## 2 Background of WUE Estimation by Remote Sensing
### Advances in Direct Estimation of Ecosystem WUE by Remote Sensing
\begin{table}
\begin{tabular}{c c c c c} \hline
**Sites or Regions** & **Ecosystem Types** & **Regression Model** & **Performance** & **Reference** \\ \hline Globe & All types & WUE = \(a\times\text{SM}+b\times(1-g^{\text{-LAI}_{\text{max}}})\) & R\({}^{2}\) = 0.64 & [6] \\ China & Afforestation & WUE = \(-0.035\times\text{Al}+2.72\times\text{LAI}-0.69\times\text{LAI}^{2}+0.98\) & R\({}^{2}\) = 0.83, RMSE = 0.44 & [92] \\ US-Bo1 & Cropland & WUE = \(7.245\times\text{EVI}-0.871\) & R\({}^{2}\) = 0.82, RMSE = 1.07 & [20] \\ US-Ro1 & Cropland & WUE = \(6.601\times\text{EVI}-0.938\) & R\({}^{2}\) = 0.90, RMSE = 0.69 & [93] \\ Toledo & Forest (oak) & WUE = \(0.19\times\text{Ta}+0.0004\times\text{LAI}\times\text{P}-0.92\) & R\({}^{2}\) = 0.78, RMSE = 0.65 & [94] \\ US-Ne2 & Cropland & WUE = \(7.570\times\text{EVI}-1.041\) & R\({}^{2}\) = 0.82 & [95] \\ US-Ne3 & Propland & WUE = \(5.565\times\text{NVDI}-1.227\) & R\({}^{2}\) = 0.75 & [95] \\ US-Ne1 & Cropland & WUE = \(7.35\times\text{EVI}-0.53\) & R\({}^{2}\) = 0.78 & [96] \\ US-Ne2 & Propland & WUE = \(7.71\times\text{EVI}+0.02\times\text{Ta}-0.89\) & R\({}^{2}\) = 0.79 & [96] \\ US-Bar & Temperate deciduous & WUE = \(8.14\pm 0.57\times\text{EVI}+0.032\pm 0.012\times\text{LST}-\) & R\({}^{2}\) = 0.74,-0.83, & [75] \\ US-Ro2 & Forest & \(10.16\pm 3.18\) & RMSE = 0.75\(-\)1.15 & [75] \\ Daman & Grassland & WUE = \(4.322\times\text{EVI}-0.559\) & R\({}^{2}\) = 0.92, RMSE = 0.19 & [97] \\ Arou & Cropland & WUE = \(7.211\times\text{EVI}-0.652\) & R\({}^{2}\) = 0.89, RMSE = 0.39 & [97] \\ Forest & U.S. & WUE–\(Lat+T_{\text{max}}+Rg_{\text{max}}\) & NA & [3] \\ \hline \end{tabular} Note: SM is soil moisture; LAI\({}_{\text{max}}\) is maximum LAI during the studied period; EVI is Enhanced Vegetation Index; Ta is air temperature; NDVI is Normalized Vegetation Index; Al is elevation; LST is land surface temperature; \(Lat\), \(T_{\text{max}}\), and \(R_{\text{max}}\) indicate the latitude, maximum temperature, and maximum global radiation, respectively.  
\end{table} 
Table 4: Referenced empirical models of WUE estimation. The optimal stomatal conductance model developed by [PERSON] et al. (2017) [100] is as follows:  
\[g_{s}=1.6\bigg{(}1+\frac{g_{1}}{\sqrt{\text{VPD}}}\bigg{)}\frac{\text{GPP}}{ \text{Ca}}, \tag{4}\]  
where, GPP is equivalent to leaf net photosynthetic carbon uptake at the leaf level; Ca is atmospheric CO\({}_{2}\) concentration (molCO\({}_{2}\) mol\({}^{-1}\) air); \(g_{1}\) is an empirical parameter fitted by the Ball stomatal conductance model (kPa\({}^{0.5}\)), which is the canopy conductance slope parameter at the ecosystem scale [101]. ET usually includes T (transpiration), E (evaporation), and \(E_{\text{i}}\) (evaporation from canopy interception). As defined in Section 2.1, WUE can be formulated as:  
\[\text{WUE}=\frac{\text{GPP}}{\text{ET}}=\frac{\text{GPP}}{\text{T}}\frac{ \text{T}}{\text{T}+\text{E}}\bigg{(}1-\frac{E_{i}}{\text{ET}}\bigg{)}, \tag{5}\]  
Based on an optimal stomatal conductance model, ecosystem WUE can be reformulated as:  
\[\text{WUE}=\frac{\text{Ca*P}_{\text{a}}}{1.6*\big{(}\text{VPD}+g_{1}*\sqrt{ \text{VPD}}\big{)}}(1-exp(-k\text{LAI}))\bigg{(}1-\frac{E_{i}}{\text{ET}} \bigg{)}, \tag{6}\]  
where, Pa is the atmospheric pressure (kPa). The first term of the equation is ecosystem transpiration WUE (GPP/T). The second term is partitioning of T and E (T/ET) estimated by Beer's Law. \(k\) is the radiation extinction coefficient. The expression of T/ET is a more complex phenomenon than that used in this model, and is a popular topic and a challenge to ecohydrology, especially by remote sensing techniques [8; 102]. The developed analytical WUE model accounts for the main physiological parameter of vegetation (\(g_{1}\)) and three key controlling factors (i.e., Ca, Pa and VPD), implying the importance of plant functional traits and the major drivers of change in ecosystem WUE at the annual time scale.