# Study on the Influence of a Soft Soil Interlayer on Spatially Varying Ground Motions
## 2 Theoretical Basis
### Simulation of Fully Non-Stationary SVEGMs  
Based on the time-varying transfer function abovementioned, the evolutionary auto/cross-power spectral density function (PSDF) on site surface can be derived by  
\[S_{jj}(\omega,t)=[a(t)]^{2}\big{|}H_{j}(i\omega,t)\big{|}^{2}S_{g}(\omega)\ \ \ \ \ j=1,\ 2,\ \ldots\ n\ \, \tag{7}\]  
\[\text{ and }S_{jk}(i\omega,t)=[a(t)]^{2}H_{j}(i\omega,t)H_{k}^{\Theta}(i\omega,t )S_{g}(\omega)\gamma_{j^{\prime}k^{\prime}}(d_{j^{\prime}k^{\prime}},i\omega) \ \ \ \ j,\ k=1,\ 2,\ \ldots\ n\ \, \tag{8}\]  
respectively, where \(H_{j}\) and \(H_{k}\) are the transfer functions at simulation points \(j\) and \(k\), respectively; \(a(t)\) is a modulating function in time; superscript \"\(\Theta\)\" denotes complex conjugate; \(S_{g}(\omega)\) is the ground motion PSDF at the base rock \(\gamma_{j^{\prime}k^{\prime}}(d_{j^{\prime}k^{\prime}},i\omega)\) is the spatial coherency function of spatial ground motions at the base rock which is related to the separation distance \(d_{j^{\prime}k^{\prime}}\) and \(\omega\).  
Then, power spectral density matrix (PSDM) can be formed. By using the classical simulation method of SVEGMs proposed by [PERSON] [18] in combination with the root decomposition [21; 54] of the PSDM, fully non-stationary SVEGMs can then be generated by  
\[f_{j}(t)=2\sum_{x=1}^{n}\sum_{y=1}^{N}\big{|}U_{jx}(\omega_{y},t)\big{|}\sqrt {\Delta\omega}\cos[\omega_{y}t-\theta_{jx}(\omega_{y},t)+\Phi_{xy}]\ \ \ \ \ j=1,\ 2,\ \ldots\ n \tag{9}\]  
\[\omega_{y}=y\Delta\omega\ \ \ \ y=1,\ 2,\ \ldots\ N \tag{10}\]  
\[\Delta\omega=\frac{\omega_{u}}{N} \tag{11}\]  
\[\theta_{jx}(\omega_{y},t)=\tan^{-1}(\frac{\text{Im}[U_{jx}(\omega_{y},t)]}{ \text{Re}[U_{jx}(\omega_{y},t)]}) \tag{12}\]With \(\Delta\omega\) denoting the bandwidth and \(N\) the number of frequency interval. \(\omega_{u}\) represents an upper cut-off frequency beyond which the elements of the PSDM may be assumed to be zero for any time instant. \(U_{jx}(\omega_{y},t)\) is an element of the matrix obtained through the decomposition of PSDM. \(\Phi_{xy}\) is a random phase angle uniformly distributed in [0, 2\(\pi\)].  
In summary, for linear soil case, 1D wave propagation theory was directly used to conduct the study, while for non-linear soil case the flow of the methodology is as follows:  
1. Obtain the time-varying trend of fundamental properties (e.g., shear modulus) of soils in the target site by using a numerical site response based on FE code ABAQUS;
2. Obtain the time-varying transfer function of the target site based on above mentioned time-varying trend of fundamental properties of soils;
3. Obtain power spectral density for surface ground motions by using time-varying transfer function and prescribed power spectral density model for bedrock and artificially simulate surface spatially varying earthquake ground motions;
4. Estimate mean coherency loss between generated spatially varying earthquake ground motions.  
Note that the simulation of quasi-stationary SVEGMs is a special case of that of fully non-stationary SVEGMs, and quasi-stationary SVEGMs can be derived by combining the transfer function that is invariable with time with the classical SRM.