## Materials and Methods
### Method
#### 2.2.2 The MPDI-Integrated SEBS Model  
Theoretically, d\({}_{0}\) + z\({}_{0\text{h}}\) defines the level where the canopy exchanges sensible and latent heat with the atmosphere [50; 51]. Hence, z\({}_{0\text{h}}\) constitutes one of the most crucial parameters for the accurate calculation of H [52]. The surface roughness height for heat (z\({}_{0\text{h}}\)) can be derived from:  
\[\text{Z}_{0\text{h}}=\
icefrac{{Z_{0}}}{{m}}\left/\exp\left(\text{kB}^{-1}\right)\right. \tag{5}\]  
ET, in arid and semi-arid regions, is limited not only by the available energy, but also by the available water [53]. When plants suffer from water stress, the leaf stomata on the top of the canopy gradually close, thus the air temperature on the top of the canopy increases due to solar radiation. However, the temperature fluctuation at the bottom of the canopy is relatively small due to the effect of leaf covers. This means the surface roughness height for heat transport moves to the canopy top, thereby z\({}_{0\text{h}}\) rises, which induces sensible heat flux to increase, in turn reducing crop transpiration derived from soil moisture [33]. Therefore, adapting z\({}_{0\text{h}}\) (via kB\({}^{-1}\)) by introducing soil moisture correction can remedy the shortcoming of the overestimation of ET using the SEBS model. Here, the modified perpendicular drought index (MPDI) proposed by [PERSON] et al. (2007) was integrated into the SEBS bulk transfer equation for H (via kB\({}^{-1}\)). MPDI is based on the spatial distribution features of soil moisture in NIR-Red spectral space, taking into account both soil moisture and vegetation growth [54]. It provides a simple method to detect soil moisture condition and crop drought and has been extensively tested and verified over different climate and hydrologic regimes in China, Iran, and Thailand [55; 56; 57]. It is written as follows:  
\[\text{MPDI}=\frac{\text{R}_{\text{Red}}+\text{MR}_{\text{NIR}}-\text{f}_{ \text{v}}(\text{R}_{\text{v,Red}}+\text{MR}_{\text{v,NIR}})}{(1-\text{f}_{ \text{v}})\sqrt{\text{M}^{2}+1}} \tag{6}\]  
where R\({}_{\text{Red}}\) and R\({}_{\text{NIR}}\) refer to the atmospherically corrected reflectance of the red and near-infrared bands, respectively; M is the slope of the soil line; R\({}_{\text{v, Red}}\) and R\({}_{\text{v, NIR}}\) are the vegetation reflectance in the red and near-infrared bands, respectively; and f\({}_{\text{v}}\) represents fraction of vegetation. More details about MPDI can be found in [PERSON] et al. (2007) [54]. The kB\({}^{-1}\) value is modified using a scaling factor (f) represented by a reverse sigmoid function:  
\[\text{kB}_{\text{u}}^{-1}=\text{SF}\times\text{kB}^{-1} \tag{7}\]  
where  
\[\text{SF}=\left[\text{a}+\frac{1}{\left(1+\exp(\text{b}-\text{c}\ /\ \text{MPDI}_{\text{rel}}) \right)}\right] \tag{8}\]  
Here, kB\({}_{\text{u}}^{-1}\) is the updated kB\({}^{-1}\); a, b, and c are the coefficients of the reverse sigmoid function. A reverse sigmoid function for SF was selected because it considerably lowers the kB\({}^{-1}\) value for relatively dry conditions (high MPDI values), while soil moisture influence on the value of kB\({}^{-1}\) is not significant for wet conditions (low MPDI values). Hence, it better fits the conceptual framework for a water-limited evapotranspiration regime [53]. The new calculation process for daily ET with the MPDI-integrated SEBS model is shown in Figure 2.  
To obtain the coefficients of the reverse sigmoid function, the authors calibrated the MPDI-integrated SEBS using EC flux observations at Daman station. Cloud-free HJ-1 data from 2015 were used to determine the coefficients (a, b, and c) of sigmoid function through an optimization by reducing the error between the observed and modeled sensible heat flux values from EC and SEBS. The parameters a, b, and c were determined as 0.024, 3.1, and 1.6, respectively. The reverse sigmoid function for SF-MPDI availably decreased the kB\({}^{-1}\) value, thereby increasing z\({}_{0\text{h}}\) at water stress conditions.