## 2 Materials and Methods
### Data Analysis  
1. WUEc Definition  
The definition of WUEc for this paper is as follows:  
\[WUEc=\frac{GPPc}{ETc} \tag{1}\]  
where \(GPPc\) is in gC\(\cdot\)m\({}^{-2}\); \(ETc\) is in mm; and \(WUEc\) is in gC\(\cdot\)mm\({}^{-1}\cdot\)m\({}^{-2}\). 2. Temporal Correlation Analysis  
Seasonal and trend decomposition using LOESS (STL) is a time-series decomposition method that uses robust locally weighted scatterplot smoothing (LOESS) as its smoothing technique [45]. It decomposes the series into trend, seasonal, and residual components using LOESS and is one of the most commonly employed methods for time-series decomposition. WUEc data are time-series data with stochastic characteristics, typically characterized by trend, seasonality, and residual components. In this study, the WUEc time-series data for Xinjiang from 2002 to 2022 are decomposed using STL into three components: seasonal, trend, and residual. The trend component of WUEc is then analyzed to better capture its nonlinear behavior. The STL method can be expressed as follows:  
\[X_{t}(j)=T_{t}(j)+S_{t}(j)+R_{t}(j) \tag{2}\]  
where \(X_{t}\) is the value of raw WUEc at moment t; \(T_{t}\) is the value of trend term at moment t; \(S_{t}\) is the value of seasonal term at moment t; \(R_{t}\) is the value of residual term at moment t, t = 1, 2, \(\ldots\), N.
3. Analysis of Spatial Trends  
The Sen + MK method is a commonly used approach for analyzing trends in spatial data, particularly in time-series data. The Sen slope method is used to estimate the magnitude of the trend, while the Mann-Kendall test is employed to assess the statistical significance of this trend [46; 47]. In this study, the Sen method is used to analyze the trend of WUEc, and the Mann-Kendall method is employed to test the significance of this trend and its associated factors. The formula for the Sen method is as follows:  
\[\beta=\text{Median}\bigg{(}\frac{S_{j}-S_{i}}{j-i}\bigg{)},\forall j>i \tag{3}\]where \(S_{i}\), \(S_{j}\) represent the data corresponding to the ith and jth years of WUEc in the years 2002-2022; the result of the trend degree \(\beta>0\) reflects an increasing trend of WUEc, while \(\beta<0\) is a decreasing trend. The MK test was calculated by the formula:  
\[Z= \begin{cases}\frac{S-1}{\sqrt{Var(S)}},S>0\\ 0,S=0\\ \frac{S+1}{\sqrt{Var(S)}},S<0\end{cases}  \tag{4}\]  
\[S=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\text{sgn}\big{(}x_{j}-x_{i}\big{)} \tag{5}\]  
\[\text{sgn}\big{(}x_{j}-x_{i}\big{)}= \begin{cases}1,x_{j}>x_{i}\\ 0,x_{j}=x_{i}\\ -1,x_{j}<x_{i}\end{cases}  \tag{6}\]  
\[Var(S)=\frac{n(n-1)(2n+5)}{18} \tag{7}\]  
where \(Z\) is the standardized test statistic; \(S\) is the test statistic; \(x_{i}\) and \(x_{j}\) are the data corresponding to years \(i\) and \(j\); n is the length of the time-series data (n = 21 in this study). In this study, the significant levels of \(\alpha\) = 0.05 and \(\alpha\) = 0.10 were selected for the significance test, and the values of \(Z_{1-\alpha/2}\) were \(\pm\)1.96 and \(\pm\)2.58, respectively. Combining the values of \(\beta\) and \(Z\), the trend of the change was categorized into five categories: significant increase, increase, no change, decrease, and significant decrease. (4) Sustainability Analysis  
The Hurst exponent is an analytical method based on the rescaled range (R/S) of long-term, non-periodic series, and it quantifies the likelihood that a series will maintain its historical trend. The exponent ranges between 0 and 1. When 0 \(<\) H \(<\) 0.5, H = 0.5, and 0.5 \(<\) H \(<\) 1, the series exhibits anti-persistence (future changes tend to reverse past trends), randomness (future changes are independent of past trends), and persistence (future changes continue past trends), respectively. The closer H is to 1, the stronger the persistence of the series; the closer it is to 0, the stronger the anti-persistence [48; 49]. In this study, the Hurst index is calculated using the R/S analytical method to predict the future evolutionary trend of WUEc in Xinjiang.