Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments

contains 23 related to the modeling and simulation of evapotranspiration (ET) and remote sensing-based energy balance determination of ET. These areas are at the forefront of technologies that quantify the highly spatial ET from the Earth's surface. The topics describe mechanics of ET simulation from partially vegetated surfaces and stomatal conductance behavior of natural and agricultural ecosystems. Estimation methods that use weather based methods, soil water balance, the Complementary Relationship, the Hargreaves and other temperature-radiation based methods, and Fuzzy-Probabilistic calculations are described. A critical review describes methods used in hydrological models. Applications describe ET patterns in alpine catchments, under water shortage, for irrigated systems, under climate change, and for grasslands and pastures. Remote sensing based approaches include Landsat and MODIS satellite-based energy balance, and the common process models SEBAL, METRIC and S-SEBS. Recommended guidelines for applying operational satellite-based energy balance models and for overcoming common challenges are made.


Introduction
Accurate modelling of evapotranspiration (ET) is required to predict the effects of climate and land use changes on water resources, agriculture and ecosystems. Significant progress has been made in estimating ET at the global and regional scale. However, further efforts are needed to improve spatial accuracy and modeling capabilities in alpine regions (Brooks & Vivoni, 2008b). This chapter will point out the components of the energy budget needed to model ET, to discuss the fundamental equations and to provide an extended review of the parametrizations available in the hydrological and land surface models literature. The second part of the chapter will explore the complexity of the energy budget with special reference to mountain environments.

The energy budget components
Evapotranspiration is controlled by the surface water and energy budget. In this section the single components of the energy budget will be discussed: radiation, soil heat flux, sensible and latent heat fluxes. The surface energy budget inside a control volume can be written as: where the energy fluxes concerning the soil-atmosphere interface in the time interval Δt are: R n net radiative flux; H sensible heat flux; LE latent heat flux; G heat flux in the soil; ΔE internal energy variation in the control volume; The control volume is assumed with a thickness of some meters, so as to include the soil layer close to the surface and the first meters of the atmosphere, including the possible vegetation cover.
Besides the surface energy budget also the mass budget must be considered in order to quantify ET, namely the water conservation inside the control volume.
where ΔS is the storing of the various supplies (underground and surface storage, soil moisture, vegetation interception, storing in channels); P is precipitation; ET = λLE is evapotranspiration; λ is the latent heat of vaporization; R is the surface runoff; R G is the runoff towards the deep water table; R S is the sub-surface runoff.
In the next sections it will be explained how the different components of the energy budget are usually described in hydrological and Land Surface Models (LSMs): radiation, sensible and latent heat flux, soil heat flux.

Radiation
The radiation is usually divided in short-wave components -indicated here as SW (including visible light, part of the ultraviolet radiation and the close infrared) and long-wave components -indicated here as LW (infrared radiation), with wavelength λ ranging respectively from 0.1 to 3 µm (98% of extraterrestrial radiation) and from 3 to 100 µm (2% of extraterrestrial radiation). In photosynthetic processes the short-wave radiation is further divided in photosynthetically active radiation (0.4 < λ < 0.7µm) and near infrared (Bonan, 1996). Moreover, there is a further distinction between diffuse radiation D (coming by diffusion in the atmosphere from any direction) and direct radiation P (coming only from the sun), and also between radiation from the sky downwards ("down" ↓) and radiation from the soil upwards ("up" ↑).
The net radiation at the soil level can be factorized as follows: with: R ↓ SW P short-wave direct radiation; R ↓ LW P long-wave direct radiation; R ↓ SW D short-wave diffuse radiation; R ↓ LW D long-wave diffuse radiation; R ↑ SW short-wave reflected radiation; R ↑ LW R long-wave reflected radiation; R ↑ LW radiation emitted by surface; R ↓ SW O +R ↓ LW O reflected radiation emitted by the surrounding surfaces; The effects of topography on the diffused radiation can be expressed through the sky view factor V, indicating the sky fraction visible in one point. In the presence of reliefs only part of the horizon is visible and this has consequences on the radiative exchanges. V is defined as: where ω is the solid angle seen from the point considered.
The presence of shadows due to surrounding mountains can be expressed through a factor sw, a function of topography and sun position, defined as: sw = 1 if the point is in the sun 0 if the point is in the shadow (5) All direct radiation terms have to be multiplied by this factor.
In the next paragraphs we analyze in detail the parametrization of the single terms composing the radiation flux.

Direct radiation
R ↓ SW P and R ↓ LW P The direct long-wave radiation R ↓ LW P is emitted directly by the sun and therefore it is negligible at the soil level (differently from the long-wave diffuse radiation). Usually the short-wave radiation R ↓ SW P is assumed as an input variable, measured or calculated by an atmospheric model. The direct radiation can be written as the product of the extraterrestrial radiation R Extr by an attenuation factor varying in time and space.
The extraterrestrial radiation can be easily calculated on the basis of geometric formulas (Iqbal, 1983). The atmospheric attenuation is due to Rayleigh diffusion, to the absorption on behalf of ozone and water vapor, to the extinction (both diffusion and absorption) due to atmospheric dust and shielding caused by the possible cloud cover. Moreover the absorption entity depends on the ray path length through the atmosphere, a function of the incidence angle and of the measurement point elevation. The effect of the latter can be very important in a mountain environment, where it is necessary to consider the shading effects. Part of the dispersed radiation is then returned as short-wave diffuse radiation (R ↓ SW D ) and part of the energy absorbed by atmosphere is then re-emitted as long-wave diffuse radiation (R ↓ LW D ). From a practical point of view, according to the application type and depending on the measured data possessed, the attenuation coefficient can be calculated with different degrees of complexity. The radiation transfer through the atmosphere is a well studied phenomenon 379 Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments www.intechopen.com and there exist many models providing the soil incident radiation spectrum in a detailed way, considering the various attenuation effects separately (Kondratyev, 1969).

Diffuse downward short-wave radiation
R ↓ SW D This term is a function of the atmospheric radiation due to Rayleigh dispersion and to the aereosols dispersion, as well as to the presence of cloud cover. The R ↓ SW D actually is not isotropic and it depends on the sun position above the horizon. For its parametrization, see, for example, Paltrige & Platt (1976).

Diffuse downward long-wave radiation
R ↓ LW D Often this term in not provided by standard meteorological measurements, and many LSMs provide expressions to calculate it. This term indicates the long-wave radiation emitted by atmosphere towards the earth. It can be calculated starting from the knowledge of the distribution of temperature, humidity and carbon dioxide of the air column above. If this information is not available, various formulas, based only on ground measurements, can be found in literature with expressions as follows: with: T a air temperature [K]; ǫ a atmosphere emissivity f (e a , T a , cloud cover); e a vapor pressure [mb]; Usually for ǫ a empirical formulas have been used, but it is also possible to provide a derivation based on physical topics like in Prata (1996). The cloud cover effect on this term is significant and not easy to consider in a simple way. Cloud cover data can be provided during the day by ground or satellite observations but, especially on night, is difficult to collect.

Reflected short-wave radiation
R ↑ SW This term indicates the short-wave energy reflection.
where a is the albedo. The albedo depends strongly on the wave length, but generally a mean value is used for the whole visible spectrum. Besides its dependance on the surface type, it is important to consider its dependence on soil water content, vegetation state and surface roughness. The albedo depends moreover on the sun rays inclination, in particular for smooth surfaces: for small angles it increases. There is very rich literature about albedo description, it being a key parameter in the radiative exchange models, see for example Kondratyev (1969). Albedo is often divided in visible, near infrared, direct and diffuse albedo, as in Bonan (1996).

Long-wave radiation emitted by the surface
R ↑ LW This term indicates the long-wave radiation emitted by the earth surface, considered as a grey body with emissivity ε s (values from 0.95 to 0.98). The surface temperature T s [K] is unknown 380 Evapotranspiration -Remote Sensing and Modeling www.intechopen.com Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments 5 and must be calculated by a LSM. σ = 5.6704 · 10 −8 W/(m 2 K 4 ) is Stefan-Boltzman constant.
2.1.6 Reflected long-wave radiation R ↑ LW R This term is small and can be subtracted by the incoming long-wave radiation, assuming surface emissivity ε s equal to surface absorptivity:

Radiation emitted and reflected by surrounding surfaces
It indicates the radiation reflected (R ↑ SW +R ↑ LW R ) and emitted (R ↑ LW ) by the surfaces adjacent to the point considered. This term is important at small scale, in the presence of artificial obstructions or in the case of a very uneven orography. To calculate it with precision it is necessary to consider reciprocal orientation, illumination, emissivity and the albedo of every element, through a recurring procedure (Helbig et al., 2009). A simple solution is proposed for example in Bertoldi et al. (2005). If the intervisible surfaces are hypothesized to be in radiative equilibrium, i.e. they absorb as much as they emit, these terms can be quantified in a simplified way:

Net radiation
Inserting expressions (7) and ( 9) in the (3), the net radiation is: with ε a = f (e a , T a , cloud cover) as for example in Brutsaert (1975). Equation (12) is not invariant with respect to the spatial scale of integration: indeed it contains non-linear terms in T a , T s , e a , consequently the same results are not obtained if the local values of these quantities are substituted by the mean values of a certain surface. Therefore, the shift from a treatment valid at local level to a distributed model valid over a certain spatial scale must be done with a certain caution.

Radiation adsorption and backscattering by vegetation
Expression (12) needs to be modified to take into account the radiation adsorption and backscattering by vegetation, as shown in Figure 2. This effect is very important to obtain a correct soil surface skin temperature (Deardorff, 1978). From Best (1998) it is possible to derive the following relationship: where T v is vegetation temperature, ε v vegetation emissivity (supposed equal to absorption), a v vegetation albedo (downward albedo supposed equal to upward albedo) and f trasm  (Dickinson et al., 1986). Bonan (1996) uses a two-layers canopy model. Law et al. (1999) explicit the relationship between leaf area distribution and radiative transfer. A first energy budget is made at the canopy cover layer, and the energy fluxes are solved to find the canopy temperature, then a second energy budget is made at the soil surface. Usually a fraction of the grid cell is supposed covered by canopy and another fraction by bare ground.  Bonan (1996), modified).

Soil heat flux
The soil heat flux G at a certain depth z depends on the temperature gradient as follows: where λ s is the soil thermal conductivity (λ s = ρ s c s κ s with ρ s density, c s specific heat and κ s soil thermal diffusivity) depending strongly on the soil saturation degree. The heat transfer inside the soil can be described in first approximation with Fourier conduction law: Equation (14) neglects the heat associated to the vapor transportation due to a vertical gradient of the soil humidity content as well as the horizontal heat conduction in the soil. The vapor transportation can be important in the case of dry climates (Saravanapavan & Salvucci, 2000). The soil heat flux can be calculated with different degrees of complexity. The most simple assumption (common in weather forecast models) is to calculate G as a fraction of net radiation (Stull (1988) suggests G = 0.1R n ). Another simple approach is to use the analytical solution for a sinusoidal temperature wave. A compromise between precision and computational work is the force restore method (Deardorff, 1978;Montaldo & Albertson, 2001), still used in many hydrological models (Mengelkamp et al., 1999). The main advantage is that only two soil layers have to be defined: a surface thin layer, and a layer getting down to a depth where the daily flux is almost zero. The method uses some results of the analytical solution for a sinusoidal forcing and therefore, in the case of days with irregular temperature trend, it provides less precise results.
The most general solution is the finite difference integration of the soil heat equation in a multilayered soil model (Daamen & Simmonds, 1997). However, this method is computationally demanding and it requires short time steps to assure numerical stability, given the non-linearity and stationarity of the surface energy budget, which is the upper boundary condition of the equation.

Snowmelt and freezing soil
In mountain environments snow-melt and freezing soil should be solved at the same time as soil heat flux. A simple snow melt model is presented in Zanotti et al. (2004), which has a lumped approach, using as state variable the internal energy of the snow-pack and of the first layer of soil. Other models consider a multi-layer parametrization of the snowpack (e.g. Bartelt & Lehning, 2002;Endrizzi et al., 2006). Snow interception by canopy is described for example in Bonan (1996). A state of the art freezing soil modeling approach can be found in Dall'Amico (2010) and Dall'Amico et al. (2011).

Turbulent fluxes
A modeling of the ground heat and vapor fluxes cannot leave out of consideration the schematization of the atmospheric boundary layer (ABL), meant as the lower part of atmosphere where the earth surface properties influence directly the characteristics of the motion, which is turbulent. For a review see Brutsaert (1982); Garratt (1992); Stull (1988). A flux of a passive tracer x in a turbulent field (as for example heat and vapor close to the ground), averaged on a suitable time interval, is composed of three terms: the first indicates the transportation due to the mean motion v, the second the turbulent transportation x ′ v ′ , the third the molecular diffusion k.
The fluxes parametrization used in LSMs usually only considers as significant the turbulent term only. The molecular flux is not negligible only in the few centimeters close the surface, and the horizontal homogeneity hypothesis makes negligible the convective term.

383
Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments www.intechopen.com

The conservation equations
The first approximation done by all hydrological and LSMs in dealing with turbulent fluxes is considering the Atmospheric Boundary Layer (ABL) as subject to a stationary, uniform motion, parallel to a plane surface.
This assumption can become limitative if the grid size becomes comparable to the vertical heterogeneity scale (for example for a grid of 10 m and a canopy height of 10 m). In this situation horizontal turbulent fluxes become relevant. A possible approach is the Large Eddy Simulation . If previous assumptions are made, then the conservation equations assume the form: • Specific humidity conservation, failing moisture sources and phase transitions: ρ is the air density [kg/m 3 ] w is the vertical velocity [m/s].
• The horizontal mean motion equations are obtained from the momentum conservation by simplifying Reynolds equations (Stull, 1988;Brutsaert, 1982 cap.3): In a turbulent motion the molecular transportation terms of the momentum, heat and vapor quantity, respectively ν, k h and k v , are several orders of magnitude smaller than Reynolds fluxes and can be neglected.

Wind, heat and vapor profile at the surface
Inside the ABL we can consider, with a good approximation, that the decrease in the fluxes intensity is linear with elevation. This means that in the first meters of the air column the fluxes and the friction velocity u * can be considered constant. Considering the momentum flux constant with elevation implies that also the wind direction does not change with elevation (in the layer closest to the soil, where the geostrofic forcing is negligible). In this way the alignment with the mean motion allows the use of only one component for the velocity vector, and the problem of mean quantities on uniform terrain becomes essentially one-dimensional, as these become functions of the only elevation z.
In the first centimeters of air the energy transportation is dominated by the molecular diffusion. Close to the soil there can be very strong temperature gradients, for example during a hot summer day. Soil can warm up much more quickly than air. The air temperature diminishes very rapidly through a very thin layer called micro layer, where the molecular processes are dominant. The strong ground gradients support the molecular conduction, while the gradients in the remaining part of the surface layer drive the turbulent diffusion.
In the remaining part of the surface layer the potential temperature diminishes slowly with elevation.
The effective turbulent flux in the interface sublayer is the sum of molecular and turbulent fluxes. At the surface, where there is no perceptible turbulent flux, the effective flux is equal to the molecular one, and above the first cm the molecular contribution is neglegible. According to Stull (1988), the turbulent flux measured at a standard height of 2 m provides a good approximation of the effective ground turbulent flux. Applying the concept of effective turbulent flux, the molecular diffusion term can be neglected, while the hypothesis of uniform and stationary limit layer leads to neglect the convective terms due to the mean vertical motion and the horizontal flux. The vertical flux at the surface can then be reduced to the turbulent term only:

385
Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments www.intechopen.com In the case of the water vapor, equation (17) shows that, if there is no condensation, the flux is: where ET is the evaporation quantity at the surface, ρ the air density and λ is the latent heat of vaporization.
Similarly, as to sensible heat, equation (18) shows that the heat flux at the surface H is: where c p is the air specific heat at constant pressure. The entity of the fluctuating terms w ′ u ′ , w ′ θ ′ and w ′ q ′ remains unknown if further hypotheses (called closing hypotheses) about the nature of the turbulent motion are not introduced. The closing model adopted by the LSMs is Bousinnesq model: it assumes that the fluctuating terms can be expressed as a function of the vertical gradients of the quantities considered (diffusive closure) .
where K M is the turbulent viscosity, K H and K W [m 2 /s] are turbulent diffusivity. Moreover a logarithmic velocity profile in atmospheric neutrality conditions is assumed: where k is the Von Karman constant, z 0 is the aerodynamic roughness, evaluated in first approximation as a function of the height of the obstacles as z 0 /h c ≃ 0.1 (for more precise estimates see Stull (1988) p.379; Brutsaert (1982) ch.5; Garratt (1992) p.87). In the case of compact obstacles (e.g. thick forests), the profile can be thought of as starting at a height d 0 , and the height z can be substituted with a fictitious height z − d 0 .  Brutsaert, 1982).
Also the other quantities θ and q have an analogous distribution. Using as scale quantities θ * 0 = −w ′ θ ′ 0 /u * 0 e q * 0 = −w ′ q ′ 0 /u * 0 and substituting them in the (25) integration is obtained: The boundary condition chosen is θ = θ 0 in z = z T and q = q 0 in z = z q . The temperature θ 0 then is not the ground temperature, but that at the elevation z T . The roughness height z T is the height where temperature assumes the value necessary to extrapolate a logarithmic profile. Analogously, z q is the elevation where the vapor concentration assumes the value necessary to extrapolate a logarithmic profile. Indeed, close to the soil (interface sublayer) the logarithmic profile is not valid and then, to estimate z T and z q , it would be necessary to study in a detailed way the dynamics of the heat and mass transfer from the soil to the first meters of air. If we consider a real surface instead of a single point, the detail requested to reconstruct accurately the air motion in the upper soil meters is impossible to obtain. Then there is a practical problem of difficult solution: on the one hand, the energy transfer mechanisms from the soil to the atmosphere operate on spatial scales of few meters and even of few cm, on the other hand models generally work with a spatial resolution ranging from tens of m (as in the case of our approach) to tens of km (in the case of mesoscale models). Models often apply to local scale the same parametrizations used for mesoscale. Therefore a careful validation test, even for established theories, is always important.
Observations and theory (Brutsaert, 1982, p.121) show that z T and z q generally have the same order of magnitude, while the ratio z T z 0 is roughly included between 1 5 − 1 10 .

The atmospheric stability
In conditions different from neutrality, when thermal stratification allows the development of buoyancies, Monin & Obukhov (1954) similarity theory is used in LSMs. The similarity theory wants to include the effects of thermal stratification in the description of turbulent transportation. The stability degree is expressed as a function of Monin-Obukhov length, defined as: where θ 0 is the potential temperature at the surface. Expressions of the stability functions can be found in many texts of Physics of the Atmosphere, for example Katul & Parlange (1992); Parlange et al. (1995). The most known formulation is to be found in Businger et al. (1971). Yet stability is often expressed as a function of bulk Richardson number Ri B between two reference heights, expresses as: where Δθ is the potential temperature difference between two reference heights, and θ is the mean potential temperature.

387
Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments www.intechopen.com If Ri B > 0 atmosphere is steady, if Ri B < 0 atmosphere is unsteady. Differently from L MO , Ri B is also a function of the dimensionless variables z/z 0 e z/z T . The use of Ri B has the advantage that it does not require an iterative scheme.
Expressions of the stability functions as a function of Ri B are provided by Louis (1979) and more recently by Kot & Song (1998). Many LSMs use empirical functions to modify the wind profile inside the canopy cover. From the soil up to an elevation h d = f (z 0 ), limit of the interface sublayer, the logarithmic universal profile and Reynolds analogy are no more valid. For smooth surfaces the interface sublayer coincides with the viscous sublayer and the molecular transport becomes important. For rough surfaces the profile depends on the distribution of the elements present, in a way which is not easy to parametrize. Particular experimental relations can be used up to elevation h d , to connect them up with the logarithmic profile (Garratt, 1992, p. 90 andBrutsaert, 1982, p. 88). These are expressions of non-easy practical application and they are still little tested.

Latent and sensible heat fluxes
As consequence of the theory explained in the previous paragraph, the turbulent latent and sensible fluxes H and LE can be expressed as: where θ 0 − θ and q 0 − q are the difference between surface and measurement height of potential temperature and specific humidity respectively. C H and C E are usually assumed to be equal and depending on the bulk Richardson number (or on Monin-Obukhov lenght): where C Hn is the heat bulk coefficient for neutral conditions: derived on Eq. 29 and depending on the wind speed u, the measurement height z, the temperature (or moisture) measurement height z a , the momentum roughness length z 0 and the heat roughness length z T . A common approach is the 'electrical resistance analogy' (Bonan, 1996), where the atmospheric resistance is expressed as:

Evapotranspiration processes
In order to convert latent heat flux in evapotranspiration the energy conservation must be solved at the same time as water mass budget. In fact, there must be a sufficient water quantity available for evaporation. Moreover, vegetation plays a key role.

Unsaturated soil evaporation
If the availability of water supply permits to reach the surface saturation level, then evaporation is potential ET = EP and then we have air saturation at the surface q(T s )= q * (T s ) (the superscript * stands here for saturation). If the soil is unsaturated, q(T s ) = q * (T s ) and different approaches are possible to quantify the water content at the surface, in dependance of the water budget scheme adopted.
1. A first possibility is to introduce then the concept of surface resistance r g to consider the moisture reduction with respect to the saturation value. As it follows from equation (34): 2. As an alternative, we can define a soil-surface relative moisture and then the expression for evaporation becomes: An expression of r h as a function of the potential ψ s [m] (work required to extract water from the soil against the capillarity forces) and of the ratio of the soil water content η to the saturation water content η s is given in Philip & Vries (1957): where R v = 461.53 [J/(kg K)] is the gas constant for water vapor, T s is the soil temperature, b an empirical constant. Tables of these parameters for different soil types can be found in Clapp & Hornberger (1978). Another more simple expression frequently applied in models to link the value r h with the soil water content η is provided by Noilhan & Planton (1989): where η is the moisture content of a soil layer with thickness d 1 , and η k is a critical value depending on the saturation water content: η k ≃ 0.75η s . 3. A third possibility, very used in large-scale models, is that of expressing the potential/real evaporation ration through a simple coefficient: The value of x can be connected to the soil water content η through the expression (Deardorff, 1978) (see Figure 4):

Transpiration
Usually transpiration takes into account the canopy resistance r c to add to the atmospheric resistance r a : The canopy resistance depends on plant type, leaf area index, solar radiation, vapor pressure deficit, temperature and water content in the root layer. There is a wide literature regarding such dependence, see for example Feddes et al. (1978); Wigmosta et al. (1994). Canopy interception and evaporation from wet leaves are important processes modeled that should be modelled, according to Deardorff (1978). It is possible to distinguish two fundamental approaches: single-layer canopy models and multi-layer canopy models.
Single-layer canopy models (or "big leaf" models) The vegetation resistance is entirely determined by stomal resistance and only one temperature value, representative of both vegetation and soil, is considered. Moreover a vegetation interception function can be defined so as to define when the foliage is wet or when the evaporation is controlled by stomal resistance.

Multi-layer canopy models
These are more complex models in which a soil temperature T g , different from the foliage temperature T f , is considered. Therefore, two pairs of equations of latent and sensible heat flux transfer, from the soil level to the foliage level, and from the latter to the free atmosphere, must be considered (Best, 1998). Moreover the equation for the net radiation calculation must consider the energy absorption and the radiation reflection by the vegetation layer. Deardorff (1978) is the first author who presents a two-layer model with a linear interpolation between zones covered with vegetation and bare soil, to be inserted into atmosphere general circulation models. Over the last years many detailed models have been developed, above all with the purpose of evaluating the CO 2 fluxes between vegetation and atmosphere. Particularly complex is the case of scattered vegetation,

390
Evapotranspiration -Remote Sensing and Modeling www.intechopen.com where evaporation is due to a combination of soil/vegetation effects, which cannot be schematized as a single layer (Scanlon & Albertson, 2003). , and with the deep soil (through evapotranspiration with resistance r s ), as well as with the soil under the vegetation (r d ), vegetation and soil surface layer are assumed to have different temperature T f and T g . P g is the rainfall reaching the soil surface (from Garratt, 1992).
Given the many uncertainties regarding the forcing data and the components involved (soil, atmosphere), and the numerous simplifying hypotheses, the detail requested in a vegetation cover scheme is not yet clear. A single-layer description of vegetation cover (big-leaf) and a two-level description of soil represent probably the minimum level of detail requested. In general, if the horizontal scale is far larger than the vegetation scale, a single-layer model is sufficient (Garratt, 1992, p. 242), as in the case of the general circulation atmospheric models or of mesoscale hydrologic models for large basins. These models determine evaporation as if the vegetation cover were but a partially humid plane at the atmosphere basis. In an approach of this kind surface resistance, friction length, albedo and vegetation interception must be specified. The surface resistance must include the dependence on solar radiation or on soil moisture, as transpiration decreases when humidity becomes smaller than the withering point (Jarvis & Morrison, 1981

Water in soils
Real evaporation is coupled to the infiltration process occurring in the soil, and its physically-based estimate cannot leave the estimation of soil water content consideration. The most simple schemes to account water in soils used in LSMs single-layer and two-layer methods. The most general approach, which allows water transport for unsaturated stratified soil, is based on the integration of Richards (1931) equation, under different degrees of approximations.

Single layer or bucket method
In this method the whole soil layer is considered as a bucket and real evaporation E 0 is a fraction x of potential evaporation E p , with x proportional to the saturation of the whole soil.
with x expressed by Eq. (44). The main problem of this method is that evaporation does not respond to short precipitation, leading to surface saturation but not to a saturation of the whole soil layer (Manabe, 1969).

Two-layer or force restore method
This method is analogous to the one developed to calculate the soil heat flux, but it requires calibration parameters which are unlikely to be known. With this method it is possible to consider the water quantity used by plants for transpiration, considering a water extraction by roots in the deepest soil layer (Deardorff, 1978).

Multilayer methods and Richards equation
Richards (1931) equation and Darcy-Buckingham law govern the unsaturated water transport in isobar and isothermal conditions: where q =( q x , q y , q z ) is the specific discharge, K is the hydraulic conductivity tensor, z is the upward vertical coordinate and ψ is the suction potential or matrix potential. The determination of the suction potential allows also a more correct schematization of the plant transpiration and it lets us describe properly flow phenomena from the water table to the surface, necessary to the maintenance of evaporation from the soils. Richards equation is, rightfully, an energy balance equation, even if this is not evident in the modes from which it has been derived. Then the solutions of the equation (48) must be searched by assigning the water retention curve which relates ψ with the soil water content η and an explicit relation of the hydraulic conductivity as a function of ψ (or η). Both relationships depend on the type of terrain and are variable in every point. K augments with η, until it reaches the maximum value K s which is reached at saturation.

392
Evapotranspiration -Remote Sensing and Modeling www.intechopen.com Although the integration of the Richards equation is the only physically based approach, it requires remarkable computational effort because of the non linearity of the water retention curve. It is difficult to find a representative water retention curve because of the high degree of spatial variability in soil properties (Cordano & Rigon, 2008).

Spatial variability in soil moisture and evapotranspiration
Topography controls the catchment-scale soil moisture distribution (Beven & Freer, 2001) and therefore water availability for ET. Two methods most frequently used to incorporate sub-grid variability in soil moisture and runoff production SVATs models are the variable infiltration capacity approach (Wood, 1991) and the topographic index approach (Beven & Kirkby, 1979). They represent computationally efficient ways to represent hydrologic processes within the context of regional and global modeling. A review and a comparison of the two methods can be found in Warrach et al. (2002). More detailed approaches need to track surface or subsurface flow within a catchment explicitly. Such approaches, which require to couple the ET model with a distributed hydrological model, are particularly useful in mountain regions, as presented in the next section.

Evapotranspiration in Alpine Regions
In alpine areas, evapotranspiration (ET) spatial distribution is controlled by the complex interplay of topography, incoming radiation and atmospheric processes, as well as soil moisture distribution, different land covers and vegetation types.
1. Elevation, slope and aspect exert a direct control on the incoming solar radiation (Dubayah et al., 1990). Moreover, elevation and the atmospheric boundary layer of the valley affect the air temperature, moisture and wind distribution (e.g., Bertoldi et al., 2008;Chow et al., 2006;Garen & Marks, 2005).
Spatially distributed hydrological and land surface models (e.g., Ivanov et al., 2004;Kunstmann & Stadler, 2005;Wigmosta et al., 1994) are able to describe land surface interactions in complex terrain, both in the temporal and spatial domains. In the next section we show an example of the simulation of the ET spatial distribution in an Alpine catchment simulated with the hydrological model GEOtop (Endrizzi & Marsh, 2010;Rigon et al., 2006).  (Bertoldi et al., 2005), used in this example. For details of the most recent numerical implementation, see Endrizzi & Marsh (2010). The model has been proved to simulate realistic values for the spatial and temporal dynamics of soil moisture, evapotranspiration, snow cover (Zanotti et al., 2004) and runoff production, depending on soil properties, land cover, land use intensity and catchment morphology (Bertoldi et al., 2010;. The model is able to simulate the following processes: (i) coupled soil vertical water and energy budgets, through the resolution of the heat and Richard's equations, with temperature and water pressure as prognostic variables (ii) surface energy balance in complex topography, including shadows, shortwave and longwave radiation, turbulent fluxes of sensible and latent heat, as well as considering the effects of vegetation as a boundary condition of the heat equation (iii) ponding, infiltration, exfiltration, root water extraction as a boundary condition of Richard's equation (iv) subsurface lateral flow, solved explicitly and considered as a source/sink term of the vertical Richard's equation (v) surface runoff by kinematic wave, and (vi) multi-layer glacier and snow cover, with a solution of snow water and energy balance fully integrated with soil. The incoming direct shortwave radiation is computed for each grid cell according to the local solar incidence angle, including shadowing (Iqbal, 1983). It is also split into a direct and diffuse component according to atmospheric and cloud transmissivity (Erbs et al., 1982). The diffuse incoming shortwave and longwave radiation is adjusted according to the theory described in Par. 2.1. The soil column is discretized in several layers of different thicknesses. The heat and Richards' equations are written respectively as:

Evapotranspiration in the GEOtop model
Where T is soil temperature, P the water pressure, C t the thermal capacity, K t the thermal conductivity, C h the specific volumetric storativity, K h the hydraulic conductivity, and q s the source term associated with lateral flow. The variables C t , K t , C h , and K h depend on water content, and, in turn, on water pressure, and are therefore a source of non-linearity. At the bottom of the soil column a boundary condition of zero fluxes has been imposed.
The boundary conditions at the surface are consistent with the infiltration and surface energy balance, and are given in terms of surface fluxes of water (Q h ) and heat (Q t ) at the surface, namely: Where p net is the net precipitation, K h1 and P 1 are the hydraulic conductivity and water pressure of the first layer, h is the pressure of ponding water, dz the thickness of the first layer, T 1 the temperature of the first layer. E is evapotranspiration (as water flux), SW in and SW out are the incoming and outgoing shortwave radiation, LW in and LW out the incoming  (Monin & Obukhov, 1954). E is partitioned by evaporation or sublimation from the soil or snow surface E G , transpiration from the vegetation E TC , evaporation of the precipitation intercepted by the vegetation E VC . Every cell has a fraction covered by vegetation and a fraction covered by bare soil. In the 0.875 version of the model, a one-level model of vegetation is employed, as in Garratt (1992) and in Mengelkamp et al. (1999): only one temperature is assumed to be representative of both soil and vegetation. In the most recent version, a two-layer canopy model has been introduced. Bare soil evaporation E g is related to the water content of the first layer through the soil resistance analogy (Bonan, 1996): where cop is fraction of soil covered by the vegetation EP is the potential ET calculated with equation 34 and r a the aerodynamic resistance: The soil resistance r s is function of the water content of the first layer.
where η 1 is the water content of the first soil layer close the surface, η r is the residual water content (defined following Van Genuchten, 1980) and η s is the saturated water content, both in the first soil layer. The evaporation from wet vegetation is calculated following Deardorff (1978): where δ W is the wet vegetation fraction. The transpiration from dry vegetation is calculated as: The root fraction f i root of each soil layer i is calculated decreasing linearly from the surface to a maximum root depth, depending from the cover type. The canopy resistance r c depends on solar radiation, vapor pressure deficit, temperature as in Best (1998) and on water content in the root zone as in Wigmosta et al. (1994).

The energy balance at small basin scale: application to the Serraia Lake.
An application of the model to a small basin is shown here, in order to bring out the problems arising when passing from local one-dimensional scale to basin-scale. The Serraia Lake basin is a mountain basin of 9 km 2 , with an elevation ranging from 900 to 1900 m, located in Trentino, Italy. Within the basin there is a lake of about 0.5 km 2 . During the year 2000 a study to calculate the yearly water balance was performed (Bertola & al., 2002).

395
Modelling Evapotranspiration and the Surface Energy Budget in Alpine Catchments www.intechopen.com The model was forced with meteorological measurement of a station located in the lower part of the basin at about 1000 m, and the stream-flow was calibrated for the sub-catchment of Foss Grand, of about 4 km 2 . Then the model was applied to the whole basin. Further details on the calibration can be found in Salvaterra (2001). Meteorological data are assumed to be constant across the basin, except for temperature, which varies linearly with elevation (0.6 o C / 100 m) and solar radiation, which slightly increases with elevation and is affected by shadow and aspect.
With the GEO TOP model it is possible to simulate the water and energy balance, aggregated for the whole basin (see figure 6 and 7) and its distribution across the basin. Figure 7 shows the map of the seasonal latent heat flux (ET) in the basin. During winter and fall ET is low (less than 40 W/m 2 ), with the lowest values in drier convex areas. During summer and spring ET increases (up to 120 W/m 2 ), with highest values in the bottom of the main valley (where indeed there are a lake and a wetland) and lowest values in north-facing, high-elevation areas. The main factors controlling the ET pattern in a mountain environment (see Figure 8) are also: elevation, which controls temperature, aspect, which influences radiation, soil thickness, which determines storage capacity, topographic convergence, which controls the moisture availability. In particular, aspect has a primary effect on net radiation and a secondary effect more on sensible rather than on latent heat flux, as in Figure 9, where south aspect locations have larger R n and H, but similar behavior for the other energy budget components). Water content changes essentially the rate between latent and heat flux, as in Figure 10 where wet locations have larger ET and lower H. Therefore, the surface fluxes distribution seems to agree with experience and current hydrology theory, but the high degree of variability poses some relevant issues because the hypothesis of homogenous turbulence at the basis of the fluxes calculation is no more valid     It can be noticed how wet locations have larger ET and lower H, but similar behavior for the other fluxes. The time lag in R n is due to differences in aspect.

398
Evapotranspiration -Remote Sensing and Modeling www.intechopen.com (Albertson & Parlange, 1999). Moreover, horizontal differences in surface fluxes can start local air circulations, which can affect temperature and wind surface values with a feedback effect. How much such processes may affect the energy and water balance of the whole basin is easy to quantify, but GEO TOP can be a powerful tool to explore these issues.

Conclusion
This chapter illustrates the components of the energy budget needed to model evapotranspiration (ET) and provides an extended review of the fundamental equations and parametrizations available in the hydrological and land surface models literature. In alpine areas, ET spatial distribution is controlled by the complex interplay of topography, incoming radiation and atmospheric processes, as well as soil moisture distribution, different land covers and vegetation types. An application of the distributed hydrological model GEOtop to a small basin is shown here, in order to bring out the problems arising when passing from local one-dimensional scale to basin-scale ET models.