## 3 Numerical Examples
### Boundary Randomness
#### 3.2.3 Effect of Permeability Coefficient on Saturated Sedimentary Valley with Random Boundary  
In this section, the ground motion of characteristic points of a saturated sedimentary valley under the change of permeability coefficient (finite value and infinity case) is calculated, and its influence on the displacement amplitude of the bedrock point and sedimentary point is obtained. For the permeability coefficient, the case of \(k=0.1,\ 0.5,\ 1,\ 5,\ \infty\) is calculated in this paper (unit: m/s). Several typical cases of permeability coefficient \(k=0.1,\ 0.5,\ \infty,\ \eta=0.5,\ 1,\ 2\) are given in Figures 10 and 11. The horizontal and vertical surface displacement amplitudes of the random boundary under vertical incidence is shown in Figure 10. The horizontal and vertical coefficients of variation of random boundaries at different incident angles is shown in Figure 11.  
As shown in Figure 10, under the incidence of the SV wave, the surface displacement response of a saturated sedimentary valley demonstrates a correlation with the permeability coefficient, incident frequency, incident angle, and other factors. When the permeability coefficient \(k>1\), the displacement amplitude is close to the pervious case, with the increase in the permeability coefficient, the amplitude slightly changes. Compared with other permeability coefficients, the confidence interval of the surface displacement amplitude of the surface point with \(k=0.1\) is wider, that is, the permeable boundary has a greater influence on the surface displacement amplitude of the random boundary. Moreover, the position of the larger confidence interval of the displacement amplitude basically corresponds to the peak position of the amplitude mean value. Within the sedimentary valley, there is the same rule. On the whole, whether in the bedrock half-space or in the sedimentary valley, the influence of the impermeable case on the displacement amplitude of the random boundary is greater than that in the permeable case. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \multicolumn{2}{c}{\(0^{\circ}\)} & \multicolumn{2}{c}{\(30^{\circ}\)} \\ \(\eta\) & \(n_{c}\) = 0.1 & \(n_{c}\) = 0.2 & \(n_{c}\) = 0.3 & \(n_{c}\) = 0.1 & \(n_{c}\) = 0.2 & \(n_{c}\) = 0.3 \\ \hline
0.25 & 0.06 & 0.09 & 0.12 & 0.07 & 0.12 & 0.14 \\
0.5 & 0.15 & 0.22 & 0.21 & 0.08 & 0.16 & 0.20 \\
1 & 0.39 & 0.37 & 0.30 & 0.64 & 0.52 & 0.61 \\
1.5 & 0.80 & 0.78 & 0.62 & 0.75 & 0.61 & 0.71 \\
2 & 0.52 & 0.41 & 0.41 & 0.34 & 0.30 & 0.27 \\ \hline \end{tabular}
\end{table} 
Table 4: Mean value of variation coefficient of horizontal displacement amplitude (with sediment porosity). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \multicolumn{2}{c}{\(0^{\circ}\)} & \multicolumn{2}{c}{\(30^{\circ}\)} \\ \(\eta\) & \(n_{c}\) = 0.1 & \(n_{c}\) = 0.2 & \(n_{c}\) = 0.3 & \(n_{c}\) = 0.1 & \(n_{c}\) = 0.2 & \(n_{c}\) = 0.3 \\ \hline
0.25 & 0.33 & 0.35 & 0.34 & 0.05 & 0.06 & 0.07 \\
0.5 & 0.23 & 0.30 & 0.27 & 0.20 & 0.18 & 0.18 \\
1 & 0.41 & 0.37 & 0.37 & 0.32 & 0.27 & 0.26 \\
1.5 & 1.32 & 1.42 & 1.26 & 1.42 & 1.05 & 1.39 \\
2 & 1.10 & 0.99 & 1.46 & 0.74 & 0.78 & 0.69 \\ \hline \end{tabular}
\end{table} 
Table 5: Mean value of variation coefficient of vertical displacement amplitude (with sediment porosity). As shown in Figure 11, the coefficient of variation at the boundary point of the sedimentary valley \(\left(\frac{x}{\delta}=\pm 1\right)\) obviously increases to form a wave peak, and this phenomenon becomes more significant with the increase in dimensionless frequency, which is consistent with the law of significant amplification at the corner of the high-frequency sedimentation valley, indicating that under the incidence of high-frequency waves, the influence of boundary randomness on the seismic response at the edge of the sedimentary valley is also very prominent. The influence of the change of permeability coefficient on the location of sedimentary domain is greater than that of bedrock.