# Study on the Influence of a Soft Soil Interlayer on Spatially Varying Ground Motions
## 4 Numerical Example 2: Non-Linear Soil Behavior
### Results and Discussion  
Mean lagged spatial coherencies for non-linear soil and linear soil cases and corresponding time-varying spectral ratios under varying shear wave velocity of soft interlayer are shown in Figure 10, and \(t_{\text{m}}\) is the time interval with respect to the lowest value of shear modulus. As a matter of fact, spectral ratio reflects the difference between the transfer functions at sites j and k, and the obvious peaks and troughs in the coherency function of surface ground motions are related to the spectral ratio [19,26]. Note that the spectral ratio at \(t=0\) is just the one under the linear soil case. Besides, the obvious peaks of spectral ratio are marked with black dashed lines crossing the top and the bottom figures to highlight the respective relation between the peaks of spectral ratio and the troughs of spatial coherency. The frequency points corresponding to the peaks of spectral ratio curve at \(t=0\) are numbered with \(f_{h}^{x}\) (\(x=1,\,2,\,\dots\) ), whereas those corresponding to the peaks of spectral ratio curve at \(t=t_{m}\) are numbered with \(f_{h}^{x}\) (\(x=1,\,2,\,\dots\) ). For varying buried depth and thickness of soft interlayer and varying PGA of input motion, the lagged spatial coherencies and corresponding time-varying spectral ratios are plotted in Figures 11-13, respectively. Figures 10-12 exhibit the same trend of spatial coherency with respect to varying parameters of soft interlayer, while Figure 13 shows that the difference of spatial coherency functions between linear soil and non-linear soil cases tends to be greater as the input PGA increases. This is because that larger PGA causes more significant non-linearity of soil and thus greater difference of frequency content of motions at sites j and k for non-linear soil case than that for linear soil case. In general, the curves of lagged spatial coherencies of SVEGMs for the cases with non-linear soil and linear soil are both lower than the curves for the base rock motions. In all study cases, the reduction of spatial coherency for the vertical motion under non-linear soil behavior is only slightly larger than that under linear soil behavior, which indicates that the influence of non-linear soil behavior may be mild for the vertical in-plane wave possessing large wavelength. For horizontal motion, more importantly, the lagged spatial coherency under non-linear soil case presents more obvious and larger reduction than that under linear soil case. This indicates that the difference between horizontal ground motions at sites j and k under non-linear soil case tends to be greater than that under linear soil case, because, as above mentioned, the spatial coherency measures the similarity between ground motions at two different locations. Due to the existence of soft soil interlayer at site k, the amplitude and vibration frequencies of transfer functions at site j and k are different at the initial time, and under the linear soil case, the lagged spatial coherencies present obvious reduction at \(f_{l}^{x}\). Under non-linear soil case, both the transfer functions at sites j and k are time-varying, but the variation trends with time are different for these two sites: (i) The peak amplitude of transfer function at site k stands greater variation due to the non-linear soft interlayer, not to mention the square amplitude of transfer function (it is the square amplitude of transfer function that is used to simulate SVEGMs); (ii) the variation ranges of frequencies corresponding to peaks are also different, which leads to that the remarkable difference of frequency content between ground motions at sites j and k under non-linear state shows up at more frequency points than that under linear soil state. So, the reduction of spatial coherency under the condition of non-linear soil is more obvious.