## 3 Validation of Effectiveness
### Ground Motion Simulation of the 1994 Northridge Earthquake
According to the \(\omega^{-2}\) model, the theoretical source spectra of the small and the large earthquake are represented by Equations (17) and (18), respectively [21]:  
\[\left|\dot{M}_{0 ij}(f)\right|=\frac{M_{0 ij}}{1+\left(f/f_{\text{cij}}^{\text{S }}\right)^{2}} \tag{17}\]  
\[\left|\dot{M}_{0}(f)\right|=\frac{M_{0}}{1+\left(f/f_{\text{c}}^{\text{S}} \right)^{2}} \tag{18}\]  
In which \(M_{0}\) and \(f_{\text{c}}^{\text{S}}\) are the seismic moment and the corner frequency of the large earthquake, respectively. The source spectrum of the large earthquake synthesized by superimposing the small earthquakes is expressed as:  
\[\dot{M}_{0}(f)=\sum_{i=1}^{NL}\sum_{j=1}^{NW}G(f)\dot{M}_{0 ij}(f)\exp\left\{2 \text{i}\pi f\left[\frac{\xi_{ij}}{V_{\text{R}}}+\frac{(\lambda-0.5)\Delta W}{ V_{\text{R}}}\right]\right\} \tag{19}\]  
where \(G(f)\) is the conversion function of the source spectrum from the small earthquake to the subfault earthquake, expressed as:  
\[G(f)=1+\frac{a\big{[}1-\exp\bigl{(}-a-2\text{i}\pi f\tau_{ij}\bigr{)}\big{]} }{n\prime[1-\exp(-a)]\Bigl{\{}1-\exp\bigl{[}\frac{-a-2\text{i}\pi f\tau_{ij}}{ (ND-1)n\prime}\bigr{]}\Bigr{\}}} \tag{20}\]  
The source spectra of the large earthquake synthesized using the above two methods are compared with the theoretical \(\omega^{-2}\) source spectrum, as shown in Figure 3.  
The results indicate that when assuming a uniform distribution of rise time on the fault plane, the sag phenomenon of the source spectrum in the intermediate frequency band becomes more and more significant as the subfault division number gradually increases, which is similar to the research results of [PERSON] [36]. However, when using the improved expression for the rise time, as the subfault division number increases, the amplitude of the source spectrum changes very little, and the sag phenomenon is relatively insignificant. In order to further verify the effectiveness of the improved expression for the rise time proposed in this study, six stations are randomly selected around the 1994 Northridge earthquake fault. The rupture distances of these six stations range from 16 km to 47 km, with two stations located on the hanging wall and four stations located on the foot wall of the fault. In the cases of subfault division numbers of 9 \(\times\) 10 and 36 \(\times\) 40, the ground motions at these stations are simulated using the two methods mentioned above. The location, rupture distance \(R_{\text{rup}}\), time-averaged shear-wave velocity of the upper 30 m \(V_{\text{s}30}\), and the National Earthquake Hazards Reduction Program (NEHRP) site classification of these stations are listed in Table 2. According to the NEHRP code [47], the site types of these stations are class C sites, which are soft rock or very dense hard soil sites. The basic information and observed seismic records of these stations are from the strong ground motion database of the Pacific Earthquake Engineering Research Center. In order to minimize the impact of randomness on the simulation results of the stochastic Green's function method and verify the effectiveness of the improvement proposed in this study, 30 acceleration time histories are randomly generated for each simulation, and the 5% damped acceleration response spectra and the displacement Fourier amplitude spectra of the 30 acceleration time histories are calculated. The average value of the 30 acceleration response spectra and the average value of the 30 displacement amplitude spectra are taken as the simulated acceleration response spectrum and the simulated displacement amplitude spectrum, respectively, and the acceleration time history corresponding to the response spectrum closest to the average response spectrum is taken as the simulated acceleration time history.