## 2 Data and Methods
### Statistical Analysis Strategy  
First, we conducted a statistical analysis using Theil-Sen median trend analysis [42] (Equation (1)) to examine the trends in NDVI\({}_{\max}\), GPP\({}_{\max}\), SM, and VPD in the TP district from 2001 to 2022. Next, we assessed the significance (\(p<0.05\)) of the trends using the Mann-Kendall test (Equations (2)-(5)). In the following, the formulas and principles are explained, taking NDVI\({}_{\max}\) as an example. \[\beta=Median\left(\frac{x_{j}-x_{i}}{j-i}\right)\qquad\forall j>i \tag{1}\]  
\[Z=\left\{ \begin{array}{c}\frac{S}{\sqrt{Var(s)}}\left(S>0\right)\\ 0\quad(S=0)\\ \frac{S+1}{\sqrt{Var(s)}}(S<0)\end{array} \right. \tag{2}\]  
\[S=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}sgn\left(x_{j}-x_{i}\right) \tag{3}\]  
\[sgn\left(x_{j}-x_{i}\right)=\left\{ \begin{array}{c}1\qquad if\left(x_{j}-x_ {i}\right)>0\\ 0\qquad if\left(x_{j}-x_{i}\right)=0\\ -1\qquad if\left(x_{j}-x_{i}\right)<0\end{array} \right. \tag{4}\]  
\[Var(S)=\frac{n(n-1)(2n+5)}{18} \tag{5}\]  
where \(x_{i}\) and \(x_{j}\) represent the values of NDVI\({}_{\max}\) for the \(i\)-th and \(j\)-th years, respectively. A value exceeding 0 denotes a tendency toward increase, whereas a value below 0 signifies a tendency toward decrease. The parameter n represents the length of the time series data, while the function \(sgn\) corresponds to the sign function. In this research, we evaluated the significance of the tendency changes at a confidence level of \(\alpha=0.05\). A trend was considered significant when \(Z>1.96\), indicating that the data trend change was statistically significant. Conversely, when \(Z<1.96\), the data trend change was not statistically significant. Subsequently, this study employed partial correlation analysis to assess the separate impacts of SM and VPD on NDVI\({}_{\text{max}}\). When multiple influencing factors were present, the partial correlation analysis enabled us to isolate the impacts of specific factors by controlling the impacts of the additional variables. To calculate the partial correlation coefficient (Equation (7)), we first needed to compute the simple correlation coefficient (Equation (6)). Using the obtained simple correlation coefficients, we calculated the first-order partial correlation coefficients, as shown below:  
\[R_{xy}=\frac{\sum_{i=1}^{n}[(x_{i}-\overline{x})\times(y_{i}-\overline{y})]}{ \sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}}} \tag{6}\]  
\[R_{xyz,z}=\frac{R_{xy}-R_{xz}R_{yz}}{\sqrt{(1-R_{xz}^{2})}\sqrt{\left(1-R_{yz} ^{2}\right)}} \tag{7}\]  
where \(\overline{x}\) and \(\overline{y}\) represent the multiyear average of data x and data y, \(R_{xy}\) represents the simple correlation coefficient between data x and data y, and \(R_{xy,z}\) represents the partial correlation coefficient between data \(x\) and data y after excluding the influence of data \(z\). \(R_{xyz,z}>0\) and \(R_{xy,z}<0\), respectively, indicate a positive and negative correlation between data x and data y. The partial correlation analysis conducted in this research comprised four main scenarios: (a) assessing the partial correlation between NDVI\({}_{\text{max}}\) and SM while excluding the impacts of VPD, precipitation, temperature, and radiation; (b) investigating the partial correlation between NDVI\({}_{\text{max}}\) and VPD while excluding the impacts of SM, precipitation, temperature, and radiation; and (c) this article also calculated the partial correlations between NDVI\({}_{\text{max}}\) and other factors. In order to comprehend the mechanisms underlying the impact of SM and VPD on NDVI\({}_{\text{max}}\), we initially examined the response of NDVI\({}_{\text{max}}\) to variations in SM. Subsequently, we compared the response of NDVI\({}_{\text{max}}\) to changes in VPD to determine the primary factor influencing NDVI\({}_{\text{max}}\). Similar analyses were conducted for GPP\({}_{\text{max}}\).