## 2 Materials and Methods
### Spatiotemporal Variation Response Analysis Method  
#### 2.5.1 GPP Trend Analysis  
Firstly, the annual mean GPP changes in northwestern Hubei from 2001 to 2020 were analyzed using the Theil-Sen median slope estimation method and Mann-Kendall trend test to classify the GPP variation trends in the study area over this period. The Theil-Sen method provides a robust estimate of trend slopes, minimizing the influence of outliers, while the Mann-Kendall test, as a non-parametric approach, assesses the significance of these trends and is well-suited for time series data that may not meet normality assumptions or contain missing values. The combination of these methods allows for a more reliable classification of significant increasing, decreasing, or stable trends in GPP. This approach supports a deeper understanding of regional carbon cycling and vegetation dynamics. Additionally, the significance levels and classification thresholds applied were adopted from the established literature to ensure scientific rigor and comparability of results [44; 45]. The Theil-Sen median slope estimator was calculated as follows:  
\[\begin{split} S_{ij}&=\frac{x_{j}-x_{i}}{t_{j}-t_{ i}},\quad\text{for all }i<j,\\ \beta&=\text{median}(S_{ij}),\end{split} \tag{6}\]  
where \(x_{i}\) and \(x_{j}\) are the GPP values at times \(t_{i}\) and \(t_{j}\), and \(\beta\) is the median slope representing the trend magnitude. The Mann-Kendall trend analysis was performed to assess the significance of the trend, with the test statistic \(S_{MK}\) defined as  
\[\begin{split} S_{MK}&=\sum_{i=1}^{n-1}\sum_{j=i+1 }^{n}\text{sgn}(x_{j}-x_{i}),\\ \text{Var}(S_{MK})&=\frac{n(n-1)(2n+5)}{18},\\ Z&=\begin{cases}\frac{S_{MK}-1}{\sqrt{\text{ Var}(S_{MK})}}&\text{if }S_{MK}>0,\\ 0&\text{if }S_{MK}=0,\\ \frac{S_{MK}+1}{\sqrt{\text{Var}(S_{MK})}}&\text{if }S_{MK}<0,\end{cases}\end{split} \tag{7}\]  
where \(n\) is the number of data points, and \(Z\) is the standardized statistic used to determine the trend significance at a 0.05 level (\(|Z|\geq 1.96\)). Based on these methods, the GPP trends in northwestern Hubei from 2001 to 2020 were classified as shown in Table 3.  
To assess the long-term trend characteristics of GPP, we calculated the Hurst exponent (\(H\)) based on rescaled range (R/S) analysis [46]. The Hurst exponent helps determine whether a time series exhibits persistent (H > 0.5), random (H = 0.5), or anti-persistent (H < 0.5) behavior, providing a statistical basis for long-term trend prediction. This method has been widely used in environmental and hydrological studies to analyze temporal dynamics. In this study, the application of the Hurst exponent to 30 m GPP time series allows for a spatially explicit evaluation of trend stability, which can be expressed as  
\[\frac{R}{S_{RS}}=c\cdot T^{H} \tag{8}\]  
Taking the logarithm, \(H\) is derived from the slope of the linear regression:  
\[\log\biggl{(}\frac{R}{S_{RS}}\biggr{)}=\log(c)+H\cdot\log(T) \tag{9}\]  
where \(R\) is the range of the cumulative deviation in the GPP difference series, \(S_{RS}\) is the adjusted standard deviation, \(T\) is the time lag (1 to 19 for the 2001-2020 GPP data), and \(c\) is a constant. The Hurst exponent (\(H\)) was then combined with the Theil-Sen median slope (\(\beta\)) to predict the future GPP trends in the study area. The classification rules for the future GPP prediction are summarized in Table 4.