## 4 Discussion  
An efficient method for calculating slope failure risk based on element failure probability is proposed in this paper, which uses \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) to judge the instability of slope, and uses velocity field obtained via the UBM to locate each failure element. The problem of solving the failure probability and failure risk coefficient of each failure mode is transformed into the problem of solving the element failure probability and failure risk coefficient of discrete elements, which greatly simplifies the calculation process. In addition, according to the definition of limit state, the element failure function is established based on \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\)[18]. It can be observed from the calculation process of \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) (as shown in Figure 1) that \(k_{F}(t_{m})\) and \(k_{\gamma}(t_{m})\) are only solved once for large-scale linear programming problems, while \(k_{m}(t_{m})\) usually requires solving \(k_{F}(t_{m})\) and \(k_{\gamma}(t_{m})\) many times and therefore the failure risk computational efficiency based on \(k_{F}(t_{m})\) and \(k_{\gamma}(t_{m})\) are greatly improved.  
The analysis of the two cases shows that the results obtained via the UBM based on \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) are very close, with a low rate of error (<5%). Theoretically, the slope failure risk coefficient under each sample should be consistent; that is, the obtained results should be equal. However, load distribution type, action form and calculation accuracy may affect the results. When using Geo-studo software for failure risk comparative analysis, the critical sliding surface obtained from the mean value of soil shear strength parameters is used to analyze all samples [5]. In addition, the LEM only considers one-dimensional spatial variability of the sliding surface and therefore its calculation results are worthy of discussion [20]. The most accurate result can be obtained by 8-node FEM based on OptumG2 on the basis of the known soil constitutive relationship. However, unfortunately, the soil constitutive relationship is complicated and the failure mode screening involves many human factors [4; 17]. It should be noted that the UBM computation program in this paper can implement parallel computation in Python and therefore the computation efficiency can be greatly improved.  
The proposed method is mainly aimed at soil slope, but the failure of rock slope is mainly affected by the cohesion, friction angle and bulk density of the structural plane [4; 35]. The study of efficient rock slope failure risk analysis will be the focus of subsequent work. In addition, earthquake and groundwater are the main causes of slope instability [5; 21; 27] and therefore it is also urgent to study efficient failure risk analysis methods under the action of earthquake and groundwater.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Method**} & \(\mathbf{R_{k}^{M-IFP}}\) **by Equation (14)** & \(\mathbf{R_{k}^{M-EFP}}\) **by Equation (20)** \\ \hline \multirow{3}{*}{UBM} & \(M=k_{F}\) & 268,403.17 & 268,403.17 \\  & \(M=k_{\gamma}\) & 276,016.29 & 276,016.29 \\  & \(M=k_{m}\) & 286,369.00 & 286,369.00 \\ \hline \multicolumn{1}{c}{LEM} & 345,574.31 & / \\ \hline \multicolumn{1}{c}{FEM} & 270,389.08 & / \\ \hline \hline \end{tabular}
\end{table}
Table 8: Statistics of heterogeneous slope failure risk.