# Quantifying Soil Moisture Impacts on Water Use Efficiency in Terrestrial Ecosystems of China
## 2 Materials and Methods
### Methods  
#### 2.3.1 Mathematical Expression between _Sm_ and _Ef_  
Previous studies have proposed an effective analysis method for the functional relationship between the evaporation fraction (_EF_) and _SM_[50; 51]. This method expressed the dependence of _EF_ on _SM_ (\(\theta\)) as follows:  
\[EF(\theta)\left\{ \begin{array}{rl}0,&\text{if }\theta<\theta_{\text{r}}\\ EF_{\text{max}}\frac{\theta-\theta_{\text{r}}}{\theta_{\text{c}}-\theta_{ \text{r}}},&\text{if }\theta_{\text{r}}\leq\theta\leq\theta_{\text{s}}\\ EF_{\text{max}},&\text{if }\theta>\theta_{\text{s}}\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & \multicolumn{6}{c}{**Soil Diameter (mm)**} \\ \hline
\begin{tabular}{c} very \\ coarse \\ sand \\ \end{tabular} & sand & sand & sand & sand & sand & sand & sand & sand \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Soil texture classification standard of soil texture dataset. where \(\theta\) is volumetric soil moisture content (m\({}^{3}\)/m\({}^{3}\)), \(\theta_{\rm c}\) is the soil moisture at the critical point, and \(\theta_{\rm r}\) is the soil wilting coefficient. \(EF\) is the evaporative fraction, which is calculated as follows:  
\[EF=\frac{LE}{R_{\rm n}} \tag{2}\]  
where \(LE\) is the latent heat flux and \(R_{\rm n}\) is the surface net radiation. When \(\theta<\theta_{\rm r}\), \(EF\) is not always zero owing to the presence of hygroscopic \(SM\)[52]. Therefore, \(EF_{\rm min}\) is defined as the \(EF\) value corresponding to a cumulative frequency percentage of 5%. In addition, when \(\theta>\theta_{\rm c}\), energy rather than \(SM\) became the dominant limiting factor for \(ET\). Thus, the study similarly defined \(EF_{\rm max}\) as the \(EF\) value corresponding to 95% of the cumulative frequency percentage. Because this study only focused on the functional relationship between \(SM\) and \(EF\) in the range of \(EF_{\rm min}\leq EF\leq EF_{\rm max}\), Equation (1) could be expressed as follows:  
\[\theta=\frac{EF}{EF_{\rm max}}(\theta_{\rm c}-\theta_{\rm r})+\theta_{\rm r} \tag{3}\]  
This study estimated the raster map of \(\theta_{\rm r}\) using soil hydraulic parameters and soil texture datasets [53]. Although the fitting between \(EF\) and \(\theta\) could obtain the parameter \(\theta_{\rm c}\), it could result in great uncertainty. Thus, this study used the soil field capacity (\(\theta_{\rm f}\)) to replace \(\theta_{\rm c}\) and introduced a dimensionless parameter A into the calculation scheme for correction. Finally, Equation (3) was deformed as follows:  
\[\theta=\frac{EF}{EF_{\rm max}}\rm{A}(\theta_{\rm f}-\theta_{\rm r})+\theta_{ \rm r} \tag{4}\]  
The gridded \(\theta_{\rm f}\) value was also calculated using soil hydraulic parameters and soil texture datasets [53]. First, we used the monthly \(SM\) and \(EF\) data from 1985 to 2000 to fit the multi-year average monthly (January to December) parameter A. Then, the monthly \(SM\) from 2001 to 2014 was estimated using the \(EF\) data during the same period and the fitted parameter A. The GLDAS \(SM\) data finally verified the estimated soil moisture data during 2001-2014. 3.2 The Response of \(W  
where \(\epsilon\) is the sensitivity coefficient; \(\overline{NPP}\) and \(\overline{\theta}\) are the long-term average values of the \(NPP\) and \(SM\) in growing season of a pixel; \(NPP_{\rm i}\) and \(\theta_{\rm i}\) are the growing season values of \(NPP\) and \(SM\) in each year, respectively. Thus, the following formula could be used to evaluate the average impact of \(SM\) variation on \(WUE\) in the past 30 years:  
\[\Delta WUE=\left|\frac{\partial WUE}{\partial\theta}\right|\Delta\theta \tag{9}\]  
where \(\Delta\theta\) is the percentage change in \(SM\) in the growing season from 1985 to 2014. Thus, \(\Delta WUE\) was the average influence (change percentage) of \(SM\) variation on \(WUE\).