## 3 Comparison of the Statistical Features between the Gaussian Model and SMART 1 Records
### Amplitude Variation of the Normalized Periodograms
Therefore, the square of the coefficient of variance (SCV) is adopted to estimate the loop variance of the normalized periodogram. Researchers have developed the distribution of the coefficient of variance of normally distributed random variables [38] and lognormal distribution random variables [39]. However, the statistical features of the coefficient of variance for the quadratic form of normal distribution remain unknown. Consequently, Monte Carlo simulations were carried out to investigate the relationship between the loop variances and the SCV. Figure 5 shows the comparison of the loop variance with the loop SCV. Both the results calculated from the HV1 and the LW models show a significant difference between the loop variance and loop SCV. The differences are frequency-dependent and, thus, cannot be replaced by a constant value. Using the loop SCV calculated with a small amount of data for approximating the loop variance can result in significant deviations. An adjustment coefficient, denoted by \(I_{cv}\), is introduced to restore the loop SCV to the loop variance. \[I_{cv}(f_{k})=\frac{\text{var}_{12}\{R_{l}(f_{k})\}}{scv_{N_{k}}\{R_{l}(f_{k})\}} \tag{19}\]  
where \(\text{var}_{12}\{R_{l}(f_{k})\}\) represents the loop variance estimated from the simulations corresponding to 12 evenly distributed stations, \(scv_{N_{k}}\{R_{l}(f_{k})\}\) means the loop SCV is calculated with the simulations corresponding to \(N_{s}\) stations. As stated in Section 2, the data obtained from the PEER center consist of unavailable records. The influence of unavailable records on the adjustment coefficient is also investigated by conducting Monte Carlo simulations. Figure 6 shows the adjustment coefficients for different unavailable record conditions. As shown in this figure, the more records that are unavailable, the more significant the adjustment coefficients are. Then, the estimated loop variances of the records' normalized periodograms for different events and their components in different directions are shown in Figure 7. The  
Figure 5: Loop variance vs. loop SCV: (**a**) HV1 model and (**b**) LW model. Figure 6: Adjustment coefficient for different loop configurations: (**a**) condition 1, (**b**) condition 2, (**c**) condition 3, (**d**) condition 4, and (**e**) condition 5.  
normalized periodogram loop variances (NPLVs) are calculated by first calculating the SCV of the corresponding loop. Next, the resulting SCVs are multiplied by the adjustment coefficient according to the unavailable record conditions. The NPLVs of the records are denoted by \(\sigma_{R,I}^{2}\)\(\sigma_{R,M}^{2}\), and \(\sigma_{R,O}^{2}\) for the inner, middle, and outer loops, respectively. The inter-event mean value of the NPLVs, which is the averaged NPLVs of all the horizontal components of the different events, is presented with bold red lines in each subgraph. In each subgraph of Figure 7, the NPLVs show features similar to that of the FAS amplitude variation reported by [PERSON] and [PERSON] [14]. As shown in this figure, the NPLVs increase with frequency and reach a relatively steady stage at 6.25 Hz, which is 1/16 of the sampling frequency. In Figure 7b, the average NPLV beyond 12.5 Hz continues to increase to an extremely high value, which can be attributed to the dividing of the small value when calculating the SCV of the normalized periodograms. The energy in the PSD of the five events beyond 12.5 Hz is small and has a low signal-to-noise ratio. The NPLVs for different loops in different events exhibit similar trends with the increase in frequency, indicating that the NPLV is a suitable index for describing spectral amplitude variation. The average of the inter-event mean value of the NPLVs between 6.75 and 12.25 Hz for each loop, denoted by \(\overline{\sigma}_{R,I}^{2}\), \(\overline{\sigma}_{R,M}^{2}\), and \(\overline{\sigma}_{R,O}^{2}\), respectively, are listed in Table 3.  
As shown in the previous section, based on the assumption of the Gaussian model, the value of an NPLV should be less than 1.