## 4 Discussion  
Besides five commonly used classical ACF models, the non-classical ACF model, which contains two or more parameters, is usually used to fit two sample path features, such as the SOF and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case No.** & **Clay** & **Weak Seam** & \(P_{f}\) \\ \hline
1 & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=2\) m & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & \(1\times 10^{-4}\) \\
2 & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=8\) m & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & \(4.7\times 10^{-3}\) \\
3 & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=2\) m & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & \(3\times 10^{-4}\) \\
4 & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=8\) m & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & \(1.22\times 10^{-2}\) \\
5 & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=2\) m & \(1\times 10^{-3}\) \\
6 & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=8\) m & \(1\times 10^{-3}\) \\
7 & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=2\) m & \(3.2\times 10^{-3}\) \\
8 & CEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=4\) m & SEF: \(\delta_{\text{h}}=20\) m, \(\delta_{\text{v}}=8\) m & \(3.2\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Reliability results for different situations for example II. Figure 11: Probability of failure for various cases for example II. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Case** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline Clay & SEF & SEF & SEF & SEF & SEF & CEF & SOEF & SMF & BNF \\ Weak seam & SEF & CEF & SQEF & SMF & BNF & SEF & SEF & SEF & SEF \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Case study. smoothness [13; 44]. [PERSON] [44] adopted the Whittle-Matern (W-M) and powered exponential (PE) models to represent the real soil data and verified that the smoothness parameter has an effect on \(P_{f}\) for some geotechnical problems. To reveal whether the non-classical ACF model would affect the reliability of the layered slope, the W-M model, which has been advocated as a general model for soil variogram [45], was used during the realizations of random fields. The Whittle-Matern model can be expressed as follows [44; 46; 47]:  
\[\rho(\tau)=\frac{2}{\Gamma(v)}\left(\frac{\sqrt{\pi}\cdot\Gamma(v+0.5)\cdot \tau}{\Gamma(v)\cdot\delta}\right)^{v}K_{v}\left(\frac{2\sqrt{\pi}\cdot\Gamma( v+0.5)\cdot\tau}{\Gamma(v)\cdot\delta}\right), \tag{3}\]  
where \(v\) is the smoothness parameter; \(\Gamma(\cdot)\) denotes the Gamma function; \(K_{v}\) denotes the modified Bessel function of the second kind with order \(v\); and \(\tau\) is the lag distances between two points. Details for constructing a two-dimension random field can be found in [PERSON] and Phoon [13]. The SEF, SMF, and SQEF are the special case of the W-M model with \(v=0.5\), \(1.5\) and \(\infty\), respectively [13; 44]. For illustration, the slope in example I was utilized to investigate the effect of \(v\) on the reliability analysis of the layered slopes. It is noted that the anisotropy of the smoothness was not considered here, which indicates that the smoothness parameters in the horizontal and vertical directions are the same within the realization of the model. Figure 12 plots the typical realization of the random fields with different \(v\). It can be observed that the random fields with a low \(v\) value (Figure 12a,b) show a relatively rough distribution of the undrained shear strength. With the increase of \(v\), the random fields become smoother (Figure 12c-e), which may be more suitable to represent the natural property of soils.