## 3 Comparison of the Statistical Features between the Gaussian Model and SMART 1 Records
### Amplitude Variation of the Normalized Periodograms  
In this study, for each earthquake event, the record from the central station is assumed to be the only seed ground motion. The other earthquake records are regarded to be conditionally generated from the seed ground motion. The Cholesky decomposition-based methods have one merit for theoretical analysis. When one seed is available, and the other locations are located the same distance from the location of the seed ground motion, the proportions of the seed ground motion in the conditional generated artificial ground motion are the same. Then, the eigenvalue decomposition-based method is used because the eigenvalue decomposition can result in symmetric coefficient matrices, while the results of the Cholesky decompositions cannot. The above process is introduced in detail as follows. The cross-PSD matrix of a random field with uniform soil conditions can be expressed as Equation (9). \[\mathbf{S}(f_{k})=S(f_{k})\mathbf{\Gamma}(f_{k}) \tag{9}\]  
The diagonal elements of \(\mathbf{\Gamma}(f_{k})\) are all one, and the other elements are the spectral coherences. Conducting the Cholesky decomposition such that \(\mathbf{\Gamma}(f_{k})=\mathbf{L}(f_{k})\mathbf{L}^{\top}(f_{k})\), Equation (10) presents the result of \(\mathbf{L}(f_{k})\). Here, \(\widetilde{\mathbf{L}}(f_{k})\) in matrix \(\mathbf{L}(f_{k})\) satisfies Equation (11)  
\[\mathbf{L}(f_{k})= \begin{pmatrix}1&0\\ \gamma_{I}(f_{k})1&\widetilde{\mathbf{L}}(f_{k})\end{pmatrix}  \tag{10}\]  
\[\widetilde{\mathbf{\Gamma}}(f_{k})=\widetilde{\mathbf{L}}(f_{k})\widetilde{ \mathbf{L}}^{T}(f_{k})=\mathbf{\Gamma}(f_{k})-|\gamma_{I}(f_{k})|^{2} \tag{11}\]  
Then, the eigenvalue decomposition to \(\widetilde{\mathbf{\Gamma}}(f_{k})\) is conducted to distribute the remaining energy among the independent random variables. The coefficients \(v_{Iki}\) can be calculated by Equation (12). \[\widetilde{\mathbf{V}}(f_{k})=\sqrt{\widetilde{\mathbf{\Gamma}}(f_{k})}= \mathbf{\Phi}(f_{k})\sqrt{\mathbf{\Lambda}(f_{k})}\mathbf{\Phi}^{-1}(f_{k}) \tag{12}\]  
\(\widetilde{v}_{Iki}\) is an element of \(\widetilde{\mathbf{V}}(f_{k})\), and \(\mathbf{\Phi}(f_{k})\) and \(\mathbf{\Lambda}(f_{k})\) are the eigenvector and eigenvalue matrix of \(\widetilde{\mathbf{\Gamma}}(f_{k})\), respectively. In Equation (12), the square root of a general matrix \(\sqrt{\widetilde{\mathbf{\Gamma}}(f_{k})}\) is defined as the matrix with eigenvalues that are the square root of \(\widetilde{\mathbf{\Gamma}}(f_{k})\), \(\sqrt{\mathbf{\Lambda}(f_{k})}\) equal to the square root of the diagonal elements. Then, the Fourier coefficients of the conditional simulated ground motion at location \(l\) can be represented as follows. \[A_{lk}=\gamma_{I}(f_{k})A_{rk}+\sqrt{S}\widetilde{\mathbf{v}}_{I}(f_{k}) \xi_{\alpha},\ B_{lk}=\gamma_{I}(f_{k})B_{rk}+\sqrt{S}\widetilde{\mathbf{v}}_{ I}(f_{k})\xi_{b} \tag{13}\]  
where \(\widetilde{\mathbf{v}}_{I}\) is the \(l^{th}\) row of matrix \(\widetilde{\mathbf{V}}(f_{k})\), \(\xi_{\alpha}\) and \(\xi_{b}\) are random vectors with elements that exhibit the standard normal distribution. \(A_{rk}\) and \(B_{rk}\) are the Fourier coefficients of the seed ground motion. After the previous decomposition approach, the periodogram of the conditional simulated ground motions can be expressed as Equation (14). \[G_{l}(f_{k})=\frac{1}{2}\bigg{[}\big{(}A_{rk}\quad\sqrt{S}\xi_{\alpha}^{*} \big{)}\mathbf{W}_{l} \begin{pmatrix}A_{rk}\\ \sqrt{S}\xi_{\alpha}\end{pmatrix} +\big{(}B_{rk}\quad\sqrt{S}\xi_{b}^{*}\big{)} \mathbf{W}_{l} \begin{pmatrix}B_{rk}\\ \sqrt{S}\xi_{b}\end{pmatrix} \bigg{]} \tag{14}\]  
where \(\square*\) represents the complex conjugate transpose of matrix \(\square\), \(\mathbf{W}_{l}= \begin{pmatrix}\gamma_{Ir}^{2}&\gamma_{Ir}\mathbf{v}_{I}\\ \gamma_{Ir}\mathbf{v}_{I}^{*}&\mathbf{v}_{I}^{*}\mathbf{v}_{I}\end{pmatrix}