## 2 Calculation Model and Method
### Calculation Method
#### 2.2.2 IBEM Method  
According to IBEM and elastic wave theory, the total displacement \(u^{(t)}\) of any point in the plane is obtained as follows:  
\[u^{(t)}=u^{(f)}+u^{(s)} \tag{5}\]  
where \(u^{(f)}\) represents the free field displacement at any point, and \(u^{(s)}\) is the scattered wave field displacement constructed by a virtual wave source. The free field displacement is calculated by using the wave reflection principle. When the SV wave is obliquely incident, three kinds of reflected waves (P\({}_{1}\) wave, P\({}_{2}\) wave, and SV wave) are generated on the ground reflector in the saturated half-space. Then the  
Figure 1: Calculation model. potential function expressions of the above three waves in a Cartesian coordinate system are as follows:  
\[\phi_{1}^{r}=a_{1}\mathrm{e}^{-\mathrm{i}k_{\alpha 1}(xsin\theta_{\alpha 1}+ycos \theta_{\alpha 1})} \tag{6}\]  
\[\phi_{2}^{r}=a_{2}\mathrm{e}^{-\mathrm{i}k_{\alpha 2}(xsin\theta_{\alpha 2}+ycos \theta_{\alpha 2})} \tag{7}\]  
\[\psi^{r}=b_{1}\mathrm{e}^{-\mathrm{i}k_{\beta}(xsin\theta_{\beta}+ycos \theta_{\beta})} \tag{8}\]  
where \(a_{1},a_{2},b_{1}\) represent the amplitude coefficient, while \(k_{\alpha 1},k_{\alpha 2},k_{\beta}\) represent the wave number. According to the relationship between the potential function and displacement, free field soil skeleton displacement \(u_{i}^{\mathrm{(f)}}\), fluid relative displacement \(w_{i}^{\mathrm{(f)}}\) and pore water pressure \(p^{\mathrm{(f)}}\) can be expressed as follows:  
\[u_{x}^{\mathrm{(f)}}=-\mathrm{i}k_{\alpha 1}\Big{(}\phi_{1}^{\mathrm{(r)}}+ \phi_{2}^{\mathrm{(r)}}\Big{)}sin\theta_{\alpha 1}+ik_{\beta}\Big{(}\psi^{ \mathrm{(i)}}-\psi^{\mathrm{(r)}}\Big{)}cos\theta_{\beta} \tag{9}\]  
\[u_{y}^{\mathrm{(f)}}=\mathrm{i}k_{\beta}\Big{(}\psi^{\mathrm{(i)}}+\psi^{ \mathrm{(r)}}\Big{)}sin\theta_{\beta}-ik_{\alpha 1}\phi_{1}^{\mathrm{(r)}}cos \theta_{\alpha 1}-\mathrm{i}k_{\alpha 2}\phi_{2}^{\mathrm{(r)}}cos\theta_{\alpha 2} \tag{10}\]  
\[w_{x}^{\mathrm{(f)}}=-\mathrm{i}k_{\alpha 1}\Big{[}\chi_{1}\phi_{1}^{\mathrm{(r)} }+\chi_{2}\phi_{2}^{\mathrm{(r)}}\Big{]}sin\theta_{\alpha 1}+ik_{\beta}\chi_{3} \Big{(}\psi^{\mathrm{(i)}}-\psi^{\mathrm{(r)}}\Big{)}cos\theta_{\beta} \tag{11}\]  
\[w_{y}^{\mathrm{(f)}}=\mathrm{i}k_{\beta}\Big{[}\chi_{3}\Big{(}\psi^{\mathrm{(i )}}+\psi^{\mathrm{(r)}}\Big{)}\Big{]}sin\theta_{\beta}-\mathrm{i}k_{\alpha 1} \chi_{1}\phi_{1}^{\mathrm{(r)}}cos\theta_{\alpha 1}-\mathrm{i}k_{\alpha 2}\chi_{2} \phi_{2}^{\mathrm{(r)}}cos\theta_{\alpha 2} \tag{12}\]  
\[p^{\mathrm{(f)}}=(\alpha+\chi_{1})Mk_{\alpha 1}^{2}\phi_{1}+(\alpha+\chi_{2})Mk_{ \alpha 2}^{2}\phi_{2} \tag{13}\]  
When the sedimentary valley exists, the scattered waves will be generated in the half-space and inside the deposit. The displacement, stress of soil skeleton and pore water pressure can be expressed by the following formula. Figure 1:0  
\[w_{i}^{\mathrm{(s)}}(x)  
sampling. It can be used to solve nonlinear, parametric excitation, and other random problems, and is widely used ([PERSON], [PERSON] et al. [43], [PERSON], [PERSON] et al. [44]). For the medium randomness, the Monte Carlo method is adopted to randomly generate multiple groups of media samples with different porosity and permeability coefficients for saturated sedimentary valleys. The seismic response of complex sites under different sample conditions is calculated based on the IBEM. The results are compared for statistical analysis to reveal the influence of media randomness on seismic response. For the boundary randomness, the previous studies are mostly based on regular boundary models, with which it is difficult to reflect the effect of irregular terrain on ground motion. In this paper, multiple groups of boundary samples are randomly generated by using the Monte Carlo method. The IBEM is used to calculate the seismic response of the samples for simulating the influence of boundary randomness on ground motion. A general conclusion is given to solve various problems of random boundary seismic response in engineering.