## 2 Materials and Methods
### Methods
#### 2.3.3 Optimal Parameters-Based Geographical Detector (OPGD)  
The geographical detector is an excellent model for the analysis of spatial heterogeneity and its underlying driving forces [41; 42]. The discretization of continuous variables in the traditional geographical detector is determined by personal experience, which is subjective and lacks accurate quantitative evaluation [43; 44]. The discretization scheme with the strongest explanatory power can be determined in OPGD, and the results are more reliable and accurate [43]. The \"GD\" package in R was applied to perform the OPGD model [43]. For geographical detection, pixels from 2001 to 2018 without land use change were employed. More than 665,000 pixels took part in the simulation. Mean values for LAI, GPP, and climate factors from 2001 to 2018 were used in the OPGD. The climate data were interpolated to 500 m through the kriging method so as to unify the spatial resolution with LAI and GPP. Data of DEM, landform, and land use were resampled to 500 m before simulation. In our investigation, we employed factor, interaction, and risk detectors. (1) The factor detector can quantify the impact of each factor (\(X\)) on the spatial heterogeneity of the variable (\(Y\)). The explanatory power (\(q\)) of \(X\) can be calculated as follows [42]:  
\[q=1-\frac{\sum\limits_{h=1}^{L}N_{h}\sigma_{h}^{2}}{N\sigma^{2}} \tag{6}\]  
In the formula, \(q\) represents the explanatory power of \(X\) on \(Y\). The higher \(q\) value indicates a higher impact of \(X\) on \(Y\). \(L\) is the stratification of \(X\) or \(Y\). \(N_{h}\) and \(N\) are the numbers of units of layer \(h\) and the whole region, respectively. \(\sigma_{h}^{2}\) and \(\sigma^{2}\) are the numbers of variance of layer \(h\) and the whole region, respectively. (2) The interaction detector can be applied to evaluate the \(q\) values after the interaction of two factors (\(X\)). When \(q\) (\(X_{1}\cap X_{2}\)) > Max (\(q\)(\(X_{1}\)), \(q\)(\(X_{2}\))), it is bivariate enhancement; when \(q\) (\(X_{1}\cap X_{2}\)) > \(q\) (\(X_{1}\)) + \(q\) (\(X_{2}\)), it is non-linear enhancement [41]. (3) The risk detector can be applied to determine whether there is an obvious variation in variables between different attributes of each factor. It can be utilized to ascertain how the variable (\(Y\))'s attribute changes as the driving factor (\(X\))'s subregions change. (4) Spatial data discretization: geographical detectors require the discretization of continuous variables into categorical variables known as spatial data discretization. Climatic elements, elevation, and slope are continuous variables that need to be discretized. Equal interval, natural breaks, quantile, and geometrical interval are the four different forms of discretization methods. The number of discretization categories was set to 4-10. The optimal combination of category numbers and discretization methods was utilized. The factors for LAI after ideal discretization are displayed in Figure 2. The climate elements were annual averages from 2001 to 2018. The temperature in the MHRB was higher in the eastern and southern regions and gradually decreased in the northern regions (Figure 2a). Similar patterns could be seen in the distribution of precipitation and humidity index, with greater values in the south of the MLHRB, MBSHRB, and MDQHRB, as well as the east of MZYHRB and the MZWHRB (Figure 2b,c). However, the northern portion of the MLHRB and MBSHRB and the MYDHRB had less precipitation and a driz climate. The value of VPD was higher in the MYDHRB and the east of the MHRB (Figure 2d). ETO was higher in the MYDHRB and MZYHRB, and the south of the MDQHRB, but lower in the MLHRB and MBSHRB (Figure 2e). In contrast to the distribution characteristics of precipitation, solar radiation was lower in the eastern and southern parts and higher in the northern part of the MHRB (Figure 2f). The MHRB's elevation was higher in the northern and western regions (Figure 2g).