# Evaluation and Prediction of Ecological Benefits in Song-Liao River Basin
## 2 Materials and Methods
### Ecosystem Service Index  
Ecosystem service refers to all the benefits humans get from the ecosystem, including supply, regulation, culture, and support [31]. Ecosystem service assessment (ESA) is a qualitative or quantitative analysis of the services provided by the ecosystem to human beings. Ecosystem service assessment can reveal the supporting role of the ecosystem in national or regional ecological security and economic and social development, as well as the evolution process of the interaction between human beings and the ecosystem [32]. This study used carbon fixation, water yield, and soil retention to evaluate ecosystem services and the changing trends in SRB from 1990 to 2020. The entropy weight method is used to determine the weight of each component of ecosystem service in each period to obtain the Ecosystem Service Index (ESI). The entropy weight method was first put forward in 1948 and was mainly used to calculate the uncertainty and randomness of information [33]. The entropy weight method is an objective weighting method, according to the variation degree of each evaluation index, which uses information entropy to calculate and correct the weight to get the more objective weight of the evaluation index. This method can explain those indexes with subjective assignment, high accuracy, and strong objectivity [34]. The weight of each component was calculated in SPSS 26, and then the ESI was built in ArcGIS 10.8. Equations (1)-(6) construct the entropy weight method.  
Firstly, the index system is constructed, and the index matrix is determined:  
\[X=\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m,1}&x_{m,2}&\cdots&x_{m,n}\end{bmatrix} \tag{1}\]  
where each column is an indicator, and each row is the value of a component in different indicators. After that, the initial matrix is standardized. If the index is positive, Equation (2) is used. Otherwise, Equation (3) is used:  
\[Z_{ij}=\frac{x_{ij}-minX_{i}}{maxX_{i}-minX_{i}} \tag{2}\]  
\[Z_{ij}=\frac{minX_{i}-x_{ij}}{maxX_{i}-minX_{i}} \tag{3}\]where \(x\) is the true value of each index in each phase element; \(minX_{i}\) is the minimum value of this index; and \(maxX_{i}\) is the maximum value of this index. After standardization, a standardized matrix is obtained:  
\[Z=\begin{bmatrix}z_{1,1}&z_{1,2}&\cdots&z_{1,n}\\ z_{2,1}&z_{2,2}&\cdots&z_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ z_{m,1}&z_{m,2}&\cdots&z_{m,n}\end{bmatrix} \tag{4}\]  
After obtaining the standardized matrix, first calculate the proportion of each phase element value in the population according to Equation (5), and then calculate the information entropy according to Equation (6):  
\[p_{ij}=\frac{z_{ij}}{\sum_{i=1}^{m}z_{ij}} \tag{5}\]  
\[H_{j}=-\frac{\sum_{i=1}^{m}p_{ij}\cdot ln(p_{ij})}{ln(m)} \tag{6}\]  
where \(p\) is the specific gravity value of each index; and \(H\) is the information entropy of each index. If \(p=0\), \(H=0\) is defined:  
\[\omega_{j}=\frac{1-H_{j}}{\sum_{j=1}^{n}(1-H_{j})} \tag{7}\]  
where \(\omega\) is the weight of each index.