## 5 Correlation Analysis  
### Selection of MS-AS Sequences  
A set of 662 real MS-AS earthquake sequences were selected from 13 earthquake events based on the NGA-West2 database ([[https://peer.berkeley.edu/research/nga-west-2](https://peer.berkeley.edu/research/nga-west-2)]([https://peer.berkeley.edu/research/nga-west-2](https://peer.berkeley.edu/research/nga-west-2))) ([PERSON] et al [58]). The NGA-West2 database has been used to develop the latest generation of ground motion prediction equations for shallow crustal earthquakes (e.g., [PERSON] and Bozorgni [59], [PERSON] et al. [32] The detailed information of the selected ground motion records and the related earthquake events can be found in [PERSON] et al. [60]. It is noteworthy that, among the aftershock sequences triggered by a specific mainshock, only aftershock record with the largest magnitude was adopted. Moreover, the selected mainshock and associated aftershock ground motions should be recorded at the same station.  
Figure 5 shows the distribution of moment magnitudes \(M_{w}\) and rupture distances \(R_{rup}\) for the selected ground motions of MSs and ASs. The magnitude of MSs ranges from \(M_{w}\) = 5.8 to \(M_{w}\) = 7.62, while the magnitude of the corresponding ASs ranges from \(M_{w}\) = 5.01 to \(M_{w}\) = 7.14. As for rupture distances, the range of \(R_{rup}\) for MSs is between 0.32 km and 208.47 km and that for ASs is between 1.98 km and 218.16 km, respectively. As seen from Figure 5b, the \(M_{w}\) values of MSs are always larger than that of AS. This is compatible with the Bath's rule [61]. Figure 6 compares the elastic response spectra for the selected ground motions of MSs and ASs. It is found that the median response spectrum  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Delta** & **PinchX** & **PinchY** & **Damage1** & **Damage2** \\ \hline Model-1 & 0 & 1.0 & 1.0 & 0 & 0 \\ Model-2 & 0.03 & 1.0 & 1.0 & 0 & 0 \\ Model-3 & 0 & 0.8 & 0.2 & 0 & 0 \\ Model-4 & 0 & 1.0 & 1.0 & 0.05 & 0.02 \\ Model-5 & 0.03 & 0.8 & 0.2 & 0 & 0 \\ Model-6 & 0.03 & 1.0 & 1.0 & 0.05 & 0.02 \\ Model-7 & 0 & 0 of MS ground motions is significantly larger than that of AS ground motions, which is expected due to the greater shaking intensities of MSs than ASs.  
For a given ground motion recorder, it could only detect seismic signals in a certain range of frequency due to its settings. This makes the recorded ground motions have a specific lowest usage frequency, i.e., \(f_{\text{low}}\). Therefore, for dynamic analysis, a ground motion can only be used for nonlinear SDOF systems with their vibration periods no larger than \(T=1\)/\(f_{\text{low}}\). As for a specific MS-AS sequence, the \(f_{\text{low}}\) values for the MS and AS ground motions are probably different. For simplicity, the representative \(f_{\text{low}}\) value for a MS-AS sequence is taken as max (\(f_{\text{low}}\)\({}^{\text{MS}}\),\(f_{\text{low}}\)\({}^{\text{AS}}\)), where \(f_{\text{low}}\)\({}^{\text{MS}}\) and \(f_{\text{low}}\)\({}^{\text{AS}}\) are the \(f_{\text{low}}\) values for the MS and AS ground motions, respectively. In this study, the highest structural period considered for the nonlinear SDOF systems is 6.0 s, and thus, the usable \(f_{\text{low}}\) values of the selected MS-AS sequences should be no less than 0.166 Hz. Figure 7 shows the number of usable earthquake sequences at different periods with the consideration of \(f_{\text{low}}\) requirement, where the number of usable sequences decreases with increasing periods of SDOF systems.  
Figure 5: The distribution of \(M_{\text{w}}\) and \(R_{\text{rup}}\) for the selected MS and AS ground motions.  
Figure 6: Elastic response spectra of the selected MS and AS ground motions. Note: \(T\) is the fundamental period of the SDOF systems, \(s\) denote the unit of \(T\), which is second.  
Figure 7: Number of usable MS–AS sequences for the SDOF systems with varying fundamental periods.