1. Introduction
Evapotranspiration (ET) linking water and energy budgets is one of the most challenging hydrometeorological processes to monitor and model due to the complex interactions among the components of the soil–plant–atmosphere system (Zhao et al. 2013). Various methods, such as empirical relationships (Kohler et al. 1955), energy budget (Fritschen 1966; Kustas and Norman 1996; Ershadi et al. 2014), bulk transfer (Harbeck 1962; Isabelle et al. 2015), combined energy budget–bulk transfer methods (Penman 1948; Duan and Bastiaanssen 2017), and eddy covariance systems (Baldocchi et al. 1988; Aubinet et al. 2012), have been used to quantify ET. The latter is certainly the most direct approach, but requires intensive technical supervision to effectively implement and operate (Shi et al. 2008).
Recent advances in contemporary nonequilibrium thermodynamics, building on the Bayesian probability and information theories, have provided a new theoretical basis for modeling ET. In this context, it was recently shown that the maximum entropy production (MEP) method may be an effective new approach for modeling ET (Wang and Bras 2009, 2011). The MEP theory is originated from the treatise of statistical mechanics pioneered by Gibbs (1902) for an ensemble of thermodynamic systems. Jaynes (1957) was the first to realize that Gibbs’s algorithm, now known as the maximum entropy (MaxEnt) principle, is a general inference algorithm based on incomplete information quantified by Shannon entropy (Shannon 1948). The reader is referred to the work of Jaynes (2003) for a comprehensive account of the MaxEnt theory. The MEP theory has received increasing attention in recent years (Ozawa et al. 2003; Kleidon and Lorenz 2004; Martyushev and Seleznev 2006; Kleidon 2009). Wang and Bras (2011) have applied the MEP theory to develop a new model of evapotranspiration and surface heat fluxes using a framework conceptually different from standard “physically based” approaches. It was formulated by combining physical principles such as conservation of energy and boundary layer turbulence, among others, with probability rules. It differs from conventional bulk transfer models in several fundamental aspects by 1) predicting surface fluxes without using temperature and humidity gradients, wind speed, and surface roughness as input variables; 2) always closing the surface energy budget, given that the energy balance equation is used as a mathematical constraint of the MEP solution of the surface fluxes; and 3) solving the surface fluxes using only net radiation, surface temperature, and surface specific humidity. Although the surface fluxes are calculated from three input variables (temperature, humidity, and net radiation), it is important to emphasize that the effects of other atmospheric and land surface conditions are taken into account through the model parameters (i.e., thermal inertia). Even with fewer input variables compared to other physically based ET models, the MEP model includes all essential physical processes behind ET under more general conditions (Wang et al. 2012). The MEP model is thus devised to make an effective use of a small number of input variables to calculate the surface fluxes.
Two different versions of the MEP model can be distinguished: MEP-Ev for soil evaporation and MEP-Tr for plant transpiration. Both have already been used successfully to estimate subdaily evaporation over bare soil (MEP-Ev) and transpiration over dense vegetation (MEP-Tr) under humid to moderately limited water availability conditions (Wang and Bras 2011). More recent studies (Shanafield et al. 2015; Wang et al. 2017) have also tested the MEP-Tr model at a humid forest site and the MEP-Ev model at an arid, nonvegetated site with encouraging results. However, further tests of the MEP model are needed to investigate its capability to estimate the seasonal variation of ET under the conditions of severe water stress and in the presence of a partially vegetated cover.
The first objective of the present study is to formulate a combined MEP model, hereafter referred to as MEP-ET, using a vegetation weighting coefficient fveg to combine the MEP-Ev and MEP-Tr estimates for partially vegetated land surfaces. The second objective is to evaluate the performance of the MEP-ET model at eight sites from different biomes and climatic conditions over multiyear periods. The third objective aims to address the MEP-ET model difficulty in estimating ET under very low water availability conditions. A stomatal regulation parameter is then introduced in this model to better represent the effect of water stress.
The present paper is organized as follows. Section 2 briefly reviews the MEP model proposed by Wang and Bras (2011), highlighting the differences between its two versions (MEP-Ev and MEP-Tr). The proposed MEP-ET model combining these versions is then formulated. Section 3 describes the test sites. Approximations made to calculate the MEP-ET input variables at these sites are also presented in this section. Section 4 presents the evaluation results of the MEP-ET model. Section 5 introduces a stomatal regulation parameter to improve the estimation of ET under severe water stress. Daily ET results obtained using the generalized MEP-ET model that includes the stomatal regulation parameter are compared with observations at the test sites, as well as with other results obtained using classic ET models.
2. Model formulation
a. MEP model
formulating the entropy production function
finding the stationary point of
numerically solving the surface fluxes given three input variables:
Depending on the application surface (soil surface or leaf surface), two versions of the MEP model can be distinguished. These versions will be presented in the following subsections. Hereafter, for the purpose of clarity, 
1) MEP-Ev
2) MEP-Tr
b. The MEP-ET model (combining MEP-Ev and MEP-Tr)
3. Data
a. Study sites
For a comprehensive evaluation of the MEP-ET model [Eq. (9)], particular attention is given to the spatial and temporal coherence of its calculations. For this purpose, FLUXNET datasets (Baldocchi et al. 2001; Agarwal et al. 2010) were selected. They provide high-quality eddy covariance measurements of surface fluxes and ancillary meteorological variables. The selected eight FLUXNET sites (http://ameriflux.ornl.gov) are from different biomes: grassland (GRA), cropland (CRO), woody savanna/shrubland (WSA), and deciduous broadleaved (DBF) and evergreen needleleaf (ENF) forests. The selected sites are located in the continental United States (Fig. 1) with a range of vegetation covers, climates, precipitation regimes, altitudes, and canopy heights (see Table 1). ENF1 (evergreen forest site) is characterized by Mediterranean climate, where ET corresponds to 45% of the annual precipitation (1294 mm). DBF1 and DBF2 (deciduous broadleaved forests sites) present a temperate climate. A humid continental climate is found at CRO1 and CRO2 (cropland sites) and GRA2 (grassland site); and a semiarid climate is found at GRA1 (grassland site) and WSA1 (woody savanna site), where ET exceeds 80% of the annual precipitation.
Location of the selected FLUXNET sites.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
Main characteristics of the test sites. Parameter zg represents the elevation above mean sea level, zc is the mean canopy height, P is the mean annual precipitation, and ET is the mean annual evapotranspiration. Koppen climate classes are humid continental (Dfa), humid subtropical (Cfa), Mediterranean (Csa), and semiarid climate (Bsk).
To characterize each site in terms of its mean limiting factors for ET, namely, available energy “demand” (net radiation) and water “supply” (precipitation), the nondimensional framework proposed by Budyko (1974) is used (Fig. 2). This framework is based on the radiative dryness (DI) and evaporative (EI) indices. DI is defined as a ratio of potential evapotranspiration (PET), estimated following Priestley and Taylor (1972), and observed precipitation. On the other hand, EI is expressed as the ratio between the observed evapotranspiration and precipitation. These indices are represented at climatological time scales by averaging over multiple years. As shown in Fig. 2, the sites are distributed in space with large demand/supply differences, while still clustered around the Budyko curve. The majority of sites are under moderate water availability, except for GRA2. Two sites (WSA1, GRA1) are located in arid regions (PET > 3P). The DBF2 site is similar to the latter sites, except it experienced some much drier years (see bottom-right corner of Fig. 2). Drought and nondrought conditions needed to be defined at this site. The transition from a nondrought to drought regime occurs when plant water potential falls below −0.7 MPa, based on an analysis of Gu et al. (2006) at the same site.
Mean EI as a function of the mean DI. The black solid curve is from Budyko (1974). Year-to-year variability is shown for DBF2 exclusively in the bottom right corner. Adapted from Williams et al. (2012).
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
b. MEP-ET input variables
In brief, model inputs needed to run MEP-ET include net radiation Rn, soil surface temperature 
c. Model evaluation criteria
Model outputs in terms of daily ET are compared against observations at the eight sites of interest throughout this analysis. Given that most evapotranspiration occurs during daytime and that the observed energy imbalance is often greater at night (Wilson et al. 2002), estimations and observations of daily evapotranspiration are determined by cumulating 30-min values (estimated or observed) from 0700 to 1800 local time. The following sets of metrics are used to assess model performance: Nash–Sutcliffe efficiency (NSE), percent bias (PB), root-mean-square error (RMSE), and the coefficient of determination R2 (Ershadi et al. 2014; Pereira et al. 2015).
4. Evaluation of the MEP-ET model
As shown in Table 2, MEP-ET model performance varies from site to site. The mean RMSE ranges from 0.53 mm day−1 at ENF1 to 0.71 mm day−1 at DBF2, with an average value of 0.62 mm day−1 for all sites. In 2007, the RMSE value exceeded 0.94 mm day−1 at DBF2. On average, the MEP-ET model explains 80% of the observed ET variations at all sites, ranging from 63% at GRA1 to a high of 87% at DBF1. An average PB of 10% was obtained for all sites and ranged from 3% at ENF to 1%–60% at GRA1. In 2009, PB exceeded 85% at GRA1. In addition, the mean NSE is highly variable, being 0.23 and 0.33 at GRA1 and WSA1, respectively, and 0.84 at ENF1 with 0.79 as an average value for all sites. High values of NSE were recorded at GRA2 (0.90 in 2010) and ENF1 (0.86 in 2003), demonstrating the capability of MEP-ET to reproduce observed ET at these sites. Occasional negative NSE values were obtained at WSA1 (−0.30 in 2004) and GRA1 (−0.52 in 2009). Sample daily values of estimated and observed ET at all sites confirm the above findings (Fig. 3). As shown in Fig. 3, the MEP-ET model captures very well the seasonal cycle of observed ET at the CRO1, GRA2, DBF1, ENF1, and CRO2 sites, which are characterized by unlimited to moderate water availability (Fig. 2). On the contrary, poor results were obtained at the GRA1 (PB = 85% in 2009), WSA1 (PB = 63% in 2007), and DBF2 (NSE = 0.49 in 2007) sites, which are characterized by low water availability (Fig. 2). The MEP-ET model tends to overestimate the daily ET at these sites (Fig. 3), particularly during low soil moisture periods (Fig. 4). Note that this issue has also been raised in other studies relying on classic ET models under similar circumstances (Hari and Kulmala 2008; Ji et al. 2017). As shown in Fig. 4, periods of very low soil moisture are characterized by the transpiration fraction of ET exceeding 80% and 90% at the GRA1 and DBF2 sites, respectively. This reveals that the overestimation of ET arises from an inaccurate treatment of transpiration. First, one could think that this is the outcome to a poor estimation of 
Performance values of the MEP-ET model in terms of NSE, R2, PB (%), and RMSE (mm day−1) at all study sites.
Daily variation of estimated (using MEP-ET) vs observed ET at the sites of interest. In the x-axis label, the three first letters represent the month, while the two numbers represent the year. In both cases (modeled and observed values), a 10-day running average was used to smooth the data.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
A close look at the issue of overestimated ET using MEP-ET under water stress conditions at the DBF2, GRA1, and WSA1 sites. (top) Model outputs from MEP-ET along with observations, (middle) the transpiration fraction, and (bottom) time series of soil moisture. The dashed lines in the bottom plots represent the upper (field capacity) and lower (wilting point) limits of soil moisture variability. In the x-axis label, the three first letters represent the month, while the two numbers represent the year. In both cases (modeled and observed values), a 10-day running average was used to smooth the data.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
Therefore, the mechanism responsible for this overestimation has to be the stomatal control of ET. Indeed, the stomata tend to close under dry conditions to prevent water loss, hence reducing transpiration. The stomatal control of water vapor loss by transpiration is increasingly accentuated as water stress increases (Hari and Kulmala 2008; Berry et al. 2010; Ji et al. 2017). Although the effect of stomatal closure is implicitly parameterized in the MEP-ET model through 
5. Modeling ET under water stress conditions: The generalized MEP-ET model
a. Water stress factor ηs
To overcome the problem of ET overestimation under water stress conditions within the framework of classic ET models, several researchers have proposed a water stress factor in the parameterization of stomatal conductance (Sala and Tenhunen 1996; Wang and Leuning 1998; Van Wijk et al. 2000; Tuzet et al. 2003; Misson et al. 2004; Uddling et al. 2005; Müller et al. 2014; Ji et al. 2017). Following this path, the effect of water stress has to be better represented in the MEP-ET model to improve its performance for water-limited environments while maintaining its advantageous features.
b. Performance of the generalized MEP-ET model
Model performance metrics for the WSA1, GRA1, and DBF2 sites are summarized in Table 3. Figure 5 compares the observed and estimated daily ET from the MEP-ET and the generalized MEP-ET models (i.e., with and without 
Performance values of the MEP-ET and the generalized MEP-ET models for critical years at sites GRA1, DBF2, and WSA1.
(a) Comparison of ET simulated by the MEP-ET and the generalized MEP-ET models along with observations at site WSA1. (b) Temporal variation of volumetric water content θ for the same site and period. In the x-axis label, the three first letters represent the month, while the two numbers represent the year. In both cases (modeled and observed values), a 10-day running average was used to smooth the data.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
Cumulative ET using observations (black dotted curve), as well as the MEP-ET and generalized MEP-ET models at site GRA1. The scores of performance of MEP-ET (black solid curve) are NSE = −0.59, PB = 65%, and RMSE = 478 mm, whereas for the generalized MEP-ET (gray solid curve), they are NSE = 0.86, PB = 18%, and RMSE = 143 mm for all years.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
(a) The 10-day averages of ET simulated by the generalized MEP-ET model, MEP-ET, and observations at site DBF2. (b) Temporal variation of soil moisture θ at site DBF2. In the x-axis label, the three first letters represent the month while the two numbers represent the year. In both cases (modeled and observed values), a 10-day running average was used to smooth the data.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
Comparison of the generalized MEP-ET estimated and observed daily averaged ET at all sites and for all years under study. The color scale indicates the number of occurrences. Scores of performance are NSE = 0.83 and PB = 3%.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
c. Performance of the generalized MEP-ET model versus classic ET models
In this section, the performance of the generalized MEP-ET model is compared with three classic ET models: the modified Penman–Monteith (PM; Brutsaert 2005; Ershadi et al. 2014), the modified Priestley–Taylor–Jet Propulsion Laboratory (PT-JPL; Fisher et al. 2008; Ershadi et al. 2014), and the air-relative-humidity-based two-source model (ARTS; Yan et al. 2012). It should be emphasized that these models take into account the effect of water stress on daily ET estimation (see appendix B for more details about the description and formulation of these models). Figure 9 shows the NSE, R2, PB (%), and RMSE (mm day−1) for all of the four models at the eight study sites. It is evident that the generalized MEP-ET model outperforms the other three models (PM, PT-JPL, ARTS) at all sites (biomes), as high values of NSE (≥0.73) and R2 (≥0.75) and low values of RMSE (≤0.6 mm day−1) and PB (≤20%) were obtained across all sites. On the contrary, using the three other models, high values of RMSE (generally larger than 0.6) and low values of NSE (generally lower than 0.7) were found. Moreover, the generalized MEP-ET model consistently performs well across all sites, while the other models do not. The NSE and R2 values associated with the generalized MEP-ET model have relatively narrow ranges of 0.73–0.84 and 0.75–0.87, respectively. In contrast, the NSE values of the other models vary from 0.32 to 0.81, and the R2 values range from 0.40 to 0.82. No model, other than the generalized MEP-ET model, is consistently robust for the different biomes studied. For instance, the ARTS model has the highest RMSE value (≈1 mm day−1) at the GRA2 and CRO2 sites, whereas it has the lowest RMSE value (≈0.42 mm day−1) at the GRA1 site. At the latter site, the worst performance is obtained with the PT-JPL model (NSE = 0.32 and R2 = 0.4), while this same model records the best performance compared to PM and ARTS models at GRA2. For the PM model, the worst performance occurred at ENF1 (NSE = 0.32), whereas the best performance (R2 = 0.85) was at the DBF1 site. The above findings concerning the PM and PT-JPL models agree with earlier studies (e.g., Ershadi et al. 2014). In addition to their fluctuating performances, the classical models generally require more input data, potentially adding large uncertainty (Massman and Lee 2002; Williams et al. 2009; Ershadi et al. 2013, 2014, 2015), which makes their application difficult. Therefore, the generalized MEP-ET stands out as a promising new method for the estimation of ET, since it uses less input data than classic ET models and provides consistently better ET estimates across diverse geographical and climatic conditions.
NSE, R2, PB (%), and RMSE (mm day−1) of the average daily estimated ET over all the years for models across different biome types.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0133.1
6. Conclusions
This study evaluated the capability of the MEP model for simulating evapotranspiration under various climatic and vegetation conditions. A weighting coefficient was introduced to combine its two versions for modeling evaporation at nonvegetated surfaces (MEP-Ev) and transpiration at fully vegetated surfaces (MEP-Tr), resulting in the MEP-ET model. This coefficient, which characterizes the vegetation cover, is calculated using NDVI data. The MEP-ET model was evaluated at eight sites in different biomes and under different climatic conditions, including canopy water stress, for which the MEP model has never been evaluated. The MEP-ET model performs well under unlimited and moderate water availability. The overestimation of ET under water-stressed conditions can be corrected by a soil-moisture-dependent water stress function representing stomatal response to water stress. The MEP model including the water stress function greatly improved the modeled ET under dry conditions (e.g., WSA1 and GRA1 sites) and much reduced the overestimation of cumulative ET. The performance of the generalized MEP-ET model is compared to three classic ET models (PM, PT-JPL, and ARTS) at the eight sites. This comparison shows that the generalized MEP-ET model consistently outperforms the other models at all test sites. The effectiveness of the MEP model, which is based on the Bayesian probability, information theories, and boundary layer turbulence theory, results from (explicit and implicit) parameterization of all important physical mechanisms at the desired space–time scales and from more efficient use of hydrometeorological data than other physically based models.
Acknowledgments
The authors would like to acknowledge Anne Morais, who performed some preliminary tests with the MEP model. This work was supported by Ouranos Consortium on Regional Climatology and Adaptation to Climate Change, Hydro-Québec, the Natural Sciences and Engineering Research Council of Canada, the MDDELCC, and Environment and Climate Change Canada through project RDC-477125-14 entitled “Modélisation hydrologique avec bilan énergétique (ÉVAP).” Jingfeng Wang’s research is sponsored by NSF CZO Project EAR-1331846 and NASA NEWS Project NNX15AT41G. The authors thank the FLUXNET community, in particular the AmeriFlux community, for allowing the use of their flux data. Finally, we acknowledge fruitful comments and suggestions from the editor and reviewers.
APPENDIX A
Thermal Inertia Parameters
APPENDIX B
Description of the Comparison Models
a. Modified PM model
b. Modified PT-JPL model
c. ARTS model
Recently, the ARTS model (Yan et al. 2012) has been proposed and tested successfully to estimate ET. Similar to the PT-JPL model, it partitions the total net radiation into soil and vegetation components and calculates potential evaporation from soil and canopy assuming no water stress. Actual ET is obtained by correcting the potential ET according to soil water availability using a soil water balance model. Then, ET is estimated following the five steps described below:
- Introduce a simple biophysical model for canopy conductance
where
where S is solar radiation (W m−2) and
- Calculate canopy transpiration
The aerodynamic conductance
- Calculate soil evaporation
- Calculate total potential evapotranspiration (ET0) as the sum of the canopy transpiration and soil evaporation, assuming the absence of soil water stress:
- Correct ET0 for soil water stress using the following soil water balance model:where
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