## 3 Numerical Examples
### Media Randomness
#### 3.1.2 Effect of Porosity on Saturated Sedimentary Valleys  
One hundred groups of random sedimentary porosity samples are generated by the Monte Carlo method, which obeys the normal distribution with mean value of 0.3 and coefficient of variation of 0.03. The dimensionless frequency \(\eta\) is defined as the ratio of the width of sedimentary valley 2a to the incident wavelength, \(\eta=2a/\lambda=\omega a/c\pi\).  
By changing the sediment porosity, the displacement response of the random samples is solved, and the mean value \(\mu\), standard deviation \(\sigma\), and variation coefficient \(\sigma/\mu\) of the surface displacement response amplitude of the saturated sediment valley under the incident plane SV wave are calculated. The confidence interval \(\mu\pm\sigma\) and variation coefficient are used to analyze the impact of porosity change on the valley site response.  
The surface response displacement amplitudes \(\mu\) and \(\mu\pm\sigma\) of the samples are shown in Figure 3, when the porosity is random while the permeability coefficient is constant (\(k=\infty\)). The solid line and the dash-dot lines represent \(\mu\) and \(\mu\pm\sigma\), respectively. The coefficients of variation of horizontal and vertical displacement amplitudes of the random samples are presented in Figure 4. Abscissa \(x/a\) represents the horizontal distance between the observation point of the surface and the center of the sedimentary surface, in unit of  
Figure 2: Sedimentary valley model (m).  
1. The ordinate represents the ratio of the amplitude of the surface displacement to the amplitude of the incident wave, in unit of 1.  
The response characteristics of surface displacement in saturated sedimentary valleys show multifactorial correlations with porosity, incident frequency, angle, incident position, and other factors. As shown in Figure 3, the confidence interval of the ground displacement amplitude in the outer half-space of the deposit is narrow, while that in the interior of the deposit is wide, indicating that the change of porosity in the sedimentary domain has a great influence on the surface response inside the deposit. It is consistent with the result of the significant increase in the coefficient of variation of the sedimentary section in Figure 4. The coefficient of variation reflects the discrete degree of the amplitude of surface displacement.  
As can be seen in Table 1, with the dimensionless frequency of the incident wave from 0.5 to 2, the coefficient of variation increases significantly, and the mean value increases by 35.87 times at most (vertical). The results show that the randomness of porosity has a large effect on the displacement at high frequency, which accords with the fluctuation law. In this paper, the horizontal direction is denoted by (H), and the vertical direction is indicated by (V). When the incident angle is different, the spatial variation of the variation coefficient of surface displacement amplitude is significantly different. On the whole, the influence on the interior of the sediment is more obvious, indicating that the influence of incident angle on seismic wave scattering cannot be ignored.  
Figure 3: Effect of porosity randomness on surface displacement amplitude in sedimentary river valleys (\(\theta=0^{\circ}\)). (**a**) Horizontal displacement; (**b**) Vertical displacement.