## 3 Illustrative Examples
### Example I: A Two-Layered Cohesive Slope
#### 3.1.1 Effects of ACF types on Slope Reliability Analysis  
This subsection describes the effects of ACFs on the slope reliability, where the ACF type was taken as the variable in the probabilistic stability analysis. Nine combinations of ACFs for different layers were considered and are listed in Table 3. In each of the nine cases, the ACF was changed among the five types given in Table 1 for clay 1 or clay 2. The cases can be divided into two groups: The change of ACF in the upper stratum (case 1-5) and the lower stratum (case 1 and 6-9). The horizontal SOF, \(\delta_{\rm h}\), and vertical SOF, \(\delta_{\rm v}\), were constant here and set as 40 m and 4 m, respectively, which follow a previous study [17] and are from common values of SOF in practice. To simplify the analysis, case 1, where both clay 1 and clay 2 take the SQEF, was considered as the baseline case, as most of the studies available in the literature prefer to use the SQEF to describe the soil spatial variability [4; 8; 17]. For case 1, the \(P_{f}\) was estimated as \(8.06\times 10^{-2}\) by the current probabilistic stability model, which is well consistent with that \((8.17\times 10^{-2})\) reported by [PERSON] [17], showing the accuracy of the model. It should be noted that the number of random field realizations plays a crucial role in the accuracy of the result. Here, the MCS size was determined based on a sensitivity study of \(P_{f}\) against the number of MCS samples, as shown in Figure 2. It was observed that \(P_{f}\) converges to around \(8.0\times 10^{-2}\) at the number of around 3000, and remained stable with the increase in the number of the MCS samples. However, a total of 5000 MCS samples were finally used to keep a balance between computation accuracy (e.g., \(COV_{pf}<0.05\)) and efficiency. For other cases, unless otherwise specified, an MCS size of 5000 was also preferably used. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Mean** & **COV** & **Distribution** \\ \hline \(S_{u_{1}}\) & 51 kPa & 0.3 & Lognormal \\ \(S_{u_{2}}\) & 34 kPa & 0.3 & Lognormal \\ \(\gamma\) & 19 kN/m\({}^{3}\) & NA & NA \\ \hline \hline \end{tabular} Note: “NA” means not applicable.  
\end{table} 
Table 2: Statistics of soil properties in example I.  
Figure 1: Slope geometry and deterministic slope stability analysis results for example I. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline Clay 1 & SQEF & SQEF & SQEF & SQEF & SQEF & SQEF & SEF & CSF & SMF & BNF \\ Clay 2 & SQEF & SEF & CEF & SMF & BNF & SQEF & SQEF & SQEF \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different cases for the study of the effects of autocorrelation function (ACF). Before directly stepping into the analysis of the effects of ACFs on slope reliability analysis, a visualization comparison of the random fields simulated based on different combinations of ACFs was conducted. Figure 3 plots the typical realizations of random fields for the nine cases, as well as the corresponding slope stability results in terms of CSS and FS. The random fields simulated for different layers are denoted by different colors. For the random field of a specific soil layer, the dark color indicates a large value of the shear strength value and vice versa. It was observed that when ACF is changed, the simulated random fields for the slope is changed accordingly, as well as the corresponding slope stability results. The random fields generated by SQEF and SMF show relatively smoother variation than those obtained from SEF, CEF, and BNF, which shows good consistency with previous studies that assume the same ACF for different soil layers [8]. Compared with the baseline case (case 1), the slope stability results for cases 2-5 where the ACF in the upper layer is respectively taking SEF, CEF, SMF, and BNF remain nearly unchanged whereas the results for cases 6-9 with the ACF in the lower layer respectively taking SEF, CEF, SMF, and BNF have changed to some extent.