# Probabilistic Analysis of Infinite Slope Stability Considering Variation in Soil Depth
## 3 Results and Discussion
### Influence of Soil Depth Variability on Probabilistic Slope Stability Analysis  
To investigate the influence of soil depth variability on probabilistic slope stability analysis, probabilistic analysis was conducted using the infinite slope model for unsaturated soil. Additionally, rainfall infiltration is a major factor that causes slope instability, and the influence of soil depth variability may vary depending on the downward movement of the wetting front due to rainfall infiltration. Therefore, the saturation depths of the upper soil layer corresponding to the downward movement of the wetting front were considered to be 0, 0.5, 1.0, 1.5, 2.0, and 2.5 m. If the saturation depth exceeded the soil depth, the soil was considered fully saturated. The statistical properties of the random variables used in the probabilistic analysis are listed in Tables 1 and 2. Figure 9 shows the changes in the probability of slope failure (\(P_{f}\)) for each case due to the downward movement of the wetting front during rainfall infiltration. In all cases, \(P_{f}\) increased as the saturation depth increased. With low saturation depth, Case 1, in which soil depth variability was not considered, showed the lowest \(P_{f}\), with a significant difference compared with the three cases where soil depth variability was considered. However, \(P_{f}\) showed a rapid increase when the saturation depth exceeded 1.5 m in Case 1, with the highest \(P_{f}\) observed when the saturation depth was 2.5 m. The differences in \(P_{f}\) based on the PDF of soil depth showed that normal and log-normal distributions exhibited nearly identical \(P_{f}\), as the shapes of the two distributions were similar, leading to a similar frequency of failure occurrence with depth. On the other hand, the uniform distribution yielded the highest \(P_{f}\). The range of the uniform distribution \(\mu\pm 2\sigma\) exhibits a higher frequency of failure occurrences compared to the normal and log-normal distributions. To compare the causes of differences in \(P_{f}\) because of the consideration of soil depth variability, the frequency distribution of slope failure by depth was compared for Case 1 and Case 2, with the results shown in Figure 10.  
Figure 9: Comparison of change in \(P_{f}\) according to saturation depth for each case. The light blue color represents the saturated zone. With a saturation depth of 1.5 m, the frequency of failure occurrences from 0 to 1.5 m appeared similar between Case 1 and Case 2. The soil depth was fixed in Case 1, with the highest failure frequency observed at 2.5 m. In contrast, in Case 2, which considered soil depth variability, a significant frequency of failure was found at soil depths greater than 2.5 m. When the saturation depth was 2.5 m, although soil depths greater than 2.5 m were considered in Case 2, slope failures were rarely observed in the range of 2.5-2.9 m, and a relatively low frequency of failure was observed at soil depths greater than 2.5 m. These findings are ascribed to the reduction in shear strength of soil owing to soil saturation resulting in a lower \(F_{\text{S}}\) at 2.5 m, and the reduction in shear strength of soil having a greater effect on the decrease in \(F_{\text{S}}\) than the increase in soil depth. Therefore, compared with Case 1, Case 2 exhibited a lower \(P_{f}\), where the soil depth was fixed at 2.5 m.  
Figures 11 and 12 show the cumulative distribution functions (CDFs) of the minimum \(F_{\text{S}}\) for saturation depths of 1.5 m and 2.5 m.  
Figure 11: CDF of the minimum \(F_{\text{S}}\) for a saturation depth of 1.5 m.  
Figure 10: Comparison of frequency distribution of failure occurrences with depth: (**a**) saturation depth = 1.5 m and (**b**) saturation depth = 2.5 m.  
Differences were observed in the CDFs of the minimum \(F_{\rm S}\) for the four cases, and when the saturation depth was 2.5 m, the differences among the cases became more pronounced for \(F_{\rm S}\) greater than 1.0.