## 2 Background of WUE Estimation by Remote Sensing
### Indirect WUE Estimation by Measuring GPP and ET at the Ecosystem Level
#### 2.2.1 Measurements Using Eddy Covariance Observations
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Equation** & **Reference** \\ \hline CASA & NPP = 0.55 \(\times\) mini/\((SR)\), 0.95] \(\times\)\(\varepsilon_{\mathrm{max}}\times f(\theta)\times f(\mathrm{Ta})\) & [42] \\ EC-LUE & GPP = PAR \(\times\) (a \(\times\) NDVI + b) \(\times\)\(\varepsilon_{\mathrm{max}}\times f(\mathrm{Ts},\mathrm{W})\) & [43] \\ C-Fix & GPP = \(\times\)\(\times\)\(\times\)\(\mathrm{NDVI}\) + b) \(\times\)\(\varepsilon_{\mathrm{max}}\times f(\mathrm{Ta})\times f(\mathrm{CO_{2}})\) & [44] \\ VPM & GPP = PAR \(\times\)\(\mathrm{EVI}\) + \(\varepsilon_{\mathrm{max}}\times f(\mathrm{Ta})\times f(\mathrm{P})\times f( \mathrm{W})\) & [45] \\ MOD17 & GPP = PAR \(\times\) (\(1-\varepsilon^{-\mathrm{HL}}\)) \(\times\)\(\varepsilon_{\mathrm{max}}\times g(\mathrm{Ta})\times f(\mathrm{VDD})\) & [41] \\ TL-LUE & GPP = (APAR \(\times\)\(\varepsilon_{\mathrm{sum}}\) + APAR \(\times\)\(\varepsilon_{\mathrm{shack}}\)) \(\times g(\mathrm{Ta})\times f(\mathrm{VDD})\) & [46] \\ VI & GPP \(\propto\) VI \(\times\) VAR & [47] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Typical LUE models for estimating GPP. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Equation** & **Reference** \\ \hline Penman-Monteith equation & \(\mathrm{LE}=\frac{\Delta\times(R_{\mathrm{e}}-G)+\mathrm{r}\times\mathrm{C}_{ \mathrm{e}}\times(\mathrm{a}ut-t)/\varepsilon_{\mathrm{s}}}{\Delta t^{+}T}\) & [55] \\ Priestley-Taylor model & \(\mathrm{LE}=\frac{\Delta(R_{\mathrm{e}}-G)}{\Delta t^{+}T}\) & [56] \\ Energy balance model & \(\mathrm{LE}=R_{\mathrm{r}}-\frac{\Delta t}{G}-H\) & [57] \\ GLASS & \(\mathrm{ET}=R_{\mathrm{r}}(0.144+0.6495\mathrm{NDVI}+0.009\mathrm{Ta}-0.0163 \mathrm{DTaR})\) & [58] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Typical process-based models of ET estimation. by soil and vegetation over wide river stretches are required to allocate water needed for agricultural and environmental sectors [63]. Remotely sensed multispectral vegetation indices are a useful tool for quantifying the variability in crop water consumption during the growing season at regional and field scales [64]. Remote sensing can also estimate crop coefficients based on spectral reflectance of vegetation indices (VIs). Crop coefficients generated from VIs determine ETc better than a tabulated \(K\)c because VIs represent the actual crop growth conditions and can capture the spatial variability among different fields [65; 66]. Moreover, net radiation is an important component of the energy balance, and its estimation accuracy has an impact on energy flux estimated from satellite data. In typical remote-sensing ET algorithms, the outgoing shortwave and longwave components of net radiation are obtained from remote sensing data, while the incoming shortwave and longwave components are typically estimated from weather data using empirical equations [67]. The surface energy balance approach, combined with empirical equations (e.g., based on MODIS VIs) for estimating shortwave and longwave radiation, is commonly used in remote sensing ET algorithms, with an error or an uncertainty of about 20% [64]. WUE is calculated as the quotient of GPP and ET as mentioned above by modeling carbon and water cycles, respectively. This method of WUE estimation is limited to the monitoring of seasonal variations in WUE, and to our understanding about the controlling factors and coupling water-carbon cycle feedback [68]. This seasonal response of WUE to climate change needs better observation-based validation to further improve predictions of ecosystem response to global change.