# Study on the Influence of a Soft Soil Interlayer on Spatially Varying Ground Motions
## 3 Numerical Example 1: Linear Soil Behavior  
### Setup of the Numerical Example  
First, a regular site without soft interlayer is set as a control (see Figure 1a), and the target site in which a soft soil layer lies beneath site k is then set up as shown in Figure 1b. In this figure, \(H\) and \(Cs\) denote the thickness and shear wave velocity of soil layer; \(\mu\) is the Poisson's ratio; \(D^{SI}\), \(H^{SI}\), and \(Cs^{SI}\) denote the buried depth, thickness, and shear wave velocity of the soft interlayer, respectively; \(\zeta\) denotes initial damping and is set to 5%. Because linear soil behavior (i.e., the parameters of soil are constant) is considered in this example, quasi-stationary SVEGMs at sites j and k are generated based on the method introduced in Section 2. The motions on the bedrock are assumed to have the same intensity and frequency contents, and the filtered Tajimi-Kanai model [55] is utilized to describe the PSDF on the bedrock:  
\[S_{g}(\omega)=\frac{\omega^{4}}{\left(\omega_{f}^{2}-\omega^{2}\right)^{2}+ \left(2\omega_{f}\omega\zeta_{f}\right)^{2}}\times\frac{\omega_{g}^{4}+4\zeta _{g}^{2}\omega_{g}^{2}\omega^{2}}{\left(\omega_{g}^{2}-\omega^{2}\right)^{2}+4 \zeta_{g}^{2}\omega_{g}^{2}\omega^{2}}S_{0} \tag{13}\]  
where \(\omega_{g}\) and \(\zeta_{g}\) are the central frequency and damping ratio of the Tajimi-Kanai PSDF, \(\omega_{f}\) and \(\zeta_{f}\) are the central frequency and damping ratio of the high pass filter; \(S_{0}\) is the scaling factor and can be calculated by [56]  
\[S_{0}=\frac{a_{\max}^{2}}{\gamma^{2}\left[\pi\omega_{g}\left(2\zeta_{g}+\frac {1}{2\zeta_{g}}\right)\right]} \tag{14}\]  
where \(a_{\max}\) denotes the PGA value and is set to 0.3 g in this example; \(\gamma\) is the peak factor. In this study, \(\gamma=2.8\), \(\omega_{g}=10\pi\text{rad}/s\), \(\zeta_{g}=0.6\), \(\omega_{f}=0.5\pi\), and \(\zeta_{f}=0.6\) are set. Note that the amplitude of the vertical motion is set to \(2/3\) of that of the horizontal component. It is also notable that the motion on the base rock is assumed to have the same intensity and frequency contents. The spatial coherency between the motions at points j' and k' (bedrock motions) is described by the Sobczyk model [57]:  
\[\gamma_{j^{\prime}k^{\prime}}(i\omega)=\exp(-\beta\omega d_{j^{\prime}k^{\prime}} ^{2}/Ve)\cdot\exp(-i\omega d_{j^{\prime}k^{\prime}}\cos\alpha/Ve)\ \ \ \ \ j^{\prime},k^{\prime}\ \ =1^{\prime},2^{\prime},3^{\prime} \tag{15}\]  
where \(\beta\) = 0.0005 is used herein, which indicates highly correlated motions; \(\alpha\) is the incident angle of the incoming wave to the site; \(Ve\) is the apparent velocity which is related to the bedrock property and the specified incident angle. The Jennings window is employed to model the temporal non-stationarity of simulated ground motions, which has the following form:  
\[a(t)=\left\{ \begin{array}{cc}(t/t_{1})^{2}&t<t_{1}\\ 1&t_{1}\leq t<t_{2}\\ \exp[-c(t-t_{2})]&t\geq t_{2}\end{array} \right. \tag{16}\]  
in which \(t_{1}\), \(t_{2}\), and \(c\) are set to 6, 10, and 0.5, respectively. The sampling frequency and upper cut-off frequency are set to 100 Hz and 25 Hz, respectively, and the duration is set to 20 s. The same set of random phase angles are used to simulate horizontal and vertical motions. Moreover, to study the influence regularity of the properties of soft interlayer itself on lagged spatial coherency, varying shear wave velocity, buried depth, and thickness of the soft interlayer are considered as shown in Table 1. Note that, in the case with thickness, \(D^{SI}\) is also varying corresponding to \(H^{SI}\) to assure the center line of soft interlayer is fixed. It is also noteworthy that the incident angle of horizontal in-plane motion is calculated according to that of vertical motion and the wave velocity ratio between shear wave and compression wave, which is demanded in 1D wave propagation theory [53].