# Research on Risk Assessment Method for Land Subsidence in Tangshan Based on Vulnerability Zoning
## 2 Materials and Methods
### Ground subsidence Hazard Assessment Methods
#### 2.2.1 Analytic Hierarchy Process (AHP)  
In this paper, the research is focused on slow land subsidence affecting large territories. We summarized the main influencing factors including natural factors, engineering factors, and inducement factors.  
The AHP is a method used to tackle complex problems and is extensively applied in the field of engineering disaster prevention. The method comprises the following four steps:  
(1) Establishing a hierarchy structure model.  
The hierarchical structure consists of three levels: the goal level, the criterion level, and the indicator level. The goal level represents the evaluation method for the urban ground subsidence hazard. The criterion level, also known as the intermediate level, is divided into three categories: natural factors, engineering factors, and triggering factors. The indicator level encompasses a total of 12 evaluation indicators.  
(2) Establishing judgment matrices.  
The scales of the judgment matrices in the AHPare depicted in Table 5.  
Figure 8: Study area selection.  
For \(n\) elements, we can obtain the corresponding judgment matrices for pairwise comparisons:  
\[A=\left(U_{ij}\right)_{n\times n} \tag{1}\]  
where \(U_{ij}>0,U_{ji}=U_{ij}^{-1},U_{ij}=1\).  
(3) Calculating the judgment matrices.  
Based on the \(\delta\) values obtained in (1), the characteristic vectors of the established judgment matrices in the ground subsidence hazard assessment system are solved to calculate the weight values of the evaluation indicators:  
(1) Normalize the column elements of the matrix:  
\[\overline{u}_{ij}=u_{ij}/\sum_{i=1}^{n}u_{ij} \tag{2}\]  
(2) Sum the normalized matrix elements:  
\[\overline{W}_{i}=\sum_{j=1}^{n}\overline{u}_{ij} \tag{3}\]  
(3) Calculate the normalized characteristic vector (\(\overline{W}=\left(\overline{W_{1}},\overline{W_{2}},\cdots,\overline{W_{n}} \right)^{T}\)) to determine the weight values of each evaluation indicator:  
\[W_{i}=\overline{W}_{i}/\sum_{i=1}^{n}\overline{W}_{i} \tag{4}\]  
(4) Perform a consistency check.  
The obtained characteristic vector needs to undergo a consistency check to verify the consistency of the process of assigning weights to the indicators. The calculation formula is as follows:  
\[CI=(\lambda_{\max}-n)/(n-1) \tag{5}\]  
\[CR=CI/RI \tag{6}\]  
In the context of the analytic hierarchy process (AHP), the Consistency Index (\(CI\)) and the Random Consistency Index (\(RI\)) are crucial parameters. The \(CI\) is used to measure how consistent the judgments are in the comparison matrix, while the \(RI\) is an average index that represents the extent of consistency that would be expected from a randomly generated matrix. The values for the \(RI\) are provided in Table 6. The Consistency Ratio (\(CR\)) is then calculated as the ratio of the \(CI\) to the \(RI\). A \(CR\) less than 0.1 signifies that the judgment matrix satisfies the consistency requirements, indicating a reasonable level of judgment consistency. A \(CR\) greater than 0.1 suggests a significant inconsistency in the judgments, necessitating an adjustment of the judgment matrix to improve its consistency.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scale** & **Meaning** & **Scale** & **Meaning** \\ \hline
1 & The impacts of U\({}_{\text{i}}\) and U\({}_{\text{j}}\) are much the same & 3 & U\({}_{\text{i}}\) is slightly stronger than U\({}_{\text{j}}\) \\
5 & U\({}_{\text{i}}\) is stronger than U\({}_{\text{j}}\) & 7 & The effect of U\({}_{\text{i}}\) on U\({}_{\text{j}}\) is significantly stronger \\
9 & The effect of U\({}_{\text{i}}\) on U\({}_{\text{j}}\) is obviously strong and reciprocal & 2, 4, 6, 8 & Intermediate value between adjacent scales, and the value of U\({}_{\text{i}}\) is less important than that of U\({}_{\text{j}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Definitions of the scales of the judgment matrices.