## 2 Materials and Methods
### Data Processing Method  
1. Model builder in ArcGIS software was used to perform coordinate conversion, mosaic, extraction, and other types of processing on the remote sensing image to obtain the monthly NDVI in the study area, which was [\(-\)0.2-1], and the format of the data was unified as tiff. 2. The one-variable linear regression analysis was adopted to simulate the annual change rate for the NDVI on the basis of each pixel. The specific calculation formula is [43; 44]: \[\theta=\frac{n\sum_{i=1}^{n}ic_{i}-\sum_{i=1}^{n}i\sum_{i=1}^{n}c_{i}}{n\sum_{i= 1}^{n}i^{2}-(\sum_{i=1}^{n}i)^{2}}\] In the formula, \(\theta\) is the slope of the NDVI; \(c_{i}\) is the value of the NDVI in year i; and \(n\) is the number of years in calculation period. 3. The coefficient of variation was adopted to characterize the fluctuation of the NDVI over the years. The specific calculation formula is: \[CV=\sqrt{\frac{1}{n-1}}\sum(x_{i}-\overline{x})^{2}\,/\overline{x}\] In the formula, \(CV\) is the coefficient of variation; \(i\) is the time series; \(x_{i}\) is the value of the NDVI in year i; and \(\overline{x}\) is the mean value of all pixels during the study period. In addition, according to previous related research [45], the fluctuation degree of the NDVI was graded. It was divided into five grades in descending order as follows: \(CV>0.2\)  
Figure 2: Locations of meteorological stations. (high fluctuation), \(0.15<CV\leq 0.2\) (relatively high fluctuation), \(0.1<CV\leq 0.15\) (moderate fluctuation), \(0.05<CV\leq 0.1\) (relatively low fluctuation), and \(CV<0.05\) (low fluctuation). See Figure 2 for details. 4. For the correlations between the NDVI and various climatic factors (P, Tmin, Tave, and Tmax), firstly, the simple correlation coefficient between the NDVI and climatic factors was calculated. Then, the partial correlation coefficient and multivariate correlation coefficient (multivariate correlation coefficients among the NDVI, P, and three types of T are represented by Mnpmin, Mnpave, and Mnpmax, respectively) were calculated [46]. The calculation formulas for the simple correlation coefficient, partial correlation coefficient, and multivariate correlation coefficient are as follows: \[R_{\text{xy}}=\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}- \overline{y}\right)}{\sqrt{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2} \sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}}}\] \[\rho_{12-3}=\frac{R_{12}-R_{13}R_{23}}{\sqrt{\left(1-R_{13}^{2}\right)\left(1- R_{23}^{2}\right)}}\] \[M_{1-23}=\sqrt{1-\left(1-R_{12}^{2}\right)\left(1-\rho_{13-2}^{2}\right)}\]  
In the formula, \(x_{i}\) and \(y_{i}\) are the values of the NDVI and P/Tmin/Tave/Tmax in year i; \(\overline{x}\) and \(\overline{y}\) are the multiyear mean values of the NDVI and P/Tmin/Tave/Tmax; \(n\) is the number of years for the calculated period; \(R_{12}\) is the simple correlation coefficient between the NDVI and P; \(R_{13}\) is a simple correlation coefficient between the NDVI and Tmin/Tave/Tmax; \(R_{23}\) is the correlation coefficient between P and Tmin/Tave/Tmax; \(\rho_{12-3}\) is the partial correlation coefficient between the NDVI and P on the basis of a fixed Tmin/Tave/Tmax; \(\rho_{13-2}\) is the partial correlation coefficient between the NDVI and Tmin/Tave/Tmax on the basis of a fixed P; and \(M_{1-23}\) are multiple correlation coefficients between NDVI, P, and Tmin/Tave/Tmax. 5. Geodetectors are a set of statistical tools that detect the spatial differentiation of geographical phenomena and reveal contributors [47]. In the study, the synergistic effect between P and T on the NDVI was explored through the interaction detector in geodetectors. This effect was also called the explanatory power of the synergistic effect between P and T for the NDVI (indicated by Q). The Q value varies between 0 and 1, the closer the Q value to zero, the weaker the explanatory power of P and T for the NDVI, and vice versa.