## 3 Numerical Examples
### Media Randomness
#### 3.1.3 Effect of Permeability Coefficient on Saturated Sedimentary Valleys  
The formation permeability coefficient determines the coupling strength between pore water and soil skeleton, which is one of the key parameters of formation-structure dynamic interaction. One hundred groups of sedimentary permeability coefficient samples are randomly generated by the Monte Carlo method, which obeys the normal distribution with a mean value 0.3 m/s and coefficient of variation 0.1, and other data are consistent with the previous section. The ground motion of the saturated sedimentary valley under the change of permeability coefficient (finite value) is calculated, and its influence law on displacement amplitude is obtained, so as to study the influence mechanism of the permeability coefficient. The surface response displacement amplitudes \(\upmu\) and \(\mu\pm\sigma\) of 100 random samples when the permeability coefficient is random and the porosity is constant \((n=0.1,\ n_{c}=0.3)\) is shown in Figure 5. \(n\) and \(n_{c}\) represent the porosity of bedrock and sediment, respectively. The coefficient of variation of the amplitude of the ground displacement response of the sample is shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(\eta\)} & \multicolumn{2}{c}{**Horizontal**} & \multicolumn{2}{c}{**Vertical**} \\ \cline{2-5}  & \(\mathbf{0}^{\circ}\) & \(\mathbf{30}^{\circ}\) & \(\mathbf{0}^{\circ}\) & \(\mathbf{30}^{\circ}\) \\ \hline
0.5 & 0.0050 & 0.0076 & 0.0069 & 0.0096 \\
1 & 0.0076 & 0.0191 & 0.0312 & 0.0467 \\
2 & 0.0885 & 0.0785 & 0.2475 & 0.1088 \\ \hline \end{tabular}
\end{table} 
Table 1: Average variation coefficient of horizontal and vertical displacement amplitude (random porosity). Figure 4: Coefficient of variation of surface displacement amplitude values for 100 sets of porosity samples. **(a)** Coefficient of variation of horizontal displacement; **(b)** Coefficient of variation in vertical displacement. Figure 5: Effect of permeability coefficient randomness at different incident angles on the surface displacement amplitude in sedimentary river valleys (\(\theta=0^{\circ}\)). (**a**) Horizontal displacement; (**b**) Vertical displacement. Figure 6: Coefficient of variation of surface displacement amplitude values for 100 sets of permeability coefficient samples. (**a**) Coefficient of variation of horizontal displacement; (**b**) Coefficient of variation in vertical displacement. Different from the effect of porosity randomness, the influence of the randomness of permeability coefficient on the displacement amplitude of the saturated sedimentary valley increases significantly. Comparing Figures 4 and 6, it can be seen that the influence of the permeability coefficient randomness on seismic wave scattering in sedimentary valley is greater than that of the porosity randomness. It can be seen from Figure 5 that the \(\pm\sigma\) confidence interval width of the surface displacement amplitude in the sedimentary valley is slightly larger than that in the outer half-space, which is consistent with the larger coefficient of variation of the surface displacement in the sediment in Figure 6. It indicates that the randomness of permeability coefficient has a more significant effect on sediment, while it has less effect on the half-space response. The incident angle still affects the spatial variation of the amplitude variation coefficient of surface displacement. When the seismic wave is obliquely incident, the variation coefficient of the left (\(x/a<0\)) surface displacement amplitude is obviously higher than that of the vertical incidence, and the change is more obvious in the half-space as a whole, and especially the difference of the vertical displacement variation coefficient distribution under the incident SV wave of \(\eta=2\) is more significant.