## 3 Comparison of the Statistical Features between the Gaussian Model and SMART 1 Records
### Amplitude Variation of the Normalized Periodograms
\), \(\widetilde{\mathbf{W}}_{l}(f_{k})=\widetilde{\mathbf{v}}_{l}^{*}(f_{k}) \widetilde{\mathbf{v}}_{l}(f_{k})\), and \(\widetilde{\mathbf{V}}(f_{k})=\sum\limits_{l=1}^{N_{l}}\widetilde{\mathbf{W}}_ {l}(f_{k})\). The expected value of \(G_{l}(f_{k})\) is equal to Equation (15). \[E\{G_{l}(f_{k})\}=\frac{1}{2}\Big{[}\gamma_{Ir}^{2}(f_{k})\Big{(}A_{rk}^{2}+B_{ rk}^{2}\Big{)}+S(f_{k})\Big{(}1-\gamma_{Ir}^{2}(f_{k})\Big{)}\Big{]} \tag{15}\]According to [PERSON] and [PERSON] [37], for a quadratic form \(y=\mathbf{x}^{*}\mathbf{W}\mathbf{x}\), the variance of \(y\) can be obtained by Equation (16). The elements in the vector \(\mathbf{x}\) exhibit independent normal distributions. \[var\{y\}=2 tr\{(\mathbf{W}\mathbf{\Sigma})^{2}\}+4\mathbf{\mu}^{*}\mathbf{W} \mathbf{\Sigma}\mathbf{W}\mathbf{\mu} \tag{16}\]  
Regarding the seed ground motion as a random process, the means of its Fourier coefficients are \(A_{rk}\) and \(B_{rk}\), and the variances are 0. Then, the variance of the periodogram \(G_{l}(f_{k})\) is derived as Equation (17). \[ \begin{array}{ll}\mathrm{var}\{G_{l}(f_{k})\}&=S^{2}tr\{\widetilde{\mathbf{W} }_{l}\widetilde{\mathbf{W}}_{l}\}+S\big{(}A_{lk}^{2}+B_{lk}^{2}\big{)}\gamma_{ l}^{2}\mathbf{v}_{l}\mathbf{v}_{l}^{T}\\ &=S\big{(}1-\gamma^{2}\big{)}\big{[}S\big{(}1-\gamma^{2}\big{)}+\gamma_{l}^{2} \big{(}A_{lk}^{2}+B_{lk}^{2}\big{)}\big{]}\end{array}  \tag{17}\]  
To eliminate the influence of spectral shape, the ground motion periodograms are divided by the PSD of the random field. The result, i.e., \(R_{l}(f_{k})\), is called the normalized periodogram. When the statistical features of several earthquakes are concerned, the inter-event variance of \(R_{l}(f_{k})\) at location \(l\) is equal to Equation (18). \[\mathrm{var}\{R_{l}(f_{k})\}=1-\gamma_{lj}^{4} \tag{18}\]  
However, the SMART 1 array records in a particular earthquake are correlated. In this study, the quantity of events is far smaller than the number of records in a specific loop of an earthquake. Therefore, the intra-event loop variance of \(R_{l}(f_{k})\), instead of the inter-event variance in Equation (18), is adopted for studying the periodograms' amplitude variation. The intra-event loop variance is a value estimated from the 12 stations in each loop and can be called the loop variance for brevity. A closed-form expression of the loop variance of \(R_{l}(f_{k})\) is not available. In this study, the loop variances of \(R_{l}(f_{k})\) were estimated by Monte Carlo simulating \(R_{l}(f_{k})\) from 12 evenly separated stations, by considering the spectral coherence between stations. The results obtained from ten thousand simulations were used for evaluating each statistic. These results are presented in Figure 4b. The theoretical inter-event variances of \(R_{l}(f_{k})\) in Equation (18) are plotted in Figure 4a for comparison. Figure 4a uses abbreviations to represent the results calculated by the different spectral models with varying separation distances. For example, HV1-200 means the results yielded by the HV1 model, and the parameter of separation distance is 200 m. When the spectral coherences between the loop stations are zero, the theoretical inter-event loop variance of \(R_{l}(f_{k})\) in Equation (18) equals the loop variance of \(R_{l}(f_{k})\). However, the spectral coherences between the loop stations and the central station are not zero. Comparing Figure 4a with Figure 4b, the loop variances of \(R_{l}(f_{k})\) are obviously more minor than its inter-event variances. Therefore, spectral coherence should be considered when analyzing the spectral amplitude variation. Comparing the results calculated by the HV1 model and the LW model, the significant differences also indicate that the spectral coherence's shape dramatically influences the shape of the loop variance. Figure 4: Variances of \(R_{l}\) calculated using HV and LW spectral coherence models: (**a**) inter-event variance of \(R_{l}\) and (**b**) loop variance of \(R_{l}\). Regarding actual ground motions, the PSD of the concerning random field is not accessible.