## 2 Study Area and Materials
### Method
#### 2.3.1 Indicator Calculation
This method is used to reflect the degree of plant growth and widening, as well as changes in land information. We use the MOD09A1 surface reflectance product, which is sensitive to land moisture information in certain bands, to calculate the humidity of the study area. The formula is as follows:  
\[Wet=0.1147\rho_{1}+0.2489\rho_{2}+0.2408\rho_{3}+0.3132\rho_{4}-0.3122\rho_{5} -0.6416\rho_{6}-0.5087\rho_{7} \tag{7}\]  
\(\rho_{1}\sim\rho_{7}\) The seven bands of reflectance in the MOD09A1 surface reflectance product are red, NIR1, Blue, Green, NIR2, SWIR1, and SWIR2. Weight Determination  
Principal component analysis (PCA) is a commonly used statistical analysis method that can combine several correlated indicators into a mutually independent composite index through a linear transformation. The advantage of PCA is that it can reduce the dimensionality of the indicators while retaining the maximum principal components, and the determination of weights is objective. The steps for determining weights based on PCA are as follows: 1 Standardize the original data. 2 Calculate the correlation coefficient matrix of each indicator. 3 Calculate the eigenvalues, contribution rates, and cumulative contribution rates of each principal component. 4 Calculate the loading values of each principal component. 5 Obtain the scores of each principal component. 6 Calculate the weight of each indicator; the formula is as follows:  
\[PC=\sum_{i=1}^{i}\frac{pc_{i}\times e_{i}}{\sqrt{k_{i}}\times E} \tag{8}\]  
where \(PC\) represents the score coefficient of each indicator, \(pc_{i}\) represents the load of each principal component, \(e_{i}\) represents the variance contribution rate of each principal component, \(k_{i}\) represents the eigenvalues of each principal component, and \(E\) represents the cumulative contribution rate of the extracted principal components. Finally, the weights of each indicator can be obtained by normalizing the \(PC\). (2) Combination weight method based on game theory  
Game theory (GT) can effectively utilize the combined information of subjective weights (SW) and objective weights (OW) and obtain the most balanced weights, also known as combined weights (CW). Supposing that the weight set generated by using a weighting method for m basic indicator vectors is \(\omega_{1}\), the weight vector obtained by n empowerment methods is then \(\omega_{i}\):  
\[\omega_{1}=[\omega_{11},\omega_{12},\omega_{13},\ldots,\omega_{1m}] \tag{9}\]  
\[\omega_{i}=[\omega_{n1},\omega_{n2},\omega_{n3},\ldots,\omega_{nm}]=[\omega_{ 1},\omega_{2},\omega_{3},\ldots,\omega_{n}],i=1,2,\ldots,n \tag{10}\]  
The arbitrary linear combination \(W\) of the \(n\) vectors is as follows:  
\[NDVI_{soil}=\frac{FVC_{max}\times NDVI_{min}-FVC_{min}\times NDVI_{max}}{FVC_{ max}-FVC_{min}} \tag{2}\]0  
Solving the optimal combination coefficient \(\gamma_{i}\) to minimize the deviation between the combination weight and the weight involved in optimization can be achieved through the following function:  
\[NDVI_{soil}=\frac{FVC_{max}\times NDVI_{min}-FVC_{min}\times NDVI_{max}}{FVC_{ max}-FVC_{min}} \tag{2}\]1  
Solving the first derivative of the above equation for optimization yields the following equivalent linear equations:  
\[\sum_{i=n}^{n}\gamma_{i}\omega_{i}\omega_{i}^{T}= \begin{bmatrix}\gamma_{1}\\ \gamma_{2}\\ \cdot\\ \cdot\\ \gamma_{n}\end{bmatrix} \times \begin{bmatrix}\omega_{1}\times\omega_{1}^{T}& \omega_{1}\times\omega_{1}^{T}&\ldots&\omega_{1}\times\omega_{1}^{T}\\ \omega_{2}\times\omega_{1}^{T}&\omega_{2}\times\omega_{2}^{T}&\ldots&\omega_ {2}\times\omega_{n}^{T}\\ \cdot&\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot&\cdot\\ \omega_{n}\times\omega_{1}^{T}&\omega_{n}\times\omega_{2}^{T}&\ldots&\omega_ {n}\times\omega_{n}^{T}\end{bmatrix} = \begin{bmatrix}\omega_{1}\times\omega_{ 1}^{T}\\ \omega_{2}\times\omega_{2}^{T}\\ \cdot\\ \cdot\\ \cdot\\ \omega_{n}\times\omega_{n}^{T}\end{bmatrix}