**2.1 Introduction**

ET is an important component in the climate system, and development of ET method has been studied by many researchers. As a result, there are many classical methods available for ET estimation based on data availability and required accuracy. One approach to estimate ET directly is the complementary relationship (CR) developed by [3]. Ref. [3] postulated that the decrease in evapotranspiration is matched by an equivalent increase in potential evapotranspiration (ETP) which is evaporation from a saturated surface, while energy and atmospheric conditions do not change. This idea has been widely tested in conjunction with the models of Priestley and Taylor [6] and Penman [1]. Among examples of widely known models, this study has focused on Granger and Gray [4] model because their model can directly estimate ET without the surface parameters or prior estimates of ETP. Furthermore, Ref. [5] extended the Granger and Gray [4] model to propose refinements to better predict regional ET especially under dry conditions and different land cover conditions. While the results of Anayah and Kaluarachchi [5] were very good, the authors also showed that further refinements can improve performance under dry conditions. In addressing the limitation of Anayah and Kaluarachchi [5] model which is named as the modified GG hereafter, this chapter is therefore to extend the modified GG model using a remote sending data, and this study is still committed to use minimal data such as meteorological data and other readily accessible information with no local calibration.

## **2.2 Methodology**

In the CR developed by [3], ET is usually calculated by Eq. (1):

$$ET + ETP = \text{2ETW} \tag{1}$$

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given in Eq. (6):

*An Advanced Evapotranspiration Method and Application*

calculate EWT instead of the Penman [1] equation:

*ETW* = *α*

mm/d.

of [3], Anayah and Kaluarachchi [5] developed their model using a three-step approach. First, they evaluated the original complementary methods under a variety of physical and climate conditions and developed 39 different model combinations. Second, three model variations were identified based on performance compared to observed data from a set of global sites. Third, a statistical analysis was conducted to contrast and compare the three models to identify the best (see detail in reference). Most importantly, the performance of the modified GG model increased by using the Priestley and Taylor [6] equation as shown in Eq. (2) to

\_\_\_\_ Δ

where ETW is in mm/d, *α* is a coefficient equal to 1.28, *Rn* is net radiation in mm/d, *γ* is the psychrometric constant in kPa/°C, *Δ* is the rate of change of saturation vapor pressure with temperature kPa/°C, and *Gsoil* is soil heat flux density in

Also, there are two parameters: relative drying power (D) and relative evapora-

*Ea* = 0.35(1 + 0.54*U*)(*es* − *ea*) (4)

where U is wind speed at 2 m above ground level that needs adjustments and conducted using the procedure described by [2], *es* is saturation vapor pressure in

where *c*1 is 1.0, *c*2 is 0.028, and *c*3 is 8.045. The effect of *Gsoil* is negligible com-

Solving Eq. (5) for ETP and substituting in Eq. (1), the modified GG model is

Therefore, the modified GG model of Anayah and Kaluarachchi [5] can estimate

In the modified GG model, the ratio of ET to ETP is defined as relative evaporation, G, as shown in Eq. (5), and parameter G was empirically derived using limited data from wet environments in Western Canada [4]. This bias towards wet region data may be the reason for relatively poor estimations with the modified GG model under dry conditions. In order to improve the ET predictions of the modified GG model, parameter G needs improvement. For this purpose, we use the theoretical framework of Budyko [8] on the basis of that the CR is consistent with the Budyko

*ETP* <sup>=</sup> \_\_\_\_\_\_\_ <sup>1</sup>

tion (G). D and G are described in Eqs. (3) and (5), respectively:

where *Ea* is drying power of air in mm/d given in Eq. (4)

pared to *Rn* when calculated at monthly or higher time scale [7].

*D* = \_\_\_\_\_\_\_\_\_\_\_ *Ea*

mmHg, and *ea* is vapor pressure of air in mmHg:

*ET* = \_\_\_\_ <sup>2</sup>*<sup>G</sup>*

ET directly without calculating ETP.

*G* = \_\_\_\_ *ET*

*<sup>γ</sup>* <sup>+</sup> <sup>Δ</sup>(*Rn* <sup>−</sup> *Gsoil*) (2)

*Ea* <sup>+</sup> (*Rn* <sup>−</sup> *Gsoil*) (3)

*<sup>c</sup>*<sup>1</sup> <sup>+</sup> *<sup>c</sup>*<sup>2</sup> *<sup>e</sup><sup>c</sup>*3*<sup>D</sup>* (5)

*<sup>G</sup>* <sup>+</sup> <sup>1</sup> *ETW* (6)

*DOI: http://dx.doi.org/10.5772/intechopen.81047*

where ETP is evaporation from a saturated surface and ETW is the value of potential evaporation when ET is equal to the potential rate. Based on the idea

*An Advanced Evapotranspiration Method and Application DOI: http://dx.doi.org/10.5772/intechopen.81047*

*Advanced Evapotranspiration Methods and Applications*

monitoring through a new drought index.

information with no local calibration.

In the CR developed by [3], ET is usually calculated by Eq. (1):

*ET* + *ETP* = 2*ETW* (1)

where ETP is evaporation from a saturated surface and ETW is the value of potential evaporation when ET is equal to the potential rate. Based on the idea

**2.2 Methodology**

**evapotranspiration**

**2.1 Introduction**

wide range of available energy to estimate ET. Bouchet [3] postulated that the decrease in ET is matched by an equivalent increase in ETP as a surface dries. Later, Granger and Gray [4] model named as the GG model is one of the widely known models using the CR because it requires only meteorological data. Recently, Ref. [5] modified the GG model with meteorological data from 34 global eddy covariance sites. While the results were very good as compared other published ET methods, they mentioned that further refinements can improve performance under dry conditions. A probable reason is that the original GG model was empirically derived from wet biased environments in Canada. Taking this limitation into account, the model development was designed to extend the latest CR model using both meteorological data and NDVI. We then will validate the proposed model with other ET methods including a remote sensing model. Finally, we will address the possibility of using ET as a proxy for drought

**2. Development of complementary relationship model for estimating** 

ET is an important component in the climate system, and development of ET method has been studied by many researchers. As a result, there are many classical methods available for ET estimation based on data availability and required accuracy. One approach to estimate ET directly is the complementary relationship (CR) developed by [3]. Ref. [3] postulated that the decrease in evapotranspiration is matched by an equivalent increase in potential evapotranspiration (ETP) which is evaporation from a saturated surface, while energy and atmospheric conditions do not change. This idea has been widely tested in conjunction with the models of Priestley and Taylor [6] and Penman [1]. Among examples of widely known models, this study has focused on Granger and Gray [4] model because their model can directly estimate ET without the surface parameters or prior estimates of ETP. Furthermore, Ref. [5] extended the Granger and Gray [4] model to propose refinements to better predict regional ET especially under dry conditions and different land cover conditions. While the results of Anayah and Kaluarachchi [5] were very good, the authors also showed that further refinements can improve performance under dry conditions. In addressing the limitation of Anayah and Kaluarachchi [5] model which is named as the modified GG hereafter, this chapter is therefore to extend the modified GG model using a remote sending data, and this study is still committed to use minimal data such as meteorological data and other readily accessible

**78**

of [3], Anayah and Kaluarachchi [5] developed their model using a three-step approach. First, they evaluated the original complementary methods under a variety of physical and climate conditions and developed 39 different model combinations. Second, three model variations were identified based on performance compared to observed data from a set of global sites. Third, a statistical analysis was conducted to contrast and compare the three models to identify the best (see detail in reference). Most importantly, the performance of the modified GG model increased by using the Priestley and Taylor [6] equation as shown in Eq. (2) to calculate EWT instead of the Penman [1] equation:

$$ETW = \alpha \frac{\Delta}{\gamma + \Delta} (R\_n - G\_{soil}) \tag{2}$$

where ETW is in mm/d, *α* is a coefficient equal to 1.28, *Rn* is net radiation in mm/d, *γ* is the psychrometric constant in kPa/°C, *Δ* is the rate of change of saturation vapor pressure with temperature kPa/°C, and *Gsoil* is soil heat flux density in mm/d.

Also, there are two parameters: relative drying power (D) and relative evaporation (G). D and G are described in Eqs. (3) and (5), respectively:

$$D = \frac{E\_a}{E\_a + (Rn - G\_{all})} \tag{3}$$

where *Ea* is drying power of air in mm/d given in Eq. (4)

$$E\_d = 0.35(1 + 0.54U)(e\_t - e\_d) \tag{4}$$

where U is wind speed at 2 m above ground level that needs adjustments and conducted using the procedure described by [2], *es* is saturation vapor pressure in mmHg, and *ea* is vapor pressure of air in mmHg:

$$\mathbf{G} = \frac{ET}{ET\mathbf{P}} = \frac{1}{c\_1 \star c\_2 e^{c\_1 D}}\tag{5}$$

where *c*1 is 1.0, *c*2 is 0.028, and *c*3 is 8.045. The effect of *Gsoil* is negligible compared to *Rn* when calculated at monthly or higher time scale [7].

Solving Eq. (5) for ETP and substituting in Eq. (1), the modified GG model is given in Eq. (6):

$$ET = \frac{2G}{G+1}ETW \tag{6}$$

Therefore, the modified GG model of Anayah and Kaluarachchi [5] can estimate ET directly without calculating ETP.

In the modified GG model, the ratio of ET to ETP is defined as relative evaporation, G, as shown in Eq. (5), and parameter G was empirically derived using limited data from wet environments in Western Canada [4]. This bias towards wet region data may be the reason for relatively poor estimations with the modified GG model under dry conditions. In order to improve the ET predictions of the modified GG model, parameter G needs improvement. For this purpose, we use the theoretical framework of Budyko [8] on the basis of that the CR is consistent with the Budyko

hypothesis through the Fu equation [9, 10]. The analytical solution of the Budyko framework is given in Eq. (7):

$$\frac{ET}{ETP} = \mathbf{1} + \frac{P}{ETP} - \left[\mathbf{1} + \left(\frac{P}{ETP}\right)^{\alpha}\right]^{1/\alpha} \tag{7}$$

where P is precipitation in mm and ETP is estimated using the Priestly and Taylor equation [6]. Parameter ω is constant and represents the land surface conditions, especially the vegetation cover [11]. Parameter ω is linearly correlated with the long-term average annual vegetation cover, and a model using NDVI can improve the estimation of ET (see details in [5]). Thus, Eq. (8) shows the Fu equation where parameter G is now defined as *Gnew*:

$$\mathbf{G}\_{new} = \frac{ET}{ETP} = \mathbf{1} + \frac{P}{ETP} - \left[\mathbf{1} + \left(\frac{P}{ETP}\right)^{ao}\right]^{1/a} \tag{8}$$

Note *Gnew* in Eq. (8) is required and can be estimated using the Penman [1] given in Eq. (9):

$$ETP = \frac{\Delta}{\Delta \star \gamma} (R\_n - G\_{soil}) + \frac{\gamma}{\gamma \star \Delta} E\_a \tag{9}$$

Having found *Gnew* from Eq. (8) and estimating ETW from Eq. (2), we can estimate ET from Eq. (10):

$$ET = \frac{2\ G\_{new}}{G\_{new} + 1} ETW \tag{10}$$

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**Figure 1.**

*Locations of 60 AmeriFlux sites used in phase 2 with number.*

*An Advanced Evapotranspiration Method and Application*

CRAE model. After we validated GG-NDVI with ground-based ET models, we also compared with a remote sensing model. Air temperature, elevation, and precipitation data were obtained from the Parameter-elevation Regressions on Independent Slopes Model (PRISM, http://www.prism.oregonstate.edu). As part of the input data for the GG-NDVI model, we used the 16-day NDVI data from MODIS (http://daac.ornl.gov/MODIS/modis.shtml). We also collected the lever 4 meteorological data including latent heat flux (LE) from 76 AmeriFlux stations, and then we excluded those stations with actual vegetation type different from the MODIS global land cover product (MOD12) at any of surrounding 500 m by 500 m spatial resolution. Also, we further excluded those stations with fewer than half a year of measurements during 2000–2007. As a result, 60 AmeriFlux stations were used in the comparison of the remote sending model as

We defined the climate class of each site using the aridity index of the United Nations Environment Programme (UNEP) proposed by [13]. The aridity index divided climate conditions to six classes: hyper-arid, arid, semiarid, dry subhumid, wet subhumid, and humid. However, this work simplified the climate class defini-

The CRAE model is considered as a simple, practical, and reliable model to estimate monthly ET [7]. The modified GG model had been validated by [5] that it showed better performance compared to the recently published works. Therefore, the phase 1 provides the opportunity to test both models compared to the proposed GG-NDVI model. The results of the comparison are given in **Table 1** and **Figure 2**. The GG-NDVI model showed the lowest mean RMSE across all models about 15 mm/month in dry sites and about 12 mm/month in wet sites. The results in general indicate that GG-NDVI can perform well in the dry conditions and even better

*DOI: http://dx.doi.org/10.5772/intechopen.81047*

shown in **Figure 1**.

**2.4 Results**

tion to two classes, dry and wet.

*2.4.1 Phase 1: validation*

Hereafter, this proposed model will be referred as the GG-NDVI model. This chapter used two phases to evaluate the performance of the proposed model. In phase 1, the GG-NDVI model compared with two CR models: the complementary relationship areal evapotranspiration (CRAE) model of [12] and the modified GG model of [5]. Moreover, comparisons are made between a commonly used remote sensing model and GG-NDVI model. In phase 2, a comparison of estimated ET from GG-NDVI with observed data from phase 1 will be performed to identify the weaknesses of the CR model, and appropriate corrections will be proposed.

#### **2.3 Data**

ET estimation from GG-NDVI was generated using meteorological data and NDVI. Meteorological data required are temperature, wind speed, precipitation, net radiation, and elevation (pressure). Among these, net radiation (*Rn*) was calculated using the equations by [2]. This chapter proposes to use data from AmeriFlux eddy covariance sites in the United States because the US sites have wide variety of climate and physical conditions and land cover especially in dry regions. In phase 1, although we selected 75 sites of Level 2 data of AmeriFlux with fewer than 50% missing data and these data were obtained from the Oak Ridge National Laboratory's website (http://ameriflux.ornl.gov/), we used only 59 sites since only these sites have incident global radiation data required by the

*An Advanced Evapotranspiration Method and Application DOI: http://dx.doi.org/10.5772/intechopen.81047*

CRAE model. After we validated GG-NDVI with ground-based ET models, we also compared with a remote sensing model. Air temperature, elevation, and precipitation data were obtained from the Parameter-elevation Regressions on Independent Slopes Model (PRISM, http://www.prism.oregonstate.edu). As part of the input data for the GG-NDVI model, we used the 16-day NDVI data from MODIS (http://daac.ornl.gov/MODIS/modis.shtml). We also collected the lever 4 meteorological data including latent heat flux (LE) from 76 AmeriFlux stations, and then we excluded those stations with actual vegetation type different from the MODIS global land cover product (MOD12) at any of surrounding 500 m by 500 m spatial resolution. Also, we further excluded those stations with fewer than half a year of measurements during 2000–2007. As a result, 60 AmeriFlux stations were used in the comparison of the remote sending model as shown in **Figure 1**.

We defined the climate class of each site using the aridity index of the United Nations Environment Programme (UNEP) proposed by [13]. The aridity index divided climate conditions to six classes: hyper-arid, arid, semiarid, dry subhumid, wet subhumid, and humid. However, this work simplified the climate class definition to two classes, dry and wet.
