## 3 Methodology
### Highway Rainfall Hazard  
Rainfall has a significant impact on the occurrence of geological disasters, particularly during intense or prolonged precipitation events. Increased soil moisture can lead to saturation, triggering landslides and debris flows. Additionally, rainfall may compromise the stability of rock formations, heightening the risk of rockfalls. The erosion and surface wash caused by rainfall can further destabilize the landscape, exacerbating the likelihood of geological disasters [11]. Thus, rainfall is a critical factor in the initiation of geological disasters, especially in areas characterized by complex topography and high human activity, necessitating close monitoring of its associated risks. The highway rainfall hazard refers to the occurrence of rockfall accidents through external rainfall. In this research, average annual rainfall and daily maximum rainfall were selected as highway rainfall hazards. The highway rainfall hazard index is shown in Figure 6.  
1. Average annual rainfall. Guizhou Province has a plateau-type subtropical climate with abundant rainfall, and the average annual rainfall is mostly between 1100 and 1400 mm. It can be seen in Figure 6a that the average annual rainfall distribution of the whole Guizhou Province is higher in the southeast and lower in the northwest. Average annual rainfall can be a good reflection of the climate of the study area; the greater the mean annual rainfall, the greater the surface runoff of the slope, and the greater the probability that the slope will be subjected to rainwater erosion. This can lead to the disruption of slope integrity and vegetation cover, ultimately leading to rockfall accidents. 2. Daily maximum rainfall. Daily maximum rainfall is the amount of rainfall in a 24 h period, which can be representative of extreme severe weather. The daily maximum rainfall in Guizhou Province is shown in Figure 6b, which shows that the daily maximum rainfall in Guizhou Province mainly occurs in the central and southern parts of the province. Excessive daily maximum rainfall causes large amounts of surface runoff from slopes, eroding the slopes, washing away the rocks on the slopes' surface, and reducing vegetation's protective effect on the slopes. It makes rockfalls much more likely to occur. After determining the rainfall hazard index, the corresponding weights were given to them through the fuzzy analytic hierarchy process (FAHP). The analytic hierarchy process (AHP) refers to a complex multi-objective decision-making problem as a system, and the objective is decomposed into multiple objectives and then into a number of levels of multi-indicators through the fuzzy quantitative method of qualitative indicators to calculate the hierarchical weights and the total ranking, in order to serve as a systematic method of the target optimization decision-making. The AHP, as one of the most widely used methods, is subject to a great deal of subjectivity, and the introduction of fuzziness can effectively reduce subjectivity. The FAHP is calculated as follows: first, a fuzzy judgment matrix is constructed, then the weights of each element are determined, and finally, a consistency test is performed. \[A= \begin{bmatrix}a_{11}&\dots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{n1}&\dots&a_{nn}\end{bmatrix}  \tag{3}\]  
The matrix parameters are selected from 0.1 to 0.9, and the matrix data parameter values are described in Table 2.  
Figure 6: Indexes of highway rainfall hazard. The weights of the index are calculated as follows:  
\[W_{i}=\frac{\sum\limits_{j=1}^{n}a_{ij}+\frac{n}{2}-1}{n(n-1)} \tag{4}\]  
The weight matrix is calculated as follows:  
\[w_{ij}=\frac{W_{i}}{W_{i}+W_{j}} \tag{5}\]  
Furthermore, fuzzy judgment matrices need to be tested for consistency, usually by means of a compatibility index \(I\) (\(A\), \(W\)). When \(I\leq\alpha\) (\(\alpha\) is the decision-maker's attitude, usually taken as 0.1), then consistency is considered to be satisfied.