## 2 Materials and Methods
### Spatiotemporal Variation Response Analysis Method
#### 2.5.2 Climate Factor Response Analysis  
The Pearson correlation coefficient (PCC or \(r\)) is a fundamental statistical tool used to quantify the strength and direction of the linear relationship between two continuous variables [47]. To analyze the relationship between the GPP variations and climatic factors in the study area, correlation analysis was employed in this research. Firstly, the correlations were calculated using the GPP and meteorological data at different temporal scales--16-day, seasonal, and annual--comparing their differences and distinctions. Secondly, the interannual correlations were computed using the 16-day data within each year, with the comparisons and analyses conducted across different years. Finally, the correlations were analyzed for  
\begin{table}
\begin{tabular}{c c c c} \hline
**Condition on \(H\)** & **Condition on \(\beta\)** & **Trend Value** & **Future Prediction** \\ \hline \(H>\) 0.5 & \(\beta>0\) & 2 & Sustained increase \\ \(H>\) 0.5 & \(\beta<0\) & \(-2\) & Sustained decrease \\ \(H<\) 0.5 & \(\beta>0\) & \(-1\) & Increase to decrease \\ \(H<\) 0.5 & \(\beta<0\) & 1 & Decrease to increase \\ \(H=\) 0.5 & – & 0 & Unpredictable \\ \hline \end{tabular}
\end{table}
Table 4: Classification rules for future GPP trend prediction based on Hurst exponent and Theil–Sen median slope.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Condition on** & **Condition on \(Z\)** & **Trend Value** & **Interpretation** \\ Slope (\(\beta\)) & \(|Z|\geq\) 1.96 & 2 & Significant positive \\ \(\beta>0\) & \(|Z|<\) 1.96 & 1 & Non-significant positive \\ \(\beta<0\) & \(|Z|\geq\) 1.96 & \(-2\) & Significant negative \\ \(\beta<0\) & \(|Z|<\) 1.96 & \(-1\) & Non-significant negative \\ \(\beta=0\) & – & 0 & No change \\ \hline \end{tabular}
\end{table}
Table 3: Classification of the GPP variation trends in northwestern Hubei (2001–2020) based on the Theil–Sen median slope and Mann–Kendall trend test.  
each 16-day period within a year, examining the intra-annual variations in the relationship between GPP and climatic factors over 20 years.  
\[r=\frac{\sum_{i=1}^{n}(GPP_{i}-\overline{GPP})(WC_{i}-\overline{WC})}{\sqrt{\sum_{ i=1}^{n}(GPP_{i}-\overline{GPP})^{2}}\sqrt{\sum_{i=1}^{n}(WC_{i}-\overline{WC})^{2}}} \tag{10}\]  
where \(\text{GPP}_{i}\) and \(\text{WC}_{i}\) represent the GPP and the climatic variable (such as precipitation or temperature) at time step \(i\) for a specific pixel, respectively, over a 20-year period from 2001 to 2020. The mean values \(\overline{\text{GPP}}\) and \(\overline{\text{WC}}\) are the averages of the GPP and the climatic variable, respectively, calculated across this period at one of the three temporal scales employed in the analysis: the 16-day scale, the seasonal scale, or the annual scale. The total number of time steps \(n\) varied depending on the chosen scale, with illustrative values being \(n=460\) for the 16-day scale (based on approximately 23 time steps per year over 20 years), \(n=80\) for the seasonal scale (reflecting four seasons per year over 20 years), and \(n=20\) for the annual scale (corresponding to one value per year over 20 years). These values were adjusted in practice to account for any missing data.