## 3 Comparison of the Statistical Features between the Gaussian Model and SMART 1 Records
### Power Variation of SMART 1 Records  
In this subsection, the ground motion power variation of the SMART 1 array records is analyzed to show, in another aspect, that the Gaussian model results in an underestimation of spectral amplitude variation. First, the power variation with the variation of separation distance is investigated. The ground motion power \(p_{I}\) is the integral of the periodogram \(G_{l}\) in the frequency domain. For loop \(r\), \(p_{I}\)'s mean and standard deviation are represented by \(\mu_{p,e}^{r}\) and \(\sigma_{p,e}^{r}\), respectively. The superscript \(r\) can be replaced by \(I\), \(M\), and \(O\). The power ratio is defined by the ground motion power divided by the power of the central station periodogram, i.e., \(\phi_{l}=p_{I}/p_{C}\). The mean power ratios in loop \(r\) of event \(e\), represented by \(\mu_{p,e}^{r}\), are plotted by the bold lines \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Variance**} & \multicolumn{3}{c}{**Records**} & \multicolumn{3}{c}{**Simulation**} \\ \cline{2-7}  & **HV1** & **HV2** & **LW** & **HV1** & **HV2** & **LW** \\ \hline \(\overline{\sigma}_{R,I}^{2}\) & \(1.1\) & 0.98 & 1.1 & 0.87 & 0.73 & 0.99 \\ \(\overline{\sigma}_{R,M}^{2}\) & \(1.42\) & 1.27 & 1.42 & 0.95 & 0.74 & 1 \\ \(\overline{\sigma}_{R,O}^{2}\) & \(1.37\) & 1.22 & 1.36 & 0.98 & 0.75 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Averaged NPLVs for different loops. Figure 7: Loop variance of \(R_{e,rj}\) in different loops: (**a**) inner loop, (**b**) middle loop, and (**c**) outer loop. with symbols in Figure 8a. The average \(\phi_{l}\) of all the horizontal components of different events in loop \(r\) is denoted by \(\mu^{r}_{\phi,a}\). The results of \(\mu^{r}_{\phi,a}\) are plotted with fine horizontal lines in this figure. The \(\mu^{r}_{\phi,a}\) for the inner, middle, and outer loops are all approximately 90%, indicating that the averaged power changes slightly with the increase in the loop radius. The power of the central station records is greater than that of all the other loops, which systematic reasons outside the scope of this study might cause, such as complex soil-layer distributions and path effects. However, the power of the central-station records remains within one standard deviation from the average power of the other loops. Thus, this difference might also be attributed to the small sample size. Next, the loop power variation is used to describe the power variation of the SMART 1 random field. To compare the loop power variation among different events, the normalized power loop standard deviation is defined as \(I^{r}_{p,\sigma}=\sigma^{r}_{p,\sigma}/\mu^{r}_{p,\sigma}\). \(I^{r}_{p,\epsilon}\) is calculated for the horizontal components of different events and is plotted with bold lines in Figure 8b. The average \(I^{r}_{p,\sigma}\) of all the events and directions is plotted with fine lines and marked with the average values. In this figure, the \(I^{r}_{p,\sigma}\) difference between events are sometimes smaller than the \(I^{r}_{p,\sigma}\) difference between different components in the same event. Therefore, the inter-event power variation is assumed to be negligible. In general, \(I^{r}_{p,\sigma}\) increases with the loop radius. Monte Carlo simulations were carried out with the conventional Gaussian model. Each estimation adopted ten thousand simulation results. The central station records were used as seeds, and the averaged periodograms of actual earthquakes in each loop were used as the PSD to generate artificial periodograms at different stations. Define the normalized power loop standard deviation, denoted by \(\hat{I}^{r}_{p,\sigma}\), as Equation (20). \(\hat{I}^{r}_{p,\epsilon}\) is the counterpart of \(I^{r}_{p,\epsilon}\) for the simulated periodograms. Figure 9 presents the results of \(\hat{I}^{r}_{p,\sigma}\). \(p_{c}\) is the power of the central station record.