## 2 Reliability Analysis of Spatially Varied Soil Slopes
### Monte Carlo Simulation for Slope Reliability Analysis  
Regarding the slope reliability analysis considering soil spatial variability, several methods are available in the literature, such as the random limit equilibrium method (RLEM) [35; 36], random finite element method (RFEM) [37; 38; 2], and random smoothed particle hydrodynamics (RSPH) [39]. In this study, the RLEM was adopted because it is conceptually simple and efficient. RLEM generally proceeds with a limit equilibrium method (LEM) within the framework of Monte Carlo simulation \begin{table}
\begin{tabular}{l c} \hline \hline
**ACF Type** & **ACF Expression** \\ \hline Squared exponential function (SQEF) & \(\rho\Big{(}\tau_{x},\tau_{y}\Big{)}=\exp\Big{[}-\pi\Big{(}\frac{\tau_{x}^{2}}{ \delta_{h}^{2}}+\frac{\tau_{y}^{2}}{\delta_{v}^{2}}\Big{)}\Big{]}\) \\ Single exponential function (SEF) & \(\rho\Big{(}\tau_{x},\tau_{y}\Big{)}=\exp\Big{[}-2(\frac{\tau_{x}}{\delta_{h}}+ \frac{\tau_{y}}{\delta_{v}})\Big{]}\) \\ Cosine exponential function (CEF) & \(\rho\Big{(}\tau_{x},\tau_{y}\Big{)}=\exp\Big{[}-(\frac{\tau_{x}}{\delta_{h}}+ \frac{\tau_{y}}{\delta_{v}})\Big{]}\) \\ Second-order Markov function (SMF) & \(\rho\Big{(}\tau_{x},\tau_{y}\Big{)}=\exp\Big{[}-4(\frac{\tau_{x}}{\delta_{h}}+ \frac{\tau_{y}}{\delta_{v}})\Big{]}(1+\frac{4\tau_{x}}{\delta_{h}})\Big{(}1+ \frac{4\tau_{y}}{\delta_{v}}\Big{)}\) \\ Binary noise function (BNF) & \(\rho\Big{(}\tau_{x},\tau_{y}\Big{)}=\left\{\begin{array}{ll}\big{(}1-\frac{ \tau_{x}}{\delta_{h}}\big{)}\big{(}1-\frac{\tau_{x}}{\delta_{v}}\big{)}&\text{ for}\tau_{x}\leq\delta_{h}\text{and}\tau_{y}\leq\delta_{v}\\ 0&\text{otherwise}\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Commonly used two-dimensional classical autocorrelation functions (ACFs) (adapted from [8]). (MCS), where LEM is repeatedly performed for vast amounts of random field and/or variable samples to obtain the corresponding slope stability results, for example, the factor of safety (FS) and critical slip surface (CSS). Note that the SLOPE/W [40], which has been verified as an effective and efficient tool for slope stability analysis [8; 15; 36], was utilized to perform the LEM analysis here. An inhouse MATLAB code, however, was used to simulate random fields and conduct MCS. Each of the samples for a soil property was generated from its probability distribution function (PDF) and correlation structure using the suggested method described in Section 2.2. The variability and non-linearity of soil properties are therefore implicitly considered by the RLEM [3]. With the MCS, the statistics of FS, such as the mean, standard deviation, and distribution of the probability, can be easily obtained. The \(P_{f}\), which is defined as the probability that the FS is less than the unity, is then calculated as:  
\[P_{f}=\frac{1}{N_{mcs}}\sum\
olimits_{i=1}^{N_{mcs}}\mathrm{I}(FS_{i}<1), \tag{1}\]  
where \(N_{mcs}\) is the number of MCS samples; \(FS_{i}\) is the FS for the \(i\)th MCS sample; and \(\mathrm{I}(\cdot)\) is an indicator function, which is equal to one when the FS is less than the unity and zero otherwise. To ensure the number of MCS samples is adequate to obtain a relatively precise result, the coefficient of variation of \(P_{f}\), \(COV_{P_{f}}\), is taken as the indicator to validate the accuracy of the \(P_{f}\), which is expressed as:  
\[COV_{P_{f}}=\sqrt{\frac{1-P_{f}}{N_{mcs}P_{f}}}. \tag{2}\]  
It can be seen from Equation (2) that the accuracy of the \(P_{f}\) is inversely proportional to the number of MCS samples \(N_{mcs}\). A small value of \(COV_{P_{f}}\) indicates a good accuracy of the estimated \(P_{f}\), and vice versa. As such, it is often faced with a dilemma of choosing between accuracy and efficiency in practical reliability analysis because \(N_{mcs}\) is closely related to the computation efficiency. A reasonable value of the \(COV_{P_{f}}\) is often desired based on the project demand [41].