# Probabilistic Analysis of Infinite Slope Stability Considering Variation in Soil Depth
## 2 Materials and Methods
### Probabilistic Slope Stability of Unsaturated Soils
#### 2.4.2 Probabilistic Analysis of Slope Stability  
Figure 4 shows the cross-section of the slope stability analysis, with soil depth treated as a random variable that varies according to the variability in soil depth.  
Figure 3: Estimation of slope depth by P-wave velocity.  
The slope angle was assumed to be \(30^{\circ}\), and soil depth was discretized into intervals of 2 cm to calculate \(F_{\text{S}}\) by soil depth. The two shear strength parameters (\(c\) and \(\phi\)) and soil depth were considered random variables for probabilistic slope stability analysis. Regarding soil properties that always have positive values, assuming a log-normal distribution is more reasonable than assuming a normal distribution [38; 39]. Therefore, the probability distribution functions (PDFs) of the two shear strength parameters were assumed to follow a log-normal distribution, and their coefficients of variation (COVs) were assumed to be \(30\) and \(20\%\), respectively, based on the literature [40; 41]. In addition, the shear strength of soil is expressed as a function of \(c\) and \(\phi\) in the Mohr-Coulomb failure criterion, and these two parameters are not independent random variables but are correlated with each other. In probabilistic slope stability analysis, their correlation has a significant impact, and it is necessary to generate correlated random variables rather than independent random variables. The correlation between \(c\) and \(\phi\) is generally known to be negative, and it was assumed to be \(-0.5\) based on the literature [42; 43; 44]. The statistical properties of shear strength parameters are listed in Table 1.  
Soil depth variability was identified using the seismic survey results from 70 slopes in South Korea, with a mean of 2.5 m and COV of 29%. The PDF of soil depth is not well known, and in this study, three PDF types were considered, namely, normal, log-normal, and uniform distributions. Regarding normal distribution, the minimum value was restricted to 0.1 m to prevent the occurrence of negative values for soil depth. A uniform distribution was applied using a \(\mu\pm 2\sigma\) range to include most data (95.4%) while excluding outliers. The probabilistic analysis cases based on the consideration of the variability of soil depth and their statistical properties are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Soil Property** & **Mean** & **Standard Deviation** & **COV (\%)** & **PDF** & **Correlation** \\ \hline Cohesion, \(c\) (kPa) & 10 & 4.5 & 30 & Log-normal & \\ \hline Friction angle, \(\phi\) (\({}^{\circ}\)) & 20 & 3.6 & 20 & Log-normal & \\ \hline \end{tabular}
\end{table}
Table 1: Statistical properties of shear strength parameters in probabilistic analysis.  
Figure 4: Cross-section for slope stability analysis.  
Slope failure is defined as the occurrence of \(F_{\text{S}}\) below 1 at any depth, and the depth at which the \(F_{\text{S}}\) first drops below 1 is considered the failure surface. Probabilistic analysis was conducted using the Monte Carlo simulation (MCS) based on the infinite slope stability equation for unsaturated soil presented in Equation (1). Slope stability analysis was repeated using a series of MCS with 200,000 runs, yielding adequately converged results for the probabilistic analysis.