## 2 Background of WUE Estimation by Remote Sensing  
### Definition of Ecosystem WUE  
The study on plant WUE began in the early 20 th century and it has been going on for nearly a century. However, the research on plant WUE is still quite active, especially considering water physiology and ecology, which is mainly due to the importance of WUE in agriculture and forestry production. In this regard, different methods have been proposed to calculate WUE, relying on the purpose of investigation [21]. WUE has a variety of different definitions according to the prospects, categories, and priorities of different disciplines. For example, plant physiology focuses on the physiological efficiency of water; irrigation agronomy focuses on crop water productivity, total farmland WUE, field WUE, irrigation WUE, and precipitation utilization efficiency. In terms of ecology, a description of WUE-related ecological parameters is shown in Table 1.  
At the single leaf or individual plant stage, there are two ways to express WUE: firstly, the ratio of net photosynthetic rate (Pn) to transpiration (T) to describe the instantaneous WUE of the plant leaf; secondly, the ratio of Pn to leaf stomatal conductance (\(g_{\mathrm{s}}\)), also known as intrinsic WUE (WUE\({}_{\mathrm{i}}\)). When \(g_{\mathrm{s}}\) becomes the dominant limiting factor for plant-leaf gaseous exchange, WUE\({}_{\mathrm{i}}\) is more suitable for describing the water use status of the plant photosynthesis process. If T and \(g_{\mathrm{s}}\) are significantly positively correlated, it leads to a significantly positive correlation between WUE and WUE\({}_{\mathrm{i}}\). The WUE and WUE\({}_{\mathrm{i}}\) indicate that the plant (s) water use efficiency is significantly indifferent. For individual plants, WUE is equal to mass of dry matter per unit of T. For plant population, WUE is equal to mass of dry matter per unit of water consumption. At the ecosystem level, definitions of WUE are basically equivalent to that of plant population. Considering that measuring ET is relatively easier than T and that T is considered to be the main component of ET during the peak growing season, ET is approximately equal to T as the denominator of fraction on the ecosystem scale [17]. The classic definition of WUE is equal to GPP/ET or WUE is equal to NPP/ET (NPP, net primary production) as shown in Table 1.  
The classical WUE is defined as GPP per unit of ET (gC kg\({}^{-1}\) H\({}_{2}\)O), which is suitable for studying the carbon-water coupling characteristics of ecosystems and their responses to environmental changes on a monthly or a yearly scale, as shown in Figure 2[6]. Recent studies have defined two new WUEs, i.e., Inherent WUE (IWUE), which is equal to GPP \(\times\) VPD/ET (gC hPa kg\({}^{-1}\) H\({}_{2}\)O) [6] and Underlying WUE (UWUE), which is equal to GPP \(\times\) VPD\({}^{0.5}\)/ET (gC hPa\({}^{0.5}\) kg\({}^{-1}\) H\({}_{2}\)O) [17; 24; 25]. These new definitions mainly consider the important influence of saturated VPD on the daily and hourly scale of carbon-water \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Definition** & **Symbol Expression** & **Reference** \\ \hline WUE & Mass of dry matter/water consumption & WUE = GPP/ET & [19; 22; 23] \\  & (transpiration + evaporation) & WUE = NPP/ET & \\ WUE\({}_{\mathrm{T}}\) & Mass of dry matter/transpiration & WUE = GPP/T & [19] \\  & & WUE = NPP/T & \\ Optical WUE & Controlling effect of VPD & GPP \(\times\) VPD\({}^{\mathrm{K}}\)/ET & [1; 6; 24; 25] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of WUE-related ecological parameters. coupling. In fact, the two definitions of WUE, i.e., IWUE and UWUE, can be unified by a common formula such as the Optical WUE (OWUE), which is equal to \(\text{GPP}\times\text{VPD}^{k}/\text{ET}\) (\(\text{gC hPa}^{k^{+}}\) kg\({}^{-1}\) H\({}_{2}\)O), and the only difference among them is the value of \(k\).