## 3 Materials and Methods
### Combination Weighting Method
#### 3.2.3 Combination Weighting Rule  
The Analytic Hierarchy Process determines the judgment matrix mainly based on the subjective experience of experts. In contrast, the evaluation process of the CRITIC method relies entirely on the own laws of objective data. In order to reflect both the researchers' intuitive understanding of debris flow in the field geological investigation stage and to take into account the regularity of objective data, the degree of difference between the weights obtained by the two methods is consistent with the degree of difference between their corresponding distribution coefficients, this paper introduces a distance function, which couples the weights obtained by the two methods together to determine the index weights comprehensively.  
Suppose the weight vector obtained by the Analytic Hierarchy Process method is \(\omega_{i}^{c}\), the weight vector obtained by the CRITIC method is \(\omega_{i}^{y}\), and the distance function between them is denoted as \(d(\omega_{i}^{c},\omega_{i}^{y})\)[33]:  
\[d(\omega_{i}^{c},\omega_{i}^{y})=\left[\frac{1}{2}\sum_{i=1}^{n}\left(\omega_ {i}^{c}-\omega_{i}^{y}\right)^{2}\right]^{\frac{1}{2}} \tag{12}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline \(1\) & Two decision factors (e.g., indicators) are equally important \\ \hline \(3\) & Two decision factors (e.g., indicators) are equally important \\ \hline \(5\) & Two decision factors (e.g., indicators) are equally important \\ \hline \(7\) & One decision factor is very strongly more important \\ \hline \(9\) & One decision factor is extremely more important \\ \hline \(2\), \(4\), \(6\), \(8\) & Intermediate values \\ \hline \multirow{2}{*}{Reciprocals} & If _ij_ is the judgement value when \(i\) is compared to \(j\), then \(U_{ji}=1/U_{ji}\) is the judgement value when \(j\) is compared to \(i\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Definition of comparative importance.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(n\) & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline _RI_ & 0 & 0 & 0.52 & 0.89 & 1.12 & 1.26 & 1.36 & 1.41 & 1.46 & 1.49 & 1.52 & 1.54 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The random average consistency index.  
Assuming that the combined weights are \(\omega_{i}^{z}\) and the linear weighting method is used to obtain \(\omega_{i}^{z}\), and assuming that the distribution coefficients of the two weights are \(a\) and \(b\), respectively, then \(\omega_{i}^{z}\) can be expressed as:  
\[\omega_{i}^{z}=a\omega_{i}^{z}+b\omega_{i}^{y} \tag{13}\]  
To ensure a consistent degree of variation in the magnitude of the weights and the distribution coefficients, make the distance function and the distribution coefficients the same:  
\[d(\omega_{i}^{z},\omega_{i}^{y})^{2}=(a-b)^{2} \tag{14}\]  
The two weight assignment coefficients also have to satisfy Equation (15). Combining Equations (14) and (15) yields the assignment coefficients \(a\) and \(b\). Substituting \(a\) and \(b\) into Equation (13) yields \(\omega_{i}^{z}\).  
\[a+b=1 \tag{15}\]