Corresponding author: S. Shang, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China. (firstname.lastname@example.org)
 Satellite remote sensing techniques are widely considered as the most promising way to estimate evapotranspiration (ET) over large geographic extents. In this study, a hybrid dual-source scheme and trapezoid framework–based evapotranspiration model (HTEM) is developed to map evapotranspiration from satellite imagery. It adopts a theoretically determined vegetation index/land surface temperature trapezoidal space to decompose bulk radiative surface temperature into component temperatures (soil and canopy) and uses a hybrid dual-source scheme of the layer approach and patch approach to partition net radiation and estimate sensible and latent fluxes separately from the soil and canopy. The proposed model was tested at the Soil Moisture-Atmosphere Coupling Experiment (SMACEX) site in central Iowa, USA, for 3 days during the campaign in 2002 using Landsat Thematic Mapper/Enhanced Thematic Mapper Plus (TM/ETM+) data, and at the Weishan flux site in the North China Plain during the main growing season of 2007 with Moderate Resolution Imaging Spectroradiometer Terra images. Results indicate that HTEM is capable of estimating latent heat flux (LE) with mean absolute percentage errors of 6.4% and 11.2% for the SMACEX and the Weishan sites, respectively. In addition, the model was found to be able to give reasonable evaporation and transpiration partitioning at both sites. Compared with other models, HTEM generally produced better sensible and latent flux estimates at the two sites and had comparable abilities in estimating net radiation and ground heat flux. Sensitivity analysis suggests that HTEM is most sensitive to temperature variables and less sensitive to other meteorological observations and parameters.
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 Land surface evapotranspiration (ET) is the second largest flux term in the terrestrial hydrological cycle, quantification of which is vital in understanding the climate and hydrology of interested areas [Oki and Kanae, 2006]. After Brown and Rosenberg  first used thermal remote sensing in retrieval surface fluxes, methods (or models) to quantify ET based on remote sensing developed rapidly [Bastiaanssen et al., 1998; Yan et al., 2012; Jiang and Islam, 1999; Kustas and Norman, 1997; Moran et al., 1994; Mu et al., 2011; Norman et al., 1995; Sánchez et al., 2008; Su, 2002; Yang, Shang, and Jiang, 2012]. Among the various models that have been proposed, two-source models, which treat soil and vegetation as independent sources of the moisture flux, are generally considered to be an advancement of single-source models. Single-source models represent the surface as a single uniform layer and, therefore, may produce significant errors when applied to partially vegetated landscapes [Timmermans et al., 2007; Verhoef et al., 1997].
 Two-source approaches require knowledge of the surface temperature of both soil and vegetation canopy, and this information is unattainable directly from satellite imagery because the land surface temperature (LST) observed by remote sensing is a single temperature over heterogeneous surfaces. The temperature difference between vegetation and soil components can be more than 20°C [Kustas and Norman, 1999]. As a result, efforts have been made to decompose remotely sensed LST into component temperatures (canopy and soil). Norman et al.  proposed a technique that uses the Priestly-Taylor relationship to estimate transpiration and canopy temperature. Soil temperature can then be estimated using the relationships between LST, fractional vegetation cover, and canopy temperature. A similar idea was adopted by Kustas and Norman  and Sánchez et al. . The difficulty in such an approach is the determination of the initial Priestley-Taylor (P-T) coefficient. In the work of Norman et al. , the initial P-T coefficient was chosen to be 1.3, while in the study of Kustas and Norman , a value of 2 was found to be a better representative of this coefficient. Generally, the P-T coefficient is given to be approximately 1.26 over moist surfaces but is much smaller for dry surfaces [Komatsu, 2003].
 Another operational way to determine component temperatures is based on the interpretation of the image (pixel) distribution in vegetation index (VI)/LST space. As reviewed by Carlson , if a sufficiently large number of pixels are present and when contaminated pixels and outliers (e.g., clouds, surface water, sloping terrain, and shading) are removed, the shape of the pixel envelope constitutes a meaningful triangle. Moran et al.  claimed that the triangle space does not account for the effect of water stresses on canopy transpiration and therefore replaced the triangle by using a trapezoid. In such a space, a higher VI value generally corresponds to a lower LST value for a pixel where higher evapotranspiration would occur and vice versa. More promisingly, isolines of surface soil wetness were found in the VI/LST space [Carlson, 2007], representing constant soil water availability. Since radiometric temperature of the soil surface is mostly affected by the soil wetness and soil texture, and the latter remains relatively constant for a certain region, it is reasonable to assume that each soil wetness isoline represents the same soil surface temperature [Carlson, 2007; Long and Singh, 2012b; Nishida et al., 2003].
 Based on the trapezoidal VI/LST space and soil wetness isolines, Long and Singh [2012b] developed a two-source evapotranspiration model named TTME (two-source trapezoid model for evapotranspiration). However, one key assumption in TTME is that the aerodynamic resistances are equal over the whole domain. This assumption is also generally invoked in other triangle/trapezoid framework–based ET models [Carlson and Ripley, 1997; Carlson et al., 1994; Gillies et al., 1997; Jiang and Islam, 1999, 2001, 2003; Jiang et al., 2009; Moran et al., 1994; Price, 1990]. These models infer evaporative fraction (EF, defined as the ratio of latent heat flux to available energy) without parameterizing the network of aerodynamic and surface resistances. As such, the effects of surface roughness and atmospheric stability are essentially ignored. Consequently, these models may work when surface roughness and LST are sufficiently uniform but will fail over more heterogeneous surfaces.
 To overcome this weakness, it is important to combine the triangle/trapezoid framework with a resistance network. In this study, a new two-source evapotranspiration model is proposed to achieve this goal. The new model is based on the trapezoidal framework to decompose LST into component temperatures and uses a dual-source modeling scheme to parameterize the resistance network and to estimate surface fluxes. Different from existing two-source models, which are based either on the “layer” approach [Kustas and Norman, 1997; Shuttleworth and Wallace, 1985] or the “patch” approach [Sánchez et al., 2008], a hybrid dual-source modeling scheme [Guan and Wilson, 2009] is adopted in the new model. As discussed by Lhomme and Chehbouni , both the layer approach and the patch approach are restricted to use within certain ranges of fractional vegetation cover: the layer model works better for uniform vegetated surfaces, while the patch model is more suitable for clumped vegetation. In addition, the layer model cannot tell the difference between under-canopy soil evaporation and inter-canopy soil evaporation, which may result in significant modeling errors when being applied to surfaces with large soil moisture heterogeneity (e.g., partially irrigated cropland [Zhang et al., 2008]). The patch model assumes a full radiation loading for each component and does not consider the attenuation of radiation by vegetation canopies. Therefore, evaporation from under-canopy soil surfaces is simply ignored in the patch model.
 The hybrid scheme adopts the layer approach to partition available energy between components and uses the patch approach to calculate energy fluxes. As a result, both under-canopy soil evaporation and inter- canopy soil evaporation are considered and distinguished. Yang, Shang, and Guan  coupled the hybrid scheme with a soil moisture simulation component to simulate actual evaporation (E) and transpiration (T) processes over a wheat field using ground-based measurements. Results indicated that the hybrid model is capable of estimating actual ET and gives reasonable partitioning between E and T.
 The objective of this study is to develop a new operational remote sensing ET model based on the hybrid dual-source modeling scheme and the trapezoid framework. Combining the two would achieve two goals: (1) to consider the surface aerodynamic characteristics in trapezoid model and (2) to incorporate remote sensing information into the hybrid dual-source modeling scheme. In the following sections, we will refer to it as the hybrid dual-source scheme and trapezoid framework–based evapotranspiration model (HTEM). The algorithm is detailed in section 2, and the data sets to validate the model are described in section 3, with results given in section 4, followed by a discussion in section 5. Conclusions are given in section 6.
2 Model Formulation
 HTEM consists of two modules. The first module is to partition the available energy for each component and to estimate the surface energy fluxes using a hybrid dual-source scheme. The second module is to decompose the bulk radiative land surface temperature into component temperatures based on a theoretically determined trapezoid framework.
2.1 The Hybrid Dual-Source Scheme
 The energy allocation and resistance network of HTEM is shown in Figure 1. In the hybrid dual-source scheme, a layer approach is used to allocate the available energy for the soil and canopy component based on Beer's law:
where A is the available energy per unit area (W m–2), subscripts c and s represent canopy and soil components, respectively. kc is the extinction coefficient of radiation attenuation within the canopy, LAI is the leaf area index (m2 m–2), and Rn is the net radiation (W m–2), which is computed from
where Sd is the downwelling shortwave radiation (W m–2), α is the surface albedo, and σ is the Stefen-Boltzmann constant (= 5.67 × 10–8 W m–2 K–4). Ta is air temperature and LST is the radiative land surface temperature observed by satellite remote sensing (K). ε is the emissivity of the bulk soil-canopy surface, and εa is the emissivity of the atmosphere, given by Brutsaert  as
where ea is the vapor pressure in hPa.
Kustas and Norman  reported that the exponential extinction of net radiation is only appropriate for canopy near full coverage and may produce systematic errors for sparse canopies. They proposed a physically based algorithm for estimating the divergence of Rn in the canopy. However, this method requires detailed information on canopy and leaf configurations and needs separate evaluations of the visible and near-infrared albedos of the soil and vegetation, which may bring further uncertainties. In HTEM, a simple linear interpolation of the value of kc between that for full vegetation cover and bare soil in terms of fractional vegetation coverage was conducted to obtain the kc value over partially vegetated surfaces, as suggested by Zhang et al. . The value of kc for full vegetation cover is set to be 0.8 for maize, 0.7 for soybean, and 0.63 for wheat following experimental studies by Lindquist et al. , Sinclair and Horie , and Thorne et al. , respectively.
 A patch approach is then used to partition available energy into the latent heat, sensible heat, and ground heat fluxes. In the patch approach, energy fluxes of each component (canopy or soil) represents an average value per unit area of component under consideration, and the average values per unit ground area are weighted by the fractional coverage of each component:
where G is the ground heat flux, LE is the latent heat flux, and H is the sensible heat flux. Fr is the fractional vegetation coverage, which is deduced from remote sensing by
where NDVImax is the normalized difference vegetation index (NDVI) for complete vegetation cover and NDVImin is the NDVI for bare soil. The coefficient n is a function of leaf orientation distribution within the canopy, the value of which typically ranges between 0.6 and 1.25 [Li et al., 2005].
 Ground heat flux (G) in HTEM is estimated from a semiempirical equation provided by Bastiaanssen :
 For each component, the sensible heat flux is calculated by the classical Ohm's law–type formulations:
where ρ is the air density (kg m–3) and Cp is the specific heat of air at constant pressure (J kg–1 K–1). Ts and Tc are soil surface temperature and canopy temperature (K), respectively. is the aerodynamic resistance to heat transfer between canopy and the reference height (m s–1); is the aerodynamic resistance to heat transfer between Zom + d (Zom is the canopy roughness length for momentum transfer, and d is zero displacement height) and the reference height (m s–1); is the aerodynamic resistance to heat flow in the boundary layer immediately above the soil surface (m s–1). The expressions to estimate these aerodynamic resistances can be found in the appendix of Sánchez et al. .
 As a result, the latent heat flux for each component is computed as a residual in equations (5) and (6):
 For the whole surface, the total latent heat flux is calculated as the sum of fluxes from each component weighted by their relative area [Lhomme and Chehbouni, 1999]:
2.2 The Trapezoid Framework and Determination of Component Temperatures
 In the hybrid dual-source scheme for evapotranspiration, soil surface temperature and canopy temperature are both required, while only the bulk surface temperature is available from remote sensing. Therefore, it is necessary to decompose the bulk surface temperature into component temperatures, which was achieved based on a theoretically determined trapezoid framework [Long et al., 2012].
 Theoretically, four critical points relating to four extreme conditions define a trapezoid (Figure 2). Point A represents the driest bare soil with the highest surface temperature (Ts_max), point B represents the fully vegetated area with the highest water stresses. As a result, point A and point B constitute the warm edge of the trapezoid space, and it is further assumed that the evaporation rate on the warm edge is zero. Points C and D represent fully vegetated surface and bare soil surface without water stress, respectively. Evaporation on the cold edge CD is assumed to be equal to the potential evaporation rate.
 Prior studies have indicated that there exist soil wetness isolines within the Fr/LST space [Carlson, 2007; Sandholt et al., 2002]. Assuming a uniform texture, the soil radiometric temperature depends solely on the soil's moisture content; based on this, it is assumed that soil sharing the same moisture content is also isothermal (Figure 2). The slope of each isoline is derived by interpolating the slope of the warm edge (βw = Tc_max – Ts_max) and that of the cold edge (βc = 0), in terms of temperature difference between the pixel and cold edge (a) and the temperature difference between the pixel and warm edge (b) (Figure 2). Then, the soil surface temperature can be computed from
 Once the soil surface temperature has been determined, the canopy temperature can be estimated from [Sánchez et al., 2008]
where εs and εc are emissivity of the soil surface and canopy surface, respectively.
 To determine the shape of the trapezoid space, surface temperatures for the four extreme points should be accurately determined. In HTEM, the algorithm proposed by Long et al.  is adopted to derive the theoretical boundaries of the trapezoid space for the given meteorological conditions and surface characteristics.
 For the cold edge, the largest evaporation rate corresponds to the lowest sensible heat flux. As a result, spatially averaged air temperature (Ta) is taken to be the horizontal cold edge. For the two extreme points on the warm edge, their temperatures are theoretically determined through solving the surface radiation budget and energy balance equations [Long et al., 2012]. As a result, temperatures for point A (Ts_max) and point B (Tc_max) are computed from the following (a detailed derivation can be found in Long et al. ):
where αs_max and αc_max are surface albedo for the two extreme points and can be estimated by extending the upper envelope of the Fr/albedo space intersecting with Fr = 0 and Fr = 1, respectively [Long and Singh, 2012a; Zhang et al., 2005]. It is noted that the vegetation height for point B (hc_max) is arbitrarily taken to be 1 m. However, sensitivity analysis in section 5.2 suggests that estimated LE is not sensitive to the changes in hc_max.
3 Description of the Study Sites and Data
 Data to validate the model come from two different sites. The first data set was collected during the Soil Moisture-Atmosphere Coupling Experiment (SMACEX) campaign conducted in the summer of 2002 in Iowa, USA. However, since the campaign was only conducted for about one month, the vegetation coverage showed little variation within the experimental period, and the second data set collected in an agricultural flux site in the North China Plain covering the whole range of vegetation coverage was used to further test the hybrid dual-source scheme (evaporation and transpiration partitioning) in HTEM.
3.1 The SMACEX Campaign
3.1.1 Site Description
 During the period from 15 June (DOY 166) through 8 July (DOY 189), the SMACEX campaign was conducted in central Iowa (41.87°N–42.05°N, 93.83°W–93.39°W), USA (Figure 3a). The region can be classified as humid, with a mean annual precipitation of about 835 mm. More than 80% of the land cover within the region was composed of rain-fed corn and soybean fields. The campaign collected an extensive measurement set, including metrological data from 14 observation towers, soil and vegetation parameters, and energy fluxes. Twelve of the meteorological towers were equipped with eddy covariance (EC) systems, and the Bowen ratio method was employed to perform the energy closure of the EC system according to Twine et al.  and Anderson et al. . Anderson et al.  reported that the observed energy fluxes after forcing with the Bowen ratio method agreed well with aircraft counterparts for the SMACEX site; the root mean square difference between the two measurements are 10 W m–2 for sensible heat and 30 W m–2 for latent heat. This suggests that the EC measurements are reasonably representative of the actual surface fluxes at the site. Detailed descriptions of the measurement during the campaign are provided by Kustas et al. .
3.1.2 Remote Sensing and Digital Elevation Model Data
 Three cloud-free Landsat Thematic Mapper (TM)/Enhanced Thematic Mapper Plus (ETM+) scenes were taken during the SMACEX campaign period. These are available from the US Geological Survey data center (http://glovis.usgs.gov/) and consist of the Landsat TM image acquired at 10:29 a.m. on DOY 174, the Landsat ETM+ image acquired at 10:42 a.m. on DOY 182, and the Landsat ETM+ image acquired at 10:48 a.m. on DOY 189. A digital elevation map of the study area with a spatial resolution of 1 arc sec (about 30 m) was obtained from the National Elevation Dataset (http://seamless.usgs.gov/).
3.1.3 Variable Derivation
 LST was derived from the infrared band (TIR, band 6) of Landsat TM/ETM+ images using a method specifically for the SMACEX site described by Li et al. . Albedo was retrieved from the visible and near-infrared bands (band 1-5, 7) of the Landsat images following Allen et al. . Fr was calculated using equation (7) with a coefficient n of 0.625, NDVImax of 0.94, and NDVImin of 0 [Li et al., 2004]. Leaf area index (LAI) and vegetation height (hc) were estimated using empirical relationships for the SMACEX site given by Anderson et al. :
where NDWI is the normalized difference water index, computed from near infrared (NIR, band 4 of TM/ETM+ images) and shortwave infrared (SWIR, band 5 of TM/ETM+ images) reflectances,
3.2 The Weishan Site in the North China Plain
3.2.1 Site Description
 The second validation study uses data from the Weishan flux site (36.65°N, 116.05°E) located in the center of the Weishan Irrigation District along the lower reaches of the Yellow River (Figure 3b), China. The elevation of the site is 30 m above the sea level, and it is characterized by subhumid climate with a mean annual precipitation of 532 mm (averaged from 1984 to 2007) and mean annual pan evaporation of 1950 mm (20 cm diameter evaporation pan; 1961 to 2005). The annual average air temperature is 13.3°C (1984 to 2007). The dominant crops at the site are winter wheat and summer maize planted in rotation; this is the main cropping system in the North China Plain. The growing season for winter wheat is from late October to early June, while summer maize is planted in mid-June and harvested in late September. However, because wheat grows very slowly or even stops during the winter season due to low temperature and frozen soil, only the main growing season of winter wheat from early March to early June and summer maize from mid-June to late September was considered in this study.
 During the growing seasons of 2007, sensible and latent heat fluxes were measured by a 10 m high flux tower mounted with an eddy covariance system placed 3.7 m above the ground at 30 min intervals. Meteorological data, including air temperature and humidity, air pressure, downward and upward solar and long-wave radiation, precipitation, and wind speed and direction were recorded at 10 min intervals. Soil heat fluxes were measured at a depth of 3 cm at two sides of the tower. The uncertainty analysis of EC measurements at Weishan site can be found in the work of Lei and Yang , and the residual method was recommended by Yang et al.  to force the energy closure of the EC system at this site. A detailed description of the site and the measurements can be found in the study by Lei and Yang .
3.2.2 Remote Sensing Data
 The Moderate Resolution Imaging Spectroradiometer (MODIS) data were used to force HTEM at the Weishan site because of its high temporal resolution (daily) and accessible spatial resolution (1 km). Good agreement between sensible heat flux measured by large aperture scintillometer (1 km) and that by EC system (100–500 m) was reported in Weishan site [Lei et al., 2011], indicating that the EC data adequately represent the area within one MODIS pixel and could provide accurate observations of actual surface fluxes. Data sets titled MOD09GA, MOD09Q1, and MOD11A1 were used; these are available from the NASA data center (http://reverb.echo.nasa.gov/). The original images were reprojected into Universal Transverse Mercator projection and resampled with a spatial resolution of 1000 m. During the study period between 1 March (DOY 60) and 30 September (DOY 273), a total of 66 cloud free MODIS images were available, with 39 days in the growing season of winter wheat and 27 days in the growing season of summer maize.
3.2.3 Variable Derivation
Liang's  method was used to calculate broadband surface albedo from seven channels recorded in the data set of MOD09GA. NDVI was derived from the red and near-infrared bands following Huete et al. . The observed maximum NDVI during the study period was 0.93 (NDVImax), and the observed NDVI for bare soil (nongrowing season) was 0.12 (NDVImin). The coefficient n was determined to be 0.7 through optimizing equation (7) based on field measurements of vegetation coverage. Because only a few measurements of LAI were available during the study period, a parametric relationship between LAI and NDVI (8 day temporal resolution and 250 m spatial resolution, MOD09Q1) was used to obtain consecutive LAI values [Lei et al., 2012], and hc was expressed as a function of LAI based on local measurement. Before vegetation height reaches its maximum value,
4.1 Validation of HTEM at SMACEX Site
 For comparison with observations from the tower network, flux estimates were averaged over an estimated upwind source area (1–2 pixels/~120 m) for each flux tower [Choi et al., 2009; Gonzalez-Dugo et al., 2009; Long and Singh, 2012b]. In addition, all fluxes and meteorological measurements were linearly interpolated to the time of satellite overpass using the two bounding measurements.
4.1.1 Model Validation with Tower-Based Measurements
 Comparisons between energy balance components (Rn, G, H, and LE) produced by HTEM with TM/ETM+ data and those from tower-based measurements for all 3 days (DOY 174, DOY182, and DOY 189) are shown in Figure 4. Generally, all four energy components estimated from HTEM agree reasonably well with flux tower-based measurements. Estimated Rn has a root mean square error (RMSE) of 19.1 W m–2 and a bias (defined as mean simulated value minus mean observed value) of 4.6 W m–2 (Table 1). The mean absolute percentage error (MAPE, defined as the ratio of mean absolute error to mean observed value) of Rn estimates for the 3 days is 2.4%. The estimated ground heat flux (G) had an RMSE of 21.6 W m–2 and MAPE of 19.7%. However, as can be seen from Figure 4b, the accuracy of G estimates is systematically lower in soybean field than in the corn field. This is possibly due to the semiempirical equation for G (equation (8)), which was derived in Gediz Basin, Turkey [Bastiaanssen, 2000], and may require local calibration.
Table 1. Summary of the Statistics of the HTEM Performances at the SMACEX Sitea
a is the mean of observed values, and is the mean of HTEM simulated values.
bNumbers inside brackets indicate the available number of measurement on each day.
 The estimated sensible heat flux (H) has an RMSE of 25.1 W m–2 and a bias of 12.88 W m–2 (Table 1). The MAPE of estimated H for all 3 days was 100% due to an extremely high MAPE value of 262% on DOY 189 (Table 1). In DOY 189, five flux sites showed negative measured H, which indicates strong advection. However, as the lower limit of surface temperature for both canopy and soil were bounded above the air temperature in the trapezoid framework, the advection effect is not considered in HTEM. In addition, studies have shown that advective conditions can greatly enhance measurement uncertainty of the EC system [e.g., Alfieri et al., 2011]. The high MAPE on DOY 189 is exacerbated by the fact that these negative measured values were small, and even though the MAPE was high on DOY 189, the RMSE for that day was 28.1 W m–2, which accounts for only 4.7% of the total available energy. On the other 2 days, the H MAPE was better: 14.5% on DOY 174 and 16.1% on DOY 182; these and the daily RMSE values are listed in Table 1.
 LE is computed by HTEM as the residual of the surface energy balance equation, and these estimates agree well with tower-based measurements (Figure 4d). The RMSE of LE for all 3 days was 31.1 W m–2, and the bias was 7.2 W m–2; the MAPE was 6.4% (Table 1). Although a high MAPE of H is found in DOY 189, the MAPE of LE for that day is only 4.1% since the true measured value was not close to zero. The RMSE on DOY 189 was 25.1 W m–2, indicating that errors in estimated H are somehow compensated by errors in the other two energy components (Rn and G). The highest RMSE of estimated LE occurred in DOY 174 with a value of 39.2 W m–2 because of a high RMSE for G on that day, which was further due to a relative large disagreement between the estimated G and the measured G in two soybean fields (the two red triangles in the upper-right part of the data clouds in Figure 4b). As a result, a negative bias of G of –18.5 W m–2 resulted in a positive bias of LE of 17.8 W m–2 on DOY 174. For DOY 182, the estimated LE showed an RMSE of 28.6 W m–2, the bias and MAPD are 1.2 W m–2 and 6.1%, respectively (Table 1).
4.1.2 Spatial Distribution of Estimated LE from HTEM
 The spatial distribution of estimated canopy transpiration (LEc) and soil evaporation (LEs) for 3 days together with the corresponding NDVI maps are shown in Figure 5. Generally, a higher NDVI value corresponds to higher LEc for all 3 days, while LEs is negatively correlated with NDVI. This phenomenon can be seen from two perspectives. On the one hand, for each day, LEc is higher and LEs is lower where there is a higher value of NDVI. An example of the relationships between NDVI and two component fluxes (LEs and LEc) on DOY 174 is shown in Figure 6. A positive relationship is clearly seen between NDVI and LEc, and a negative relationship is found between NDVI and LEs. On the other hand, the overall NDVI was increasing through DOY 174 to 189, which resulted in an increase in overall LEc and a decrease in overall LEs from DOY 174 to 189 (a relative high LEs was observed in DOY 189 due to a rainfall event 3 days before) (Figure 5).
 On DOY 174, when NDVI and soil moisture had the widest ranges within the study area, the estimated LEc and LEs showed the largest variations across the whole domain (Figure 5). The coefficient of variation (CV) is 0.41 and 0.47 for LEc and LEs on DOY 174, respectively. On the contrary, on DOY 189, NDVI reached its maximum value for most pixels, and soil moisture was relatively constant across the region due to the rainfall event in DOY 185. As a result, both LEc and LEs on DOY 189 showed the smallest spatial variations throughout the region across the 3 days (Figure 5) (CV values on these 2 days are 0.26 for LEc and 0.32 for LEs on DOY 182, and 0.22 for LEc and 0.18 for LEs on DOY 189). These results are consistent with those reported by Choi et al.  and Long and Singh [2012b].
4.2 Validation of HTEM at Weishan Site
4.2.1 Compared with Tower-Based Measurement
 Comparisons between measured energy fluxes (Rn, G, H, and LE) and estimated ones from HTEM at the Weishan site during the growing season of 2007 are shown in Figure 7, and statistics of the HTEM performances are summarized in Table 2. The estimated Rn had an RMSE of 24.1 W m–2 and a bias of –1.3 W m–2over both growing seasons (winter wheat and summer maize). However, slightly better agreement was found during the maize season than in the wheat season (Table 2). The estimated G had an overall bias of 3 W m–2 (6.6 W m–2 for the wheat season and –2.2 W m–2 in the maize season) and an overall RMSE of 20.3 W m–2 (23.8 W m–2 in the wheat season and 15.2 W m–2 in the maize season). Similar to the SMACEX site, a local calibration of equation (11) might be beneficial to improve the accuracy of the G estimates.
Table 2. Summary of the Statistics of the HTEM Performances at the Weishan Sitea
Bias (W m–2)
RMSE (W m–2)
a is the mean of observed values, and is the mean of HETM simulated values.
 The sensible heat flux had an overall RMSE of 27.3 W m–2 and a bias of 5.7 W m–2, which accounts for 5.9% of the mean observed H. However, the sensible heat flux was systematically overestimated in the wheat growing season and underestimated in the maize growing season. This was likely a result of the semiempirical estimates of LAI and hc (equations (26) and (27)).
 Despite the systematic errors in estimated H, the simulated LE showed high consistency with measurements, suggesting that errors in the estimates of available energy and the sensible heat flux were somehow canceled each other out. The overall RMSE of estimated LE for both seasons was 45 W m–2, and the bias was –10 W m–2. The LE RMSE was lower for winter wheat than for summer maize; however, the MAPE was higher for maize than for winter wheat.
 In general, HTEM performed better in estimating all four energy components during the maize season than in the wheat season. The fact that only a few measurements of LAI and hc were available during this experiment may result in inaccurate parameter estimates for equations (26) and (27). A more realistic relationship between LAI and hc may improve predictions of H and LE at this site.
4.2.2 Processes of Evaporation and Transpiration
 The measured total LE and estimated canopy transpiration (LEc) and soil evaporation (LEs) during the study period are shown in Figure 8. All three variables showed a bimodal process within one year due to the crop rotation cycle. The estimated LEc increases with the greening of crops and decreases with crop senescence (as indicated by changes in NDVI). On the contrary, LEs decreases during the crop greening stages and increases during senescence. This phenomenon suggests that HTEM could reasonably reflect the vegetation coverage effect on evaporation and transpiration partitioning. To further test the vegetation control on LEc and LEs partitioning in HTEM, the ratio of LEc (LEs) to equilibrium evaporation against LAI (Fr) are shown in Figure 9. The equilibrium evaporation is calculated from [Eichinger et al., 1996]
where Δ is the slope of relation between saturated vapor pressure and temperature, and γ is the psychometric constant.
 For both growing seasons of winter wheat and summer maize, the ratio of LEc to LEeq has a positive relationship with both LAI and Fr (Figures 9a and 9b). Theoretically, this is because a higher LAI value corresponds to a higher canopy light interception and a higher Fr value represents a larger proportion of surfaces occupied by vegetation, and therefore, as defined in the patch approach, a higher LEc would occur. Not surprisingly, the ratio of LEs to LEeq is negatively correlated with LAI and Fr, as can be seen from Figures 9c and 9d. It is noted that in Figures 9c and 9d, some points (within the oval) are significantly lower than the fitted lines; this was due to the fact that the surface soil moisture for these days was significantly lower than during the other days. Since the equilibrium evaporation only reflects atmospheric controls on evapotranspiration, the variation of soil moisture availability could lead to the scatter of the point clouds in Figure 9.
5.1 Comparison with Other Models
5.1.1 SMACEX site
 Extensive validation and intercomparison studies of remote sensing–based evapotranspiration models have been conducted using the SMACEX data [e.g., Choi et al., 2009; Li et al., 2005; Long and Singh, 2012a, 2012b]. Figure 10 shows the comparison of statistics of the discrepancies between surface flux retrievals and flux tower measurements from published studies and the present study. The main features of each model and the remote sensing data being used in each study are summarized in Table 3.
Table 3. Main Characteristics of Models Used for Comparison in This Studya
aClosure techniques include Bowen ratio (BR) and residual (RE) methods. Hyphen (-) indicates that the corresponding model does not need LST decomposition.
 As can be seen from Figure 10, the performances of HTEM in estimating H and LE are generally better than all other models in terms of both RMSE and bias with reference to tower-based measurements. The RMSE and bias of HTEM-estimated Rn and G lie in the ranges of those found in other models. The comparison between HTEM and TTME could be a convincing evidence of the importance of considering the surface aerodynamic characteristics in the trapezoid model, as the major difference between the two models is whether a resistance network was incorporated in the trapezoidal Fr/LST space or not. Although the SMACEX site has a relatively homogeneous landscape (dominate by crops), the comparison result does show a better H and LE estimates from HTEM than those from TTME.
 As for other models, SEBAL (surface energy balance algorithm for land) showed a comparable performance with other two-source models (i.e., TTME and TSEB), with an RMSE of LE of about 50 W m–2 in both of its applications ((1) in Long and Singh [2012a], and (2) in Choi et al. ), while the bias from SEBAL is on a magnitude of ~10 W m–2 but differs in direction between the two applications; this could possibly be due to a different selection of hot and cold pixels between applications. Long and Singh [2012a] modified SEBAL by introducing the trapezoid framework to avoid the subjectivity in selecting extreme points. The modified SEBAL model performed better than its ancestor with an RSME of 41.1 W m–2 and a bias of –4.4 W m–2 in estimated LE. In the two applications of the TSEB model ((1) in Li et al. , and (2) in Choi et al. ), the initial P-T coefficient was respectively set to be 1.3 and 1.26 for the whole study area, including both stressed and unstressed natural vegetation and crops. However, theoretically, the P-T coefficient should be a function of vegetation type and density, soil water status, and vapor pressure deficit [Agam et al., 2010]. A P-T coefficient larger than 1.26 typically represents unstressed full coverage conditions. Hence, TSEB has a tendency to overestimate LE under less soil wetness and larger drying power of air conditions [Agam et al., 2010; Choi et al., 2009; Fisher et al., 2008; Kustas and Norman, 1999; Long and Singh, 2012b].
5.1.2 Weishan site
Yang et al.  estimated energy fluxes using the surface energy balance system (SEBS) model based on the MODIS Terra image for the main growing season of winter wheat and summer maize of 2006–2008 at the Weishan flux site, and their model performances are compared with those from HTEM (Figure 11).
 For both growing seasons of winter wheat and summer maize, the HTEM RMSE of H and LE were slightly lower than those from SEBS. The benefit of HTEM is its hybrid dual-source scheme; the single-source scheme used by SEBS does not distinguish evaporation and transpiration, which results in great errors under low LAI and Fr conditions (i.e., the greening and senescence stages of both wheat and maize). However, both HTEM and SEBS showed systematic biases in H and LE during the wheat season. Since both Rn and G were accurately estimated during the wheat season, these systematic errors were most likely due to nonoptimal vegetation parameters, determined from remote sensing–based LAI estimates.
5.2 Sensitivity Analysis
 A local sensitivity analysis was conducted to examine how the uncertainties in the HTEM estimated LE could be apportioned to different sources of uncertainty in the model input. The sensitivity to the ith forcing variable or parameter is assessed by calculating LE with a set of baseline parameters (LE0) and comparing this with LE calculated by varying the ith parameter (LE±); the sensitivity index is
 The variation ranges and steps of each input variable are set the same as those in Long and Singh [2012b], which are 2 K for temperature variables, with a step of 0.5 K, and 20% for other variables, with a step of 5%. As suggested by Long and Singh [2012b], the sensitivity analysis was conducted using the data from the SMACEX site on DOY 174, which showed a wider range of soil moisture and vegetation coverage conditions. For DOY 182 and 189 in the SMACEX site, the soil moisture and vegetation coverage had less variability, which resulted in conservative estimates of sensitivity.
 As shown in Figure 12, LE is most sensitive to changes in temperature variables. The estimated LE showed positive correlations with Ta but negative correlations with LST (Figure 12a). An increase of 2 K in LST and Ta resulted in a 23.2% decrease and a 15.3% increase in estimated LE, while a 2 K decrease in LST and Ta could lead to a 17.3% increase and a 21% decrease in LE estimates, respectively (Table 4). However, compared with the TTME model, which is also based on the theoretical trapezoid, HTEM showed less sensitivity to temperatures. Long and Singh [2012b] reported that a 2 K increase in LST and Ta would respectively result in a 28.6% decrease and a 27.6% increase in estimated LE from the TTME model.
Table 4. Relative Sensitivity of Estimated LE from HTEM to Input Variable at the SMACEX Site on DOY 174
 The surface albedo plays a fundamental role in determining the total available energy, and LE is sensitive to changes in albedo. A 20% increase in α would result in a 7% decrease in LE. However, the changes in albedo of two extreme surfaces (equations (18)–(21)) have an insubstantial effect on the final estimation of LE (Table 4). Therefore, even if the process of determining αc_max and αs_max suffers from certain subjectivities, it will not greatly affect the accuracy of LE estimates.
 For other input variables, results indicate that wind speed (u), ea, and hc are all negatively correlated with LE estimates. A 20% increase in u, ea, and hc would result in a 7.4%, 1.6%, and 2.1% decrease in LE estimates, respectively. The reason why LE and u are negatively correlated is that an increase in u would lead to decreases in aerodynamic resistances and thus decreases in both Ts_max and Ta_max (equations (18) and (19)), which is indicative of the warm edge moving downward and therefore resulting in less latent heat flux. Moreover, it is shown that the hypothesized vegetation height for the theoretical fully vegetated surface with largest water stresses (hc_max) would not result in large uncertainty in the model.
5.3 Further Discussions on HTEM
 As implied in its name, there are two major advantages to HTEM. The first advantage lies in its trapezoid framework, in which the extreme boundaries were determined theoretically. The trapezoid framework allows HTEM to take soil water stresses on vegetation transpiration into consideration, which are neglected in triangle-based ET models and not fully addressed in the TSEB model. In triangle models, such as those in Batra et al. , Carlson , and Jiang and Islam , there is no difference between the wettest and driest fully vegetated surfaces, suggesting that vegetation is considered transpiring at its potential rate for the whole scene. In the TSEB model, an initial P-T coefficient of about 1.26 is a typical indicator of nonstressed conditions, while there are still uncertainties in determining the value of this coefficient in the model [Agam et al., 2010; Kustas and Norman, 1999; Long and Singh, 2012b].
 In addition, the warm and cold edges of the trapezoid are determined theoretically based on ground measurements and some hypothetic vegetation conditions in HTEM, which can avoid subjectivity and uncertainties in determining these extreme boundaries based on visual interpretation of satellite imagery being used [Long et al., 2012]. However, similar with the TTME model [Long and Singh, 2012b], the derivation of the theoretical boundaries in HTEM requires relative homogeneous meteorological conditions (i.e., Ta, Sd, and u) over the entire scene. For the application in the SMACEX site, spatially averaged Ta, Sd, and u measured at each flux tower were used to determine the extreme boundary conditions for the whole study area. This is because the SMACEX site is a relatively simple landscape dominated by crop species, and all measured meteorological variables showed small variations among sites. However, for more heterogeneous surfaces, especially with a high heterogeneity in the temperature field, this spatially average method of meteorological variable should be avoided. Alternatively, since these extreme conditions theoretically exist and the estimation of energy fluxes is pixel independent, the extreme boundaries and, therefore, the energy fluxes for each pixel can still be accurately estimated if the spatial distributions of meteorological variables are well defined.
 The second advantage of HTEM benefits from its hybrid dual-source scheme, which allows the trapezoid model to consider the surface aerodynamic characteristics. In addition, the dual-source scheme is generally superior to the single-source scheme in its ability to separate evaporation and transpiration, while single-source models are considered inappropriate for estimating evapotranspiration over sparse vegetated surfaces (Verhoef et al., 1997). An intercomparison of three remote sensing models (TSEB, METRIC, and TIM) conducted by Choi et al.  suggested that the largest difference of estimated fluxes between single-source models (METRIC and TIM) and a two-source model (TSEB) was observed over partially vegetated areas with LAI < 2.
 For dual-source models, both layer or “series” approach and patch or “parallel” approach have been widely used [Kustas and Norman, 1997; Lhomme and Chehbouni, 1999; Sánchez et al., 2008]. However, as indicated in the introduction, a layer approach works better for more uniform vegetated surfaces, while the patch approach performs better over more clumpy vegetation. In addition, both approaches have some limitations in estimating evaporation from soil component. The layer approach cannot distinguish evaporation from the soil surfaces under or between canopies, while the patch approach just simply neglects evaporation from under-canopy soil surfaces. The hybrid dual-source scheme used in HTEM, which partitions available energy between components based on the layer approach and route energy fluxes based on the patch approach, allows the model to estimate evaporation from both under-canopy and inter-canopy soils and distinguish between them.
 More importantly, both LAI and Fr are considered in the hybrid dual-source scheme in determining and partitioning energy fluxes between components, while the layer approach only considers the LAI and the patch approach only considers the Fr. It is important to note that both LAI and Fr are critical variables in determining and separating surface energy fluxes. However, LAI and Fr depict different characteristics of vegetation distribution. LAI focuses on the description of vertical density and distribution of leaves, while Fr explains more on the horizontal development of vegetation canopies. Therefore, both variables showed strong, but different controls on evaporation and transpiration processes, as shown in Figure 9. Although the value of both variables would change synchronously in some situations (i.e., in the farmland ecosystem), they function differently in determining evaporation and transpiration processes. In addition, this synchronized change in both LAI and Fr is unusual in natural ecosystems.
 Overall, HTEM showed reasonable ability in partitioning canopy transpiration and soil evaporation for a wide range of vegetation coverage conditions. However, without direct measurements of evaporation and transpiration, it is difficult to evaluate the accuracy of the partitioning results. The accuracy of partitioning estimates is highly dependent on that of energy allocating between soil and canopy, as it defines the amount of energy available for each component. The Beer's law (equations (1) and (2)) used in this study, as well as in other layer models, represents a simple but reasonable approximation of more sophisticated radiation transfer models over relatively uniform vegetated surfaces (e.g., croplands, grassland, and forests). However, it should be noted that the linear interpolation of the extinction coefficient kc between full vegetation cover and bare soil conditions or even the exponential extinction function itself may fail over more complex landscapes, such as shrublands and savannas. More efforts are needed to improve the radiation transfer module in the model.
 In the current study, a new remote sensing evapotranspiration model (HTEM) based on the hybrid dual-source scheme and the theoretical trapezoid framework is proposed. This model, designed to estimate energy fluxes using remotely sensed data, was inspired by Guan and Wilson , who partitioned potential evaporation and potential transpiration using the hybrid dual-source scheme, and by Long et al. , who theoretically determined the boundaries of the trapezoidal Fr/LST space. HTEM employs the layer approach to partition available energy and the patch approach to estimate sensible and latent fluxes separately from the soil and vegetation canopy. HTEM is different from a layer model in that it distinguishes the difference in evaporation from inter-canopy soil and from under-canopy soil, and limit convective transfer contribution to transpiration only from vegetated fractions. HTEM is also different from a patch model in that it allows soil evaporation from under-canopy soil, and the vegetation effect on both evaporation and transpiration is somehow considered. These features suggest a high potential of HTEM to be used for a wide range of surfaces with different vegetation coverage patterns.
 Soil wetness isolines within a theoretically determined trapezoid Fr/LST space are used in HTEM to decompose bulk radiative surface temperature into canopy temperature and soil temperature. In such a way, additional assumptions such as the P-T approximation used in the TSEB model and the complementary relationship used by Nishida et al.  are no longer needed. However, ignoring the advection effect on turbulent transport is still a limitation of HTEM; a more realistic cold edge should be the focus of further efforts.
 The performance of HTEM was tested at both the humid SMACEX site in Iowa with Landsat TM/ETM+ data and the subhumid Weishan site in North China Plain with MODIS Terra data. Results showed that energy fluxes from HTEM agree well with tower-based measurements and are generally better than other remote sensing evapotranspiration models applied with the same data sets. Additionally, HTEM could provide reasonable partitioning between evaporation and transpiration. Sensitivity analysis suggests that HTEM is mostly sensitive to temperature variables and less sensitive to other meteorological observations and the hypothetic vegetation parameters.
 We greatly thank the National Snow and Ice Center for providing the SMACEX data set; all people involved in the field campaign are greatly acknowledged. We thank Prof. Dawen Yang and Dr. Huimin Lei from Tsinhua University for sharing the Weishan data. We are also grateful to three anonymous reviewers, whose comments and suggestions were very helpful to us in improving the manuscript. This study is financially supported by the National Key Technology R&D Program of China (Grant No. 2011BAD25B05) and the National Natural Science Foundation of China (Grant Nos. 51279077 and 50939004).