Improving evapotranspiration estimates in Mediterranean drylands: The role of soil evaporation
Estación Experimental de Zonas Áridas, Consejo Superior de Investigaciones Científicas (CSIC), Ctra. de Sacramento s/n La Cañada de San Urbano, Almería, Spain
Corresponding author: L. Morillas, Estación Experimental de Zonas Áridas, Consejo Superior de Investigaciones Científicas (CSIC), Ctra. de Sacramento s/n La Cañada de San Urbano, Almería ES-04120, Spain. (firstname.lastname@example.org)
 An adaptation of a simple model for evapotranspiration (E) estimations in drylands based on remotely sensed leaf area index and the Penman-Monteith equation (PML model) (Leuning et al., 2008) is presented. Three methods for improving the consideration of soil evaporation influence in total evapotranspiration estimates for these ecosystems are proposed. The original PML model considered evaporation as a constant fraction (f) of soil equilibrium evaporation. We propose an adaptation that considers f as a variable primarily related to soil water availability. In order to estimate daily f values, the first proposed method (fSWC) uses rescaled soil water content measurements, the second (fZhang) uses the ratio of 16 days antecedent precipitation and soil equilibrium evaporation, and the third (fdrying), includes a soil drying simulation factor for periods after a rainfall event. E estimates were validated using E measurements from eddy covariance systems located in two functionally different sparsely vegetated drylands sites: a littoral Mediterranean semiarid steppe and a dry-subhumid Mediterranean montane site. The method providing the best results in both areas was fdrying (mean absolute error of 0.17 mm day−1) which was capable of reproducing the pulse-behavior characteristic of soil evaporation in drylands strongly linked to water availability. This proposed model adaptation, fdrying, improved the PML model performance in sparsely vegetated drylands where a more accurate consideration of soil evaporation is necessary.
 Evapotranspiration (E), is the largest term in the terrestrial water balance after precipitation. Additionally, its energetic equivalent, the latent heat flux (λE), plays an important role in the surface energy balance affecting terrestrial weather dynamics and vice versa. The importance of E in drylands, covering 45% of the Earth surface [Asner et al., 2003; Schlesinger et al., 1990], is critical since it accounts for 90–100% of the total annual precipitation [Glenn et al., 2007]. Therefore, an accurate regional estimation of E is crucial for many operational applications in drylands: irrigation planning, management of watersheds and aquifers, meteorological predictions, and detection of droughts and climate change.
 Remote sensing has been recognized as the most feasible technique for E estimation at regional scales with a reasonable degree of accuracy [Kustas and Norman, 1999; Mu et al., 2011]. Several methods have been developed for estimating regional E in the last decades. Many of them are based on the indirect estimation of E as a residual of the surface energy balance equation (SEB) using direct estimates of the sensible heat flux (H) derived from remotely sensed surface temperatures [Glenn et al., 2007; Kalma et al., 2008]. However, residual estimation of E in Mediterranean drylands remains problematic due to the reduced magnitude of λE in conditions where H is the dominant flux [Morillas et al., 2013]. Reduced inaccuracies affecting estimates of Rn and H derived from surface temperature measurements (∼10 and ∼30%, respectively) strongly affected the residually estimated values of λE (∼90% of error) in such conditions [Morillas et al., 2013]. This suggests that direct estimation of E might be more advisable in Mediterranean drylands.
Cleugh et al.  presented a method for direct estimation of E based on regional application of the Penman-Monteith (PM) equation [Monteith, 1964] using leaf area index (LAI) from MODIS (Moderate Resolution Imaging Spectrometer) and gridded meteorological data. This work stimulated a number of later studies [Leuning et al., 2008; Mu et al., 2007, 2011; Zhang et al., 2010, 2008] that have demonstrated the potential of the PM equation as a robust and biophysically based framework for E direct estimation using remote-sensing inputs [Leuning et al., 2008].
 The key parameter of the PM equation is the surface conductance (Gs), the inverse of the resistance of the soil-canopy system to lose water. A simple linear relationship between Gs and LAI was initially proposed by Cleugh et al.  to estimate E at two field sites in Australia. Mu et al. [2007, 2011] took one step forward with separate estimations for the two major components of E: canopy transpiration (Ec) and soil evaporation (Es), both controlled by different biotic and physical processes in sparse vegetated areas [Hu et al., 2009]. Mu et al. [2007, 2011] included a formulation for Ec considering the effects of vapor pressure deficit (Da) and air temperature (Ta) on canopy conductance (Gc) but assumed constant parameters for each vegetation type. Based on these studies, Leuning et al.  developed a less empirical formulation for Gs to apply the PM equation regionally. This new formulation also considers both Ec and Es. For Gc, a more biophysical algorithm based on radiation absorption and Da was proposed by Leuning et al.  based on Kelliher et al. . In this case, Es is estimated as a constant fraction, f, of soil equilibrium or potential evaporation [Priestley and Taylor, 1972] defined as the evaporation occurring under given meteorological conditions from a continuously saturated soil surface [Donohue et al., 2010; Thornthwaite, 1948]. Application of the Penman-Monteith-Leuning, PML model, as it was named by Zhang et al. , requires commonly available meteorological data (more details in section 2), LAI data from MODIS or other remote-sensing platforms and two main parameters, considered by Leuning et al.  to be constants: gsx, maximum stomatal conductance of leaves at the top of the canopy and f, representing the ratio of soil evaporation to the equilibrium rate. The potential of the PML for global estimates of E is promising as shown by accurate estimates (systematic root-mean-square error of 0.27 mm day−1) found in 15 Fluxnet sites located across a wide range of climatic conditions, from wetlands to woody savannas [Leuning et al., 2008]. Nonetheless, the latter model has not been tested in Mediterranean drylands characterized by strongly reduced magnitudes of E (mean annual E values ranging 0.5 mm day−1) resulting from the typical asynchrony of energy and water availability in these environments [Serrano-Ortiz et al., 2007].
 In drylands, where water availability is the main controlling factor of biological and physical processes [Noy-Meir, 1973], evaporation from soil can exceed 80% of total E [Mu et al., 2007]. Soil water availability, the main factor controlling Es in water-limited areas [McVicar et al., 2012], is highly variable in these ecosystems and, therefore, assuming f as constant, as the original PML model of Leuning et al.  did, is inadequate. Leuning et al.  acknowledged this limitation and recommended that remote-sensing or other techniques should be developed to treat f as a variable instead of a parameter, especially for sparsely vegetated sites (LAI < 3). Many authors have also claimed the necessity to increase the efforts to carefully quantify the Es contribution to total E in low LAI ecosystems as semiarid grasslands and shrublands [Hu et al., 2009; Kurc and Small, 2004]. Numerous E models that include specific methods for Es estimation, from the simplest to the most complex formulations, exist [Allen et al., 1998; Fisher et al., 2008; Kite, 2000; Mu et al., 2007; Shuttleworth and Wallace, 1985]. Special attention has been paid to this topic in the agronomy sector because from an agricultural point of view, soil evaporation is considered an unproductive use of water that requires quantification [Kite and Droogers, 2000]. Thus, many efforts have been devoted to improve Es formulation in croplands [Kite, 2000; Lagos et al., 2009; Snyder et al., 2000; Torres and Calera, 2010; Ventura et al., 2006]. The FAO 56 methodology [Allen et al., 1998] is one of the most used methods in agricultural areas due to its capacity to estimate both Es and Ec beyond standard conditions (well-watered conditions) and some subsequent refinements have been proposed [Snyder et al., 2000; Torres and Calera, 2010; Ventura et al., 2006]. However, when applying this method, detailed local soil characteristics, such as depth of soil or soil texture, are needed for estimating Es. This limits the regional application of this model beyond agricultural areas where little detailed soil information is available. There are other types of models partitioning the total E by considering a different number of layers or sources like the sparse-crop model of Shuttleworth and Wallace  or the model from Brenner and Incoll . The layers are defined depending on the site-specific surface heterogeneity (i.e., canopy, bare soil, under plant soil, residue covered soil, etc.). These models have provided successful results in sparsely vegetated areas such as irrigated agricultural scenarios [Lagos et al., 2009; Ortega-Farias et al., 2007] and natural conditions [Domingo et al., 1999; Hu et al., 2009]. Yet, they require specific information regarding the vegetation physiology and the substrate. Furthermore, complex modeling of aerodynamic and surface resistances governing the flux from each layer is necessary, limiting its regional application. From another perspective, the distributed hydrological models also deal with Es estimation. These models consider all the water reservoirs, modeling runoff and infiltration processes in a basin scale using satellite data [Kite, 2000; Kite and Droogers, 2000] to offer E estimates at macroscale basins. However, these models require the measurements of all the terms of the hydrological balance to be validated. Those measurements are not routinely available for many macroscale basins.
 From a more regionally operative point of view, several models designed for global E estimation have also successfully estimated Es as a fraction, f, of soil equilibrium evaporation, as the PML model proposed. That soil equilibrium evaporation rate has been estimated using the PM equation [Mu et al., 2007, 2011] or the Priestley-Taylor equation [Fisher et al., 2008; García et al., 2013; Zhang et al., 2010] but all these models considered f as temporally variable. f has been estimated as a function of Da, relative humidity and a locally calibrated parameter β (which indicates the relative sensitivity of soil moisture to Da) every month or 8 days periods [Fisher et al., 2008; Mu et al., 2007, 2011]. Garcia et al.  proved that such approach is very sensitive to β parameter in a daily time basis and consequently proposed an alternative formulation for f based on Apparent Thermal Inertia using surface temperature and albedo observations. Finally, Zhang et al.  used the ratio between precipitation and equilibrium evaporation rate as an indicator of soil water availability to obtain f values over successive 8 days intervals.
 Because Mediterranean drylands are characterized by irregular precipitation which causes rapid increases in soil moisture during rain followed by extended drying periods, we considered it important to develop a specific formulation for f that models the soil drying process after precipitation. Black et al.  and Ritchie  presented a simple formulation to model the soil drying process as a function of the time (in days) following precipitation that we adapted for daily f estimation.
 The objective of this paper was to adapt and evaluate the PML model for estimating daily E in Mediterranean drylands where a more precise consideration of Es is necessary. To achieve this goal, we tested three different approaches to estimate the temporal variation of f: (i) using direct soil water content measurements; (ii) adapting Zhanget al.'s  method for daily application; and (iii) including a simple model for modeling the soil drying after precipitation based on Black et al.  and Ritchie . The PML model performance using the three f approaches was evaluated by comparison with E measurements obtained from eddy covariance systems at two functionally different Mediterranean drylands: (i) a littoral semiarid steppe and (ii) a shrubland montane site.
2. Model Description
2.1. Penman-Monteith-Leuning Model (PML) Description
 Actual evapotranspiration (E) is the sum of canopy transpiration (Ec), soil evaporation (Es), and evaporation of precipitation intercepted by canopy and litter (Ei) [D'odorico and Porporato, 2006]. Despite the fact that Ei has been shown to account for up to 30% of the annual rainfall in some arid communities [Dunkerley and Booth, 1999], the magnitude of Ei is considered a small amount of the total water losses in areas with low ecosystem LAI or short vegetation because of the reduced fraction cover of plants and the lower aerodynamic conductance of these areas in comparison with forests [Mu et al., 2007; Muzylo et al., 2009]. Moreover, in Mediterranean areas, a reduced relative magnitude of Ei can be expected because precipitation events are intense and they occur mainly in the lower available energy seasons (autumn and winter), both factors decreasing the interception fraction [Domingo et al., 1998]. In this regard, Garcia et al.  reported no improvements on actual E estimation by considering Ei in two natural semiarid sites, one of them included in this work. Therefore, in the present work, only Ec and Es were considered for actual E estimation following the expression,
 The fluxes of latent heat associated with Ec and Es were written by Leuning et al.  as
where the first term is the PM equation written for the plant canopy and the second term is the flux of latent heat from the soil expressed as a fraction of potential. The variables Ac and As (W m−2) are the energy absorbed by the canopy and soil, respectively. Ga and Gc (m s−1) are the aerodynamic and canopy conductances, as defined below. ε (kPa K−1) is the slope (s) of the curve relating saturation water vapor pressure to air temperature divided by the psychrometric constant (γ), ρ (kg m−3) is air density, cp (J kg K−1) is the specific heat of air at constant pressure, and Da (kPa) is the vapor pressure deficit of the air, computed as the difference between the saturation vapor pressure at air temperature, esat, and the actual vapor pressure, e (Da = esat − e). The factor f in the second term of equation (2) modulates potential evaporation rate at the soil surface expressed by the Priestley-Taylor equation, , by f = 0 when the soil is dry, to f = 1 when the soil is completely wet. In spite of the Priestley-Taylor formulation was designed to estimate potential evaporation in energy-limited ecosystems [Priestley and Taylor, 1972], recent works have demonstrated that accurate estimates of actual E can be determined in water-limited conditions by downscaling Priestley-Taylor potential evapotranspiration according to multiple stresses at daily time scale [Fisher et al., 2008; Garcia et al., 2013] as the PML model does through f.
 To estimate partitioning of available energy between soil and canopy surfaces, the Beer-Lambert law has been applied by many authors even in sparse vegetated areas [Hu et al., 2009; Leuning et al., 2008; Zhang et al., 2010]. Based on Beer-Lambert law, soil available energy can be estimated as As = Aτ and canopy available energy is Ac = A(1 − τ), where τ = exp(-kALAI) and kA is the extinction coefficient for total available energy A. When eddy covariance data are used for validation, A = H +λE can be assumed in order to ensure internal consistency in relation to eddy covariance closure error [Leuning et al., 2008]. Kustas and Norman  have, however, questioned the reliability of the Beer-Lambert approach in sparse vegetation. Alternatively, they proposed a more complex method for energy partitioning based on surface temperature and shortwave incoming radiation retrievals that accounts for the different behavior of soil and canopy for the visible and near infrared regions of spectrum. Preliminary analyses included in Appendix A showed that mean absolute differences between daytime averages of Ac and As estimated by those two energy partitioning approaches were minor (18 and 32 W m−2 for Ac and As, respectively) over 144 days in 2011 when infrared sensors were available to measure surface temperature and shortwave incoming radiation. Because of these reduced differences (Figure A1) at daytime scale, the Beer-Lambert method was used to maintain the reduced number of PML model inputs. Of far greater importance is correctly estimating f, as discussed below.
 Aerodynamic conductance Ga is estimated using [Monteith and Unsworth, 1990]
where k is Von Karman's constant (0.40), u (m s−1) is wind speed, d (m) is zero plane displacement height, zom and zov (m) are roughness lengths governing transfer of momentum and water vapor and zr (m) is the reference height where u is measured. In this version of equation (3), the influence of atmospheric stability conditions over Ga has been neglected for two reasons: (i) in dry surfaces where Gc << Ga, E is relatively insensitive to errors in Ga [Leuning et al., 2008; Zhang et al., 2010, 2008] and (ii) in semiarid areas, where highly negative temperature gradients between surface and air temperature are found, correction for atmospheric stability can cause more problems than it solves for estimating Ga [Villagarcia et al., 2007]. The variables d, zom, and zov were estimated via the canopy height (h) in m, using the general relations given by Allen : d = 0.66 h, zom = 0.123 h, and zov = 0.1.
 Canopy conductance was estimated using Leuning et al.  formulation, based on Kelliher et al. , as follows,
where kQ, is the extinction coefficient of visible radiation, gsx (m s−1) is the maximum conductance of the leaves at the top of the canopy, Qh (W m−2) is the visible radiation reaching the canopy surface that can be approximated as Qh = 0.8A [Leuning et al., 2008] and Q50 (W m−2) and D50 (kPa) are values of visible radiation flux and water deficit, respectively, when the stomatal conductance is half of its maximum value. We used Q50 = 30 Wm−2, D50 = 0.7 kPa, and kQ = kA = 0.6 following the sensitivity analysis presented in Leuning et al. .
 The PML model (equations (2)-(4)) includes factors controlling canopy transpiration and soil evaporation but accurate estimation of gsx and f is crucial for model success. Three methods for estimating f, with increasing complexity, presented in section 2.2 were evaluated for improving PML performance in drylands.
2.2. Methods for f Estimation
 Evaporation from soil surfaces is mainly controlled by volumetric soil water content in the top soil layer [Anadranistakis et al., 2000; Farahani and Bausch, 1995] and has been traditionally described occurring in three stages. An energy-limited stage (Stage 1) when enough soil water is available to satisfy the potential evaporation rate (f = 1), a falling-rate stage (Stage 2) when soil is drying and water availability limits the soil evaporation rate (0 < f < 1) and a third stage (Stage 3) when soil is dry and it can be considered negligible (f = 0) [Idso et al., 1974; Ventura et al., 2006]. We tested three different methods to capture this dynamic of f.
2.2.1. f As a Function of Soil Water Content Data (fSWC)
 We used measured values of volumetric soil water content measured at 4 cm depth (θobs) rescaled between a minimum (θmin) and a maximum (θmax) threshold value to estimate f following the expression,
θmin was experimentally estimated as the minimum value of the dry season and θmax as the value of θ in the 24 h after a strong rainfall event, which can be considered as an estimate of the field capacity [Garcia et al., 2013], using data measured during the study period.
 We tested the method proposed by Zhang et al.  to estimate f using the ratio of accumulated values of precipitation (P) and Eeq,s, both in mm day−1, over N days. While the original formulation of Zhang et al.  was designed to estimate the averaged value of f over successive 8 day intervals using accumulated values of P and Eeq,s in N = 32 days (covering 16 days prior and 16 days after the current day i), we adapted this method for daily estimates of f. After a sensitivity analysis, included in Appendix B, here we set N = 16, between day i and 15 preceding days (i − 15), to estimate daily f using measured values of P and Eeq,s and it is expressed as,
where Pi is the accumulated daily precipitation and Eeq,s,i is the daily soil equilibrium evaporation rate for day i.
2.2.3. f as a Function of Soil Drying After Precipitation (fdrying)
Black et al.  formulated the cumulative evaporation in terms of the square root of time after precipitation considering the soil drying process after rain and Ritchie  used the same approach for modeling the Stage 2 of soil evaporation. Thus for daily f estimation, we proposed to add use a similar formulation for the soil drying periods during dry days in combination with the fZhang method (equation (7)) used here to estimate f during the effective precipitation days (Pi > Pmin = 0.5 mm day−1). This is,
where fLP is the f value for the last effective precipitation day, Δt is number of days between this and the current day i and α (day−1) is a parameter controlling the rate of soil drying, higher α values reflecting higher soil drying speed. For simplicity, α was considered a constant estimated by optimization, even though it is known that α is related to air temperature, wind speed, vapor pressure deficit, and soil hydraulic properties [Ritchie, 1972].
3. Material and Methods
3.1. Validation Field Sites and Measurements
 The PML model was evaluated at two experimental sites located in southeast Spain characterized by Mediterranean climate, sparse vegetation (LAI < 1) and winter rainfall (see Table 1). Both sites are water-limited areas, following the classification proposed by McVicar et al. , with dryness index [Budyko, 1974] of 2.8 and 2.3, respectively, during the study period. These are stronger aridity conditions than where the PML model has been previously tested [Leuning et al., 2008; Zhang et al., 2010].
 Water vapor fluxes were measured at each site using eddy covariance (EC) systems consisting of a three axis sonic anemometer (CSAT3, Campbell Scientific Inc., USA) for wind speed and sonic temperature measurement and an open-path infrared gas analyzer (Li-Cor 7500, Campbell Scientific Inc., USA) for variations in H2O density. EC sensors were located above horizontally uniform vegetation at 3.5 m at Balsa Blanca and at 2.5 m at Llano de los Juanes (zr = 3.5 and zr = 2.5, respectively). Data were sampled at 10 Hz and fluxes were calculated and recorded every 30 min. Corrections for density perturbations [Webb et al., 1980] and coordinate rotation [Kowalski et al., 1997; McMillen, 1988] were carried out in postprocessing, as was the conversion to half-hour means following Reynolds' rules [Moncrieff et al., 1997]. The slope of the linear regressions between available energy (Rn − G) and the sum of the surface fluxes (H + λE) yields a slope ∼0.8 in Balsa Blanca and ∼0.7 in Llano de los Juanes. This is consistent with the ∼20% of energy imbalance found in the European FLUXNET stations [Franssen et al., 2010].
 Complementary meteorological measurements were also made at each field site. An NR-Lite radiometer (Kipp & Zonen, Netherlands) measured net radiation over representative surfaces at 1.9 m height at Balsa Blanca and 1.5 m at Llano de los Juanes. Soil heat flux was calculated at both sites following the combination method [Fuchs, 1986; Massman, 1992], as the sum of averaged soil heat flux measured by two flux plates (HFT-3; REBS, Seattle,Wa, USA) located at 0.08 m depth, plus heat stored in upper soil measured by two thermocouples (TCAV; Campbell Scientific LTD) located at two depths 0.02 and 0.06 m. Air temperature and relative humidity were measured by thermohygrometers located at 2.5 m height at Balsa Blanca field site and 1.5 m at Llano de los Juanes (HMP45C, Campbell Scientific Ltd., USA). A 0.25 mm resolution pluviometer (model ARG100 Campbell Scientific INC., USA) was used to measure precipitation at Balsa Blanca and a 0.2 mm resolution pluviometer was used at Llano de los Juanes (model 785, Davis Instruments Corp. Hayward, California, USA). Soil water content was measured at both sites using water content reflectometers (model CS616, Campbell Scientific INC., USA) located at 0.04 m depth with a reported accuracy by the manufacturer of ±2.5% volumetric water content. Due to the high soil heterogeneity, three randomly located sensors were averaged to obtain a representative SWC value at Llano de los Juanes, while at Balsa Blanca, one sensor located in bare soil was used. All complementary measurements were recorded every 30 min using data loggers (Campbell CR1000 and Campbell CR3000 data loggers, Campbell Scientific Inc., USA) and daytime (from sunrise to sunset) averages were used for model running.
Table 1. Details of Field Sites Used to Evaluate the PML Model Performancea
Llano de los Juanes
Quantitative data were derived using data from the entire study period (Table 2).
Dryness index calculated as the average of the annual dryness index (Eeq/P) [Budyko, 1974] for the total study period (Table 2).
LAI estimates were level 4 Moderate Resolution Imaging Spectrometers (MODIS) composite products provided by the ORNL-DAAC (http://daac.ornl.gov): (i) MOD15A (collection 5) from the Terra satellite and (ii) MYD15A2 from the Aqua satellite, both with a temporal resolution of 8 days. The averaged value of LAI reported from MOD15A and MYD15A2 for the 3 km × 3 km area centered on each site EC tower was computed. Filtering was performed according to MODIS quality assessment (QA) flags to eliminate poor quality data (affecting five and three observations at Balsa Blanca site and Llano de los Juanes, respectively) which were replaced by the average of previous and subsequent LAI values.
3.3. Model Performance Evaluation
 Average daytime E measurements were used to validate daily estimates of E derived from the PML model run using average daytime micrometeorological data [Cleugh et al., 2007; Leuning et al., 2008; Zhang et al., 2010]. The measurement data sets were divided into an optimization period, to estimate locally specific gsx and α values using the rgenoud package for the R software environment [Mebane and Sekhon, 2011], and a validation period, to validate PML model outputs at both field sites (see Table 2). The optimization was performed to find the values of gsx and α that minimized the cost function F for the total sample number, N, included in the optimization period (See N values in Tables 2 and 4), that is:
where Eest,i is estimated E for day i and Eobs,i is observed E for same day.
Table 2. Optimization and Validation Periods Used in Both Field Sites
Experimental Field Site
18 Oct 2006
19 Oct 2007
18 Oct 2007
31 Dec 2008
N = 365 days
N = 440 days
Llano de los Juanes
27 Mar 2007
4 Apr 2005
31 Dec 2007
24 Mar 2006
N = 279 days
N = 355 days
Standarized Major Axis Regression (SMA) type II [Warton et al., 2006] was used for comparing daily measurements and model estimates of E during the validation period. SMA regression attributes error in the regression line to both the X and Y variables, a method which is recommended when the X variable is subject to measurement errors, as is assumed for the EC system measurements used in this work. Slope, intercept, and coefficient of determination (R2) computed using SMA regression were reported in XY plots. Mean absolute difference (MAD) [Willmott and Matsuura, 2005] is used for quantitative evaluation of PML model results, while root mean square difference (RMSD) is also presented for comparison with previous works. Systematic and unsystematic components of RMSD [Willmott, 1982] are also reported. A low systematic difference indicates model structure adequately captures the system dynamics [Choler et al., 2010].
 The two studied sites are Mediterranean drylands with clear functional differences (Figures 1a and 1b). Both sites presented a very different temporal pattern in phenology (LAI) with an early spring maximum at Balsa Blanca and a late-spring maximum at Llano de los Juanes. Balsa Blanca presented intermittent rainfall throughout the year causing a more fluctuating SWC pattern than at Llano de los Juanes which had distinct wet and dry seasons. These functional differences were also found in the temporal E pattern, that was more fluctuating at Balsa Blanca where E was more strongly linked to the SWC (Figures 1a and 1c), than at Llano de los Juanes where phenology was the main factor controlling E (Figures 1b and 1d).
 Optimized values of gsx were similar for both field sites under the three proposed formulations for f (gsx ranging from 0.0067 to 0.0109 m s−1) (Table 3). On the other hand, α = 0.137 day−1 at Balsa Blanca was considerably lower than α = 0.478 day−1 at Llano de los Juanes, which indicates the model considered a faster drying rate for Llano de los Juanes than for Balsa Blanca. Experimental values of θmax and θmin for applying fSWC were θmax = 0.20 m3 m−3 and θmin = 0.05 m3 m−3 at Balsa Blanca, and θmax = 0.35 m3 m−3 and θmin = 0.10 m3 m−3 at Llano de los Juanes.
Table 3. Optimized Model Parameters and Statistic of Model Performance for the Whole Validation Period (N = 440 Days in Balsa Blanca and N = 355 Days in Llano de los Juanes)
Eavg mean observed value of daily evapotranspiration (mm day−1) during the validation period in brackets and mean estimated values from each f approach. N/A, not applicable parameter.
 Predictions of E obtained using the PML model with fdrying were superior to both fSWC and fZhang, yielding the lowest values of MAD (0.17 mm day−1) and RMSD (0.22–0.24 mm day−1) at both study sites (Table 3). The percentage systematic difference was low using fdrying especially at Balsa Blanca site (18%), where fZhang also presented a low percentage systematic difference (5%). However, percentages of systematic difference remained higher at Llano de los Juanes using any of the three proposed methods for estimate f (40–42%).
 Using fSWC the PML model resulted in strong overestimations of E following heavy rainfall at both field sites (Figures 1e and 1f). A similar tendency was observed running the PML model using fZhang but not using fdrying that clearly reduced that tendency reaching a closer agreement with observations. However, all three methods for estimating f overestimated E when observed E was lower than 0.2 mm day−1 at Balsa Blanca, but systematically underestimated E at the beginning of the dry season at Llano de los Juanes mountain site coinciding with great part of the growing season (April to July of 2005). Reasons for this are discussed in section 5.
 Estimated values of daily E from the PML model are compared to observations at both field sites in Figure 2. Using fdrying in the PML model resulted in the best slope (0.98) and intercept (0.01) for linear correlation versus observed E, though the coefficient of determination (R2 = 0.47) using fdrying was slightly lower than with fSWC (R2 = 0.54) at Balsa Blanca. Despite the better correlation achieved using fSWC, this method tended to overestimate E values (Figure 2a), a problem not found using fdrying (Figure 2e). The highest correlation at Llano de los Juanes was again obtained using fdrying (R2 = 0.59), whereas using fSWC and fZhang produced two clusters of high and low predictions (Figures 2b and 2d) and hence poor coefficients of determination (R2 = 0.24 and R2 = 0.31, respectively). However, the tendency of the PML model with fdrying to underestimate E during the growing season at this site (when E > 1.10 mm day−1) reduced the linear agreement resulting in a linear regression slope of 0.79 (Figure 2f).
 Additional analyses were performed to determine if the systematic underestimation of E found at Llano de los Juanes during the growing season (Figure 2f) using the three f methods was caused by a too low gsx value reducing Ec. To evaluate if underestimates of gsx were being obtained by including in the optimization data set periods showing a very different vegetation activity at this strongly seasonal site (the growing and the nongrowing season) (Figure 1b), parameters optimizations were performed using specific periods (Table 4).
Table 4. Estimated Model Parameters by Optimizing Using the Original Optimization Period, the Growing Season or the Nongrowing Seasona
f Estimation Method
Abreviations as follows: gsx, maximum conductance of leaves; α, soil drying speed; and N/A, not applicable parameter.
27 Mar 2007
31 Dec 2007
18 Apr 2007
5 Aug 2007
10 Aug 2007
22 Dec 2007
 Our results showed that estimates of model parameters (gsx and α) did not significantly differ using different optimization periods (Table 4). Only optimized values of the gsx parameter for the non growing season using fSWC and fZhang were clearly lower. These lower values of gsx generated a better fit of model output during the nongrowing season using fSWC and fZhang but strongly increased the underestimates of E for the growing season (Figure 3c). Thus, improvement of model performance during the dry and growing season of the validation period was not found using model parameters optimized specifically for those conditions (Figure 3b). This test also showed a low sensitivity of the optimization method to the time period used especially using fdrying (Table 4).
 Important functional differences were observed between the two field sites, with an E pattern more strongly linked to SWC at Balsa Blanca but better explained by phenology in Llano de los Juanes (Figure 1). These results can be understood considering the vegetation composition and geomorphological characteristics of both field sites.
 At Balsa Blanca, the vegetation is dominated by the perennial grass S. tenacissima (57.2%) that is well adapted to aridity and shows opportunistic growth patterns with leaf conductance and photosynthetic rates largely dependent on water availability in the upper soil layer [Haase et al., 1999; Pugnaire and Haase, 1996]. This explains the observed link between E and SWC pattern here, where both Es and Ec are controlled by water availability in the upper soil layer. In contrast, the vegetation at Llano de los Juanes is codominated by perennial grasses, Festuca scariosa (Lag.) Hackel (19%), and shrubs, Genista pumila ssp pumila (11.5%) and Hormatophylla spinosa (L). P. Küpfer (6.3%) [Serrano-Ortiz et al., 2007, 2009]. At this montane site, extraction of water by shrubs from deep cracks and fissures in the bedrock has been previously detailed [Cantón et al., 2010] explaining the phenological control of E during the dry period and the coincidence of the dry and growing seasons. These functional considerations of the sites help to understand the performance of the three proposed methods to improve E estimates by the PML model.
5.1. Using Soil Water Content Data to Estimate Soil Evaporation (fSWC)
 Despite the fact that the energy consumed by Es mainly depends on the moisture content of the soil near the surface in water-limited areas [Leuning et al., 2008; McVicar et al., 2012], the PML model using fSWC (equation (5)) provided unsatisfactory estimates of E (Table 3). This method tended to systematically overestimate E at Balsa Blanca (Figure 2a) and presented a poor linear agreement with measured E at Llano de los Juanes (Figure 2b). A similar approach to fSWC was used by Garcia et al.  to estimate Es at Balsa Blanca and another woody savanna site but using a different approach to estimate daily Ec based on Fisher et al. . These authors found better E estimates using fSWC with R2 values ranging from 0.74 to 0.86. Our poorer results may be due to inaccuracies affecting the experimental threshold values θmin and θmax. In the present study, these values were estimated using data from the study period (Table 2), whereas Garcia et al.  used a more extended study period (6 years) to estimate θmin and θmax. Nevertheless, as only estimates of total E were evaluated in both studies, it is difficult to conclude that the disparity between both studies derives from better Es estimates, since more accurate estimates of Ec obtained through their daily adapted version of Fisher et al.  model may also explain these differences. At the mountain site Llano de los Juanes, different reasons may explain the poor performance of fSWC. E underestimates found during the growing season using fSWC were a consequence of an underestimated value of gsx (gsx = 0.0076 m s−1) found from optimization using fSWC. This gsx value was lower than the one obtained using fZhang and fdrying (Table 3) resulting in stronger underestimates of E during this period than the other two methods (Figure 1f). As Figure 3b shows, a higher gsx value (gsx = 0.0088 m s−1) derived from optimization in the growing season (Table 3) reduced the aforementioned underestimates during that period using fSWC (Figure 3b). In contrast, during the wet season (November to March 2006) using fSWC led to overestimates of E (Figure 1f) that we attributed to an effect of the high stoniness and frequent rock outcrops (30–40% rock fragment content) found in this field site [Serrano-Ortiz et al., 2007]. This high percentage of rock coverage reduces the effective soil surface described by the SWC data and results in Es overestimations. Consequently, our results suggest the necessity to adjust the fraction of transpiring soil surface in order to use SWC measurements to estimate Es as a portion of the equilibrium rate in areas with an important percentage of rocks.
5.2. Using Precipitation and Equilibrium Evaporation to Estimate Soil Evaporation (fZhang)
 Use of fZhang in the PML model resulted in a strong overestimation of E during periods following heavy or intermittent rain events (Figures 1e and 1f). Thus, we found generally low correlations with observations at both field sites (Figures 2c and 2d). This occurred because fZhang (equation (6)) assumes that the effect of rain over the soil water availability is limited to a time period of N days (N = 16). As a result, after precipitation the model predicts that f reaches high values remaining high for “N” days, after which an artificial drop takes place or, when rainfall is heavy and intermittent, the model predicts f = 1 during maintained periods of time. This is not an accurate representation of the real SWC pattern, which actually increases during rain and decreases progressively after rain events. Originally Zhang et al.  used this approach to estimate f over 32 day intervals for which a coarse resolution could be effective. They obtained an RMSD of 0.56 mm day−1 for a sparsely vegetated savanna site in Australia (Virginia Park) where the mean annual E (1.20 mm day−1) was higher than that of our field sites. When we applied our proposed daily version of fZhang to our sites, we obtained an RMSD of 0.34–0.31 mm day−1. Since the mean annual was 0.49 mm day−1 at Balsa Blanca and 0.56 mm day−1 at Llano de los Juanes (Table 3) this RMSD is relatively larger than the reported by Zhang et al. . In other words, these results showed that the fZhang method did not improve PML model performance in Mediterranean drylands. The increase of SWC as a result of a rain event depends on the prior rain SWC level. Zhang et al.  tried to incorporate this concept using the ratio of accumulated values of P and Eeq,s during N previous days for modeling f. However, this method is unable to record rapid decreases of SWC following rain in Mediterranean drylands where a higher temporal resolution is necessary to capture the daily variation of SWC.
5.3. Modeling the Soil Drying Process to Estimate Soil Evaporation (fdrying)
 Adoption of the fdrying method clearly improved PML model performance at both sites (Table 3), outperforming the other two approaches (fSWC and fZhang) (Figures 1e and 1f). E estimated using fdrying did not show the strong overestimation obtained using fSWC or fZhang after rainfall, showing a better capacity to describe the gradual drying of soil following rainfall. This method uses the formulation based on Zhang et al.  to estimate the increment of SWC as result of each precipitation event but it included a simple method to model the decrease of SWC during Stage 2 as a function of time after the last precipitation (equation (7)). Considering the difficulties associated with E-modeling in Mediterranean drylands, where measured E rates are especially low, often not exceeding the error range of methods for estimating E from remote sensing [Domingo et al., 2011], using fdrying the PML model achieved reasonable agreement with EC-derived daily E rates. This method showed an RMSD of 0.22–0.24 mm day−1 and R2 from 0.47 to 0.59 (Figures 2e and 2f). This accuracy level is similar or slightly better than the results found by Leuning et al.  and Zhang et al.  in the Australian woody savanna sites Tonzi and Virginia Park. Fisher et al.  found better correlation between estimates and EC-derived monthly averages of λE (R2 ∼0.8) at those two same Australian sites. However, their model overpredicted λE during low λE periods [Fisher et al., 2008] similarly to the overestimations that we found at Balsa Blanca site (when E was lower than 0.2 mm day−1) (Figure 1e). Garcia et al.  found R2 values of 0.58 and 0.82 at two drylands (including Balsa Blanca site) using the same approach to estimate Ec than Fisher et al.  but including a different approach for f based on Apparent Thermal Inertia derived from in situ surface temperature and albedo measurements. However, their results deteriorated further than ours (R2 = 0.32) when remote sensed surface temperature and albedo from SEVIRI (Spinning Enhanced Visible and Infared Imager) were used to estimate f at Balsa Blanca site. Improved MODIS global terrestrial E algorithm combined with tower meteorological data found RMSD values of 0.67–0.91 mm day−1 and R2 values ranging from 0.24 to 0.78 in three woody savannas (including Tonzi site) where observed E was 0.94–2.08 mm day−1 [Mu et al., 2011]. Eventhough, in two shrubland sites, where observed E was 1.04 and 0.19 mm day−1, respectively, the same model reached higher inaccuracies than ours, with RMSD values of 1.10 mm day−1 (R2 = 0.02) and 0.31 mm day−1 (R2 = 0.35). These previous results demonstrate that the accuracy level found by the PML model using fdrying was similar or even outperformed previous models to estimate E using remote-sensing data in drylands where E modeling is still a challenging task [Domingo et al., 2011].
 Like fZhang, fdrying shares the advantage of only requiring widely available precipitation and equilibrium evaporation data, with the expense of a single additional parameter α. With the use of fdrying, the PML model was able to capture the varying controls on Es at both field sites (Figures 1e and 1f). Thus, the optimized value of the α parameter, representing the speed at which soil reduces the capacity to evaporate water, was lower at Balsa Blanca (α = 0.137 day−1) than at Llano de los Juanes (α = 0.478 day−1). This implies that Es at Balsa Blanca has a longer period of influence on total E than at Llano de los Juanes where the soil is assumed to dry more quickly. This is in agreement with the fact that Llano de los Juanes is a karstic area characterized by infiltration occurring in preferential flows through the abundant cracks, joints and fissures [Cantón et al., 2010; Contreras, 2006].
 Overall, the stronger phenological control over E, the reduction of effective evaporative soil surface due to stoniness and rocky soil features and the importance of infiltration at Llano de los Juanes, contribute to Es having a less important role in total E dynamics than at Balsa Blanca. This explains the higher systematically percentage differences found at Llano de los Juanes (Table 3) where all three adapted model versions, including fdrying, were less effective at capturing the system dynamics because they were designed to improve Es, a less crucial factor at this site.
 The systematic underestimation of E by the PML model at the beginning of the dry season observed at Llano de los Juanes (Figure 2d) using fdrying (and also with fZhang) was proven not to be a consequence of underestimates of Ec resulting from failed optimized values of gsx (Figure 3). In fact, tests optimizing model parameters using different optimization periods showed consistency for gsx, especially using fdrying, the method less sensitive to changes in the optimization period (Table 4). Therefore, underestimates of E by the PML model using fdrying (and fZhang) at the beginning of the dry season were explained instead by errors in Es caused by low f values. During this period, the effect of precipitation from the preceding wet season (finishing 20 days before our validation period) was not considered by fdrying (or fZhang) because these methods assume that the effects of rain over SWC only persist during N days (N = 16, in this case). In summary, underestimates of E along the dry and growing seasons at our montane site showed the limitation of fdrying, and fZhang to capture high soil water availability levels originated by the cumulative effect of a long prior wet season.
 The capacity of Penman-Monteith-Leuning model (PML model) to estimate daily evaporation in sparsely vegetated drylands is demonstrated through the development of methods for temporal estimation of the soil evaporation parameter f. We advanced Leuning et al.  who found that estimating soil evaporation parameter f as a local time constant produced poor results in sparsely vegetated areas (LAI < 2.5). Out of three proposed methods, fdrying showed the best results for PML model adaptation at two experimental sites and was able to capture the daily pattern of near surface soil moisture content. This proposed method considers the soil water availability conditions previous to rainfall to estimate the SWC increment derived from rain and explicitly models the progressive soil drying process following precipitation. This way, the fdrying method avoided the strong overestimates of E obtained with two other f estimation approaches, fSWC and fZhang. Nevertheless, the fdrying method showed some limitations in its ability to model the soil evaporation rate when this was influenced by high soil water availability levels during the growing season from the cumulative effect of a long prior wet season at Llano de los Juanes.
 The use of time-invariant parameters for evaporation modeling is a delicate issue in drylands and other extreme ecosystems where vegetation and soil are exposed to strong fluctuations in environmental conditions. Where a simplifying compromise is required in the design of operational and regionally applicable models, we showed here that reasonable results can be obtained using temporally constant estimates of gsx and α in the PML model and the robustness of optimization period to estimate model parameters.
 To evaluate the differences in available energy (A) partitioning between soil (As) and canopy (Ac) using the Beer-Lambert law (BL) or the method proposed Kustas and Norman  specifically designed for sparse vegetation (K&N), daytime estimates of As and Ac obtained following these two different methods were compared at Balsa Blanca during a 144 days (15 January to 8 June 2011). During this time period, one Pyranometer (LPO2, Campbell Scientific, Inc., USA) and two broadband thermal infrared thermometers (Apogee IRT-S, Campbell Scientific, Inc., USA) were available at this site to measure incoming short-wave radiation and surface temperatures necessary for K&N method application. Measurements of: (i) composite soil-vegetation surface (TR) and (ii) pure bare soil surface (Ts) at the field site were obtained using Apogee IRT-S, and canopy temperature (Tc) was derived from both applying the nonlinear relation between TR, Ts, and Tc based on vegetation cover fraction proposed by Norman et al. . Further details can be found in Morillas et al. .
 To estimate As and Ac using the Beer-Lambert law, AsBL and AcBL were estimated as follows
where kA = 0.6 and A = H +λE using daytime measured averages of H and λE [Leuning et al., 2008].
 To estimate As and Ac using the method proposed Kustas and Norman , AsK&N and AcK&N where estimated following equations (A(3)) and (A4)
where Rns and Rnc are daytime averaged estimates from equations (A(5)) and (A(6)) and G is daytime averaged soil heat flux from measurements (section 3.1).
where S (W m−2) is the incoming shortwave radiation, τs is solar transmittance through the canopy, αs is soil albedo, αc is the canopy albedo. Estimates of τs, αs, and αc are computed following the equations (15.4)–(15.11) in Campbell and Norman  and based on LAI, the reflectances and trasmittances of soil and a single leaf, and the proportion of diffuse irradiation, assuming that the canopy has a spherical leaf angle distribution.
Lns and Lnc (W m−2) are the net soil and canopy long-wave radiation, respectively, estimated using the following expressions:
where kL (kL ≈ 0.95) is the long-wave radiation extinction coefficient, which is similar to the extinction coefficient for diffuse radiation with low vegetation, i.e., LAI lower than 0.5 [Campbell and Norman, 1998]. Ω is the vegetation clumping factor proposed by Kustas and Norman  for sparsely vegetated areas, which can be set to one when measured LAI implicitly includes the clumping effect (i.e., LAI from the Moderate Resolution Imaging Spectroradiometer, MODIS) [Anderson et al., 1997; Norman et al., 1995; Timmermans et al., 2007], and Ls, Lc, and Lsky (W m−2) are the long-wave emissions from soil, canopy and sky computed by the Stefan-Boltzman equation based on measured Ts, derived Tc and measured air temperature and vapor pressure [Brutsaert, 1982]. For further details about Kustas and Norman  partitioning of Rn used here see Morillas et al. .
 The linear agreement between daytime estimates of As and Ac from both methods was high with a determination coefficient (R2) of 0.92 for As and 0.79 for Ac (Figures A1a and A1b). Mean absolute differences between estimates of As and Ac from both methods were 31.74 and 17.97 Wm−2 for As and Ac, respectively, during the 144 days tested. Considering these small differences, the higher complexity of K&N method and the increment of model inputs that this method implies, we decided that using the LB method for A partitioning between As and Ac at daytime scale was efficient and consistent.
 To determine the optimal number of days, N, to consider in equations (6) and (7) for estimating fZhang and fdrying a sensitivity analysis was performed using data of Balsa Blanca field site. We obtained statistics of PML model performance using fZhang and fdrying estimated using N values from 4 to 25 and also considering the time period including four 4 days previous and the 4 days after the current one (signed as 4_4), the latter an approach more similar to the originally proposed by Zhang et al. .
 The PML model using fZhang presented a better performance using high N values, with similar results using N from 16 to 25 (Table B1). Using that range of N values, the lowest values of MAD (0.23–0.25 mm day−1) and RMSD (0.30–0.34 mm day−1) (Figure B1a) coincided with the better linear agreement showing a R2 range of 0.40–0.42, slope range of 1.32–1.51, and intercept values from −0.07 to −0.16 (Figure B1b).
Table B1. Statistics of PML Model Performance With fZhang Using Different N Valuesa
The value of gsx obtained by optimization in the optimization period (Table 2) is also presented for each N value used for fZhang estimation. RMSD and MAD values in mm day−1.
N = 4_4 considers the time period including four 4 days previous and the 4 days after the current one.
 Considering the modeling of the soil drying process included in fdrying, the PML model performance also obtained better results using high values of N (Table B2), but the lowest mean inaccuracies were obtained using N values from 16 to 20 (Figure B2a) with MAD ∼0.17 mm day−1 and RMSD ∼0.21 mm day−1. Using N from 16 to 20 also the linear agreement between model outputs and measured E was improved (Figure B2b) but the best linear agreement was obtained using N = 16 showing R2 = 0.47 and the best slope and intercept values (slope = 0.97 and intercept = 0.02).
Table B2. Statistics of PML Model Performance With fdrying Using Different N Valuesa
The value of gsx and α obtained by optimization in the optimization period (Table 2) is also presented for each N value used for fdrying estimation. RMSD and MAD values in mm day−1.
N = 4_4 considers the time period including four 4 days previous and the 4 days after the current one.
 Based on these results, we decided that N = 16 was the more suited value for daily estimation of fZhang and fdrying for PML model performance included in this paper. However, it is important to notice that the model accuracy did not showed a strong variation of model accuracy under the range of N values studied (Tables B1 and B2) with a maximum difference on accuracy of ±0.1 mm day−1 depending on N. This suggests a low sensitivity of the PML model using fZhang and fdrying to the N value and that alternative N values could be used without a strong effect on model performance.
 This research was funded by the Andalusian regional government projects AQUASEM (P06-RNM-01732), GEOCARBO (P08-RNM-3721), RNM-6685, and GLOCHARID, including European Union ERDF funds, with support from Spanish Ministry of Science and Innovation projects CARBORAD (CGL2011–27493) and CARBORED-2 (CGL2010–22193-C04-02). L. Morillas received a PhD grant and funding for a visit to CSIRO Marine and Atmospheric Research from the Andalusian regional government. The authors would like to thank Philippe Choler for his assistance with programming parameter optimization in R, Helen Cleugh for her comments and support during the stay at CSIRO Marine and Atmospheric Research, Peter Briggs for his help on the edition of this paper and the anonymous reviewers for providing helpful and constructive suggestions to improve the manuscript.