## 4 Discussion and Analysis
### Exploration of the Correlation between Pore Characteristics and Macroscopic Indicators
However, the accuracy and sensitivity of different weathering indicators in assessing the extent of rock damage differ. In this study, the entropy weight-TOPSIS method (approaching the ideal solution) was utilized for visual analysis based on physical property indicators. The fundamental principle of the TOPSIS method is to identify the optimal and worst solutions within a finite scheme derived from the normalized original matrix. By calculating the relative proximity degree based on the distance between each target and the optimal and worst solutions, a comprehensive evaluation result can be obtained. This approach allows for the assessment of the research objects' advantages and disadvantages. Firstly, five indicators,  
Figure 14: Relationship curves between the pore damage factor and macroscopic indicators of sandstone and limestone. (**a**) Relationship curves between the pore damage factor and macroscopic weathering indicators of sandstone; (**b**) relationship curves between the pore damage factor and macroscopic weathering indicators of limestone. including mass change rate, dry density, and porosity, are normalized. Secondly, the data for each indicator are multiplied by the corresponding weight to assign weights to different weathering indicators. Finally, the distance between the evaluation object and the optimal scheme and the worst scheme is calculated, respectively, and the relative proximity between the evaluation object and the optimal scheme is obtained [36; 37]. \[x_{ij}^{*}=x_{max}-x_{ij};\;i=1,\;2,\cdots\;m;\;1\leq j\leq n \tag{8}\]  
\[z_{ij}= \begin{cases}\frac{x_{ij}}{\sqrt{\sum x_{ij}}}\;;j\;\text{is listed as positive indicators},\;i=1,\;2,\cdots\;m;\;1\leq j\leq n\\ \frac{x_{ij}}{\sqrt{\sum x_{ij}^{*}}}\;;j\;\text{is listed as negative indicators},\;i=1,\;2,\cdots\;m;\;1\leq j\leq n\end{cases}  \tag{9}\]  
\[Z_{ij}^{*}= \begin{bmatrix}w_{1}z_{11}&w_{2}z_{12}&\cdots&w_{n}z_{1n}\\ w_{1}z_{21}&w_{2}z_{22}&\cdots&w_{n}z_{2n}\\ \cdots&\cdots&\ddots&\cdots\\ w_{1}z_{m1}&w_{2}z_{m2}&\cdots&w_{n}z_{mn}\end{bmatrix}  \tag{10}\]  
\[z^{+}=\left(z_{1}^{+},z_{2}^{+},\cdots,z_{n}^{+}\right)=\left\{\max(w_{j}z_{ ij}^{-})|i=1,2,\cdots,m\right\} \tag{11}\]  
\[z^{-}=\left(z_{1}^{-},z_{2}^{-},\cdots,z_{n}^{-}\right)=\left\{\min\!\left(w _{j}z_{ij}\right)\!\left|i=1,2,\cdots,m\right\} \tag{12}\]  
\[D_{i}^{+}=\sqrt{\sum_{j=1}^{n}\left(z_{j}^{+}-z_{ij}\right)^{2}};\;i=1,\;2, \cdots\;m \tag{13}\]  
\[D_{i}^{-}=\sqrt{\sum_{j=1}^{n}\left(z_{j}^{-}-z_{ij}\right)^{2}};\;i=1,\;2, \cdots\;m \tag{14}\]  
\[C_{i}=\frac{D_{i}^{-}}{D_{i}^{-}+D_{i}^{+}};\;(i=1,\;2,\cdots\;m) \tag{15}\]  
\[V_{i}=\frac{C_{i}}{\sum_{1}^{m}C_{i}};\;(i=1,\;2,\cdots\;m) \tag{16}\]  
where \(x_{ij}^{*}\) represents the value of the negative indicator after positive normalization, \(x_{max}\) is the maximum value of the column indicator, and \(z_{ij}\) represents the normalized value after standardization. Equation (8) is the positive normalization function; Equation (9) represents the normalization function; Equation (10) is the weighted matrix; Equations (10) and (11) indicate the function for seeking the optimal solution; Equations (12) and (13) can be used to calculate the Euclidean distance of each evaluation object from the optimal solution and the worst solution; and Equations (14) and (15) indicate the function for indicator re-normalization. Under normal circumstances, \(C_{i}\) can be used as the final evaluation score, but in order to reflect the vulnerability more directly, this paper adopts Equation (16) to normalize the relative proximity degree, and takes the processed \(V_{i}\) as the final vulnerability evaluation score of the evaluation object, which then makes the maximum vulnerability 1 and the minimum vulnerability 0, so the quantization result is more intuitive. Upon analyzing the comparison of weight factors for different weathering indicators in Figure 15a regarding the two types of rock, it can be concluded that the weight factor for the mass change rate is the highest, reaching approximately 0.25, under sulfate action.