# Probabilistic Analysis of Infinite Slope Stability Considering Variation in Soil Depth
## 3 Results and Discussion  
### Soil Depth Variability  
Figure 6 illustrates the process of extracting soil depth at regular intervals to assess soil depth variability using seismic refraction survey results. The extracted soil depth refers to the gravity-directional soil depth. \begin{table}
\begin{tabular}{c c c c c} \hline
**Soil Property** & **Case** & \multicolumn{2}{c}{**Statistical Properties**} & **PDF** \\ \hline \multirow{6}{*}{Soil depth, \(d_{g}\) (m)} & 1 & & 2.5 & Deterministic \\ \cline{3-5}  & \multirow{2}{*}{2} & Mean & Standard deviation & COV (\%) & Normal \\ \cline{3-5}  & & 2.5 & 0.725 & 29 & (truncated) \\ \cline{3-5}  & \multirow{2}{*}{3} & Mean & Standard deviation & COV (\%) & \\ \cline{3-5}  & & 2.5 & 0.725 & 29 & Log-normal \\ \cline{3-5}  & \multirow{2}{*}{4} & Lower bound & Upper bound & \\ \cline{3-5}  & & 1.05 & 3.95 & Uniform \\ \hline \end{tabular}
\end{table} 
Table 2: Probabilistic analysis conditions for soil depth. Figure 5: Flowchart for the probabilistic slope stability analysis. The slope was divided into 5 m horizontal intervals, and soil depth was calculated using the difference in elevation between the surface and the corresponding elevation for a seismic wave velocity of 700 m/s. Soil depth was extracted, and its statistical characteristics were analyzed in the same manner for 70 slopes where seismic surveys were conducted. Figure 7 presents examples of representative cross-sections of the slope categorized by the average soil depth and its COV. Figure 6: Soil depth extraction at 5 m intervals to determine the variability in depth along the slope. Figure 7: Examples of slopes based on the average soil depth and its COV. In cases where the slope section includes areas with relatively shallow or deep soil depths, a higher COV was observed. The average and COV of soil depth across the 70 slopes exhibited a notably wide range. To determine the distribution shape of soil depth within a single slope, the skewness and kurtosis were calculated. The box plots of each statistical value for soil depth are shown in Figure 8.  
The range of average soil depth for each slope was 0.3-5.2 m. The average soil depth across 70 slopes was 2.5 m, with an average COV of approximately 28.9%. These findings indicate that significant variation in soil depth could occur within a single slope, with some slopes exhibiting a high COV in soil depth that exceeds 50%. The average skewness of soil depth was 0.19, which is a relatively small positive skew. This finding suggests that although the data distribution is skewed slightly right, it does not deviate significantly from a normal distribution. The kurtosis of soil depth had an average value of \(-0.32\), which is lower than that of normal distribution (kurtosis = 0). This indicates that the distribution of soil depth had shorter and thinner tails compared with normal distribution, and the data were concentrated around the center, resulting in a flatter peak. Generally, a skewness range between \(-1\) and \(+1\) and a kurtosis range between \(-2\) and \(+2\) can be accepted for normal distribution. As a result, the skewness and kurtosis of soil depth were determined to be 0.19 and \(-0.32\), respectively, supporting the assumption that the data followed a normal distribution. Pearson correlation coefficients among the statistics of soil depth were calculated, and their correlation matrix is presented in Table 3.  
The average soil depth and its COV showed a negative correlation, with a \(p\)-value of 0.0016, indicating significance at the 0.01 level. Accordingly, lower soil depth could be  
Figure 8: Box plots of statistics of soil depth.