## 4 Scarp Topography Site
### Modal Truncation Error for Time-History Analysis  
As shown in Figure 8, points A, B, C, D, E, and F near the step are taken as the research objects. Points C and D are the toe and crest of the slope, respectively; points A and B are at the distances of \(2h\) and \(h\) from the toe, respectively; and points E and F are at the distances of \(h\) and \(2h\) from the crest, respectively. Three seismic waves listed in Table 1 are selected as  
Figure 8: Finite element model of scarp topography site. Figure 7: Mean cumulative errors vary with \(N\). the horizontal seismic inputs. The response obtained by the 1000-th mode is considered the exact solution, and the cumulative error varying with \(N\) is shown in Table 4. Table 4 also presents partial frequencies and modal truncation indexes, where \(e_{ai}\) (\(i\) = A, B, \(\ldots\), F) is the error of cumulative modal contribution factor of acceleration; \(e_{ai}\) (\(i\) = A, B, \(\ldots\), F) is the error of cumulative modal contribution factor. Figures 9 and 10 show the mean cumulative error of displacement, velocity, and acceleration responses obtained by the central difference method varying with different numbers of mode. From Table 4 and Figures 9 and 10, the following can be seen:  
1. If the threshold is set to 10% for modal truncation, the mode number selected based on \(e_{m}\) is 63, the mode number selected based on \(e_{uA}\), \(e_{uB}\), \(e_{uC}\), \(e_{uD}\), \(e_{uE}\), \(e_{uF}\) are 22, 63, 63, 22, 22, 22, respectively, and the mode number selected based on \(e_{aA}\), \(e_{bB}\), \(e_{aC}\), \(e_{aD}\), \(e_{aE}\), \(e_{aF}\) are 317, 277, 194, 135, 100, and 135, respectively. 2. For displacement response, when \(N\geq 63\), except for \(u_{B}\) whose calculation error is slightly greater than 10%, the cumulative error of displacement at all other points is less than 10%, indicating that the mode number selected based on modal participation mass for displacement response analysis is reasonable. The required mode numbers for \(\bar{u_{i}}\) (\(i\) = A, B, \(\ldots\), F) are 243, 243, 243, 243, 194, and 277 respectively. The truncation numbers based on the \(e_{ai}\) are much larger than the \(e_{m}\) and \(e_{ui}\). This demonstrates once again that the cumulative modal contribution factor of acceleration should be used as the modal truncation criterion for acceleration response. 3. The cumulative error of acceleration is the largest, followed by velocity, and the smallest is displacement. 4. The cumulative errors of displacement, velocity, and acceleration at the lower step are all greater than those at the upper step. The reason is that the contributions of higher modes to the lower step are larger than to the upper step. For the points A, B, and C on the lower step, 277 modes obtained by \(e_{aB}\) would make the cumulative error at all points less than 10%. Therefore, the modal contribution factor of acceleration at the position with a distance of one step height from the toe of the slope at the lower step can be used as the basis for modal truncation. 5. When \(N\geq 277\), the velocity and acceleration response errors obtained by the central difference method are less than 10%, which indicates that taking the cumulative modal contribution factor of acceleration as the basis for modal truncation, and calculating acceleration by the central difference method, shows high accuracy. Figure 9: Mean cumulative error curves of displacement of scarp topography site. Figure 10 shows that the maximum error occurs at point B. Figure 11 shows the acceleration time history and Fourier spectra at point B obtained by the exact solution and by the central difference method through the use of the displacement based on 277 modes. The exact solution and the central difference approximate solution are almost coincident, indicating that after obtaining high-precision displacement response, the acceleration response obtained by using the central difference method has very high accuracy.