Research Article | Open Access
Wind Speed-Independent Two-Source Energy Balance Model Based on a Theoretical Trapezoidal Relationship between Land Surface Temperature and Fractional Vegetation Cover for Evapotranspiration Estimation
An accurate estimation of terrestrial evapotranspiration over heterogeneous surfaces using satellite imagery and few meteorological observations remains a challenging task. Wind speed (u), which is known to exhibit high temporal-spatial variation, is a significant constraint in the abovementioned task. In this study, a wind speed-independent two-source energy balance (WiTSEB) model is proposed on the basis of a theoretical land surface temperature (Tr)-fractional vegetation coverage (fc) trapezoidal space and a two-stage evapotranspiration decomposing method. The temperatures in theoretically driest boundaries of the Tr-fc trapezoid are iteratively calculated without u by using an assumption of the absence of sensible heat exchange between water-saturated surface and atmosphere in the vertical direction under the given atmospheric condition. The WiTSEB was conducted in HiWATER-MUSOEXE-12 in the middle reaches of the Heihe watershed across eight landscapes by using ASTER images. Results indicate that WiTSEB provides reliable estimates in latent heat flux (LE), with root-mean-square-errors (RMSE) and coefficient of determination of 68.6 W m−2 and 0.88, respectively. The RMSE of the ratio of the vegetation transpiration component to LE is 5.7%. Sensitivity analysis indicates WiTSEB does not aggravate the sensitivity on meteorological and remote sensing inputs in comparison with other two-source models. The errors of estimated Tr and observed soil heat flux result in LE overestimation/underestimation over parts of landscapes. The two-stage evapotranspiration decomposing method is carefully verified by ground observation.
Evapotranspiration (ET) plays a significant role in modeling the terrestrial hydrological cycle and energy exchange in a soil-vegetation-atmosphere system. Satellite remote sensing technology potentially provides regional and global ET in an efficient way due to the routine and large spatial scale of observations of land surface properties (e.g., albedo, land surface temperature, Tr, and vegetation index, VI). Numerous remote sensing models that imply different theoretical complexity and assumptions have been proposed [1–4]. Among them, models using flux-profile relationship (expressed as vast temperature gradients and aerodynamic resistances) on the basis of the Monin–Obukhov Similarity (MOS) theory and energy balance theory to physically calculate LE (e.g., SEBAL , METRIC , MOD16 [7, 8], TSEB , HTEM , and STSEB ) or evaporation ratio (e.g., SEBS  and TTME ), are primarily proposed. Considering the different treatment of land surface, those physically based methods can be further divided into one-source methods (e.g., SEBAL, METRIC, and SEBS) and two-source methods (e.g., TSEB, TTME, HTEM, MOD16, and STSEB). One-source methods treat land surface as a “big leaf” and insist turbulent exchange between land surface and atmosphere to occur at a certain height. Although one-source algorithms reported reliable heat fluxes estimate [6, 14, 15], they often require a precise priori calibration which cannot be guaranteed by diverse range of surface conditions, especially in sparse canopy areas [1–3, 16–20]. Two-source models treat soil and vegetation as different “sources” in heat and water exchange and simulate sensible heat flux, H, and LE components differently by component temperatures (i.e., the soil and canopy temperatures), thereby representing an advancement to avoid the priori local calibration in the one-source model . The two-source model is demonstrated to be robust for a wide range of landscape and hydrometeorological conditions [20–24].
Generally, wind speed u (or friction velocity ) is the core parameter in calculation of aerodynamic resistance, ra in both one- and two-source physical models is based on flux-profile relationship on the basis of MOS theory, and u (or ) considerably affects H and LE estimate. Sánchez et al.  emphasized that a 10% perturbation of u causes the H and LE estimates by the STSEB model perturbing 17% and 4%. Long and Singh  reported that a 20% perturbation of brings an approximately 12.0% variation in the LE estimate in the TTME model. A 20% perturbation of u in the HTEM model has resulted in a 7.4% variation of LE . Wang et al.  reported a 25% increase or decrease in u leads to a 8.6% increase or a 11.5% decrease in H estimates in a modified SEBAL model. Webster et al.  found that u and Ta are nearly as influential as Tr for the HTEM and SEBS LE estimates.
However, high-quality grid u (or ) is not routinely available. It cannot be remotely sensed. Although reanalysis datasets (e.g., the National Centers for Environmental Prediction Department of Energy (NCEP/DOE), NCEP National Center for Atmospheric Research (NCEP/NCAR), Global Land Data Assimilation System (GLDAS), and China Meteorological Administration Land Data Assimilation System (CLDAS) provide grid u, the accuracy essentially depends on the ground observation network, which needs to be improved to match the requirement of ET estimation. Decker et al.  indicated that six reanalysis products had RMSE between 1.5 m s−1 and 4.5 m s−1 against observation of 33 stations in global. In addition, the temporal-spatial scale of the reanalysis data is usually too coarse to be combined with the finer-scale remote-sensing image .
There are many studies which attempt to reduce the dependence on u. Some researchers transformed ra to substitutive u-independent resistances, and Mu et al. [7, 8] and Yao et al.  assumed that the ra was parallel to radiative transfer resistance and convective transfer resistance and made ra quantitatively relate to Ta and atmospheric pressure or only Ta. This assumption seems too simple in simulating the significant variation of turbulent flux of heat and momentum . Others introduced a strategy to build a bridge between the actual pixel and a reference site. Qiu et al.  assumed that the actual ra was equal to that over a referenced dry bare soil surface. This assumption was inconsistent with the observations by Liu et al.  over bare soil and over the maize canopy. Sun et al.  subsequently proposed an assumption that u of the actual pixel was equal to that of referenced dry bare soil within the atmospheric surface layer to modify Qiu’s method’s lack of considering aerodynamic characters of land surface. Nishida et al.  employed similar hypothesis and found that the estimation of u was one of the largest error source. The estimated error of u was deduced from the unstable estimation of variables (e.g., Tr, Rn, and soil heat flux, Gs) over referenced dry bare soil surface. Furthermore, equivalent u assumption not takes into account the effect of surface roughness on u .
To overcome the dependence on u, this paper simulates aerodynamic resistances at given vegetation coverage by means of a referenced water-saturated site with the assumption of the absence of sensible heat exchange between referenced site and upper atmosphere in the vertical direction and proposes a u-independent two-source energy balance (WiTSEB) model. Section 2 introduces the WiTSEB model. Section 3 describes the study area and data. Section 4 presents the results. Section 5 discusses the sensitivity, error source, uncertainty analyses, and comparison with other models. Finally, Section 6 provides the conclusions drawn from this study.
2. Materials and Methods
The WiTSEB model includes three major modules, i.e., a two-source model framework (Section 2.1), the decomposition of Tr via a two-stage ET method [32, 33] (Section 2.2), and an iterative process of calculating LEc and LEs (Section 2.3). The second module is primarily based on a theoretical u independent Tr-fc trapezoidal space modified from a Tr-VI trapezoidal space proposed by Wang et al.  and Wang et al. .
2.1. Two-Source Model Framework
The two-source model normally contains layer configuration  and patch configuration [11, 35]. Compared to the layer approach, the patch approach provided comparable estimate accuracy  and has an advantage of requiring fewer additional information of the canopy structure to allocate net radiation, Rn, for the soil and canopy component . Therefore, the WiTSEB adopts the patch framework  to simulate energy fluxes:where subscripts c and s represent the vegetation and soil components hereafter, respectively; Rn denotes the net radiation (W·m−2); H is the sensible heat flux (W·m−2); LE is the latent heat flux (W·m−2);G and Gs are the soil heat flux and soil heat flux of soil component (W·m−2), respectively; Gs can be estimated as a fraction (CG) of Rns , where CG varies from 0.2 to 0.5 depending on the soil type and soil moisture, and a constant value of CG (=0.35) is used like other two-source models [9–11, 13]; fc is the fractional vegetation coverage and is calculated by the method recommended by Choudhury et al.  but replaces the normalized differential vegetation index (NDVI) by the enhanced vegetation index (EVI) because NDVI results in asymptotic (saturated) signals and scaling problems during high biomass conditions .where EVImax and EVImin are the EVI of the complete vegetation and bare soil surface, respectively, and n is an empirical coefficient which is related to leaf orientation and distribution within a canopy.
R nc and Rns are Calculated aswhere αc and αs represent albedos (dimensionless) of vegetation and bare soil, respectively; we assume that the actual pixel α is a weighted composite of αc and αs with fc, and αc is fixed as 0.20 ; Rd is the downward shortwave radiation (W·m−2) and is calculated by the method proposed by Allen et al. ; Ta is the air temperature (K); Tc and Ts are the temperatures of canopy and soil (K), respectively; ε is the land surface emissivity (dimensionless); εa is the atmospheric emissivity (dimensionless) and is calculated by the method recommended in Brutsaert ; and σ is the Boltzmann constant.
H c and Hs are calculated as follows:where ρ is the air density (kg⋅m−3), Cp is the air specific heat at a constant pressure (1004 J K−1 kg−1), and rac is the aerodynamic resistance (s m−1) to heat transfer between meteorological observation height z (m) and vegetation surface; it is expressed aswhere k is the Von Karman constant (=0.41); d is the zero-displacement height (m) and is equal to 2/3 h, where h is the vegetation height (m); is the friction wind speed of the canopy surface; L is the Monin–Obukhov length (m); φh and φm are the stability functions for heat and momentum, calculated according to the value of L; and z0hc is the canopy roughness length for heat (m) and is calculated by the canopy roughness length for momentum (z0mc) and a dimensionless parameter (), i.e., , where the is calculated with the approach proposed by Brutsaert :where Cd is the drag coefficient of the foliage elements and sets to 0.2 , Ct is the heat transfer coefficient of the leaf, and Ux is a function of nondimensional drag area density.
r as is the aerodynamic resistance (s m−1) to heat transfer between z and bare soil surface and is calculated aswhere is the friction wind speed of the soil surface in the canopy and z0hs is the soil roughness length for heat (m) and is calculated by the soil roughness length for momentum (z0ms) and , where we set z0ms as 0.01 m; is calculated as where Re is the roughness Reynolds number (, with being kinematic viscosity of the air).
r ss is the aerodynamic resistance (s m−1) between soil surface in canopy and z0ms + d, which is calculated by the method recommended in Zeng et al. :where cs is the turbulent transfer coefficient.
2.2. Decomposition of Tr Based on the Trapezoidal Space
2.2.1. Construction of the Theorical Tr-fc Trapezoidal Space without u
Wang et al.  constructed a theoretical Tr-VI trapezoidal space (Figure 1(a)) on a pixel basis. Wang et al.  further refined the Tr-VI trapezoidal space to estimate the regional ET of a semiarid watershed. Theoretically, there are four vertexes representing extreme conditions within the trapezoid space; i.e., Point 1 represents a well-watered vegetation in which the soil moisture of the root zone is sufficient and has a minimum canopy temperature, Tc,min; Point 2 represents the water-stressed vegetation in which the soil moisture of the root zone is considerably deficient and has the maximum canopy temperature, Tc,max; Point 3 represents the well-watered bare soil that has a minimal soil temperature, Ts,min; and Point 4 represents the dry bare soil in which soil evaporation is seriously inhibited and has a maximum soil temperature, Ts,max. The temperatures at the four vertices of the Ts-VI trapezoid are calculated aswhere subscripts 1, 2, 3, and 4 represent the values of vertices plotted in Figure 1, Gf denotes the ratios of G to Rn, Gf3 and Gf4 are set to 0.25 and 0.35, respectively, rcm and rcx are the minimum and maximum canopy resistances (s m−1) and set to 12.5 and 625, respectively, Δ is the slope of saturated vapor pressure to air temperature (kPa C−1), γ is the constant (KPa⋅C−1), and VPD is the vapor pressure deficit of the air (hPa).
The Tr-VI trapezoid method has robust theoretical basis, but it requires grid u to calculate ras and rac, which may show uncertainty especially in heterogeneous landscapes . Here, we further modify the Tr-VI trapezoid in two aspects to construct a u-independent Tr-fc trapezoid (Figure 1(b)).
First, we use fc to replace EVI in constructing the trapezoid space to be consistent with the decomposition of Tr. Tr is typically noted as a mean of the Tc and Ts weighted by fc :
Second, we simplify the calculation of the extreme temperatures by employing an assumption that sensible heat flux in the vertical direction at the well-watered edge under given meteorological conditions is negligible. It is equivalent to the assumption in previous studies [13, 41] that well-watered edge in theory is at equilibrium ET rate when it ignores horizontal advection. This assumption implies the neutral atmospheric conditions of the well-watered edge on basis of MOS theory. Then, we can calculate the rac2 and ras4 of neutral atmospheric conditions (i.e., rac0 and ras0) according to equations (16a) and (16c) without employing u as the input when the Tc,min and Ts,min are known. In practice, we employ the averaged Ta of well-watered landscapes as Tc,min and Ts,min. The calculation of ras4 (or rac2) in equation (13) (or equation (11)) can be split into two parts: the ras0 (or rac0) and the atmospheric stability correction function, f(φms4, φhs4), or f(φmc2, φhc2) :
The procedure for calculating Tc,max and Ts,max on a pixel basis is summarized in the left dash line box of Figure 2, which is divided into two major steps:(1)Calculate the initial rac2 and ras4 (i.e., the rac0 and ras0 by means of equations (16a) and (16c), respectively) assuming f(φmc2, φhc2) = 0 and f(φms4, φhs4) = 0.(2)Iteratively calculate Tc,max and Ts,max (equations (16b) and (16d); Rnc2 (equation (8a)), Rns4 (equation (8b)), Hc2 (=0.9Rnc), and Hs4 (=Rns4(1 − Gf4)); Lc2 and Lc4; φmc2, φhc2, φms4, and φhs4; (combining equation (9) with equation (11)), (combining equation (10) with equation (13)), (equation (12)), and (equation (14)); and f(φmc2, φhc2) (the equation similar to equation (18)), f(φms4, φhs4) (equation (18)), rac2 (equation (11)), and ras4 (equation (13)) until the rac2 and ras4 are stable, i.e., the difference between two adjacent calculations are smaller than five percent. Normally, the stability can be satisfied within 10 times.
It should be noted that the Hc2 is set to 0.9 Rnc2 because there exists the epidermal cuticle transpiration under extreme drought conditions .
2.2.2. Calculation of Component Temperatures
We adopt a two-stage ET method with the Tr-fc trapezoidal to estimate Ts and Tc. The two-stage ET method divides the dynamic variation of ET versus soil moisture into two successive phases under a given atmospheric and vegetation conditions, i.e., the soil evaporation stressed stages and the vegetation transpiration stressed stages.
In the soil evaporation stressed stage, where the surface soil moisture (0–5 cm) gradually decreases, LEs vary from the maxima to 0. By contrast, the soil moisture of the root zone remains in the bound that can maintain a canopy transpiring nearly in the potential rate. These variations reflected in the Tr-fc trapezoid (Figure 1(b)) are the temperature variation versus fc of the under triangle. Ts increases from the minima (Ts,min) to the maxima (Ts,max), and Tc remains invariant, which are given by
In the vegetation transpiration stressed stage, the soil moisture of the root zone substantially decreases and restrains LEc and Tc varies from the minima (Tc,min) to the maxima (Tc,max). Moreover, Ts maintains Ts,max under extreme deficiency of surface soil moisture. The variation in temperature versus fc in this stage can be reflected by the upper triangle depicted in Figure 1(b). Ts and Tc can be calculated as
2.3. Calculation of LEs, LEc, and LE
The aforementioned assumption is also employed in this part to calculate Hc and Hs, i.e., the u-independent rac0 and ras0 combining with the Tc and Ts are used to estimate Hc and Hs by an iterative process, respectively.
The iterative process of calculation of Hc and Hs contains two major parts (Figure 2, right dotted box):(1)Calculate the initial rac and ras, i.e., the u-independent rac0 and ras0(2)Iteratively calculate Hc (equation (9)) and Hs (equation (10)); Lc and Ls; φmc, φhc, φms, and φhs; (combining equation (9) with equation (11)) and (combining equation (10) with equation (13)); (equation (12)), (equation (14)), f(φmc, φhc), f(φms, φhs), rac (equation (11)), and ras (equation (13)); and rss (equation (15)) until rac, ras, and rss are stable
3. Study Area and Data
3.1. Study Area and Ground Observations
The multiscale observation experiment on evapotranspiration over the heterogeneity of the Heihe Watershed Allied Telemetry Experimental Research in 2012 called HiWATER-MUSOEXE-12  was conducted between May and September 2012 in the desert-oasis transition zone in the middle reaches of the Heihe watershed (Figure 3). The mean annual (1961–2010) air temperature and precipitation of the area are 7.4°C and 128.7 mm .
HiWATER-MUSOEXE-12 contains two nested observation matrices, namely, a large area of 30 km × 30 km and a core area 5.5 km × 5.5 km located in the Yingke and Daman irrigation district. HiWATER-MUSOEXE-12 equipped 21 automatic weather stations (AWS) and 22 eddy-covariance (EC) stations (two in the DM superstation) and 1 water isotope station (DM superstation) over different land covers (including corn, desert, Gobi, desert steppe, orchard, vegetable, residential, and wetland). Meteorological data, including Ta, relative humdity rh, upward/downward shortwave radiation, upward/downward longwave radiation, net radiation, multilayer soil moisture, and soil temperature, were observed at 21 AWS with an interval of 10 min. Ta and rh were spatially interpolated to the study area with the inverse distance weighting method (IDW) . The soil heat flux G is calculated via the Plate Cal method . This method comprises two parts as follows: the heat-plate flux, which is calculated using fc weighted average of the three heat-plates (6 cm below the ground around each flux tower) measurements, and the change of the heat storage in soil, which is calculated by soil temperature change rate, soil moisture, and soil porosity at depths of 2 and 4 cm. G was not calculated at S4, HZZ, and ZY sites given the lack of soil porosity measurement. Quality control on G was conducted primarily on the basis of the rationality of temporal variation of soil temperatures. Vegetation height, h, is measured routinely at each AWS station, and the interval of h between two measurements is calculated by linear interpolation.
The preprocessed fluxes from the EC were averaged in 30 min and typically divided into three quality levels as follows: Level 0 (the quality assessment method for stationarity, Δst < 30, and the integral turbulent characteristics test, ITC < 30), Level 1 (Δst < 100 and ITC < 100), and Level 2 (Δst < 100 and ITC > 100). To ensure quality, EC fluxes from Level 2, from suspected instrument drift, and from abnormal G (mainly refers to soil temperature measurement abnormality) were rejected. EC fluxes are generally considered as an energy imbalance. Bowen ratio (BR) and residual energy (RE) are two common methods used for solving the energy imbalance problem in EC systems . The BR method was used in the present study. At the DM station, the ratios of the transpiration component to LE (LEcanopy/LE, where LEcanopy is the canopy transpiration component in a mixed pixel, = LEc × fc) and the evaporation component to LE (LEsoil/LE, where LEsoil is the soil evaporation component in a mixed pixel, = LEs × (1 − fc)) were measured using a cavity ring-down spectroscopy (CRDS) water vapor isotope (Model L1102-I, Picarro, Inc.) and represented an average of 13:00–15:00 (local time) ; the ET and its evaporation component were calculated via a gradient diffusion method and Craig–Gordon model, respectively.
3.2. Remote-Sensing Data
ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer) level 1B images on eight cloud-free dates (June 15, June 24, July 10, August 2, August 11, August 18, August 27, and September 3) were acquired from the US Geological Survey website (https://earthexplorer.usgs.gov/). The ASTER provides visible and near-infrared band observations with spatial resolutions of 15 m between 12:10 and 12:20 (local time). ASTER L1B visible-infrared data were resampled to 90 m via a nearest neighbor method. α is calculated by fitting narrow-band reflectances :where ρ1, ρ2, and ρ3 represent the reflectances of band1, band2, and band3 covering the visible and near-infrared spectrum, respectively. The EVI was calculated via a two-band method proposed by Jiang et al.  who considered that the ASTER sensor did not provide a blue band:
Land cover images are acquired via visual interpretation. Tr and ε products were provided by the Cold and Arid Regions Science Data Center; these products were estimated on the basis of temperature-emissivity separation (TES) algorithm  using the ASTER thermal infrared band with the spatial resolution of 90 m.
4.1. Validation of WiTSEB Outputs
The scatterplots of WiTSEB outputs (Rn, G, H, and LE) versus ground observations are displayed in Figure 4. The accuracy of WiTSEB outputs are further measured using the mean bias error (MBE), root mean square error (RMSE), and coefficient of determination (r2) in comparison with ground observations, as listed in Table 1.
Rn was best modeled based on equations (1), (8a), and (8b) in terms of the overall RMSE of 31.9 W m−2 among the four components of land surface energy balance equations. There was averagely a little underestimation by 5.3 W m−2 with an r2 of 0.82. The performance of estimated Rn varied with landscapes. That is, it was underestimated at corn, vegetable, and orchard landscapes, but was overestimated at desert steppe, Gobi, residential, and wetland sites. The largest error was occurring over the wetland landscape with an MBE of 64.6 W m−2 and an RMSE of 65.1 W m−2, which would be probably due to the significant underestimation of α.
Due to lack of soil porosity measurement, the accuracy of estimated G was only evaluated at corn, vegetable, orchard, desert, and Gobi sites. G was estimated as a constant of Rns (=0.35), overall yielding an MBE of −6.4 W m−2, an RMSE of 43.1 W m−2, and an r2 of 0.22, compared with tower-based measurements. The fixed fraction strategy might be the principal reason for the low performance in the G estimates.
Reasonable agreement was obtained between the estimated H and the tower-based observations with the MBE and RMSE of −11.7 and 63.5 W m−2, respectively. Vegetable and corn sites had relatively small bias of H estimates over in contrast to other landscapes. Negative H measurements are performed at several individual corn sites (Figure 4(c)), whereas their H estimates are positive because the assumption is absence of accounting the effects of horizontal convection.
The estimated LE by WiTSEB has an MBE of 2.7 W m−2, an RMSE of 68.6 W m−2, and a r2 of 0.88 (Table 1). Corn and orchard sites show closer agreement with the ground observation than dry sites, i.e., Gobi, desert, and desert steppe sites, in terms of a smaller bias and a lower RMSE.
Furthermore, we also estimated LE by a model (we named it as TSEBu for convenience) that is identical to WiTSEB, i.e., the patch frame and the two-stage decomposing method, except that TSEBu employes u as the input to calculate the aerodynamic resistances (e.g., rac2, ras4, rac, ras, and rss). The results in Table 2 are the LE-estimated accuracy of TSEBu by using ASTER images from eight dates at the HiWATER-MUSOEXE-12 against the 21 tower observations. Although TSEBu outperforms WiTSEB in terms of RMSE over vegetable, wetland, desert, and desert steppe, it overall shows slightly lower performance, yielding an MBE of 8.0 W m−2, an RMSE of 70.7 W m−2, and an r2 of 0.87.
4.2. Comparison with Previous Studies
The HiWATER-MUSOEXE-12 dataset has been used to evaluate different ET estimation methods in previous studies (Table 3). The RMSE of LE via different methods varies between 17.9 and 133 W m−2, with a median of 74.8 W m−2. Where, TD-TSEB  and the nonparametric model  have the advantage of removing the dependence on u, which provide an RMSE of 89.8 and 133 W m−2, respectively. The WiTSEB uses similar inputs and two-source framework as the TD-TSEB but has a lower RMSE (i.e., 41.9 W m−2 at the DM station). Although this accuracy comparison is lacking absolute meanings because of assumptions difference, it can illustrate the acceptable accuracy.
Superscripts b and c represent the observed LE corrected via the BR and RE method, respectively.
4.3. LE Partition
Five days (June 24, July 10, August 2, August 11, and August 27) within the eight dates of ASTER have CRDS water vapor isotope observations at the DM station. The five-day average of canopy transpiration component ratio (LEcanopy/LE) estimation is 84.6%, which is 3.2% lower than the observation (Table 4). A previous study indicated that the CRDS slightly overestimates the ratio of LEcanopy/LE . Therefore, the accuracy of LEcanopy/LE estimates via the WiTSEB is expected to be high. By contrast, Song et al.  adopted a TSEB model using an eight-date observed Tr from a Fluke Ti55 thermal infrared imager and other observations (Table 3) to estimate LEcanopy/LE and obtained the MBE and RMSE of 1% and 2%, respectively. Yang et al.  evaluated the performances of the HTEM, TSEB, and MOD16 models using ASTER images from six dates and determined the MBE of LEcanopy/LE at −1.1%, 4.1%, and −20.5%, respectively. Yao et al.  used Landsat images from five dates to assess the TD-TSEB model and obtained the MBE of −11.1%.
We assessed the performance of the WiTSEB (Table 4) in LEsoil (the soil evaporation component in a mixed pixel, = LEs × (1 − fc)) and LEcanopy estimates at the DM station. The results indicated that the MBE (RMSE) of LEsoil, LEcanopy, and LE are 21.9 (33.7), −0.9 (49.3), and 20.8 (41.9) W m−2, respectively. By contrast, Song et al.  obtained the RMSE of LE at 61 W m−2 by using the TSEB model and ground observations. The values of LEsoil, LEcanopy, and LE underestimated by the TD-TSEB model  were 4.2, 60.1, and 63.6 W m−2 with the RMSE of 16.5, 91.3, and 89.8 W m−2, respectively.
4.4. Spatial-Temporal Variation
The estimated LE maps for eight dates are exhibited in Figure 5. These maps show significant spatial variations dominated by the spatial distribution of land covers. Generally, LE is large at places of high fc (i.e., wetland, corn, vegetable, and orchard) and low at places with sparse vegetation (i.e., Gobi, desert, and desert steppe). Wetland has the largest LE given sufficient water supply, with an average of 485.4 W m−2 over eight dates. Although precipitation is scarce (the cumulative precipitation between June and September in 2012 is ∼100 mm), the orchard, corn, and vegetable-covered areas produce a high LE because of irrigation, with averages of 452.1, 422.1, and 393.3 W m−2, respectively. Furthermore, regional irrigation can amplify the spatial difference in soil moisture to increase the spatial variation in LE despite the same land cover type; for example, the LE in June 24 is higher in the northeastern region of the core area (blue box) than in other areas (Figure 5, left column). The soil moisture at a 4 cm depth is significantly higher in S3, S9, and S10 (36–39%) than in other corn-covered sites (17–29%). The LE of the residential area has an average of 262.6 W m−2, which is mainly derived from canopy transpiration (174.6 W m−2). Owing to water stress (i.e., lack of precipitation and no irrigation), desert, Gobi, and desert steppe provide a small LE, with an average of 44.5, 66.3, and 69.0 W m−2, respectively. August 18 is the only date with a two-day antecedent effective precipitation (i.e., daily precipitation is greater than 0.5 mm) of 3–5 mm; therefore, Gobi, desert, and desert steppe consistently have a significantly higher LE on August 18 than other dates.
The comparison of LE maps in different dates indicates a temporal variation in LE estimates in areas with high fc due to phenological changes. The averages of LE at the corn and vegetable areas are significantly higher in July and August than in June and September. The orchard and wetland areas also show a temporal variation, but their ranges are smaller than those of the corn and vegetable areas.
In Figure 6, the ratios of LEcanopy to LE (LEcanopy/LE) on the desert and desert steppe surfaces are the highest with the average of 0.95 and 0.92, respectively, over eight dates. This result is because the soil evaporation is extremely stressed given the severe dryness on the soil surface, whereas the canopy can use the deep soil water transpiration by maintaining a slight transpiration. LEcanopy/LE on the desert and desert steppe surfaces are lower on August 18 than on other dates given soil evaporation increase in relation to precipitation. Crops (including corn and vegetable) and orchard areas have analogous LEcanopy/LE with an average of approximately 0.86. Residential has the lowest LEcanopy/LE with an average of 0.72.
5.1. Sensitivity Analysis
The WiTSEB model inputs include the following details: (1) the ground meteorological observations: Ta, rh, and h; (2) remotely sensed data: Tr, α, ε, and EVI; and (3) the parameters: Gf3, Gf4, α1, α2, α3, α4, z0mc, rcm, and rcx. Model sensitivity to inputs is analyzed to understand the sources of error and mechanisms of error propagation of the WiTSEB. The sensitivity of LE to the ith input, Si, is defined aswhere LE0 represents the averaged LE estimated by actual inputs, LE± represents the average LE of eight dates of the study area estimated by the ith input increase (+) or decrease (−) of a value or a ratio with respect to the actual LE, and the variation ranges and steps are 2 K and 0.5 K for temperature inputs (Ta and Tr) and 20% and 5% for other inputs, respectively.
In Table 5, the WiTSEB is sensitive to temperature variables. Tr and Ta are negatively and positively correlated with LE, respectively. A 2 (1) K decrease or increase in Tr may result in a 15.9 (8.1)% increase or a 17.2 (8.5)% decrease in LE estimates, respectively. A perturbation of −2 (−1) K or 2 (1) K in Ta will cause an LE perturbation of −22.7 (−10.8)% or 18.9 (9.9)%, respectively. The WiTSEB shows a similar magnitude of sensitivity to other remotely sensed variables (α, ε, and EVI) and is lower than the sensitivity to Tr. LE estimates are positive to EVI, and a 20% increase or decrease in the EVI causes a 1.9% increase or a 4.5% decrease in LE estimates, respectively. LE estimates are negative to α, ε, and rh. A 20% increase or decrease in α will result in a 5.1% decrease or a 5.3% increase. The WiTSEB is insensitive to parameters. The perturbation of LE is less than 3% when the parameters perturb 20%. LE estimates show a positive relationship with Gf3, α1, α3, z0mc, and rcx but are negatively related to Gf4, α4, h, and rcm. A −20 (20)% perturbation of α2, α4, and h can cause LE perturbations of 0.1 (−0.1)%, 1.9 (−2.3)%, and 0.2 (−0.4)%.
Variations of Tr and Ta in K and other variables in percentage (%).
Similar sensitivity analyses have been conducted for many trapezoid-based two-source models (Table 6), such as HTEM , TTME , and ESVEP . In comparison with these models, the WiTSEB avoids using u as the input and does not increase the dependence on other inputs.
Furthermore, if the ground Rd and Rld observation is sufficient, we can employ them as inputs to WiTSEB. This way, the performance of WiTSEB could be comparatively improved, i.e., the RMSE of LE decreases from 68.6 W m−2 to 62.9 W m−2 and the r2 increases from 0.88 to 0.89. We conducted the sensitivity analysis. Both Rd and Rld are positive to LE estimates. A 5% perturbation in Rd and Rld may result in 6.6% and 3.1% LE estimate variation, which are almost the same as the ESVEP  and slightly lower than that of STSEB .
5.2. Error Source Analysis
5.2.1. Error in Tr and α Estimation
The Tr observation is computed aswhere Rlu and Rld are the observed upward and downward longwave radiation (W m−2), respectively. Land surface emissivity ε is derived from the Tr product, which is estimated by the TES algorithm.
In Figure 7(a), the Tr product shows consistency with the observation, with MBE, MAE, RMSE, and r2 of 1.1 K, 2.0 K, 2.9 K, and 0.89, respectively. The Tr product shows high accuracy at the corn and orchard area. Similarly, LE estimates at both areas yield a high accuracy (Figure 4(d) and Table 1). Tr is overestimated in the Gobi, desert steppe, and desert areas, with an MBE of 2.1, 4.6, and 5.4 K, respectively. LE estimates at the sparse vegetation areas are negatively biased by 42.1, 108.3, and 73.9 W m−2, respectively. On June 15, June 24, August 2, August 11, August 27, and September 3, Tr at the desert steppe site is overestimated by 4.5, 4.6, 6.0, 3.4, 5.6, and 5.6 K, respectively, thus potentially rendering the point (fc, Tr) out of the envelope displayed in Figure 1(b) and resulting in LE estimates at approximately 0. At the residential site, Tr is significantly underestimated, which is consistent with an LE overestimation of 34.9 W m−2 presented in Figure 4(d) and Table 1.
Similar to Tr, α is also estimated from the ASTER image. The estimated α has an RMSE of 0.021 and an r2 of 0.65. α shows slight overestimation in general compared with the observation from AWS (Figure 7(b)) with an MBE of 0.01. However, α gives significantly underestimated over wetland, with an MBE of −0.05, which creates the comparatively overestimations of Rn (Figure 4(a) and Table 1) and LE (i.e., an MBE of 70.7 W m−2) even though Tr is overestimated (i.e., an MBE of 0.64 K).
5.2.2. Error in Ta Interpolation
We calculate the standard deviations (SD) of grids in the 90 m × 90 m resolution to quantify a spatial variability (Table 7) and employ the leave-one-out crossvalidation (LOOCV) method to analyze interpolation accuracy (Figure 8). The SD varies between 0.1 K and 0.3 K with an average of 0.2 K in eight dates, thereby indicating low spatial variability. The interpolated Ta shows a favorable relationship with the observation given the r2 of 0.97. The interpolated Ta is lower than the observation Ta of 0.2 K and has an MAE and RMSE of 0.4 K and 0.5 K, respectively. Those two indices imply that Ta may not be the main error source of the WiTSEB model in this study area.
However, researchers need to be cautious when to use the method to complex topography and rare stations area because the performance of the interpolation methods is sensitive to the density and variability of meteorological data. Webster et al.  evaluated seven interpolation methods at a diverse topography and high landscape heterogeneity area in Southeastern Australia and found that no single spatial interpolation method can provide a reliable performance across various conditions and meteorological data.
5.2.3. Errors Introduced by the Assumption
To circumvent the dependence on u, an assumption, i.e., the water-saturated surface is under neutral conditions, is used to construct the Tr-fc trapezoid (Figure 1(b)) and to calculate the LE components by means of averaged Ta of vegetable, corn, and wetland landscapes. Using a certain condition of Ta as Tc,min and Ts,min is a common practice for constructing a theoretical Tr-VI or Tr-fc trapezoidal space [10, 13, 27, 32]. Tang and Li  found that the estimated Tc,min and Ts,min via Moran’s method  were slightly different from the Ta at the Yucheng station covered by croplands over the 58 clear-sky days from March 2010 to October 2011. Wang et al.  found that the calculated Tc,min via a revised Moran’s method is lower than the pixel Ta of 2.9 K, However, the discrepancy is related to the semiarid spare vegetation condition. Nevertheless, Prince and Goward  believed that the magnitude of the difference between the two temperatures is approximately 2 K.
Here, we analyze error propagation in the WiTSEB variables (i.e., Rnc1, Rns3, ras0, rac0, ras, rac, Ts,max, Tc,max, Ts, Tc, Hc, Hs, H, and LE) by introducing the decrease in Tc,min, Ts,min, and both at 3 K by considering the research of Wang et al. . In Figure 9, a 3 K decrease in Ts,min causes the magnitude of increase or decrease in variables less than 4.5%, where the magnitudes are 0 in the Rnc1, rac0, rac, and Tc,max estimate. A 3 K decrease in Tc,min and in Tc,min and Ts,min results in an increase in Hc estimates of 15.1% and 16.3%, respectively, whereas the effects on other variables are not obvious with 5.5% and 6.9% in H estimates and −3.3% and −4.3% in LE estimates, respectively.
5.3. Uncertainty in G
EC flux measurements are extensively known as an energy imbalance, i.e., the sum of the observed fluxes (H + LE) is less than the available energy (Rn − G), thereby yielding an approximately 16% uncertainty for LE during HiWATER-MUSOEXE-12 . Reasons that cause energy unclosure include mismatch of measurement footprints , horizontal advection, and ignoring heat storage in measuring G . Here, we emphatically analyze the G measurement error and its effect on the LE observation. The Plate Cal method  is used to observe G. Sites (including S4 residential, HZZ desert steppe, and ZY wetland) are not used in calculating the heat storage changes given a lack of soil porosity measurements, thus resulting in the G observations less than the actual value. Consequently, the corrected LE via the BR method will be higher than the actual value. The HZZ desert steppe has a less systemic error of Tr estimate (MBE of 4.6 K vs. 5.4 K) but has a larger LE underestimation (−108.3 W m−2 vs. −73.9 W m−2) than the desert site. This contrast can be interpreted by the observed error of LE caused by G observation, i.e., the underobserved G augments the BR-corrected LE and aggravates the underestimation of LE estimates. Furthermore, the underobserved G at the residual site mitigates the overestimation caused by the significant underestimation of Tr.
The result shown in Figure 4(b) and Table 1 indicates a relatively large disagreement between the estimated G and the measured G. This disagreement is likely due to the fixed value of CG throughout the study area across all the eight days. CG was reported to vary with time of day, soil type, and soil moisture . Specifically, CG shows diurnal variation  but normally is constant at the time range around local solar noon  and the ASTER overpass time is within the time range; therefore, the strategy of set CG as constant in the given soil type and soil moisture condition on a day is acceptable. There has been a systematically overestimated G at Gobi and desert sites, which contrasts to the performance in the vegetable site. This discrepancy can be explained by soil properties difference , i.e., the dry rock and sandy soil has lower soil heat capacity than vegetable soil due to the lower soil moisture and higher soil porosity . As to the effect of soil moisture on CG, it is hard to quantitatively evaluate in this study because we lack ground data to isolate the effects of soil properties over corn which is the only landscape showing statistical significance. Importantly, local calibration of CG for the sites of the study area would improve the agreement; however, the purpose of this study is to build a broadly applicated u-independent two-source energy balance model rather than to tune it to those specific sites.
5.4. Comparison of Tr Decomposition Methods
The decomposition of Tr into Tc and Ts is a key process for two-source models. Four direct methods, namely, multiangle method , empirical method , wetness isoline method , and two-stage ET method [32, 33], are adopted; furthermore, one indirect method, namely, the Priestley–Taylor iteration method , is used. Given a lack of two-view angle images, here, except for the multiangle method, we mainly discuss three direct methods, i.e., the empirical, wetness isoline, and two-stage ET methods.
The empirical method uses the empirical relationship between Ts − Tc and Tr − Ta (i.e., Ts − Tc = Ca(Tr − Ta)m, where Ca = 0.1 and m = 2)  to segment Tr. We drew the relationship between Tc and Tr under a given meteorological condition (Ta = 293 K) based on the Lhomme relationship. In Figure 10, Tc increases approximately linearly with Tr in the high vegetation cover (fc > 0.7) area, whereas Tc decreases with the increase in Tr under the fc < 0.7 condition. This phenomenon is opposite to the natural state, in which Tr and Tc increase with water stress and vegetation transpiration constraint. Furthermore, Tc is even smaller than 0 under the condition of Tr > 315 K (i.e., Tr − Ta > 20 K) in sparse areas. In fact, an average Tr − Ta in eight dates on the ASTER overpass time at the desert, desert steppe, and Gobi areas is 20.9, 18.2, and 18.1 K, respectively. Thus, Tc in those areas is erroneously estimated. These phenomena verify the limitation of empirical relationships, especially at sparse areas . Moreover, Lhomme empirical coefficients vary with regions. Zhan et al.  reported that the coefficient Ca fluctuates between 0.07 and 0.27 when m is set to 2 based on the measured data from FIFE’87, Monsoon’90, and Washita’92 sites.
The wetness isoline method was originally proposed in the PCACA model  with the assumption of a nearly straight isopiestic wetness line with equivalent Ts and Tc under a uniform atmospheric environment and homogeneous soil surface. This method has similarity with the two-stage ET method in using a trapezoidal space to segment Tr. However, the decomposed component temperatures between the two methods are considerably different. Here, the Tc and Ts decomposed from the wetness isoline method denote the Tr-fc trapezoidal space as Tc,Zhang and Ts,Zhang to make a distinction. In Figure 11, Tc is nearly equal to Ta with an average of 297.9 K in eight dates at the DM station. By contrast, Tc,Zhang is larger than Tc by 3.5 K. Carlson  asserted that Tc is primarily driven by the soil moisture in the root zone. Wu et al.  reported that the depth of the corn root zone is 20–80 cm in the growing period. Here, we calculated the soil moisture in the root zone by the average soil moisture of 20, 40, and 80 cm and found the soil moisture in the corn root zone is slightly greater than the field capacity in all dates, thereby indicating that the vegetation transpiration is not stressed. Therefore, Tc in theory is approximately equal to Ta. The average of Ts is 307.7 K and is clearly higher than Ts,Zhang of 5.6 K. Ts is driven primarily by the soil surface moisture (∼2 cm) by considering that Tr in actual is “skin temperature” at depths in which electromagnetic radiation at the given wavelengths can penetrate .
Moreover, the significant differences are also embodied in Ts − Tc. The average of Ts − Tc in eight dates is 9.7 K, and the Ts,Zhang − Tc,Zhang is only 0.8 K. On June 24, significant water stress occurred with the moisture of soil surface only 10.5%. Ts in theory must be clearly larger than Tc; however, the Ts,Zhang − Tc,Zhang is only 2.2 K (Figure 11), which may be relatively small compared to observations. Previous studies have reported that the observed Ts − Tc may exceed 10 K. For example, Colaizzi et al.  found that Ts − Tc is over 10 K at noon in the early- and midseason at the upland cotton area, Bushland, Texas, USA, in accordance with the observations from IRT (model IRT/c, Exergen Corp., Watertown, Massachusetts, USA). Tian et al.  reported that soybeans Ts − Tc has an extremum of more than 20 K at noon by using a thermal infrared imager (Fluke IR FlexCam Ti55, Fluke Crop., USA). Some researchers observed the Ts − Tc often exceeding 20 K in semiarid environments (e.g., Chehbouni et al. , Humes et al. , and Kustas et al. ). Kustas and Norman  emphasized that the average Ts − Tc has an extreme of 25 K in cases. Allen et al.  indicated that Ts − Tc can be 20–30 K at the exposed and dry surface.
The ET methods is based on the wetness isoline estimate soil evaporation fraction ((LEs/(Rns − Gs)) and vegetation transpiration fraction (LEc/Rnc) with the assumption that they vary quasilinear with Ts and Tc in the trapezoid space, respectively; these methods include PCACA , TTME , and estimated potential evaporation fraction (LEs/LEsp, LEsp is the potential evaporation of soil surface) and vegetation potential transpiration fraction (LEc/LEcp) which are quasilinear with Ts and Tc, respectively, (e.g., ETEML (Yang et al. ). Based on our assumption, we could deduce the rough equality between evaporation and transpiration fractions  and the equality between the soil and vegetation potential evaporation ratios. Here, we calculated the potential evaporation and transpiration fractions using the observed LEsoil/LE, LE, and meteorological data (i.e., Ta, rh, and u) at the DM station to assess the deduction, in which the observed LEs is calculated by LEsoil/LE × LE/(1 − fc); fc is calculated via the method proposed by Anderson et al. , in which the LEsp and LEcP are calculated using the equations recommended by Mu et al.  and Guan and Wilson :where rs, ra, and rtot are the aerodynamic resistances of the soil surface, canopy surface, and the total, respectively, and rs,min is the minimum surface resistance.
In Figure 12, the potential evaporation fraction is obviously lower and fluctuates more dramatically than the potential transpiration fraction. The potential evaporation fraction ranges between 0.09 and 1.32, with an average and a standard deviation of 0.47 and 0.24, respectively. Notably, the potential evaporation fraction greater than 1 on July 30 is mainly due to a small EsP that resulted from a high rh value (79%). The potential transpiration fraction fluctuates between 0.81 and 1.37, with an average and a standard deviation of 0.95 and 0.1, respectively. Moreover, the potential transpiration fraction at the grain filling period (roughly before August) has an average of 0.98, which is higher than that at the mature period (roughly August and later), at 0.91. Therefore, the wetness isoline method potentially overestimates LEsoil and underestimates LEcanopy. The MBE of LEsoil and LEcanopy via the wetness isoline method is 53.4 and −11.1 W m−2, respectively, thereby confirming the conclusion.
Two preconditions are implied in the two-stage ET method. The first precondition is that the soil moisture decreases at the root zone lag behind that on the surface (∼2 cm). The second precondition is that the soil moisture variation is the driver of radiative temperature and ET changes. Figure 12 illustrates that surface soil moisture shows a dramatic decrease in the intervals between two instances of flood irrigation (DM exhibited flood irrigation on July 2, July 28, and August 28 ). The soil moisture in the root zone decreases during the intervals and is maintained in a scope that is greater than the field capacity, thus indicating no water stress. Tc has been recognized and used extensively as an indicator of water availability such as the calculation of the crop water stress index (CWSI) . Moran et al.  developed the concept of CWSI to bare soil and proposed the water deficit index, indicating that Ts is approximately linearly related to surface soil moisture and soil evaporation under a given Ta, Rn, VPD, and ra, as verified by Vidal and Perrier . Therefore, applying the two-stage ET method is appropriate for segmenting Tr under a given meteorological condition.
The reduction in the dependence on the ground meteorological data, especially the data known as high spatial and temporal variation, i.e., wind speed u, is significant for mitigating uncertainties of remotely sensed ET in large-scale and heterogeneous surfaces. In this study, we propose a WiTSEB model by using a simplified theoretical surface temperature (Tr)-vegetation coverage (fc) trapezoid, a two-stage ET Tr decomposing method, and the two-source patch framework.
The WiTSEB was conducted in HiWATER-MUSOEXE-12 sites at the desert-oasis transition zone of Zhangye City in the middle reaches of the Heihe watershed using ASTER images over eight dates. Rn, H, and LE estimates agree well with the observations from 21 flux towers. The RMSE and r2 of LE estimates are 68.6 W m−2 and 0.88, respectively. The accuracies of Rn, G, H, and LE estimates significantly vary with land cover types (including corn, desert, Gobi, desert steppe, orchard, vegetable, residential, and wetland). The MBE and RMSE of LEcanopy/LE estimates are −3.2% and 5.7%, respectively, in comparison with the CRDS water vapor isotope measurement. The LE estimate shows a high spatial variation across landscapes. The LE estimate is generally large in high vegetation cover areas and low in sparse vegetation areas. Furthermore, irrigation strategy amplifies spatial difference. The LE estimates at cropland (corn and vegetable) are high in July and August and low in June and September. The LE estimates at the orchard and wetland areas show less temporal variation between June and August but are conspicuously larger than the value in September. Precipitation is the primary reason for the LE temporal variation at low vegetation cover areas.
The WiTSEB is most sensitive to Ta and Ts but insensitive to other meteorological, remote sensing, and other parameters. A 2 K increase in Tr and Ta results in a 17.2% decrease and a 18.9% increase in LE estimates, respectively. A 20% perturbation of other inputs causes a sensitivity that reaches 6.2%. In comparison with the other two-source models, the WiTSEB does not aggravate the sensitivity on meteorological and remote sensing inputs. Error analysis shows that the estimate error of Tr contributes to the LE overestimation/underestimation at certain land cover types. The observed error of G can exacerbate LE underestimation and mitigate LE overestimation. Rationality assessment indicates that the assumption, which is employed to avoid u in constructing a Tr-fc trapezoidal space and iteratively calculating H components, yields minimal influence on core variable estimates.
The rationality of a two-stage ET pattern in decomposing Tr is comprehensively verified by using the ground meteorological observations in the DM station. Compared to the empirical and wetness isoline, the two-stage ET pattern provides more flexibility to the natural environment and gives higher accuracy in LE estimation.
As with other satellite-based models, WiTSEB has a certain limitation because of some not routinely available inputs, such as the canopy height (h) , which is one of necessary factors on simulation of surface roughness and aerodynamic resistance. The uncertainty of the parameterization (i.e., rcx, rcm, Gf, and α) under extreme conditions for constructing the Tr-fc trapezoidal space also needs to be worked on further.
The ground observed data used to support the findings of this study are released upon application to the Cold and Arid Regions Science Data Center at Lanzhou (http://card.westgis.ac.cn/). The L1B ASTER image is freely downloadable at US Geological Survey website (https://earthexplorer.usgs.gov/). The remotely sensed products, including albedo and enhanced vegetation index, are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The authors thank the Heihe Plan Science Data Center, National Natural Science Foundation of China, for providing ground and near-surface measurements obtained in the Heihe Watershed Allied Telemetry Experimental Research (HiWATER, http://westdc.westgis.ac.cn/hiwater). This work was supported by the China National Key Research and Development Program (grant no. 2017YFC0405801-02), the Technologic Innovation Foundation of Pearl River Hydraulic Research Institute (grant no.  ky015), CRSRI Open Research Program (no. CKWV2017529/KY), Fundamental Research Funds for the Central Universities (no. 2017B614X14), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (no. KYCX17_0419).
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