## 3 Materials and Methods
### Combination Weighting Method  
#### 3.2.1 CRITIC Method  
The CRITIC method is an objective weighting that reflects the discreteness and factor conflict between samples through standard deviation and correlation coefficient. The size of the standard deviation is directly proportional to the degree of discreteness and factor weight, and the size of the correlation coefficient is also directly proportional to the conflict between factors. The larger the correlation coefficient, the smaller the weight [34]. The CRITIC method takes into account sample information and factor correlation. Also, it utilizes the coefficient of variation to make the dispersion reflected by standard deviation more realistic, with significant advantages [35; 36; 37]. The specific calculation steps are as follows:  
(1) Assuming \(m\) samples containing \(n\) indicators, construct the original data matrix using the indicators:  
\[X=\begin{bmatrix}\alpha_{11}&\alpha_{12}&\ldots&\alpha_{1n}\\ \alpha_{21}&\alpha_{22}&\ldots&\alpha_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ \alpha_{m1}&\alpha_{m2}&\ldots&\alpha_{mn}\end{bmatrix} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Number** & **Debris Flow** & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** & **F7** & **F8** & **F9** \\ \hline
1 & Xiaoniuquan & 1.11 & 0.66 & 16.20 & 2350 & 1.1711 & 39.46 & 70 & 5 & 119.5 \\
2 & Lianshan & 2.15 & 1.36 & 33.56 & 1430 & 1.1250 & 11.06 & 60 & 5 & 25.50 \\
3 & Feishuiyan & 2.20 & 1.52 & 33.66 & 1420 & 1.1620 & 12.32 & 60 & 5 & 21.33 \\
4 & Huilong & 1.25 & 0.49 & 20.09 & 2130 & 1.1535 & 38.72 & 70 & 5 & 122.70 \\
5 & Yanzidong & 1.19 & 0.58 & 14.28 & 2245 & 1.1816 & 40.90 & 70 & 5 & 113.20 \\
6 & Shiliangzi & 1.47 & 0.78 & 30.26 & 1500 & 1.2731 & 21.66 & 55 & 1 & 35.16 \\
7 & Machang & 1.92 & 1.33 & 35.58 & 1430 & 1.1947 & 19.71 & 55 & 1 & 38.99 \\
8 & Manban & 1.73 & 1.40 & 29.02 & 1400 & 1.0588 & 22.72 & 55 & 1 & 36.47 \\
9 & Henghe & 0.80 & 0.50 & 18.53 & 2137 & 1.1615 & 39.13 & 50 & 1 & 135.30 \\
10 & Yusshi & 0.94 & 0.45 & 15.57 & 2500 & 1.2020 & 44.05 & 80 & 150 & 259.34 \\
11 & Longcao & 1.80 & 0.75 & 16.62 & 2300 & 1.2160 & 42.45 & 80 & 150 & 277.80 \\
12 & Meizilin & 0.80 & 0.33 & 15.59 & 1700 & 1.2429 & 43.22 & 80 & 150 & 252.57 \\
13 & Xujia & 1.70 & 1.22 & 18.48 & 1130 & 1.1236 & 14.66 & 60 & 50 & 35.16 \\
14 & Baiyan & 2.53 & 2.19 & 33.98 & 1320 & 1.1841 & 13.45 & 60 & 30 & 24.86 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Evaluation index values of debris flow risk in the research area.  
(2) Normalization of indicators:  
\[q_{ij}=\frac{\alpha_{ij}-\min_{j}(\alpha_{ij})}{\max_{j}(\alpha_{ij})-\min_{j}( \alpha_{ij})} \tag{2}\]  
(3) Calculate coefficient of variation:  
\[\overline{\alpha_{j}}=\frac{\sum_{i=1}^{m}\alpha_{ij}}{m} \tag{3}\]  
\[S_{j}=\sqrt{\frac{1}{m}{\sum_{i=1}^{m}\left(\alpha_{ij}-\overline{\alpha_{j}} \right)^{2}}} \tag{4}\]  
\[\mu_{j}=\frac{S_{j}}{\overline{\alpha_{j}}} \tag{5}\]  
In the formula, \(\overline{\alpha_{j}}\) is the average value of each indicator; \(S_{j}\) is the standard deviation; \(\mu_{j}\) is the coefficient of variation.  
(4) Calculate the correlation coefficient matrix:  
\[\kappa_{ij}=\frac{cov(y_{k},y_{l})}{(s_{k}s_{j})},k=1,2,\ldots,n;l=1,2,\ldots,n \tag{6}\]  
In the formula, \(\kappa_{ij}\) represents the correlation coefficient between indicators, and \(cov(y_{k},y_{l})\) represents the covariance between indicators.  
(5) Calculation of indicator information quantity:  
\[\omega_{j}=\mu_{j}{\sum_{i=1}^{n}(1-\kappa_{ij})},j=1,2,\ldots,n \tag{7}\]  
The weights of each indicator are:  
\[y_{j}=\frac{\omega_{j}}{\sum_{i=1}^{n}\omega_{j}}j=1,2,\ldots,n \tag{8}\]