## 2 Materials and Methods
### Research Method
#### 2.3.2 Analysis of the NPP Change Volatility and Trends  
The coefficient of variation (\(CV\)) of NPP can reflect the interannual volatility of the grassland growth status. The calculation of the \(CV\) was conducted on the GEE platform, of which the formula is expressed as follows:  
\[CV=\frac{\sqrt{\frac{1}{\Xi-1}\times\sum_{i=1}^{n}\left( NPP_{i}-\overline{ NPP}\right)^{2}}}{\overline{ NPP}} \tag{5}\]  
where \(CV\) is the CV of NPP; \(n\) was 21 in this study because of the years from 2000 to 2020; \(i\) is the index of years; \(NPP_{i}\) is the value of NPP in the \(i\)-th year; and \(\overline{NPP}\) is the average NPP from 2000 to 2020.  
The ordinary least squares (OLS) method was selected to estimate the linear trend of NPP over the long time series, which can reflect the change trend of grassland NPP [13]. The OLS method also was completed on the GEE platform. The formula for OLS is expressed as follows:  
\[\theta_{OLS}=\frac{\frac{n\times\sum_{i=1}^{n}i\times NPP_{i}-\left(\sum_{i=1 }^{n}i\right)\times\left(\sum_{i=1}^{n} NPP_{i}\right)}{n\times\sum_{i=1}^{n}i^{2}-\left(\sum_{i=1}^{n}i \right)^{2}}}{\theta_{OLS}} \tag{6}\]  
where \(\theta_{OLS}\) is the linear trend of NPP, and the other parameters are the same as above.  
Theil-Sen median trend analysis, combined with the Mann-Kendall (MK) test, can be applied to analyze long time series of vegetation indicators, and reflect the change trends of each pixel in a time series [73; 74; 75]. Theil-Sen median trend analysis is free from the interference of outliers. The Theil-Sen median trend analysis and MK test were conducted in Python 3, using pyMann-Kendall open-source software [76]. The detailed formulae are expressed as follows:  
\[\theta_{TS}=\text{median}\Big{(}\frac{ NPP_{j}- NPP_{j}}{j-i}\Big{)},\,2000\leq i<j\leq 2020 \tag{7}\]  
where \(\theta_{TS}\) is the Theil-Sen median result; and \(NPP_{i}\) and \(NPP_{j}\) represent the NPP in the years of \(i\) and \(j\), respectively.  
The MK test, which a non-parametric statistical test, is widely applied to measure the significance of Theil-Sen median trends [77; 78]. It has also been used to analyze vegetation growth trends over long time series [75; 79].  
The MK statistic, i.e., \(Z\), is defined as follows:  
\[Z=f(x)=\left\{\begin{array}{cc}\frac{S-1}{\sqrt{s}(S)},&S>0\\ 0,&S=0\\ \frac{S+1}{\sqrt{s}(S)},&S<0\end{array}\right. \tag{8}\]where  
\[S=\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}\text{sgn}\big{(}NPP_{j}-NPP_{i} \big{)} \tag{9}\]  
\[\text{sgn}\big{(}NPP_{j}-NPP_{i}\big{)}=\left\{\begin{array}{cc}1,&NPP_{j}- NPP_{i}>0\\ 0,&NPP_{j}-NPP_{i}=0\\ -1,&NPP_{j}-NPP_{i}<0\end{array}\right. \tag{10}\]  
\[s(S)=\frac{n(n-1)(2n+5)}{18} \tag{11}\]  
where \(NPP_{i}\) and \(NPP_{j}\) represent the same as above, \(n\) is the length of the time series, and sgn is a sign function. The \(Z\) statistic value ranges from \(-\infty\) to \(+\infty\), and \(|Z|>u_{1-a/2}\) reflects whether the long time series shows significance at the level of \(\alpha\). In this study, we set \(\alpha=0.05\), so that \(u_{1-a/2}=u_{0.975}=1.96\). The NPP change trends were reclassified into four classes, according to the detected change trend and its significance test result, as shown in Table 3.