## 3 Comparison of the Statistical Features between the Gaussian Model and SMART 1 Records
### Description of the Spatially Varying Ground Motions  
The cross-PSD matrix describes the second-order statistical feature of a random field, with elements that can be expressed as follows. \[S_{lj}(f_{k})=\gamma_{lj}(f_{k})\sqrt{S_{ll}(f_{k})S_{jj}(f_{k})} \tag{1}\]  
where \(S_{lj}\) and \(\gamma_{lj}\) are the cross-PSD function and the coherence function between the ground displacement \(l\) and \(j\), respectively. The phase of \(\gamma_{ij}\) is mainly composed of the wave-passage and site-response effect. In this study, the variation of the periodograms is of concern. Therefore, this study considers only the spectral coherence function's amplitude. \(|\gamma_{ij}|\) is the amplitude of the spectral coherence function. Two spectral coherence models are used, including the Harichandran-Vanmarcke model [33] (HV model), shown in Equation (2),  
\[\gamma_{ij}^{l}(\omega)=Ae^{\frac{\delta}{\delta}}+(1-A)e^{\delta} \tag{2}\]  
where \(\delta=-\frac{2d_{ij}}{\theta(\omega)}(1-A+\alpha A)\) and \(\theta(\omega)=k[1+(\frac{\omega}{2\pi f_{0}})^{b}]^{-\frac{1}{2}}\), and the Luco-Wong model [34] (LW model), shown in Equation (3). \[\gamma_{ij}^{l}=\exp[-\left(\frac{\lambda_{x}\omega d_{ij}}{v_{\rm s}}\right) ^{2}] \tag{3}\]  
The HV model with two parameter settings, denoted by HV1 and HV2, respectively, and the LW model with one parameter setting are chosen for studying the influence of coherence shape on the analysis results. The HV model is empirical. Table 2 lists the two-parameter settings, which [PERSON] et al. [35] fitted with different earthquake records from the SMART 1 array. The LW model is semi-empirical. [PERSON] [36] recommended adopting a universal parameter of \(\lambda_{x}/v_{s}=2.5\times 10^{-4}\) in this model. The three spectral coherence model for separation distance of 200m and 1 km are demonstrated and compared in Figure 3.  
Figure 2: Conditions of unavailable records in the SMART 1 record. A ground-motion time series can be represented by the Fourier series expansion in Equation (4). \[u_{l}(t)=\sum_{k=1}^{N_{k}}\left[A_{lk}cos(f_{k}t)+B_{lk}sin(f_{k}t)\right],l=1, \ldots,L \tag{4}\]  
where \(A_{lk}\) and \(B_{lk}\) are the Fourier coefficients. When the ground motion belongs to a random field, the Fourier coefficients of the ground motions should satisfy Equations (5) and (6). \[E\{A_{lk}A_{jk}\}=E\{B_{lk}B_{jk}\}= \begin{cases}S_{ll}(f_{k})\Delta f,&l=j\\ Re\{S_{lj}(f_{k})\Delta f\},&l\
eq j\end{cases}  \tag{5}\]  
\[E\{A_{lk}B_{jk}\}=-E\{B_{lk}A_{jk}\}= \begin{cases}0,&l=j\\ -Im\{S_{lj}(f_{k})\Delta f\},&l\
eq j\end{cases}  \tag{6}\]  
where \(A_{lj}\) and \(B_{lj}\) are independent, random Fourier coefficients exhibiting the zero-mean Gaussian distribution. To guarantee the Fourier coefficients satisfy Equations (5) and (6), some researchers apply Cholesky decomposition, while some apply eigenvalue decomposition. These methods assume that \(2N_{l}\) random components constitute a ground motion, as in Equation (7). The number \(N_{l}\) counts in both the seed and the target ground motions. \[A_{lk}=\sum_{i=1}^{N_{l}}q_{lki}\tilde{\xi}_{a,lki},\,B_{lk}=\sum_{i=1}^{N_{l}} q_{lki}\tilde{\xi}_{b,lki} \tag{7}\]  
where \(\tilde{\xi}_{a,lki}\) and \(\tilde{\xi}_{b,lki}\) are independent random variables exhibiting the standard normal distribution. The seed ground motion is also assumed to be realized by random variables. \(q_{lki}\) satisfies Equation (8). \[S_{ll}(f_{k})=\sum_{i=1}^{N_{l}}|q_{lki}|^{2},S_{lj}(f_{k})=\sum_{i=1}^{N_{l}} q_{lki}v_{lki}^{*} \tag{8}\]  
The differences between the different conditional simulation methods are how to calculate \(q_{lki}\), in other words, how to distribute the energy among the random variables.