)=E0 For SFC >s>s\* (33) On the other hand, Rodriguez-Iturbe et al. (1999) model the leakage component is represented by the exponential decay Gardner model. This model was also adopted by Guswa et al. (2002). Leakage component is assumed as exponential decay function of the effective degree of soil saturation, as well as soil characteristics (saturated hydraulic

The daily bucket with bottom hole model (BBH) proposed by Kobayashi et al. (2001) is also based on Manabe model involving one layer bucket but including gravity drainage (leakage) as well as capillary rise. Kobayashi et al. (2001) outlined that the soil moisture dynamics is better simulated by BBH than by Bucket (Manabe) model. Kobayashi et al. (2007) developed a new version of BBH named BBH-B including a second soil layer in order to take into account for the variability of the soil profile when the root zone is rather deep (1

In the following, we focus on BBH model where forcing variables are precipitation P and

Ea= M' E0 For s≤s\*

(34)

potential evapotranspiration E0. The actual evapotranspiration is assumed as:

Where M' is a water stress factor updated at each time step and expressed as:

the ratio s/s\* while the evapotranspiration is at potential value for s> s\*.

completely opened (Eq. 32 and Eq. 33).

**3.4.3 Bottom hole bucket model** 

m or more).

conductivity, drainage curve parameter and field capacity).

Ea= E0 For s>s\*

According to the previous presentation and model comparison, bucket type models involves one parameter in Manabe model (Wk) up to six parameters in BBH (Wmax,a,b,c,). The minimum level of model complexity for bucket type models is discussed using a daily time step by Atkinson et al. (2002). These authors introduced the permanent wilting point θpwp to refine the bucket capacity Sbc = (n-θpwp)Za. Also, complexity is raised by the inclusion of a separation between transpiration and evaporation from bare soil. Hence a parameter which represents the fraction of basin area covered by forests is incorporated. A linear piecewise function is assumed similarly to Rodriguez-Iturbe et al. (1999) in both cases (bare soil areas and forest areas). They suppose that storage at field capacity Sfc is the bucket capacity Sbc scaled by a threshold storage parameter fc with Sfc = fc Sbc and fc =(θfc- θpwp)/ (n-θpwp) where θfc is volumetric water content corresponding to field capacity. In addition, they assume that saturation excess runoff occurs when the storage exceeds Sbc and that subsurface runoff occurs when the storage exceeds Sfc with a piecewise non linear drainage function involving two recession parameters. These parameters are further calibrated using observed discharge recession curves while the other parameters are adapted from soil properties (via field data interpretation). Under wet, energy limited catchments authors conclude that the threshold storage parameter fc has a little control on runoff. Conversely, under drier catchments they conclude that the threshold storage parameter fc controls runoff volumes. Either, Kalma & Boulet (1998) compared simulation results of the hydrological model VIC which assumes a bucket representation including spatial variability of soil parameters to the one dimensional physically based model SiSPAT. Using soil moisture profile data for calibration, they conclude that catchment scale wetness index for very dry and very wet periods are misrepresented by SiSPAT while VIC model may better capture the water flux near and by the land surface. However, they outlined that the difficulty of physical interpretation of the bucket VIC model parameters (maximum and minimum storage capacity) constitutes a major drawbacks of the bucket approach.

Guswa et al. (2002) also compared simulations of Richards (1D) and daily bucket model for African Savanna. They outlined that the differences between models outputs are mainly in the relationship between evapotranspiration and average root zone saturation, timing and intensity of transpiration as well as uptake separation between transpiration and evaporation. Vrugt et al. (2004) as well compared the daily Bucket model to a 3-D model (MODHMS) based on Richards equation while taking into account drainage observations. They concluded that Bucket model results are similar to MODHMS results. They also noticed that physical interpretation of MODHMS parameters is difficult since they represent effective properties. Moreover it is noticed that soil control on evapotranspiration is important in dry conditions. Besides, the introduction of a threshold parameter for evapotranspiration uptake is much recommended under arid conditions. Else, according to Rodriguez-Iturbe et al. (1999) under dry conditions, the spatial variation in soil properties has very little impact on the mean soil moisture. DeMaria et al. (2007) analyzed VIC parameter identifiability using stream flows data. Classifying four basins according to their climatic conditions (driest, dry, wet, wettest) they concluded that parameter sensitivity was more strongly dictated by climatic gradients than by changes in soil properties.

## **4. Rainfall runoff hydrological models**

Soil water balance represents a key component of the structure of many Rainfall-runoff (R-R) models. Rainfall-runoff models are primarily tools for runoff prediction for water infrastructure sizing, water management and water quality management. On the basis of rainfall and temperature information, they aim to simulate the water balance at local and regional scales often adopting daily time step. In the majority of cases, model structure is a conceptual representation of the water balance, model parameters having to be adjusted using climatic and soil information as well as hydrological data, in order to match model outputs to observed outputs (Wagener et al., 2003). R-R models have two main components: a soil moisture-accounting module (also named production function) and a routine module (also named transfer function). In the former, the soil moisture status is up-dated while in the latter the runoff hydrograph is simulated. Models differ by the sub-models which are used for each hydrological process in both modules. The way of computing infiltration, evapotranspiration and leakage is of amount importance in the moisture-accounting module which simulates the soil moisture dynamics. It is worth noting that the Rainfall-Runoff Modelling Toolkit (RRMT), developed at Imperial College offers a generic modeling covering to the user to help him (her) to implement different lumped model structures to built his (her) own model (http://www3.imperial.ac.uk/ewre/research/software/toolkit). The system architecture of RRMT is composed by the production and transfer functions modules, and either an off-line data processing module, a visual analysis module and optimization tools module for calibration purposes (Wagener et al. 2001). In this section, we focus on evapotranspiration sub-models of two well-used R-R models (section 4.1). Then, we review the main steps of the calibration process required to estimate the model parameters (section 4.2). Finally three case studies are reported (section 4.3).

### **4.1 Evapotranspiration sub models**

Despite the focus on runoff results in R-R modeling, evapotranspiration computation is a key part of R-R models. As an example, we emphasize the evapotranspiration sub-model of

the difficulty of physical interpretation of the bucket VIC model parameters (maximum and

Guswa et al. (2002) also compared simulations of Richards (1D) and daily bucket model for African Savanna. They outlined that the differences between models outputs are mainly in the relationship between evapotranspiration and average root zone saturation, timing and intensity of transpiration as well as uptake separation between transpiration and evaporation. Vrugt et al. (2004) as well compared the daily Bucket model to a 3-D model (MODHMS) based on Richards equation while taking into account drainage observations. They concluded that Bucket model results are similar to MODHMS results. They also noticed that physical interpretation of MODHMS parameters is difficult since they represent effective properties. Moreover it is noticed that soil control on evapotranspiration is important in dry conditions. Besides, the introduction of a threshold parameter for evapotranspiration uptake is much recommended under arid conditions. Else, according to Rodriguez-Iturbe et al. (1999) under dry conditions, the spatial variation in soil properties has very little impact on the mean soil moisture. DeMaria et al. (2007) analyzed VIC parameter identifiability using stream flows data. Classifying four basins according to their climatic conditions (driest, dry, wet, wettest) they concluded that parameter sensitivity was

minimum storage capacity) constitutes a major drawbacks of the bucket approach.

more strongly dictated by climatic gradients than by changes in soil properties.

parameters (section 4.2). Finally three case studies are reported (section 4.3).

Despite the focus on runoff results in R-R modeling, evapotranspiration computation is a key part of R-R models. As an example, we emphasize the evapotranspiration sub-model of

Soil water balance represents a key component of the structure of many Rainfall-runoff (R-R) models. Rainfall-runoff models are primarily tools for runoff prediction for water infrastructure sizing, water management and water quality management. On the basis of rainfall and temperature information, they aim to simulate the water balance at local and regional scales often adopting daily time step. In the majority of cases, model structure is a conceptual representation of the water balance, model parameters having to be adjusted using climatic and soil information as well as hydrological data, in order to match model outputs to observed outputs (Wagener et al., 2003). R-R models have two main components: a soil moisture-accounting module (also named production function) and a routine module (also named transfer function). In the former, the soil moisture status is up-dated while in the latter the runoff hydrograph is simulated. Models differ by the sub-models which are used for each hydrological process in both modules. The way of computing infiltration, evapotranspiration and leakage is of amount importance in the moisture-accounting module which simulates the soil moisture dynamics. It is worth noting that the Rainfall-Runoff Modelling Toolkit (RRMT), developed at Imperial College offers a generic modeling covering to the user to help him (her) to implement different lumped model structures to built his (her) own model (http://www3.imperial.ac.uk/ewre/research/software/toolkit). The system architecture of RRMT is composed by the production and transfer functions modules, and either an off-line data processing module, a visual analysis module and optimization tools module for calibration purposes (Wagener et al. 2001). In this section, we focus on evapotranspiration sub-models of two well-used R-R models (section 4.1). Then, we review the main steps of the calibration process required to estimate the model

**4. Rainfall runoff hydrological models** 

**4.1 Evapotranspiration sub models** 

*GR4* model which is a parsimonious lumped model proposed by CEMAGREF (France) and running at the daily step with four parameters. A full model description is available in (Perrin et al., 2003). At each time step, a balance of daily rainfall and daily potential evapotranspiration is performed. Consequently, a net evapotranspiration capacity En and a net rainfall Pn are computed. If Pn ≠ 0, a part Ps of Pn fills up the soil reservoir (so, Ps represents infiltration). It is noticeable that this quantity Ps depends on the actual soil moisture content w according to a non linear decreasing function of the w/x1 where x1 is the maximum capacity of the reservoir soil (which might represent the field capacity). On the other hand, if the net evapotranspiration capacity En ≠ 0, actual evapotranspiration Es is computed as a non linear increasing function of the water content involving the ratio w/x1. Also, this function is parameterized through the ratio En/x1 which refers to the characteristics of climate-soil complex. Furthermore, a leakage component is assumed with a power law function of the reservoir water content w.

For P ≥ E0; Pn = P –E0 and En = 0 (38)

$$\text{For } \mathbf{P} \prec \mathbf{E}\_0; \quad \mathbf{P}\_\mathbf{n} = \mathbf{0} \qquad\qquad \text{and} \qquad \mathbf{E}\_\mathbf{n} = \mathbf{E}\_0 - \mathbf{P} \tag{39}$$

$$\mathbf{E\_{v}} = \mathbf{w} \left( 2 \text{-} (\mathbf{w} / \mathbf{x\_{l}}) \right) \tanh(\mathbf{E\_{n}} / \mathbf{x\_{l}}) / \left\{ 1 \text{-} [(1 \text{-} \mathbf{w} \mathbf{x\_{l}}) \tanh(\mathbf{E\_{n}} / \mathbf{x\_{l}})] \right\} \tag{40}$$

Where tanh(.) stands for hyperbolic tangent.

As second example, we underline the sub-models adopted in the *HBV* conceptual semidistributed model proposed by the Swedish hydrological institute (Begström, 1976). The fraction Q of precipitation entering the soil reservoir is assumed as power law function of the ratio (w/FC) of reservoir water content w to a parameter FC representing soil field capacity in HBV model.

$$
\Lambda \mathbf{Q} = \mathbf{P}\_{\text{el}} [1 \text{-} (\text{w/FC})^{\text{f}}] \tag{41}
$$

Where ' is a calibration parameter usually estimated by fitting observed and simulated runoff data. Also, Pe is effective precipitation. In addition, the actual evapotranspiration is a piecewise linear function. The control of actual evapotranspiration rates is performed using a parameter PWP representing a threshold water content. If w< PWP, the evapotranspiration uptake is a fraction of the potential evapotranspiration otherwise it is at potential rate.

$$\begin{aligned} \mathbf{E\_a/E\_0 = w/PWP} & \text{for } w \lhd \text{PWP};\\ \text{and } \mathbf{E\_a = E\_0} \text{ for } w \rhd \mathbf{P} \mathbf{Q} \mathbf{P} \end{aligned} \tag{42}$$

#### **4.2 Model calibration issues**

As runoff has been for long time the main targeted response of rainfall-runoff modeling, rainfall-runoff models were often adjusted according to runoff observations. So far, observations from other control variables such as soil moisture content (Lamb et al., 1998), water table levels (Seibert, 2000) and either low flows (Dunne, 1999) have been adopted to enhance runoff predictions. Calibration of model parameters against runoff data is often performed using criteria such as bias and Root Mean Square Error **(**RMSE), which helps quantifying the discrepancy between observed discharges y0 and simulated discharges yi over a fixed time period with N observations.

$$\text{RMSE} = \left(\frac{1}{N} \sum\_{i=1}^{i=N} \left(y\_{si} - y\_{oi}\right)^2\right)^{\frac{1}{2}} \tag{43}$$

The difficulty in the calibration process is that various parameter sets and even model structures might result in similarly good levels of performance, which constitutes a source of ambiguity as out pointed by Wagener et al. (2003) and many other authors before them (see the literature review of Wagener et al. (2003)). Also, it is noticeable that this problem of ability of various model structures and model parameters to perform equal quality with respect to matching observations is not dependent of the calibration process itself. In other words, the use of a performing optimisation tool does not prevent the problem. Another question is related to the single versus multi objective optimization. Wagener et al. (2003) reported that "single objective function is sufficient to identify only between three and five parameters" while lumped R-R models usually adopt far superior number of parameters. Multi-objective approach of calibration using additional output variables such as water table levels or soil moisture observations has been introduced to deal with the problem. Yet, inadequate model structure may be responsible of mismatching between observed and simulated outputs, as related by Boyle et al. (2000).

#### **4.3 Case studies**

Three case studies are presented in this section. In the first case, we propose a method for calibrating the empirical parameter k' of Hsuen Chun (1988) (Eq. 28). In the second case, we propose as example of calibrating HBV model using both runoff data and regional evapotranspiration information. In the third case, calibration of BBH model is performed using both runoff data and regional evapotranspiration information.

#### **4.3.1 Fitting empirical models of regional evapotranspiration**

This case study is presented in Bargaoui et al. (2008) and Bargaoui & Houcine (2010). It is aimed to calibrate the H.C. model using climatic, rainfall and runoff data from gauged watersheds. Monthly temperature and solar radiation data as well as annual rainfall and runoff data from various locations in Tunisia listed in Table 1 are adopted to calibrate the parameter k' of the empirical Hsuen Chen model (Eq. 28). To this end, 18 rainfall stations and 20 river discharge stations are considered, as well as 8 meteorological stations (Table 1). On the other hand, the potential evapotranspiration E0 is computed at monthly scale using Turc formula.

$$\mathbf{E}\_{\rm 0} = 0.4 \text{ T}\_{\rm m} \left[ (\mathbf{R}\_{\rm 0} / \mathbf{N}\_{\rm l}) + 50 \right] / \left[ \mathbf{R} \rm \rm + 15 \right] \tag{44}$$

Tm : monthly average temperature in (°C); Rg : global solar radiation (cal.cm-2 month-1); Nj : number of days by month

For each river basin, simulated average (long term) annual evapotranspiration is computed using Eq. ( 28). Then, simulated mean annual runoff is computed as the difference between observed mean annual precipitation and simulated average annual evapotranspiration. The fitting of annual simulated runoff to annual observed runoff using the 20 river discharge stations results in k'= 1.5. The good adequacy of the model is well reflected in the plot of average simulated versus average observed annual runoff (Fig. 1).

1 1 *i N*

The difficulty in the calibration process is that various parameter sets and even model structures might result in similarly good levels of performance, which constitutes a source of ambiguity as out pointed by Wagener et al. (2003) and many other authors before them (see the literature review of Wagener et al. (2003)). Also, it is noticeable that this problem of ability of various model structures and model parameters to perform equal quality with respect to matching observations is not dependent of the calibration process itself. In other words, the use of a performing optimisation tool does not prevent the problem. Another question is related to the single versus multi objective optimization. Wagener et al. (2003) reported that "single objective function is sufficient to identify only between three and five parameters" while lumped R-R models usually adopt far superior number of parameters. Multi-objective approach of calibration using additional output variables such as water table levels or soil moisture observations has been introduced to deal with the problem. Yet, inadequate model structure may be responsible of mismatching between observed and

Three case studies are presented in this section. In the first case, we propose a method for calibrating the empirical parameter k' of Hsuen Chun (1988) (Eq. 28). In the second case, we propose as example of calibrating HBV model using both runoff data and regional evapotranspiration information. In the third case, calibration of BBH model is performed

This case study is presented in Bargaoui et al. (2008) and Bargaoui & Houcine (2010). It is aimed to calibrate the H.C. model using climatic, rainfall and runoff data from gauged watersheds. Monthly temperature and solar radiation data as well as annual rainfall and runoff data from various locations in Tunisia listed in Table 1 are adopted to calibrate the parameter k' of the empirical Hsuen Chen model (Eq. 28). To this end, 18 rainfall stations and 20 river discharge stations are considered, as well as 8 meteorological stations (Table 1). On the other hand, the potential evapotranspiration E0 is computed at monthly scale using

 E0= 0.4 Tm [(Rg/Nj )+50] / [Rg+15] (44) Tm : monthly average temperature in (°C); Rg : global solar radiation (cal.cm-2 month-1); Nj :

For each river basin, simulated average (long term) annual evapotranspiration is computed using Eq. ( 28). Then, simulated mean annual runoff is computed as the difference between observed mean annual precipitation and simulated average annual evapotranspiration. The fitting of annual simulated runoff to annual observed runoff using the 20 river discharge stations results in k'= 1.5. The good adequacy of the model is well reflected in the plot of

*i y y <sup>N</sup>* 

*si oi*

 

1 <sup>2</sup> <sup>2</sup>

(43)

RMSE=

simulated outputs, as related by Boyle et al. (2000).

using both runoff data and regional evapotranspiration information.

**4.3.1 Fitting empirical models of regional evapotranspiration** 

average simulated versus average observed annual runoff (Fig. 1).

**4.3 Case studies** 

Turc formula.

number of days by month


Table 1. Location of stations to calibrate H.C. model (after Bargaoui &Houcine, 2010)

Fig. 1. Comparison of observed and simulated runoff for 20 river basins

#### **4.3.2 Multicriteria calibration of HBV model using regional evapotranspiration information**

This application is presented in Bargaoui et al. (2008). The idea is to use the information about the climatic regime as a driver for runoff prediction. Effectively, for a large number of basins with areas in the interval 50 à 1000 km², Wagener et al., (2007) suggested that there is a significant correlation between annual runoff and the ratio of forcing variables P/E0. In the same way, we seek to use information about average (regional) actual evapotranspiration which is a bio-climatic indicator as means to improve accuracy of runoff predictions. To develop these ideas, the HBV rainfall-runoff model was adopted, coupled to a SCE-UA optimization tool. The calibration method adopts an objective function combining three criteria: minimisation of runoff root mean square error, minimisation of water budget simulation error, minimisation of the difference between mean annual simulated evapotranspiration Ea and regional Em. The case study is a mountainous watershed of Wadi Sejnane (Tunisia). Mean daily runoff observations from September 1964 to August 1969 are available for a hydrometric station controlling an area of 378 km². Average basin annual rainfall is 931 mm/year. Over 8 years of rainfall observations, the minimum value of the series is 628 mm/year while the maximum value is 1141 mm/year denoting an important rainfall inter annual variability. Mean annual discharge is 2.43 m3/s. Average evapotranspiration computed using HC model (Eq. 28) with k'=1.5 results in Em=643 mm/year. To calibrate the HBV model parameters, we adopt the period 1964/1967 for calibration and the period 1967/1969 for validation. The minimization of the objective function is performed using SCE-UA algorithm (Duan et al., 1994) in order to adjust 10 parameters (while 7 other HBV parameters have been set constant because they were found insensitive). First, the Nash coefficient of mean daily discharges is chosen as objective function F0=NashR. The resulting value F0=0.81 is quite good. However, for the validation

0 50 100 150 200 250 300 350 400 450 Observed mean annual runoff (mm/year)

This application is presented in Bargaoui et al. (2008). The idea is to use the information about the climatic regime as a driver for runoff prediction. Effectively, for a large number of basins with areas in the interval 50 à 1000 km², Wagener et al., (2007) suggested that there is a significant correlation between annual runoff and the ratio of forcing variables P/E0. In the same way, we seek to use information about average (regional) actual evapotranspiration which is a bio-climatic indicator as means to improve accuracy of runoff predictions. To develop these ideas, the HBV rainfall-runoff model was adopted, coupled to a SCE-UA optimization tool. The calibration method adopts an objective function combining three criteria: minimisation of runoff root mean square error, minimisation of water budget simulation error, minimisation of the difference between mean annual simulated evapotranspiration Ea and regional Em. The case study is a mountainous watershed of Wadi Sejnane (Tunisia). Mean daily runoff observations from September 1964 to August 1969 are available for a hydrometric station controlling an area of 378 km². Average basin annual rainfall is 931 mm/year. Over 8 years of rainfall observations, the minimum value of the series is 628 mm/year while the maximum value is 1141 mm/year denoting an important rainfall inter annual variability. Mean annual discharge is 2.43 m3/s. Average evapotranspiration computed using HC model (Eq. 28) with k'=1.5 results in Em=643 mm/year. To calibrate the HBV model parameters, we adopt the period 1964/1967 for calibration and the period 1967/1969 for validation. The minimization of the objective function is performed using SCE-UA algorithm (Duan et al., 1994) in order to adjust 10 parameters (while 7 other HBV parameters have been set constant because they were found insensitive). First, the Nash coefficient of mean daily discharges is chosen as objective function F0=NashR. The resulting value F0=0.81 is quite good. However, for the validation

Fig. 1. Comparison of observed and simulated runoff for 20 river basins

**4.3.2 Multicriteria calibration of HBV model using regional evapotranspiration** 

0

**information** 

50

100

150

200

250

H.C estimated annual runoff (mm/year) k=1.5

300

350

400

450

period the ensuing optimal parameter set results in very poor fitting with a negative value of the Nash coefficient (NashR = -0.084). Consequently, the objective function was modified to F1 integrating the average model error (bias) of runoff output. Hence,

$$\mathbf{F\_{l} = Nashh\_{R} - w' \to R\_{RA}} \tag{45}$$

Where ERRA is the absolute relative error with respect to annual discharge. The weight coefficient *w'* = 0.1 is adopted according to Lindström and al. (1997) and helps aggregate the two criteria NashR and ERRA. In fact, the adoption of ERRA aims to consider climatic zonality during the calibration process. Resulting optimal solution corresponds to NashR =0.81 and ERRA = 5%, which is believed good performance. It is worth noting that this modification of the objective function greatly improved NashR also for the validation period (NashR =0.55). The mean annual simulated evapotranspiration using HBV model is equal to 728 mm/ year while the H.C. model with k'=1.5 results in 643mm/year. To try to overcome such overestimation, it was proposed to directly include the information about evapotranspiration by adopting a new objective function F2.

$$\mathbf{F\_2 = Nashh\_R - 0, 1 \text{ ER}\_{RA} - 0, 1 \text{ ER}\_{\text{ETRG}}} \tag{46}$$

Where ERETRG is the absolute relative error with respect to mean annual evapotranspiration (simulated by HBV versus estimated by H.C with k'=1.5). The resulting runoff Nash is a little smaller (NashR =0.79) than for F1, but a real improvement is obtained during the validation period (NashR = 0.68). Fig. 2 reports HBV estimated annual evapotranspiration obtained with the optimal HBV solution (squares) versus annual rainfall. Comparatively, we also report annual evapotranspiration as evaluated using H.C model with k'=1.5 (interrupted line). Effect of year to year rainfall fluctuation on HBV estimations is well seen in the graph.

Fig. 2. Comparison of evapotranspiration estimates from HBV and HC models in relation with rainfall

#### **4.3.3 Multicriteria calibration of BBH model using regional evapotranspiration information**

In the third application it is aimed to compare BBH model results using the decadal time step. A part of this case study is presented in Bargaoui & Houcine (2011) using monthly data for model evaluation. Here will report results of decadal evaluations. Data are from the Wadi Chaffar watershed (250 km²) situated under arid climate, South Tunisia. Vegetation cover comprises mainly olives. Meteorological data (solar radiation, air temperature and humidity, sky cloudiness, wind speed and Piche evaporation) are available from September 1989 to August 1999 for computing the daily reference evapotranspiration E0 according to Allen et al. (1998). E0 is multiplied by the crop coefficient Kc of olives trees to obtain daily potential evapotranspiration (Allen et al., 1998). Daily average basin rainfalls are available from September 1985 to August 1999. Stream discharge data are available for the basin outlet at the daily time step from September 1985 to August 1999. In the period September 1985 to August 1989, meteorological data are missing and the used E0 values are the daily long term average computed for September 1989- August 1999. The H.C. model results in an average annual evapotranspiration Em = 213 mm/year (Bargaoui & Houcine, 2010). BBH model inputs are precipitation and potential evapotranspiration and seven parameters are to be calibrated. To reduce the number of calibrated parameters, we first fix the thickness of active soil layer Za (in mm) and the effective soil porosity n (unit less). Also, we undertake a reformulation of leakage component L(s) by using the model of Guswa et al. (2002) where

$$\text{KL}(\mathbf{s}) = \mathbf{K}\_{\text{S}} \frac{\mathbf{e}^{\text{B}\left(\mathbf{s} \cdot \mathbf{S}\_{\text{FC}}\right)} - 1}{e^{\text{B}\left(1 - \mathbf{S}\_{\text{FC}}\right)} - 1} \tag{47}$$

where s is the degree of saturation (unit less); KS saturated hydraulic conductivity at soil surface (mm/day); *B* is the soil water retention curve shape parameter; SFC (unit less) is the field capacity; *Wmax* = *n*Za (*Wmax* is the total water-holding capacity in mm).

Coupling this expression with pedo-transfer functions it makes it possible after Bargaoui & Houcine ( 2010), to derive the parameters (a, b, c) as following using pedo-transfer parameters KS , *B* and SFC:

$$a = \mathcal{W}\_{\text{max}} \left[ S\_{FC} - \frac{1}{B} L n \left( K\_S \frac{1}{e^{B\left(1 - S\_{FC}\right)} - 1} \right) \right] \tag{48}$$

$$b = \mathbf{W}\_{\text{max}} \, \frac{1}{B} \tag{49}$$

$$\mathcal{L} = \left(\frac{1}{e^{B\left(1-S\_{FC}\right)} - 1}\right) K\_S \tag{50}$$

In this case, the model by Rawls et al. (1982) is adopted for *KS* estimation while SFC is derived according to the Cosby (1984) and Saxton et al. (1986) models recently adopted by Zhan et al., (2008). Finally *B* = 9 is assumed in agreement with Rodriguez-Iturbe et al. (1999). The dominant soil type is considered to represent the soil characteristics. So, the value n=0.34 corresponding to a sandy soil was adopted; these assumptions result in KS = 3634 mm/d and SFC= 0.166. Also, after many trials the value Za= 0.5 m was adopted. The two remaining parameters and (0< <1; 0< *η* <1) represent respectively the resistance of vegetation to

In the third application it is aimed to compare BBH model results using the decadal time step. A part of this case study is presented in Bargaoui & Houcine (2011) using monthly data for model evaluation. Here will report results of decadal evaluations. Data are from the Wadi Chaffar watershed (250 km²) situated under arid climate, South Tunisia. Vegetation cover comprises mainly olives. Meteorological data (solar radiation, air temperature and humidity, sky cloudiness, wind speed and Piche evaporation) are available from September 1989 to August 1999 for computing the daily reference evapotranspiration E0 according to Allen et al. (1998). E0 is multiplied by the crop coefficient Kc of olives trees to obtain daily potential evapotranspiration (Allen et al., 1998). Daily average basin rainfalls are available from September 1985 to August 1999. Stream discharge data are available for the basin outlet at the daily time step from September 1985 to August 1999. In the period September 1985 to August 1989, meteorological data are missing and the used E0 values are the daily long term average computed for September 1989- August 1999. The H.C. model results in an average annual evapotranspiration Em = 213 mm/year (Bargaoui & Houcine, 2010). BBH model inputs are precipitation and potential evapotranspiration and seven parameters are to be calibrated. To reduce the number of calibrated parameters, we first fix the thickness of active soil layer Za (in mm) and the effective soil porosity n (unit less). Also, we undertake a reformulation of leakage component L(s) by using the model of Guswa et al. (2002) where

**4.3.3 Multicriteria calibration of BBH model using regional evapotranspiration** 

field capacity; *Wmax* = *n*Za (*Wmax* is the total water-holding capacity in mm).

 FC s-S

1 *SFC e*

*B*

 <sup>1</sup> <sup>1</sup> 1 Wmax *<sup>B</sup>* <sup>1</sup> *SFC FC <sup>S</sup> <sup>e</sup> Ln <sup>K</sup>*

*<sup>B</sup> <sup>S</sup> KS <sup>e</sup> <sup>c</sup> FC*

 <sup>1</sup> 1

 

In this case, the model by Rawls et al. (1982) is adopted for *KS* estimation while SFC is derived according to the Cosby (1984) and Saxton et al. (1986) models recently adopted by Zhan et al., (2008). Finally *B* = 9 is assumed in agreement with Rodriguez-Iturbe et al. (1999). The dominant soil type is considered to represent the soil characteristics. So, the value n=0.34 corresponding to a sandy soil was adopted; these assumptions result in KS = 3634 mm/d and SFC= 0.166. Also, after many trials the value Za= 0.5 m was adopted. The two remaining

<1; 0< *η* <1) represent respectively the resistance of vegetation to

*<sup>B</sup> <sup>a</sup> <sup>S</sup>* (48)

*<sup>b</sup>* <sup>1</sup> Wmax (49)

<sup>1</sup> (50)

(47)

S 1 e 1 Ls K

 

where s is the degree of saturation (unit less); KS saturated hydraulic conductivity at soil surface (mm/day); *B* is the soil water retention curve shape parameter; SFC (unit less) is the

Coupling this expression with pedo-transfer functions it makes it possible after Bargaoui & Houcine ( 2010), to derive the parameters (a, b, c) as following using pedo-transfer

**information** 

parameters KS , *B* and SFC:

parameters

and (0<  evapotranspiration and the moisture retaining capacity. The problem is now to fit the parameters and They are adjusted using two different methods: i.e. using only observed runoff (method 1) and using both observed runoff and regional evapotranspiration information (method 2). Also BBH model has been completed adopting a , contributing area sub-model (Betson, 1964); Dunne et Black (1970). According to this assumption, runoff originates from a part of the watershed (contributing area) contrarily to the assumption of runoff occurring from the entire watershed. For a fixed day j, the contributing area CAj is herein assumed linked to the soil moisture content according to Dickinson & Whiteley (1969). Additionally, a logistic Probability Density Function as a function of humidity index IHj is adopted with parameters ac and bc (Eq. 51). It means that the mean contributing area is ac and that the variance of the contributing area is (bc)²/3. The humidity index takes account for the rainfall accumulated during the actual day and the IX previous days (Eq. 52).

$$\text{CA}\_{\text{j}} = \frac{\mathbf{e}^{\{ (\text{IH}\_{\text{j}} \cdot \text{a}\_{\text{c}})/\text{b}\_{\text{c}} \}}}{\text{(1} + \mathbf{e}^{\{ (\text{IH}\_{\text{j}} \cdot \text{a}\_{\text{c}})/\text{b}\_{\text{c}} \}}} \tag{51}$$

$$IH\_j = \mathcal{W}\_{j-1} + o\prime \sum\_{l=0}^{lX} P\_{j-l} \tag{52}$$

where '' is a fixed weight ('' =0.1). Then, two cases are considered: case (a) when the total basin area contributes to runoff at the basin outlet; case (b) when only a contributive area gives rise to runoff at the outlet.

After many trials and errors we assumed IX= 90 days, ac = 20 and bc = 10 in case (b). The model was calibrated for and using daily hydro meteorological data (solar radiation, air temperature, air humidity, mean areal rainfall) as well as daily runoff records and also average annual evapotranspiration. The decadal, monthly and annual totals are adopted to evaluate model performance.

In each case (a) and (b), a first criterion based on the matching of decadal runoff Eq. 53 is adopted to delineate adequate solutions for and *η* (0< <1; 0< *η* <1). A supplementary criterion is based on the matching of long term annual evapotranspiration (Eq. 54).

$$C\_{\mathbf{y}}(\sigma,\eta) = \frac{1}{N} \sum\_{i=1}^{N} \left| (\mathbf{y}\_{si} - \mathbf{y}\_{oi}) / \mathbf{y}\_{oi} \right| \tag{53}$$

$$C\_E\left(\sigma,\eta\right) = \frac{1}{N'} \sum\_{i=1}^{N'} \left| \left(E\_{si} - E\_m\right) / E\_m \right| \tag{54}$$

In Eq. (53), yoi and ysi are respectively decadal observed and simulated volume runoff and *N* is the number of simulated decades. In Eq. (54), Esi is simulated annual evapotranspiration and *N'* is the number of simulated years.

For each pair of simulated ) (0< <1; 0< *η* <1), candidate solutions verifying the criterion Cy ) < Eq. 53with =20% the Nash coefficient RN is then evaluated. Pairs for which it is found that RN>0.5, are thus selected. Also, introducing Em for calibration method *2*, the absolute value CE ) of the relative error between mean annual simulated evapotranspiration and Em, is used through the additional selection criterion of Eq. 54.

Fig. 3. Estimated decadal runoff versus decadal precipitation with the assumption of total watershed contributing to runoff (+ represent observed volumes and squares represent simulated volume for the selected pairs of ))

Fig. 4. Estimated decadal runoff versus decadal precipitation with the assumption of contributive area (+ represent observed volumes and squares represent simulated volume for the selected pairs of )).

Pairs of simulated ) (0< <1; 0< *η* <1) which satisfy both Cy < RN> 0.5 and CE ) < ' with '=30%are finally selected as adequate solutions.

Fig. 3 and 4 report model outputs for sets of ) fulfilling the above conditions under the assumptions of cases (a) and (b) in case where Em information is included. Estimated decadal volumes (squares) for the selected pairs of ) are compared to observed decadal volumes (+ ) and are reported versus precipitation data. Fig. 3 is related to case (a) corresponding to the assumption of total watershed contributing to runoff. Fig. 4 is related to case (b) assuming a contributing area. The results suggest that the introduction of contributing area outcomes produce outputs which result in a better fitting of the rainfallrunoff evolution. In effect, in the case of total area contributing no solution is found able to simulate the most rainy decade,(squares are far from the symbol + for the Rainiest decade). Conversely, some solutions are found able to reproduce the most rainy decade if we consider contributing area scheme (some squares are located near the +). Also, evapotranspiration information has greatly reduced the interval of acceptable solutions. Effectively, selected solutions are such that 0.15 < <0.35 and 0.15 < <0.25.

## **5. Conclusions**

168 Evapotranspiration – Remote Sensing and Modeling

**0 20 40 60 80 100 120 140 160** Decadal rainfall (mm)

**0 20 40 60 80 100 120 140 160** Decadal rainfall (mm)

Fig. 4. Estimated decadal runoff versus decadal precipitation with the assumption of contributive area (+ represent observed volumes and squares represent simulated volume

Fig. 3. Estimated decadal runoff versus decadal precipitation with the assumption of total watershed contributing to runoff (+ represent observed volumes and squares represent

simulated volume for the selected pairs of ))

IX=90; ac=20; bc=10

**0**

0,00E+00

for the selected pairs of )).

2,00E+06

4,00E+06

6,00E+06

Decadal volumes (m3)

8,00E+06

1,00E+07

1,20E+07

**5**

**10**

**15**

**20**

**25**

Decadal runoff volumes (h m3)

**30**

**35**

**40**

**45**

The simulation of evapotranspiration using the water balance equation is part of hydrological modelling (rainfall-runoff models) and is also important in the framework of global circulation models (Land surface models). A lot of models are now functioning and their formulation is based on different assumptions on soil characteristics in relation with soil moisture, transpiration schemes, as well as infiltration and runoff schemes.

Empirical models for estimating regional evapotranspiration are worth noting for estimating average long term evapotranspiration. They are generally based on climatic information (rainfall and potential evapotranspiration). They often require the adjustment of a single empirical parameter. Under particular climate and soil vegetation, evapotranspiration is controlled by soil moisture dynamics. Thus, Bucket type soil water budget models are worth noting for estimating time series of actual evapotranspiration at smaller time scales (daily to monthly). They involve from one parameter such as in the Manabe model (with parameter Wk) up to six parameters such as in BBH model (with parameters Wmax,a,b,c,). Parameters are linked to soil, climatic and vegetation characteristics. However, it is generally believed that the temporal variability of soil moisture series is mostly dependent on the rainfall variability especially under conditions of low precipitations. On the other hand, soil parameters such as field capacity, hydraulic conductivity at saturation and wilting point potential are key parameters controlling the evapotranspiration model outputs. One way to derive soil parameters is to adopt pedo transfer functions. Transpiration which corresponds to vegetation uptake is regulated by stomata and driven by atmospheric demand. It is widely represented by a linear piecewise function with parameters depending on vegetation characteristics. Thus, in computing evapotranspiration, a main assumption is the linear piecewise function of evapotranspiration in relation with potential evapotranspiration for taking account for soil water stress. Such an assumption is underlined in several rainfall runoff models (for example the two models GR4 and HBV studied here adopt such analytical form). Model adequacies introduce the question of the choice of the objective function as well as the output variables adopted for model evaluation. In the case studies presented here, results suggest that the introduction of the information about average (long term) annual evapotranspiration may help improving the accuracy of the water balance simulation results. In effect the runoff Nash coefficient is found to be improved during the validation period in the case where long term evapotranspiration is taking account during the calibration process.

## **6. Annexe**

## **6.1 Glossary**

a: parameter related to the field capacity (mm)

*a'* : pore size distribution parameter

ac: Logistic density distribution parameter

(: the root efficiency function.

A(t) : the fraction of area for which the infiltration capacity is less than i(t)

*B* : the soil water retention curve shape parameter;

b: parameter representing the decay of soil moisture (mm);

b'' is the infiltration shape parameter.

bc: Logistic density distribution parameter

surface slope angle

' : a calibration parameter in HBV model

c' : pore disconnectedness index

c: parameter representing the daily maximal capillary rise (mm)

C% : soil percent of clay

CAj: : the contributing area

cos: the cosinus function

d1 : depth of near surface soil layer

d2 : depth of root zone soil layer;

D:soil-water diffusivity parameter

: difference in average matrix potential before and after wetting

: difference in average soil water content before and after wetting

∆w : the change in soil water content

z : soil depth.

C1 : parameter,

C2 : parameter,

E(s,t): : evapotranspiration

Ea : actual evapotranspiration

EDI : dryness index

Ee exfiltration parameter as function of initial degree of saturation s0

Eg : bare soil evaporation rate at the surface

Em average annual evapotranspiration

En: net evapotranspiration capacity

Er: reference evapotranspiration according to FAO model

Etr : transpiration rate from the root zone of depth d2

ERETRG : the absolute relative error with respect to mean annual evapotranspiration

Epa average annual potential evapotranspiration

evaluation. In the case studies presented here, results suggest that the introduction of the information about average (long term) annual evapotranspiration may help improving the accuracy of the water balance simulation results. In effect the runoff Nash coefficient is found to be improved during the validation period in the case where long term

evapotranspiration is taking account during the calibration process.

A(t) : the fraction of area for which the infiltration capacity is less than i(t)

a: parameter related to the field capacity (mm)

*B* : the soil water retention curve shape parameter;

b: parameter representing the decay of soil moisture (mm);

c: parameter representing the daily maximal capillary rise (mm)

: difference in average matrix potential before and after wetting : difference in average soil water content before and after wetting

Ee exfiltration parameter as function of initial degree of saturation s0

ERETRG : the absolute relative error with respect to mean annual evapotranspiration

Er: reference evapotranspiration according to FAO model Etr : transpiration rate from the root zone of depth d2

*a'* : pore size distribution parameter ac: Logistic density distribution parameter

(: the root efficiency function.

b'' is the infiltration shape parameter. bc: Logistic density distribution parameter

c' : pore disconnectedness index

d1 : depth of near surface soil layer d2 : depth of root zone soil layer; D:soil-water diffusivity parameter

∆w : the change in soil water content

Eg : bare soil evaporation rate at the surface Em average annual evapotranspiration En: net evapotranspiration capacity

Epa average annual potential evapotranspiration

E(s,t): : evapotranspiration Ea : actual evapotranspiration

EDI : dryness index

' : a calibration parameter in HBV model

surface slope angle

C% : soil percent of clay CAj: : the contributing area cos: the cosinus function

z : soil depth. C1 : parameter, C2 : parameter,

**6. Annexe 6.1 Glossary**  ERRA : the absolute relative error with respect to annual discharge ETurc : monthly potential evapotranspiration (mm); Era average annual surface retention f : infiltration capacity F cumulative infiltration for a rainfall event fc: threshold storage parameter FC: representing soil field capacity in HBV model Gd(t): Daily percolation and capillary rise term gr(,z): vegetation uptake of soil moisture I(s,t): infiltration into the soil Inf(,z0) : precipitation infiltrating into the soil i(t): infiltration capacity at time t. imax : maximum value of infiltration capacity f (t) : the cumulative infiltration J(.): evapotranspiration function k : intrinsic permeability k(1): intrinsic permeability at saturation K: hydraulic conductivity K (1) hydraulic conductivity at saturation kv : plant coefficient av : average saturated hydraulic conductivity k' : parameter of HC model Kc : crop coefficient *KS* : the saturated hydraulic conductivity; K's : correction coefficient of the crop coefficient κ: shape parameter of the Gamma distribution *l*: factor linked to soil matrix tortuosity L(s,t) :leakage LAI : Leaf area index mean storm arrival rate Mv : vegetation fraction of surface. dynamic viscosity of water; n: soil effective porosity : parameter volumetric water content θf :the volumetric moisture contents of the soil at field capacity θw: the volumetric moisture contents at wilting point θpwp: permanent wilting point g : volumetric water contents of near surface soil layer; s : saturated soil moisture content 2 : volumetric water contents of root zone soil layer; geq : equilibrium surface volumetric soil moisture content 1 : specific value of soil moisture content 0 : specific value of soil moiqture content N: number of observations

Nj : number of days by month

NashR: Nash coefficient of mean daily discharges

P : average annual precipitation

Pe : effective precipitation

PWP : parameter representing a threshold water content in HBV model.

Pg : precipitation infiltrating into the soil;

Pn: net rainfall

q2 : rate of drainage out of the bottom of the root zone;

R1 : (s cm-1) a resistance to moisture flow in soil

R2 : (s cm-1) is vegetation resistance to moisture flow;

Rn : average annual net radiation

Rs : surface runoff

Rg : global solar radiation (cal.cm-2 month-1)

r(z) : a root density function (cm-1)

w : density of the water;

s: relative soil moisture content or degree of saturation

s\*: saturation threshold

s1 :threshold value of soil saturation

sw: soil moisture at wilting point.

s0: initial degree of saturation

S : sorptivity

S% : soil percent of sand

Sbc: bucket capacity

SFC : soil field capacity

Sfc: storage at field capacity

parameter representing the resistance of vegetation to evapotranspiration;

t: time

Tm : monthly average temperature in (°C);

u(z,t) : local transpiration uptake

w : the actual soil water storage

w0 : critical soil water storage in Budyko model

W0: water holding capacity

Wk : a fraction of the soil field capacity

Wmax : total water-holding capacity (mm);

Wm0: mean water holding capacity

: a fixed weight

: soil moisture potential (bars)

p : leaf moisture potential (bars)

: the wilting point potential

(1) : the bubbling pressure head which represents matrix potential at saturation.

x1 : maximum capacity of the reservoir soil

yi: simulated discharges

y0: observed discharges

z: the vertical coordinate (z>0 downward from surface)

Za: thickness of active soil layer (mm);

z0 : the vertical coordinate at the surface

## **7. References**

172 Evapotranspiration – Remote Sensing and Modeling

Nj : number of days by month

P : average annual precipitation Pe : effective precipitation

Rn : average annual net radiation

r(z) : a root density function (cm-1)

s1 :threshold value of soil saturation sw: soil moisture at wilting point. s0: initial degree of saturation

Tm : monthly average temperature in (°C);

Wk : a fraction of the soil field capacity Wmax : total water-holding capacity (mm);

x1 : maximum capacity of the reservoir soil

Za: thickness of active soil layer (mm); z0 : the vertical coordinate at the surface

z: the vertical coordinate (z>0 downward from surface)

Wm0: mean water holding capacity

: soil moisture potential (bars) p : leaf moisture potential (bars) : the wilting point potential

w0 : critical soil water storage in Budyko model

u(z,t) : local transpiration uptake w : the actual soil water storage

W0: water holding capacity

: a fixed weight

yi: simulated discharges y0: observed discharges

Pn: net rainfall

Rs : surface runoff

w : density of the water;

s\*: saturation threshold

S% : soil percent of sand Sbc: bucket capacity SFC : soil field capacity Sfc: storage at field capacity

S : sorptivity

t: time

Pg : precipitation infiltrating into the soil;

Rg : global solar radiation (cal.cm-2 month-1)

NashR: Nash coefficient of mean daily discharges

q2 : rate of drainage out of the bottom of the root zone; R1 : (s cm-1) a resistance to moisture flow in soil R2 : (s cm-1) is vegetation resistance to moisture flow;

s: relative soil moisture content or degree of saturation

PWP : parameter representing a threshold water content in HBV model.

parameter representing the resistance of vegetation to evapotranspiration;

(1) : the bubbling pressure head which represents matrix potential at saturation.


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## **Evapotranspiration of Grasslands and Pastures in North-Eastern Part of Poland**

Daniel Szejba

*Warsaw University of Life Sciences – SGGW Poland* 

## **1. Introduction**

178 Evapotranspiration – Remote Sensing and Modeling

Zhao, R. J., and X. R. Liu (1995), The Xinjiang model, in Computer Models of Watershed

Colo.

Hydrology, edited by V. P. Singh, chap. 7, *Water Resour. Publ.*, Highlands Ranch,

The problem of plant water requirements and supply is of great importance to agricultural water management. It is crucial to determine and provide the water amount required in a certain region to support the plants assimilation function. The quantity of water required on a specific farm can be determined by analyzing the water balance, where precipitation and evapotranspiration are basic elements. Evapotranspiration data is also indispensable when mathematically modelling the water balance. The values of evapotranspiration can be obtained from lysimeter measurements. However, this measurement is labour intensive and also requires special equipment; thus, it is not widely applied. To address this problem, a number of methods of evapotranspiration estimation based on physical and empirical equations are available, where the quantity of evapotranspiration depends on other measured factors. Penman (1948) developed a method for determination of the potential evapotranspiration as a product of the crop coefficient for a certain crop in a certain development stage and the reference evapotranspiration (Łabędzki et al., 1996). Open water surface evaporation is the reference evapotranspiration used in this method. Currently, the method most widely applied in Poland for evapotranspiration estimation is a method called the "French Modified Penman method", which is a version of FAO Modified Penman method (Doorenbos & Pruitt, 1977), with the net radiation flux calculated by Podogrodzki (Roguski et al., 1988). Name of "Modified Penman method" is using in further part of this text. On the other hand, the Food and Agriculture Organization (FAO) recommends the Penman-Monteith method for evapotranspiration estimation (Allen et al., 1998). The aforementioned methods require relevant crop coefficients to estimate the potential evapotranspiration. Although crop coefficients for grasslands and pastures applicable to the modified Penman are available for Polish conditions (Roguski et al., 1988; Brandyk et al., 1996; Szuniewicz & Chrzanowski, 1996), the problem occurs when the potential evapotranspiration has to be calculated according to the FAO standards which require the Penman-Monteith method to be used. Both the methods (Modified Penman and Penman-Monteith) require meteorological data including: air temperature, humidity, cloudiness or sunshine and wind speed. If one or more of the required inputs are not available, then applying any of the two methods is difficult, perhaps even impossible. In such cases, the Thornthwaite method, developed in 1931, can be a viable alternative (Byczkowski, 1979; Skaags, 1980; Newman, 1981; Pereira & Pruitt, 2004). The Thornthwaite method is commonly used in the USA. This method requires only two basic climatic inputs that determine the solar energy supply and are necessary to estimate the potential evapotranspiration: air temperature and day length.

There are two objectives of this chapter. The first objective is to determine the crop coefficient needed when estimating the potential evapotranspiration with the Penman-Monteith method. The second objective is a comparative analysis of the potential evapotranspiration estimates obtained from the Thornthwaite method and the crop coefficient approach with Penman-type formula as a reference evapotranspiration.

### **2. Reviewing the selected methods for evapotranspiration estimation: Modified Penman, Penman-Monteith and Thornthwaite**

It can be assumed, that the amount of a farm plants evapotranspiration depends on such factors as atmosphere condition, plants development stage and soil moisture. The interdependence of these factors is complex and difficult to describe mathematically. This dependence can be expressed as a product of following functions:

$$\rm ET = f\_1(M) \cdot f\_2(P) \cdot f\_3(S) \tag{1}$$

where:

M – atmosphere factors,

P – plant factors,

S – soil moisture factors.

Groups of atmosphere factors can be formulated as a reference evapotranspiration (ET0), which characterises meteorological conditions in the evapotranspiration process and describes evaporation ability in the atmosphere. This factor determines the intensity of evapotranspiration process in the case of unlimited access to a water source, that is deplete of soil water:

$$\text{If } \mathbf{f}\_1(\mathbf{M}) = \mathbf{E}\mathbf{T}\_0 \tag{2}$$

f2(P) function describes the influence of plant parameters such as: plant species, development stage, mass of above ground and underground parts, leaf area index (LAI), growth dynamics, nutrients supply, yield and frequency of harvesting. A group of these parameters is expressed as a crop coefficient (kc), which is empirically determined in independently by soil moisture conditions:

$$\mathbf{f\_2(P)} = \mathbf{k\_c} \tag{3}$$

f3(S) function describes the influence of soil moisture and the availability of soil water for plants (as a soil water potential) on evapotranspiration amount. With our knowledge of soil physics and plant physiology knowledge, it can be assumed that evapotranspiration during sufficient water supply does not depend or slightly depend on soil moisture (Łabędzki et al., 1996, as cited in: Kowalik, 1973; Salisbury & Ross, 1975; Feddes et al., 1978; Rewut, 1980; Olszta, 1981; Korohoda, 1985; Więckowski, 1985; Brandyk, 1990). Sufficient water supply does not limit evapotranspiration and plant yield is defined as a soil moisture range between optimum water content (when air content equals at least 8 – 10% in root zone) and refill point (pF 2.7 – 3.0). In other words, sufficient water supply means easily available water or readily available water (RAW). Evapotranspiration reductions has a place, when RAW becomes consumed by plants. The deciding factor of evapotranspiration reduction amounts is the difference between actual soil moisture content and soil moisture content when the evapotranspiration process fades (wilting point). Thus, it can be showed in general (Łabędzki et al., 1996, as cited in: Olszta et al., 1990; Łabędzki & Kasperska, 1994; Łabędzki, 1995):

$$\mathbf{f\_3(S)} = \mathbf{k\_s(\theta)}\tag{4}$$

where:

180 Evapotranspiration – Remote Sensing and Modeling

determine the solar energy supply and are necessary to estimate the potential

There are two objectives of this chapter. The first objective is to determine the crop coefficient needed when estimating the potential evapotranspiration with the Penman-Monteith method. The second objective is a comparative analysis of the potential evapotranspiration estimates obtained from the Thornthwaite method and the crop

It can be assumed, that the amount of a farm plants evapotranspiration depends on such factors as atmosphere condition, plants development stage and soil moisture. The interdependence of these factors is complex and difficult to describe mathematically. This

Groups of atmosphere factors can be formulated as a reference evapotranspiration (ET0), which characterises meteorological conditions in the evapotranspiration process and describes evaporation ability in the atmosphere. This factor determines the intensity of evapotranspiration process in the case of unlimited access to a water source, that is deplete

f2(P) function describes the influence of plant parameters such as: plant species, development stage, mass of above ground and underground parts, leaf area index (LAI), growth dynamics, nutrients supply, yield and frequency of harvesting. A group of these parameters is expressed as a crop coefficient (kc), which is empirically determined in

f3(S) function describes the influence of soil moisture and the availability of soil water for plants (as a soil water potential) on evapotranspiration amount. With our knowledge of soil physics and plant physiology knowledge, it can be assumed that evapotranspiration during sufficient water supply does not depend or slightly depend on soil moisture (Łabędzki et al., 1996, as cited in: Kowalik, 1973; Salisbury & Ross, 1975; Feddes et al., 1978; Rewut, 1980; Olszta, 1981; Korohoda, 1985; Więckowski, 1985; Brandyk, 1990). Sufficient water supply does not limit evapotranspiration and plant yield is defined as a soil moisture range between optimum water content (when air content equals at least 8 – 10% in root zone) and refill point (pF 2.7 – 3.0). In other words, sufficient water supply means easily available water or readily available water (RAW). Evapotranspiration reductions has a place, when

ET f M f P f S 1 23 (1)

f M ET 1 0 (2)

fP k 2 c (3)

coefficient approach with Penman-type formula as a reference evapotranspiration.

**2. Reviewing the selected methods for evapotranspiration estimation:** 

**Modified Penman, Penman-Monteith and Thornthwaite** 

dependence can be expressed as a product of following functions:

where:

of soil water:

M – atmosphere factors, P – plant factors,

S – soil moisture factors.

independently by soil moisture conditions:

evapotranspiration: air temperature and day length.

ks() – soil coefficient as a function of soil moisture.

Summarizing, equation (1) can be noted as below, where ETa is called actual evapotranspiration:

$$\text{ETa} = \text{ET}\_0 \cdot \text{k}\_c \cdot \text{k}\_s \tag{5}$$

In cases when sufficient water supply does not limiting evapotranspiration (ks = 1), actual evapotranspiration (ETa) equals potential evapotranspiration (ETp):

$$\text{ETp} = \text{ET}\_0 \cdot \text{k}\_c \tag{6}$$

The problem becomes how to determine a reference evapotranspiration and a crop coefficient.

#### **2.1 The reference evapotranspiration computing by the Modified Penman method**

Penman (1948) estimated the evaporation from an open water surface, and than used that as a reference evaporation. This method requires measured climatic data on temperature, humidity, solar radiation and wind speed. Analyzing a range of lysimeter data worldwide, Doorenbos and Pruitt (1977) proposed the FAO Modified Penman method. These authors adopted the same approach as Penman to estimate reference evapotranspiration. They replaced Penman's open water evaporation with evapotranspiration from a reference crop. The reference crop was defined as "an extended surface of an 8 to 15 cm tall green grass cover of uniform height, actively growing, completely shading the ground, and not short of water". The reference evapotranspiration according to Modified Penman method commonly applied in Poland was calculated by the following algorithm. This algorithm was developed according to following literature: Roguski et al. (1988); Feddes & Lenselink (1994), Kowalik (1995), Kędziora (1999), Woś (1995), Łabędzki et al. (1996), Łabędzki (1997), Feddes et al. (1997) and van Dam et al. (1997). The parameters are as follows:


J – day number [-],

T – daily average air temperature [C],

RH - daily average relative humidity [%],

hi - anemometer level above ground level [m],

vhi – average wind speed on 10 m level [m s-1],

c – average daily cloudiness in 11 degree scale,

n – duration of direct sunshine [h],

Ra - solar radiation at the external atmosphere border [W m-2],



 - latent heat of vaporization equals to 2.45 [MJ kg-1], - Stefan – Boltzmann constant equals to 4.903\*10-9 [MJ m-2 K-4 d-1], Gsc – solar constant equals to 0.082 [MJ m-2 min-1]. Saturation vapour pressure (ed) [kPa]:

$$\mathbf{e}\_{\rm d} = 0.6108 \cdot \exp\left(\frac{17.27 \cdot \rm T}{\rm T + 237.3}\right) \tag{7}$$

Actual vapour pressure (ea) [kPa]:

$$\mathbf{e}\_{\mathbf{a}} = \frac{\mathbf{R}\mathbf{H}}{100} \cdot \mathbf{e}\_{\mathbf{d}} \tag{8}$$

The slope of the vapour pressure curve ( [kPa C-1]:

$$
\Delta = \frac{4098 \cdot \text{e}\_d}{\left(\text{T} + 237.3\right)^2} \tag{9}
$$

Wind speed on 10 m level above ground level (v10) [m s-1]:

$$\mathbf{v}\_{10} = \frac{\mathbf{v}\_{\text{hi}}}{\left(\frac{\mathbf{h}\_{\text{i}}}{10}\right)^{\mathbf{J}\_{\text{T}}}} \tag{10}$$

Solar declinations ([rad]:

$$\delta = 0.409 \cdot \sin\left(\frac{2\pi}{365} \cdot \text{J} - 1.39\right) \tag{11}$$

Relative distance to the Sun (dr) [-]:

$$\mathbf{d}\_{\mathbf{r}} = 1 + 0.033 \cdot \cos\left(\frac{2\pi}{365} \cdot \mathbf{J}\right) \tag{12}$$

Time from sunrise to noon (ws) [rad]:

$$\mathbf{w}\_s = \mathbf{a} \cos \left( -\tan \boldsymbol{\uprho} \cdot \tan \boldsymbol{\updelta} \right) \tag{13}$$

Possible sunshine (N) [h]:

$$\mathbf{N} = \frac{24}{\pi} \cdot \mathbf{w}\_s \tag{14}$$

Solar radiation at the external atmosphere border (Ra) [W m-2]:

$$\mathbf{R\_a = \frac{24 \cdot 60}{\pi} \cdot G\_{\text{sc}} \cdot d\_{\text{r}} \cdot \left(\mathbf{w\_s \cdot \sin \phi \cdot \sin \delta + \cos \phi \cdot \cos \delta \cdot \sin \mathbf{w\_s}}\right)}\tag{15}$$

Relation between real radiation to possible radiation – in case when sunshine value is not available there is calculated according to Angstöm criteria:

$$\frac{\text{m}}{\text{N}} = 1 - \frac{\text{c}}{\text{10}} \tag{16}$$

The net incoming short wave radiation flux (Rns) [W m-2]:

$$\mathbf{R\_{ns}} = \mathbf{R\_a} \cdot (1 - \alpha) \cdot \left( 0.209 + 0.565 \cdot \frac{\mathbf{n}}{\mathbf{N}} \right) \tag{17}$$

The net outgoing long wave radiation flux (Rnl) [W m-2]:

$$\mathbf{R}\_{\rm nl} = \sigma \cdot \left(\mathbf{T} + 273.2\right)^4 \cdot \left(0.56 - 0.08 \cdot \sqrt{10 \cdot \mathbf{e\_a}}\right) \cdot \left(0.1 + 0.9 \cdot \frac{\mathbf{n}}{\mathbf{N}}\right) \tag{18}$$

The net radiation flux (Rn) [W m-2]:

182 Evapotranspiration – Remote Sensing and Modeling

17.27 T e 0.6108 exp T 237.3 

> a d RH e e

hi 10 1 <sup>7</sup> <sup>i</sup>

v

h 10

<sup>2</sup> 0.409 sin J 1.39 <sup>365</sup> 

<sup>2</sup> d 1 0.033 cos J <sup>365</sup> 

<sup>24</sup> N w

<sup>a</sup> sc r s <sup>s</sup> 24 60 R G d w sin sin cos cos sin w

Relation between real radiation to possible radiation – in case when sunshine value is not

s

4098 e T 237.3

 

> 

v

r

Solar radiation at the external atmosphere border (Ra) [W m-2]:

available there is calculated according to Angstöm criteria:

d 2

(7)

(9)

(10)

(11)

(12)

(14)

(15)

w acos tan tan <sup>s</sup> (13)

<sup>100</sup> (8)


The slope of the vapour pressure curve ( [kPa C-1]:

Wind speed on 10 m level above ground level (v10) [m s-1]:

Gsc – solar constant equals to 0.082 [MJ m-2 min-1].

Saturation vapour pressure (ed) [kPa]:

Actual vapour pressure (ea) [kPa]:

Solar declinations ([rad]:

Relative distance to the Sun (dr) [-]:

Time from sunrise to noon (ws) [rad]:

Possible sunshine (N) [h]:


d

$$\mathbf{R\_n = R\_{ns} - R\_{nl}}\tag{19}$$

The aerodynamic factor (Ea) [mm d-1]:

$$\mathbf{E\_a = 2.6 \cdot (e\_d - e\_a) \cdot (1 + 0.4 \cdot v\_{10})} \tag{20}$$

Modified Penman reference evapotranspiration (ETMP) [mm d-1]:

$$\rm ET\_{nr} = \frac{\Delta}{\chi + \Delta} \cdot \frac{\rm R\_n}{\chi} + \frac{\gamma}{\chi + \Delta} \cdot \rm E\_\* \tag{21}$$

#### **2.2 The reference evapotranspiration computing by the Penman-Monteith method**

Among scientists is unanimous the consensus is that the best method of evapotranspiration calculation is a method proposed and developed by John Monteith (1965). Monteith's derivation was built upon that of Penman (1948) in the now well-known combination equation (combination of an energy balance and an aerodynamic formula). The equation describes the evapotranspiration from a dry, extensive, horizontally uniform vegetated surface, which is optimally supplied with water. This equation is known as the Penman-Monteith equation and it is currently recommending by FAO. Potential and even actual evapotranspiration estimates are possible with the Penman-Monteith equation, through the introduction of canopy and air resistance to water vapour diffusion. Nevertheless, since accepted canopy and air resistance may not be available for many crops, a two-step approach is still recommended under field conditions. The first step is the calculation of the reference evapotranspiration as an evapotranspiration of a reference crop for some steady parameters and soil moisture conditions. In the second step the actual evapotranspiration is calculated using the root water uptake reduction due to water stress. The reference crop is defined as "a hypothetical crop which is grass, with a constant, uniform canopy 12 cm tall, constant canopy resistance equals to 70 s m-1, constant albedo equals to 0.23, in conditions of active development and optimally supplied with water" (Łabędzki et al., 1996; Feddes et al., 1997; van Dam et al., 1997; Allen et al., 1998; Howell & Evett, 2004, as cited in: Monteith, 1965). The Penman-Monteith reference evapotranspiration recommended by FAO was calculated by a similar algorithm shown in point 2.1. The difference between the Modified Penman and Penman-Monteith methods bases on solar radiation and an aerodynamic formula calculation in general. Named factors were calculated according to following formulas shown below (Feddes & Lenselink, 1994).

The following parameters were used:


TKmin – daily minimum air temperature [K],

TKmax – daily maximum air temperature [K],

v – average wind speed on 2 m level [m s-1],


Solar radiation at the external atmosphere border (Ra) [W m-2]:

$$\mathbf{R\_a = 435 \cdot d\_r \cdot \left(\mathbf{w\_s \cdot \sin\phi \cdot \sin\delta + \cos\phi \cdot \cos\delta \cdot \sin\mathbf{w\_s}\right)}} \tag{22}$$

Solar radiation (Rs) [W m-2]:

$$\mathbf{R\_s = R\_a \cdot \left[0.25 + \left(0.5 \cdot \frac{\mathbf{n}}{\mathbf{N}}\right)\right]}\tag{23}$$

The net incoming short wave radiation flux (Rns) [W m-2]:

$$\mathbf{R\_{rs}} = \left(1 - \alpha\right) \cdot \mathbf{R\_s} \tag{24}$$

The net outgoing long wave radiation flux (Rnl) [W m-2]:

$$\mathbf{R\_{nl}} = \left(0.9 \cdot \frac{\mathbf{n}}{\mathbf{N}} + 0.1\right) \cdot \left(0.34 - 0.139 \cdot \sqrt{\mathbf{e\_a}}\right) \cdot \sigma \cdot \frac{\left(\mathbf{T\_{K\max}^4} + \mathbf{T\_{K\min}^4}\right)}{2} \tag{25}$$

The radiation factor (Rn' ) [mm d-1]:

$$\mathbf{R}\_{\rm n}^{\prime} = 86400 \cdot \frac{\left(\mathbf{R}\_{\rm rs} - \mathbf{R}\_{\rm nl}\right)}{\lambda} \tag{26}$$

The atmospheric pressure [pa] [kPa]:

$$\mathbf{p\_a} = 101.3 \cdot \frac{\left(\text{T} + 273.16 - 0.0065 \cdot \text{H}\right)}{\text{T} + 273.16} \tag{27}$$

The psychrometric constant () [kPa C]:

$$\gamma = 1615 \cdot \frac{\text{P}\_{\text{a}}}{\text{A}} \tag{28}$$

Modified psychrometric constant (' ) [kPa C]:

$$
\gamma \stackrel{\cdot}{=} \left(1 + 0.337 \cdot \mathbf{v}\right) \cdot \gamma \tag{29}
$$

The aerodynamic factor (Ea) [mm d-1]:

$$\mathbf{E\_a} = \frac{900}{\left(\mathbf{T} + 275\right)} \cdot \mathbf{v} \cdot \left(\mathbf{e\_d} - \mathbf{e\_a}\right) \tag{30}$$

And finally Penman-Monteith reference evapotranspiration (ETP-M) [mm d-1]:

$$\mathbf{ET\_{P-M}} = \frac{\boldsymbol{\Delta}}{\boldsymbol{\Delta} + \boldsymbol{\gamma}} \cdot \mathbf{R\_n}^{\cdot} + \frac{\boldsymbol{\gamma}}{\boldsymbol{\Delta} + \boldsymbol{\gamma}} \cdot \mathbf{E\_a} \tag{31}$$

#### **2.3 Crop coefficient**

184 Evapotranspiration – Remote Sensing and Modeling

formula calculation in general. Named factors were calculated according to following

R 435 d w sin sin cos cos sin w a rs <sup>s</sup> (22)

N

4 4

(25)

R1 R ns s (24)

K max K min

R 86400 (26)

' 1 0.337 v (29)

(23)

(27)

(28)

(30)

 

formulas shown below (Feddes & Lenselink, 1994).


The net incoming short wave radiation flux (Rns) [W m-2]:

The net outgoing long wave radiation flux (Rnl) [W m-2]:

) [mm d-1]:

a


s a

nl a

n

<sup>n</sup> R R 0.25 0.5

T T <sup>n</sup> R 0.9 0.1 0.34 0.139 e

p 101.3 T 273.16 

) [kPa C]:

pa <sup>1615</sup>

 <sup>a</sup> d a <sup>900</sup> <sup>E</sup> ve e T 275

N 2 

ns nl '

R R

T 273.16 0.0065 H

The following parameters were used:

Solar radiation (Rs) [W m-2]:

The radiation factor (Rn'

The atmospheric pressure [pa] [kPa]:

The psychrometric constant () [kPa C]:

Modified psychrometric constant ('

The aerodynamic factor (Ea) [mm d-1]:

TKmin – daily minimum air temperature [K], TKmax – daily maximum air temperature [K], v – average wind speed on 2 m level [m s-1],

Potential evapotranspiration is calculated by multiplying ETo by kc, a coefficient expressing the difference in evapotranspiration between the cropped and reference grass surface. The difference can be combined into a single coefficient, or it can be split into two factors describing separately the differences in evaporation and transpiration between both surfaces. The selection of the approach depends on the purpose of the calculation, the accuracy required, the climatic data available and the time step with which the calculations are executed (Allen et al., 1998). Due to the purpose of this chapter, only the single coefficient approach is taken under consideration. The single crop coefficient combined the effect of crop transpiration and soil evaporation. The crop coefficient expresses crop actual mass and development stage influence on the evapotranspiration value, in sufficient soil moisture content. It is dependant on crop type, development stage and yield. The generalized crop coefficient curve is shown in Figure 1. Shortly after the planting of annuals or shortly after the initiation of new leaves for perennials, the value for kc is small, often less than 0.4. The kc begins to increase from the initial kc value, kc ini, at the beginning of rapid plant development and reaches a maximum value, kc mid, at the time of maximum or near maximum plant development. During the late season period, as leaves begin to age and senesce due to natural or cultural practices, the kc begins to decrease until it reaches a lower value at the end of the growing period equal to kc end (Roguski et al., 1988; Allen et al., 1998).

Fig. 1. Crop coefficient due to plant development stage

The objective of this work is to determine the crop coefficient needed when estimating the potential evapotranspiration with the Penman-Monteith method, when the potential evapotranspiration calculated as a product of Modified Penman reference evapotranspiration and appropriate crop coefficient for this method is known. Based on procedures proposed by Feddes et al. (1997), the conversion of the Modified Penman crop coefficient kc MP to the Penman-Monteith crop coefficient kc P-M can be write as:

$$\mathbf{ETp} = \mathbf{ET}\_{\mathbf{MP}} \cdot \mathbf{k}\_{\mathbf{c}\,\mathrm{MP}} = \mathbf{ET}\_{\mathrm{P}-\mathbf{M}} \cdot \mathbf{k}\_{\mathbf{c}\,\mathrm{P}-\mathbf{M}} \tag{32}$$

from which:

$$\mathbf{k}\_{\rm cP-M} = \frac{\mathbf{E} \mathbf{T}\_{\rm MP} \cdot \mathbf{k}\_{\rm cMP}}{\mathbf{E} \mathbf{T}\_{\rm P-M}} \tag{33}$$

#### **2.4 Potential evapotranspiration estimation by the Thornthwaite method**

Both Modified Penman and Penman-Monteith methods required many climatic inputs like: air temperature, relative humidity, wind speed and solar radiation or at least daily sunshine. These are limited or even not available for many regions. Another problem is noncontinuous data series for some periods. Thus using the Modified Penman and Penman-Monteith methods for evapotranspiration calculation is not so easy and problematic in some cases. An alternative commonly used in the United States is the Thornthwaite method, because it requires only air temperature as a input data (Skaags, 1980; Newman, 1981). This method is based on determination of available energy required for the evaporation process. The relationship between average monthly air temperature and potential evapotranspiration is calculated based on a standard 30 days month with 12 hours of daylight each day according to the following equation (Byczkowski, 1979; Newman, 1981; Pereira & Pruitt, 2004):

$$\text{ETp}\_{\text{T}} = 16.2 \cdot \left(\frac{10 \cdot \text{T}\_{\text{j}}}{\text{I}}\right)^{\text{a}} \tag{34}$$

where:

ETpT – Thornthwaite monthly potential evapotranspiration (mm),

df – correction factor for daylight hours and days in month (-),

Tj – average monthly air temperature (C),

I – annual heat index as a sum of monthly heat index Ii:

$$\mathbf{I} = \sum\_{i=1}^{12} \mathbf{I}\_i = \sum\_{i=1}^{12} \left(\frac{\mathbf{T}\_i}{5}\right)^{1.514} \tag{35}$$

a – coefficient derived from climatological data:

$$\mathbf{a} = 6.75 \cdot 10^{-7} \cdot \mathbf{I}^3 - 7.71 \cdot 10^{-5} \cdot \mathbf{I}^2 + 1.79 \cdot 10^{-2} \cdot \mathbf{I} + 0.492 \tag{36}$$

In order to convert the estimates from a standard monthly ETpT to a decade of evapotranspiration the following correction factor for daylight hours and days in month df (-) was used:

$$\mathbf{d}\_{\mathbf{f}} = \frac{\mathbf{N}\_{\text{dec}}}{360} \tag{37}$$

where:

186 Evapotranspiration – Remote Sensing and Modeling

procedures proposed by Feddes et al. (1997), the conversion of the Modified Penman crop

MP cMP

ET k

ET

Both Modified Penman and Penman-Monteith methods required many climatic inputs like: air temperature, relative humidity, wind speed and solar radiation or at least daily sunshine. These are limited or even not available for many regions. Another problem is noncontinuous data series for some periods. Thus using the Modified Penman and Penman-Monteith methods for evapotranspiration calculation is not so easy and problematic in some cases. An alternative commonly used in the United States is the Thornthwaite method, because it requires only air temperature as a input data (Skaags, 1980; Newman, 1981). This method is based on determination of available energy required for the evaporation process. The relationship between average monthly air temperature and potential evapotranspiration is calculated based on a standard 30 days month with 12 hours of daylight each day according to the following equation (Byczkowski, 1979;

P M

a j

10 T

1.514 <sup>12</sup>

j

 

7 3 5 2 <sup>2</sup> a 6.75 10 I 7.71 10 I 1.79 10 I 0.492 (36)

(35)

<sup>360</sup> (37)

5 T

12 i 1

In order to convert the estimates from a standard monthly ETpT to a decade of evapotranspiration the following correction factor for daylight hours and days in month df

> f <sup>N</sup> <sup>d</sup>

dec

ETp ET k ET k MP cMP P M cP M (32)

(33)

(34)

coefficient kc MP to the Penman-Monteith crop coefficient kc P-M can be write as:

cP M

**2.4 Potential evapotranspiration estimation by the Thornthwaite method** 

T

i 1

i

<sup>I</sup> <sup>I</sup> 

ETpT – Thornthwaite monthly potential evapotranspiration (mm), df – correction factor for daylight hours and days in month (-),

ETp 16.2 <sup>I</sup>

k

from which:

where:

(-) was used:

Newman, 1981; Pereira & Pruitt, 2004):

Tj – average monthly air temperature (C),

a – coefficient derived from climatological data:

I – annual heat index as a sum of monthly heat index Ii:

Ndec - possible sunshine for decade (h)

It must to be noted, that the Thornthwaite method is valid for average monthly air temperature from 0 to 26.5 °C.

## **3. Grasslands and pastures in the north-eastern part of Poland and local condition climate data**

As Statistical Yearbook of Agriculture and Rural Areas (2009) presents, grasslands and pastures occupy about 3271.2 thousand hectares which is 20% of the total agricultural land in Poland. According to administrative division, the north-eastern part of Poland are Podlaskie and the eastern part of Warmińsko-Mazurskie voivodships. Grasslands and pastures occupy 393.5 thousand hectares (35%) and 290 thousand hectares (28.1%) of these voivodships agricultural land respectively. The valley of the River Biebrza, (22° 30′–23° 60′ E and 53° 30′–53° 75′ N) (Fig. 2) is one of the last extensive undrained valley mires in Central Europe. The Biebrza features several types of mires. The dominant types are fens, which account for some 75.9% of the wetland area (Okruszko, 1990). The altitude of the valley ranges from 100 to 130 m above mean sea level and the catchment area of approximately 7000 km2 has a maximum altitude of 160 m (Byczkowski & Kicinski, 1984). The mean yearly rainfall is 583 mm, of which 244 mm falls in the wet summers. Mean annual temperature is rather low (6.8 °C), and the growing season is quite short (around 200 days) (Kossowska-Cezak, 1984). The part of Warmińsko-Mazurskie voivodship is Warmia region. Main town (former capital of Warmia region) situated on the north part of Warmia region (Fig. 2) is Lidzbark Warmiński (20° 35′ E, 54° 08′ N).

Fig. 2. An approximate location of considered regions in Poland

The altitude of the region ranges from 80 to 100 m above mean sea level on the borders and falls down from 40 to 50 m above mean sea level to the center. Brown Soils and Mollic Gleysols developed from silt and clay dominate in this. These soils are situated on sloping areas with partly well surface water outflow. In the study region average yearly air temperature is equal to 7.1°C and average yearly sum of precipitation equal to 624 mm. The highest amount of rainfall is usually observed in July and August. The vegetation period lasts about 200 days. The snow cover occurs during 60–65 days (Nowicka et al., 1994). The needed meteorological data are available for the 1989-2004 grassland growing seasons derived from the Biebrza meteorological station located in the Middle Biebrza River Basin. The estimation of the pasture evapotranspiration will be based on the meteorological data collected in the Warmia region during the 1999 through 2010 period.

## **4. Results and discussion**

The decade Modified Penman and Penman-Monteith reference evapotranspiration values were calculated both for Warmia Region and Middle Biebrza River Basin. The relationship between reference evapotranspiration values of two kinds of Penman methods was shown on Fig. 3.

Fig. 3. The relationship between the Modified Penman and the Penman-Monteith reference evapotranspiration for: a) Middle Biebrza River Basin, b) Warmia Region

The relationship was fitted by linear regression through origin. Obtained linear equations indicates there is not significant difference between reference evapotranspiration calculated with Modified Penman and Penman-Monteith methods in both cases. It must to be noted that there is very good correlation between Modified Penman and Penman-Monteith methods. The coefficient of determination r2 is equall to 99.7% and 99.8% respectively. Due to linear equation, Penman-Monteith reference evapotranspiration values are about 2% lower than values calculated by Modified Penman method for Middle Biebrza River Basin case (Fig. 3a). Whereas, an opposite situation was observed for Warmia Region. Reference evapotranspiration values calculated by the Modified Penman are 1.6% lower than values obtained by the Penman-Monteith method (Fig. 3b).

areas with partly well surface water outflow. In the study region average yearly air temperature is equal to 7.1°C and average yearly sum of precipitation equal to 624 mm. The highest amount of rainfall is usually observed in July and August. The vegetation period lasts about 200 days. The snow cover occurs during 60–65 days (Nowicka et al., 1994). The needed meteorological data are available for the 1989-2004 grassland growing seasons derived from the Biebrza meteorological station located in the Middle Biebrza River Basin. The estimation of the pasture evapotranspiration will be based on the meteorological data

The decade Modified Penman and Penman-Monteith reference evapotranspiration values were calculated both for Warmia Region and Middle Biebrza River Basin. The relationship between reference evapotranspiration values of two kinds of Penman methods was shown

Fig. 3. The relationship between the Modified Penman and the Penman-Monteith reference

The relationship was fitted by linear regression through origin. Obtained linear equations indicates there is not significant difference between reference evapotranspiration calculated with Modified Penman and Penman-Monteith methods in both cases. It must to be noted that there is very good correlation between Modified Penman and Penman-Monteith methods. The coefficient of determination r2 is equall to 99.7% and 99.8% respectively. Due to linear equation, Penman-Monteith reference evapotranspiration values are about 2% lower than values calculated by Modified Penman method for Middle Biebrza River Basin case (Fig. 3a). Whereas, an opposite situation was observed for Warmia Region. Reference evapotranspiration values calculated by the Modified Penman are 1.6% lower than values

evapotranspiration for: a) Middle Biebrza River Basin, b) Warmia Region

Y = 1.0161 X r = 0.998

0 5 10 15 20 25 30 35 40 45 50 55 60 Reference evapotranspiration calculated with Modiffied Penman method (mm)

calculated fitted

collected in the Warmia region during the 1999 through 2010 period.

0 5 10 15 20 25 30 35 40 45 50 55 60 Reference evapotranspiration calculated with Modiffied Penman method (mm)

obtained by the Penman-Monteith method (Fig. 3b).

a) b)

**4. Results and discussion** 

Y = 0.979 X r = 0.997

on Fig. 3.

calulated with

Reference

evapotranspiration

Penman-Monteith

 method (mm) Consequently, an attempt was made for crop coefficient calculation (Eq. 33) proper for determination of potential evapotranspiration with the Penman-Monteith method. The following croplands were taken under consideration: pasture located in Warmia Region and intensive meadow, extensive meadow and natural wetland plant communities characteristic of Middle Biebrza River Basin. The calculation was conducted for vegetation period decade values of Modified Penman and Penman-Monteith reference evapotranspiration and crop coefficient for the Modified Penman method elaborated by Roguski et al. (1988), Brandyk et al. (1996) and Szuniewicz & Chrzanowski (1996). Considered values of crop coefficient both for Modified Penman (kc MP) and Penman-Monteith (kc P-M) for pasture was presented on Table 1. It can be maintain that kc P-M values for April are about 0.05 lower than kc MP values. The values for May, June and July are the same or almost the same – the difference does not exceed 0.02. The most significant differences are present in September, where kc P-M is lower than kc MP from 0.09 to 0.21.


Table 1. Crop coefficient of pasture for Modified Penman and Penman-Monteith methods

Modified Penman crop coefficient for extensive meadows (EM) and natural wetlands plant communities (NWPC) was published by Brandyk et al. (1996) as cited in: Roguski (1985) and Łabędzki & Kasperska (1994). Values of these crop coefficients as well as values of calculated Penman-Monteith crop coefficients was presented on Table 2. It can be maintain that kc P-M values are higher than kc MP values from 0.01 to 0.12 for extensive meadow in general. An exception to this rule is the last five decades, when kc P-M values are lower then kc MP values from 0.01 to 0.23. A similar tendency can be observed for natural wetland plant communities. But wider differences occur between kc P-M and kc MP. A value of kc P-M is higher up to 0.08 than kc MP value for a few decades and lower until 0.31 for the last decade of September.


Table 2. Crop coefficient of extensive meadow and natural wetland plant communities for Modified Penman and Penman-Monteith methods

The Modified Penman crop coefficient for intensive meadow located in Middle Biebrza River Basin was elaborated by Szuniewicz & Chrzanowski (1996). They based the research on lysimeter experiments conducted on peat –moorsh soil with a ground water level of 35 – 90 cm (optimum soil moisture) during the 1982-1991 period. Researchers had established conditions for 3-cut meadows with different hay yields: 0.10, 0.20, 0.30, 0.40 and 0.50 Mg ha-1. The climate of the considered region is more severe compared to other plain regions in Poland, thus the vegetation period starts about two weeks later. Elaborated by Szuniewicz & Chrzanowski crop coefficients for the Modified Penman method as well as calculated crop coefficients for Penman-Monteith was presented on Table 2. There are not significant differences between kc P-M and kc MP values for the first two decades of the vegetation period.

general. An exception to this rule is the last five decades, when kc P-M values are lower then kc MP values from 0.01 to 0.23. A similar tendency can be observed for natural wetland plant communities. But wider differences occur between kc P-M and kc MP. A value of kc P-M is higher up to 0.08 than kc MP value for a few decades and lower until 0.31 for the last decade

> Crop coefficient EM NWPC kc MP kc P-M kc MP kc P-M

1 0.93 1.05 0.62 0.70 2 0.93 0.97 0.79 0.83 3 0.85 0.84 0.75 0.74

1 0.88 0.90 0.77 0.79 2 1.04 1.09 1.06 1.10 3 1.03 1.08 1.21 1.27

1 0.76 0.79 1.24 1.30 2 0.91 0.96 1.28 1.36 3 0.98 1.04 1.40 1.48

1 0.99 1.03 1.32 1.37 2 1.01 1.06 1.18 1.23 3 0.98 1.04 1.40 1.48

1 0.97 0.98 1.30 1.31 2 1.07 1.07 1.40 1.39 3 1.18 1.15 1.40 1.36

1 1.34 1.27 1.63 1.55 2 1.41 1.27 1.85 1.66 3 1.41 1.18 1.60

Table 2. Crop coefficient of extensive meadow and natural wetland plant communities for

The Modified Penman crop coefficient for intensive meadow located in Middle Biebrza River Basin was elaborated by Szuniewicz & Chrzanowski (1996). They based the research on lysimeter experiments conducted on peat –moorsh soil with a ground water level of 35 – 90 cm (optimum soil moisture) during the 1982-1991 period. Researchers had established conditions for 3-cut meadows with different hay yields: 0.10, 0.20, 0.30, 0.40 and 0.50 Mg ha-1. The climate of the considered region is more severe compared to other plain regions in Poland, thus the vegetation period starts about two weeks later. Elaborated by Szuniewicz & Chrzanowski crop coefficients for the Modified Penman method as well as calculated crop coefficients for Penman-Monteith was presented on Table 2. There are not significant differences between kc P-M and kc MP values for the first two decades of the vegetation period.

Month Decade

April

May

June

July

August

September

Modified Penman and Penman-Monteith methods

of September.

The differences increase during successive decades of May and June from 0.02 up to 0.07. Next, they decrease from 0.04 to 0.02 in July. There are not significant differences again for first and second decades of July. The difference begins it's increase from the third decade of July up to the second decade of September. The values of kc P-M are even 0.12 – 0.18 lower than kc MP for the second decade of September. There is also a clear tendency towards an increase of differences between crop coefficients kc P-M and kc MP values due to an increase of potential hay yield. The kc P-M values get higher from 0.02 to 0.07 in May and June. However, the opposite tendency can be observed in September, when kc P-M get lower from 0.06 to even 0.18.


Table 3. Crop coefficient of 3-cut meadow for Modified Penman and Penman-Monteith methods

The next step of this work use to be an comparison potential evapotranspiration calculated as a product of Penman-Monteith reference evapotranspiration and determined crop coefficient (kc P-M) with alternative potential evapotranspiration by Thornthwaite. In order to solve the problem, decade values of Thornthwaite potential evapotranspiration was calculated (Eq. 34-37) and Penman-Monteith potential evapotranpiration applying crop coefficient for proper land use. The relationship between Thornthwaite potential evapotranspiration and Penman-Monteith potential evapotranspiration was presented on Fig. 4. The relationship was fitted by linear regression through origin. Analyzing obtained results, it can be maintain that Penman-Monteith evapotranspiration values are lower by about 25% for pasture (Fig. 4a) and 8% for extensive meadow than the Thornthwaite method

Fig. 4. The relationship between Thornthwaite potential evapotranspiration and Penman potential evapotranspiration for: pasture (a), extensive meadow (b) and natural wetland plant communities (c)

calculated with Thornthwaite method (mm)

evapotranspiration and Penman-Monteith potential evapotranspiration was presented on Fig. 4. The relationship was fitted by linear regression through origin. Analyzing obtained results, it can be maintain that Penman-Monteith evapotranspiration values are lower by about 25% for pasture (Fig. 4a) and 8% for extensive meadow than the Thornthwaite method

1:1

Y = 0.752 X r = 0.961

Y = 1.172 X r = 0.969

Fig. 4. The relationship between Thornthwaite potential evapotranspiration and Penman potential evapotranspiration for: pasture (a), extensive meadow (b) and natural wetland

1:1

a) b)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Potential evapotranspiration calculated with Thornthwaite method (mm)

> calculated fitted

1:1

Y = 0.922 X r = 0.966

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Potential evapotranspiration calculated with Thornthwaite method (mm)

c)

plant communities (c)

calculated with

Potential

evapotranspiration

Penman-Monteith

 method (mm)

calulated with

Potential

evapotranspiration

Penman-Monteith

 method (mm)

Fig. 5. The relationship between Thornthwaite potential evapotranspiration and Penman potential evapotranspiration of 3-cut meadow for hay yield Mg ha-1: 0.10 (a), 0.20 (b), 0.30 (c) and 0.40 (d)

(Fig. 4b). Whereas in case of natural wetland plan community evapotranspiration, values calculated with Penman-Monteith method are of about 17% higher then values calculated with Thornthwaite method. It must to be noted, that coefficient of determination is almost equal (r2 ≈ 97%) for all three cases. The relationship between Thornthwaite potential evapotranspiration and Penman-Monteith potential evapotranspiration for 3-cut meadow was presented on Fig. 5. Analyzing obtained results, it can be maintained that Penman-Monteith evapotranspiration values are very close to Thornthwaite evapotranspiration values for 0.30 Mg ha-1 hay yield. An evapotranspiration calculated with the Thornthwaite method is just about 2% higher than Penman-Monteith evapotranspiration. The highest overestimation (20%) of the Thornthwaite method is observed for the lowest hay yield (0.10 Mg ha-1). The case of 0.20 Mg ha-1 hay yield characterizes about a 10% overestimation of the Thornthwaite method. An opposite case is the case of 0.40 Mg ha-1 hay yield, where the Thornthwaite method underestimates evapotranspiration by about 5%. Coefficients of determination vary between 94.3% (0.40 Mg ha-1 hay yield) and 96.6% (0.10 Mg ha-1 hay yield).

## **5. Conclusion**

Based on the performed research the following conclusions can be formulated:

There are not significant differences between reference evapotranspiration calculated with the Modified Penman and Penman-Monteith methods of the Warmia Region as well as Middle Biebrza River Basin for entire vegetation period (April – September). Due to linear equation, Penman-Monteith reference evapotranspiration values are about 1.6 % higher than values calculated by the Modified Penman method for the Warmia Region case. Whereas, values of Modified Penman reference evapotranspiration are about 2.0% lower than values obtained with the Penman-Monteith method. From a practical point of view, the difference of total vegetation period reference evapotranspiration equals about 8 mm for the Warmia Region and 10 mm for Middle Biebrza River Basin due to 513 mm (Warmia Region) and 486 mm (Middle Biebrza River Basin) of average vegetation period reference evapotranspiration assumption.

Crop coefficients calculated for the Penman-Monteith evapotranspiration method are comparable or lower than crop coefficients for the Modified Penman method in case of pasture. Taking under consideration crop coefficient differences for extensive meadow and natural wetland plant communities it can be found that kc P-M values are higher than kc MP values from 0.01 to 0.12 for most of the vegetation period in general. An exception to this rule is the last five decades, when kc P-M values were lower then kc MP values from 0.01 even to 0.31. There are not significant differences between kc P-M and kc MP values for the first and second decades of vegetation period as well as for the first and second decades of July in the case of 3-cut meadow. The difference begins to from the third decade of July up to the second decade of September. The values of kc P-M are even 0.12 – 0.18 lower than kc MP for the second decade of September. Summarizing, crop coefficients calculated for Penman-Monteith method are almost equal or slightly higher compare to Modified Penman crop coefficients for most of a vegetation period in all considered land use. An exception are last three to four decades of vegetation period when values of kc P-M are clearly lower compared to kc MP values. These differences are equal during the entire vegetation period. But they can have essential meaning in certain parts (decades) of vegetation period when a crop water requirement is determined.

Potential evapotranspiration values calculated with the Thornthwaite method are overestimated in ratio to values calculated with the Penman-Monteith method in the following cases by about: 25% for pasture, 20% for 3-cut meadow (0.10 Mg ha-1 hay yield), 10% for 3-cut meadow (0.20 Mg ha-1 hay yield) and 8% for extensive meadow. Whereas, one time Thornthwaite potential evapotranspiration values were lower by about 5% for 3-cut meadow (0.40 Mg ha-1 hay yield). The best convergence of the considered methods is observed for 3-cut meadow in case of 0.30 Mg ha-1. It has to be said, that coefficient of determination r2 exceeds 94% of the value for all cases. Summarized, the Thornthwaite potential evapotranspiration method is comparable with the Penman-Monteith method for 3-cut meadow with a high value of hay yield and extensive meadow.

Future research should be focused on trials to find correlations between Thornthwaite and Penman-Monteith potential evapotranspiration for individual months of vegetation period. Another aim could be crop coefficient calculation for the Penman-Monteith method for field crops like grains, potatoes or sugar beets.

## **6. Acknowledgment**

194 Evapotranspiration – Remote Sensing and Modeling

(0.10 Mg ha-1). The case of 0.20 Mg ha-1 hay yield characterizes about a 10% overestimation of the Thornthwaite method. An opposite case is the case of 0.40 Mg ha-1 hay yield, where the Thornthwaite method underestimates evapotranspiration by about 5%. Coefficients of determination vary between 94.3% (0.40 Mg ha-1 hay yield) and 96.6%

There are not significant differences between reference evapotranspiration calculated with the Modified Penman and Penman-Monteith methods of the Warmia Region as well as Middle Biebrza River Basin for entire vegetation period (April – September). Due to linear equation, Penman-Monteith reference evapotranspiration values are about 1.6 % higher than values calculated by the Modified Penman method for the Warmia Region case. Whereas, values of Modified Penman reference evapotranspiration are about 2.0% lower than values obtained with the Penman-Monteith method. From a practical point of view, the difference of total vegetation period reference evapotranspiration equals about 8 mm for the Warmia Region and 10 mm for Middle Biebrza River Basin due to 513 mm (Warmia Region) and 486 mm (Middle Biebrza River Basin) of average vegetation period reference evapotranspiration

Crop coefficients calculated for the Penman-Monteith evapotranspiration method are comparable or lower than crop coefficients for the Modified Penman method in case of pasture. Taking under consideration crop coefficient differences for extensive meadow and natural wetland plant communities it can be found that kc P-M values are higher than kc MP values from 0.01 to 0.12 for most of the vegetation period in general. An exception to this rule is the last five decades, when kc P-M values were lower then kc MP values from 0.01 even to 0.31. There are not significant differences between kc P-M and kc MP values for the first and second decades of vegetation period as well as for the first and second decades of July in the case of 3-cut meadow. The difference begins to from the third decade of July up to the second decade of September. The values of kc P-M are even 0.12 – 0.18 lower than kc MP for the second decade of September. Summarizing, crop coefficients calculated for Penman-Monteith method are almost equal or slightly higher compare to Modified Penman crop coefficients for most of a vegetation period in all considered land use. An exception are last three to four decades of vegetation period when values of kc P-M are clearly lower compared to kc MP values. These differences are equal during the entire vegetation period. But they can have essential meaning in certain parts (decades) of vegetation period when a crop water

Potential evapotranspiration values calculated with the Thornthwaite method are overestimated in ratio to values calculated with the Penman-Monteith method in the following cases by about: 25% for pasture, 20% for 3-cut meadow (0.10 Mg ha-1 hay yield), 10% for 3-cut meadow (0.20 Mg ha-1 hay yield) and 8% for extensive meadow. Whereas, one time Thornthwaite potential evapotranspiration values were lower by about 5% for 3-cut meadow (0.40 Mg ha-1 hay yield). The best convergence of the considered methods is observed for 3-cut meadow in case of 0.30 Mg ha-1. It has to be said, that coefficient of determination r2 exceeds 94% of the value for all cases. Summarized, the Thornthwaite potential evapotranspiration method is comparable with the Penman-Monteith method for

3-cut meadow with a high value of hay yield and extensive meadow.

Based on the performed research the following conclusions can be formulated:

(0.10 Mg ha-1 hay yield).

**5. Conclusion** 

assumption.

requirement is determined.

A part of this work considered to evapotranspiration calculation of Warmia Region was supported by the grant of Polish Ministry of Science and Higher Education No N N305 039234.

Special thanks to friend of mine Dr Jan Szatyłowicz for help with Penman's methods evapotranspiration calculation for Middle Biebrza River Basin.

## **7. References**

Allen R.G., Pereira L.S., Raes D. & Smith M. (1998). Crop evapotranspiration - Guidelines for computing crop water requirements. *FAO Irrigation and Drainage Paper,* No. 56, pp. 290, ISBN 92-5-104219-5, FAO, Rome, Retrieved from:

http://www.fao.org/docrep/x0490e/x0490e00.htm#Contents


## **The Role of Evapotranspiration in the Framework of Water Resource Management and Planning Under Shortage Conditions**

Giuseppe Mendicino and Alfonso Senatore *Department of Soil Conservation, University of Calabria, Arcavacata di Rende (CS) Italy* 

## **1. Introduction**

196 Evapotranspiration – Remote Sensing and Modeling

Newman J.E. (1981). Weekly Water Use Estimates by Crops and Natural Vegetation in

Nowicka A., Banaszkiewicz B. & Grabowska K. (1994). The selected meteorological elements

Okruszko H. (1990). *Wetlands of the Biebrza Valley, their Value and Future Management*, ISBN

Pereira A.R. & Pruitt W.O. (2004). Adaptation of the Thornthwaite scheme for estimating

Roguski W., Sarnacka S. & Drupka S. (1988). Instrukcja wyznaczania potrzeb i niedoborów

Skaggs R.W., (1980). Drainmod Reference Report. U.S. Department of Agriculture, Soil

Statistical Yearbook of Agriculture and Rural Areas (2009). ISSN 1895-121X, Zakład

Szuniewicz J. & Chrzanowski S. (1996). Współczynniki roślinne do obliczania

Van Dam J.C., Huygen J., Wesseling J.G., Feddes R.A., Kabat P., Van Walsum P.E.V.,

0944, Technical Document 45 DLO Winand Staring Centre, Wageningen

wschodniej. *Wiad. IMUZ XVIII(4),* pp. 109-118, ISBN 83-85735-28-3

Experimental Station Purdue University. West Lafayette, Indiana

83-00-03461, Polish Academy of Science, Warszawa

ISSN 0860-0813, Wyd. IMUZ, Falenty

Wydawnictw Statystycznych, Warszawa

Woś A. (1995). *ABC meteorologii*. ISBN 8323207097, U.A.M. Poznań

29.09.1994

3774

19–23

Indiana. *Station Bulletin* No. 344, pp. 1-2, Department of Agronomy, Agricultural

for Olsztyn region in 1951–1990 years with comparison to averages for 1881–1930 period. *Mat. Konf. XXV zjazd agrometeorologow*, Olsztyn-Mierki, Poland, 27-

daily reference evapotranspiration. *Agric. Water Manag*. 66, pp. 251-257, ISSN: 0378-

wodnych roślin uprawnych i użytków zielonych. *Materiały Instruktażowe* 66, pp. 90.,

Conservation Service, North Carolina State University. Raleigh, North Carolina, pp.

ewapotranspiracji łąki trzykośnej na glebie torfowo-murszowej w Polsce północno-

Groenendijk P. & Van Diepen C.A. (1997). *Theory of SWAP version 2.0*, ISSN 0928-

The increased availability of observed data and of advanced techniques for the analysis of meteo-hydrological information allows an even more detailed description of the evolution of global climate. The results showed by the Fourth Assessment Report (FAR) of the International Panel on Climate Change (IPCC, 2007) about the changes that, starting from 1950, are affecting the atmosphere, the cryosphere and the oceans, confirm global warming. The global average surface temperature has increased in the last 100 years by 0.74°C ± 0.18°C, accelerating in the last 50 years (0.13°C ± 0.03°C per decade), especially over land (about 0.27 °C per decade) and at higher northern latitudes. As a consequence, the higher available energy on the surface has speeded up the hydrological cycle. The concentration of the water vapor in the troposphere has increased (1.2 ± 0.3% per decade from 1988 to 2004), while long-period precipitation trends (both positive and negative) in many regions have been observed by analyzing time series from the year 1900 to the year 2005. Changes in temperature and precipitation regimes strongly affect the hydrological cycle. As an example, the increase in temperature has produced a substantial reduction in snow cover in several regions, mainly in spring, and a reduction in the areas covered by seasonal frozen ground (reduction of about 7% in the northern hemisphere over the latter half of the 20th century). Direct long-term measurements of all the main components of the hydrological cycle are not widely available: in order to assess soil moisture long-term changes, due to the lack of direct measurements the primary approach is to calculate Palmer Drought Severity Index, while long-term stream flow gauge records do not cover entirely and uniformly the world, and they present gaps and different record lengths. However, generally stream flow trends are positively correlated to precipitation, while a common effect of climate change is arising independently on precipitation trends: starting from the '70s a considerable increase of the frequency of extreme hydrological events (floods and droughts) has been observed. Also concerning actual evapotranspiration, direct measurements over global land areas are still very limited, but already the Third Assessment Report (TAR) reported that actual evapotranspiration increased during the second half of the 20th century over most dry regions of the USA and Russia, and, by means of observed precipitation, temperature, cloudiness-based surface solar radiation and a land surface model, Qian et al. (2006) found that global land evapotranspiration closely follows variations in land precipitation.

Following the FAR, it is extremely unlikely (<5% probability) that the global warming trend observed in the last half century, whose remarkable characteristics in the history of the Earth seem to be confirmed even by paleo-climatic studies, could be explained without considering external forcings, and is very likely (>90%) that the production of greenhouse gases is the main cause of the observed increase in temperature.

Human activities negatively impact on water resource availability, not only contributing to the water cycle changes on a global scale, but also in a more direct way, through the pollution of water courses and aquifers. This pollution is specifically generated by the overexploitation of the soil and chemical contaminants due to agriculture and forestry, by urban waste, transportation and building, and by the over-exploitation of the coastal aquifers, which generates saline water intrusion.

Many of the problems connected to water shortage and to bad water quality are due to not efficient or even inexistent water resources planning and management. Recently, most advanced planning studies have adopted tools for integrated water resources management. Specifically, by now among planners the idea is diffused that a reactive approach, based on the implementation of actions after a drought event has occurred and is perceived, is not adequate and a proactive approach is needed (Yevjevich et al., 1983; Rossi, 2003), based on the development of plans allowing the identification of long- and short-term actions to face drought, and the implementation of such plans, on the basis of timely information provided by a drought monitoring system.

Different measures can be used to cope with water resource crises due to drought. Rossi et al. (2007) show several classifications of these measures: first, the one suggested by Yevjevich et al. (1978) that distinguishes among measures aimed at increasing water supply, reducing demand and minimizing impacts; next, considering the one differentiating reactive and proactive measures (Yevjevich et al., 1983); and finally, the one between long- and short-term measures. The Water Scarcity Drafting Group (2006) disseminated a document specifying a series of mitigation measures that can be adopted in the EU countries. Pereira (2007), starting from a conceptual distinction between water conservation (referred to the measures for the conservation and preservation of water resource) and water saving (referred to the measures aimed at limiting and/or controlling water demand), points out a set of actions that can be adopted in agriculture to reduce the impacts of drought resulting economically, socially and environmentally more competitive than the "classical" proposal of realizing artificial reservoirs, the latter being an alternative preferred in even fewer cases in the countries where water resource planning is more advanced (e.g. Cowie et al., 2002). Finally, the European Commission in the Communication "Addressing the challenge of water scarcity and droughts in the European Union", adopted on July 18, 2007 (COM, 2007), while stating the necessity of progressing towards the full implementation of the Water Framework Directive 2000/60/EC (WFD), underlines the huge potential for water saving across Europe, where people continue to waste at least 20% of water due to inefficiency, indeed leakages greater than 50% have been recorded in the irrigation networks. A report connected to the EU Communication (Dworak et al., 2007) estimates a potential water saving in the EU of about 40%. Regarding the strategic paths for future interventions, the enhancement of drought risk management should be achieved also through: developing drought risk management plans; developing an observatory (an European Drought

cloudiness-based surface solar radiation and a land surface model, Qian et al. (2006) found

Following the FAR, it is extremely unlikely (<5% probability) that the global warming trend observed in the last half century, whose remarkable characteristics in the history of the Earth seem to be confirmed even by paleo-climatic studies, could be explained without considering external forcings, and is very likely (>90%) that the production of greenhouse

Human activities negatively impact on water resource availability, not only contributing to the water cycle changes on a global scale, but also in a more direct way, through the pollution of water courses and aquifers. This pollution is specifically generated by the overexploitation of the soil and chemical contaminants due to agriculture and forestry, by urban waste, transportation and building, and by the over-exploitation of the coastal aquifers,

Many of the problems connected to water shortage and to bad water quality are due to not efficient or even inexistent water resources planning and management. Recently, most advanced planning studies have adopted tools for integrated water resources management. Specifically, by now among planners the idea is diffused that a reactive approach, based on the implementation of actions after a drought event has occurred and is perceived, is not adequate and a proactive approach is needed (Yevjevich et al., 1983; Rossi, 2003), based on the development of plans allowing the identification of long- and short-term actions to face drought, and the implementation of such plans, on the basis of timely information provided

Different measures can be used to cope with water resource crises due to drought. Rossi et al. (2007) show several classifications of these measures: first, the one suggested by Yevjevich et al. (1978) that distinguishes among measures aimed at increasing water supply, reducing demand and minimizing impacts; next, considering the one differentiating reactive and proactive measures (Yevjevich et al., 1983); and finally, the one between long- and short-term measures. The Water Scarcity Drafting Group (2006) disseminated a document specifying a series of mitigation measures that can be adopted in the EU countries. Pereira (2007), starting from a conceptual distinction between water conservation (referred to the measures for the conservation and preservation of water resource) and water saving (referred to the measures aimed at limiting and/or controlling water demand), points out a set of actions that can be adopted in agriculture to reduce the impacts of drought resulting economically, socially and environmentally more competitive than the "classical" proposal of realizing artificial reservoirs, the latter being an alternative preferred in even fewer cases in the countries where water resource planning is more advanced (e.g. Cowie et al., 2002). Finally, the European Commission in the Communication "Addressing the challenge of water scarcity and droughts in the European Union", adopted on July 18, 2007 (COM, 2007), while stating the necessity of progressing towards the full implementation of the Water Framework Directive 2000/60/EC (WFD), underlines the huge potential for water saving across Europe, where people continue to waste at least 20% of water due to inefficiency, indeed leakages greater than 50% have been recorded in the irrigation networks. A report connected to the EU Communication (Dworak et al., 2007) estimates a potential water saving in the EU of about 40%. Regarding the strategic paths for future interventions, the enhancement of drought risk management should be achieved also through: developing drought risk management plans; developing an observatory (an European Drought

that global land evapotranspiration closely follows variations in land precipitation.

gases is the main cause of the observed increase in temperature.

which generates saline water intrusion.

by a drought monitoring system.

Observatory is now available at http://edo.jrc.ec.europa.eu) and an early drought warning system; further optimizing the use of the EU Solidarity Fund and European Mechanism for Civil Protection; fostering water efficient technologies and practices; fostering the emergence of a water-saving culture in Europe.

In this framework, evapotranspiration assessment is of outstanding importance both for planning and monitoring purposes. Its magnitude (mainly referring to potential evapotranspiration) is comparable to the main forcing of the water balance, i.e. precipitation, and for this reason several climatic classifications are based on comparisons between these two quantities, with the aim of determining specific climate conditions for different areas (e.g. Rivas-Martinez, 1995). Furthermore, evapotranspiration is the only component of the water balance with a central role also in the energy and carbon balance, since it directly accounts for hydrological, agricultural and ecological effects of drought events. Specifically, in agriculture evapotranspiration can be closely related to water demand. This means that the role of evapotranspiration, and losses due to evapotranspiration in agriculture (which are foreseeable to a certain extent) can be handled in a way allowing to assure the best conditions for agricultural needs, if water resources management is correctly planned and implemented. Hence, in this chapter evapotranspiration assessment/water demand fulfillment will be considered within the wider framework of water resources management and planning, both for a correct evaluation of the water balance (considering both the hydrological balance and the differences between water requirements and availability), and for determining incoming drought events through appropriate indices (drought monitoring). The issue of reducing water requirements, meaning loss reductions and/or evapotranspiration reductions (mainly in agriculture) will only be touched on, while dealing with methods and tools for water resource management under shortage conditions.

In the next sections, after an analysis of the available water resource and water demand in a southern Italian region (Calabria), the chapter highlights some weaknesses of the regional water system in rainfall deficit conditions, drafting the main strategies of intervention to be adopted to face the different aspects of drought. Then, some guidelines for the proactive management of drought in agriculture are proposed and specifically, by means of a casestudy related to one of the most important agricultural areas in southern Italy (the Sibari Plain), the development of the three most important operational management tools is shown, i.e. the Strategic Plan for long-term interventions, the Management Plan for shortterm interventions and the Contingency Plan for emergency conditions. Drought indices are important tools for correctly drafting these plans: a specific section will provide some insight about them. Finally, some climatologic and hydrologic scenarios over a specific basin are hypothesized, with the aim of assessing water resource availability in the second half of the present century and of verifying whether the intense and prolonged drought periods currently affecting the Calabria region will become ordinary situations in the near future.

## **2. Natural water resource**

Since no useful information is available for an estimate of the direct runoff volume on the whole region, natural water resource was determined using a distributed monthly water balance model described by Mendicino & Versace (2007) and Mendicino et al. (2008a), which extends the approach proposed by Thornthwaite & Mather (1955) and simulates soil moisture variations, evapotranspiration, and runoff on a 5 km regular grid (Fig. 1) using data sets that include climatic drivers, vegetation, and soil properties. This model does not consider the horizontal motion of water on the land surface, or in the soil (hence no flow routing algorithms are required), and it is based on a simplified mass balance:

$$
\Delta \mathcal{W} = P + SM - SA - ET - Q \tag{1}
$$

where ΔW is the change in soil moisture storage, P the precipitation, SM the snow melt, SA the snow accumulation, ET the actual evapotranspiration, and Q is the runoff (all the quantities are evaluated in mm month-1). In the model, potential evapotranspiration PET is estimated through the Priestley-Taylor method (Priestley & Taylor, 1972), requiring only temperature, air pressure and net radiation data, overcoming the lack of observed wind speed and air humidity data in the analyzed area before the year 2000. In the case of net radiation, monthly values were obtained starting from a modified version of the model originally suggested by Moore et al. (1993). Actual evapotranspiration ET is calculated starting from PET and considering the Accumulated Potential Water Loss (APWL), such as suggested by Thornthwaite & Mather (1955), which represents the total amount of unsatisfied potential evapotranspiration to which the soil has been subjected.

Because of the significant reforestation campaigns carried out in Calabria after the Second World War, whose results were evident already at the end of 1950s, the starting period for the analysis was assumed to be 1957. The assumption of constant soil use (derived by the Corine Land Cover 2000 project) is justified by the coarse resolution of the model (5 km grid). The model schematized in figure 1 was improved also considering: i) that a portion of the rainfall is directly transformed into "instantaneous" runoff (depending on the ratio between actual soil moisture and soil water holding capacity WHC, in its turn derived by combining soil use with a detailed soil texture map of Calabria); ii) an additional very simple snow module, which partitions snow and rain precipitation and regulates snow melt just referring to the current monthly temperature in the cell; iii) that the hydraulic subsoil characteristics are simulated with reservoirs whose rates of depletion vary with the predominating geo-lithological characteristics in the single cells of the model (Mendicino et

Fig. 1. Schematization of the water balance model and overlay of the 5 km regular grid in the analyzed region.

consider the horizontal motion of water on the land surface, or in the soil (hence no flow

where ΔW is the change in soil moisture storage, P the precipitation, SM the snow melt, SA the snow accumulation, ET the actual evapotranspiration, and Q is the runoff (all the quantities are evaluated in mm month-1). In the model, potential evapotranspiration PET is estimated through the Priestley-Taylor method (Priestley & Taylor, 1972), requiring only temperature, air pressure and net radiation data, overcoming the lack of observed wind speed and air humidity data in the analyzed area before the year 2000. In the case of net radiation, monthly values were obtained starting from a modified version of the model originally suggested by Moore et al. (1993). Actual evapotranspiration ET is calculated starting from PET and considering the Accumulated Potential Water Loss (APWL), such as suggested by Thornthwaite & Mather (1955), which represents the total amount of

Because of the significant reforestation campaigns carried out in Calabria after the Second World War, whose results were evident already at the end of 1950s, the starting period for the analysis was assumed to be 1957. The assumption of constant soil use (derived by the Corine Land Cover 2000 project) is justified by the coarse resolution of the model (5 km grid). The model schematized in figure 1 was improved also considering: i) that a portion of the rainfall is directly transformed into "instantaneous" runoff (depending on the ratio between actual soil moisture and soil water holding capacity WHC, in its turn derived by combining soil use with a detailed soil texture map of Calabria); ii) an additional very simple snow module, which partitions snow and rain precipitation and regulates snow melt just referring to the current monthly temperature in the cell; iii) that the hydraulic subsoil characteristics are simulated with reservoirs whose rates of depletion vary with the predominating geo-lithological characteristics in the single cells of the model (Mendicino et

Fig. 1. Schematization of the water balance model and overlay of the 5 km regular grid in the

analyzed region.

*W P SM SA ET Q* (1)

routing algorithms are required), and it is based on a simplified mass balance:

unsatisfied potential evapotranspiration to which the soil has been subjected.

al., 2005). The different characteristics of subsoil leaded to the subdivision of the region into three categories: I) areas with a high capability of producing perennial flow (rocks with high permeability not in the plain); II) areas with mean capability of producing perennial flow (rocks with mean permeability); III) areas with low capability of producing perennial flow (rocks with low permeability or with high permeability in the plain).

The monthly water balance model was validated considering about 2900 monthly runoff values observed in 14 Calabrian catchments during the period 1955-2006 (Fig. 2). Figure 2 also shows the quite satisfactory performance of the model that, besides reproducing the monthly average behaviour of each considered catchment, provided values of the slopes of the regression curves obtained comparing observed and simulated runoff values varying from a minimum of 0.791 (Alli Orso) to a maximum of 1.135 (Esaro La Musica), while the correlation coefficients *r* varied from 0.447 (Coscile Camerata) to 0.939 (Corace Grascio).

Fig. 2. Spatial distribution of the gauged catchments and comparison between all observed and simulated runoff during the period 1960-2006.

The monthly water balance model was applied on the whole territory of Calabria for the period 1960-2006 on a 5 km regular grid, where each cell was independent from the others, determining the main components of the hydrological balance in the whole region: precipitation, actual evapotranspiration, soil moisture storage, groundwater volume and instantaneous, surface and subsurface runoff. In several areas of the region a negative trend was observed for many of these variables. Specifically, while the potential evapotranspiration trend was strongly related to increasing temperature, actual evapotranspiration was affected also by changes (reduction) in precipitation. Considering the whole region, the average annual actual evapotranspiration estimated in the analyzed period is 581 mm, equal to about 57.8% of the average cumulated annual rainfall (potential evapotranspiration is about 110%). Figure 3 (left side) shows the average monthly values in the whole region for actual and potential evapotranspiration. The months where a significant difference can be observed are the months from May to September. In these months (the less rainy and warmest ones), evaporation of soil moisture accumulated in wintertime exceeds rainfall, requiring irrigation in most of the agricultural areas. Figure 3 (right side) also shows the trend of cumulated annual actual evapotranspiration. The decrease in time of this quantity due to rainfall reduction is partly balanced by the increasing temperatures, hence the negative trend is not significant. It is noteworthy that peaks and troughs are generally dependent on rainy (e.g. 2005) or not rainy (e.g. 2001) years, even though rainfall distribution during the single year also affects the evapotranspirative phenomenon. The correlation coefficient between cumulated annual actual evapotranspiration and precipitation in the period 1960-2006 was 0.638.

Fig. 3. Left: average monthly values in Calabria of actual (AE) and potential evapotranspiration (PE), precipitation (P) and temperature (T) during the period 1960-2006. Right: trend of cumulated annual actual evapotranspiration.

## **3. Water demand and availability**

The water balance between available water resource and water demand is the starting point for a correct water management. One of the main problems occurring in this phase is the general lack of observed data, obliging to synthetic estimates of water availability and several levels of approximation in the assessment of water needs, mainly for irrigation and for determining the management rules of the reservoirs.

In this context, the water balance on the Calabrian region was carried out considering also withdrawals from springs, streams, reservoirs and wells for irrigation and for potable uses, adopting two sequential simulation models. The former is a modified version of the distributed hydrological model, where the natural water balance is integrated with the withdrawal for irrigation and potable uses, producing (output variable) a residual availability. This water availability is used in a second GIS-based model considering the effects of diversions and reservoirs.

In the first model, inside a single 5 km squared cell can co-exist both wells and springs used to feed small irrigation systems or few users, located in the same cell, and wells and springs used for water mains collecting water outside the cell. If both the points where the water is withdrawn and used are inside the same cell (this happens only for wells for irrigation purposes), the schematization shown in figure 4a is adopted, hypothesizing that inside the cell a known volume is transferred monthly from the subsoil reservoir to the surface as an "added" precipitation (owing to the irrigation). This volume has to be summed to the meteorological precipitation and is subjected to the cycle simulated by the water balance, increasing the soil moisture and actual evapotranspiration and eventually feeding the

months (the less rainy and warmest ones), evaporation of soil moisture accumulated in wintertime exceeds rainfall, requiring irrigation in most of the agricultural areas. Figure 3 (right side) also shows the trend of cumulated annual actual evapotranspiration. The decrease in time of this quantity due to rainfall reduction is partly balanced by the increasing temperatures, hence the negative trend is not significant. It is noteworthy that peaks and troughs are generally dependent on rainy (e.g. 2005) or not rainy (e.g. 2001) years, even though rainfall distribution during the single year also affects the evapotranspirative phenomenon. The correlation coefficient between cumulated annual actual

evapotranspiration and precipitation in the period 1960-2006 was 0.638.

Fig. 3. Left: average monthly values in Calabria of actual (AE) and potential

Right: trend of cumulated annual actual evapotranspiration.

for determining the management rules of the reservoirs.

**3. Water demand and availability** 

effects of diversions and reservoirs.

evapotranspiration (PE), precipitation (P) and temperature (T) during the period 1960-2006.

The water balance between available water resource and water demand is the starting point for a correct water management. One of the main problems occurring in this phase is the general lack of observed data, obliging to synthetic estimates of water availability and several levels of approximation in the assessment of water needs, mainly for irrigation and

In this context, the water balance on the Calabrian region was carried out considering also withdrawals from springs, streams, reservoirs and wells for irrigation and for potable uses, adopting two sequential simulation models. The former is a modified version of the distributed hydrological model, where the natural water balance is integrated with the withdrawal for irrigation and potable uses, producing (output variable) a residual availability. This water availability is used in a second GIS-based model considering the

In the first model, inside a single 5 km squared cell can co-exist both wells and springs used to feed small irrigation systems or few users, located in the same cell, and wells and springs used for water mains collecting water outside the cell. If both the points where the water is withdrawn and used are inside the same cell (this happens only for wells for irrigation purposes), the schematization shown in figure 4a is adopted, hypothesizing that inside the cell a known volume is transferred monthly from the subsoil reservoir to the surface as an "added" precipitation (owing to the irrigation). This volume has to be summed to the meteorological precipitation and is subjected to the cycle simulated by the water balance, increasing the soil moisture and actual evapotranspiration and eventually feeding the aquifer from which it has been withdrawn. Instead, if the cell where the water is withdrawn does not coincide with the cell where it is used (that is only the case of regional water mains) then the schematization shown in figure 4b is adopted. The source cell is subjected to a reduction of the volume of the subsoil reservoir, while the water is hypothesized to reach directly the water stream in the destination cell, feeding the surface runoff with a restitution coefficient equal to 0.7.

Fig. 4. Schematization of the modified water balance model considering withdrawals for irrigation and potable uses.

Summarizing, the proposed model allows that every month for each cell a volume *cellout* can be extracted from the subsoil reservoir, which is equal to the withdrawals for irrigation and potable purposes, that a volume *cellin\_irr* can be added like a supplementary precipitation representing the water derived from the same cell and used for irrigation, and finally, that a volume *cellin\_pot* can be added like a supplementary surface runoff accounting for the water come in the cell to satisfy the potable uses. All the data related to potable and irrigation withdrawals were derived from several official sources, even if sometimes incomplete, and were aggregated at the resolution of the water balance model. Figure 5 shows the distribution of the regional water mains and of the local water distribution systems.

The modified natural water balance is the input of the commercial GIS-based model Mike Basin (DHI Software), accounting for the effects of diversions and reservoirs aimed at satisfying irrigation, hydro-power, civil and industrial requirements (Fig. 6). The lack of actual information about the management rules of reservoirs led to hypothesize several working schemes for the definition of the optimal water balance. Finally, for all the analyzed reservoirs the minimum flow requirements were considered following two different approaches: the former proposed by the Regional Basin Authority (very conservative, especially for some typical Calabrian rivers, called *fiumare*¸ characterized by no flow conditions for a relevant part of the year) and; the latter based on the Q7,10 flow, i.e. the lowest 7 consecutive-day average flow characterized by a 10 years time period.

In the case of the irrigation demand (i.e. water requirements for balancing evapotranspiration losses), a detailed analysis was carried out on each irrigation district (Fig. 7) during the irrigation season April – September. Specifically, the assessment of the effective water consumption was determined by considering different seasonal (spring, summer, autumn) soil use spatial distributions (e.g. in Table 1). For each soil use the seasonal irrigation demand (m3/ha) of the crops (Table 2) was achieved. The same was split monthly taking into account that the highest request is obtained during the trimester June – August (Table 3). An adequately detailed knowledge of the irrigation network allowed the correct estimate of the possible uptake of volumes to/from other cells. It is noteworthy to highlight that all the information related to soil use and water requirements were aggregated at the resolution of the model, i.e. 5×5 km2, for the whole region.

In the proposed analysis the quite small volumes related to industrial areas were neglected.

Fig. 5. Left: regional water mains (479 springs, 281 wells and about 2000 conveying pipes). Right: local water distribution systems (over 1200 springs and wells).

Water balance results showed that, for average conditions, the residual annual water availability is great, even if some weaknesses arise. Among these, the strong differences in the seasonal precipitation, which is mainly concentrated in the wet winter period (80-90%), require an accurate management of the volumes stored in natural and artificial reservoirs for facing the hot and dry Mediterranean summer. Furthermore, the decrepitude of several conveying pipes has to be considered with remarkable water losses, and the negative precipitation trend due to climate change that seems to be relevant in Calabria (a preliminary analysis about future climate scenarios in Calabria is shown in the 6th section). The weaknesses pointed out in normal conditions suggested water resources availability should be analyzed when drought conditions occur. Specifically, through the use of the Standardized Precipitation Index (SPI, McKee et al., 1993) intensity and duration of

droughts were determined on the whole Calabrian region.

(Fig. 7) during the irrigation season April – September. Specifically, the assessment of the effective water consumption was determined by considering different seasonal (spring, summer, autumn) soil use spatial distributions (e.g. in Table 1). For each soil use the seasonal irrigation demand (m3/ha) of the crops (Table 2) was achieved. The same was split monthly taking into account that the highest request is obtained during the trimester June – August (Table 3). An adequately detailed knowledge of the irrigation network allowed the correct estimate of the possible uptake of volumes to/from other cells. It is noteworthy to highlight that all the information related to soil use and water requirements were

In the proposed analysis the quite small volumes related to industrial areas were neglected.

Fig. 5. Left: regional water mains (479 springs, 281 wells and about 2000 conveying pipes).

Water balance results showed that, for average conditions, the residual annual water availability is great, even if some weaknesses arise. Among these, the strong differences in the seasonal precipitation, which is mainly concentrated in the wet winter period (80-90%), require an accurate management of the volumes stored in natural and artificial reservoirs for facing the hot and dry Mediterranean summer. Furthermore, the decrepitude of several conveying pipes has to be considered with remarkable water losses, and the negative precipitation trend due to climate change that seems to be relevant in Calabria (a preliminary analysis about future climate scenarios in Calabria is shown in the 6th section). The weaknesses pointed out in normal conditions suggested water resources availability should be analyzed when drought conditions occur. Specifically, through the use of the Standardized Precipitation Index (SPI, McKee et al., 1993) intensity and duration of

Right: local water distribution systems (over 1200 springs and wells).

droughts were determined on the whole Calabrian region.

aggregated at the resolution of the model, i.e. 5×5 km2, for the whole region.

Fig. 6. Example of water system schematization realized within the GIS-based model Mike Basin.


Table 1. Seasonal soil use for a generic irrigation district.

For each of the most significant Calabrian basins, and for each month of the period 1960- 2006, a mean SPI areal value was calculated for different time scales (1-, 3-, 6-, 12-, 24- and 48-months), with the aim of highlighting the longest and most intense drought periods (Fig. 8). Drought indices are essential at all levels of the planning process. The reader is referred to section 5 for a brief review of the most diffused ones.

Usually, the beginning of a drought period can be defined when SPI values are lower than -1.0, and its end when the values come back positive. Nevertheless, based on a historical analysis of the official declarations of "natural disaster" in Calabria due to drought, even a 12-month SPI value equal to -0.7 was observed to be adequate as a drought threshold. Hence, when a generic month presented a 12-month SPI value lower than -0.7, it was considered a drought month, and the correspondent total runoff simulated with the water balance model was taken into account. The aggregation, from January to December, of the average runoff estimated during the drought months leaded to the definition of a so-called "scarce year" whose runoff values, even if statistically less probable than the ones of the single months, pointed out the possibility of extremely critical situations in Calabria, with a reduction of total runoff up to 43%. This analysis introduces issues related both to the management of water shortage and to the mitigation of drought through the use of restrictive measures. The development and implementation of strategic and emergency plans are primary tools to face the different aspects of drought phenomenon, as it is shown in the next paragraph.

Fig. 7. Calabrian irrigation districts and network systems.


Table 2. Seasonal irrigation demand (m3/ha) of the crops.

even if statistically less probable than the ones of the single months, pointed out the possibility of extremely critical situations in Calabria, with a reduction of total runoff up to 43%. This analysis introduces issues related both to the management of water shortage and to the mitigation of drought through the use of restrictive measures. The development and implementation of strategic and emergency plans are primary tools to face the different

aspects of drought phenomenon, as it is shown in the next paragraph.

Fig. 7. Calabrian irrigation districts and network systems.

Table 2. Seasonal irrigation demand (m3/ha) of the crops.

Code Description Irrigation demand (m3/ha)

2121 Spring-summer herbaceous crops 7000 2122 Summer-autumn/spring horticultural crops 7600 2123 Spring-summer horticultural crops 5000 2125 Greenhouse crops 9000 213 Rice fields 15000 2211 Irrigated vineyards 3500 2221 Irrigated orchards 5000 2231 Irrigated olive groves 3000


Table 3. Monthly irrigation demand (m3/ha) of the crops.

Fig. 8. Temporal evolution of SPI values in a generic Calabrian river basin. Red squares correspond to drier periods.

## **4. Water resource management under shortage conditions**

In its 2007 Communication (COM, 2007) the European Commission stated that the challenge of water scarcity and droughts needs to be addressed both as an essential environmental issue and as a precondition for sustainable economic growth in Europe, and highlighted the necessity of progressing towards full implementation of the EU Water Framework Directive (WFD) 2000/60. The WFD is the EU's flagship Directive on water policy, explicitly defining long-term planning as the main tool for ensuring good status of water resources. Nevertheless, it does not indicate criteria and actions to face risk of drought, delegating National Legislations to concretely realize its framework (after a series of yearly follow-up reports, a policy review is foreseen for 2012 at the EU level).

In Italy the EU WFD was taken into account with the Legislative Decree 152/2006 on environmental protection. Though this act is quite recent, it seems to be far from being adequate to actually cope with drought, mainly because it does not stress the necessity of passing from a reactive to a proactive approach, based on preparedness and mitigation actions planned in advance with the contribution of all the involved stakeholders, ready to be implemented when drought phenomena occur.

Within a comprehensive drought management planning process, Rossi et al. (2007) proposed the identification of three main tools: Strategic Water Shortage Preparedness Plan, Water Supply System Management Plan and Drought Contingency Plan. Following, an example of application of the proposed guidelines is shown for the planning of the best mix of measures needed for coping with drought phenomena on one of the most important agricultural areas in southern Italy, the Low Esaro and Sibari Plain (Mendicino et al., 2008b). It is noteworthy that in the proposed example (water shortage planning in the agricultural sector) water demand is strictly correlated to the amount of water needed from crops for facing lack of precipitation and high potential evapotranspiration during summer (see Table 3). Hence, in this case the planning process is triggered by the need of coping with the high water loss due to evapotranspiration in a particularly dry period of the year. As it is explained in the next sections, this objective can be reached by means of demand reduction, water supply increase or impacts minimization measures, and considering long-, mediumand short-term actions.

#### **4.1 Methods and tools**

The Agricultural Strategic Water Shortage Preparedness Plan (ASP) is aimed at obtaining the reduction of drought vulnerability in the analyzed area through the implementation in normal conditions of long term mitigation measures, consisting in a series of structural and non-structural actions applied in the water supply system. Usually, structural measures are economically expensive and require the use of many human resources. However, their effects are easier to be foreseen than the effects produced by the nonstructural mitigation actions, in their turn usually more accepted by all the stakeholders. The long term mitigation measures are specifically indicated in the systems characterized by a low level of reliability and are oriented at improving the water balance in the analyzed system. These actions not only enhance the reliability of the system through fulfilling water requirements, but also reduce its vulnerability with respect to future drought events, fulfilling three main objectives: i) water demand reduction; ii) water supply increase and improvement of the efficiency of the system; iii) minimization of the impacts. Within the actions reducing water demand, some are directly aimed at reducing evapotranspiration by adopting appropriate agronomic techniques, such as e.g. irrigating during non windy periods for minimizing wind drift losses, or early defoliation to reduce crop transpiration surface (for a deeper description, the reader is referred to Pereira, 2007). In table 4 the long term measures that can be potentially adopted in agriculture are listed, subdivided considering their main objectives.

long-term planning as the main tool for ensuring good status of water resources. Nevertheless, it does not indicate criteria and actions to face risk of drought, delegating National Legislations to concretely realize its framework (after a series of yearly follow-up

In Italy the EU WFD was taken into account with the Legislative Decree 152/2006 on environmental protection. Though this act is quite recent, it seems to be far from being adequate to actually cope with drought, mainly because it does not stress the necessity of passing from a reactive to a proactive approach, based on preparedness and mitigation actions planned in advance with the contribution of all the involved stakeholders, ready to

Within a comprehensive drought management planning process, Rossi et al. (2007) proposed the identification of three main tools: Strategic Water Shortage Preparedness Plan, Water Supply System Management Plan and Drought Contingency Plan. Following, an example of application of the proposed guidelines is shown for the planning of the best mix of measures needed for coping with drought phenomena on one of the most important agricultural areas in southern Italy, the Low Esaro and Sibari Plain (Mendicino et al., 2008b). It is noteworthy that in the proposed example (water shortage planning in the agricultural sector) water demand is strictly correlated to the amount of water needed from crops for facing lack of precipitation and high potential evapotranspiration during summer (see Table 3). Hence, in this case the planning process is triggered by the need of coping with the high water loss due to evapotranspiration in a particularly dry period of the year. As it is explained in the next sections, this objective can be reached by means of demand reduction, water supply increase or impacts minimization measures, and considering long-, medium-

The Agricultural Strategic Water Shortage Preparedness Plan (ASP) is aimed at obtaining the reduction of drought vulnerability in the analyzed area through the implementation in normal conditions of long term mitigation measures, consisting in a series of structural and non-structural actions applied in the water supply system. Usually, structural measures are economically expensive and require the use of many human resources. However, their effects are easier to be foreseen than the effects produced by the nonstructural mitigation actions, in their turn usually more accepted by all the stakeholders. The long term mitigation measures are specifically indicated in the systems characterized by a low level of reliability and are oriented at improving the water balance in the analyzed system. These actions not only enhance the reliability of the system through fulfilling water requirements, but also reduce its vulnerability with respect to future drought events, fulfilling three main objectives: i) water demand reduction; ii) water supply increase and improvement of the efficiency of the system; iii) minimization of the impacts. Within the actions reducing water demand, some are directly aimed at reducing evapotranspiration by adopting appropriate agronomic techniques, such as e.g. irrigating during non windy periods for minimizing wind drift losses, or early defoliation to reduce crop transpiration surface (for a deeper description, the reader is referred to Pereira, 2007). In table 4 the long term measures that can be potentially adopted in agriculture are

reports, a policy review is foreseen for 2012 at the EU level).

be implemented when drought phenomena occur.

listed, subdivided considering their main objectives.

and short-term actions.

**4.1 Methods and tools** 


Table 4. Main long term drought mitigation measures in agriculture (adapted from Rossi et al., 2007, and Georgia Dept. Of Natural Resources, 2003).

Since the ASP has to be drawn up choosing among several combinations of long-term mitigation measures, a suitable evaluation procedure has to be adopted. A multi-criteria technique could provide an as objective as possible comparison among different alternatives, according to a series of economic, environmental and social criteria, and taking into account the point of view of all the stakeholders. The tool adopted in this study for multi-criteria analysis is the software NAIADE (Munda, 1995).

The ASP should be prepared by the Basin or Hydrographic District Authorities, which are the bodies responsible for planning, and corresponds to the Drought Management Plan included into the River Basin Management Plan provided in the WFD.

Once the long-term mitigation measures are defined, an Agricultural Water Supply System Management Plan (AMP) has to be developed with the aim of: defining the best mix of long and short-term measures to avoid the beginning of a real water emergency; estimating the costs and the financing sources for the chosen mitigation measures, and; fostering the stakeholder participation and exchanges. It is prepared by the authority responsible for agricultural water management (i.e. the Land Reclamation Consortium), and the operative measures defined have to be adopted according to the values of early warning indicators, showing Normal, Pre-Alert or Alert conditions. The threshold values of the indicators can be chosen through an objective function or, if several aspects have to be accounted for, through a multi-criteria analysis.

In table 5 the short term measures that can be potentially adopted in agriculture are shown, subdivided on the basis of their principal objectives. With respect to the long-term mitigation measures, in this case the actions in the "demand reduction" category implicitly accept a certain percentage of water stress for the crops, because they are only aimed to reduce water consumption, without taking into account crop conditions. On the contrary, former long-term mitigation measures suggested some structural actions (i.e. actions to be adopted always) aimed at limiting some additional evapotranspiration due, e.g., to not correct irrigation practices, and that could be avoided without consequences for the crop. In brief, adopting the AMP evapotranspiration losses could be not completely compensated, and the farmers should be supported in assessing how to minimize water stress effects adopting even more specific agronomic techniques.


Table 5. Main short term mitigation measures in agriculture (adapted from Rossi et al., 2007).

If a particularly severe drought occurs, and the indicators signal Alarm conditions, the Agricultural Drought Contingency Plan (ACP) has to be adopted, defining the most appropriate short-term measures to reduce the impact of emergency situations. In this case the efforts are turned to protect the essential activities of the agricultural system, and the threshold values of the indicators have to be chosen taking into account this objective, preferably using a probabilistic approach, that allows the decision-makers to evaluate the effective risk of having water deficit for different scenarios. The ACP should be prepared by the Basin or Hydrographic District Authorities, with the collaboration of the Civil Protection.

Such as in the AMP, also in the ACP the assessment of crop losses can be made through production functions. In the case of extreme and particularly prolonged drought also the damage to perennial crops, the excessive decrease of the water tables of the aquifers, sea water intrusion, ecological damages to aquatic flora and fauna have to be considered. Some of this damage can be irreversible and can also influence crop production in the following years.

## **4.2 Case study**

210 Evapotranspiration – Remote Sensing and Modeling

In table 5 the short term measures that can be potentially adopted in agriculture are shown, subdivided on the basis of their principal objectives. With respect to the long-term mitigation measures, in this case the actions in the "demand reduction" category implicitly accept a certain percentage of water stress for the crops, because they are only aimed to reduce water consumption, without taking into account crop conditions. On the contrary, former long-term mitigation measures suggested some structural actions (i.e. actions to be adopted always) aimed at limiting some additional evapotranspiration due, e.g., to not correct irrigation practices, and that could be avoided without consequences for the crop. In brief, adopting the AMP evapotranspiration losses could be not completely compensated, and the farmers should be supported in assessing how to minimize water stress effects

> Public information campaign for water saving Restriction of irrigation of annual crops Pricing (discourage excessive water use) Mandatory rationing

Improvement of existing water systems efficiency (leak detection programs, new operating rules, etc.) Use of emergency sources (additional sources of low quality and/or high exploitation cost) Over exploitation of aquifers (use of strategic reserves) Increased diversion by relaxing ecological or recreational use constraints

> Temporary reallocation of water resources Public aids to compensate income losses Tax reduction or delay of payment deadline Public aids for crops insurance

**Category Short-term actions** 

Table 5. Main short term mitigation measures in agriculture (adapted from Rossi et al., 2007). If a particularly severe drought occurs, and the indicators signal Alarm conditions, the Agricultural Drought Contingency Plan (ACP) has to be adopted, defining the most appropriate short-term measures to reduce the impact of emergency situations. In this case the efforts are turned to protect the essential activities of the agricultural system, and the threshold values of the indicators have to be chosen taking into account this objective, preferably using a probabilistic approach, that allows the decision-makers to evaluate the effective risk of having water deficit for different scenarios. The ACP should be prepared by the Basin or

Such as in the AMP, also in the ACP the assessment of crop losses can be made through production functions. In the case of extreme and particularly prolonged drought also the damage to perennial crops, the excessive decrease of the water tables of the aquifers, sea water intrusion, ecological damages to aquatic flora and fauna have to be considered. Some of this damage can be irreversible and can also influence crop production in the

Hydrographic District Authorities, with the collaboration of the Civil Protection.

adopting even more specific agronomic techniques.

Demand reduction

Water supply increase

Impacts minimization

following years.

The core of the analyzed water supply system is the Farneto Dam (Fig. 9), closing the Esaro Catchment (about 245.4 km2) in southern Italy. The dam is aimed at: (i) containing the ordinary floods and mitigating the extraordinary ones, according to the condition that the reservoir level is maintained almost empty from October to March; (ii) supplying water (about 30 hm3 from April to September) to the downstream agricultural area (about 85 km2), sited in the Low Esaro and Sibari Plain. At present about 63% of the irrigable area is based on open channel irrigation systems.

Fig. 9. Study area for the development of the planning process.

## **4.3 Applying the Agricultural Strategic Plan**

Table 6 shows 13 selected alternatives (from A to M), obtained combining the following six long-term mitigation measures: 0) System in current configuration; 1) Modernization of the irrigation network for reducing water losses and evaporation (it has been calculated that the efficiency of the actual scenario is equal to 67%, while the efficiency of the "modernized" scenario will be 80%; Mendicino et al., 2008b); 2) Construction of farm ponds; 3) Construction of a new upstream dam; 4) Economic incentives and educational activities for water saving; 5) Allowing the dam to store a little volume during the winter (i.e. dam not empty in March).


Table 6. Long-term mitigation measures and alternatives.

The alternatives were compared within the DSS tool NAIADE according to 4 economic criteria (construction costs of infrastructures, operation and maintenance costs, crop yield losses and amount of public aids needed), 2 environmental criteria (failures to meet ecological requirements and reversibility of the alternatives) and 4 social criteria (system vulnerability, temporal reliability, realization time of the infrastructures and employment increase). Since the observed period is short in order to evaluate the criteria and is characterized by few drought events, two monthly synthetic temperature and precipitation series of 1000 years were generated as input of the water balance model providing the corresponding runoff values.

Within the analysis carried out with NAIADE the final ranking of the alternatives comes from the intersection of two separate rankings. The former + is based on the "better" and "much better" preference relations, hence it points out how an alternative is "better" than the others. The latter is based on the "worse" and "much worse" preference relations, and indicates how an alternative is "worse" than the others.

The two rankings are different, since one alternative could result slightly better than the others with respect to few criteria and at the same time could result worse with respect to many criteria, or vice versa. In figure 10 the partial rankings and the final ranking are shown. The most efficient alternative is the "J", where measures 1, 4 and 5 are considered together. The alternative "M", mainly characterized by the construction of a new upstream dam, is the best only in the + ranking. A sensitivity analysis, carried out to assess the robustness of the achieved solution, showed a substantial stability of the ranking, constantly confirming alternative J as the optimal one. It is pointed out that alternative J is made up also by measure 1), allowing a reduction of evaporation losses.

Fig. 10. Partial and final ranking of the drought mitigation alternatives in the Esaro River Basin.

## **4.4 Applying the Agricultural Management Plan**

212 Evapotranspiration – Remote Sensing and Modeling

losses and amount of public aids needed), 2 environmental criteria (failures to meet ecological requirements and reversibility of the alternatives) and 4 social criteria (system vulnerability, temporal reliability, realization time of the infrastructures and employment increase). Since the observed period is short in order to evaluate the criteria and is characterized by few drought events, two monthly synthetic temperature and precipitation series of 1000 years were generated as input of the water balance model providing the

Within the analysis carried out with NAIADE the final ranking of the alternatives comes from the intersection of two separate rankings. The former + is based on the "better" and "much better" preference relations, hence it points out how an alternative is "better" than

The two rankings are different, since one alternative could result slightly better than the others with respect to few criteria and at the same time could result worse with respect to many criteria, or vice versa. In figure 10 the partial rankings and the final ranking are shown. The most efficient alternative is the "J", where measures 1, 4 and 5 are considered together. The alternative "M", mainly characterized by the construction of a new upstream dam, is the best only in the + ranking. A sensitivity analysis, carried out to assess the robustness of the achieved solution, showed a substantial stability of the ranking, constantly confirming alternative J as the optimal one. It is pointed out that alternative J is made up

Fig. 10. Partial and final ranking of the drought mitigation alternatives in the Esaro River

is based on the "worse" and "much worse" preference relations, and

corresponding runoff values.

indicates how an alternative is "worse" than the others.

also by measure 1), allowing a reduction of evaporation losses.

the others. The latter -

Basin.

The AMP is aimed at defining the indicators and the triggers for establishing the Normal, Pre-Alert and Alert conditions for the agricultural areas of the system. It has to take into account the guidelines provided by the ASP. In fact, it has to select the best combination among the optimal long-term mitigation measure previously determined (J) and the several short-term measures that can be adopted to manage water deficits on the analyzed area. Whereas the long-term measure J is adopted continuously, the short-term measures vary following the status of the system. Specifically, for this case study:


With the aim of determining the threshold values of the indices indicating the passage from one status to another, for every month from April to September a multicriteria analysis of the effects through NAIADE was carried out. The conflicting objectives to minimize are:


For each month, starting from April, an impact matrix was achieved where, on the basis of the criteria selected for the fulfillment of the objectives, the optimal combination of the thresholds triggering the Pre-Alert and Alert status was selected (Fig. 11). The selected index for the definition of the drought thresholds is the volume stored in the dam from May to

Fig. 11. Pre-Alert and Alert thresholds defined in the AMP.

September, while for the month of April a meteorological index was chosen, since owing to the rules adopted for dam management, at the end of March the dam level is not a significant index. For the month of April an analysis was carried out relating the yearly irrigation deficit to the 6 month-SPI calculated in March (considering in this way the first six months of the hydrologic year, from October to March). In the selection of the threshold values a rule was followed considering that, if the multicriteria analysis provides more optimal solutions, the one with the lowest irrigation deficit is selected.

#### **4.5 Applying the Agricultural Contingency Plan**

The first objective of the ACP is the definition of indices and their thresholds for univocally establishing the beginning of an emergency situation. Since the hydrologic analysis in April shows that the water demand is always less than the water availability in the Farneto del Principe Dam, and that every year the volume stored increases during this month, the thresholds are selected starting from May, choosing as an index, such as in the AMP, the volume stored in the dam. Furthermore, since using the 1000-year series of generated meteorological data the application of the two previous Plans determined a very high temporal reliability of the system (98.7%), it is not useful to evaluate the emergency thresholds considering the few residual years. Hence, the adopted approach was based on a probabilistic analysis of the system failures and deficit percentage of the demand.

Specifically, hypothesizing that all the short-term measures were already adopted, the 1000 year series of generated meteorological data, for every month and for different fixed initial volumes stored, were used to assess the probability of having failures in fulfilling demand either in the same month or in the subsequent irrigation period, and the deficit percentage with respect to demand. The results, allowing the decision-makers to evaluate the effective risk of having water deficit for a specific storage in a specific month, are shown (from May to August) in figure 12.

Fig. 12. Monthly risk of having failures and deficit percentage with respect to demand (from May to August).

## **5. Drought indices**

214 Evapotranspiration – Remote Sensing and Modeling

September, while for the month of April a meteorological index was chosen, since owing to the rules adopted for dam management, at the end of March the dam level is not a significant index. For the month of April an analysis was carried out relating the yearly irrigation deficit to the 6 month-SPI calculated in March (considering in this way the first six months of the hydrologic year, from October to March). In the selection of the threshold values a rule was followed considering that, if the multicriteria analysis provides more

The first objective of the ACP is the definition of indices and their thresholds for univocally establishing the beginning of an emergency situation. Since the hydrologic analysis in April shows that the water demand is always less than the water availability in the Farneto del Principe Dam, and that every year the volume stored increases during this month, the thresholds are selected starting from May, choosing as an index, such as in the AMP, the volume stored in the dam. Furthermore, since using the 1000-year series of generated meteorological data the application of the two previous Plans determined a very high temporal reliability of the system (98.7%), it is not useful to evaluate the emergency thresholds considering the few residual years. Hence, the adopted approach was based on a

probabilistic analysis of the system failures and deficit percentage of the demand.

Specifically, hypothesizing that all the short-term measures were already adopted, the 1000 year series of generated meteorological data, for every month and for different fixed initial volumes stored, were used to assess the probability of having failures in fulfilling demand either in the same month or in the subsequent irrigation period, and the deficit percentage with respect to demand. The results, allowing the decision-makers to evaluate the effective risk of having water deficit for a specific storage in a specific month, are shown (from May

Fig. 12. Monthly risk of having failures and deficit percentage with respect to demand (from

optimal solutions, the one with the lowest irrigation deficit is selected.

**4.5 Applying the Agricultural Contingency Plan** 

to August) in figure 12.

May to August).

Drought indices are tools necessary at all levels of the planning process: as it was shown in the previous sections, in the Strategic Plan they are used to identify the zones most exposed to drought risk in the analyzed areas, whereas in the Management Plan and in the Contingency Plan they are used to define trigger values for the activation of the measures for impact prevention or mitigation.

Most of the proposed methodologies for the characterization and the monitoring of drought phenomena are based on drought indices with the capability of synthetically summarizing drought conditions in a specific moment for a particular area. Nevertheless, drought is difficult to represent through a single index, hence frequently more indices or aggregate indices are used.

In rainfed agriculture meteorological indices are particularly suitable, because they give the opportunity of establishing a direct spatial correlation between the drought event and the agricultural production, allowing drought risk maps to be drawn.

Many authors provide lists describing the characteristics of the main drought indices (e.g. Ntale & Gan, 2003; Tsakiris et al., 2007a). Among them, the most widely used are the Palmer Drought Severity Index (PDSI; Palmer, 1965), the most "classical" drought index formulated to evaluate prolonged periods of both abnormally wet and abnormally dry weather conditions, and the Standardized Precipitation Index (SPI; McKee et al., 1993), a meteorological drought index based on the precipitation amount in a period of n months. Since SPI just needs precipitation data to be calculated, it has found widespread application. Guttman (1998) shows that the PDSI has a complex structure with an exceptionally long memory, while the SPI is an easily interpreted, simple moving average process. Hayes et al. (1999) describe the three main advantages in using SPI: the first and primary is its simplicity, the second is its variable time scale, and the third is its standardization. Nevertheless, the SPI is a meteorological index unable to take into account the effects of aquifers, soil, land use characteristics, crop growth and temperature anomalies, which influence agricultural and hydrological droughts.

Besides SPI, in the process of drought identification the MEDROPLAN Guidelines (Tsakiris et al., 2007a) suggest using also: the Reconnaissance Drought Index (RDI, Tsakiris et al., 2007b), also accounting for temperature anomalies (therefore for an eventual excessive evapotranspiration); deciles (Gibbs & Maher, 1967), used by the Australian Drought Watch System, which compare monthly observed precipitation values with the quantiles corresponding to the not exceeded frequencies of 10%, 20%,… 100% achieved from a long enough monthly precipitation series; the Surface Water Supply Index (SWSI, Shafer & Dezman, 1982), aggregating information about precipitation, runoff, volumes stored in the reservoirs and snowpack, and expressing drought conditions in a standardized way. Furthermore, owing to their diffusion, other two indices are recalled: the run method (Yevjevich, 1967), based on the comparison between the time series of the analyzed hydrological index and a representative threshold of "normal" conditions, and the Palmer Hydrological Drought Index (Karl, 1986), a modified version of the PDSI for real-time monitoring.

An interesting way to account for soil and land use effects (in some respects, the way followed by Palmer to calculate PDSI) is to derive the drought indices starting from hydrological modeling. These indices can be called "comprehensive" drought indices, because they allow a more comprehensive picture of the water cycle and its elements (Niemeyer, 2008). A typical example of comprehensive drought index is the Groundwater Resource Index (GRI) derived by Mendicino et al. (2008a) using the monthly water balance model shown in figure 1. For each single element where the model was applied (5 km regular cell), the monthly values of groundwater detention (i.e. the storage *D*) were standardized (for almost all the cells and months the skewness test of normality showed that the series were normally distributed) through the following equation:

$$\text{GRT}\_{y,m} = \frac{D\_{y,m} - \mu\_{y,m}}{\sigma\_{y,m}} \tag{2}$$

where *GRIy,m* and *Dy,m* are respectively the values of the index and of the groundwater detention for the year *y* and the month *m*, while *D,m* and *D,m* are respectively the mean and the standard deviation of groundwater detention values *D* simulated for the month *m* in a defined number of years (at least 30). This simple index, but based on several pieces of information provided by the water balance model, allows assessment of the deviation from the mean values of the available groundwater in a spatially-distributed way for the whole territory where the model is applied. Figure 13 shows the maps of the GRI distribution in northern Calabria for the months of April from 1979 to 2006. Examining the maps immediately the years with lower GRI values (the driest years, with brighter colors) are recognizable, as are the wettest years (darkest colors).

Fig. 13. Boundaries of the selected study area in Calabria and GRI distribution in northeastern Calabria for the months of April from 1979 to 2006 (from Mendicino et al., 2008a).

Other comprehensive indices were developed by Narasimhan & Srinivasan (2005), who using the Soil and Water Assessment Tool (SWAT) model, derived two drought indices

(Niemeyer, 2008). A typical example of comprehensive drought index is the Groundwater Resource Index (GRI) derived by Mendicino et al. (2008a) using the monthly water balance model shown in figure 1. For each single element where the model was applied (5 km regular cell), the monthly values of groundwater detention (i.e. the storage *D*) were standardized (for almost all the cells and months the skewness test of normality showed that

*D*

where *GRIy,m* and *Dy,m* are respectively the values of the index and of the groundwater

the standard deviation of groundwater detention values *D* simulated for the month *m* in a defined number of years (at least 30). This simple index, but based on several pieces of information provided by the water balance model, allows assessment of the deviation from the mean values of the available groundwater in a spatially-distributed way for the whole territory where the model is applied. Figure 13 shows the maps of the GRI distribution in northern Calabria for the months of April from 1979 to 2006. Examining the maps immediately the years with lower GRI values (the driest years, with brighter colors) are

Fig. 13. Boundaries of the selected study area in Calabria and GRI distribution in northeastern Calabria for the months of April from 1979 to 2006 (from Mendicino et al., 2008a).

Other comprehensive indices were developed by Narasimhan & Srinivasan (2005), who using the Soil and Water Assessment Tool (SWAT) model, derived two drought indices

, ,

(2)

*D,m* are respectively the mean and

, *y m y m*

*D,m* and

*y m*

the series were normally distributed) through the following equation:

detention for the year *y* and the month *m*, while

recognizable, as are the wettest years (darkest colors).

,

*GRI*

*y m*

for agricultural drought monitoring, the Soil Moisture Deficit Index (SMDI) and the Evapotranspiration Deficit Index (ETDI), based respectively on weekly soil moisture and evapotranspiration (ET) deficit. Also Matera et al. (2007) derived a new agricultural drought index, called DTx, based on the daily transpiration deficit calculated by a water balance model.

In the few last years the possibility of using long data series coming from remote sensing has opened new and promising perspectives to satellite-derived drought indices, which have the advantage of being intrinsically spatially distributed. Anderson et al. (2007) provide a brief presentation of TIR-based drought indices, while a list of many NOAA-AVHRR images-derived drought indices is presented by Bayarjargal et al. (2006). Zhang et al. (2005) exploit the capabilities of the MODerate resolution Imaging Spectroradiometer (MODIS) for monitoring and forecasting crop production using a satellite-based Climate-Variability Impact Index.

Several remote sensing-derived drought indices depend on the ratio *ET*/*PET*, where *ET* is actual evapotranspiration and *PET* potential evapotranspiration (e.g. Crop Water Stress Index (CWSI), Jackson et al., 1981; Drought Severity Index (DSI), Su et al., 2003; Evaporative Drought Index (EDI), Anderson et al., 2007; Yao et al., 2010). While *PET* is generally calculated by means of ground based measurements, *ET* is easily estimated through "residual" methods (e.g. SEBAL, Bastiaanssen et al., 1998; and Bastiaanssen, 2000; SEBI, Menenti & Choudhury, 1993; S-SEBI, Roerink et al., 2000; SEBS, Su, 2002; TSEB, Norman et al., 1995; DisAlexi, Anderson et al., 1997; METRIC, Allen et al., 2007), where the evapotranspirative term is the residual term of the energy balance equation:

$$
\lambda \mathcal{E} = \mathcal{R}\_n - G - H \tag{3}
$$

with *Rn* net radiation, *G* soil heat flux, *H* sensible heat flux and *E* latent heat flux, from which *ET* is derived.

Even though at this stage very seldom they are used as operational tools, remote sensingderived indices are potentially very useful because they intrinsically provide space-time variation of drought phenomena, and the ratio *ET*/*PET* can be reasonably related to soil water content. For instance, the relative evaporation *r* can be directly linked to the soil degree of saturation /*<sup>s</sup>* (Su et al., 2003). As an example, figure 14 shows the space-time evolution of the DSI, derived from SEBS and MODIS images, during summer 2006 in Northern Calabria. DSI is equal to 1 - *E* / *Ewet* (where *Ewet* is the latent heat flux estimated for the so-called "wet" pixel), hence higher DSI values indicate low actual evapotranspiration. A graph shown at the top of the figure provides information about precipitation in a micrometeorological station placed almost in the middle of the area (these data are only roughly representative, owing to the extension of the whole area). Figure 14 shows that the maps with the highest DSI values (e.g. July 20, but also September 4 and October 31), indicating drought stress conditions, are related to some of the most distant days from antecedent significant precipitation events.

To complete this brief review, a much-discussed issue is mentioned, i.e. the possibility of using the drought indices (especially SPI) to forecast stochastically the possible evolution of an ongoing drought (Cancelliere et al., 1996; Lohani et al., 1998; Bordi et al., 2005; Cancelliere et al., 2007). Several studies are also aimed at explaining and predicting possible drought conditions through the analysis of sea surface temperature (SST) and atmospheric circulation patterns (e.g. Wilby et al., 2004; Kim et al., 2006; Cook et al., 2007).

Fig. 14. Evolution of the DSI derived from SEBS in northern Calabria, from May 22 to October 31 2006. Top graph shows precipitation events on the representative micrometeorological station, placed approximately in the middle of the analyzed area.

However, when dealing with complex systems, where irrigated agriculture assumes a greater importance, one single index is often not able to capture the different features of drought and to take in account the effects of human activities (use of irrigation, water from reservoirs, wells, etc.) on the hydrological cycle. On the other hand, it is more practical to declare drought condition considering only one indicator. Thus, there is a growing interest in aggregating more indices. Keyantash & Dracup (2004) use an Aggregate Drought Index that considers all relevant variables of the hydrological cycle through Principal Component Analysis (but they do not include groundwater in the suite of variables); instead Steinemann & Cavalcanti (2006) use the probabilities of different indicators of drought and shortage, selecting the trigger levels on the basis of the most severe level of the indicator or the level of the majority of the indicators.

## **6. Future scenarios**

The most critical scenarios discussed in the previous paragraphs could become "normal" circumstances if global climate change increases the prolonged and intense drought periods. At the end of the proposed analysis, it is useful to hypothesize some future climatic scenarios, with the aim of steering decision makers towards suitable water management policies, as it is suggested by the European Commission (COM, 2009).

The methodology usually followed to assess the hydrological consequences of climate change basically consists of a three-step process (Xu et al., 2005): (1) the development and use of general circulation models (GCMs) to provide future global climate scenarios under the effect of increasing greenhouse gases, (2) the development and use of downscaling techniques (both statistical methods and nested regional climate models, RCMs, which are being continuously improved) for "downscaling" the GCM output to the scales compatible

Fig. 14. Evolution of the DSI derived from SEBS in northern Calabria, from May 22 to

micrometeorological station, placed approximately in the middle of the analyzed area.

However, when dealing with complex systems, where irrigated agriculture assumes a greater importance, one single index is often not able to capture the different features of drought and to take in account the effects of human activities (use of irrigation, water from reservoirs, wells, etc.) on the hydrological cycle. On the other hand, it is more practical to declare drought condition considering only one indicator. Thus, there is a growing interest in aggregating more indices. Keyantash & Dracup (2004) use an Aggregate Drought Index that considers all relevant variables of the hydrological cycle through Principal Component Analysis (but they do not include groundwater in the suite of variables); instead Steinemann & Cavalcanti (2006) use the probabilities of different indicators of drought and shortage, selecting the trigger levels on the basis of the most severe level of the indicator or the level of

The most critical scenarios discussed in the previous paragraphs could become "normal" circumstances if global climate change increases the prolonged and intense drought periods. At the end of the proposed analysis, it is useful to hypothesize some future climatic scenarios, with the aim of steering decision makers towards suitable water management

The methodology usually followed to assess the hydrological consequences of climate change basically consists of a three-step process (Xu et al., 2005): (1) the development and use of general circulation models (GCMs) to provide future global climate scenarios under the effect of increasing greenhouse gases, (2) the development and use of downscaling techniques (both statistical methods and nested regional climate models, RCMs, which are being continuously improved) for "downscaling" the GCM output to the scales compatible

policies, as it is suggested by the European Commission (COM, 2009).

October 31 2006. Top graph shows precipitation events on the representative

the majority of the indicators.

**6. Future scenarios** 

with hydrological models, and (3) the development and use of hydrological models to simulate the effects of climate change on hydrological regimes at various scales. However, uncertainties within this framework have to be taken into account such as the internal variability of the climate system, model structure and parameterizations at different spatial and temporal scales, the downscaling techniques and bias correction methods and the choice of future climate scenarios. Several different approaches were chosen for providing operational solutions to these drawbacks (Xu et al., 2005). However, numerous GCM simulations show almost univocal trends for global climate evolution. Giorgi & Lionello (2008) highlight a robust and consistent description specifically for the Mediterranean area, with a significant reduction in precipitation, mainly in summertime. In the same area, according to Giorgi (2006), a major increase in climatic variability is also expected.

Below, some results obtained by Senatore et al. (2011) are shown related to future water availability in the main basin of northern Calabria (Crati River Basin, 1332 km2, Fig. 15) at the end of the XXI century. Future scenarios were made by applying the outputs of three Regional Climate Models (RCMs) RegCM, HIRHAM and COSMO-CLM to the newly developed Intermediate Space Time Resolution Hydrological Model (In-STRHyM). The analysis was performed using two time slices (1961–1990 and 2070–2099) with the SRES A2 (GCM HAD3AM) and A1B (GCM ECHAM5/MPI-OM) scenarios. Observed biases in simulated precipitation and temperature fields during the control period (1961-1990) were corrected before using meteorological outputs from each RCM as input for In-STRHyM.

In-STRHyM is a fully distributed hydrological model detailed enough to describe the hydrological processes of several small-medium sized Mediterranean basins. It has a relatively simple structure and is suitable for long period simulations to be undertaken within acceptable time frames. Specifically, In-STRHyM calculates separately transpiration and evaporation, depending on a remote sensing-derived vegetation fraction. Both transpiration and bare soil evaporation are estimated through the crop coefficient approach suggested by Allen et al. (1998), considering a water stress coefficient of the canopy depending on soil moisture conditions, and the reference values calculated through the Priestley & Taylor (1972) equation.

The RCMs predict an increase in mean annual temperature from 3.5 °C to 3.9 °C, and a decrease in mean annual precipitation from 9% to 21%. The effects of the changes in the forcing meteorological variables are relevant for all the hydrological output variables. Here we highlight results achieved for actual evapotranspiration (*ET*). This variable tends to decrease with reduced precipitation, but it increases with higher temperatures. Lower decrease in precipitation predicted by HIRHAM, together with the higher temperatures, leads to an average year *ET* increase of +2.5%, while for RegCM and CLM the annual mean reduction is equal to -5.1% (Fig. 15) and -8.3%, respectively. However, in the summer period, that is the irrigation period, in all cases an *ET* reduction is achieved (from -1.0% with HIRHAM to -9.1% with RegCM, Fig. 15), indicating a decrease in water availability for plants and soil. This water stress is better highlighted when considering simulated root zone soil moisture. For this variable a reduction is predicted, differently from *ET*, during the whole year (-20.7%±1.9%, -12.8%±1.9% and -17.6%±1.8% with RegCM, HIRHAM and CLM, respectively). Figure 16 shows as an example the daily changes computed using RegCM (the behavior considering the other RCMs is similar): they are less relevant in winter and spring, but the reduction is dramatic in summer and early autumn, due to the increased evaporative demand (up to -40% with RegCM).

Fig. 15. Location of the Crati River Basin (left) and spatially distributed percentage changes in annual actual evapotranspiration (middle) and in actual evapotranspiration during the April–September irrigation period (right) simulated using RegCM (2070-2099 vs 1961-1990) (adapted from Senatore et al., 2011).

Fig. 16. Daily changes in root zone soil moisture computed using RegCM. RCM values are rescaled over 360 days, with the first day being October the 1st (readapted from Senatore et al., 2011).

## **7. Summary and conclusions**

Evapotranspiration deeply affects the water resources availability in Calabria (average annual actual evapotranspiration estimated equal to almost 60% of the average cumulated annual rainfall). Highest water requirements come from agriculture, where losses due to evapotranspiration demand have to be re-equilibrated by huge amounts of water, mainly in the summer hot and dry period. The analysis of the comparison between the available water resource and the water demand was carried out considering both the "normal" conditions due to meteorological forcing, and the most critical derived by intense and prolonged drought periods. In the first case, neglecting the very conservative constraints proposed by the Regional Basin Authority for the minimum flow requirements, specific issues are not observed, the residual water availability being sufficient. Several problems arise instead when drought conditions occur: in these cases the development of guidelines is essential to define operative aspects about the individuation of the water use priorities, to characterize different drought levels, to individuate the main objectives of water management related to these levels, and to determine and apply the mitigation measures.

The proposed example of water resource management under shortage conditions in the agricultural area of the Sibari Plain shows the benefits that a proactive approach may provide with respect to the classical approaches based on emergency measures, which are usually expensive and not efficient. Within a proactive approach, specific care should be taken into account for reducing evapotranspiration losses through appropriate agronomic techniques. This action has to be considered as a strategic measure, with an impact on water scarcity reduction comparable to the effect of structural measures.

The review of drought indices showed that evapotranspiration could provide useful insights: i) when adopted within comprehensive indices, considering the effects of the whole water balance, and not only of some components, on water resources availability; ii) and mainly, when dealing with optical remote sensing techniques, because these allow to estimate in a relatively easy way the spatially distributed actual evapotranspiration over a specific area, and then they can relate this quantity to soil moisture and to the incoming of drought events.

Finally, applying some future scenarios with different GCMs and RCMs, it was observed that in Calabria the issues related to water resource management under shortage conditions in the next few years will be more frequent and intense, affecting wider areas. Evapotranspiration will be "tied down" by reduced precipitation (reducing its magnitude) and by higher temperatures (providing an opposite effect). It will not clearly increase or decrease on an annual basis, but in any case it will contribute to reduce useable water from the soil, needed for agricultural purposes. The hypothesized scenarios of climate change, though subject to uncertainty, have to be intended as an important part of knowledge for the planning of future interventions on the water resource by the Public Authorities, and for defining the optimal criteria to evaluate the amount of public investments.

## **8. References**

220 Evapotranspiration – Remote Sensing and Modeling

Fig. 15. Location of the Crati River Basin (left) and spatially distributed percentage changes in annual actual evapotranspiration (middle) and in actual evapotranspiration during the April–September irrigation period (right) simulated using RegCM (2070-2099 vs 1961-1990)

Fig. 16. Daily changes in root zone soil moisture computed using RegCM. RCM values are rescaled over 360 days, with the first day being October the 1st (readapted from Senatore et

Evapotranspiration deeply affects the water resources availability in Calabria (average annual actual evapotranspiration estimated equal to almost 60% of the average cumulated annual rainfall). Highest water requirements come from agriculture, where losses due to evapotranspiration demand have to be re-equilibrated by huge amounts of water, mainly in the summer hot and dry period. The analysis of the comparison between the available water resource and the water demand was carried out considering both the "normal" conditions due to meteorological forcing, and the most critical derived by intense and prolonged drought periods. In the first case, neglecting the very conservative constraints proposed by the Regional Basin Authority for the minimum flow requirements, specific issues are not observed, the residual water availability being sufficient. Several problems arise instead when drought conditions occur: in these cases the development of guidelines is essential to define operative aspects about the individuation of the water use priorities, to characterize different drought levels, to individuate the main objectives of water management related to

these levels, and to determine and apply the mitigation measures.

(adapted from Senatore et al., 2011).

**7. Summary and conclusions** 

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## **Guidelines for Remote Sensing of Evapotranspiration**

Christiaan van der Tol and Gabriel Norberto Parodi *University of Twente, Faculty of ITC The Netherlands* 

## **1. Introduction**

226 Evapotranspiration – Remote Sensing and Modeling

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This chapter describes the possibilities, the limitations and the future of remote sensing of evapotranspiration (ET). The principles behind the techniques of remote sensing of ET are presented systematically. The mathematical formulations of the key equations are used to highlight the critical parts and the variables that remote ET is most sensitive to. The focus will be on the input data. Which input data do we definitively need, and with what accuracy? How can we select the best methodology to estimate ET spatially? A number of new developments will be introduced, and priorities for the near future formulated.

There is no global, validated ET product available today. We can find products of other components of the terrestrial water cycle, like rainfall and soil moisture, but not of ET. This means that remote estimation of ET is custom made, and that it requires specific skills. At first glance, this is surprising, because the idea of remote sensing of evapotranspiration is more than three decades old (Jackson et al., 1977; Jackson et al., 1987; Seguin, 1988). In this chapter we hope to clarify the reasons why the operational dissemination of remote sensing evapotranspiration products lags behind.

The one fundamental problem with estimating ET is that it cannot be measured directly. This is well illustrated by borrowing an allegory from the evangelist Billy Graham: "I've seen the effects of the wind but I've never seen the wind". This quote is certainly true, in a literal sense, for evapotranspiration. Evapotranspiration affects the water and energy balances, and it is these effects of evapotranspiration which are observed. For example, evapotranspiration reduces soil moisture content and its cools the land surface. By studying changes in soil moisture with lysimeters, or by studying patterns in land surface temperature with remote sensing, ET is estimated. The problem is that ET is not the only factor affecting the water and energy budgets of the surface. Other processes and physical properties play important roles as well. For this reason, quite a large number of input variables are needed to achieve reasonably accurate estimates of ET; estimates that are at least better than rule-of-thumb a priori estimates. This is the case for field techniques, and even more so for remote techniques. In remote sensing an additional issue is that the input data are not from just one source. Often data of different satellite platforms are needed in conjunction with meteorological data from ground stations.

In the process of collecting the data, the user faces practical and scientific problems. The practical problems of the user include the data collection, merging the data in a GIS database, and programming the algorithm for the calculation of evapotranspiration. Each of these steps requires expertise and access to dedicated GIS software packages, because most algorithms are not available 'on the shelf'. The scientific problem is that the procedure requires merging of data measured at different time and spatial scales. The merging of data for the calculation of evapotranspiration is a typical example of a data assimilation problem. Ideally, both the accuracy and the representativeness of each of the input data are taken into account in the calculation of the final product: the spatial map of evapotranspiration. Although algorithms for the calculation of evapotranspiration are available in the literature and on the internet, there is no consistent data assimilation procedure attached that calculates the accuracy and reliability of the final product.

In recent years there have been a number of initiatives to build global products of evapotranspiration (Vinukullu et al., 2011; www.wacmos.org), addressing the above mentioned issues. The priorities for the near future are to establish a consistent way of merging the input data, improve the data assimilation techniques and to validate algorithms. In addition, new sources of data may be introduced. A promising tool is the use of satellite based laser altimetry for surface roughness estimation (Rosette et al., 2008).

It is the aim of this chapter to focus on principles that the algorithms have in common, and on the input data. Reviews of the history of remote sensing of ET can be found in the scientific literature (Courault et al., 2005; Glenn et al., 2007; Gowda et al., 2007; Kalma et al., 2008). In addition to these reviews, we would like to provide some anchor points and guidelines for the selection of a methodology for estimating ET in basin hydrology. We will quantify and evaluate the error of each of the input data, and show how this error propagates into the final result. For this analysis we will use theoretical considerations, a remote sensing model, and a selection of field data.

## **2. Principles of remote sensing algorithms for evapotranspiration**

Although remote sensing of evapotranspiration has evolved since the first initiatives in the 1970's, the fundamental principle has remained the same. All remote sensing based evapotranspiration estimates make use of the thermal and visible bands and the formulation of the energy balance of the surface. The instantaneous latent heat flux of evaporation is calculated as a residual of the energy balance, and this latent heat flux is in turn converted into an evapotranspiration rate after time integration. An inherent problem of this approach is that the errors in the various terms of the energy balance are affecting the latent heat flux in a manner that is difficult to predict. For this reason it is necessary to evaluate the different terms of the energy balance individually.

In the evaluation of the remote sensing algorithms presented in this section, we will discuss the terms, and indicate at what spatial and temporal resolution the data can be collected. It will become clear that the land surface temperature is the most important state variable. It plays a crucial role in sensible heat flux, ground heat flux and the balance of long wave radiation. Apart from the collection of accurate land surface temperature data, important selection criteria for a methodology are the heterogeneity of the land cover, the topography and the spatial resolution (sampling) of the remote data.

Neglecting the energy used in the process of photosynthesis, the instantaneous energy balance equation (EBE) over crops reads:

$$
\mathcal{R}\_n = \mathcal{G} + \mathcal{H} + \mathcal{\lambda}E \tag{1}
$$

*R*n is the net radiation remaining in the system, *G* the ground heat flux, *H* the sensible heat flux and *E* is the latent heat flux that is the energy consumed in evapotranspiration (all in W m-2). Radiation fluxes are positive when directed towards the land surface, the other fluxes are positive when pointed away from the surface. The partition of energy between the terms is largely controlled by the availability of water or moisture in the system. When moisture is not restricted, λ*E* reaches a maximum and *H* is small.

In order to estimate ET, Eq. 1 is solved for λ*E*. When applied to remote sensor retrievals, *R*<sup>n</sup> is solved entirely from a combination of radiation counting at sensor level and few ground information. Ground or soil heat flux is a minor component in densely vegetated areas, but a large term in non-vegetated or sparsely vegetated areas (Heusinkveld et al., 2004). The importance of a better evaluation of the soil heat flux is gaining attention, mainly to ensure the EBE closure in such areas. The evaluation of *H* is the major difficulty. There are several models and approaches to solve for *H* (SEB models) and a number of parameters and assumptions are still under debate. The remote sensing models for ET mainly differ in the way *H* is treated.

In the following sections, the individual terms of the EBE (Eq. 1) will be discussed in further detail. A theoretical description is presented for each term in the EBE, in combination with a discussion on the feasibility of data acquisition from remote and ground sources.

#### **2.1 Net radiation**

228 Evapotranspiration – Remote Sensing and Modeling

these steps requires expertise and access to dedicated GIS software packages, because most algorithms are not available 'on the shelf'. The scientific problem is that the procedure requires merging of data measured at different time and spatial scales. The merging of data for the calculation of evapotranspiration is a typical example of a data assimilation problem. Ideally, both the accuracy and the representativeness of each of the input data are taken into account in the calculation of the final product: the spatial map of evapotranspiration. Although algorithms for the calculation of evapotranspiration are available in the literature and on the internet, there is no consistent data assimilation procedure attached that

In recent years there have been a number of initiatives to build global products of evapotranspiration (Vinukullu et al., 2011; www.wacmos.org), addressing the above mentioned issues. The priorities for the near future are to establish a consistent way of merging the input data, improve the data assimilation techniques and to validate algorithms. In addition, new sources of data may be introduced. A promising tool is the use of satellite based laser altimetry for surface roughness estimation (Rosette et al., 2008). It is the aim of this chapter to focus on principles that the algorithms have in common, and on the input data. Reviews of the history of remote sensing of ET can be found in the scientific literature (Courault et al., 2005; Glenn et al., 2007; Gowda et al., 2007; Kalma et al., 2008). In addition to these reviews, we would like to provide some anchor points and guidelines for the selection of a methodology for estimating ET in basin hydrology. We will quantify and evaluate the error of each of the input data, and show how this error propagates into the final result. For this analysis we will use theoretical considerations, a

calculates the accuracy and reliability of the final product.

remote sensing model, and a selection of field data.

terms of the energy balance individually.

balance equation (EBE) over crops reads:

and the spatial resolution (sampling) of the remote data.

**2. Principles of remote sensing algorithms for evapotranspiration** 

Although remote sensing of evapotranspiration has evolved since the first initiatives in the 1970's, the fundamental principle has remained the same. All remote sensing based evapotranspiration estimates make use of the thermal and visible bands and the formulation of the energy balance of the surface. The instantaneous latent heat flux of evaporation is calculated as a residual of the energy balance, and this latent heat flux is in turn converted into an evapotranspiration rate after time integration. An inherent problem of this approach is that the errors in the various terms of the energy balance are affecting the latent heat flux in a manner that is difficult to predict. For this reason it is necessary to evaluate the different

In the evaluation of the remote sensing algorithms presented in this section, we will discuss the terms, and indicate at what spatial and temporal resolution the data can be collected. It will become clear that the land surface temperature is the most important state variable. It plays a crucial role in sensible heat flux, ground heat flux and the balance of long wave radiation. Apart from the collection of accurate land surface temperature data, important selection criteria for a methodology are the heterogeneity of the land cover, the topography

Neglecting the energy used in the process of photosynthesis, the instantaneous energy

*R GH E <sup>n</sup>*

*R*n is the net radiation remaining in the system, *G* the ground heat flux, *H* the sensible heat flux and *E* is the latent heat flux that is the energy consumed in evapotranspiration (all in

(1)

Net radiation *R*n is the dominant term in the EBE, since it represents the source of energy that must be balanced by the thermodynamic equilibrium of the other terms. The net radiation can also be expressed as an electromagnetic balance of all incoming and outgoing radiation reaching and leaving a flat horizontal and homogeneous surface as:

$$R\_n = S\downarrow - S\uparrow + L\downarrow - L\uparrow \tag{2}$$

Where *S* is the shortwave radiation, nominally between 0.25 to 3m and *L* is the long wave radiation, nominally between 3 to 100 m. The arrows show the direction of the flux entering '' or leaving '' the system.

Equation 2 is very convenient from the data acquisition point of view since each term can either be obtained from available models, or directly from instruments at ground stations or remote platforms. As remote sensors are positioned looking to Earth, they measure outgoing radiation only. The incoming fluxes must be either modelled or derived through alternative methodologies.

The instantaneous incoming shortwave radiation (also called global radiation), *S*, is commonly measured at ground stations by means of pyranometers or solarimeters. These instruments usually work in the shortwave broadband range (usually 0.305 - 2.4 m). This range comprises almost 96% of the spectral interval of the solar irradiance. Recently there are remote sensing products and clearinghouses that account for the incoming and outgoing shortwave and long wave radiation. The use of them may reduce the need of permanently operational ground radiometers.

The outgoing shortwave radiation is the portion of the shortwave reflected back to the atmosphere. It is characterized by the albedo. The reflectance is the ratio between the reflected and the incoming radiation in a certain wavelength over an arbitrary horizontal plane. The integrated value over all visible bands defines the albedo, *r*0. Since albedo is a reflective property of the material, it can be evaluated from remote sensors multi-spectral bands, and the integration to full shortwave range is approached by a linear model that might include the atmospheric correction. The shortwave radiation balance reads:

$$
\Delta S = S \downarrow - S \uparrow = (1 - r\_0) \cdot S \downarrow \tag{3}
$$

For all bodies, the total incident radiation is either reflected by the body, absorbed by it or transmitted through. This is expressed by the Kirchoff's law:

$$1 = \rho\_{\lambda} + \tau\_{\lambda} + \alpha\_{\lambda} \tag{4}$$

where is the reflectivity, the transmissivity and the absorptivity. A blackbody is defined as a body that absorbs all the radiation that receives. A blackbody is a physical abstraction that does not exist in nature. To keep a body temperature constant, it should emit the same radiation that absorbs. As a consequence a property of blackbodies, the absorptivity, is equal to the emissivity, and both are equal to 1, while reflectivity and transmissivity are equal to zero.

Terrestrial materials behave more as grey bodies, meaning that part of the received radiation is reflected back to the atmosphere, or in other words, not all the energy that receives is absorbed. In order to keep the temperature constant, the absorbed radiation should equal the emission, so again emissivity is equal to absorptivity. Because the reflectivity is not zero, emissivity of real bodies is smaller than 1.

The longwave radiation terms are calculated with Planck's equation extended to real bodies. A blackbody having a kinetic temperature *T*0 [K] emits in a single wavelength a radiation that corresponds to:

$$L\_{\lambda}^{bb} = \frac{3.74 \cdot 10^8}{\lambda^5} \cdot \frac{1}{\left[ \exp\left(\frac{1.44 \cdot 10^4}{\lambda \cdot T\_0}\right) - 1 \right]}\tag{5}$$

Where *L* bb is the blackbody energy emission [W m-2 m-1] and is the wavelength [m]. The kinetic temperature is the temperature as it would be measured by a standard thermometer in contact with the surface of the body. Emissivity ελ at a chosen wavelength is the ratio of the radiation emitted by a real body at temperature *T*0 to the radiation emitted by a blackbody at the same temperature. By definition, a blackbody has a constant emissivity equal to one for all wavelengths, whereas the real emissivity varies with wavelength. For natural bodies, the thermal emission can then be written as:

$$L\_{\vec{\lambda}}(T\_0) = \varepsilon\_{\vec{\lambda}} \cdot L\_{\vec{\lambda}}^{bb}(T\_0) \tag{6}$$

Integration of *L* over all wavelengths leads to:

$$L(T\_0) \uparrow = \bigcap\_{0}^{\omega} \varepsilon\_{\lambda} \cdot L\_{\lambda}^{bb}(T\_0) \cdot d\lambda = \sigma \cdot \varepsilon\_0 \cdot T\_0^4 \tag{7}$$

where = 5.67 x10-8 W m-2 K-4 is the Stefan-Boltzmann constant and 0 is a broadband surface emissivity. A remote sensor working within a spectral range of the thermal channels measures only a portion of *L*(*T*0). The outgoing longwave radiation at any sensor channel is calculated by integration over the spectral range of the sensor:

$$L\_i^{sat}(T\_0) \uparrow = \int\_i \boldsymbol{\varepsilon}\_{\lambda} \cdot L\_{\lambda}^{bb}(T\_0) \cdot d\lambda \tag{8}$$

For all bodies, the total incident radiation is either reflected by the body, absorbed by it or

 

defined as a body that absorbs all the radiation that receives. A blackbody is a physical abstraction that does not exist in nature. To keep a body temperature constant, it should emit the same radiation that absorbs. As a consequence a property of blackbodies, the absorptivity, is equal to the emissivity, and both are equal to 1, while reflectivity and

Terrestrial materials behave more as grey bodies, meaning that part of the received radiation is reflected back to the atmosphere, or in other words, not all the energy that receives is absorbed. In order to keep the temperature constant, the absorbed radiation should equal the emission, so again emissivity is equal to absorptivity. Because the reflectivity is not zero,

The longwave radiation terms are calculated with Planck's equation extended to real bodies. A blackbody having a kinetic temperature *T*0 [K] emits in a single wavelength a radiation

5 4

The kinetic temperature is the temperature as it would be measured by a standard thermometer in contact with the surface of the body. Emissivity ελ at a chosen wavelength is the ratio of the radiation emitted by a real body at temperature *T*0 to the radiation emitted by a blackbody at the same temperature. By definition, a blackbody has a constant emissivity equal to one for all wavelengths, whereas the real emissivity varies with

0 0 () () *bb LT L T*

 

0 0 0 0

surface emissivity. A remote sensor working within a spectral range of the thermal channels measures only a portion of *L*(*T*0). The outgoing longwave radiation at any sensor channel is

0 0 () () *sat bb*

*i L T LT d* 

 

() () *bb LT L T d T*

0

4

(7)

*T*

1.44 10 exp 1

8

bb is the blackbody energy emission [W m-2 m-1] and

wavelength. For natural bodies, the thermal emission can then be written as:

0

calculated by integration over the spectral range of the sensor:

*i*

 

= 5.67 x10-8 W m-2 K-4 is the Stefan-Boltzmann constant and

3.74 10 1

the transmissivity and

 

1 

transmitted through. This is expressed by the Kirchoff's law:

*bb L*

is the reflectivity,

transmissivity are equal to zero.

that corresponds to:

Where *L*

where

emissivity of real bodies is smaller than 1.

Integration of *L* over all wavelengths leads to:

where  <sup>0</sup> *SS S r S* (1 ) (3)

(4)

the absorptivity. A blackbody is

(5)

is the wavelength [m].

(6)

(8)

0 is a broadband

The surface temperature *T*0 is retrieved from Eq (8), once the surface emissivity in the considered thermal channel is estimated. Once '*T*0' is obtained and 0 estimated*, L* is retrieved from Eq 7. Before the application of Eq (8), an atmospheric correction process is needed to derive Li sur(*T*0) at the surface, because *L*<sup>i</sup> sat(*T*0) as measured at the satellite sensor is affected by atmospheric interference. Atmospheric correction in the thermal range and in the shortwave is out of the scope of this chapter. We only mention that Li sur(*T*0) can be obtained from Li sat(*T*0) and using atmospheric correction model, in which water vapour and aerosol concentrations are the main input variables.

The incoming long wave radiation cannot be derived directly from remote sensors. It can either be determined from ground data or derived after atmospheric modelling. It varies with cloudiness (water vapour), air temperature and atmospheric constituents. For clear skies, the notion of effective thermal infrared emissivity of the atmosphere or apparent emissivity of the atmosphere ( ' *a* ) introduces an overall emission value for all constituents. If the air temperature *T*a at screen level is available, *L* is estimated as:

$$L \cdot \mathbb{L} = \sigma \cdot \boldsymbol{\varepsilon}\_a^\cdot \cdot \boldsymbol{T}\_a^4 \tag{9}$$

There are several models simple to evaluate ' *a* . The apparent emissivity of the atmosphere is usually estimated with equations based on vapour pressure and temperature at standard meteorological stations. For clear skies a common formulation, among others, is (Brutsaert, 1975):

$$
\omega\_a^\circ = \mathbf{1.24} \left( \frac{e\_a}{T\_a} \right)^{\frac{1}{7}} \tag{10}
$$

Where *T*a is the air temperature [K] and *e*a is the vapour pressure [mbar], everything measured at screen level. A portion *L* reaching the Earth surface is reflected back to the atmosphere. Since the surface is opaque the transmissivity is zero, the reflection of *L*↓ can be evaluated with Kirchoff law. As 0 describes the emissivity of a body in the thermal range, (1- 0) accounts for the reflection. The final expression for *R*n becomes:

$$R\_n = (1 - r\_o) \cdot S \stackrel{\downarrow}{\rightsquigarrow} + \varepsilon\_a^{\uparrow} \cdot \sigma \cdot T\_a^4 - (1 - \varepsilon\_0) \cdot \varepsilon\_a^{\uparrow} \cdot \sigma \cdot T\_a^4 - \varepsilon\_0 \cdot \sigma \cdot T\_0^4 \tag{11}$$

Eq. 11 is valid for instantaneous observations. The conversion to a daily value is briefly discussed in Sect 2.4.

It is not always necessary to carry out the calculations of Eqs 3-11 manually. Some organizations provide atmospherically corrected components of the radiation platforms directly. For example LandSaf (landsaf.meteo.pt) provides MeteoSat Second Generation (MSG) products of atmospherically corrected *S* and *L* with a 15 minute resolution and daily albedo for South America, Africa and Europe. An emissivity product will be released soon. Validation over ground based measurements for a site in Spain over sparse vegetation shows that these products are rather reliable (Fig 1.).

As an alternative to the use of satellite data, a computation of the radiation terms from synoptic weather stations is also possible. The recommendations by the FAO (Doorenbos and Pruitt, 1977; Allen et al., 1998) could be followed. The daily short wave radiation *S*↓day [MJm-2day-1], is measured at agrometeorological stations with pyranometers and integrated to daytime hours. In most areas in the world, only sunshine hours are measured with periheliometers. In that case, the daily incoming shortwave radiation *S*↓day can be obtained from the following empirical relationship:

$$S\_{\rm day} \cdot \downarrow = (a\_s + b\_s \cdot \frac{m}{N}) \cdot S\_{0,\rm day} \cdot \downarrow \tag{12}$$

where *a*s is the fraction of the extraterrestrial radiation reaching the ground in a complete overcast day (when *n*=0), *a*s + *b*s the fraction of the extraterrestrial radiation reaching the ground in a complete clear day (*n*=*N*), *n* the duration of bright sunshine per day [hours], *N* the total daytime length [hours], *S*0, day is the terrestrial radiation [MJm-2 day-1]. Local instrumentation can be used to calibrate *a*s and *b*s for local conditions.

Fig. 1. Comparison between MeteoSat Second Generation radiation products (symbols) with 5-minute interval ground based measurements (lines) for a pixel with sparse vegetation in central Spain, for 5 July 2010.

The net daily shortwave radiation *S*day is estimated as in Eq. (3), assuming an average daily (sun hours only) albedo *r*0day. The daily longwave radiation exchange between the surface and the atmosphere is very significant. Since on average the surface is warmer than the atmosphere and 0>a', there is usually a net loss of energy as thermal radiation from the ground. The daily net shortwave radiation *L*day [W m-2] between vegetation and soil on the one hand, and atmosphere and clouds on the other, can be represented by the following radiation law:

$$
\Delta L\_{\text{day}} = -f \cdot \varepsilon\_{a,\text{day}}^{\text{'}} \cdot \sigma \cdot \left(T\_{a,\text{mean}} + 273.15\right)^{4} \tag{13}
$$

periheliometers. In that case, the daily incoming shortwave radiation *S*↓day can be obtained

day 0,day ( ) *s s <sup>n</sup> S ab S*

where *a*s is the fraction of the extraterrestrial radiation reaching the ground in a complete overcast day (when *n*=0), *a*s + *b*s the fraction of the extraterrestrial radiation reaching the ground in a complete clear day (*n*=*N*), *n* the duration of bright sunshine per day [hours], *N* the total daytime length [hours], *S*0, day is the terrestrial radiation [MJm-2 day-1]. Local

Fig. 1. Comparison between MeteoSat Second Generation radiation products (symbols) with 5-minute interval ground based measurements (lines) for a pixel with sparse vegetation in

The net daily shortwave radiation *S*day is estimated as in Eq. (3), assuming an average daily (sun hours only) albedo *r*0day. The daily longwave radiation exchange between the surface and the atmosphere is very significant. Since on average the surface is warmer than the

ground. The daily net shortwave radiation *L*day [W m-2] between vegetation and soil on the one hand, and atmosphere and clouds on the other, can be represented by the following

day ,da ,mean ( 273.15) *Lf T ay a*

 

a', there is usually a net loss of energy as thermal radiation from the

(13)

' 4

instrumentation can be used to calibrate *a*s and *b*s for local conditions.

*N*

(12)

from the following empirical relationship:

central Spain, for 5 July 2010.

0>

atmosphere and

radiation law:

Where '*a*,day [-] is the daily net emissivity between the atmosphere and the ground, *f* a cloudiness factor and *T*a,mean is the mean daily air temperature at screen level [°C]. Parameter '*a,*day can be estimated from data from meteorological stations as:

$$
\stackrel{\circ}{\varepsilon\_{a,\text{day}}} = a\_{\varepsilon} + b\_{\varepsilon} \cdot \sqrt{\stackrel{\varepsilon\_{d,\text{mean}}}{10}} \tag{14}
$$

Where *a*e is a correlation coefficient (ranging from 0.34 to 0.44, with a default of 0.34), *b*e a correlation coefficient (ranging from -0.14 to –0.25 with a default of -0.14), *e*d,mean the average vapour pressure at temperature [mbar]. If true *e*d,mean is not available, then it can be calculated from daily average relative humidity *RH*mean and mean air temperature *T*a,mean [°C]:

$$e\_{d, \text{mean}} = \frac{RH\_{\text{mean}}}{100} \cdot e\_{s, \text{mean}} \text{ and } \ e\_{s, \text{mean}} = 6.108 \cdot \exp\left|\frac{17.27 \cdot T\_{\text{o,mean}}}{T\_{\text{o,mean}} + 237.15}\right|\tag{15}$$

The cloudiness factor *f* is equal to 1 in case of a perfect clear day and 0 in a complete overcast day. In case the station has solar radiation data from pyranometers *f* can be calculated as:

$$f = a\_c \cdot \frac{S \downarrow\_{\text{day}}}{S \downarrow\_{\text{clear,day}}} + b\_c \cdot \text{or} \cdot f = a\_c \cdot \frac{S \downarrow\_{\text{day}}}{(a\_s + b\_s \cdot) \cdot S\_0 \downarrow\_{\text{day}}} + b\_c \tag{16}$$

Where *as*, *bs*, *a*c and *i*c are calibration values to be estimated through specialized local studies which involve measuring longwave radiation values. Average values for *a*c and *b*c in arid and humid environments can be found in Table 1:


Table 1. Typical values the coefficients *a*c, *b*c, *a*s and *b*s for arid and humid climates (Maidment, 1992).

If only data on sunshine hours data are available, then:

$$f = \left(a\_c \cdot \frac{b\_s}{a\_s + b\_s}\right) \cdot \frac{n}{N} + \left(b\_c + \frac{a\_s}{a\_s + b\_s} \cdot a\_c\right) \tag{17}$$

#### **2.2 Sensible heat flux**

The sensible heat flux (*H*) is the exchange of heat through air as a result of a temperature gradient between the surface the atmosphere. Since the surface temperature during the day is usually higher than the air temperature, the sensible heat flux is normally directed upwards. During the night the situation may be reversed. Close to the surface, the sensible heat transport takes place mostly by diffusive processes, whereas at some distance away from the surface turbulent transport becomes more important.

The mathematical formulation of the sensible heat flux is based on the theory of mass transport of heat and momentum between the surface and the near-surface atmospheric environment (surface boundary layer). All existing remote sensing algorithms for turbulent sensible heat flux use the analogy of Ohm law of resistance driven by a gradient of temperature:

$$H = \rho\_a \cdot c\_p \cdot \frac{T\_s - T\_a}{r\_{\text{ah}}} \tag{18}$$

where a is the density of moist air [kg m-3], *c*p is the air specific heat at constant pressure [J kg-1 K-1], *r*ah is the aerodynamic resistance to heat transport between the surface and the reference level [s m-1] and *T*s – *T*a is the driving temperature gradient between the surface (with temperature *T*s) and the reference height (with temperature *T*a).

Equation 18 shows that the estimation of sensible heat flux has two main elements: a temperature difference between two heights and the corresponding resistance. As a first approximation we can conclude that the error in the sensible heat is linearly proportional to the error in the temperature gradient, and linearly proportional to the error in the inverse of the resistance. Equation 1 shows that this error (W m-2) is directly transferred to the latent heat flux. We will now show that this is only approximately true, because the equation is not linear and the aerodynamic resistance itself depends on the temperature gradient. We will show that because of this, *r*ah can only be solved iteratively.

Understanding the physical concepts involved in the calculation of sensible heat flux, and in particular the aerodynamic resistance, is essential for an evaluation of remote sensing techniques. The evaluation of *r*ah is the most complicated issue of all in the whole EBE procedure for AET estimates. It is our experience that lack of or incomplete knowledge of the entire formulation, image pre-processing and atmospheric correction processes leads to severe flaws in the intermediate and final outputs. Many researchers are still seeking for alternatives, procedures and methods to improve the accuracy of ET estimates form the EBE – RS approach. The actual parameterization is not optimal in the sense that some sensitive information can only be strictly evaluated under controlled experimental research, and not in a routine fashion.

Near the ground two phenomena take place simultaneously in the transfer of heat between the surface and the atmosphere: free convection produced by temperature gradient *T*s-*T*<sup>a</sup> and forced convection by the dragging forces of the wind. Then, the estimation of the turbulent heat fluxes requires a description of the turbulent wind profile near the surface. The starting point of the analysis is the wind profile in a neutral atmosphere (no convection, and *T*s=*T*a). In this situation and for an open site, the horizontal wind speed *u* [m s-1] varies logarithmically with height above the ground *z* [m]:

$$
\ln(z) = A \cdot \ln(z) + B \tag{19}
$$

*B* is usually replaced by *A* .ln (zom) where *z*0m is the aerodynamic roughness length of the surface for momentum transport and represents the value of *z* for which Eq 19 predicts *u*(*z*) = 0 (see also Fig 2):

$$
\mu(z) = A \cdot \ln \left( \frac{z}{z\_{0\text{m}}} \right) \tag{20}
$$

In Eq 20, *A* must have the dimension of velocity and it should be independent of *z* since the profile description is given by the logarithmic term. Over plant communities of uniform height *h*, the turbulent boundary layer behaves as if the vertically distributed elements of the community were located at a certain distance *d* from the ground. Parameter *d* is called

sensible heat flux use the analogy of Ohm law of resistance driven by a gradient of

kg-1 K-1], *r*ah is the aerodynamic resistance to heat transport between the surface and the reference level [s m-1] and *T*s – *T*a is the driving temperature gradient between the surface

Equation 18 shows that the estimation of sensible heat flux has two main elements: a temperature difference between two heights and the corresponding resistance. As a first approximation we can conclude that the error in the sensible heat is linearly proportional to the error in the temperature gradient, and linearly proportional to the error in the inverse of the resistance. Equation 1 shows that this error (W m-2) is directly transferred to the latent heat flux. We will now show that this is only approximately true, because the equation is not linear and the aerodynamic resistance itself depends on the temperature gradient. We will

Understanding the physical concepts involved in the calculation of sensible heat flux, and in particular the aerodynamic resistance, is essential for an evaluation of remote sensing techniques. The evaluation of *r*ah is the most complicated issue of all in the whole EBE procedure for AET estimates. It is our experience that lack of or incomplete knowledge of the entire formulation, image pre-processing and atmospheric correction processes leads to severe flaws in the intermediate and final outputs. Many researchers are still seeking for alternatives, procedures and methods to improve the accuracy of ET estimates form the EBE – RS approach. The actual parameterization is not optimal in the sense that some sensitive information can only be strictly evaluated under controlled experimental research, and not

Near the ground two phenomena take place simultaneously in the transfer of heat between the surface and the atmosphere: free convection produced by temperature gradient *T*s-*T*<sup>a</sup> and forced convection by the dragging forces of the wind. Then, the estimation of the turbulent heat fluxes requires a description of the turbulent wind profile near the surface. The starting point of the analysis is the wind profile in a neutral atmosphere (no convection, and *T*s=*T*a). In this situation and for an open site, the horizontal wind speed *u* [m s-1] varies

*B* is usually replaced by *A* .ln (zom) where *z*0m is the aerodynamic roughness length of the surface for momentum transport and represents the value of *z* for which Eq 19 predicts *u*(*z*)

( ) ln *<sup>z</sup> uz A*

In Eq 20, *A* must have the dimension of velocity and it should be independent of *z* since the profile description is given by the logarithmic term. Over plant communities of uniform height *h*, the turbulent boundary layer behaves as if the vertically distributed elements of the community were located at a certain distance *d* from the ground. Parameter *d* is called

0m

*z* 

(with temperature *T*s) and the reference height (with temperature *T*a).

show that because of this, *r*ah can only be solved iteratively.

logarithmically with height above the ground *z* [m]:

ah *s a a p T T H c*

(18)

*uz A z B* ( ) ln( ) (19)

(20)

*r*

a is the density of moist air [kg m-3], *c*p is the air specific heat at constant pressure [J

temperature:

in a routine fashion.

= 0 (see also Fig 2):

where  the zero plane displacement or displacement height level of the flow. It acts as a correction to the level where *z*=0, and thus *z* in Eqs 19 and 20 should be directly replaced by *z*-*d* in vegetated areas:

shift *z zd* (21) <sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> 2.5 3 3.5 4 u (m s-1) ln (z)

Fig. 2. Example plot of mean wind speed *u* against ln(*z*), measured above a 31 m tall forest in The Netherlands in August 2009. The intercept with the vertical axis leads to *z*0m+*d* = 17.9 m.

The displacement height *d* is usually rather large: it ranges between 60 to 80 percent of the plant community height. Common values for well developed wheat are found in Verma and Bartfield (1979). A relation *d*= 0.67*h* is usually adopted for other vegetation types (Allen et al., 1998). An exact estimation can be carried out when a wind speed measurements at three or more heights are available. A plot of ln(*z*-*d*) versus wind speed should then give a straight line for the correctly calibrated value of *d*. Strictly speaking, *d* also depends on plant density; for sparse vegetation, *d* is often neglected. In many occasions, insufficient data are available to accurately predict its values for discrete crop canopies (Verma and Bartfield, 1979). If no valid field data are present, then it is suggested to leave *d* out of the equations altogether.

The logarithmic wind profile in neutral conditions forms the basis for calculating the aerodynamic resistance for heat transport (and transport of water vapour) by mechanical turbulence. To include thermal turbulence (or convection or buoyancy), the Monin-Obukhov theory is needed in addition (Obukhov, 1946). We will now summarize the equations leading to the aerodynamic resistance of mechanical transport, followed by the modification for non-neutral conditions, when convection plays a role.

The transfer of momentum in the direction of the flux takes place through molecular and turbulent eddy activity. Random vertical movements of the air cause air with different horizontal wind speeds to mix. This causes a momentum sink at the surface in a form of shear stress, . For convenience the shear stress is expressed as a function of a scalar *u*\*, usually called eddy velocity or friction velocity [m s-1]:

$$
\pi = \rho\_d \mu\_\*^2 \tag{22}
$$

Since wind is produced by turbulent eddy motion, it is postulated that *A* in Eq 20 is proportional to the speed of the internal eddies. Then it can be demonstrated that:

$$A = \frac{\mu\_\*}{k} \tag{23}$$

Where *k* is Von Kármán's constant, experimentally found to be 0.41. The final expression of the wind profile under neutral atmosphere is:

$$
\mu(z) = \frac{\mu\_\*}{k} \cdot \ln\left(\frac{z}{z\_{0\text{m}}}\right) \tag{24}
$$

Then the gradient of wind speed with height can be expressed as:

$$\frac{d\mathbf{u}}{dz} = \frac{\mathbf{u} \cdot \mathbf{s}}{k \cdot z} \tag{25}$$

In the theory of estimating the aerodynamic resistance from the wind profile, only vertical transport is considered. In the atmosphere the steepest gradients of heat, wind speed and humidity are found in the vertical direction. Horizontal variation is present in the order of tens of kilometres (Brutsaert, 2005), but these are considered negligible. This implies that horizontal advection effects (for example between pixels) are not considered in the remote sensing approach, a serious restrictions in patchy (wet and dry) environments.

In analogy to horizontal wind speed, heat and water vapour also have vertical profiles near the surface. Vertical mixing then causes a transport of heat and vapour too, resulting in vertical fluxes of sensible and latent heat. The three fluxes, of momentum (*F*u), heat (*F*h) and vapour (*F*v), can be expressed as the covariance of vertical wind speed (*w*') and concentration of the admixture (*u*', *T*' and *q*'):

$$\begin{aligned} F\_u &\equiv -\tau = \rho \cdot \overline{w' \cdot u'}\\ F\_h &\equiv -H = \rho \cdot c\_p \cdot \overline{w' \cdot T'}\\ F\_v &\equiv -\lambda E = \rho \cdot \overline{w' \cdot q'} \end{aligned} \tag{26}$$

These equations can be linked with the approach of electrical analogy (Eq 18) by approximating the covariance to simply the product of the vertical gradient of the quantities at two different heights (Brutsaert, 2005). A dimensionless parameter *C* is needed to fit the equality. This can be done for all three quantities. For example, for heat:

$$
\overline{w' \cdot T'} = \mathcal{C}\_h \cdot \left(\overline{\mu\_2} - \overline{\mu\_1}\right) \cdot \left(\overline{T\_4} - \overline{T\_3}\right) \tag{27}
$$

In this case, *Ch* will depend on the heights 1, 2, 3 and 4. It is convenient to choose the heights 4 and 3 equal to 2 and 1:

$$\overline{w' \cdot T'} = \mathbb{C}\_h \cdot \left(\overline{u\_2} - \overline{u\_1}\right) \cdot \left(\overline{T\_2} - \overline{T\_1}\right) \tag{28}$$

Similarly, for momentum transport (shear stress):

$$
\overline{uw'u'} = \mathbb{C}\_d \cdot \left(\overline{\mu\_2} - \overline{\mu\_1}\right)^2 \tag{29}
$$

The coefficient *Cd* can be calculated by combining Eqs 22, 24, 26 and 29. Using *z*1=*z*0m, and considering that at *z*= *z*0m , the wind speed is zero:

*<sup>u</sup>*\* *<sup>A</sup>*

Where *k* is Von Kármán's constant, experimentally found to be 0.41. The final expression of

\*

*u z*

sensing approach, a serious restrictions in patchy (wet and dry) environments.

*u*

*v*

equality. This can be done for all three quantities. For example, for heat:

*h p*

 

These equations can be linked with the approach of electrical analogy (Eq 18) by approximating the covariance to simply the product of the vertical gradient of the quantities at two different heights (Brutsaert, 2005). A dimensionless parameter *C* is needed to fit the

In this case, *Ch* will depend on the heights 1, 2, 3 and 4. It is convenient to choose the heights

The coefficient *Cd* can be calculated by combining Eqs 22, 24, 26 and 29. Using *z*1=*z*0m, and

2

*F wu F H c wT F E wq*

 

 

Then the gradient of wind speed with height can be expressed as:

( ) ln *u z*

\* *du u*

In the theory of estimating the aerodynamic resistance from the wind profile, only vertical transport is considered. In the atmosphere the steepest gradients of heat, wind speed and humidity are found in the vertical direction. Horizontal variation is present in the order of tens of kilometres (Brutsaert, 2005), but these are considered negligible. This implies that horizontal advection effects (for example between pixels) are not considered in the remote

In analogy to horizontal wind speed, heat and water vapour also have vertical profiles near the surface. Vertical mixing then causes a transport of heat and vapour too, resulting in vertical fluxes of sensible and latent heat. The three fluxes, of momentum (*F*u), heat (*F*h) and vapour (*F*v), can be expressed as the covariance of vertical wind speed (*w*') and

' '

' '

' '

*wT C u u T T* ' ' *<sup>h</sup>* 21 43 (27)

*wT C u u T T* ' ' *<sup>h</sup>* 21 21 (28)

2 1 ' ' *wu C u u <sup>d</sup>* (29)

*k z* 

0m

the wind profile under neutral atmosphere is:

concentration of the admixture (*u*', *T*' and *q*'):

Similarly, for momentum transport (shear stress):

considering that at *z*= *z*0m , the wind speed is zero:

4 and 3 equal to 2 and 1:

*<sup>k</sup>* (23)

*dz k z* (25)

(24)

(26)

$$\mathbf{C}\_d = \left[ k \sqrt{\ln \left( \frac{z\_2}{z\_{0\text{m}}} \right)} \right]^2 \tag{30}$$

In neutral conditions it can be assumed that *C*h=*C*v=*C*d, and thus:

$$H = -\rho \cdot c\_p \mathbb{C}\_d \cdot \mu\_2 \cdot \left(\overline{T\_2} - \overline{T(z\_{0m})}\right) \tag{31}$$

The appearance of the average temperature at height *z*0m in Eq 31 is inconvenient. It can be eliminated by assuming a logarithmic wind profile for temperature too, by defining a scalar roughness height for heat transfer *z*0h at which the extrapolated temperature profile fitted through *T*2 and 0 *T z*( ) *<sup>m</sup>* becomes *T*0, i.e. the kinematic surface temperature. Using this we finally express the aerodynamic resistance *rah* in neutral conditions as:

$$r\_{\rm ah} = \frac{\ln\left(\frac{z\_2}{z\_{0\rm m}}\right)\ln\left(\frac{z\_2}{z\_{0\rm h}}\right)}{k^2 \cdot \nu} = \frac{\ln\left(\frac{z\_2}{z\_{0\rm h}}\right)}{k\nu\_\*}\tag{32}$$

The roughness height, *z*0h, changes with surface characteristics, atmospheric flow and thermal dynamic state of the surface (Blümel, 1999; Massman, 1999). It can be shown that:

$$z\_{0h} = z\_{0m} \;/\exp\left(kB^{-1}\right) \tag{33}$$

where *B*-1 is the inverse Stanton number, a dimensionless heat transfer coefficient.

Free convection might alter the forced convective eddies generated by wind turbulence. During daytime or when temperature decreases with height, convection amplifies the vertical eddy motions (unstable condition). During the night or when inversion conditions occur, and temperature increases with height, the horizontal eddy motions are enhanced (stable conditions).

Mechanical turbulence and buoyancy coexists in a form of a hybrid regime known as mixconvection. Monin and Obukhov showed that these conditions eventually lead to an alteration of the wind and temperature profiles (Brutsaert, 1982). The Monin-Obukhov similarity theory uses dimensional analysis to correct the wind profile produced by buoyancy effects in such conditions. A non-dimensional correction factor for momentum transfer m() is introduced to correct the wind profile gradient for conditions different from neutral, in which is the ratio of thermal to mechanical turbulence:

$$\frac{du}{dz} = \frac{u\_\*}{kz} \cdot \varphi\_m\left(\xi\right) \tag{34}$$

They introduced semi-empirical functions to correct the wind profile depending on the stability, based on dimensional analysis, of the form:

$$
\mu\_\* = \frac{\overline{u(z)}}{k} \cdot \left[ \ln \left( \frac{z - d\_0}{z\_{0\text{m}}} \right) - \Psi\_m \left( \frac{z - d\_0}{L} \right) + \Psi\_m \left( \frac{z\_{0\text{m}}}{L} \right) \right] \tag{35}
$$

$$H = \frac{T\_s - \overline{T(z)}}{k \cdot \mu\_\* \cdot \rho \cdot c\_p} \cdot \left[ \ln \left( \frac{z - d\_0}{z\_{0\text{h}}} \right) - \Psi\_h \left( \frac{z - d\_0}{L} \right) + \Psi\_h \left( \frac{z\_{0\text{h}}}{L} \right) \right] \tag{36}$$

Where *L* is defined as the Monin-Obukhov length (*L* = *z* ) [m], calculated as:

$$L = -\frac{\rho\_a \cdot \mathbb{C}\_p \cdot u\_\*^{\overline{3}} \cdot T(z)}{k \cdot \mathbb{g} \cdot H} \tag{37}$$

Where *g* is the gravity constant (9.81 m s-2). Semi-empirical expressions for the stability corrections *<sup>h</sup>* and *m* can be found in the literature, for example Paulson (1970) and Brutsaert (1982). It is important to realize that *L* depends on air temperature and sensible heat flux, while sensible heat flux and air temperature in turn depend on *L*. For this reason, an iterative procedure is needed to calculate *L*, *u\** and *H* using Eqs 35-37.

#### **2.3 Ground heat flux**

The ground heat flux has received relatively little attention compared to the other terms. This is often justified, because ground heat flux is usually the smallest of all terms. Moreover, the 24-hour sum of ground heat flux is close to zero, because the heat absorbed during the day is released during the night.

At the moment of a satellite overpass, ground heat flux is not necessarily negligible. At midday it usually varies from 10% of net radiation for dense vegetation to 45% of net radiation for bare soil (Clothier et al., 1986). Often a vegetation cover dependent ratio between *G* and *R*n is assumed at satellite overpass (Kustas et al., 1990).

If more accurate estimates of ground heat flux are required, for example in areas with sparse vegetation, then remote estimates of ground heat flux are possible with the method of Van Wijk and De Vries (1963). For this method, diurnal cycles of land surface temperature and net radiation are needed (Verhoef, 2004; Murray and Verhoef, 2007); this means that time series of data of a geostationary satellite are required.

An equation for ground heat flux can be derived the thermal diffusion equation, assuming a periodic land surface temperature:

$$G(t) = \Gamma \sum\_{k=0}^{n} \sqrt{k o / 2} \cdot \left( A\_k \cdot \sin(\alpha kt) + B\_k \cdot \cos(\alpha kt) \right) \tag{38}$$

where is the thermal inertia of the soil (J m-2 K-1 s-1/2), which depends on texture and soil moisture, *t* is time (s), = (2/*N*) is the radial frequency (s-1), *N* the length of the time series [s], *A* and *B* integration coefficients [C], and *n* the number of harmonics. The coefficients *A* and *B* are fitted against the observed land surface temperature time series, for a chosen number of harmonics. The thermal inertia can be estimated from soil texture and soil moisture, or calibrated against night time radiation, by assuming that night time radiation equals the night-time ground heat flux.

#### **2.4 Latent heat flux**

Latent heat flux is finally calculated as a residual of the energy balance (Eq. 1). Because *H*, *G* and *R*n are instantaneous measurements, it is necessary to find a procedure to integrate to daily totals. A common way to carry out this integration, is by making use of the evaporative fraction, . The evaporative fraction (Brutsaert and Sugita, 1992) is the energy used for the evaporation process divided by the total amount of energy available for the evaporation process:

3 \* ( ) *a p C u Tz*

*kgH*

Where *g* is the gravity constant (9.81 m s-2). Semi-empirical expressions for the stability corrections *<sup>h</sup>* and *m* can be found in the literature, for example Paulson (1970) and Brutsaert (1982). It is important to realize that *L* depends on air temperature and sensible heat flux, while sensible heat flux and air temperature in turn depend on *L*. For this reason,

The ground heat flux has received relatively little attention compared to the other terms. This is often justified, because ground heat flux is usually the smallest of all terms. Moreover, the 24-hour sum of ground heat flux is close to zero, because the heat absorbed

At the moment of a satellite overpass, ground heat flux is not necessarily negligible. At midday it usually varies from 10% of net radiation for dense vegetation to 45% of net radiation for bare soil (Clothier et al., 1986). Often a vegetation cover dependent ratio

If more accurate estimates of ground heat flux are required, for example in areas with sparse vegetation, then remote estimates of ground heat flux are possible with the method of Van Wijk and De Vries (1963). For this method, diurnal cycles of land surface temperature and net radiation are needed (Verhoef, 2004; Murray and Verhoef, 2007); this means that time

An equation for ground heat flux can be derived the thermal diffusion equation, assuming a

( ) 2 sin( ) cos( )

where is the thermal inertia of the soil (J m-2 K-1 s-1/2), which depends on texture and soil

[s], *A* and *B* integration coefficients [C], and *n* the number of harmonics. The coefficients *A* and *B* are fitted against the observed land surface temperature time series, for a chosen number of harmonics. The thermal inertia can be estimated from soil texture and soil moisture, or calibrated against night time radiation, by assuming that night time radiation

Latent heat flux is finally calculated as a residual of the energy balance (Eq. 1). Because *H*, *G* and *R*n are instantaneous measurements, it is necessary to find a procedure to integrate to daily totals. A common way to carry out this integration, is by making use of the evaporative fraction, . The evaporative fraction (Brutsaert and Sugita, 1992) is the energy used for the evaporation process divided by the total amount of energy available for the

*G t k A kt B kt*

(38)

= (2/*N*) is the radial frequency (s-1), *N* the length of the time series

*k k*

 

) [m], calculated as:

(37)

Where *L* is defined as the Monin-Obukhov length (*L* = *z*

**2.3 Ground heat flux** 

during the day is released during the night.

series of data of a geostationary satellite are required.

equals the night-time ground heat flux.

periodic land surface temperature:

moisture, *t* is time (s),

**2.4 Latent heat flux** 

evaporation process:

*L*

an iterative procedure is needed to calculate *L*, *u\** and *H* using Eqs 35-37.

between *G* and *R*n is assumed at satellite overpass (Kustas et al., 1990).

0

*k*

*n*

$$
\Lambda = \frac{\lambda E}{\lambda E + H} = \frac{\lambda E}{R\_n - G} \tag{39}
$$

It is assumed that the evaporative fraction remains constant throughout the day.

$$
\Lambda\_{inst} = \Lambda\_{24\,hrs} \tag{40}
$$

Assuming that the ground heat flux integrated over 24-hours is negligible, the evapotranspiration rate over 24 hours can be calculated as:

$$E\_{24hrs} = \frac{8.64 \cdot 10^7 \,\Lambda\_{inst}}{\lambda \rho\_w} R\_{n,24hrs} \tag{41}$$

Where = 2.0501-0.00236 Twater MJ kg-1 (T in C), w= 1000 kg m-3 and *R*n24 is the average net radiation over 24 hours [W m-2].

The assumption of a constant evaporative fraction may lead to underestimates of daily evaporation, because the evaporative fraction in reality has a diurnal cycle with a concave shape (Gentine, et al., 2007). The concave shape is caused by changes in weather conditions (wind, advection, humidity), a phase difference between ground heat flux and net radiation, and stomatal regulation. There is an alternative to the assumption of constant evaporative fraction if hourly weather data are available. It may then be assumed that the ratio of actual to reference evaporation is constant over the day; hourly values of reference evaporation can be calculated (Allen et al., 2007). The ratio of actual to reference evaporation is more stable, because it eliminates the effects of diurnal variations in weather conditions.

#### **3. Data requirements and sensitivity**

Every remote sensing based SEB model requires a sequence of dedicated ground and remote sensing data to properly operate. Efforts increasingly focus on the remote estimation of the necessary variables, but ground data are still needed in addition.

All models require net radiation and land surface temperature retrieved from remote sensing. The additional required information varies among algorithms. As an example we list the input needed for the remote sensing model SEBS (Su, 2002). This model explicitly solves Eqs 35-37. It also includes an algorithm to estimate *kB*-1 from vegetation cover fraction.

SEBS requires the following data, most of which cannot be retrieved from remote sensing, but is obtained from ground-based meteorological data instead:


All meteorological input must be instantaneous information collected at the time of satellite overpass, interpolated and re-sampled to the pixel size. Other models require similar input. Two source models do not use the concept of *kB*-1, but require separate resistances for soil and vegetation. Although the exact input data varies per algorithm, the most important are those related to the calculation of sensible heat flux, in particular the surface-air gradient and the corresponding aerodynamic resistance *r*ah. The success or failure of a SEB relies on the skills of the research team to extract realistic values for these two variables. For this reason, we will discuss these in more detail in the following sections.

### **3.1 The temperature gradient**

For the temperature gradient we need to estimate both the air and the land surface temperature. The issue is that sensible heat flux is proportional to a difference between two temperatures which are obtained from two different sources in the same vertical. For this reason great care should be taken to retrieve both temperatures accurately.

For the air temperature at reference height, interpolated data of meteorological stations are commonly used. We need the air temperature well above the canopy, in the atmospheric surface layer, for which the aerodynamic resistance is defined. The standard measurement height in meteorological stations of 2 m cannot be used for vegetation taller than this height. Thus a conversion of temperatures from the meteorological stations to a higher reference height is needed. Another option is to use temperature profiles disseminated by organizations like EUMetsaf (www.eumetsat.int).

For the surface temperature it is necessary to take a closer look at the concepts first. As discussed before, the radiometric temperature is the temperature as it is retrieved from a remote radiometer by inverting Stefan-Boltzmann's law, assuming a bulk emissivity for the thermal spectrum range of the radiometer. The kinematic temperature is the real, contact temperature. A third definition is needed here: the aerodynamic temperature, which is hypothetic temperature obtained when extrapolating the vertical profile of air temperature to the depth *z*0h. The aerodynamic temperature is a conceptual model parameter that is close to the kinetic temperature, but they are not equal. The reason is that kinematic temperature varies between the elements of the surface within a remote sensing pixel. For example, sunlit and shaded parts of the soil and canopy may have rather different temperatures. This is particularly the case in a heterogeneous landscape, where bare soil, vegetated and paved areas are mixed. It is even the case in a homogeneous land cover, where leaf temperatures may differ depending on their vertical position in the crown. This is illustrated in Fig 3, showing the diurnal variations of contact temperatures of a needle forest in the Netherlands, the Speulderbos site, measured during a field campaign in The Netherlands on 16 June 2006 (Su et al., 2009). The lines are ensembles of 8 soil surface and 9 needle temperature sensors, mounted at different heights of the canopy of just a few trees.

The heterogeneity of the soil and canopy temperatures will affect the radiometric surface temperature. The radiometric temperature is predominantly affected by the upper, visible, part of the canopy. Lower canopy layers also contribute to the outgoing upward radiation, but their contribution will be relatively low due to re-absorption of radiation. The radiometric temperature also depends on the solar angle and the observation angle of the satellite.

A new model to analyse these effects is the model SCOPE (**S**oil **C**anopy **O**bvservation of **P**hotosynthesis and the **E**nergy balance). This model is a radiative transfer model combined with an energy balance model for homogeneous vegetation (Van der Tol et al., 2009). With SCOPE one can analyse the relation between the sensible heat flux, the kinematic

All meteorological input must be instantaneous information collected at the time of satellite overpass, interpolated and re-sampled to the pixel size. Other models require similar input. Two source models do not use the concept of *kB*-1, but require separate resistances for soil and vegetation. Although the exact input data varies per algorithm, the most important are those related to the calculation of sensible heat flux, in particular the surface-air gradient and the corresponding aerodynamic resistance *r*ah. The success or failure of a SEB relies on the skills of the research team to extract realistic values for these two variables. For this

For the temperature gradient we need to estimate both the air and the land surface temperature. The issue is that sensible heat flux is proportional to a difference between two temperatures which are obtained from two different sources in the same vertical. For this

For the air temperature at reference height, interpolated data of meteorological stations are commonly used. We need the air temperature well above the canopy, in the atmospheric surface layer, for which the aerodynamic resistance is defined. The standard measurement height in meteorological stations of 2 m cannot be used for vegetation taller than this height. Thus a conversion of temperatures from the meteorological stations to a higher reference height is needed. Another option is to use temperature profiles disseminated by

For the surface temperature it is necessary to take a closer look at the concepts first. As discussed before, the radiometric temperature is the temperature as it is retrieved from a remote radiometer by inverting Stefan-Boltzmann's law, assuming a bulk emissivity for the thermal spectrum range of the radiometer. The kinematic temperature is the real, contact temperature. A third definition is needed here: the aerodynamic temperature, which is hypothetic temperature obtained when extrapolating the vertical profile of air temperature to the depth *z*0h. The aerodynamic temperature is a conceptual model parameter that is close to the kinetic temperature, but they are not equal. The reason is that kinematic temperature varies between the elements of the surface within a remote sensing pixel. For example, sunlit and shaded parts of the soil and canopy may have rather different temperatures. This is particularly the case in a heterogeneous landscape, where bare soil, vegetated and paved areas are mixed. It is even the case in a homogeneous land cover, where leaf temperatures may differ depending on their vertical position in the crown. This is illustrated in Fig 3, showing the diurnal variations of contact temperatures of a needle forest in the Netherlands, the Speulderbos site, measured during a field campaign in The Netherlands on 16 June 2006 (Su et al., 2009). The lines are ensembles of 8 soil surface and 9 needle temperature sensors,

The heterogeneity of the soil and canopy temperatures will affect the radiometric surface temperature. The radiometric temperature is predominantly affected by the upper, visible, part of the canopy. Lower canopy layers also contribute to the outgoing upward radiation, but their contribution will be relatively low due to re-absorption of radiation. The radiometric

A new model to analyse these effects is the model SCOPE (**S**oil **C**anopy **O**bvservation of **P**hotosynthesis and the **E**nergy balance). This model is a radiative transfer model combined with an energy balance model for homogeneous vegetation (Van der Tol et al., 2009). With SCOPE one can analyse the relation between the sensible heat flux, the kinematic

temperature also depends on the solar angle and the observation angle of the satellite.

reason, we will discuss these in more detail in the following sections.

reason great care should be taken to retrieve both temperatures accurately.

**3.1 The temperature gradient** 

organizations like EUMetsaf (www.eumetsat.int).

mounted at different heights of the canopy of just a few trees.

Fig. 3. Diurnal cycle of 8 soil and 9 needle contact temperature measurements (with NTC sensors) at the Speulderbos needle forest site in the Netherlands, on 16 June 2006.

temperatures of different elements in the canopy and the radiometric temperature. An example of an analysis carried out with SCOPE is shown in Fig 4. This figure shows simulated radiometric temperature of sparse but homogeneous crop as a function of the satellite observation azimuth (the counter-clockwise rotation angle from the top in the graph) and zenith angle (distance from the centre of the graph). We can see pronounced differences in the observed radiometric temperatures. Cleary visible is the hotspot, the situation where the solar zenith and azimuth angles equal those of the sensor. In the hotspot the radiometric temperature is higher than outside the hotspot. Radiometric temperatures are also higher at lower zenith angles compared to NADIR observations (vertically downward, in the centre of the graph). The differences in temperature are up to 2 C, indicating that care should be taken of the observation angle relative to the solar angle. It is also possible to exploit the differences in radiometric surface temperature observed at different angles in order to separate soil and canopy kinetic temperatures (Timmermans et al, 2009).

How severe is an error of 2 C for the estimation of sensible heat flux? Equation 18 shows that the error in sensible heat flux is proportional to the ratio of the temperature gradient to the resistance. This means that for the same error in the temperature gradient, the error in the sensible heat flux will be larger if the aerodynamic resistance is low than if the aerodynamic resistance is high.

In order to consider the sensitivity more precisely, we take the example of a situation where the aerodynamic resistance is low: the Speulderbos forest site in The Netherlands. This site is equipped with a 46-m tall eddy covariance measurement tower. Because of the low aerodynamic resistance, the sensitivity to temperature is expected to be relatively high. For this site, we calculated the friction velocity and the sensible heat flux with Eqs 35-37, using a canopy height of 30 m, the measured wind speed and temperature at 45 m height, radiometric temperature measured with a long-wave radiometer, and assuming that *z*0m = 0.12 *h* and *d* = 0.67 *h*.

Figure 5 shows the results for 15-18 July 2009. The day-time friction velocity matches well with the measurements, showing that the calculation of aerodynamic resistance was

Fig. 4. Hemispherical graph of simulated radiometric surface temperature of a thinned maize crop with a LAI of 0.25, as a function of viewing zenith angle and viewing azimuth angle (relative to the solar azimuth). Zenith angle varies with the radius, the azimuth angle (in italic) increases while rotating anticlockwise from north. The solar zenith angle was 48° (after Van der Tol et al., 2009).

Fig. 5. Measured (symbols) and modeled (line) friction velocity *u*\* and sensible heat *H* flux versus Julian day number (14-19 July 2009) for an eddy covariance tower in the Speulderbos forest site, The Netherlands.

Fig. 4. Hemispherical graph of simulated radiometric surface temperature of a thinned maize crop with a LAI of 0.25, as a function of viewing zenith angle and viewing azimuth angle (relative to the solar azimuth). Zenith angle varies with the radius, the azimuth angle (in italic) increases while rotating anticlockwise from north. The solar zenith angle was 48°

196 197 198 199 200

196 197 198 199 200

DOY 2009

Fig. 5. Measured (symbols) and modeled (line) friction velocity *u*\* and sensible heat *H* flux versus Julian day number (14-19 July 2009) for an eddy covariance tower in the Speulderbos

(after Van der Tol et al., 2009).

0


forest site, The Netherlands.

0

200

H (W m-2)

400

600

0.5

u\* (m s-1)

1

1.5

accurate. During the night (stable conditions), the performance is worse, but this is not a large problem because the night-time sensible heat flux proved very small. However, there is a 50% error in afternoon sensible heat flux. Can this be related to an error in the surface temperature? Figure 6 shows the result of a sensitivity analysis to surface temperature. A consistent bias was added to the measured time series (*x*-axis), and the resulting root mean square error (RMSE) of friction velocity and sensible heat flux was calculated (*y*-axis). The RMSE reaches a minimum when surface temperature is 0.5 C above the measured value, but it rises to unacceptably high values of the absolute temperature bias is greater than 2 C. In this example, field data of radiometric surface temperature were used. What if remote sensing data are available? Satellite products are available at either high temporal (geostationary satellites) or at high spatial resolution (polar orbiting satellites). Data are available at a spatial resolution of 3-5 km and a temporal resolution of 15 minutes (MeteoSat or GOES) to 1 km resolution at a daily time scale (AVHRR, MODIS or MERIS), or 60 m with a repetition time of weeks to months (LANDSAT, ASTER). The low temporal resolutions are not really useful, because of the dynamic nature of the turbulent heat fluxes. The daily revisits are useful provided that reasonable assumptions are made about the diurnal cycles of the fluxes (see Sect 2.4). The orbits are designed to overpass at the same solar time every day. The 15-minute intervals are ideal, but the spatial resolution makes the estimation of an effective aerodynamic resistance difficult, as we will see later.

Fig. 6. Root mean square error of modelled sensible heat flux and friction velocity versus a forced bias in the observed radiometric surface temperatures for the Speulderbos forest site, for 14-19 July 2009.

A final issue that needs to be considered is the topography. In areas with large elevation differences, the interpolation technique for air temperature data is crucial. Errors of several degrees in the air temperature are easily introduced if an incorrect adiabatic lapse rate is used.

It is possible to circumvent the problem of estimating the temperature gradient by using an image based calibration (Bastiaanssen et al., 1998), in which assumptions are made for the energy balance state at the hottest and the coolest pixel in the image. In the first versions of this approach the calculated fluxes depended on the size of the image that was selected and the on assumption that the hottest pixel is dry, but more recent developments do not suffer from this drawback. The Mapping Evapotranspiration with Internalized Calibration (METRIC) model (Allen et al., 2007) uses reference evapotranspiration of alfalfa to calibrate the relation between the temperature gradient and the measured surface temperature. In the METRIC model it is assumed that the evaporation in the wettest pixel is 5% above the reference evapotranspiration, and the evaporation of the driest pixel is estimated with a soilvegetation-atmosphere model. This has the additional advantage that the evaporation values are bound to a realistic minimum and a realistic maximum rate. The METRIC model also accounts for topography by correcting radiation for slope and aspect and temperature for elevation using a local lapse rate.

#### **3.2 Sensitivity to the aerodynamic resistance**

The roughness length *z*0m (and often displacement height is linked to it) is recognized as the main source of error in the remote estimate of ET. Currently, there are several methods that can be used to approach a good z0m (see Table 2).

When near surface wind speed and vegetation parameters (height and leaf area index) are available, the within-canopy turbulence model proposed by Massman (1999) can be used to estimate aerodynamic parameters, *d*, the displacement height, and, z0m, the roughness height for momentum. This model has been shown by Su et al. (2001) to produce reliable estimates of the aerodynamic parameters. If only the height of the vegetation is available, the relationships proposed by Brutsaert (1982) can be used. If a detailed land use classification is available, for example based on LandSat images, the tabulated values of Wieringa (1993) can be used. By using literature values, errors in the canopy height of the order of decimetres to several metres are likely to occur, and errors in the roughness length in the order of decimetres.


Table 2. Methods for the estimation of z0m (After: Su, 2002).

When all of the above information is not available, then the aerodynamic parameters can be related to vegetation indices derived from satellite data. However in this case, care must be taken, because the vegetation indices saturate at higher vegetation densities and the relationships are vegetation type dependent. For example, characteristic of the land surface are sometimes calculated from indices like the Normalized Difference Vegetation Index (NDVI), but there is no reason why NDVI should have a universal relation with surface roughness. A grass field may have a similar NDVI to that of a forest, but a roughness length that is an order of magnitude smaller. For this reason, literature data or ground-truth data are indispensible for an accurate estimate of the surface roughness. A relation between NDVI and surface roughness can only be made for low vegetation, normally irrigated. In

the on assumption that the hottest pixel is dry, but more recent developments do not suffer from this drawback. The Mapping Evapotranspiration with Internalized Calibration (METRIC) model (Allen et al., 2007) uses reference evapotranspiration of alfalfa to calibrate the relation between the temperature gradient and the measured surface temperature. In the METRIC model it is assumed that the evaporation in the wettest pixel is 5% above the reference evapotranspiration, and the evaporation of the driest pixel is estimated with a soilvegetation-atmosphere model. This has the additional advantage that the evaporation values are bound to a realistic minimum and a realistic maximum rate. The METRIC model also accounts for topography by correcting radiation for slope and aspect and temperature

The roughness length *z*0m (and often displacement height is linked to it) is recognized as the main source of error in the remote estimate of ET. Currently, there are several methods that

When near surface wind speed and vegetation parameters (height and leaf area index) are available, the within-canopy turbulence model proposed by Massman (1999) can be used to estimate aerodynamic parameters, *d*, the displacement height, and, z0m, the roughness height for momentum. This model has been shown by Su et al. (2001) to produce reliable estimates of the aerodynamic parameters. If only the height of the vegetation is available, the relationships proposed by Brutsaert (1982) can be used. If a detailed land use classification is available, for example based on LandSat images, the tabulated values of Wieringa (1993) can be used. By using literature values, errors in the canopy height of the order of decimetres to several metres are likely to occur, and errors in the roughness length

**Method Input needed Remark** 

When all of the above information is not available, then the aerodynamic parameters can be related to vegetation indices derived from satellite data. However in this case, care must be taken, because the vegetation indices saturate at higher vegetation densities and the relationships are vegetation type dependent. For example, characteristic of the land surface are sometimes calculated from indices like the Normalized Difference Vegetation Index (NDVI), but there is no reason why NDVI should have a universal relation with surface roughness. A grass field may have a similar NDVI to that of a forest, but a roughness length that is an order of magnitude smaller. For this reason, literature data or ground-truth data are indispensible for an accurate estimate of the surface roughness. A relation between NDVI and surface roughness can only be made for low vegetation, normally irrigated. In

Wind speed profiles Point values only

LIDAR Experimental. Costly. Costly method

for elevation using a local lapse rate.

in the order of decimetres.

Retrievals from wind

profiles

**3.2 Sensitivity to the aerodynamic resistance** 

can be used to approach a good z0m (see Table 2).

z0m = 0.136 h Vegetation height map (h) from Lookup table (LUT) Vegetation map & z0m LUT From vegetation index Vegetation index maps z0m from modelling Landuse & veg. structure

Table 2. Methods for the estimation of z0m (After: Su, 2002).

that case a non-linear relation with vegetation structure is first established by assigning a maximum value and a minimum value of height corresponding to values of NDVI.

We illustrate the sensitivity of the aerodynamic resistance model with the data set of the Dutch forest site introduced in the previous section. Now, the height of the forest was varied between 5 and 45 m, and the RMSE of sensible heat flux and friction velocity evaluated (Fig 7). Note that the vertical scale in Fig 7 is much smaller than in Fig 6, which indicates that for forest, the model is less sensitive to errors in the canopy height than errors in surface temperature.

Sparse canopies require special attention. In sparse canopies the temperature differences between canopy and soil may be over 20 ºC. In addition, no canopy height can be defined, which makes it difficult to estimate the roughness length *z*0m, and normally *d* is neglected. A solution to these problems is to program a two-source model (e.g. Norman et al., 1995). An alternative solution is to modify the parameter *kB*-1 to incorporate the differences in surface temperature implicitly in the value of *z*0h (Verhoef et al., 1997). We will illustrate the latter solution with a simple example.

Fig. 7. Root mean square error of modelled sensible heat flux *H* and friction velocity *u*\* velocity versus assumed canopy height of the Speulderbos forest site, for 14-19 July 2009.

The sparse canopy of our example is a study site in the province of León, Spain. The vegetation cover fraction is 11%, consisting of patches of 6-m tall *Quercus ilex* and *Quercus pyrenaica*. Data of an eddy covariance flux tower are used for validation of the satellite product. For this site, a roughness length of *z*0m=0.2 m was assumed, and a displacement height of *d*=0. The friction velocity and sensible heat flux were again calculated from Eqs 35- 37. For *z*0h, a value of 0.02 m was initially assumed (*kB*-1 = 2.3), and for wind speed, the field measurements at the flux tower were used. For net radiation and surface temperature, 15 minute interval MeteoSat Second Generation (MSG) satellite data were used. The top panels in Fig 8 show the results of the satellite based algorithm. The friction velocity observations are accurately reproduced, but the modelled sensible heat flux is extremely high, even double the net radiation. The overestimate is solved when we reduce *z*0h by four orders of magnitude (*kB*-1 = 11.5). The reduction in *z*0h needed to match the model with the observations is large. This problem was discussed earlier after the Hapex-Sahel measurement campaign (Verhoef et al., 1997). It was then concluded that the whole concept of *kB*-1 is questionable. It is indeed recommended to avoid the use of *kB*-1, and this can be done in two ways: (1) by using more complicated two-source models for sparse vegetation, or (2) to use image-based calibration to relate surface temperature to a temperature gradient between two heights well above *z*0h. The second approach is used in models such as SEBAL (Bastiaanssen et al., 1998) or METRIC (Allen et al., 2007).

A model exists to estimate the *kB*-1 from vegetation density (Su et al., 2001). This model is used in the remote sensing algorithm SEBS (Su, 2002). However, care should be taken with any *kB*-1 model for areas where no detailed information on cover or other field data are available for calibration.

In the future, global maps of surface roughness may become timely available. Through synthesis of LiDAR with high resolution optical remote sensing, the roughness parameters have been successfully estimated spatially (Tian et al., 2011). Surface maps produced with laser satellites (NASA's ICESat and the future ICESat2) are also promising tools for estimating roughness (Roxette et al., 2008).

Fig. 8. Measured (symbols) and modeled (line) friction velocity *u*\* and sensible heat *H* flux versus Julian day number (14-19 July 2010) for an eddy covariance tower in the sparsely vegetated area of Sardon, Spain. Top graphs: using *z*0m = 0.2 and *kB*-1 = 2.3. Bottom graphs: using *z*0m = 0.2 and *kB*-1 = 11.5.

### **4. Conclusions**

All remote sensing algorithms for ET make use of the energy balance equation (EBE). In this equation, latent heat flux is calculated as a residual of the energy balance. Net radiation can be estimated from remote sensing products relatively easily. Ground heat flux can only be retrieved with geostationary satellites for sparsely vegetated areas or bare land. It is usually a minor term in vegetated areas that causes relatively small errors in the final ET product.

The most critical component of the energy balance is the sensible heat flux. In the calculation of the sensible heat flux, both the temperature difference (land surface temperature minus the air temperature) and the aerodynamic resistance need careful attention.

In areas with high elevation differences, the errors in temperature are usually so high, and temperature correction using local lapse rates is necessary. In flat areas, a local sensitivity analysis is recommended. For forest, the accuracy of the temperature gradient should be better than 2 C in order to achieve reasonable results. In sparse vegetation two source models are preferred over single-source models, because in the latter, parameterization of *z*0h on operational basis is no better than a wild guess. If a two-source model is not an option, then image based calibration using reference evaporation is a good alternative in these areas. Accurate roughness information (*z*0m) is required; the information is preferably verified and monitored on the ground. Satellite laser altimetry provides a promising tool for better roughness estimates in the near future.

### **5. References**

246 Evapotranspiration – Remote Sensing and Modeling

done in two ways: (1) by using more complicated two-source models for sparse vegetation, or (2) to use image-based calibration to relate surface temperature to a temperature gradient between two heights well above *z*0h. The second approach is used in models such as SEBAL

A model exists to estimate the *kB*-1 from vegetation density (Su et al., 2001). This model is used in the remote sensing algorithm SEBS (Su, 2002). However, care should be taken with any *kB*-1 model for areas where no detailed information on cover or other field data are

In the future, global maps of surface roughness may become timely available. Through synthesis of LiDAR with high resolution optical remote sensing, the roughness parameters have been successfully estimated spatially (Tian et al., 2011). Surface maps produced with laser satellites (NASA's ICESat and the future ICESat2) are also promising tools for

Fig. 8. Measured (symbols) and modeled (line) friction velocity *u*\* and sensible heat *H* flux versus Julian day number (14-19 July 2010) for an eddy covariance tower in the sparsely vegetated area of Sardon, Spain. Top graphs: using *z*0m = 0.2 and *kB*-1 = 2.3. Bottom graphs:

All remote sensing algorithms for ET make use of the energy balance equation (EBE). In this equation, latent heat flux is calculated as a residual of the energy balance. Net radiation can

(Bastiaanssen et al., 1998) or METRIC (Allen et al., 2007).

estimating roughness (Roxette et al., 2008).

using *z*0m = 0.2 and *kB*-1 = 11.5.

**4. Conclusions** 

available for calibration.


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## **Estimation of the Annual and Interannual Variation of Potential Evapotranspiration**

## Georgeta Bandoc

*University of Bucharest, Department of Meteorology and Hydrology Center for Coastal Research and Environmental Protection Romania* 

## **1. Introduction**

250 Evapotranspiration – Remote Sensing and Modeling

Wieringa, J. (1993). Representative Roughness Parameters for Homogeneous Terrain,

*Boundary-Layer Meteorol.* 63, pp. 323–363

823

Evaluation of three process-based approaches. *Remote Sens. Environ.* 115, pp. 801-

Knowledge of ecological factors for all natural systems, including human-modified natural systems, is essential for determining the nature of changes in these systems and to establish interventions that must be achieved to ensure optimal functioning of these systems.

The purpose of this chapter is to identify annual and interannual variations of potential evapotranspiration, in conjunction with climate changes in recent years, on the coastal region of Sfântu Gheorghe – Danube Delta. Under natural conditions, evapotranspiration flows continuously throughout the year, representing a main link in the water cycle and an important heat exchange factor affecting ecosystems. Potential evapotranspiration is the maximum amount of water likely to be produced by a soil evaporation and perspiration of plants in a climate.

Real balance between the amount of precipitation fallen named *P* and the amount of water taken from the atmosphere as vapour, called potential evapotranspiration *PET* is of particular importance in characterizing climate, representing an expression of power absorption by the atmosphere and expressing quantity water on soil and vegetation that request (Henning & Henning, 1981).

The difference between precipitation (*P*) and potential evapotranspiration (*PET*), i.e. *P PET* known as *P* is denoted by excess precipitation to *PET* (*E*) or deficit of precipitation to *PET* (*D*) if the difference is positive or, respectively, negative. The intensity of water loss through evaporation from the soil or by transpiration from the leaf surface is largely determined by vapour pressure gradient, i.e. the vapour pressure difference between leaf and soil surface and atmospheric vapour pressure (Berbecel et al, 1970).

The vapour pressure gradient is determined, in turn, by the characteristics of air and soil factors, such as: radiant energy, air temperature, vertical and horizontal movements of the air saturation deficit, the degree of surface water supply evaporation, plant biology and soil characteristics.

Heat factor also has a significant influence on evapotranspiration as temperature, on one hand, intensify of water vapour increases and, on the other hand, increases air capacity to maintain water vapour saturation state, reducing atmosphere's evaporated power (Eagleman, 1967).

## **2. General issues related to estimate**

Potential evapotranspiration evidence and interannual variations of *PET* potential evapotranspiration and water balance, climate charts are used based on measurements from weather stations hydrothermal (Walter & Lieth, 1960; Walter, 1955, 1999; Köppen, 1900, 1936 etc.).

In 2005, Oudin et al. compile lists 25 methods for estimating potential evapotranspiration based on a series of meteorological parameters (Douglas et al, 2009).

Estimation of potential evapotranspiration can be done using the indirect method based on air temperature readings and diagrams and on Thornthwaite's tables (Thorntwaite, 1948; Donciu, 1958; Walter & Lieth, 1960).

Recent studies use Penman's equation for this purpose, Penman (Penman, 1946), Penman - Monteith (Thomas, 2000a, 200b; Choudhury, 1997; Allen et al., 1998; Chen et al, 2005). Also, in determining the potential evapotranspiration, other formulas have been used with results almost similar with the ones of direct measurements, such as formulas of Bouchet (Bouchet, 1964), Turc (Turc, 1954), Hargreaves (Hargreaves & Samani, 1982), Papadakis (1966), Hamon (1963), Priestley – Taylor (1972), Makkink (1957) (Lu et al, 2005) and Blaney-Criddle (1950) (Ponce, 1989).

Potential evapotranspiration *PET* is of great temporal variability and thus an estimation can be done based on heat and water vapour from the atmosphere (Dugas et al, 1991; Celliar and Brunet, 1992; Rana & Katerji, 1996; Droogers at al, 1996; Frangi et al, 1996; Linda et al, 2002 as cited in Chuanyan et al, 2004).

Another model of estimation for *PET* is based on soil moisture and rainfall (model Century) (Metherell et al 1992; Zhou et al, 2008 as cited in Liang et al, 2010). On the interaction of global precipitation and air temperature estimates can be done for potential evapotranspiration (Raich & Schlesinger, 1992; Buchmann, 2000; Andréassian et al, 2004; Li et al, 2008a, 2008b; Casals et al, 2009).

Estimation of potential evapotranspiration can be achieved also based on satellite measurements related to air humidity and wind characteristics, but only in case of highresolution satellite images (Irmak, 2009).

In the estimation of *PET* remote sensing methods are applied (Chaudhury, 1997; Granger, 1997; Stefano & Ferro, 1997; Caselles et al, 1998; Stewarta et al, 1999). These methods are based using geographic information system using GIS spatial modeling (Baxter et al, 1996; Srinivasan et al 1996; Moore, 1996; Cleugh et al, 2007, Tang et al, 2010).

Other studies use numerical modeling to simulate various weather variables in a particular location, variables used to calculate potential evapotranspiration (Kumar et al, 2002; Smith et al, 2006; Torres et al 2011).

## **3. Research on characteristics of coastal area of potential evapotranspiration**

The location where this study has been made is the south-east of Salt and marine field is bordered by the Black Sea coast in the east, marine low deltaic plain in the west and northwest and the arm of Sfântu Gheorghe –Danube Delta (fig. 1). To the south of arm of Sfântu Gheorghe is the marine plain Dranov , Sfântu Gheorghe secondary delta and Sacalin Island. In the context of global climate change, interannual evolution analysis, annual and multiannual magnitudes that characterize the climate of a region are of particular interest (Palutikov et al, 1994; Chattopadhyay et al, 1997; Kouzmov, 2002; Oguz et al, 2006). This interest increases when it is a coastal region where sea atmosphere - interactions induce very specific issues.

Fig. 1. Study area location

Potential evapotranspiration evidence and interannual variations of *PET* potential evapotranspiration and water balance, climate charts are used based on measurements from weather stations hydrothermal (Walter & Lieth, 1960; Walter, 1955, 1999; Köppen, 1900, 1936

In 2005, Oudin et al. compile lists 25 methods for estimating potential evapotranspiration

Estimation of potential evapotranspiration can be done using the indirect method based on air temperature readings and diagrams and on Thornthwaite's tables (Thorntwaite, 1948;

Recent studies use Penman's equation for this purpose, Penman (Penman, 1946), Penman - Monteith (Thomas, 2000a, 200b; Choudhury, 1997; Allen et al., 1998; Chen et al, 2005). Also, in determining the potential evapotranspiration, other formulas have been used with results almost similar with the ones of direct measurements, such as formulas of Bouchet (Bouchet, 1964), Turc (Turc, 1954), Hargreaves (Hargreaves & Samani, 1982), Papadakis (1966), Hamon (1963), Priestley – Taylor (1972), Makkink (1957) (Lu et al, 2005) and Blaney-Criddle (1950)

Potential evapotranspiration *PET* is of great temporal variability and thus an estimation can be done based on heat and water vapour from the atmosphere (Dugas et al, 1991; Celliar and Brunet, 1992; Rana & Katerji, 1996; Droogers at al, 1996; Frangi et al, 1996; Linda et al,

Another model of estimation for *PET* is based on soil moisture and rainfall (model Century) (Metherell et al 1992; Zhou et al, 2008 as cited in Liang et al, 2010). On the interaction of global precipitation and air temperature estimates can be done for potential evapotranspiration (Raich & Schlesinger, 1992; Buchmann, 2000; Andréassian et al, 2004; Li

Estimation of potential evapotranspiration can be achieved also based on satellite measurements related to air humidity and wind characteristics, but only in case of high-

In the estimation of *PET* remote sensing methods are applied (Chaudhury, 1997; Granger, 1997; Stefano & Ferro, 1997; Caselles et al, 1998; Stewarta et al, 1999). These methods are based using geographic information system using GIS spatial modeling (Baxter et al, 1996;

Other studies use numerical modeling to simulate various weather variables in a particular location, variables used to calculate potential evapotranspiration (Kumar et al, 2002; Smith

**3. Research on characteristics of coastal area of potential evapotranspiration**  The location where this study has been made is the south-east of Salt and marine field is bordered by the Black Sea coast in the east, marine low deltaic plain in the west and northwest and the arm of Sfântu Gheorghe –Danube Delta (fig. 1). To the south of arm of Sfântu Gheorghe is the marine plain Dranov , Sfântu Gheorghe secondary delta and Sacalin Island. In the context of global climate change, interannual evolution analysis, annual and multiannual magnitudes that characterize the climate of a region are of particular interest (Palutikov et al, 1994; Chattopadhyay et al, 1997; Kouzmov, 2002; Oguz et al, 2006). This interest increases when it is a coastal region where sea atmosphere - interactions induce very

Srinivasan et al 1996; Moore, 1996; Cleugh et al, 2007, Tang et al, 2010).

based on a series of meteorological parameters (Douglas et al, 2009).

**2. General issues related to estimate** 

Donciu, 1958; Walter & Lieth, 1960).

2002 as cited in Chuanyan et al, 2004).

et al, 2008a, 2008b; Casals et al, 2009).

resolution satellite images (Irmak, 2009).

et al, 2006; Torres et al 2011).

specific issues.

etc.).

(Ponce, 1989).

The Danube Delta combines the temperate semi-arid climate space typical for the Pontic steppes. The aquatic very wide plane spaces, differently covered by vegetation and intrerrupted by the sandy islands of the marine fields, make up an active area specific to the delta and to the adjacent lagoons but totally different from that belonging to the Pontic stepps. This active area reacts upon the total radiation intercepted by the general circulation of atmosphere, resulting in a mosaic of microclimates (Vespremeanu, 2000, 2004).

For determining how climate changes affect the interannual potential evapotranspiration in the Sfântu Gheorghe costal area it was started, primarily from the fact that *PET* potential evapotranspiration has strong fluctuations in time and space as a direct consequence of the variation factors leads. Thus, in order to achieve the intended purpose of this chapter, interannual and annual potential evapotranspiration values were determined according to Thornthwaite's method, both for the period 1961 - 1990, taken as a reference period and analyzed for the studied period 2000 - 2009. Interannual differences *P PET* as well as annual amounts of the differences of the same sign, *P PET* and *P PET* as well as the annual review, are important climatic indicators. The determination of the efficiency of precipitation was done by calculating the difference *P PET* taken as reference period 1961 - 1990 and for the period under study from 2000 to 2009. Positive differences indicate excess water from rainfall, water shortages *P* and the negative ones indicate deficit of precipitation, water requirements from the atmosphere *<sup>P</sup>* .

It was also determined the precipitation deficit offset by previously accumulated surpluses and deficits of precipitation uncompensated by previous surpluses.

To identify climate changes in coastal Sfântu Gheorghe area and deviations from the average annual values of air temperature and precipitation, diagrams were drawn, type Walter and Leith, to identify dry periods and also different indices and specific factors were calculated such as: Martonne arid index ( *Iar* ), retention index offset ( *I hc* ), the amount of rainfall in the period with t ≥ 10 ° C temp ( <sup>0</sup> <sup>10</sup> *<sup>P</sup> t C* ) rainfall amount of soil loading in the months from November to March ( *PXI III* ), the amount of summer rainfall in July and August ( *VII VIII <sup>P</sup>* ), Lang precipitation index for the period with t ≥ 10 ° C ( <sup>0</sup> <sup>10</sup> *Lt C* ), precipitation index for summer Lang ( *LVI VIII* ) and Lang precipitation index for spring season ( *LIII V* ) and annual and interannual precipitation deficits (*D*) and excess (*E*) respectively , comparing to potential evapotranspiration of 10 mm, 20 mm, 30 mm etc. These indices and ratios were calculated based on meteorological measurements for the period 1961 - 1990, taken as a reference period for the 2000 - 2009 period under study. In this chapter, climate charts are playing an important role in the knowledge of the climate changes in the studied area and also helps in determining the precipitation – evapotranspiration, and hence the temperature deficit or surplus in the form of precipitation from evapotranspiration. Dryness site layout is determined in this study. Curve surplus or deficit of precipitation from evapotranspiration is crucial in environmental hydrothermal annual and interannual knowledge of an area.

Climate chart includes curved surfaces and values of temperature, precipitation at the scale 1/5 and 1/3 and potential evapotranspiration after Thornthwaite, interannual, annual and for certain periods (the amount of rainfall during the period from November - March, yet soil load, and the summer period July - August). The diagram also contains interannual surpluses and deficits and total rainfall to *PET*, the deficits in compensated and uncompensated previous surpluses, Walter - Lieth dry period, the annual aridity index, retention index offset, Lang rainfall index, calculated for the period temperature <sup>0</sup> *t C* 10 , for summer and spring time.

At the bottom of the chart months of the year and intra-annual valu*e*s *P* are indicated to express the character of moisture or dryness of the climate in different months, the monthly differences in classification categories *E* and *D* for each 10 mm, 20 mm, 30 mm etc. On the diagram, for 1 degree of temperature correspond 5 mm, 3 mm respectively of precipitation. Scale 1/5 was chosen in order to maximally achieve principle of the rainfall curve to be above the temperature when precipitation *PET* outperforms, and below it, when *PET* exceeds precipitation. Scale 1/3 was chosen to determine the dry period after the Walter - Lieth, which lasts as long as the rainfall curve is well below that of temperature.

It is important to know to what extent and interannual deficit of precipitation to *PET* during the growing season is offset by the surplus of precipitation to *PET* during the loading of the soil with water from precipitation (late autumn - winter). In this way deficit or surplus annual and interannual of effective precipitation is obtained comparing to *PET*.

In case of no loss of water through surface runoff and water infiltration or gains, the excess water is retained during loading or accumulated in the soil, and it is called full hydrologic soil (Chiriţă et al, 1977).

The entire accumulated surplus of precipitation is the main reserve of soil water in the vegetation is gradually consumed and evapotranspiration together with new fallen rains (Donciu, 1983).

For the studied area, where the climate is characterized by periods of dryness, the water reserve accumulated in the soil is gradually depleted by evapotranspiration and biomass formation. This last amount of water should be considered as an element of water balance, important in the quantitative ratio (Chiriţă et al, 1977).

Until finishing the accumulated precipitation of the soil in each month, water loss through evapotranspiration and precipitation is compensated by the previous reserve accumulation. Once this reserve is ended, precipitation deficit starts for the area studied. Evapotranspiration consumes current rainfall, leaving an additional demand of the atmosphere, dissatisfied with the precipitation.

The deficit of precipitation in this period presents the quantitative nature of *PET*'s dry climate and soil and thus the existence of a period of severe water available to vegetation.

## **4. Results and discussion**

254 Evapotranspiration – Remote Sensing and Modeling

It was also determined the precipitation deficit offset by previously accumulated surpluses

To identify climate changes in coastal Sfântu Gheorghe area and deviations from the average annual values of air temperature and precipitation, diagrams were drawn, type Walter and Leith, to identify dry periods and also different indices and specific factors were

months from November to March ( *PXI III* ), the amount of summer rainfall in July and August ( *VII VIII <sup>P</sup>* ), Lang precipitation index for the period with t ≥ 10 ° C ( <sup>0</sup> <sup>10</sup> *Lt C* ), precipitation index for summer Lang ( *LVI VIII* ) and Lang precipitation index for spring season ( *LIII V* ) and annual and interannual precipitation deficits (*D*) and excess (*E*) respectively , comparing to potential evapotranspiration of 10 mm, 20 mm, 30 mm etc. These indices and ratios were calculated based on meteorological measurements for the period 1961 - 1990, taken as a reference period for the 2000 - 2009 period under study. In this chapter, climate charts are playing an important role in the knowledge of the climate changes in the studied area and also helps in determining the precipitation – evapotranspiration, and hence the temperature deficit or surplus in the form of precipitation from evapotranspiration. Dryness site layout is determined in this study. Curve surplus or deficit of precipitation from evapotranspiration is crucial in environmental hydrothermal

Climate chart includes curved surfaces and values of temperature, precipitation at the scale 1/5 and 1/3 and potential evapotranspiration after Thornthwaite, interannual, annual and for certain periods (the amount of rainfall during the period from November - March, yet soil load, and the summer period July - August). The diagram also contains interannual surpluses and deficits and total rainfall to *PET*, the deficits in compensated and uncompensated previous surpluses, Walter - Lieth dry period, the annual aridity index, retention index offset, Lang rainfall index, calculated for the period temperature <sup>0</sup> *t C* 10 ,

At the bottom of the chart months of the year and intra-annual valu*e*s *P* are indicated to express the character of moisture or dryness of the climate in different months, the monthly differences in classification categories *E* and *D* for each 10 mm, 20 mm, 30 mm etc. On the diagram, for 1 degree of temperature correspond 5 mm, 3 mm respectively of precipitation. Scale 1/5 was chosen in order to maximally achieve principle of the rainfall curve to be above the temperature when precipitation *PET* outperforms, and below it, when *PET* exceeds precipitation. Scale 1/3 was chosen to determine the dry period after the Walter - Lieth, which lasts as long as the rainfall curve is well below that of

It is important to know to what extent and interannual deficit of precipitation to *PET* during the growing season is offset by the surplus of precipitation to *PET* during the loading of the soil with water from precipitation (late autumn - winter). In this way deficit or surplus

In case of no loss of water through surface runoff and water infiltration or gains, the excess water is retained during loading or accumulated in the soil, and it is called full hydrologic

annual and interannual of effective precipitation is obtained comparing to *PET*.

*hc* ), the amount of

*t C* ) rainfall amount of soil loading in the

and deficits of precipitation uncompensated by previous surpluses.

rainfall in the period with t ≥ 10 ° C temp ( <sup>0</sup> <sup>10</sup> *<sup>P</sup>*

annual and interannual knowledge of an area.

for summer and spring time.

temperature.

soil (Chiriţă et al, 1977).

calculated such as: Martonne arid index ( *Iar* ), retention index offset ( *I*

The analysis of average monthly *PET* value as obtained for Sfântu Gheorghe, was a functional correlation of these values with the mean monthly air temperature *T* (Bandoc & Golumbeanu, 2010). For both analyzed periods, the correlations are straightforward.

From the calculation of correlation coefficient *r* and determination coefficient <sup>2</sup> *r* between potential evapotranspiration air temperature values of this coefficient *r* 0,98 , *r* 0, 97 and <sup>2</sup> *r* 96,04% , <sup>2</sup> *r* 94,04% resulted, for the reference period 1961 - 1990 and for 2000 - 2009 period under study (fig. 2).

From the climate charts made for coastal Sfântu Gheorghe area (fig. 2, fig. 3, fig. 4, fig. 5, fig. 6, fig. 7, fig. 8 and fig. 9) for the analyzed periods, the result is a series of changes comparing to the duration of dryness reference interval 1961 - 1990.

Analizing the data the drought period for 2000-2009 was found to be 7 months which compared to the reference 1961-1990 (average drought) period of 6 months shows a increase of one month of drought per year.

Arid annual index Martonne calculation to determine the ratio between the amount of rainfall and temperatures 10 *<sup>P</sup> Iar <sup>T</sup>* showed that for the period 2000 - 2009, there was a

decrease in the value of the index with 17,71 % which leads to increased awareness of dryness for the studied area (fig. 10).

Rain index called Lang index or Lang factor of the period with temperatures ≥ 10 0C ( <sup>0</sup> <sup>10</sup> *Lt C* ), spring ( *LIII V* ) and summer ( *LVI VIII* ) determined as a ratio of the average

monthly precipitation values and *P* values of monthly average air temperature T *<sup>P</sup> <sup>L</sup> T* .

The results obtained for these intervals revealed that the index 0 <sup>10</sup> *Lt C* values decreased by 20,22 % for the period with t ≥ 10 0C, for spring period *LIII V* rainfall index fell 26,05 %, while during the summer *LVI VIII* value of this index was 37,20 % compared to the reference period 1961 - 1990 (fig. 10).

*Offset fluid index <sup>P</sup> <sup>I</sup> hc P* expresses the extent of precipitation deficits are

compensated by the surpluses. Values lower than the 1 *<sup>I</sup>* <sup>1</sup> *hc* expressed precipitation deficits unabated. Following determination of the index for the two periods analyzed, that index values are 0,24 for the reference period 1961 - 1990 and 0,15 for the period 2000 - 2009. From the two values determined using the formula (0,24 and 0,15), for the past 10 years interval, results that the fluid compensation index decreased by 37,5 % compared to the reference period 1961 - 1990.

Fig. 2. Correlation between the potential evapotranspiration *PET* and air temperature *T* in the coastal region Sfântu Gheorghe for reference period 1961 - 1990

Fig. 3. Correlation between the potential evapotranspiration *PET* and air temperature *T* in the coastal region Sfântu Gheorghe for the period 2000 – 2009

compensated by the surpluses. Values lower than the 1 *<sup>I</sup>* <sup>1</sup> *hc* expressed precipitation deficits unabated. Following determination of the index for the two periods analyzed, that index values are 0,24 for the reference period 1961 - 1990 and 0,15 for the period 2000 - 2009. From the two values determined using the formula (0,24 and 0,15), for the past 10 years interval, results that the fluid compensation index decreased by 37,5 % compared to the

Fig. 2. Correlation between the potential evapotranspiration *PET* and air temperature *T* in

Fig. 3. Correlation between the potential evapotranspiration *PET* and air temperature *T* in

the coastal region Sfântu Gheorghe for reference period 1961 - 1990

the coastal region Sfântu Gheorghe for the period 2000 – 2009

expresses the extent of precipitation deficits are

*Offset fluid index <sup>P</sup> <sup>I</sup>*

reference period 1961 - 1990.

*hc P*

Fig. 4. Climate diagrams for reference interval 1960 – 1990 and interval 2000-2009 with characteristics sizes determined for reviewed site

Fig. 5. Climate charts for years 2000 and 2001 and characteristics sizes determined for reviewed site

Fig. 5. Climate charts for years 2000 and 2001 and characteristics sizes determined for

reviewed site

Fig. 6. Climate charts for years 2002 and 2003 and characteristics sizes determined for reviewed site

Fig. 7. Climate charts for years 2004 and 2005 and characteristic sizes determined for reviewed site

Fig. 7. Climate charts for years 2004 and 2005 and characteristic sizes determined for

reviewed site

Fig. 8. Climate charts for years 2006 and 2007 and characteristic sizes determined for reviewed site

Fig. 9. Climate charts for years 2008 and 2009 and characteristic sizes determined for reviewed site

Fig. 9. Climate charts for years 2008 and 2009 and characteristic sizes determined for

reviewed site

All the obtained values places the deltaic coast Sfântu Gheorghe in area with a dry climate (Bandoc, 2009).

Regarding the average annual values of the variation of potential evapotranspiration, we can say that, for the period 2000 - 2009 is an increase *PET* value to the annual average of the reference period 1961 - 1990 at a rate of 7 %. Highest increases were registered in 2002, 2007 and 2009, years in which temperatures were recorded over annual average values of the reference period.

The observed values of *PET* in these years are on average 11 % higher than the reference period 1961 - 1990, while during other years the annual increases are in the range 0,07 ... 1 6 % for the period 2000 - 2009 (fig. 11).

Concluding, it can be stated that for Sfântu Gheorghe coastal region there is a significant increase in the potential evapotranspiration *PET* for the last 10 years compared to the reference 1961-1990.

The method used to calculate potential evapotranspiration is Thorntwaite's method, using average monthly air temperature values. Based on the values obtained for *PET* using the method of Thornthwaite (Thornthwaite diagram), one can say that there are significant variations in *PET* for the period under study from 2000 to 2009 compared with the reference period 1961 - 1990, both as annual values and mean interannual values (fig. 12).

The interannual distribution of *PET* in the period 2000 - 2009 shows that these values were, in most months in each year of the analyzed interval over the average interannual values of the reference period 1961 - 1990. It appears that for the months of July and August all *PET* values are over the annual average calculated for the same month of the reference period 1961 - 1990. For instance, for the months of July in 2000-2009 period compared to the the reference values in 1961-1990, PET values are above the multiannual July average (fig.12). Notable years for July values are 2001, 2007 and 2009 where the increase above the multiannual monthly average were 20.14%, 13.66% and 17.98% respectively.

In the same time the following indices were calculated: monthly differences *P PET* , annual amounts of differences with the same sign *P PET* and *P PET* , as well as the yearly balance *P PET<sup>A</sup>* , all these being important climatic indices. Calculations for the two analyzed periods led to the following results regarding water deficit and excess from precipitation presented below:

Fig. 10. Increases of the average annual percentage values of main indices for the period 2000 - 2009 for the studied site comparing to the specific values of the reference period 1961 – 1990.

Fig. 11. Changes in annual and multiannual average values of *PET* for the period 2000 - 2009. Comparison with the 1961 - 1990 annual average for the chosen location.

$$
\Sigma \begin{pmatrix} P - PET \end{pmatrix}\_{1961-1990}^{-} = 430,4mm \text{ : } \Sigma \begin{pmatrix} P -PET \end{pmatrix}\_{2000-2009}^{-} = 515,2mm \text{ : } \Sigma \begin{pmatrix} P -PET \end{pmatrix}\_{2000-2009}^{+} = 100,4mm
$$
 
$$
\Sigma \begin{pmatrix} P -PET \end{pmatrix}\_{1961-1990}^{+} = 106,2mm \text{ : } \Sigma \begin{pmatrix} P -PET \end{pmatrix}\_{2000-2009}^{+} = 80,8mm
$$

The annual balance sheet :2000 2009 *P PET <sup>A</sup>* shows a significant increase, with 31,6 % of the water deficit comparing to the period 1961 - 1990 for which the balance reference value is 330, 2 :1961 1990 *P PET mm <sup>A</sup>* .

The obtained values show that there is an increase in the deficit for the last 10 years by 19,7 % compared to the reference period and a decrease of 23,9 % in terms of excess rainfall for the period 2000 - 2009 (fig. 13 ).

For emphasizing very clear each month's character, at the bottom of the chart climate values *P* were given indicating each month's category in terms of surplus *E* or deficit *D* of precipitation versus potential evapotranspiration. Thus, there are determined the interannual values for the period 2000 - 2009 as well as average multiannual values for the two periods under study.

Based on measurements one could build a mosaic of surpluses *E* and deficits *D* of precipitation variation comparing to potential evapotranspirationfor in the period 2000- 2009, comparison with average multianual of *E* and *D* of the periods 2000-2009 and 1961- 1990 intervals (fig. 14).

Values for excess precipitation comparing to potential evapotranspiration reached a maximum of *E9* (>80 mm) and *E7* (>60 mm) in February and November 2007 respectively, values much higher than multiannual average of the reference period when the values were *E3* and *E2* (see fig. 14).

Fig. 11. Changes in annual and multiannual average values of *PET* for the period 2000 -

430, 4 1961 1990 *P PET mm* **;**  515, 2 2000 2009 *P PET mm*

106, 2 1961 1990 *P PET mm* **;**  80, 8 2000 2009 *P PET mm*

The annual balance sheet :2000 2009 *P PET <sup>A</sup>* shows a significant increase, with 31,6 % of the water deficit comparing to the period 1961 - 1990 for which the balance reference

The obtained values show that there is an increase in the deficit for the last 10 years by 19,7 % compared to the reference period and a decrease of 23,9 % in terms of excess rainfall

For emphasizing very clear each month's character, at the bottom of the chart climate values *P* were given indicating each month's category in terms of surplus *E* or deficit *D* of precipitation versus potential evapotranspiration. Thus, there are determined the interannual values for the period 2000 - 2009 as well as average multiannual values for the

Based on measurements one could build a mosaic of surpluses *E* and deficits *D* of precipitation variation comparing to potential evapotranspirationfor in the period 2000- 2009, comparison with average multianual of *E* and *D* of the periods 2000-2009 and 1961-

Values for excess precipitation comparing to potential evapotranspiration reached a maximum of *E9* (>80 mm) and *E7* (>60 mm) in February and November 2007 respectively, values much higher than multiannual average of the reference period when the values were

**;** 

2009. Comparison with the 1961 - 1990 annual average for the chosen location.

value is 330, 2 :1961 1990 *P PET mm <sup>A</sup>* .

for the period 2000 - 2009 (fig. 13 ).

two periods under study.

1990 intervals (fig. 14).

*E3* and *E2* (see fig. 14).

Fig. 12. Interannual distribution of *PET* in the period 2000 - 2009 comparing to the annual average of the reference period 1961 - 1990 for the studied area.

In addition, a reduction of the months with surplus between 2000 - 2009 for the years 2000, 2001, 2003 and 2004 can be seen. Also, there is a reduction in the number of months with a precipitation surplus for 2000, 2001, 2003 and 2004. In these years the precipitation excedent over *PET* period narrowed to 2 months in 2000 and 3 months in 2001, 2002, 2003 compared to 5 months in the reference 1961-1990 period (fig. 14).

As for the precipitation - potential evapotranspiration deficit it can be stated that the deficits suffered a significant increase compared to the reference period. Thus, there can be noticed maximum values of deficits *D17* (>160 mm) to be recorded in 2001 and 2002.

Fig. 13. Percent interannual variations of deficits *D* and surpluses *E* of precipitation to potential evapotranspiration for the period 2000 - 2009.

It appears that while the deficit intervals of the average multiannual values is seven months, the interannual period with deficit intervals is a few months longer between 2000 - 2009. Thus, in 2000, 2001 and 2004 this period has increased by three months and two months respectively compared to that of reference period (fig. 14).


Fig. 14. Distribution of surpluses *E* and deficits *D* of precipitation comparing to potential evapotranspiration in the period 2000 - 2009; comparison with average multiannual of *E* and *D* of the periods 2000 - 2009 and 1961 – 1990.

Analysis of reference period in terms of deficit and surplus, highlights that the studied area is characterized by a lack of *D3* compared to the same period last years when the average value increased to a deficit of *D4*, which means a 17,06 % increase in the deficit.

## **5. Conclusions**

266 Evapotranspiration – Remote Sensing and Modeling

Fig. 13. Percent interannual variations of deficits *D* and surpluses *E* of precipitation to

It appears that while the deficit intervals of the average multiannual values is seven months, the interannual period with deficit intervals is a few months longer between 2000 - 2009. Thus, in 2000, 2001 and 2004 this period has increased by three months and two months

 I II III IV V VI VII VIII IX X XI XII 2000 E5 E1 D1 D4 D9 D9 D14 D12 D4 D5 D2 D1 2001 D1 E3 D1 D2 D7 D8 D17 D13 D7 D5 E2 E1 2002 E2 D2 E4 D4 D10 D12 D17 D10 D7 D4 D2 E2 2003 E3 E2 D1 D2 D10 D11 D10 D12 D1 E1 E3 E1 2004 E4 E1 D2 D3 D3 D11 D11 D12 D6 D3 D1 E3 2005 E4 E4 E3 D3 D8 D7 D9 D12 D2 D4 E6 E4 2006 E2 E2 E4 D3 D5 D13 D13 D7 D5 D5 D1 E1 2007 E2 E9 E1 D4 D10 D12 D16 D14 D6 E2 E7 E3 2008 E3 E3 E1 D4 D8 D11 D13 D12 D4 D3 E2 E1 2009 E2 E2 D1 D5 D9 D13 D12 D13 D5 D3 E1 E3 1961-1990 E3 E3 E1 D5 D7 D9 D11 D9 D6 D2 E2 E3 2000-2009 E3 E3 E1 D3 D8 D11 D13 D12 D5 D3 E2 E2 Fig. 14. Distribution of surpluses *E* and deficits *D* of precipitation comparing to potential evapotranspiration in the period 2000 - 2009; comparison with average multiannual of *E* and

potential evapotranspiration for the period 2000 - 2009.

respectively compared to that of reference period (fig. 14).

*D* of the periods 2000 - 2009 and 1961 – 1990.

The research results concerning yearly and monthly potential evapotranspiration in the Sfantu Gheorghe coastal area, synthetized in this chapter revealed for years 2001 to 2009 changes in the humidity periods, an increase in air temperature (Busuioc et al, 2010), a diminished atmospheric precipitation amount and also an increase of precipitation to potential evapotranspiration deficit compared to 1961-1990 reference period.

All these changes lead to high vulnerability and low adaptive capacity to adverse impacts from climate change of this area (Liubimtseva & Henebry, 2009).

Thus, by drawing Walter and Leith diagrams, significant increase of dryness periods and decrease of moisture periods were observed with implications upon potential evapotranspiration and upon the shore phytocoenoses.

There are also changes in the length of the periods with precipitation surplus and deficit compared to potential evapotranspiration that means increasing periods of deficit and decreasing periods of surplus.

The following calculated characteristic measurements include the delta coast in Sfântu Gheorghe in arid climate and climatic changes show that the period 2000 - 2009 led to a trend towards increasing aridity: *Martonne arid index ( Iar ), retention index offset ( Ihc ), the amount of rainfall in the period with temperature T ≥ 10 ° C (* <sup>0</sup> <sup>10</sup> *<sup>P</sup> t C ), the amount of rainfall the soil load in the months from November to March ( PXI III ), the amount of summer rainfall July and August* ( *VII VIII <sup>P</sup>* ), *Lang precipitation index for the period with t ≥ 10 °C* ( <sup>0</sup> <sup>10</sup> *Lt C* ), *Lang precipitation index for the summer season* ( *LVI VIII* ) *and Lang precipitation index for the spring season* ( *LIII V* ).

From the differences in monthly *P PET* calculation of amounts *P PET* , *P PET* of the precipitation deficit offset by previously accumulated *P* , surpluses and deficits of precipitation uncompensated by previous surpluses *Puc* and the annual balance *P PET <sup>A</sup>* for the period under study year 2000 - 2009 and for the reference period 1961 - 1990, there was a deficit increase and a decrease of excess water from precipitation, an extension of periods of water shortage against period with excess of water and a significant increase by about 23,9 % for deficit of water that gathers negative differences uncompensated during periods of surplus.

Therefore, the research presented in this article have highlighted significant changes in potential evapotranspiration in relation to climate changes for the 2000 - 2009 studied period, in Sfântu Gheorghe area - Danube Delta, showing an increase of precipitation deficit and an increase of climate aridity .

Indirect method used in this paper work to determine the potential evapotranspiration was based on the values of air temperature and Thornthwaite's diagrams and tables. In this way a general view of a time variation of *PET* for Sfântu Gheorghe area - Danube Delta, has been created.

The advantages of this indirect method results from the fact that it doesn't require a large number of measured meteorological parameters and that it can be easily applied obtaining good estimates.

In the future it is intended that research should continue in order to see whether the growth trend of a interannual and annual potential evaporation is kept over the period 2000 - 2009. No doubt that climate change is underway affecting Earth's biodiversity.

Biggest challenge in this respect is related to the marine area, but it is unclear to what extent these changes in climate will affect ecosystems.

What is known is that the temperatures that rise steadily and increasingly frequent extreme weather events are those that have influence on migrating wildlife and also causes invasive species.

Coastal areas offer considerable benefits to society while human activities are exerting considerable pressure on coastal ecosystems. Therefore, these benefits to society are in danger (Nobre, 2009).

## **6. Acknowledgment**

Research carried out were conducted at the Center for Coastal Research and Environmental Protection, Department of Meteorology and Hydrology at the University of Bucharest, Romania.

## **7. References**


The advantages of this indirect method results from the fact that it doesn't require a large number of measured meteorological parameters and that it can be easily applied obtaining

In the future it is intended that research should continue in order to see whether the growth trend of a interannual and annual potential evaporation is kept over the period 2000 - 2009.

Biggest challenge in this respect is related to the marine area, but it is unclear to what extent

What is known is that the temperatures that rise steadily and increasingly frequent extreme weather events are those that have influence on migrating wildlife and also causes invasive

Coastal areas offer considerable benefits to society while human activities are exerting considerable pressure on coastal ecosystems. Therefore, these benefits to society are in

Research carried out were conducted at the Center for Coastal Research and Environmental Protection, Department of Meteorology and Hydrology at the University of Bucharest,

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## **Evapotranspiration of Partially Vegetated Surfaces**

L.O. Lagos1,2, G. Merino1, D. Martin2, S. Verma2 and A. Suyker2 *1Universidad de Concepción Chile 2University of Nebraska-Lincoln 1Chile 2USA* 

## **1. Introduction**

272 Evapotranspiration – Remote Sensing and Modeling

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Latent heat flux equivalent to Evapotranspiration (ET) is the total amount of water lost via transpiration and evaporation from plant surfaces and the soil in an area where a crop is growing. Since 80-90% of precipitation received in semiarid and subhumid climates is commonly used in evapotranspiration, accurate estimations of ET are very important for hydrologic studies and crop water requirements. ET determination and modelling is not straightforward due to the natural heterogeneity and complexity of agricultural and natural land surfaces. In evapotranspiration modelling it is very common to represent vegetation assuming a single source of energy flux at an effective height within the canopy. However, when crops are sparse, the single source/sink of energy assumption in such models is not entirely satisfied. Improvements using multiple source models have been developed to estimate ET from crop transpiration and soil evaporation. Soil evaporation on partially vegetated surfaces over natural vegetation and orchards includes not only the soil under the canopy but also areas of bare soil between vegetation that contribute to ET. Soil evaporation can account for 25-45% of annual ET in agricultural systems. In irrigated agriculture, partially vegetated surfaces include fruit orchards (i.e. apples, oranges, vineyards, avocados, blueberries, and lemons among others), which cover a significant portion of the total area under irrigation.

In semiarid regions, direct soil evaporation from sparse barley or millet crops can account for 30% to 60% of rainfall (Wallace et al., 1999). On a seasonal basis, sparse canopy soil evaporation can account for half of total rainfall (Lund & Soegaard, 2003). Allen (1990) estimated the soil evaporation under a sparse barley crop in northern Syria and found that about 70% of the total evaporation originated from the soil. Lagos (2008) estimated that under irrigated maize conditions soil evaporation accounted for around 26-36% of annual evapotranspiration. Under rain-fed maize conditions annual evaporation accounted for 36- 39% of total ET. Under irrigated soybean the percentage was 41%, and under rainfed soybean conditions annual evaporation accounted for 45-47% of annual ET. Massman (1992) estimated that the soil contribution to total ET was about 30% for a short grass steppe measurement site in northeast Colorado. In a sparse canopy at the middle of the growing season, and after a rain event, more than 50% of the daily ET corresponds to directly soil evaporation (Lund & Soegaard, 2003). Soil evaporation can be maximized under frequent rainfall or irrigation events, common conditions in agricultural systems for orchard with drip or micro sprinklers systems. If some of this unproductive loss of water could be retained in the soil and used as transpiration, yields could be increased without increased rainfall or the use of supplemental irrigation (Wallace et al., 1999). The measurement and modelling of soil evaporation on partially vegetated surfaces is crucial to estimate how much water is lost to the atmosphere via soil evaporation. Consequently, better water management can be proposed for water savings.

Partially vegetated surface accounts for a significant portion of land surface. It occurs seasonally in all agricultural areas and throughout the year in or chard and natural land covers. Predictions of ET for these conditions have not been thoroughly researched. In Chile, agricultural orchards with partially vegetated surfaces include apples, oranges, avocados, cherries, vineyards, blueberries, and berries, among others. According to the agricultural census (INE, 2007) the national orchard surface covers more than 324,000 ha, representing 30% of the total surface under irrigation.

Similar to the Shuttleworth and Wallace (1985), Choudhury and Monteith (1988) and Lagos (2008) models, the modelling of evapotranspiration for partially vegetated surfaces can be accomplished using explicit solutions of the equations that define the conservation of heat and water vapor fluxes for partially vegetated surfaces and soil. Multiple-layer models offer the possibility to represent these conditions to solve the surface energy balance and consequently, estimate evapotranspiration. Modelling is essential to predict long-term trends and to quantify expected outcomes. Since ET is such a large component of the hydrologic cycle in areas with partially vegetated surfaces, small changes in the calculation of ET can result in significant changes in simulated water budgets. Thus, good data and accurate modelling of ET is essential for predicting not only water requirements for agricultural crops but also to predict the significance of irrigation management decisions and land use changes to the entire hydrologic cycle.

Currently, several methods and models exist to predict natural environments under different conditions. More complex models have been developed to account for more variables affecting model performance. However, the applicability of these models has been limited by the difficulties and tedious algorithms needed to complete estimations. Mathematical algorithms used by multiple-layer models can be programmed in a software package to facilitate and optimize ET estimation by any user. User-friendly software facilitates the use of these improved methods; users (i.e. students) can use the computer model to study the behaviour of the system from a set of parameters and initial conditions.

Accordingly, in this chapter, a review of models that estimate ET for partially covered surfaces that occur normally in agricultural systems (i.e. orchards or vineyards) is presented, and the needs for further research are assessed.

## **2. ET modelling review**

Evapotranspiration (ET) is the total amount of water lost via transpiration and evaporation from plant surfaces and the soil in an area where a crop is growing. Traditionally, ET from agricultural fields has been estimated using the two-step approach by multiplying the weather-based reference ET (Jensen et al., 1971; Allen et al., 1998 and ASCE, 2002) by crop coefficients (Kc) to make an approximate allowance for crop differences. Crop coefficients are determined according to the crop type and the crop growth stage (Allen et al., 1998). However, there is typically some question regarding whether the crops grown compare with the conditions represented by the idealized Kc values (Parkes et al., 2005; Rana et al.,

rainfall or irrigation events, common conditions in agricultural systems for orchard with drip or micro sprinklers systems. If some of this unproductive loss of water could be retained in the soil and used as transpiration, yields could be increased without increased rainfall or the use of supplemental irrigation (Wallace et al., 1999). The measurement and modelling of soil evaporation on partially vegetated surfaces is crucial to estimate how much water is lost to the atmosphere via soil evaporation. Consequently, better water

Partially vegetated surface accounts for a significant portion of land surface. It occurs seasonally in all agricultural areas and throughout the year in or chard and natural land covers. Predictions of ET for these conditions have not been thoroughly researched. In Chile, agricultural orchards with partially vegetated surfaces include apples, oranges, avocados, cherries, vineyards, blueberries, and berries, among others. According to the agricultural census (INE, 2007) the national orchard surface covers more than 324,000 ha, representing

Similar to the Shuttleworth and Wallace (1985), Choudhury and Monteith (1988) and Lagos (2008) models, the modelling of evapotranspiration for partially vegetated surfaces can be accomplished using explicit solutions of the equations that define the conservation of heat and water vapor fluxes for partially vegetated surfaces and soil. Multiple-layer models offer the possibility to represent these conditions to solve the surface energy balance and consequently, estimate evapotranspiration. Modelling is essential to predict long-term trends and to quantify expected outcomes. Since ET is such a large component of the hydrologic cycle in areas with partially vegetated surfaces, small changes in the calculation of ET can result in significant changes in simulated water budgets. Thus, good data and accurate modelling of ET is essential for predicting not only water requirements for agricultural crops but also to predict the significance of irrigation management decisions

Currently, several methods and models exist to predict natural environments under different conditions. More complex models have been developed to account for more variables affecting model performance. However, the applicability of these models has been limited by the difficulties and tedious algorithms needed to complete estimations. Mathematical algorithms used by multiple-layer models can be programmed in a software package to facilitate and optimize ET estimation by any user. User-friendly software facilitates the use of these improved methods; users (i.e. students) can use the computer model to study the behaviour of the system from a set of parameters and initial conditions. Accordingly, in this chapter, a review of models that estimate ET for partially covered surfaces that occur normally in agricultural systems (i.e. orchards or vineyards) is presented,

Evapotranspiration (ET) is the total amount of water lost via transpiration and evaporation from plant surfaces and the soil in an area where a crop is growing. Traditionally, ET from agricultural fields has been estimated using the two-step approach by multiplying the weather-based reference ET (Jensen et al., 1971; Allen et al., 1998 and ASCE, 2002) by crop coefficients (Kc) to make an approximate allowance for crop differences. Crop coefficients are determined according to the crop type and the crop growth stage (Allen et al., 1998). However, there is typically some question regarding whether the crops grown compare with the conditions represented by the idealized Kc values (Parkes et al., 2005; Rana et al.,

management can be proposed for water savings.

30% of the total surface under irrigation.

and land use changes to the entire hydrologic cycle.

and the needs for further research are assessed.

**2. ET modelling review** 

2005; Katerji & Rana, 2006; Flores, 2007). In addition, it is difficult to predict the correct crop growth stage dates for large populations of crops and fields (Allen et al., 2007).

A second method is to make a one-step estimate of ET based on the Penman-Monteith (P-M) equation (Monteith, 1965), with crop-to-crop differences represented by the use of cropspecific values of surface and aerodynamic resistances (Shuttleworth, 2006). ET estimations using the one-step approach with the P-M model have been studied by several authors (Stannard, 1993; Farahani & Bausch, 1995; Rana et al., 1997; Alves & Pereira, 2000; Kjelgaard & Stockle, 2001; Ortega-Farias et al., 2004; Shuttleworth, 2006; Katerji & Rana, 2006; Flores, 2007; Irmak et al., 2008). Although different degrees of success have been achieved, the model has generally performed more satisfactorily when the leaf area index (LAI) is large (LAI>2). Results shows that the "big leaf" assumption used by the P-M model is not satisfied for sparse vegetation and crops with partial canopy cover.

A third approach consists of extending the P-M single-layer model to a multiple-layer model (i.e. two layers in the Shuttleworth-Wallace (S-W) model (Shuttleworth-Wallace, 1985) and four layers in the Choudhury-Monteith (C-M) model (Choudhury & Monteith, 1988). Shuttleworth and Wallace (1985) combined a one-dimensional model of crop transpiration and a one-dimensional model of soil evaporation. Surface resistances regulate the heat and mass transfer in plant and soil surfaces, and aerodynamic resistances regulate fluxes between the surface and the atmospheric boundary layer. Several studies have evaluated the performance of the S-W model to estimate evapotranspiration (Farahani & Baush,1995; Stannard, 1993; Lafleur & Rouse, 1990; Farahani & Ahuja, 1996; Iritz et al. 2001; Tourula & Heikinheimo, 1998; Anadranistakis et al., 2000; Ortega-Farias et al., 2007). Field tests of the model have shown promising results for a wide range of both agricultural and non-agricultural vegetation.

Farahani and Baush (1995) evaluated the performance of the P-M model and the S-W model for irrigated maize. Their main conclusion was that the Penman-Monteith model performed poorly when the leaf area index was less than 2 because soil evaporation was neglected in calculating surface resistance. Results of the S-W model were encouraging as it performed satisfactorily for the entire range of canopy cover. Stannard (1993) compared the P-M, S-W and Priestley-Taylor ET models for sparsely vegetated, semiarid rangeland. The P-M model was not sufficiently accurate (hourly r2 =0.56, daily r2=0.60); however, the S-W model performs significantly better for hourly (r2=0.78) and daily data (r2=0.85). Lafleur and Rouse (1990) compared the S-W model with evapotranspiration calculated from the Bowen Ratio Energy Balance technique over a range of LAI from non-vegetated to fully vegetated conditions. The results showed that the S-W model was in excellent agreement with the measured evapotranspiration for hourly and day-time totals for all values of LAI. Using the potential of the S-W model to partition transpiration and evaporation, Farahani and Ahuja (1996) extended the model to include the effects of crop residues on soil evaporation by the inclusion of a partially covered soil area and partitioning evaporation between the bare and residue-covered areas. Iritz et al. (2001) applied a modified version of the S-W model to estimate evapotranspiration for a forest. The main modification consisted of a two-layer soil module, which enabled soil surface resistance to be calculated as a function of the wetness of the top soil. They found that the general seasonal dynamics of evaporation were fairly well simulated with the model. Tourula and Heikinheimo (1998) evaluated a modified version of the S-W model in a barley field. A modification of soil surface resistance and aerodynamic resistance, over two growing seasons, produced daily and hourly ET estimates in good agreement with the measured evapotranspiration. The performance of the S-W model was evaluated against two eddy covariance systems by Ortega-Farias et al. (2007) over a Cabernet Sauvignon vineyard. Model performance was good under arid atmospheric conditions with a correlation coefficient (r2) of 0.77 and a root mean square error (RMSE) of 29 Wm-2.

Although good results have been found using the Shuttleworth-Wallace approach, the model still needs an estimation or measurement of soil heat flux (G) to estimate ET. Commonly, G is calculated as a fixed percentage of net radiation (Rn). Shuttleworth and Wallace (1985) estimated G as 20% of the net radiation reaching the soil surface. In the FAO56 method, Allen et al. (1998) estimated daily reference ET (ETr and ETo), assuming that the soil heat flux beneath a fully vegetated grass or alfalfa reference surface is small in comparison with Rn (i.e. G=0). For hourly estimations, soil heat flux was estimated as one tenth of the Rn during the daytime and as half of the Rn for the night time when grass was used as the reference surface. Similarly, G was assumed to be 0.04xRn for the daytime and 0.2xRn during the night time for an alfalfa reference surface. A more complete surface energy balance was presented by Choudhury and Monteith (1988). The proposed method developed a four-layer model for the heat budget of homogeneous land surfaces. The model is an explicit solution of the equations which define the conservation of heat and water vapor in a system consisting of uniform vegetation and soil. An important feature was the interaction of evaporation from the soil and transpiration from the canopy expressed by changes in the vapor pressure deficit of the air in the canopy. A second feature was the ability of the model to partition the available energy into sensible heat, latent heat, and soil heat flux for the canopy/soil system.

Similar to Shuttleworth-Wallace (1985), the Choudhury-Monteith model included a soil surface resistance to regulate the heat and mass transfer at the soil surface. However, residue effects on the surface energy balance are not included in the model. Crop residue generally increases infiltration and reduces soil evaporation. Surface residue affects many of the variables that determine the evaporation rate. These variables include Rn, G, aerodynamic resistance and surface resistances to transport of heat and water vapor fluxes (Steiner, 1994; Horton et al., 1996; Steiner et al., 2000).

Caprio et al. (1985) compared evaporation from three mini-lysimeters installed in bare soil and in a 14 and 28 cm tall standing wheat stubble. After nine days of measurements, evaporation from the lysimeter with stubble was 60% of the evaporation measured from bare soil. Enz et al. (1988) evaluated daily evaporation for bare soil and stubble-covered soil surfaces. Evaporation was always greater from the bare soil surface until it was dry, then evaporation was greater from the stubble covered-surface because more water was available. Evaporation from a bare soil surface has been described in three stages. An initial, energy-limited stage occurs when enough soil water is available to satisfy the potential evaporation rates. A second, falling rate stage is limited by water flow to the soil surface, while the third stage has a very low, nearly constant evaporative rate from very dry soil (Jalota & Prihar, 1998). Steiner (1989) evaluated the effect of residue (from cotton, sorghum and wheat) on the initial, energy-limited rate of evaporation. The evaporation rate relative to bare soil evaporation was described by a logarithmic relationship. Increasing the amount of residue on the soil surface reduced the relative evaporation rate during the initial stage. Bristow et al. (1986) developed a model to predict soil heat and water budgets in a soilresidue-atmosphere system. Results from application of the model indicate that surface residues decreased evaporation by roughly 36% compared with simulations from bare soil.

With the recognition of the potential of multiple-layer models to estimate ET, a modified surface energy balance model (SEB) was developed by Lagos (2008) and Lagos et al. (2009) to include the effect of crop residue on evapotranspiration. The model relies mainly on the Schuttleworth-Wallace (1985) and Choudhury and Monteith (1988) approaches and has the potential to predict evapotranspiration for varying soil cover ranging from partially residue-covered soil to closed canopy surfaces. Improvements to aerodynamic resistance, surface canopy resistance and soil resistances for the transport of heat and water vapor were also suggested.

## **2.1 The SEB model**

276 Evapotranspiration – Remote Sensing and Modeling

Sauvignon vineyard. Model performance was good under arid atmospheric conditions with a

Although good results have been found using the Shuttleworth-Wallace approach, the model still needs an estimation or measurement of soil heat flux (G) to estimate ET. Commonly, G is calculated as a fixed percentage of net radiation (Rn). Shuttleworth and Wallace (1985) estimated G as 20% of the net radiation reaching the soil surface. In the FAO56 method, Allen et al. (1998) estimated daily reference ET (ETr and ETo), assuming that the soil heat flux beneath a fully vegetated grass or alfalfa reference surface is small in comparison with Rn (i.e. G=0). For hourly estimations, soil heat flux was estimated as one tenth of the Rn during the daytime and as half of the Rn for the night time when grass was used as the reference surface. Similarly, G was assumed to be 0.04xRn for the daytime and 0.2xRn during the night time for an alfalfa reference surface. A more complete surface energy balance was presented by Choudhury and Monteith (1988). The proposed method developed a four-layer model for the heat budget of homogeneous land surfaces. The model is an explicit solution of the equations which define the conservation of heat and water vapor in a system consisting of uniform vegetation and soil. An important feature was the interaction of evaporation from the soil and transpiration from the canopy expressed by changes in the vapor pressure deficit of the air in the canopy. A second feature was the ability of the model to partition the available energy into

Similar to Shuttleworth-Wallace (1985), the Choudhury-Monteith model included a soil surface resistance to regulate the heat and mass transfer at the soil surface. However, residue effects on the surface energy balance are not included in the model. Crop residue generally increases infiltration and reduces soil evaporation. Surface residue affects many of the variables that determine the evaporation rate. These variables include Rn, G, aerodynamic resistance and surface resistances to transport of heat and water vapor fluxes

Caprio et al. (1985) compared evaporation from three mini-lysimeters installed in bare soil and in a 14 and 28 cm tall standing wheat stubble. After nine days of measurements, evaporation from the lysimeter with stubble was 60% of the evaporation measured from bare soil. Enz et al. (1988) evaluated daily evaporation for bare soil and stubble-covered soil surfaces. Evaporation was always greater from the bare soil surface until it was dry, then evaporation was greater from the stubble covered-surface because more water was available. Evaporation from a bare soil surface has been described in three stages. An initial, energy-limited stage occurs when enough soil water is available to satisfy the potential evaporation rates. A second, falling rate stage is limited by water flow to the soil surface, while the third stage has a very low, nearly constant evaporative rate from very dry soil (Jalota & Prihar, 1998). Steiner (1989) evaluated the effect of residue (from cotton, sorghum and wheat) on the initial, energy-limited rate of evaporation. The evaporation rate relative to bare soil evaporation was described by a logarithmic relationship. Increasing the amount of residue on the soil surface reduced the relative evaporation rate during the initial stage. Bristow et al. (1986) developed a model to predict soil heat and water budgets in a soilresidue-atmosphere system. Results from application of the model indicate that surface residues decreased evaporation by roughly 36% compared with simulations from bare soil. With the recognition of the potential of multiple-layer models to estimate ET, a modified surface energy balance model (SEB) was developed by Lagos (2008) and Lagos et al. (2009) to include the effect of crop residue on evapotranspiration. The model relies mainly on the Schuttleworth-Wallace (1985) and Choudhury and Monteith (1988) approaches and has the potential to predict

correlation coefficient (r2) of 0.77 and a root mean square error (RMSE) of 29 Wm-2.

sensible heat, latent heat, and soil heat flux for the canopy/soil system.

(Steiner, 1994; Horton et al., 1996; Steiner et al., 2000).

The modified surface energy balance (SEB) model has four layers (Figure 1), the first extended from the reference height above the vegetation and the sink for momentum within the canopy, a second layer between the canopy level and the soil surface, a third layer corresponding to the top soil layer and a lower soil layer where the soil atmosphere is saturated with water vapor. The soil temperature at the bottom of the lower level was held constant for at least a 24h period.

The SEB model distributes net radiation (Rn), sensible heat (H), latent heat (E), and soil heat fluxes (G) through the soil/residue/canopy system. Horizontal gradients of the potentials are assumed to be small enough for lateral fluxes to be ignored, and physical and biochemical energy storage terms in the canopy/residue/soil system are assumed to be negligible. The evaporation of water on plant leaves due to rain, irrigation or dew is also ignored.

The SEB model distributes net radiation (Rn) into sensible heat (H), latent heat (λE), and soil heat fluxes (G) through the soil-canopy system (Figure 2). Total latent heat (λE) is the sum of latent heat from the canopy (λEc), latent heat from the soil (λEs) and latent heat from the residue-covered soil (λEr). Similarly, sensible heat is calculated as the sum of sensible heat from the canopy (Hc), sensible heat from the soil (Hs) and sensible heat from the residue covered soil (Hr).

Fig. 1. Fluxes of the surface energy balance model (SEB).

The total net radiation is divided into that absorbed by the canopy (Rnc) and the soil (Rns) and is given by Rn = Rnc + Rns. The net radiation absorbed by the canopy is divided into latent heat and sensible heat fluxes as Rnc = λEc +Hc. Similarly, for the soil Rns = Gos + Hs, where Gos is a conduction term downwards from the soil surface and is expressed as Gos = λEs + Gs, where Gs is the soil heat flux for bare soil. Similarly, for the residue-covered soil Rns = Gor + Hr where Gor is the conduction downwards from the soil covered by residue. The conduction is given by Gor = λEr + Gr where Gr is the soil heat flux for residue-covered soil.

Total latent heat flux from the canopy/residue/soil system is the sum of the latent heat from the canopy (transpiration), latent heat from the soil and latent heat from the residue-covered soil (evaporation), calculated as:

$$
\lambda \mathbf{E} = \lambda \mathbf{E\_c} + (\mathbf{1} - \mathbf{f} \mathbf{r}) \cdot \lambda \mathbf{E\_s} + \mathbf{f} \mathbf{r} \cdot \lambda \mathbf{E\_r} \tag{1}
$$

where fr is the fraction of the soil affected by residue. Similarly, the total sensible heat is given by:

$$\mathbf{H} = \mathbf{H\_c} + (\mathbf{1} - \mathbf{f}\mathbf{r}) \cdot \mathbf{H\_s} + \mathbf{f}\mathbf{r} \cdot \mathbf{H\_r} \tag{2}$$

The differences in vapor pressure and temperature between levels can be expressed with an Ohm's law analogy using appropriate resistance and flux terms (Figure 2). The sensible and latent heat fluxes from the canopy, from bare soil and soil covered by residue are expressed by (Shuttleworth & Wallace, 1985):

$$\mathbf{H}\_{\mathbf{c}} = \frac{\rho \cdot \mathbf{c}\_{\mathbf{p}} \cdot (\mathbf{T}\_{\mathbf{l}} - \mathbf{T}\_{\mathbf{b}})}{\mathbf{r}\_{\mathbf{l}}} \qquad \text{and} \qquad \lambda \mathbf{E}\_{\mathbf{c}} = \frac{\rho \cdot \mathbf{C}\_{\mathbf{p}} \cdot (\mathbf{e}\_{\mathbf{l}}^{\*} - \mathbf{e}\_{\mathbf{b}})}{\chi \cdot (\mathbf{r}\_{\mathbf{l}} + \mathbf{r}\_{\mathbf{c}})} \tag{3}$$

$$\mathbf{H}\_{\rm s} = \frac{\rho \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{T}\_2 - \mathbf{T}\_b)}{\mathbf{r}\_2} \qquad \text{and} \qquad \lambda \mathbf{E}\_{\rm s} = \frac{\rho \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{e}\_{\rm L}^\* - \mathbf{e}\_{\rm b})}{\chi \cdot (\mathbf{r}\_2 + \mathbf{r}\_s)} \tag{4}$$

$$\mathbf{H}\_{\rm r} = \frac{\rho \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{T}\_{2\rm r} - \mathbf{T}\_{\rm b})}{\mathbf{r}\_2 + \mathbf{r}\_{\rm rh}} \qquad \text{and} \qquad \lambda \mathbf{E}\_{\rm r} = \frac{\rho \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{e}\_{\rm Lr}^\* - \mathbf{e}\_{\rm b})}{\chi \cdot (\mathbf{r}\_2 + \mathbf{r}\_s + \mathbf{r}\_{\rm r})} \tag{5}$$

where, ρ is the density of moist air, Cp is the specific heat of air, γ is the psychrometric constant, T1 is the mean canopy temperature, T2 is the temperature at the soil surface, Tb is the air temperature within the canopy, T2r is the temperature of the soil covered by residue, r1 is an aerodynamic resistance between the canopy and the air, rc is the surface canopy resistance, r2 is the aerodynamic resistance between the soil and the canopy, rs is the resistance to the diffusion of water vapor at the top soil layer, rrh is the residue resistance to transfer of heat, rr is the residue resistance to the transfer of vapor acting in series with the soil resistance rs, eb is the vapor pressure of the atmosphere at the canopy level, e1 \* is the saturation vapor pressure in the canopy, eL\* is the saturation vapor pressure at the top of the wet layer, and eLr\* is the saturation vapor pressure at the top of the wet layer for the soil covered by residue. Conduction of heat for the bare-soil and residue-covered surfaces are given by:

$$\mathbf{G\_{os}} = \frac{\rho \cdot \mathbf{C\_p} \cdot (\mathbf{T\_2} - \mathbf{T\_L})}{\mathbf{r\_u}} \qquad \text{and} \qquad \mathbf{G\_s} = \frac{\rho \cdot \mathbf{C\_p} \cdot (\mathbf{T\_L} - \mathbf{T\_m})}{\mathbf{r\_L}} \tag{6}$$

$$\mathbf{G}\_{\rm{or}} = \frac{\rho \cdot \mathbf{C}\_{\rm{p}} \cdot (\mathbf{T}\_{2\rm{r}} - \mathbf{T}\_{\rm{L}r})}{\mathbf{r}\_{\rm{u}}} \qquad \text{and} \qquad \mathbf{G}\_{\rm{r}} = \frac{\rho \cdot \mathbf{C}\_{\rm{p}} \cdot (\mathbf{T}\_{\rm{L}r} - \mathbf{T}\_{\rm{m}})}{\mathbf{r}\_{\rm{L}}} \tag{7}$$

where; ru and rL are resistance to the transport of heat for the upper and lower soil layers, respectively, TL and TLr are the temperatures at the interface between the upper and lower layers for the bare soil and the residue-covered soil, and Tm is the temperature at the bottom of the lower layer which was assumed to be constant on a daily basis.

where Gos is a conduction term downwards from the soil surface and is expressed as Gos = λEs + Gs, where Gs is the soil heat flux for bare soil. Similarly, for the residue-covered soil Rns = Gor + Hr where Gor is the conduction downwards from the soil covered by residue. The conduction is given by Gor = λEr + Gr where Gr is the soil heat flux for residue-covered soil. Total latent heat flux from the canopy/residue/soil system is the sum of the latent heat from the canopy (transpiration), latent heat from the soil and latent heat from the residue-covered

where fr is the fraction of the soil affected by residue. Similarly, the total sensible heat is given by:

The differences in vapor pressure and temperature between levels can be expressed with an Ohm's law analogy using appropriate resistance and flux terms (Figure 2). The sensible and latent heat fluxes from the canopy, from bare soil and soil covered by residue are expressed

where, ρ is the density of moist air, Cp is the specific heat of air, γ is the psychrometric constant, T1 is the mean canopy temperature, T2 is the temperature at the soil surface, Tb is the air temperature within the canopy, T2r is the temperature of the soil covered by residue, r1 is an aerodynamic resistance between the canopy and the air, rc is the surface canopy resistance, r2 is the aerodynamic resistance between the soil and the canopy, rs is the resistance to the diffusion of water vapor at the top soil layer, rrh is the residue resistance to transfer of heat, rr is the residue resistance to the transfer of vapor acting in series with the soil resistance rs, eb is

the canopy, eL\* is the saturation vapor pressure at the top of the wet layer, and eLr\* is the

where; ru and rL are resistance to the transport of heat for the upper and lower soil layers, respectively, TL and TLr are the temperatures at the interface between the upper and lower layers for the bare soil and the residue-covered soil, and Tm is the temperature at the bottom

saturation vapor pressure at the top of the wet layer for the soil covered by residue. Conduction of heat for the bare-soil and residue-covered surfaces are given by:

of the lower layer which was assumed to be constant on a daily basis.

and λE� <sup>=</sup> ρ∙C� ∙ (e�

and λE� <sup>=</sup> ρ∙C� ∙ (e�

and λE� <sup>=</sup> ρ∙C� ∙ (e��

and G� <sup>=</sup> ρ∙C� ∙ (T� − T�)

and G� <sup>=</sup> ρ∙C� ∙ (T�� − T�)

r�

r�

λE = λE� + (1 − fr) ∙ λE� + fr ∙ λE� (1)

H=H� + (1 − fr) ∙ H� + fr ∙ H� (2)

<sup>∗</sup> − e�)

<sup>∗</sup> − e�)

<sup>∗</sup> − e�)

γ ∙ (r� + r�) (3)

γ ∙ (r� + r�) (4)

γ ∙ (r� + r� + r�) (5)

\* is the saturation vapor pressure in

(6)

(7)

soil (evaporation), calculated as:

by (Shuttleworth & Wallace, 1985):

H� <sup>=</sup> ρ∙c� ∙ (T� − T�) r�

H� <sup>=</sup> ρ∙C� <sup>∙</sup> (T� − T�) r�

H� <sup>=</sup> ρ∙C� ∙ (T�� − T�) r� + r��

the vapor pressure of the atmosphere at the canopy level, e1

G�� <sup>=</sup> ρ∙C� <sup>∙</sup> (T� − T�) r�

 G�� <sup>=</sup> ρ∙C� ∙ (T�� − T��) r�

Choudhury and Monteith (1988) expressed differences in saturation vapor pressure between points in the system as linear functions of the corresponding temperature differences. They found that a single value of the slope of the saturation vapor pressure, Δ, when evaluated at the air temperature, Ta, gave acceptable results for the components of the heat balance. The vapor pressure differences were given by:

$$\mathbf{e}\_1^\* - \mathbf{e}\_\mathbf{b}^\* = \Delta \cdot (\mathbf{T}\_\mathbf{l} - \mathbf{T}\_\mathbf{b}) \mathbf{e}\_\mathbf{L}^\* - \mathbf{e}\_\mathbf{b}^\* = \Delta \cdot (\mathbf{T}\_\mathbf{L} - \mathbf{T}\_\mathbf{b}) \mathbf{e}\_\mathbf{b}^\* - \mathbf{e}\_\mathbf{a}^\* = \Delta \cdot (\mathbf{T}\_\mathbf{b} - \mathbf{T}\_\mathbf{a}) \tag{8}$$

$$\dots \qquad \dots \qquad \vdots \qquad \vdots \qquad \mathbf{e}\_{\mathbf{L}r}^\* - \mathbf{e}\_\mathbf{b}^\* = \Delta \cdot (\mathbf{T}\_{\mathbf{L}r} - \mathbf{T}\_\mathbf{b}) \tag{9}$$

The above equations were combined and solved to estimate fluxes. Details are provided by Lagos (2008). The solution gives the latent and sensible heat fluxes from the canopy as:

$$\lambda \mathbf{E}\_{\mathbf{c}} = \frac{\Delta \cdot \mathbf{r}\_1 \cdot \mathbf{R} \mathbf{n}\_{\mathbf{c}} + \rho \cdot \mathbf{C}\_{\mathbf{p}} \cdot (\mathbf{e}\_{\mathbf{b}}^\ast - \mathbf{e}\_{\mathbf{b}})}{\Delta \cdot \mathbf{r}\_1 + \chi \cdot (\mathbf{r}\_1 + \mathbf{r}\_c)} \quad \text{and} \quad \mathbf{H}\_{\mathbf{c}} = \frac{\chi \cdot (\mathbf{r}\_1 - \mathbf{r}\_c) \cdot \mathbf{R} \mathbf{n}\_{\mathbf{c}} - \rho \cdot \mathbf{C}\_{\mathbf{p}} \cdot (\mathbf{e}\_{\mathbf{b}}^\ast - \mathbf{e}\_{\mathbf{b}})}{\Delta \cdot \mathbf{r}\_1 + \chi \cdot (\mathbf{r}\_1 + \mathbf{r}\_c)} \tag{9}$$

Fig. 2. Schematic resistance network of the Surface Energy Balance (SEB) model a) Latent heat flux and b) Sensible heat flux.

$$\lambda \mathbf{E}\_{\mathbf{s}} = \frac{\mathbf{R} \mathbf{n}\_{\mathbf{s}} \cdot \Delta \cdot \mathbf{r}\_2 \cdot \mathbf{r}\_\mathrm{L} + \rho \cdot \mathbf{C}\_\mathrm{p} \cdot \left[ \left( \mathbf{e}\_\mathrm{b}^\* - \mathbf{e}\_\mathrm{b} \right) \cdot \left( \mathbf{r}\_\mathrm{u} + \mathbf{r}\_\mathrm{L} + \mathbf{r}\_\mathrm{2} \right) + \left( \mathbf{T}\_\mathrm{m} - \mathbf{T}\_\mathrm{b} \right) \cdot \Delta \cdot \left( \mathbf{r}\_\mathrm{u} + \mathbf{r}\_\mathrm{2} \right) \right]}{\chi \cdot \left( \mathbf{r}\_\mathrm{2} + \mathbf{r}\_\mathrm{s} \right) \cdot \left( \mathbf{r}\_\mathrm{u} + \mathbf{r}\_\mathrm{L} + \mathbf{r}\_\mathrm{2} \right) + \Delta \cdot \mathbf{r}\_\mathrm{L} \cdot \left( \mathbf{r}\_\mathrm{u} + \mathbf{r}\_\mathrm{2} \right)}} \tag{10}$$

$$\mathbf{H}\_{\rm s} = \frac{\mathbf{R}\mathbf{n}\_{\rm s} \cdot \mathbf{r}\_{\rm L} \cdot \Delta - \lambda \mathbf{E}\_{\rm s} \cdot [\mathbf{r}\_{\rm L} \cdot \Delta + \mathbf{y} \cdot (\mathbf{r}\_2 + \mathbf{r}\_{\rm s})] + \boldsymbol{\rho} \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{e}\_{\rm b}^\* - \mathbf{e}\_{\rm b}) - \boldsymbol{\rho} \cdot \mathbf{C}\_{\rm p} \cdot \Delta \cdot (\mathbf{T}\_{\rm b} - \mathbf{T}\_{\rm m})}{\mathbf{r}\_{\rm L} \cdot \Delta} \tag{11}$$

$$\lambda \mathbf{E}\_{\rm r} = \frac{\text{R} \mathbf{n}\_{\rm s} \cdot \Delta \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{\rm rh}) \cdot \mathbf{r}\_{\rm L} + \rho \cdot \mathbb{C}\_{\rm p} \cdot [(\mathbf{e}\_{\rm b}^{\ast} - \mathbf{e}\_{\rm b}) \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{\rm L} + \mathbf{r}\_{2} + \mathbf{r}\_{\rm rh}) + (\mathbf{T}\_{\rm m} - \mathbf{T}\_{\rm b}) \cdot \Delta \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{2} + \mathbf{r}\_{\rm r})]}{\chi \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{\rm s} + \mathbf{r}\_{\rm r}) \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{\rm L} + \mathbf{r}\_{2} + \mathbf{r}\_{\rm rh}) + \Delta \cdot \mathbf{r}\_{\rm L} \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{\rm r} + \mathbf{r}\_{\rm rh})}} \tag{12}$$

$$\mathbf{H}\_{\mathbf{r}} = \frac{\mathbf{R}\mathbf{n}\_{\mathrm{s}} \cdot \mathbf{r}\_{\mathrm{L}} \cdot \Delta - \lambda \mathbf{E}\_{\mathrm{r}} \cdot \left[\mathbf{r}\_{\mathrm{L}} \cdot \Delta + \mathbf{y} \cdot (\mathbf{r}\_{\mathrm{2}} + \mathbf{r}\_{\mathrm{s}} + \mathbf{r}\_{\mathrm{I}})\right] + \boldsymbol{\rho} \cdot \mathbf{C}\_{\mathrm{p}} \cdot (\mathbf{e}\_{\mathrm{b}}^{\*} - \mathbf{e}\_{\mathrm{b}}) - \boldsymbol{\rho} \cdot \mathbf{C}\_{\mathrm{p}} \cdot \Delta \cdot (\mathbf{T}\_{\mathrm{b}} - \mathbf{T}\_{\mathrm{m}})}{\mathbf{r}\_{\mathrm{L}} \cdot \Delta} \tag{13}$$

$$\mathbf{e}\_{\mathbf{b}} = \left( \mathbf{T}\_{\mathbf{b}} \cdot (\boldsymbol{\Delta} \cdot \mathbf{A}\_2 - \mathbf{A}\_3) + \frac{\mathbf{A}\_1}{\rho \cdot \mathbf{C}\_{\mathbf{p}}} - \boldsymbol{\Delta} \cdot \mathbf{A}\_2 \cdot \mathbf{T}\_{\mathbf{a}} + \mathbf{A}\_2 \cdot \mathbf{e}\_{\mathbf{a}}^\* + \mathbf{T}\_{\mathbf{m}} \cdot \mathbf{A}\_3 + \frac{\mathbf{e}\_{\mathbf{a}}}{\mathbf{y} \cdot \mathbf{r}\_{\mathbf{a}\mathbf{w}}} \right) \cdot \left( \frac{\mathbf{y} \cdot \mathbf{r}\_{\mathbf{a}\mathbf{w}}}{1 + \mathbf{A}\_2 \cdot \mathbf{y} \cdot \mathbf{r}\_{\mathbf{a}\mathbf{w}}} \right) \tag{14}$$

$$\mathbf{T\_b = \left[\frac{\mathbf{B\_1}}{\rho \cdot \mathbf{C\_p}} + \mathbf{T\_a} \cdot \left(\frac{1}{\mathbf{r\_{ah}}} - \Delta \cdot \mathbf{B\_2}\right) + \left(\mathbf{e\_a^\*} - \mathbf{e\_b}\right) \cdot \mathbf{B\_2} + \mathbf{T\_m} \cdot \mathbf{B\_3}\right] \cdot \left(\frac{\mathbf{r\_{ah}}}{1 - \Delta \cdot \mathbf{B\_2} \cdot \mathbf{r\_{ah}} + \mathbf{B\_3} \cdot \mathbf{r\_{ah}}}\right) \tag{15}$$

$$\mathbf{A}\_{1} = \frac{\Delta \cdot \mathbf{r}\_{1} \cdot \text{R} \mathbf{n}\_{c}}{\Delta \cdot \mathbf{r}\_{1} + \mathbf{y} \cdot (\mathbf{r}\_{1} + \mathbf{r}\_{c})} + (\mathbf{1} - \mathbf{f}\_{r}) \cdot \frac{\text{R} \mathbf{n}\_{s} \cdot \Delta \cdot \mathbf{r}\_{2} \cdot \mathbf{r}\_{L}}{\mathbf{y} \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{s}) \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{L} + \mathbf{r}\_{2}) + \Delta \cdot \mathbf{r}\_{L} \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{2})} + 1} + 1$$

$$\mathbf{f}\_{\text{f}} \cdot \frac{\text{R} \mathbf{n}\_{s} \cdot \Delta \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{\text{th}}) \cdot \mathbf{r}\_{L}}{\mathbf{y} \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{s} + \mathbf{r}\_{\text{f}}) \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{L} + \mathbf{r}\_{2} + \mathbf{r}\_{\text{th}}) + \Delta \cdot \mathbf{r}\_{L} \cdot (\mathbf{r}\_{u} + \mathbf{r}\_{2} + \mathbf{r}\_{\text{th}})}$$

$$\mathbf{A}\_{2} = \frac{1}{\Delta \cdot \mathbf{r}\_{1} + \mathbf{y} \cdot (\mathbf{r}\_{1} + \mathbf{r}\_{c})} + (\mathbf{1} - \mathbf{f}\_{r}) \cdot \frac{(\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{2})}{\mathbf{y} \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{s}) \cdot (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{2}) + \Delta \cdot \mathbf{r}\_{\mathrm{L}} \cdot (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{2})} + \tag{17}$$

$$\mathbf{f}\_{\mathrm{r}} \cdot \frac{(\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{2} + \mathbf{r}\_{\mathrm{rh}})}{\mathbf{y} \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{\mathrm{s}} + \mathbf{r}\_{\mathrm{r}}) \cdot (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{2} + \mathbf{r}\_{\mathrm{rh}}) + \Delta \cdot \mathbf{r}\_{\mathrm{L}} \cdot (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{2} + \mathbf{r}\_{\mathrm{rh}})}$$

$$\begin{aligned} \mathbf{A}\_{3} &= \left| (\mathbf{1} - \mathbf{f\_{r}}) \cdot \frac{\Delta \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2}})}{\chi \cdot (\mathbf{r\_{2}} + \mathbf{r\_{s}}) \cdot (\mathbf{r\_{u}} + \mathbf{r\_{L}} + \mathbf{r\_{2}}) + \Delta \cdot \mathbf{r\_{L}} \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2}})} \right. \\ &+ \mathbf{f\_{r}} \cdot \frac{\Delta \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2}} + \mathbf{r\_{rh}})}{\chi \cdot (\mathbf{r\_{2}} + \mathbf{r\_{s}} + \mathbf{r\_{r}}) \cdot (\mathbf{r\_{u}} + \mathbf{r\_{L}} + \mathbf{r\_{2}} + \mathbf{r\_{rh}}) + \Delta \cdot \mathbf{r\_{L}} \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2}} + \mathbf{r\_{rh}})} \right| \end{aligned} \tag{18}$$

$$\mathbf{B}\_{1} = \left[ \mathbf{R} \mathbf{n}\_{\rm c} \cdot \frac{\mathbf{y} \cdot (\mathbf{r}\_{1} + \mathbf{r}\_{\rm c})}{\Delta \cdot \mathbf{r}\_{1} + \mathbf{y} \cdot (\mathbf{r}\_{1} + \mathbf{r}\_{\rm c})} + \mathbf{R} \mathbf{n}\_{\rm s} \cdot \begin{pmatrix} (\mathbf{1} - \mathbf{f}\_{\rm r}) \cdot (\mathbf{1} - \Delta \cdot \mathbf{r}\_{2} \cdot \mathbf{r}\_{\rm L} \cdot \mathbf{X}\_{\rm s}) + \\ \mathbf{f}\_{\rm r} \cdot (\mathbf{1} - \Delta \cdot (\mathbf{r}\_{2} + \mathbf{r}\_{\rm rh}) \cdot \mathbf{r}\_{\rm L} \cdot \mathbf{X}\_{\rm r}) \end{pmatrix} \right] \tag{19}$$

$$\begin{aligned} \mathbf{B}\_{2} &= \frac{-1}{\Delta \cdot \mathbf{r}\_{1} + \mathbf{y} \cdot (\mathbf{r}\_{1} + \mathbf{r}\_{c})} + (\mathbf{1} - \mathbf{f}\_{\mathrm{r}}) \cdot \left(\frac{1}{\mathbf{r}\_{\mathrm{L}}\Delta} - (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{\mathrm{2}}) \cdot \mathbf{X}\_{\mathrm{s}}\right) \\ &+ f\_{\mathrm{r}} \cdot \left(\frac{1}{r\_{\mathrm{L}}\Delta} - (\mathbf{r}\_{\mathrm{u}} + \mathbf{r}\_{\mathrm{L}} + \mathbf{r}\_{\mathrm{2}} + \mathbf{r}\_{\mathrm{rh}}) \cdot \mathbf{X}\_{\mathrm{r}}\right) \end{aligned} \tag{20}$$

$$\mathbf{B}\_3 = \left[ (\mathbf{1} - \mathbf{f\_r}) \cdot \left( \frac{\mathbf{1}}{\mathbf{r\_L}} - \Delta \cdot (\mathbf{r\_u} + \mathbf{r\_2}) \cdot \mathbf{X\_s} \right) + \mathbf{f\_r} \cdot \left( \frac{\mathbf{1}}{\mathbf{r\_L}} - \Delta \cdot (\mathbf{r\_u} + \mathbf{r\_2} + \mathbf{r\_{rh}}) \cdot \mathbf{X\_r} \right) \right] \tag{21}$$

$$\mathbf{X\_{s}} = \left(\frac{1}{\mathbf{y} \cdot (\mathbf{r\_{2}} + \mathbf{r\_{s}}) \cdot (\mathbf{r\_{u}} + \mathbf{r\_{L}} + \mathbf{r\_{2}}) + \Delta \cdot \mathbf{r\_{L}} \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2}})}\right) \left(\frac{\{\mathbf{r\_{L}} \cdot \Delta + \mathbf{y} \cdot \{\mathbf{r\_{2}} + \mathbf{r\_{s}}\}\}}{\mathbf{r\_{L}} \cdot \Delta}\right) \text{ and} \tag{22}$$
 
$$\mathbf{r\_{s}} = \frac{\{\mathbf{r\_{s}} \cdot \Delta + \mathbf{y\_{s}} \cdot \{\mathbf{r\_{s}} + \mathbf{r\_{s}} \cdot \mathbf{r\_{s}}\}}{\mathbf{r\_{s}} \cdot \Delta + \mathbf{y\_{s}} \cdot \{\mathbf{r\_{s}} + \mathbf{r\_{s}} \cdot \mathbf{r\_{s}}\}} \tag{23}$$

$$\mathbf{X}\_{\mathbf{r}} = \left(\frac{1}{\mathbf{y} \cdot (\mathbf{r}\_2 + \mathbf{r}\_s + \mathbf{r}\_r) \cdot (\mathbf{r}\_u + \mathbf{r}\_L + \mathbf{r}\_2 + \mathbf{r}\_{\rm rh}) + \Delta \cdot \mathbf{r}\_{\rm L} \cdot (\mathbf{r}\_u + \mathbf{r}\_2 + \mathbf{r}\_{\rm rh})}\right) \left(\frac{(\mathbf{r}\_L \cdot \Delta + \mathbf{y} \cdot (\mathbf{r}\_2 + \mathbf{r}\_s + \mathbf{r}\_r))}{\mathbf{r}\_L \cdot \Delta}\right)^{-1}$$

$$\mathbf{r\_{ah}} = \mathbf{r\_{am}} + \mathbf{r\_{bh}} \qquad \qquad \text{and} \qquad \qquad \mathbf{r\_{aw}} = \mathbf{r\_{am}} + \mathbf{r\_{bw}} \tag{25}$$

$$\mathbf{r}\_{\rm am} = \frac{1}{\mathbf{k} \cdot \mathbf{u}^\*} \cdot \mathrm{Ln} \left( \frac{\mathbf{z}\_\mathbf{r} - \mathbf{d}}{\mathbf{h} - \mathbf{d}} \right) + \frac{\mathbf{h}}{\mathbf{a} \cdot \mathbf{K}\_\mathbf{h}} \cdot \left[ \exp \left( \mathbf{a} \cdot \left( \mathbf{1} - \frac{\mathbf{z}\_\mathbf{o} + \mathbf{d}}{\mathbf{h}} \right) \right) - \mathbf{1} \right] \tag{24}$$

Verma (1989) expressed the excess resistance for heat transfer as:

$$\mathbf{r}\_{\rm bh} = \frac{\mathbf{k} \cdot \mathbf{B}^{-1}}{\mathbf{k} \cdot \mathbf{u}^\*} \tag{25}$$

where B-1 represents a dimensionless bulk parameter. Thom (1972) suggests that the product kB-1 equal approximately 2 for most arable crops.

Excess resistance was derived primarily from heat transfer observations (Weseley & Hicks 1977). Aerodynamic resistance to water vapor was modified by the ratio of thermal and water vapor diffusivity:

$$\mathbf{r}\_{\rm bw} = \frac{\mathbf{k} \cdot \mathbf{B}^{-1}}{\mathbf{k} \cdot \mathbf{u}^\*} \left(\frac{\mathbf{k}\_1}{\mathbf{D}\_\mathbf{V}}\right)^{2/3} \tag{26}$$

where, k1 is the thermal diffusivity and Dv is the molecular diffusivity of water vapor in air. Similarly, Shuttleworth and Gurney (1990) expressed the aerodynamic resistance (r2) by integrating the eddy diffusion coefficient between the soil surface and the sink of momentum in the canopy to yield:

$$\mathbf{r}\_2 = \frac{\mathbf{h} \cdot \exp(\mathbf{a})}{\mathbf{a} \cdot \mathbf{K}\_\mathbf{h}} \cdot \left[ \exp\left(\frac{-\mathbf{a} \cdot \mathbf{z}\_\mathbf{o}}{\mathbf{h}}\right) - \exp\left(\frac{-\mathbf{a} \cdot (\mathbf{d} + \mathbf{z}\_\mathbf{o})}{\mathbf{h}}\right) \right] \tag{27}$$

where zo' is the roughness length of the soil surface. Values of surface roughness (zo) and displacement height (d) are functions of leaf area index (LAI) and can be estimated using the expressions given by Shaw and Pereira (1982).

The diffusion coefficients between the soil surface and the canopy, and therefore the resistance for momentum, heat, and vapor transport are assumed equal although it is recognized that this is a weakness in the use of the K theory to describe through-canopy transfer (Shuttleworth & Gurney, 1990). Stability is not considered.

#### **2.1.1.2 Canopy resistances**

The mean boundary layer resistance of the canopy r1, for latent and sensible heat flux, is influenced by the surface area of vegetation (Shuttleworth & Wallace, 1985):

$$\mathbf{r\_1} = \frac{\mathbf{r\_b}}{\mathbf{2} \cdot \mathbf{LAI}} \tag{28}$$

where rb is the resistance of the leaf boundary layer, which is proportional to the temperature difference between the leaf and surrounding air divided by the associated flux (Choudhury & Monteith, 1988). Shuttleworth and Wallace (1985) noted that resistance rb exhibits some dependence on in-canopy wind speed, with typical values of 25 s m-1. Shuttleworth and Gurney (1990) represented rb as:

$$\mathbf{r\_b = \frac{100}{\alpha} \cdot \left(\frac{\mathbf{w}}{\mathbf{u\_h}}\right)^{1/2} \cdot \left(1 - \exp\left(\frac{-\alpha}{2}\right)\right)^{-1}}\tag{29}$$

where w is the representative leaf width and uh is the wind speed at the top of the canopy. This resistance is only significant when acting in combination with a much larger canopy surface resistance, and Shuttleworth and Gurney (1990) suggest that r1 could be neglected for foliage completely covering the ground. Using rb = 25 s m-1 with an LAI = 4, the corresponding canopy boundary layer resistance is r1 = 3 s m-1.

Canopy surface resistance, rc, can be calculated by dividing the minimum surface resistance for a single leaf (rl) by the effective canopy leaf area index (LAI). Five environmental factors have been found to affect stomata resistance: solar radiation, air temperature, humidity, CO2 concentration and soil water potential (Yu et al., 2004). Several models have been developed to estimate stomata conductance and canopy resistance. Stannard (1993) estimated rc as a function of vapor pressure deficit, leaf area index, and solar radiation as:

$$\mathbf{r\_c = \left[\mathbf{C\_1} \cdot \frac{\text{LAI}}{\text{LAI}\_{\text{max}}} \cdot \frac{\mathbf{C\_2}}{\mathbf{C\_2} + \text{VPD}\_{\text{a}}} \cdot \frac{\text{Rad} \cdot (\text{Rad}\_{\text{max}} + \text{C\_3})}{\text{Rad}\_{\text{max}} \cdot (\text{Rad} + \text{C\_3})}\right]^{-1} \tag{30}$$

where LAImax is the maximum value of leaf area index, VPDa is vapor pressure deficit, Rad is solar radiation, Radmax is the maximum value of solar radiation (estimated at 1000 W m-2) and C1, C2 and C3 are regression coefficients. Canopy resistance does not account for soil water stress effects.

#### **2.1.1.3 Soil resistances**

282 Evapotranspiration – Remote Sensing and Modeling

r�� <sup>=</sup> k∙B�� k∙u<sup>∗</sup>

where B-1 represents a dimensionless bulk parameter. Thom (1972) suggests that the product

Excess resistance was derived primarily from heat transfer observations (Weseley & Hicks 1977). Aerodynamic resistance to water vapor was modified by the ratio of thermal and

where, k1 is the thermal diffusivity and Dv is the molecular diffusivity of water vapor in air. Similarly, Shuttleworth and Gurney (1990) expressed the aerodynamic resistance (r2) by integrating the eddy diffusion coefficient between the soil surface and the sink of

 ´

where zo' is the roughness length of the soil surface. Values of surface roughness (zo) and displacement height (d) are functions of leaf area index (LAI) and can be estimated using the

The diffusion coefficients between the soil surface and the canopy, and therefore the resistance for momentum, heat, and vapor transport are assumed equal although it is recognized that this is a weakness in the use of the K theory to describe through-canopy

The mean boundary layer resistance of the canopy r1, for latent and sensible heat flux, is

where rb is the resistance of the leaf boundary layer, which is proportional to the temperature difference between the leaf and surrounding air divided by the associated flux (Choudhury & Monteith, 1988). Shuttleworth and Wallace (1985) noted that resistance rb exhibits some dependence on in-canopy wind speed, with typical values of 25 s m-1.

where w is the representative leaf width and uh is the wind speed at the top of the canopy. This resistance is only significant when acting in combination with a much larger canopy surface resistance, and Shuttleworth and Gurney (1990) suggest that r1 could be neglected

∙ �1 − exp �−α

<sup>2</sup> ��

��

r� <sup>=</sup> r�

<sup>h</sup> � − exp �−α ∙ (d + z�)

k� D� � � ��

r�� <sup>=</sup> k∙B�� k∙u<sup>∗</sup> �

∙ �exp �−α ∙ z�

transfer (Shuttleworth & Gurney, 1990). Stability is not considered.

influenced by the surface area of vegetation (Shuttleworth & Wallace, 1985):

(25)

(26)

(29)

<sup>h</sup> �� (27)

2 ∙ LAI (28)

Verma (1989) expressed the excess resistance for heat transfer as:

kB-1 equal approximately 2 for most arable crops.

r� <sup>=</sup> h ∙ exp(α) α ∙ K�

expressions given by Shaw and Pereira (1982).

Shuttleworth and Gurney (1990) represented rb as:

r� <sup>=</sup> <sup>100</sup>

<sup>α</sup> ∙ �<sup>w</sup> u� � � ��

water vapor diffusivity:

momentum in the canopy to yield:

**2.1.1.2 Canopy resistances** 

Farahani and Bausch (1995), Anadranistakis et al. (2000) and Lindburg (2002) found that soil resistance (rs) can be related to volumetric soil water content in the top soil layer. Farahani and Ahuja (1996) found that the ratio of soil resistance when the surface layer is wet relative to its upper limit depends on the degree of saturation (θ/θs) and can be described by an exponential function as:

$$\mathbf{r}\_{\rm s} = \mathbf{r}\_{\rm so} \cdot \exp\left(-\boldsymbol{\mathfrak{\boldsymbol{\beta}}} \cdot \frac{\boldsymbol{\mathfrak{\boldsymbol{\theta}}}}{\boldsymbol{\mathfrak{\boldsymbol{\theta}}}\_{\rm s}}\right) \qquad\qquad\text{and}\qquad\qquad\qquad\mathbf{r}\_{\rm so} = \frac{\mathbf{r}\_{\rm t} \cdot \mathbf{r}\_{\rm s}}{\mathbf{D}\_{\rm v} \cdot \boldsymbol{\mathfrak{\boldsymbol{\theta}}}}\tag{31}$$

where Lt is the thickness of the surface soil layer, τs is a soil tortuosity factor, Dv is the water vapor diffusion coefficient and ∅ is soil porosity, θ is the average volumetric water content in the surface layer, θs is the saturation water content, and β is a fitting parameter. Measurements of θ from the top 0.05 m soil layer were more effective in modeling rs than θ for thinner layers.

Choudhury and Monteith (1988) expressed the soil resistance for heat flux (rL) in the soil layer extending from depth Lt to Lm as:

$$\mathbf{r}\_{\rm L} = \frac{\rho \cdot \mathbf{C}\_{\rm p} \cdot (\mathbf{L}\_{\rm m} - \mathbf{L}\_{\rm t})}{\rm K} \tag{32}$$

where K is the thermal conductivity of the soil. Similarly, the corresponding resistance for the upper layer (ru) of depth Lt and conductivity K*'* as:

$$\mathbf{r}\_{\mathbf{u}} = \frac{\rho \cdot \mathbf{C}\_{\mathbf{p}} \cdot \mathbf{L}\_{\mathbf{t}}}{\mathbf{K}'} \tag{33}$$

#### **2.1.1.4 Residue resistances**

Surface residue is an integral part of many cropping systems. Bristow and Horton (1996) showed that partial surface mulch cover can have dramatic effects on the soil physical environment. The vapor conductance through residue has been described as a linear function of wind speed. Farahani and Ahuja (1996) used results from Tanner and Shen (1990) to develop the resistance of surface residue (rr) as:

$$\mathbf{r}\_{\mathbf{r}} = \frac{\mathbf{L}\_{\mathbf{r}} \cdot \mathbf{r}\_{\mathbf{r}}}{\mathbf{D}\_{\mathbf{v}} \cdot \boldsymbol{\mathfrak{G}}\_{\mathbf{r}}} (1 + \mathbf{0}.\mathbf{7} \cdot \mathbf{u}\_2)^{-1} \tag{34}$$

where Lr is residue thickness, τr is residue tortuosity, Dv is vapor diffusivity in still air, ∅�is residue porosity and u2 is wind speed measured two meters above the surface. Due to the porous nature of field crop residue layers, the ratio τr/∅� is about one (Farahani & Ahuja, 1996).

Similar to the soil resistance, Bristow and Horton (1996) and Horton et al. (1996) expressed the resistance of residue for heat transfer, rrh, as:

$$\mathbf{r\_{rh}} = \frac{\rho \cdot \mathbf{C\_p} \cdot \mathbf{L\_r}}{\mathbf{K\_r}} \tag{35}$$

where Kr is the residue thermal conductivity.

The fraction of the soil covered by residue (fr) can be estimated using the amount and type of residue (Steiner et al., 2000). The soil covered by residue and the residue thickness are estimated using the expressions developed by Gregory (1982).

#### **2.1.2 SEB model inputs**

Inputs required to solve multiple layer models (i.e. Shuttleworth and Wallace (1985), Choudhury and Monteith (1988) and Lagos (2008) models) are net radiation, solar radiation, air temperature, relative humidity, wind speed, LAI, crop height, soil texture, soil temperature, soil water content, residue type, and residue amount. In particular, net radiation, leaf area index, soil temperatures and residue amount are variables rarely measured in the field, other than at research sites. Net radiation and soil temperature models can be incorporated into surface energy balance models to predict evapotranspiration from environmental variables typically measured by automatic weather stations.

Similar to the Shuttleworth and Wallace (1985) and Choudhury and Monteith (1988) models, measurements of net radiation and estimations of net radiation absorbed by the canopy are necessary for the SEB model. Beer's law is used to estimate the penetration of radiation through the canopy and estimates the net radiation reaching the surface (Rns) as:

$$\mathbf{Rn\_s} = \mathbf{Rn} \cdot \exp(-\mathbf{C\_{ext}} \cdot \mathbf{LAI}) \tag{36}$$

where Cext is the extinction coefficient of the crop for net radiation. Consequently, net radiation absorbed by the canopy (Rnc) can be estimated as Rnc = Rn – Rns.

#### **2.1.3 SEB model evaluation**

An irrigated maize field site located at the University of Nebraska Agricultural Research and Development Center near Mead, NE (41o09'53.5"N, 96o28'12.3"W, elevation 362 m) was used for model evaluation. This site is a 49 ha production field that provides sufficient upwind fetch of uniform cover required for adequately measuring mass and energy fluxes using eddy covariance systems. The area has a humid continental climate and the soil corresponds to a deep silty clay loam (Suyker & Verma, 2009). The field has not been tilled since 2001. Detailed information about planting densities and crop management is provided by Verma et al. (2005) and Suyker and Verma (2009).

where Lr is residue thickness, τr is residue tortuosity, Dv is vapor diffusivity in still air, ∅�is residue porosity and u2 is wind speed measured two meters above the surface. Due to the porous nature of field crop residue layers, the ratio τr/∅� is about one (Farahani & Ahuja,

Similar to the soil resistance, Bristow and Horton (1996) and Horton et al. (1996) expressed

r�� <sup>=</sup> ρ∙C� ∙ L� K�

The fraction of the soil covered by residue (fr) can be estimated using the amount and type of residue (Steiner et al., 2000). The soil covered by residue and the residue thickness are

Inputs required to solve multiple layer models (i.e. Shuttleworth and Wallace (1985), Choudhury and Monteith (1988) and Lagos (2008) models) are net radiation, solar radiation, air temperature, relative humidity, wind speed, LAI, crop height, soil texture, soil temperature, soil water content, residue type, and residue amount. In particular, net radiation, leaf area index, soil temperatures and residue amount are variables rarely measured in the field, other than at research sites. Net radiation and soil temperature models can be incorporated into surface energy balance models to predict evapotranspiration from environmental variables typically measured by automatic weather

Similar to the Shuttleworth and Wallace (1985) and Choudhury and Monteith (1988) models, measurements of net radiation and estimations of net radiation absorbed by the canopy are necessary for the SEB model. Beer's law is used to estimate the penetration of radiation

where Cext is the extinction coefficient of the crop for net radiation. Consequently, net

An irrigated maize field site located at the University of Nebraska Agricultural Research and Development Center near Mead, NE (41o09'53.5"N, 96o28'12.3"W, elevation 362 m) was used for model evaluation. This site is a 49 ha production field that provides sufficient upwind fetch of uniform cover required for adequately measuring mass and energy fluxes using eddy covariance systems. The area has a humid continental climate and the soil corresponds to a deep silty clay loam (Suyker & Verma, 2009). The field has not been tilled since 2001. Detailed information about planting densities and crop management is provided

Rn� = Rn ∙ exp(−C��� ∙ LAI) (36)

through the canopy and estimates the net radiation reaching the surface (Rns) as:

radiation absorbed by the canopy (Rnc) can be estimated as Rnc = Rn – Rns.

(1 + 0.7 ∙ u�)�� (34)

(35)

r� <sup>=</sup> L� ∙ τ� D� ∙ ∅�

the resistance of residue for heat transfer, rrh, as:

where Kr is the residue thermal conductivity.

**2.1.2 SEB model inputs** 

**2.1.3 SEB model evaluation** 

by Verma et al. (2005) and Suyker and Verma (2009).

estimated using the expressions developed by Gregory (1982).

1996).

stations.

Soil water content was measured continuously at four depths (0.10, 0.25, 0.5 and 1.0 m) with Theta probes (Delta-T Device, Cambridge, UK). Destructive green leaf area index and biomass measurements were taken bi-monthly during the growing season. The eddy covariance measurements of latent heat, sensible heat and momentum fluxes were made using an omnidirectional three dimensional sonic anemometer (Model R3, Gill Instruments Ltd., Lymington, UK ) and an open-path infrared CO2/H2O gas analyzer system (Model LI7500, Li-Cor Inc, Lincoln, NE). Fluxes were corrected for sensor frequency response and variations in air density. More details of measurements and calculations are given in Verma et al. (2005). Air temperature and humidity were measured at 3 and 6 meters (Humitter 50Y, Vaisala, Helsinki, Finland), net radiation at 5.5 m (CNR1, Kipp and Zonen, Delft, NLD) and soil heat flux at 0.06 m (Radiation and Energy Balance Systems Inc, Seattle, WA). Soil temperature was measured at 0.06, 0.1, 0.2 and 0.5 m depths (Platinum RTD, Omega Engineering, Stamford, CT). More details are given in Verma et al. (2005) and Suyker and Verma ( 2009).

Evapotranspiration predictions from the SEB model were compared with eddy covariance flux measurements during 2003 for an irrigated maize field. To evaluate the energy balance closure of eddy covariance measurements, net radiation was compared against the sum of latent heat, sensible heat, soil heat flux and storage terms. Storage terms include soil heat storage, canopy heat storage, and energy used in photosynthesis. Storage terms were calculated by Suyker and Verma (2009) following Meyers and Hollinger (2004). During these days, the regression slope for energy balance closure was 0.89 with a correlation coefficient of r2 = 0.98.

For model evaluation, 15 days under different LAI conditions were selected to initially test the model, however further work is needed to test the model for entire growing seasons and during longer periods. Hourly data for three 5-day periods with varying LAI conditions (LAI = 0, 1.5 and 5.4) were used to compare measured ET to model predictions. Input data of the model included hourly values for: net radiation, air temperature, relative humidity, soil temperature at 50 cm, wind speed, solar radiation and soil water content. During the first 5-day period, which was prior to germination, the maximum net radiation ranged from 240 to 720 W m-2, air temperature ranged from 10 to 30°C, soil temperature was fairly constant at 16°C and wind speed ranged from 1 to 9 m s-1 but was generally less than 6 m s-1 (Figure 3). Soil water content in the evaporation zone averaged 0.34 m3 m-3and the residue density was 12.5 ton/ha on June 6, 2003. Precipitation occurred on the second and fifth days, totaling 17 mm.

Evapotranspiration estimated with the SEB model and measured using the eddy covariance system is given in Figure 4. ET fluxes were the highest at midday on June 6, reaching approximately 350 W m-2. The lowest ET rates occurred on the second day. Estimated ET tracked measured latent heat fluxes reasonably well. Estimates were better for days without precipitation than for days when rainfall occurred. The effect of crop residue on evaporation from the soil is shown in Figure 4 for this period. Residue reduced cumulative evaporation by approximately 17% during this five-day period. Evaporation estimated with the SEB model on June 6 and 9 was approximately 3.5 mm/day, totaling approximately half of the total evaporation for the five days.

During the second five-day period, when plants partially shaded the soil surface (LAI = 1.5), the maximum net radiation ranged from 350 to 720 W m-2 and air temperature ranged from 10 to 33°C (Figure 5). The soil temperature was nearly constant at 20°C. Wind speed ranged from 0.3 to 8 m s-1 but was generally less than 6 m s-1. The soil water content was about 0.31 m3 m-3 and the residue density was 12.2 ton/ha on June 24, 2003. Precipitation totaling 3mm occurred on the fifth day. The predicted rate of ET estimated with the SEB model was close to the observed data (Figure 6). Estimates were smaller than measured values for June 24, which was the hottest and windiest day of the period. The ability of the model to partition ET into evaporation and transpiration for partial canopy conditions is also illustrated in Figure 6. Evaporation from the soil represented the majority of the water used during the night, and early or late in the day. During the middle of the day transpiration represented approximately half of the hourly ET flux.

Fig. 3. Environmental conditions during a five-day period without canopy cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec.), vapor pressure deficit (VPD), and wind speed (u).

m3 m-3 and the residue density was 12.2 ton/ha on June 24, 2003. Precipitation totaling 3mm occurred on the fifth day. The predicted rate of ET estimated with the SEB model was close to the observed data (Figure 6). Estimates were smaller than measured values for June 24, which was the hottest and windiest day of the period. The ability of the model to partition ET into evaporation and transpiration for partial canopy conditions is also illustrated in Figure 6. Evaporation from the soil represented the majority of the water used during the night, and early or late in the day. During the middle of the day transpiration represented

*LAI = 0*

0 8

6/6 6/7 6/8 6/9 6/10 6/11

**Date**

**Rn**

0

1

2

3

4

**Precipitation, mm**

5

6

7

0

10

20

**Temperature, oC** 

30

40

Fig. 3. Environmental conditions during a five-day period without canopy cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec.), vapor

**Tm**

**Ta**

6/6 6/7 6/8 6/9 6/10 6/11

**Date**

pressure deficit (VPD), and wind speed (u).

**W m-2**

approximately half of the hourly ET flux.

Prec. u VPD

5

10

15

20

**VPD (mb) and Wind Speed (m s-1)**

25

30

35

40

The last period represents a fully developed maize canopy that completely shaded the soil surface. The crop height was 2.3 m and the LAI was 5.4. Environmental conditions for the period are given in Figure 7. The maximum net radiation ranged from 700 to 740 W m-2 and air temperature ranged from 15 to 36 ºC during the period. Soil temperature was fairly constant during the five days at 21.5°C and wind speed ranged from 0.3 to 4 m s-1. The soil water content was about 0.25 m3 m-3 and the residue density was 11.8 ton/ha on July 16, 2003. Precipitation totaling 29 mm occurred on the third day. Observed and predicted ET fluxes agreed for most days with some differences early in the morning during the first day and during the middle of several days (Figure 8). Transpiration simulated with the SEB model was nearly equal to the simulated ET for the period as evaporation rates from the soil was very small.

Fig. 4. Evapotranspiration estimated by the Surface Energy Balance (SEB) model and measured by an eddy covariance system and simulated cumulative evaporation from bare and residue-covered soil for a period without plant canopy cover.

Fig. 5. Environmental conditions for a five-day period with partial crop cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec), vapor pressure deficit (VPD), and wind speed (u).

*LAI = 1.5*

0 8

6/24 6/25 6/26 6/27 6/28 6/29

**Date**

*LAI = 1.5* **Rn**

**Tm**

**Ta**

0

1

2

3

4

**Precipitation, mm**

5

6

7

0

10

20

**Temperature, oC** 

30

40

Fig. 5. Environmental conditions for a five-day period with partial crop cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec), vapor

6/24 6/25 6/26 6/27 6/28 6/29

**Date**

pressure deficit (VPD), and wind speed (u).

**W m-2**

5

10

15

20

**VPD (mb) and Wind Speed (m s-1)**

25

30

35

Prec. u VPD

40

Fig. 6. Evapotranspiration and transpiration estimated by the Surface Energy Balance (SEB) model and ET measured by an eddy covariance system for a 5-day period with partial canopy cover.

Hourly measurements and SEB predictions for the three five-day periods were combined to evaluate the overall performance of the model (Figure 9). Results show variation about the 1:1 line; however, there is a strong correlation and the data are reasonably well distributed about the line. Modeled ET is less than measured for latent heat fluxes above 450 W m-2. The model underestimates ET during hours with high values of vapor pressure deficit (Figure 6 and 8), this suggests that the linear effect of vapor pressure deficit in canopy resistance estimated with equation (30) produce a reduction on ET estimations. Further work is required to evaluate and explore if different canopy resistance models improve the performance of ET predictions under these conditions. Various statistical techniques were used to evaluate the performance of the model. The coefficient of determination, Nash-Sutcliffe coefficient, index of agreement, root mean square error and the mean absolute error were used for model evaluation (Legates & McCabe 1999; Krause et al., 2005; Moriasi et al., 2007; Coffey et al. 2004). The coefficient of determination was 0.92 with a slope of 0.90 over the range of hourly ET values. The root mean square error was 41.4 W m-2, the mean absolute error was 29.9 W m-2, the Nash-Sutcliffe coefficient was 0.92 and the index of agreement was 0.97. The statistical parameters show that the model represents field measurements reasonably well. Similar performance was obtained for daily ET estimations (Table 1). Analysis is underway to evaluate the model for more conditions and longer periods. Simulations reported here relied on literature-reported parameter values. We are also exploring calibration methods to improve model performance.

Fig. 7. Environmental conditions for 5-day period with full canopy cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec), vapor pressure deficit (VPD) and wind speed (u).

0 40

7/16 7/17 7/18 7/19 7/20 7/21

**Date**

**Ta**

**Rn**

0

5

*LAI = 5.4*

10

15

20

**Precipitation, mm**

25

30

35

**Temperature (oC)** 

Fig. 7. Environmental conditions for 5-day period with full canopy cover for net radiation (Rn), air temperature (Ta), soil temperature (Tm), precipitation (Prec), vapor pressure deficit

7/16 7/17 7/18 7/19 7/20 7/21

**Date**

(VPD) and wind speed (u).

**W m-2**

**Tm**

5

10

15

20

**VPD (mb) and Wind Speed (m s-1)**

25

30

35

40

Prec u VPD

Fig. 8. Evapotranspiration and transpiration estimated by the Surface Energy Balance (SEB) model and ET measured by an eddy covariance system during a period with full canopy cover.

Fig. 9. Measured versus modeled hourly latent heat fluxes.


Table 1. Daily evapotranspiration estimated with the Surface Energy Balance (SEB) model and measured from the Eddy Covariance (EC) system.

#### **2.2 The modified SEB model for Partially Vegetated surfaces (SEB-PV)**

Although good performance of multiple-layer models has been recognized, multiple-layer models estimate more accurate ET values under high LAI conditions. Lagos (2008) evaluated the SEB model for maize and soybean under rainfed and irrigated conditions; results indicate that during the growing season, the model more accurately predicted ET after canopy closure (after LAI=4) than for low LAI conditions. The SEB model, similar to S-W and C-M models, is based on homogeneous land surfaces. Under low LAI conditions, the land surface is partially covered by the canopy and soil evaporation takes place from soil below the canopy and areas of bare soil directly exposed to net radiation. However, in multiple-layer models, evaporation from the soil has been only considered below the canopy and hourly variations in the partitioning of net radiation between the canopy and the soil is often disregarded. Soil evaporation on partially vegetated surfaces & inorchards and natural vegetation include not only soil evaporation beneath the canopy but also evaporation from areas of bare soil that contribute directly to total ET.

Recognizing the need to separate vegetation from soil and considering the effect of residue on evaporation, we extended the SEB model to represent those common conditions. The modified model, hereafter the SEB-PV model, distributes net radiation (Rn), sensible heat (H), latent heat (E), and soil heat fluxes (G) through the soil/residue/canopy system. Similar to the SEB model, horizontal gradients of the potentials are assumed to be small enough for lateral fluxes to be ignored, and physical and biochemical energy storage terms in the canopy/residue/soil system are assumed to be negligible. The evaporation of water on plant leaves due to rain, irrigation or dew is also ignored.

The SEB-PV model has the same four layers described previously for SEB (Figure 10):the first extended from the reference height above the vegetation and the sink for momentum within the canopy, a second layer between the canopy level and the soil surface, a third

Date m2 m-2 SEB EC 6-Jun 0 3.2 3.7 7-Jun 0 0.7 1.4 8-Jun 0 2.3 3.2 9-Jun 0 3.5 2.7 10-Jun 0 2.4 3.5 24-Jun 1.5 2.9 4.4 25-Jun 1.5 1.7 2.1 26-Jun 1.5 4.1 4.3 27-Jun 1.5 4.0 5.0 28-Jun 1.5 3.8 4.7 16-Jul 5.4 5.1 5.1 17-Jul 5.4 5.8 6.8 18-Jul 5.4 5.2 5.0 19-Jul 5.4 5.0 4.1 20-Jul 5.4 5.1 5.4 Table 1. Daily evapotranspiration estimated with the Surface Energy Balance (SEB) model

and measured from the Eddy Covariance (EC) system.

**2.2 The modified SEB model for Partially Vegetated surfaces (SEB-PV)** 

evaporation from areas of bare soil that contribute directly to total ET.

on plant leaves due to rain, irrigation or dew is also ignored.

Although good performance of multiple-layer models has been recognized, multiple-layer models estimate more accurate ET values under high LAI conditions. Lagos (2008) evaluated the SEB model for maize and soybean under rainfed and irrigated conditions; results indicate that during the growing season, the model more accurately predicted ET after canopy closure (after LAI=4) than for low LAI conditions. The SEB model, similar to S-W and C-M models, is based on homogeneous land surfaces. Under low LAI conditions, the land surface is partially covered by the canopy and soil evaporation takes place from soil below the canopy and areas of bare soil directly exposed to net radiation. However, in multiple-layer models, evaporation from the soil has been only considered below the canopy and hourly variations in the partitioning of net radiation between the canopy and the soil is often disregarded. Soil evaporation on partially vegetated surfaces & inorchards and natural vegetation include not only soil evaporation beneath the canopy but also

Recognizing the need to separate vegetation from soil and considering the effect of residue on evaporation, we extended the SEB model to represent those common conditions. The modified model, hereafter the SEB-PV model, distributes net radiation (Rn), sensible heat (H), latent heat (E), and soil heat fluxes (G) through the soil/residue/canopy system. Similar to the SEB model, horizontal gradients of the potentials are assumed to be small enough for lateral fluxes to be ignored, and physical and biochemical energy storage terms in the canopy/residue/soil system are assumed to be negligible. The evaporation of water

The SEB-PV model has the same four layers described previously for SEB (Figure 10):the first extended from the reference height above the vegetation and the sink for momentum within the canopy, a second layer between the canopy level and the soil surface, a third

LAI Evapotranspiration (mm day-1)

layer corresponding to the top soil layer and a lower soil layer where the soil atmosphere is saturated with water vapor.

Total latent heat (E) is the sum of latent heat from the canopy (Ec), latent heat from the soil (Es) beneath the canopy, latent heat from the residue-covered soil (Er) beneath the canopy, latent heat from the soil (Ebs) directly exposed to net radiation and latent heat from the residue-covered soil (Ebr) directly exposed to net radiation.

$$
\lambda \mathbf{E} = [\lambda \mathbf{E}\_{\rm c} + \lambda \mathbf{E}\_{\rm s} (\mathbf{1} - \mathbf{f}\_{\rm r}) + \lambda \mathbf{E}\_{\rm r} \mathbf{f}\_{\rm r}] \mathbf{F}\_{\rm V} + [\lambda \mathbf{E}\_{\rm bs} (\mathbf{1} - \mathbf{f}\_{\rm r})] (\mathbf{1} - \mathbf{F}\_{\rm V}) \tag{37}
$$

Where fr is the fraction of the soil affected by residue and Fv is the fraction of the soil covered by vegetation. Similarly, sensible heat is calculated as the sum of sensible heat from the canopy (Hc), sensible heat from the soil (Hs) and sensible heat from the residue covered soil (Hr), sensible heat from the soil (bs) directly exposed to net radiation and latent heat from the residue-covered soil (Hbr) directly exposed to net radiation.

$$\mathbf{H} = \begin{bmatrix} \mathbf{H}\mathbf{c} + \mathbf{H}\mathbf{s}\left(\mathbf{1} - \mathbf{f}\mathbf{r}\right) + \mathbf{H}\mathbf{r}\mathbf{f} \end{bmatrix} \mathbf{F}\mathbf{v} + \begin{bmatrix} \mathbf{H}\mathbf{b}\mathbf{s}\left(\mathbf{1} - \mathbf{f}\mathbf{r}\right) + \mathbf{H}\mathbf{b}\mathbf{r}\mathbf{f}\mathbf{f} \end{bmatrix} \begin{Bmatrix} \mathbf{1} - \mathbf{F}\mathbf{v} \end{Bmatrix} \tag{38}$$

For the fraction of the soil covered by vegetation, the total net radiation is divided into that absorbed by the canopy (Rnc) and the soil beneath the canopy (Rns) and is given by Rn = Rnc + Rns. The net radiation absorbed by the canopy is divided into latent heat and sensible heat fluxes as Rnc = Ec + Hc. Similarly, for the soil Rns = Gos + Hs, where Gos is a conduction term downwards from the soil surface and is expressed as Gos = Es + Gs, where Gs is the soil heat flux for bare soil. Similarly, for the residue covered soil Rns = Gor + Hr where Gor is the conduction downwards from the soil covered by residue. The conduction is given by Gor = Er + Gr where Gr is the soil heat flux for residue-covered soil. For the area without vegetation, total net radiation is divided into latent and sensible heat fluxes as Rn = Ebs +Ebr + Hbs + Hbr.

The differences in vapor pressure and temperature between levels can be expressed with an Ohm's law analogy using appropriate resistance and flux terms (Figure 10). Latent and sensible flux terms with in the resistance network were combined and solved to estimate total fluxes. The solution gives the latent and sensible heat fluxes from the canopy, the soil beneath the canopy and the soil covered by residue beneath the canopy similar to equations (9), (10), (11), (12) and (13).

The new expressions for latent heat flux of bare soil and soil covered by residue, both directly exposed to net radiation are:

For bare soil:

$$\lambda \mathbf{E\_{bs}} = \frac{\{\mathbf{R\_{n}} \cdot \Delta \cdot (\mathbf{r\_{2b}}) \cdot \mathbf{r\_{L}} + \mathbf{p} \cdot \mathbb{C\_{p}} \cdot \left\{ (\mathbf{e\_{b}}^{\*} - \mathbf{e\_{b}}) \cdot \mathbf{r\_{u}} + \mathbf{r\_{L}} + \mathbf{r\_{2b}} \right\} + (\mathbf{T\_{m}} - \mathbf{T\_{b}}) \cdot \Delta \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2b}})}{\chi \cdot (\mathbf{r\_{2b}} + \mathbf{r\_{s}}) \cdot (\mathbf{r\_{u}} + \mathbf{r\_{L}} + \mathbf{r\_{2b}}) + \Delta \cdot \mathbf{r\_{L}} \cdot (\mathbf{r\_{u}} + \mathbf{r\_{2b}})} \tag{39}$$

For residue covered soil:

$$\lambda \mathbf{E}\_{\rm tr} = \frac{\mathbf{R}\_{\rm n} \cdot \Delta \cdot \left(\mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm th}\right) \cdot \mathbf{r}\_{\rm L} + \rho \cdot \mathbf{C}\_{\rm p} \cdot \left(\left(\mathbf{e}\_{\rm b}^{\ast} - \mathbf{e}\_{\rm b}\right) \cdot \left(\mathbf{r}\_{\rm u} + \mathbf{r}\_{\rm L} + \mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm th}\right) + \left(\mathbf{T}\_{\rm m} - \mathbf{T}\_{\rm b}\right) \cdot \Delta \cdot \left(\mathbf{r}\_{\rm u} + \mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm r}\right)\right)}{\sqrt{\cdot \left(\mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm s} + \mathbf{r}\_{\rm r}\right) \cdot \left(\mathbf{r}\_{\rm u} + \mathbf{r}\_{\rm L} + \mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm th}\right) + \Delta \cdot \mathbf{r}\_{\rm L} \cdot \left(\mathbf{r}\_{\rm u} + \mathbf{r}\_{\rm 2b} + \mathbf{r}\_{\rm th}\right)}} \tag{40}$$

These relationships define the surface energy balance model, which is applicable to conditions ranging from closed canopies to surfaces partially covered by vegetation. If Fv = 1 the model SEB-PV is similar to the original SEB model and with Fv=1 without residue, the model is similar to that by Choudhury and Monteith (1988).

Fig. 10. Schematic resistance network of the modified Surface Energy Balance (SEB - PV) model for partially vegetated surfaces a) Sensible heat flux and b) Latent heat flux.

#### **2.2.1 Model resistances**

294 Evapotranspiration – Remote Sensing and Modeling

Fig. 10. Schematic resistance network of the modified Surface Energy Balance (SEB - PV) model for partially vegetated surfaces a) Sensible heat flux and b) Latent heat flux.

Model resistances are similar to those described by the SEB model; however, a new aerodynamic resistance (r2b) for the transfer of heat and water flux is required for the surface without vegetation.

The aerodynamic resistance between the soil surface and Zm (r2b) could be calculated by assuming that the soil directly exposed to net radiation is totally unaffected by adjacent vegetation as:

$$\mathbf{r}\_{\rm as} = \frac{\ln\left(\frac{\mathbf{Z\_m}}{\mathbf{Z\_0}}\right)^2}{\mathbf{k^2 u}}\tag{41}$$

According to Brenner and Incoll (1997), actual aerodynamic resistance (r2b) will vary between ras for Fv=0 and r2 when the fractional vegetative cover Fv=1. The form of the functional relationship of this change is not known, r2b was varied linearly between ras and r2 as:

$$\mathbf{r\_{2b}} = \text{FV(r\_2)} + (\mathbf{1} - \text{FV}) \text{(r\_{as})} \tag{42}$$

#### **2.2.2 Model inputs**

The proposed SEB-PV model requires the same inputs of the SEB model plus the fraction of the surface covered by vegetation (Fv).

#### **2.3 Sensitivity analysis**

A sensitivity analysis was performed to evaluate the response of the SEB model to changes in resistances and model parameters. Meteorological conditions, crop characteristics and soil/residue characteristics used in these calculations are given in Table 2. Such conditions are typical for midday during the growing season of maize in southeastern Nebraska. The sensitivity of total latent heat from the system was explored when model resistances and model parameters were changed under different LAI conditions. The effect of the changes in model parameters and resistances were expressed as changes in total ET (λE) and changes in the crop transpiration ratio. The transpiration ratio is the ratio between crop transpiration (Ec) over total ET (transpiration ratio= Ec / E).

The response of the SEB model was evaluated for three values of the extinction coefficient (Cext = 0.4, 0.6 and 0.8), three conditions of vapor pressure deficit (VPDa = 0.5 kPa, 0.1 kPa and 0.25 kPa) three soil temperatures (Tm=21°C, 0.8xTm=16.8 °C and 1.2xTm=25.2 °C) (Figure 11), changes in the parameterization of aerodynamic resistances (the attenuation coefficient, = 1, 2.5 and 3.5), the mean boundary layer resistance, rb (±40% ) the crop height, h (±30%)), selected conditions for the soil surface resistance, rs ( 0, 227, and 1500 s m-1) (Figure 12), four values for residue resistance, rr (0, 400, 1000, and 2500 s m-1), and changes of ±30% in surface canopy resistance, rc (Figure 13).

In general, the sensitivity analysis of model resistances showed that simulated ET was most sensitive to changes in surface canopy resistance for LAI > 0.5 values, and soil surface resistance and residue surface resistance for small LAI values (LAI < ~3). The model was less sensitive to changes in the other parameters evaluated.


Table 2. Predefined conditions for the sensitivity analysis.

**Variable Symbol Value Unit**  Net Radiation Rn 500 W m-2 Air temperature Ta 25 oC Relative humidity RH 68 % Wind speed U 2 m s-1 Soil Temperature at 0.5 m Tm 21 oC Solar radiation Rad 700 W m-2

Canopy resistance coeff. C1, C2, C3 5, 0.005, 300

Soil tortuosity s 1.5 Residue fraction Fr 0.5

Residue tortuosity r 1 Residue porosity r 1

Drag coefficient Cd 0.07

Attenuation coefficient 2.5

Extinction coefficient Cext 0.6

Fitting parameter 6.5

Table 2. Predefined conditions for the sensitivity analysis.

Soil thermal conductivity, upper

Soil thermal conductivity, lower

Maximum leaf area index LAImax 6 m2 m-2 Soil water content 0.25 m3 m-3 Saturation soil water content s 0.5 m3 m-3 Soil porosity 0.5 m3 m-3

Thickness of the residue layer Lr 0.02 m

Upper layer thickness Lt 0.05 m Lower layer depth Lm 0.5 m Soil roughness length Zo' 0.01 m

Reference height Z 3 m

Mean leaf width W 0.08 m Water vapor diffusion coefficient Dv 2.56x10-5 m2 s-1

Maximum solar radiation Radmax 1000 W m-2

layer K 2.8 W m-1oC-1

layer K' 3.8 W m-1oC-1

Fig. 11. Sensitivity analysis of the SEB-PV model for Fv=1 (left) and Fv=0,5 (right) under different soil temperatures Tm, and soil resistance conditions.

Fig. 12. Sensitivity analysis of the SEB-PV model for Fv=1 (left) and Fv=0,5 (right) under different residue and canopy conditions.

## **3. Conclusions**

298 Evapotranspiration – Remote Sensing and Modeling

Fig. 12. Sensitivity analysis of the SEB-PV model for Fv=1 (left) and Fv=0,5 (right) under

different residue and canopy conditions.

A surface energy balance model (SEB) based on the Shuttleworth-Wallace and Choudhury-Monteith models was developed to account for the effect of residue, soil evaporation and canopy transpiration on ET. The model describes the energy balance of vegetated and residue-covered surfaces in terms of driving potential and resistances to flux. Improvements in the SEB model were the incorporation of residue into the energy balance and modification of aerodynamic resistances for heat and water transfer, canopy resistance for water flux, residue resistance for heat and water flux, and soil resistance for water transfer. The model requires hourly data for net radiation, solar radiation, air temperature, relative humidity, and wind speed. Leaf area index and crop height plus soil texture, temperature and water content as well as the type and amount of crop residue are also required. An important feature of the model is the ability to estimate latent, sensible and soil heat fluxes. The model provides a method for partitioning ET into soil/residue evaporation and plant transpiration, and a tool to estimate the effect of residue ET on water balance studies. Comparison between estimated ET and measurements from an irrigated maize field provide support for the validity of the surface energy balance model. Further evaluation of the model is underway for agricultural and natural ecosystems during growing seasons and dormant periods. We are developing calibration procedures to refine parameters and improve model results.

The SEB model was modified for modeling evapotranspiration of partially vegetated surfaces given place to the SEB-PV model. The SEB-PV model can be used for partitioning total ET on canopy transpiration and soil evaporation beneath the canopy and soil directly exposed to net radiation. The model can be used for partitioning net radiation into not only latent heat fluxes but also sensible heat fluxes from each surface. A preliminary sensitivity analysis shows that similar to the SEB model, the proposed modification was sensitive to soil surface resistance, residue resistance, canopy resistance and vapor pressure deficit. Further model evaluation is needed to test this approach. A model to estimate Rn and a model to estimate soil temperature Tm from air temperature and soil conditions are also required to reduce the required inputs of the model.

## **4. List of variables**



TL Soil temperature at the interface between the upper and lower layers for bare soil (oC). TLr Soil temperature at the interface between the upper and lower layers for residue-

eLr\* Saturated vapor pressure at the top of the wet layer for the residue-covered soil (mb).

r1 Aerodynamic resistance between the canopy and the air at the canopy level (s m-1).

ras Aerodynamic resistance between the soil surface and Zm totally unaffected by

r2 Aerodynamic resistance between the soil and the air at the canopy level (s m-1). r2b Actual aerodynamic resistance between the soil surface and Zm (s m-1).

Gr Soil heat flux for residue-covered soil (W m-2). fr Fraction of the soil covered by residue (0-1).

T2r Soil surface temperature below the residue (oC).

Tm Soil temperature at the bottom of the lower layer (oC).

eL\* Saturated vapor pressure at the top of the wet layer (mb). eb\* Saturated vapor pressure at the canopy level (mb).

ram Aerodynamic resistance for momentum transfer (s m-1). rah Aerodynamic resistance for heat transfer (s m-1). raw Aerodynamic resistance for water vapor (s m-1). rbh Excess resistance term for heat transfer (s m-1). rbw Excess resistance term for water vapor (s m-1).

eb Vapor pressure of the air at the canopy level (mb). e1\* Saturated vapor pressure at the canopy (mb).

ea\* Saturated vapor pressure of the air (mb).

rb Boundary layer resistance (s m-1).

rr Residue resistance for water vapor flux (s m-1). rs Soil surface resistance for water vapor flux (s m-1). rrh Residue resistance to transfer of heat (s m-1). rr Residue resistance for heat flux (s m-1).

ru Soil heat flux resistance for the upper layer (s m-1). rL Soil heat flux resistance for the lower layer (s m-1). ∆ Slope of the saturation vapor pressure (mb oC-1).

LAImax Maximum value of leaf area index (m2 m-2).

zr Reference height above the canopy (m).

zo' Roughness length of the soil surface (m).

 adjacent vegetation (s m-1). rc Surface canopy resistance (s m-1).

h Vegetation height (m). LAI Leaf area index (m2 m-2).

Zm Reference height (m).

d Zero plane displacement (m).

zo Surface roughness length (m).

ρ Density of moist air (Kg m-3). Cp Specific heat of air (J Kg-1 oC-1). γ Psychrometric constant (Kpa °C-1).

Tb Air temperature at canopy height (oC).

Ta Air temperature (oC).

covered soil (oC).

T1 Canopy temperature (oC). T2 Soil surface temperature (oC).

ea Vapor pressure of the air (mb).


Hbr Latent heat from the residue-covered soil (W m-2).

## **5. Acknowledgments**

We thank the University of Nebraska Program of Excellence, the University of Nebraska-Lincoln Institute of Agriculture and Natural Resources, Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT 11100083) and Fondo de Fomento al Desarrollo Cientifico y Tecnologico (FONDEF D09I1146) Their support is gratefully recognized.

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