# Saturation Effect on the Coherency Loss of Spatially Varying Vertical Ground Motions
## 2 Materials and Methods  
The transfer function in the one-dimension wave propagation theory ([PERSON] et al., 2017) was utilized to describe the amplification and filtering effects of soil on the incident wave. The formulas for estimating the P-wave velocity of a porous medium, proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2017), were employed to represent the effect of water saturation on the site amplification. The establishment method of the dynamic-stiffness matrix of the water layer, derived by [PERSON] et al. [PERSON] et al. (2017), was adopted in the study case with WLT to facilitate the establishment of the total dynamic-stiffness matrix of site under water. Based on the methods or parameters mentioned above, the transfer function of target site considering the saturation, or surface water layer, can be established. Then, the auto/cross-power spectral density function (PSDF) on site surface can be derived ([PERSON] et al., 2017; [PERSON] et al., 2017):  
\[S_{jj}(\omega)=\big{|}H_{j}(i\omega)\big{|}^{2}S_{g}(\omega)\quad j=1,2,\ldots n, \tag{1}\]  
and  
\[S_{jk}(i\omega)=H_{j}(i\omega)H_{k}^{\Theta}(i\omega)S_{g}(\omega)\gamma_{j^{ \prime}k^{\prime}}(d_{j^{\prime}k^{\prime}},i\omega)\quad j,k=1,2,\ldots n, \tag{2}\]  
respectively, where \(H_{j}\) and \(H_{k}\) are the transfer functions at simulation points \(j\) and \(k\), respectively; superscript '\(\Theta\)' denotes the complex conjugate; \(S_{g}(\omega)\) is the ground motion PSDF at the base rock; \(\gamma_{f^{\prime}k^{\prime}}(d_{j^{\prime}k^{\prime}},i\omega)\) is the coherency loss function between bedrock motions, which is related to the separation distance \(d_{j^{\prime}k^{\prime}}\) and \(\omega\). By using the classical spectral representation method [19] in combination with Jennings window, non-stationary SVEGMs can be generated by  
\[f_{j}(t)=2a(t)\sum_{x=1}^{n}\sum_{y=1}^{N}\left|U_{jx}(\omega_{y})\middle|\sqrt {\Delta\omega}\cos[\omega_{y}t-\theta_{jx}(\omega_{y})+\Phi_{xy}]\right.\quad j =1,2,\ldots n \tag{3}\]  
\[\omega_{y}=y\Delta\omega\quad y=1,2,\ldots N \tag{4}\]  
\[\Delta\omega=\frac{\omega_{u}}{N} \tag{5}\]  
\[\theta_{jx}(\omega_{y})=\tan^{-1}(\frac{\text{Im}[U_{jx}(\omega_{y})]}{\text {Re}[U_{jx}(\omega_{y})]}) \tag{6}\]  
with \(\Delta\omega\) denoting the bandwidth and \(N\) the number of frequency intervals. \(\omega_{u}\) represents an upper cut-off frequency beyond which the elements of the power spectral density matrix (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 \([0,2\pi]\). \(a(t)\) is a modulating function in time. The filtered Tajimi-Kanai model [28] is utilized to describe the PSDF on the bedrock:  
\[S_{g}(\omega)=\frac{\omega^{4}}{\left(\omega_{f}^{2}-\omega^{2}\right)^{2}+ \left(2\omega_{f}\omega\zeta_{f}\right)^{2}}\times\frac{\omega_{g}^{4}+4\zeta _{g}^{2}\omega_{g}^{2}\omega^{2}}{\left(\omega_{g}^{2}-\omega^{2}\right)^{2}+4 \zeta_{g}^{2}\omega_{g}^{2}\omega^{2}}S_{0} \tag{7}\]  
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; and \(S_{0}\) is the scaling factor. According to [PERSON] and [PERSON] [19], \(\omega_{g}=10\pi\text{rad}/s\), \(\zeta_{g}=0.6\), \(\omega_{f}=0.5\pi\), \(\zeta_{f}=0.6\) and \(S_{0}=0.0034\) m\({}^{2}\)/s\({}^{3}\).