# Study on the Influence of a Soft Soil Interlayer on Spatially Varying Ground Motions
## 3 Numerical Example 1: Linear Soil Behavior
### Results and Discussion  
To illustrate, for the case with Cs\({}^{SI}\) = 100 m/s, \(D^{SI}\) = 5 m, and \(H^{SI}\) = 5 m, the transfer functions and one set of SVEGMs samples are plotted in Figures 2 and 3, respectively. Although the same set of random phase angles were used to simulate horizontal and vertical motions, different characteristics are shown, which is attributed to the difference between vibration modes in horizontal and vertical directions. Figure 4 gives the mean lagged spatial coherency between surface horizontal out-of-plane motions under different shear velocities, buried depths, and thicknesses of the soft interlayer, where the corresponding modulus of spectral ratio between sites j and k are also given (here, spectral ratio is defined as the ratio between the transfer functions at sites j and k). The mean lagged spatial coherency between bedrock motions and the one for the regular site without soft interlayer are shown in the same figure as well. Figures 5 and 6 present the results for horizontal and vertical in-plane motions, respectively. Comparing the spatial coherency functions on the site with soft interlayer (Figure 1b) with those on the regular site (Figure 1a), it can be found that the existence of a soft interlayer can further reduce the spatial coherency. Besides, the spatial coherency functions decrease as the shear wave velocity of the soft interlayer declines; by contrast, higher buried depth and thickness can bring about lower spatial coherency. This is because that the decreasing shear wave velocity or increasing buried depth and thickness of soft interlayer can significantly differ the transfer function at site k from that at site j, i.e., changing the amplitude and vibration frequencies, which can be justified by observing the spectral ratio (see Figures 4-6) and can bring about obvious differences between the frequency contents of ground motions at sites j and k. Therefore, the spatial coherency tends to be lower. Figure 3: Simulated ground motions for: (**a**) horizontal out-of-plane motion; (**b**) horizontal in-plane motion; and (**c**) vertical in-plane motion \(Cs^{SI}\) = 100 m/s, \(D^{SI}\) = 5 m, and \(H^{SI}\) = 5 m.  
Figure 2: Transfer functions at sites j and k for: (**a**) horizontal out-of-plane motion; (**b**) horizontal in-plane motion; and (**c**) vertical in-plane motion when \(Cs^{SI}\) = 100 m/s, \(D^{SI}\) = 5 m, and \(H^{SI}\) = 5 m.  
Figure 4: Mean lagged coherency between horizontal out-of-plane motions and corresponding spectral ratios for varying (**a**) shear wave velocity; (**b**) buried depth; and (**c**) thickness of interlayer. Figure 5: Mean lagged coherency between horizontal in-plane motions and corresponding spectral ratios for varying (**a**) shear wave velocity; (**b**) buried depth; and (**c**) thickness of interlayer. In the cases with buried depth, spatial coherency is more likely to dramatically decrease in long period range in general, which is different from the results for the cases with shear wave velocity and thickness to some extent. With varying buried depth and certain shear wave velocity and thickness, the soft interlayer always amplifies the motion in lower frequency range, while attenuating the motion in higher frequency range. In other words, obvious pikes of modulus of spectral ratio that are larger than 1 always appear in lower frequency range for the case with buried depth, indicating that in lower frequency range the transfer functions at sites j and k present more remarkable difference. By contrast, in the cases with shear wave velocity and thickness, the peaks of spectral ratios tend to more \"scattered\" or \"discrete\". Thus, the reduction of lagged spatial coherency caused by varying buried depth may be more inclined to concentrate in the lower frequency range than that caused by shear wave velocity or thickness. In comparison with horizontal motions, the reduction of spatial coherency for vertical motion is much smaller.