## 3 Calibration and Application
### Case 1: A Homogeneous Slope
#### 3.1.2 Result of Homogeneous Slope
From the perspective of calculation efficiency, the above three methods in this paper can carry out large-scale parallel calculation. However, it required 0.52 h (about 20.8 h for single thread calculation), 0.53 h (about 21.2 h for single thread calculation) and 4.16 h (about 166.4 h for single thread calculation), respectively, to calculate the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) of 1000 random field samples. It required 28.4, 29.1 and 127.2 h, respectively, to calculate the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) using the FEM method for the same sample. It is clear that the calculation efficiency of slope reliability analysis based on \(k_{F}(t_{m})\) and \(k_{\gamma}(t_{m})\) is much higher than that \(k_{m}(t_{m})\). The main reason is that solving the \(k_{m}(t_{m})\) requires repeated calculation of \(k_{F}(t_{m})\) or \(k_{\gamma}(t_{m})\) and judging whether the slope is in the limit state, as shown in Figure 1, and therefore the calculation efficiency is low. 3. Figures 3 and 4 reflect the process of gradually increasing the top load and bulk density to cause the slope to reach the limit state. It can be observed intuitively that with the gradual increase of top load \(P\) and bulk density \(\gamma\), the slope stability coefficient obtained by the UBM, LEM and FEM respectively shows a decreasing trend and the results obtained by the UBM and FEM are close to each other and are both greater than the results calculated by the LEM. The main reason is that the LEM is based on the assumed sliding surface for slope stability analysis. It only considers the one-dimensional soil shear strength parameter random field on the critical sliding surface and therefore there is a significant gap between the results obtained by LEM and those obtained by the UBM and FEM. 4. Figure 5 provides the histogram of \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\). It can be intuitively observed that \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) all present the characteristics of lognormal distribution that are high in the center and low on both sides. A MATLAB software program titled Geo-studio was prepared and the slope stability was calculated with the Bishop method of slope/W module. Only one slope failure mode existed by default, depicted by the blue line in Figure 6; the slope stability was calculated with the 8-node FEM method based on OptumG2. Only three slope failure modes existed, depicted by the black line in Figure 6. The most significant advantage of the proposed method is that it can capture all slope failure modes simultaneously. By calculating 1000 groups of slope random field samples, a total of three slope failure modes are obtained, as depicted by the pink area in Figure 6. The histogram of the slope failure area is shown in Figure 7. Table 3 lists the statistics of slope failure modes by IFP method. Table 4 lists the statistics of the slope failure risk. The calculation results show that:  
1. Failure mode 1 is obtained by the UBM, LEM and FEM; however, the other two failure modes are not obtained by the LEM. This is because the LEM calculates slope stability under random samples of shear strength parameter based on the pre-assumed critical sliding surface and therefore its failure mode is not a real failure mode. The FEM obtained similar calculation results as the proposed method; however, the FEM method needed to build a finite element model satisfying the static equilibrium condition, strain compatibility condition and constitutive relation of soil mass when