**Stomatal Conductance Modeling to Estimate the Evapotranspiration of Natural and Agricultural Ecosystems**

Giacomo Gerosa1, Simone Mereu2, Angelo Finco3 and Riccardo Marzuoli1,4 *1Dipartimento Matematica e Fisica, Università Cattolica del Sacro Cuore, via Musei 41, Brescia 2Dipartimento di Economia e Sistemi Arborei, Università di Sassari, 3Ecometrics s.r.l., Environmental Monitoring and Assessment, via Musei 41, Brescia 4Fondazione Lombardia per l'Ambiente, Piazza Diaz 7, Milano Italy* 

## **1. Introduction**

26 Will-be-set-by-IN-TECH

402 Evapotranspiration – Remote Sensing and Modeling

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Canopy structure versus physiology effects on net photosynthesis of mountain

This chapter presents some of the available modelling techniques to predict stomatal conductance at leaf and canopy level, the key driver of the transpiration component in the evapotranspiration process of vegetated surfaces. The process-based models reported, are able to predict fast variations of stomatal conductance and the related transpiration and evapotranspiration rates, e.g. at hourly scale. This high–time resolution is essential for applications which couple the transpiration process with carbon assimilation or air pollutants uptake by plants.

## **2. Stomata as key drivers of plant's transpiration**

Evapotranspiration from vegetated areas, as suggested by the name, has two different components: evaporation and transpiration. Evaporation refers to the exchange of water from the liquid to the gaseous phase over living and non-living surfaces of an ecosystem, while transpiration indicates the process of water vaporisation from leaf tissues, i.e. the mesophyll cells of leaves. Both processes are driven by the available energy and the drying potential of the surrounding air, but transpiration depends also on the capacity of plants to replenish the leaf tissues with water coming from the roots through their hydraulic conduction system, the xylem. This capacity depends directly on soil water availability (i.e. soil water potential), which contributes to the onset of the water potential gradient within the soil-plant-atmosphere continuum.

Moreover, since the cuticle -a waxy coating covering the leaf surface- is nearly impermeable to water, the main part of leaf transpiration (about 95%) results from the diffusion of water vapour through the stomata. Stomata are little pores in the leaf lamina which provide lowresistance pathways to the diffusional movement of gases (CO2, H2O, air pollutants) from outside to inside the leaf and vice versa. Following complex signal pathways, environmental, osmotic and hormonal, stomata regulate their opening area and thus the water vapour loss from leaves. When the evaporative demand is bigger than the water replenishing capability from the xylem, stomata closes partially or even totally. High evaporative demands can be due to elevated air temperature, high leaf-to-air vapour pressure deficit (VPD), and intense winds. Stomatal closure can also be caused by high concentration of carbon dioxide in the mesophyll space.

Stomata, thus, directly control plant transpiration preventing plants from excessive drying, and acting as key drivers of water vapour movements from vegetated surfaces to the atmosphere.

This chapter illustrates the modelling techniques to predict the stomatal behaviour of vegetation at high-resolution time scale, and the related water fluxes.

## **3. Modelling stomatal behaviour: The Jarvis-Stewart model and the Ball-Berry model**

Stomata play an essential role in the regulation of both water losses by transpiration and CO2 uptake for photosynthesis and plant growth. Stomatal aperture is controlled by the turgor pressure difference between the guard cells surrounding the pore and the bulk leaf epidermis. In order to optimize CO2 uptake and water losses in rapidly changing environmental conditions, plants have evolved the ability to control stomatal aperture in the order of seconds. Stomatal aperture responds to multiple environmental factors such as, solar radiation, temperature, drought, VPD, wind speed, and sub-stomatal CO2 concentrations.

The availability of modern physiological instrumentation (diffusion porometers, gasexchange analyzers) has allowed to measure leaf stomatal conductance (*g*s) in field conditions and to study how environmental variables influence this parameter.

However, measurements of *g*s by porometers and gas-exchange analyzers can be made only when foliage is dry, and long-term enclosure in measuring chamber may lead to changes in the physiological state of the leaves. Consequently measurements in the field are usually made intensively over selected periods of a few hours in selected days.

Furthermore, stomatal conductance values depend also upon the physiological condition of the plant, which relates to the weather of the previous days as well as to the previous season for perennial species.

Therefore it is important to have continuous *g*s measurements over the whole vegetative season in order to improve the interpretation of other physiological data such as photosynthesis rate and carbon assimilation.

An alternative to very frequent measurements of *g*s in the field is to predict them from models that describe its dependency on environmental factors. These models can be parameterized using the available field measurements conducted on occasional periods.

Furthermore, modeling appears the most effective tool for integration, simulation and prediction purposes concerning the effects of climatic global change on vegetation.

Stomatal conductance is among the processes that have been most extensively modeled during the last decades. In their excellent review, Damour et al. (2010) describe 35 stomatal conductance models classified as:


The next paragraphs provides information on two early developed gs models which are currently among the most widely used: the multiplicative model of Jarvis (1976) based on climatic control and later modified by Stewart (1988), and the Ball Berry model (1988), based on *g*s-photosynthesis relationship.

### **3.1 The Jarvis-Stewart model**

404 Evapotranspiration – Remote Sensing and Modeling

outside to inside the leaf and vice versa. Following complex signal pathways, environmental, osmotic and hormonal, stomata regulate their opening area and thus the water vapour loss from leaves. When the evaporative demand is bigger than the water replenishing capability from the xylem, stomata closes partially or even totally. High evaporative demands can be due to elevated air temperature, high leaf-to-air vapour pressure deficit (VPD), and intense winds. Stomatal closure can also be caused by high

Stomata, thus, directly control plant transpiration preventing plants from excessive drying, and acting as key drivers of water vapour movements from vegetated surfaces to the

This chapter illustrates the modelling techniques to predict the stomatal behaviour of

**3. Modelling stomatal behaviour: The Jarvis-Stewart model and the Ball-Berry** 

Stomata play an essential role in the regulation of both water losses by transpiration and CO2 uptake for photosynthesis and plant growth. Stomatal aperture is controlled by the turgor pressure difference between the guard cells surrounding the pore and the bulk leaf epidermis. In order to optimize CO2 uptake and water losses in rapidly changing environmental conditions, plants have evolved the ability to control stomatal aperture in the order of seconds. Stomatal aperture responds to multiple environmental factors such as, solar radiation, temperature, drought, VPD, wind speed, and sub-stomatal CO2

The availability of modern physiological instrumentation (diffusion porometers, gasexchange analyzers) has allowed to measure leaf stomatal conductance (*g*s) in field

However, measurements of *g*s by porometers and gas-exchange analyzers can be made only when foliage is dry, and long-term enclosure in measuring chamber may lead to changes in the physiological state of the leaves. Consequently measurements in the field are usually

Furthermore, stomatal conductance values depend also upon the physiological condition of the plant, which relates to the weather of the previous days as well as to the previous season

Therefore it is important to have continuous *g*s measurements over the whole vegetative season in order to improve the interpretation of other physiological data such as

An alternative to very frequent measurements of *g*s in the field is to predict them from models that describe its dependency on environmental factors. These models can be parameterized using the available field measurements conducted on occasional periods. Furthermore, modeling appears the most effective tool for integration, simulation and

Stomatal conductance is among the processes that have been most extensively modeled during the last decades. In their excellent review, Damour et al. (2010) describe 35 stomatal

prediction purposes concerning the effects of climatic global change on vegetation.

conditions and to study how environmental variables influence this parameter.

made intensively over selected periods of a few hours in selected days.

concentration of carbon dioxide in the mesophyll space.

vegetation at high-resolution time scale, and the related water fluxes.

atmosphere.

**model** 

concentrations.

for perennial species.

photosynthesis rate and carbon assimilation.

conductance models classified as:

The stomatal conductance model developed by Jarvis (1976) can be defined as an empirical multiplicative model based on the observed responses of *g*s to environmental factors. The assumption of this model is that the influence of each environmental factor on *g*s is independent of the others and can be determined by boundary line analysis (Webb 1972).

The Jarvis model, in its first form, integrates the responses of *g*s to light intensity, leaf temperature, vapour pressure deficit, ambient CO2 concentration and leaf water potential, according to the following equation:

$$\mathbf{g}\_s = f(\mathbf{Q}) \cdot f(T\_l) \cdot f(VPD\_l) \cdot f(\mathbf{C\_a}) \cdot f(\mathbf{P}) \tag{1}$$

where *Q* is the quantum flux density (E m-2s-1), *T*l is the leaf temperature (°C), *VPD*l is the leaf-to-air vapour pressure deficit calculated at leaf temperature (kPa), *C*a is the ambient CO2 concentration (ppm) and is leaf water potential (MPa).

Stewart (1988) further implemented this model adopting the assumption that the functions of environmental variables have values between zero and unit and exert their influence reducing the maximum stomatal conductance of the plant (*g*smax), a species-specific value depending on leaf stomatal density, that can be defined as the largest value of conductance observed in fully developed leaves – but not senescent – of well-watered plants under optimal climatic conditions (Körner et al., 1979). This value can be derived from field measurements conducted under the above mentioned optimal conditions.

Furthermore, in Stewart formulation, quantum flux density is replaced by global solar radiation, leaf temperature by air temperature, leaf-to-air vapour pressure deficit by air vapour pressure deficit and leaf water potential by soil moisture deficit measured in the first meter of soil (i.e. soil water content, SWC). Stewart also omitted *f*(Ca) because the effect of CO2 ambient concentrations was considered negligible: this simplification allows for an easier data collection to run the model, but it must be kept in mind that Ca change considerably among seasons and thus the simplification may lead to a considerable error, especially when the model is used for annual *g*s behavior of evergreen species.

The model is defined by the following equation:

$$\mathcal{g}\_s = \mathcal{g}\_{\text{smax}} \cdot f(\mathbf{Q}) \cdot f(T\_\mathbf{a}) \cdot f(VPD\_\mathbf{a}) \cdot f(\text{SVC}) \tag{2}$$

It is important to notice that *f*(Q) can also be replaced by the more specific *f*(PAR), based on the photosynthetically active radiation.

Each function has a characteristic shape described by the following equations:

$$f(\underline{Q}) = 1 - \exp^{-u\underline{Q}}\tag{3}$$

$$f\left(T\_a\right) = \frac{\left(T - T\_{\rm min}\right)}{\left(T\_{opt} - T\_{\rm min}\right)} \left[\frac{\left(T\_{\rm max} - T\right)}{\left(T\_{\rm max} - T\_{opt}\right)}\right]^\rho \tag{4}$$

where max min ( ) ( ) *opt opt T T b T T* , *f*(*T*a) = 0.1 when *<sup>T</sup>* <sup>≤</sup> *<sup>T</sup>*min or *<sup>T</sup>* <sup>≥</sup>*T*max and *f*(*T*a) = 1 when *T* = *T*opt

$$f(VPD) = \frac{(1 - 0.1) \cdot (c - VPD)}{(c - d)} \tag{5}$$

where *f*(*VPD)*= 1 when VPD ≤ *d* and *f*(*VPD*) = 0.1 when VPD ≥ *c*

$$f\left(SWC\right) = 1 - \exp\left[k\left(SWC - SWC\_{\text{max}}\right)\right] \tag{6}$$

where *f*(SWC) = 1 when SWC=SWCmax

Since *g*s depends on four major variables, field measurements do not usually show a clear relationship with any of the considered variables. Often, *g*s is reduced below the value expected for a value of a single independent variable, as the result of the influences of the other variables. As a consequence, the coefficients of each function must be derived with boundary-line analysis, plotting all field measurements of relative gs (gsrel = gs/gsmax) against each environmental variable considered separately.

Provided that enough measurements have been adequately performed to cover variable space, the upper limit of the scatter diagram indicates the response of *g*s to the particular independent variable, when the other variables are not limiting.

An example of boundary-line analysis is reported in figure taken from Gerosa et al. (2009):

The main criticism formulated against this kind of approach is that the interactive effects between environmental factors are not properly taken into account, since interactions are only partially explained by the multiplicative nature of the model which simply multiplies concomitant effects, avoiding any synergistic interaction.

where max

*b*

min

where *f*(SWC) = 1 when SWC=SWCmax

each environmental variable considered separately.

where *f*(*VPD)*= 1 when VPD ≤ *d* and *f*(*VPD*) = 0.1 when VPD ≥ *c*

independent variable, when the other variables are not limiting.

concomitant effects, avoiding any synergistic interaction.

0

0.2

0.4

0.6

**g/gmax**

0.8

1

1.2

( ) ( ) *opt*

*opt T T*

min max min max ( ) ( ) ( ) ( )( ) *<sup>a</sup> opt opt* 

*T T* , *f*(*T*a) = 0.1 when *<sup>T</sup>* <sup>≤</sup> *<sup>T</sup>*min or *<sup>T</sup>* <sup>≥</sup>*T*max and *f*(*T*a) = 1 when *T* = *T*opt

(1 0.1) ( ) ( ) ( ) 

Since *g*s depends on four major variables, field measurements do not usually show a clear relationship with any of the considered variables. Often, *g*s is reduced below the value expected for a value of a single independent variable, as the result of the influences of the other variables. As a consequence, the coefficients of each function must be derived with boundary-line analysis, plotting all field measurements of relative gs (gsrel = gs/gsmax) against

Provided that enough measurements have been adequately performed to cover variable space, the upper limit of the scatter diagram indicates the response of *g*s to the particular

An example of boundary-line analysis is reported in figure taken from Gerosa et al. (2009): The main criticism formulated against this kind of approach is that the interactive effects between environmental factors are not properly taken into account, since interactions are only partially explained by the multiplicative nature of the model which simply multiplies

> 0 500 1000 1500 2000 **PAR [umol m-2 s-1]**

*b T T T T f T TT T T* (4)

*c VPD f VPD c d* (5)

*f SWC exp k SWC SWC* ( )1 ( max ) (6)

Fig. 1. Boundary-line analysis for the definition of *g*s limiting function parameters (modified from Gerosa et al. 2008)

#### **3.2 The Ball-Berry model**

The Ball-Berry empirical model describes the behaviour of *g*s as a function of environmental conditions and net photosynthetic rate. In its simplest form (Ball et al., 1987) the model states:

$$\mathbf{g}\_s = \mathbf{g}\_0 + a\_1 A n \frac{\mathbf{R}H}{\mathbf{C}\_s} \tag{7}$$

Where *g*s is the stomatal conductance to water vapour, *g*0 is the stomatal conductance at the light compensation point, *a*1 is a fitting parameter representing the slope of the equation, *An* is photosynthesis, *RH* is relative humidity and *C*s is the molar fraction of CO2 at the leaf surface. The model takes advantage of the feedback loop that exists between *A* and *g*<sup>s</sup> (Farquhar at al., 1978) implying that they are interdependent. Additionally, *An* and *g*s can respond independently to environmental variables and so they cannot be considered driving variables but rather state variables. The empirical relationship emerges from optimized vegetation behaviour that maximizes productivity (Patwardhan et al., 2006): the relationship, actually, corresponds roughly to the value of maximum surface conductance that maximizes productivity.

In order to derive *g*s, the model needs to be coupled with a photosynthesis model (most often the Farquhar biochemical model) from which *An* is calculated. In order to derive *g*<sup>s</sup> two equations must be solved simultaneously:

$$\mathbf{g}\_s = \mathbf{g}\_0 + a\_1 A n \frac{\mathcal{R} H}{\mathcal{C}\_s} \tag{8}$$

$$A = (\mathbb{C}\_s - \mathbb{C}\_i)\mathbb{g}\_s\tag{9}$$

The problem is often solved by reiteration of the two equations where *g*s at time *t*n+1 is computed with *An* at time *t*n and *An* at time *t*n+2 is computed using *g*s at time *t*+1. The reiteration approach however can give birth to oscillations in time of *g*s and *An* due to chaotic solution in particular conditions. However, Baldocchi et al. (1994) found an analytical solution for the set of equations that bypasses this problem.

The original Ball-Berry model (Ball et al., 1987) was further implemented by Leuning (1995), considering that stomata respond to vapour pressure deficit (VPD) rather than humidity. In its modified version the equation takes the form:

$$\mathbf{g}\_s = \mathbf{g}\_0 + \frac{a\_1 A n}{(\mathbf{C}\_s - \Gamma) \left(1 + \frac{VPD\_0}{VPD\_s}\right)}\tag{10}$$

Where, is the CO2 compensation point, *C*s and *VPD*s are the CO2 concentration and vapour pressure deficit at the leaf surface, and *VPD*0 is an empirical coefficient.

This model encapsulates two empirical trends reported in the literature. First, through the correlation between *g*s and *An* the equation predicts that the ratio (*C*i-)/(*C*s-) is largely independent of leaf irradiance and *C*s, except near the light and CO2 compensation points. It also predicts that *g*s declines linearly as *VPD*s increases, in fact through the relation

$$\mathbf{E} = \mathbf{1.6} \cdot \mathbf{g}\_s \cdot \text{VPD}\_s \tag{11}$$

The Ball-Berry empirical model describes the behaviour of *g*s as a function of environmental conditions and net photosynthetic rate. In its simplest form (Ball et al., 1987) the model

*s*

*s*

(7)

(8)

( ) *A C Cg s s <sup>i</sup>* (9)

E=1.6 · *g*s · *VPDs* (11)

(10)

*s* 0 1

*RH g g a An <sup>C</sup>*

Where *g*s is the stomatal conductance to water vapour, *g*0 is the stomatal conductance at the light compensation point, *a*1 is a fitting parameter representing the slope of the equation, *An* is photosynthesis, *RH* is relative humidity and *C*s is the molar fraction of CO2 at the leaf surface. The model takes advantage of the feedback loop that exists between *A* and *g*<sup>s</sup> (Farquhar at al., 1978) implying that they are interdependent. Additionally, *An* and *g*s can respond independently to environmental variables and so they cannot be considered driving variables but rather state variables. The empirical relationship emerges from optimized vegetation behaviour that maximizes productivity (Patwardhan et al., 2006): the relationship, actually, corresponds roughly to the value of maximum surface conductance

In order to derive *g*s, the model needs to be coupled with a photosynthesis model (most often the Farquhar biochemical model) from which *An* is calculated. In order to derive *g*<sup>s</sup>

*s* 0 1

analytical solution for the set of equations that bypasses this problem.

*s*

0

vapour pressure deficit at the leaf surface, and *VPD*0 is an empirical coefficient.

also predicts that *g*s declines linearly as *VPD*s increases, in fact through the relation

*RH g g a An <sup>C</sup>*

The problem is often solved by reiteration of the two equations where *g*s at time *t*n+1 is computed with *An* at time *t*n and *An* at time *t*n+2 is computed using *g*s at time *t*+1. The reiteration approach however can give birth to oscillations in time of *g*s and *An* due to chaotic solution in particular conditions. However, Baldocchi et al. (1994) found an

The original Ball-Berry model (Ball et al., 1987) was further implemented by Leuning (1995), considering that stomata respond to vapour pressure deficit (VPD) rather than humidity. In

*g g VPD <sup>C</sup>*

Where, is the CO2 compensation point, *C*s and *VPD*s are the CO2 concentration and

This model encapsulates two empirical trends reported in the literature. First, through the correlation between *g*s and *An* the equation predicts that the ratio (*C*i-)/(*C*s-) is largely independent of leaf irradiance and *C*s, except near the light and CO2 compensation points. It

1

*a An*

<sup>0</sup> ( )1

*<sup>s</sup> <sup>s</sup>*

*VPD*

**3.2 The Ball-Berry model** 

that maximizes productivity.

two equations must be solved simultaneously:

its modified version the equation takes the form:

states:

for the transpiration rate (*E*), the hyperbolic function of *VPD*s is equivalent to a linear decline of *g*s with increasing *E*.

The main limitation of the Ball-Berry-Leuning (BBL) model is its failure in describing stomatal closure in drought conditions. The model has been further implemented by Dewar (2002) to take SWC in consideration by coupling the BBL model with Tardieu model for stomatal response to drought. The coupled model takes the form:

$$\mathbf{g}\_s = \frac{a\_1(An + Rd)}{\mathrm{CI}\left(1 + \frac{VPD\_0}{VPDs}\right)} \exp\left\{-\left[ABA\right]\beta \exp(\delta\,\Psi)\right\} \tag{12}$$

Where *Rd* is dark respiration, [*ABA*] is the concentration of abscisic acid in the leaf xylem, is the leaf water potential, is the basal sensitivity of ion diffusion to [*ABA*] at zero leaf water potential, and describes the increase in the sensitivity of ion diffusion to [*ABA*] as declines.

The model has the advantage of describing stomatal responses to both atmospheric and soil variables and has proven to reproduce a number of common water use trends reported in the literature as, for example, isohydric and anisohydric behaviour.

## **4. Modelling water vapour exchange between leaves and atmosphere and scaling it up to plant and ecosystem level: The** *big-leaf* **approach and the resistive analogy**

The exchange of water vapour through stomata is a molecular diffusion process since air in the sub-stomatal cavities is motionless as well as the air in the first layer outside the stomata directly in contact with the outer leaf surface, i.e. the leaf boundary-layer,. Outside the leaf boundary-layer, it is the turbulent movement of air that removes water vapour, and this process is two orders of magnitude more efficient than the molecular diffusion. The exchange of water between the plant and the atmosphere is further complicated by the physiological control that stomatal resistance exerts on the diffusion of water vapour to the atmosphere.

Transpiration is modelled through an electric analogy (Ohm's law) introduced by Chamberlain and Chadwick (1953). Transpiration behaves analogously to an electric current, which originates from an electric potential difference and flows through a conductor of a given resistance from the high to the low potential end (Figure 2).

The driving potential of the water flux *E* is assumed to be the difference between the water vapour pressure in ambient air *e(Ta)* and the water vapour pressure inside the sub-stomatal cavity *es(Tl)*, the latter being considered at saturation. The resistances that water vapour encounters from within the leaf to the atmosphere is given by the resistance of the stomatal openings (rs) and the resistance of the leaf boundary laminar sub-layer (rb). This process can be represented by the following equation:

$$E = \frac{\left[e\_s(T\_l) - e(T\_a)\right]}{r\_b + r\_s} \cdot \frac{\rho c\_p}{\lambda \gamma} \tag{13}$$

where *Ta* is air temperature (°K), *Tl* is leaf temperature (°K), *e* is water vapour pressure in the ambient air (Pa), *e*s is water vapour pressure of saturated air (Pa) and the term *cp/* is a factor to express *E* in mass density units (kg m-2 s-1), equivalent to mm of water per second, being *cp* the heat capacity of air at constant pressure (1005 J K-1 kg-1), the air density (kg m-3), the vaporisation heat of water (2.5x106 J kg-1), and =cp/ the psychrometric constant (67 Pa K-1). Despite the apparent difference with the well-known Penman-Monteith equation (Monteith, 1981), Eq. 13 is an equivalent formulation of this latter, as demonstrated by Gerosa et al. (2007).

Fig. 2. Schematic picture of the transpirative process form a leaf. The symbols are explained in the text.

While the water vapour pressure deficit [*es*(*Tl*)-*e*(*Ta*)] driving the water exchange is determined by temperature difference, the amount of water flux is regulated by the resistances along the path of the flux.

The stomatal resistance *rs*, reciprocal of the stomatal conductance *gs*, is obtained applying one of the stomatal prediction models presented in the previous paragraph, which are fed by meteorological and agrometeorological data.

The quasi-laminar sub-layer resistance *rb* depends on the molecular properties of the diffusive substance and on the thickness of the layer. The resistance against the diffusion of a gas through air is defined as:

$$r = \int\_{z\_1}^{z\_2} \frac{1}{D\_{H\_2O}} dz\tag{14}$$

for the leaf boundary-layer the equation gives:

$$\mathbf{r}\_{\mathsf{B}} = (\mathbf{z}\_{\mathsf{Z}} \mathbf{z}\_{\mathsf{I}}) / \mathbf{D}\_{\mathsf{H}\to\mathsf{D}} \tag{15}$$

where DH2O is the diffusion coefficient of water vapour in the air, *z1* and *z2* representing the lower and upper height of the leaf boundary-layer.

(67 Pa K-1). Despite the apparent difference with the well-known Penman-Monteith equation (Monteith, 1981), Eq. 13 is an equivalent formulation of this latter, as demonstrated

*E*

Fig. 2. Schematic picture of the transpirative process form a leaf. The symbols are explained

*Substomatal cavity* 

*es*(*Tf*)

*e*(*Ta*)

*rb*

*rs*

*Stomata Epidermis with tricoms* 

While the water vapour pressure deficit [*es*(*Tl*)-*e*(*Ta*)] driving the water exchange is determined by temperature difference, the amount of water flux is regulated by the

The stomatal resistance *rs*, reciprocal of the stomatal conductance *gs*, is obtained applying one of the stomatal prediction models presented in the previous paragraph, which are fed

The quasi-laminar sub-layer resistance *rb* depends on the molecular properties of the diffusive substance and on the thickness of the layer. The resistance against the diffusion of

2

1 *<sup>z</sup>*

2 1

*z H O r dz*

where DH2O is the diffusion coefficient of water vapour in the air, *z1* and *z2* representing the

*<sup>D</sup>* (14)

rb=(z2-z1)/DH2O (15)

 =cp/ *Turbulent surface layer* 

*Leaf boundary layer* 

the air density (kg m-

*z2*

*z1*

the psychrometric constant

being *cp* the heat capacity of air at constant pressure (1005 J K-1 kg-1),

the vaporisation heat of water (2.5x106 J kg-1), and

3), 

in the text.

resistances along the path of the flux.

a gas through air is defined as:

by meteorological and agrometeorological data.

for the leaf boundary-layer the equation gives:

lower and upper height of the leaf boundary-layer.

by Gerosa et al. (2007).

However, the thickness of the leaf boundary-layer depends on leaf geometry, wind intensity and atmospheric turbulence. In order to take these factors in consideration, a more practical formulation, proposed by Unsworth et al. (1984), can be used:

$$r\_b = k \left( d \;/\; u \right)^{1/2} \tag{16}$$

where *k* is an empirical coefficient set to a value of 132 (Thom 1975), *d* is the downwind leaf dimension, and *u* is the horizontal wind speed near the leaves.

The transpiration of a whole plant, or of a vegetated surface with closed canopy, may be modelled using a similar approach referred to as the *big-leaf*. The *big-leaf* assumes the canopy vegetation as an ideal big-leaf lying at a virtual height *z*=*d*+*z0* above ground (Figure 3). The *d*  parameter is the displacement height, i.e. the height of the zero-plane of the canopy, equal to 2/3 of the canopy height, *z0* is the roughness length, i.e. the additional height above *d* where the wind extinguishes inside the canopy (sink for momentum), around 1/10 of the canopy height, and *d*+*z0'* is the apparent height of water vapour source.

Fig. 3. The *big-leaf* approach to model water vapour exchange of a vegetated surface. Left side a real canopy; right side its big-leaf representation. The laminar sub-layer has been enlarged and the stomatal resistance is not shown. Please note the upper case notation of the resistances.

This transpiring big-leaf has a bulk stomatal resistance *Rs* equal to the sum of the stomatal resistances *rs* of all the *n* leaves of the canopy. Recalling the rules of composition for parallel resistances:

$$1\ \,\,\mathbf{1}\ /\,\,R\_s = \sum\_{1}^{n} \mathbf{1}\ /\,\,r\_s = n\ /\,\,r\_s\tag{17}$$

Since the number of leaf is rarely known, a practical way of upscaling *rs* is to consider the thickness of the "big-leaf" equal to the leaf area index of the canopy (*LAI*= m-2leaf/m-2ground)

i.e. the square meters of leaf area projected on each square meter of ground surface. This assumption is equivalent to stating that the light extinction coefficient of the big-leaf is equal to the light extinction of the canopy.

The transpiration rate of the "big-leaf", the whole canopy, is then obtained in a way very similar to those above developed for the leaves:

$$E = \frac{\left[e\_s(T\_l) - e(T\_a)\right]}{R\_a + R\_b + R\_s} \cdot \frac{\rho c\_p}{\lambda \gamma} \tag{18}$$

It is worth noticing the upper case notation for the "bulk" resistances and the introduction of the aerodynamic resistance *Ra*.

The aerodynamic resistance depends on the turbulent features of the atmospheric surface layer, and it is introduced to account for the distance *zm* at which the atmospheric water potential is measured above the canopy. It is formally the vertical integration of the reciprocals of the turbulent diffusion coefficients for all scalars, which in turn depends on the friction velocity *u\** and the atmospheric stability. The integrated version of *Ra* is given by

$$R\_a = \frac{1}{k \cdot \mu^\*} \left| \ln(\frac{z\_m - d}{z\_0}) - \Psi\_M \right| \tag{19}$$

where *k* is the von Kármán dimensionless constant (0.41), *u\** is the friction velocity (m s-1), a quantity indicating the turbulent characteristic of the atmosphere, and *M* is the integrated form of the atmospheric stability function for momentum (non-dimensional).

The friction velocity, if not available, can be derived with the following equation:

$$
\mu^\* = \frac{k \, u\_{zm}}{\ln(\frac{z\_m - d}{z\_0}) - \Psi\_M} \tag{20}
$$

where *uzm* is the wind velocity measured at *zm*, and *<sup>M</sup>* is a function defined as:

$$\mathcal{W}\_M = \begin{cases} 2\ln(\frac{1+y^2}{2}) & \text{with} \quad y = \left(1 - 16(z-d) / L\right)^{1/4} & \text{if } (z-d) / L < 0 \quad \text{unstable condition} \\ & \text{o} \\ & \text{-5(z-d) / L} \qquad \text{if } (z-d) / L = 0 \quad \text{neutral condition} \\ & \text{-5(z-d) / L} \qquad \text{if } (z-d) / L > 0 \quad \text{stable condition} \end{cases} \tag{21}$$

*L* is the length (m) of Monin-Obukhov (1954) indicating the atmospheric stability:

$$L = \frac{-\rho \, c\_p \, \, T\_0 \,\mu \,\text{s}^3}{k \,\text{g} \,\text{H}} \tag{22}$$

with *T0* the reference temperature (273.16 K), *g* the gravity acceleration (9.81 m s-2) and *H* the sensible heat flux (W m-2).

Since *L* is a function of *u\** and *H*, and vice versa, concurrent determination of *u\** and *M* from routine weather data would normally require an iterative procedure (Holtslag and van Ulden, 1983).

i.e. the square meters of leaf area projected on each square meter of ground surface. This assumption is equivalent to stating that the light extinction coefficient of the big-leaf is equal

The transpiration rate of the "big-leaf", the whole canopy, is then obtained in a way very

 *sl a* () ( ) *<sup>p</sup> abs*

(18)

(19)

(20)

*M*

*<sup>M</sup>* is a function defined as:

(22)

*if z d L neutral condition*

*RRR*

It is worth noticing the upper case notation for the "bulk" resistances and the introduction

The aerodynamic resistance depends on the turbulent features of the atmospheric surface layer, and it is introduced to account for the distance *zm* at which the atmospheric water potential is measured above the canopy. It is formally the vertical integration of the reciprocals of the turbulent diffusion coefficients for all scalars, which in turn depends on the friction velocity *u\** and the atmospheric stability. The integrated version of *Ra* is given by

> 0 <sup>1</sup> ln( ) \* *<sup>m</sup> a M z d <sup>R</sup> ku z*

> > *zm <sup>m</sup> <sup>M</sup>*

*<sup>y</sup> with y z d L if z d L unstable condition*

3 *<sup>p</sup>* 0 \* *c Tu*

*zd L if z d L stable condition*

where *k* is the von Kármán dimensionless constant (0.41), *u\** is the friction velocity (m s-1), a quantity indicating the turbulent characteristic of the atmosphere, and *M* is the integrated

0

0 ( )/ 0 5( ) / ( )/ 0

*<sup>M</sup>* (21)

*kgH*

with *T0* the reference temperature (273.16 K), *g* the gravity acceleration (9.81 m s-2) and *H* the

from routine weather data would normally require an iterative procedure (Holtslag and van

Since *L* is a function of *u\** and *H*, and vice versa, concurrent determination of *u\** and

*k u*

ln( )

*z* 

*<sup>u</sup> z d*

form of the atmospheric stability function for momentum (non-dimensional). The friction velocity, if not available, can be derived with the following equation:

\*

<sup>2</sup> <sup>1</sup> 1/4 2 ln( ) (1 16( ) / ) ( ) / 0 <sup>2</sup>

*L* is the length (m) of Monin-Obukhov (1954) indicating the atmospheric stability:

*L*

 

where *uzm* is the wind velocity measured at *zm*, and

Ulden, 1983).

sensible heat flux (W m-2).

*E*

*e T eT c*

to the light extinction of the canopy.

of the aerodynamic resistance *Ra*.

similar to those above developed for the leaves:

If the atmospheric stability is not known as well as the sensible heat flux, and the water potential in the atmosphere is measured near the canopy, a neutral stability can be assumed by setting *<sup>M</sup>*=0 in the *u\** equation with fairly good approximation.

The laminar sub-layer resistance *Rb* can by computed with a general purpose formulation proposed by Hicks et al. (1987) which involves the Schmidt and Prandtl numbers, being *Sc*=0.62 for water vapour and *Pr*=0.72 respectively:

$$R\_b = \frac{2}{k\mu} (\text{Sc } / \text{Pr})^{2/3} \tag{23}$$

where *k* is here the von Kármán constant.

Modelling canopy transpiration using only three resistances in series might seem an oversemplification; however the approach has proven valid in different cases in predicting fast variations of water exchange over a vegetated surface following the stomatal behaviour, as well as to predict the total amount of transpired water (Grunhage et al., 2000).

To obtain a higher modelling performance, the resistive network of the "big-leaf" model can be implemented for specific needs. For example, multiple vegetation layers can be included in order to account for the transpiration of the understory vegetation below a forest, or the canopy can be decomposed in several layers, each with its own properties (De Pury and Farquhar, 1997) In such cases the models take the name of multi-layer models. Other improvements are required when multiple sources of water vapour have to be considered, for example when the evaporation from a water catchment, or evaporation from bare soil in ecosystems with sparse vegetation. .

All these models are collectively known as *1-D* SVAT models (one-dimensional Soil Vegetation Atmosphere Transfer models).

In the following paragraph a multi-layer dual-source model to predict the evapotranspiration from a poplar plantation ecosystem with understory vegetation is presented.

## **5. Example and applications - a multi-layer model for the transpiration of a mature poplar plantation ecosystem - comparison with eddy covariance measurements**

The poplar plantation used for this modelling exercise was located in the Po valley near the city of Pavia. The ecosystem was made by mature poplar trees of about 27 m height with the soil below the plant mainly covered by poplar saplings and perennial grasses. Since the canopy was completely closed, most of the evapotranspiration was due to plants transpiration i.e. evaporation from other surfaces can be considered negligible. According to Choudhury and Monteith (1988), less than 5% of the water vapour flux is due to evaporation from soil for a closed canopy. In this case study evaporation from soil was strongly limited by the absence of tillage and by the coverage of understory vegetation. Moreover the upper soil layer resulted very dry and acted as a screen against water vapour transport from wetter underlying soil layers.

The water exchange was modelled using only two water sources, both of them transpirative: the poplar crown and the understory vegetation. Thus this example model includes only two layers (Figure 4).

The model is composed of three different sub-models: one stomatal sub-model for the stomatal conductance of the transpiring plants, one soil sub-model for the soil water content, and one atmospheric sub-model to describe the water vapour exchange dynamic at canopy level following the adopted resistive network.

Fig. 4. A multi-layer multiple source model to estimate the water exchange between a poplar plantation ecosystem and the atmosphere.

### **5.1 The stomatal conductance sub-model**

To describe the physiological behaviour of the bulk stomatal conductance (*G*s) a Jarvis-Stewart multiplicative model was used, according to the following formulation:

$$G\_{\sf s} = g\_{\sf smax} \cdot \left[ f(\sf PHEN) \cdot f(T) \cdot f(\sf PAR) \cdot f(\sf VPD) \cdot f(\sf SVC) \right] \tag{24}$$

where *gsmax* is the maximum stomatal conductance expressed by the poplar trees in nonlimiting conditions. A maximum value of 1.87 cm s-1 (referred to the Projected Leaf Area) has been found in the literature for *gsmax* of poplar leaves located at 2 meter of height in Italian climatic condition (Marzuoli et al., 2009). This value has been reduced to 57% to account for the decreasing of *gsmax* with the canopy height, as proposed by Schafer et al. (2000). Thus a *gsmax* value of 0.8 cm s-1 was assumed for the canopy.

The phenology function *f(PHEN)* has been assumed equal to zero when the vegetation was without leaves and equal to one after the leaf burst when the leaves were fully expanded. This was fixed to the 110th day of the year (*DOY*).

Compared to Eq. 2, Eq. 24 includes a limiting function based on phenology *f*(*PHEN*) which grows linearly from 0 to 1 during the first 10 days after leaves emergence, and decreases linearly in the last 10 days, starting from *DOY* 285th, simulating leaf's senescence:

$$f\text{(}PHEN\text{)} = \begin{cases} 0 & \forall DOY \le SGS \quad \text{or} \quad \forall DOY \ge EGS\\ 1 & \forall (SGS + DayUp \text{Lp}) < DOY < (EGS - DayDown) \\ \text{((DOY - SGS) / DayUp)} & \forall \text{SGS $$

Fig. 4. A multi-layer multiple source model to estimate the water exchange between a poplar

To describe the physiological behaviour of the bulk stomatal conductance (*G*s) a Jarvis-

 *G*s = *g*smax · [*f*(*PHEN*) · *f*(*T*) · *f*(*PAR*) · *f*(*VPD*) · *f*(*SWC*)] (24) where *gsmax* is the maximum stomatal conductance expressed by the poplar trees in nonlimiting conditions. A maximum value of 1.87 cm s-1 (referred to the Projected Leaf Area) has been found in the literature for *gsmax* of poplar leaves located at 2 meter of height in Italian climatic condition (Marzuoli et al., 2009). This value has been reduced to 57% to account for the decreasing of *gsmax* with the canopy height, as proposed by Schafer et al.

The phenology function *f(PHEN)* has been assumed equal to zero when the vegetation was without leaves and equal to one after the leaf burst when the leaves were fully expanded.

Compared to Eq. 2, Eq. 24 includes a limiting function based on phenology *f*(*PHEN*) which grows linearly from 0 to 1 during the first 10 days after leaves emergence, and decreases

1 ( )( ) ( ) ((DOY - SGS) / DayUp) DOY ( )

((EGS - DOY) / DayDown) ( ) DOY EGS

*f PHEN* (25)

*DOY SGS or DOY EGS SGS DayUp DOY EGS DayDown SGS SGS DayUp*

*EGS DayDown*

linearly in the last 10 days, starting from *DOY* 285th, simulating leaf's senescence:

Stewart multiplicative model was used, according to the following formulation:

(2000). Thus a *gsmax* value of 0.8 cm s-1 was assumed for the canopy.

plantation ecosystem and the atmosphere.

**5.1 The stomatal conductance sub-model** 

This was fixed to the 110th day of the year (*DOY*).

0

*SGS* and *EGS* are the days for the start and the end of the growing season respectively. . *DayUp* and *DayDown* are the number of days necessary to complete the new leaves expansion and to complete the leaves senescence, respectively.

The *G*s dependence on light was modelled according to Eq. 3 form:

$$f\left(PAR\right) = 1 - \exp^{-aPAR} \tag{26}$$

where *a* represents a specie-specific coefficient (0.006 in this study) and *PAR* is the Photosynthetically Active Radiation expressed as mol photons m-2 s-1.

Eq. 4 and Eq. 5 were used for *Gs* dependence on temperature and VPD, respectively.

For soil water content *SWC* a different limiting function, from that reported by Sterwart (1988), was used. The boundary-line analysis revealed that SWC exerted its influence on *gsmax* according to the following equation:

$$f(SWC) = \max\left\{0.1; \min\left[1; \mathbf{g} \cdot SWC^{\left(h/SWC\right)}\right]\right\} \tag{27}$$

where *SWC* is expressed as fraction of soil field capacity while *g* and *h* are two coefficients whose values are respectively 1.0654 and 0.2951.

The bulk stomatal conductance of the understory vegetation was modelled using the same parameterization but assuming a *gsmax* value equal to 1.87 cm s-1. The inherent approximation is that the understory vegetation was entirely composed of young poplar plantlets.


Table 1. Values of the *f* limiting functions coefficients and *gsmax* for the stomatal conductance model of *Populus nigra*.

#### **5.2 The soil sub-model**

The water availability in the soil was modelled using a simple "bucket" model. In this paradigm the soil is considered as a bucket and the water content is assessed dynamically, step by step, via the hydrological balance between the water inputs (rains) and outputs (plant consumption) occurred in the previous time step. The model was initialised assuming the soil water saturated at the beginning of the season and assuming a root depth for soil exploitation of 3 m:

$$\text{AVHCC} = (\theta\_{\text{FC}} \cdot \theta\_{\text{NP}}) \cdot 1000 \text{ } \cdot \text{RootDepth} = 243 \text{ mm H}\_2\text{O} \text{ } / \text{ m}^3 \text{ soil} \tag{28}$$

$$\text{AVV}\_{\overline{t}} \mathbf{0} \; \text{ = AVWHC \; (mm)}\tag{29}$$

where *AWHC* is the available water holding capability of the sandy soil between the wilting point (*WP*= 0.114 m3 m-3 for our sandy loam soil) and the field capacity (*FC*=0.195 m3 m-3). The running equations were:

$$ET\_{t\cdot 1} = F\_{H20, t\cdot 1} \cdot \text{3600 / } \downarrow \text{ (mm)} \tag{30}$$

$$A\text{V}\_{l} = A\text{V}\_{l}\text{-}1 + \text{Rain}\_{l\text{-1}}\text{-}ET\_{l\text{-1}} \pmod{\text{ (mm)}}\tag{31}$$

$$\text{SWC}\_{l} = \text{AW}\_{l} / \text{AWHC} \quad \text{(\% of FC)} \tag{32}$$

Eq. 32 represents the water loss of plant ecosystem through the transpiration of the two layers (*FH20, t-1*) in the previous time step. Since water fluxes are expressed as rates (mm s-1), for an hourly time step, as in our cases, their values must be multiplied by 3600 in order to get the water consumed in one hour.

*AWt* is the available water in the soil after water inputs and consumptions. The effects of runoff and groundwater level rising have been neglected due to the flatness of the ecosystem and the groundwater level which were deeper than the root exploration depth.

*SWC* represents the soil water content expressed as percentage of field capacity, as requested by the *f*(*SWC*) function of the stomatal sub-models.

#### **5.3 The atmospheric sub-model and the resistive network**

The resistance *Ra* was calculated by using Eq. 19 and Eq. 21, with *zm*=33 m the measurement height, *h*= 26.3 m the canopy height, *u\** the friction velocity, *u* the horizontal wind speed, *L* the Monin-Obhukhov length, *d*=2/3·*h* the zero-plane displacement height and *z0*=1/10·*h* the roughness length.

The laminar sub-layer resistances of the layers 1 and 2 (*Rb1* and *Rb2*) were both calculated using the Eq. 23 given *u\**.

The stomatal resistances of the layers 1 and 2 (*Rstom1* and *Rstom2*) were calculated using the stomatal sub-model after having estimated the leaf temperatures from the air temperature *T* and the heat fluxes *H*:

$$T\_l = T + H \cdot \left(R\_a + R\_{b, \text{heat}}\right) / \left(\rho \cdot c\_p\right) \tag{33}$$

where *Rb,heat* was calculated using the Eq. 23 with *Sc=0.67* and *Pr=0.71*.

Then the vapour pressure deficit *VPD* = *es*(*Tl*) - *e*(*T*) was derived from the *Tl* for the calculation of *es*(*Tl*) and from the air temperature *T* and the relative humidity *RH* for the actual *e*: *e*(*T*)=*UR* · *es*(*T*).

The vapour pressure of the saturated air can be calculated from the well-known Teten-Murray empirical equation:

$$e\_s(T) = 0.611 \cdot \exp(17.269 \cdot (T - 273) / \text{ (T - 36)}) \tag{34}$$

which gives *es* in kPa when *T* is expressed as °K.

The stomatal resistance of the crown *Rstom1* was obtained as the reciprocal of the stomatal conductance obtained by the Jarvis–Stewart sub-model fed with *PAR*, *Tleaf*, *VPD* and *SWCt*, the latter being the soil water content calculated with the Eq. 32 .

The understory *Rstom2* was obtained in a similar way but considering a understory gmax (=1.87 cm s-1) and the *PAR fraction* reaching the below canopy vegetation instead of the original *PAR*:

$$\text{PAR}\_{\text{fraction}} = \exp(\cdot \mathbf{k} \cdot \text{LAI}\_1) \tag{35}$$

where *k* is the light extinction factor within the canopy, set to 0.54, and *LAI1* is the leaf area index of the crown, assumed to be equal to 2 at maximum leaf expansion. The in-canopy resistance *Rinc* was calculated following Erisman et al. (1994):

$$R\_{inc} \text{ = (14 } \cdot \text{LAI}\_1 \cdot \text{h) / } \text{ } u^\* \tag{36}$$

where *h* is the canopy height and *LAI1* the leaf area index of the crown.

The stomata of the big leaves of the two layers of Figure 4 (*G1* and *G2*) were assumed as water generators driven by the difference of water concentration between the leaves (*χsat*), assumed water saturated al leaf temperature *Tl*, and the air (*χair*):

$$G\_1 = G\_2 = \chi\_{\rm sat} - \chi\_{\rm air} \quad \text{(g } \mathbf{m}^{\rm 3}\text{)}\tag{37}$$

where

416 Evapotranspiration – Remote Sensing and Modeling

where *AWHC* is the available water holding capability of the sandy soil between the wilting

*AWt* = *AWt-1* + *Raint-1* –*ETt-1* (mm) (31)

 *SWCt* = *AWt* / *AWHC* (% of FC) (32) Eq. 32 represents the water loss of plant ecosystem through the transpiration of the two layers (*FH20, t-1*) in the previous time step. Since water fluxes are expressed as rates (mm s-1), for an hourly time step, as in our cases, their values must be multiplied by 3600 in order to

*AWt* is the available water in the soil after water inputs and consumptions. The effects of runoff and groundwater level rising have been neglected due to the flatness of the ecosystem and the groundwater level which were deeper than the root exploration depth. *SWC* represents the soil water content expressed as percentage of field capacity, as

The resistance *Ra* was calculated by using Eq. 19 and Eq. 21, with *zm*=33 m the measurement height, *h*= 26.3 m the canopy height, *u\** the friction velocity, *u* the horizontal wind speed, *L* the Monin-Obhukhov length, *d*=2/3·*h* the zero-plane displacement height and *z0*=1/10·*h* the

The laminar sub-layer resistances of the layers 1 and 2 (*Rb1* and *Rb2*) were both calculated

The stomatal resistances of the layers 1 and 2 (*Rstom1* and *Rstom2*) were calculated using the stomatal sub-model after having estimated the leaf temperatures from the air temperature *T*

Then the vapour pressure deficit *VPD* = *es*(*Tl*) - *e*(*T*) was derived from the *Tl* for the calculation of *es*(*Tl*) and from the air temperature *T* and the relative humidity *RH* for the

The vapour pressure of the saturated air can be calculated from the well-known Teten-

The stomatal resistance of the crown *Rstom1* was obtained as the reciprocal of the stomatal conductance obtained by the Jarvis–Stewart sub-model fed with *PAR*, *Tleaf*, *VPD* and *SWCt*,

*e*s(*T*) = 0.611 · *exp*(17.269 · (*T* - 273) / (*T* - 36)) (34)

*Tl* = *T* + *H* · (*Ra* + *Rb, heat*) / (

where *Rb,heat* was calculated using the Eq. 23 with *Sc=0.67* and *Pr=0.71*.

the latter being the soil water content calculated with the Eq. 32 .

*WP*= 0.114 m3 m-3 for our sandy loam soil) and the field capacity (

*WP*) · 1000 · *RootDepth* = 243 mm H2O / m3 soil (28)

*AWt*=0 = *AWHC* (mm) (29)

(mm) (30)

· *cp*) (33)

*FC*=0.195 m3 m-3).

*AWHC* = (

The running equations were:

get the water consumed in one hour.

roughness length.

using the Eq. 23 given *u\**.

and the heat fluxes *H*:

actual *e*: *e*(*T*)=*UR* · *es*(*T*).

Murray empirical equation:

which gives *es* in kPa when *T* is expressed as °K.

point ( *FC*-

*ETt-1* = *FH20, t-1* · 3600 /

requested by the *f*(*SWC*) function of the stomatal sub-models.

**5.3 The atmospheric sub-model and the resistive network** 

$$\chi\_{\rm sat} = 2.165 \cdot e\_s(T\_l) \ne T\_l \quad \text{(g m-3)}$$

$$\chi\_{air} = 2.165 \cdot e(lIR\_\prime \ T) \ne T \quad \text{(g m-3)}$$

being 2.165 the ratio between the molar weight of water molecules *Mw* (18 g mol-1) and the gas constant *R* (8.314 J mol-1 K-1) if *e* and *es* are expressed in Pa (multiplied by 1000 if expressed in kPa).

Then the total water flux of the ecosystem *FH2O* could be calculated by composing all the resistances and the generators within the modelled resistive network, following the electrical composition rules for resistances and generators in series and in parallel, and applying the scaling strategy according to the *LAI*:

$$R\_1 = \left(R\_{b1} + R\_{stm1} \; / \; LAI\_1\right) \quad \text{(s/m)}\tag{38}$$

$$R\_2 = \left(R\_{b2} + R\_{stm2} \; / \; LAI\_2\right) \quad \text{(s/m)}\tag{39}$$

$$R\_3 = R\_{inc} + R\_2 \quad \text{(s/m)}\tag{40}$$

$$\mathbf{G\_{eq}} = \mathbf{G\_2} \cdot \begin{pmatrix} \mathbf{G\_2} \ \mathbf{-} \mathbf{G\_1} \end{pmatrix} \cdot \mathbf{R\_3} \Big/ \begin{pmatrix} \mathbf{R\_1} + \mathbf{R\_3} \end{pmatrix} \tag{9.10.3}$$

$$R\_{eq} = R\_1 \cdot R\_3 \text{ / (R}\_1 + R\_3\text{)} \quad \text{(s/m)}\tag{42}$$

*FH2O* = *Geq* / (*Req* + *Ra*) / 1000 (kg m-2 s-1 = mm s-1) (43)

where *LAI2* is the leaf area index of the understory vegetation (=0.5)

#### **5.4 Comparison with EC measurements**

Concurrent measurements of *E* were performed over the same ecosystem by means of eddy covariance technique with instrumentation set-up according to Gerosa et al. (2005). The comparison between the direct *E* measurements and the modelled ones allowed the evaluation of model performance.

The model performance was very good in predicting the hourly variation of *E* both during the summer season (*Modeled* = 0.885 · *Measured* + 8.4389; *R2*=0.85, *p*<0.001, *n*=1872) with a slight tendency to underestimate the peaks.

An example of the comparison exercise for a summer week is shown in Figure 5

**DOY**

Fig. 5. Comparison between modelled and measured *E*, expressed as W m-2 unit

Fig. 6. Mean daily course of the modeled *E* compared to the measured one. All the available hourly measurements were considered (*n*=3914)

For the whole year the performance was less good (*Modeled* = 0.876 · *Measured* + 21.443; *R2*=0.68, *p*<0.001, *n*=3914) but still acceptable, especially in reproducing the average daily course of *E* (Figure 6).

## **6. Conclusions**

418 Evapotranspiration – Remote Sensing and Modeling

the summer season (*Modeled* = 0.885 · *Measured* + 8.4389; *R2*=0.85, *p*<0.001, *n*=1872) with a

195 196 197 198 199 200

0 3 6 9 12 15 18 21

**hours**

**DOY**

*E* measurements and the modelled ones allowed the

LE modeled LE measured (W m-2)

*E*, expressed as W m-2 unit

*E* compared to the measured one. All the

LE modeled LE measured (W/m2)

*E* both during

The model performance was very good in predicting the hourly variation of

An example of the comparison exercise for a summer week is shown in Figure 5

The comparison between the direct

slight tendency to underestimate the peaks.

Fig. 5. Comparison between modelled and measured

evaluation of model performance.


0


Fig. 6. Mean daily course of the modeled

available hourly measurements were considered (*n*=3914)

0

50

100

**LE (W/m2)**

150

200

250

100

200

300

**LE (W/m2)**

400

500

600

700

Scientific literature provides many ways (e.g. FAO) to estimate the evapotranspiration of a vegetated surface. Sometimes there is the need to predict this process at a very high-time resolution (e.g. hourly means). Hourly estimations of evapotranspiration, for example, are important in all the applications and the methodologies which couple transpiration process with carbon assimilation or air pollutants uptake by plants.

In these cases, the *big-leaf* approach, together with the resistive analogy which simulates the gas-exchange between vegetation and atmosphere, is a simple but valid example of a process-based model which includes the stomatal conductance behaviour, as well as a basic representation of the canopy features.

## **7. Acknowledgements**

This publication was partially funded by the Catholic University's program for promotion and divulgation of scientific research.

## **8. References**


and ozone exposure. An Open-Top Chambers experiment in South Alpine environmental conditions. *Environmental Pollution* 152, 274–284.

