## 2 Materials and Methods
### Methods
#### 2.2.1 The Permanent Displacement Analysis
The formula is as follows (Figure 5) [37]:  
\[a_{d}=a_{E}\cos\delta\cos\varphi_{s}-a_{N}\cos\delta\sin\varphi_{s}-a_{V}\sin\delta \tag{4}\]  
\[a_{s}=a_{E}\sin\varphi_{s}-a_{N}\cos\varphi_{s} \tag{5}\]  
\[a_{n}=a_{E}\sin\delta\cos\varphi_{s}-a_{N}\sin\delta\sin\varphi_{s}+a_{V}\sin\delta \tag{6}\]  
The slider generates inertial force with the ground movement, if not, consider inertia, and the slider and ground acceleration directions are the same when ground acceleration decreases or moves in the opposite direction (Figure 6). However, the slider and ground acceleration directions are opposite when considering inertia (Figure 7). The inertia force drives the slider movement, which helps produce displacement between the slider and the ground. The safety coefficients of the slider with and without considering the inertia force are as follows:  
\[F_{s}=\left[\mu_{s}(g\cos\delta+a_{n})+\frac{cA}{m}\right]/(g\sin\delta-a_{d}) \tag{7}\]  
\[{F_{s}}^{*}=\left[\mu_{s}(g\cos\delta+a_{n})+\frac{cA}{m}\right]/(g\sin\delta+ a_{d}) \tag{8}\]  
Figure 4: Force model of the sliding block in critical equilibrium before the earthquake [16]. \[D=\iint\limits_{0}^{t}\Bigl{[}a_{(t)}-a_{c}g\Bigr{]}dt^{2} \tag{2}\]0 Schematic diagram of force decomposition on the potential sliding surface, (**a**) projection of horizontal acceleration along the cross-sectional direction, (**b**) projection of horizontal and vertical accelerations along the vertical and downslope directions, (**c**) seismic loading on the potential sliding surface [38]. \[D=\iint\limits_{0}^{t}\Bigl{[}a_{(t)}-a_{c}g\Bigr{]}dt^{2} \tag{2}\]1 Schematic diagram of the acceleration direction of the slider and the base considering the non-inertia (**a**) and inertia forces (**b**) [39]. As defined by [PERSON], the slider is stable when the sliding force is less than the resistance force, i.e., \(F_{\text{s}}\) > 1. When \(F_{\text{s}}\) < 1, the slider will be destabilized along the sliding surface. Assuming that the slider is rigid, the acceleration along the sliding surface is calculated by the three directional acceleration components. The calculation formulas of acceleration with or without considering inertia force are as follows:  
\[D=\iint\limits_{0}^{t}\Bigl{[}a_{(t)}-a_{c}g\Bigr{]}dt^{2} \tag{2}\]2  
\[D=\iint\limits_{0}^{t}\Bigl{[}a_{(t)}-a_{c}g\Bigr{]}dt^{2} \tag{2}\]3  
When \(S(S^{\ast})\) < 0, the slider is at rest, and when \(S(S^{\ast})\) > 0, the slider moves. According to [PERSON] and [PERSON] [40], \(S(S^{\ast})\) is used instead \(\left[a_{(t)}-a_{c}g\right]\) in Equation (2) to obtain the permanent displacement induced by the earthquake. The DGB landslide was generalized to a wedge-shaped slider of about 400 m height, 600 m width and 2200 m length (Figure 8). The orientation of the rock strata is N88\({}^{\circ}\)W/NE\(\angle\)32\({}^{\circ}\), and the apparent dip angle is calculated to be 17\({}^{\circ}\). The landslide is primarily composed of carbonate rock, with the sliding plane being dominated by rigid dolomite, a type of hard rock. The downward acceleration and permanent displacement calculated by considering and ignoring vertical acceleration are \(S\), \(SI\), \(D\) and \(DI\), respectively. Under the ignored inertial frame reference, the downward acceleration and permanent displacement calculated by considering and ignoring vertical acceleration are \(S^{\ast}\), \(SI^{\ast}\), \(D^{\ast}\) and \(DI^{\ast}\), respectively.