## 3 Numerical Examples
### Boundary Randomness
#### 3.2.2 Effect of Porosity of Sedimentary Domain on Saturated Sedimentary Valley with Random Boundary  
Porosity is a factor affecting the scattering of elastic waves in saturated sedimentary valleys. Based on [PERSON]'s theory and the IBEM, this section solves the displacement response of random boundary samples by changing the porosity of the half-space. Comparing the results of elastic sedimentary valleys with those of saturated sedimentary valleys, the necessity of considering the saturated sedimentary valleys and its influence on the amplitude of random boundary surface displacement is discussed. Then, by changing the sedimentary porosity, the mean value \(\mu\), standard deviation \(\sigma\) and coefficient of variation \(\sigma/\mu\) of the surface displacement response amplitude of the saturated sedimentary valley under the incident plane SV wave are calculated, and the confidence interval \(\mu\pm\sigma\) and coefficient of variation are used to analyze the influence of the change of porosity on the valley site response under the random boundary. Five dimensionless frequencies of \(\eta=0.25,\ 0.5,\ 1,\ 1.5,\ 2\) are selected to calculate the surface displacement amplitude and coefficient of variation of the random boundary when the bedrock half-space porosity \(n\) is 0.001 and the sedimentary porosity \(n_{\text{c}}\) is 0.1, 0.2, 0.3, respectively. The physical parameters of saturated soils with different porosity are shown in Table 3:  
The images of the amplitude and coefficient of variation of the ground surface at the time of \(\eta=0.5,\ 1,\ 2,\ n_{\text{c}}=0.1,\ 0.2,\ 0.3\) are shown in Figures 8 and 9. In the figure, HDA represents the horizontal surface displacement amplitude. The horizontal surface response displacement amplitudes in the case of vertical incidence is shown in Figure 8. In order to facilitate the comparison, the calculation results of the average radius are also given. The average value of the sample amplitude is consistent with the result of the average radius, which shows that the calculation is reasonable. The coefficients of variation of horizontal displacement amplitudes of the random samples are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Damping** & **Poisson** & **Porosity** & **Critical** & **SV Wave** \\  & **Ratio** & **Ratio** & **Ratio** & **Porosity** & **Velocity (m/s)** \\ \hline Bedrock & 0.02 & 0.25 & 0.001 & 0.36 & 792.6 \\ Overburden & & & 0.1 & 0.36 & 706.3 \\ layer & 0.05 & 0.25 & 0.2 & 0.36 & 599.8 \\  & & & 0.3 & 0.36 & 444.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation parameters of saturated sedimentary valley. Figure 7: Random boundary sedimentary valley samples (m). When the SV wave is incident, the spatial distribution characteristics of displacement are significantly different at different frequencies. As shown in Figure 8, when the dimensionless frequency increases, the change of the amplitude of surface displacement is more complex. When the dimensionless frequency \(\eta\) is from 0.5 to 2, the amplitude of sedimentary displacement is magnified by 4.67 times (\(n_{\rm c}=0.1\)), 5.85 times (\(n_{\rm c}=0.2\)), and 12.52 times (\(n_{\rm c}=0.3\)), respectively. The results show that the change of sedimentary porosity has little effect on the amplitude at low frequency. At high frequency, with the increase in sedimentary porosity, the amplitude of surface displacement increases, and the amplitude of sedimentary displacement oscillates and increases. When the porosity of the sedimentary medium is close to the critical state, the variation of surface displacement is more complex. When porosity \(n_{\rm c}=0.2\) or 0.3, the magnification also increases gradually. The results show that with the increase in frequency and porosity, the confidence interval of displacement amplitude becomes wider and the discreteness becomes larger. Figure 8: Horizontal surface displacement amplitude (\(\theta=0^{\circ}\)). (**a**) \(n_{\rm c}=0.1\); (**b**) \(n_{\rm c}=0.2\); (**c**) \(n_{\rm c}=0.3\).