## 3 Validation of Effectiveness
### Ground Motion Simulation of the 2013 Lushan Earthquake
(The red solid line represents the source spectra of the large earthquake synthesized using the constant rise time, the blue solid line represents the source spectra of the large earthquake synthesized using the modified rise time expression, the black dashed line represents the theoretical \(\omega^{-2}\) source spectrum.). In order to measure the simulation accuracy of the improved method, the simulation misfit of the ground motion of the Lushan earthquake in the NS, EW, and UD directions is calculated, as shown in Figure 19.  
The results show that when the subfault division number is 11 \(\times\) 7, the simulation accuracy of the NS and EW components is improved in the period range longer than 1.5 s after using the improved method, and the simulation accuracy of ground motion in the UD direction is improved in the whole period range. In the period range of 0.01 s to 6 s, the misfit values of the NS and EW components using the improved method are within \(\pm\)0.2, and the simulation misfit of the UD component using the improved method is within \(\pm\)0.3. When the subfault division number is 44 \(\times\) 28, the simulation accuracy of the three components is significantly improved in most period ranges after using the improved method. In the period range of 0.01 s to 6 s, the misfit values of the NS, EW, and UD components using the improved method are within \(\pm\)0.35, \(\pm\)0.4, and \(\pm\)0.6, respectively. Overall, the simulation accuracy is higher when the subfault division number is 11 \(\times\) 7 than when the subfault division number is 44 \(\times\) 28. Figure 13: Simulation results of the 51 BXZ station. (The comparisons of acceleration time histories are shown in the top row, the comparisons of acceleration response spectra are shown in the middle row, and the comparisons of displacement amplitude spectra are shown in the bottom row. The simulation results in the NS, EW, and UD directions are shown from left to right. The black line represents the observed record, the yellow line represents the simulation result with the constant rise time when the subfault division number is 11 \(\times\) 7, the green line represents the simulation result with the modified rise time expression when the subfault division number is 11 \(\times\) 7, the brown line represents the simulation result with the constant rise time when the subfault division number is 44 \(\times\) 28, and the blue line represents the simulation result with the modified rise time expression when the subfault division number is 44 \(\times\) 28). Overall, after using the improved rise time expression proposed in this study, the accuracy of the simulated response spectra and amplitude spectra has been improved in the broadband frequency range. When the subfault division number of the Northridge earthquake fault is 9 \(\times\) 10, and the subfault division number of the Lushan earthquake fault is 11 \(\times\) 7, the simulation results are in good agreement with the observed records. Therefore, the improved stochastic Green's function method proposed in this study can better reproduce the ground motion in the near-fault region. However, due to the uncertainty of source parameters, propagation paths, and local site conditions, as well as the approximation of the simulation method, there are still some differences between the simulation results and the observed records [52; 53]. In addition, it seems that the simulation results show stronger sensitivity to the rise time expression at stations with forward directivity effects, such as L09 station in the Northridge earthquake and 51 LSF station in the Lushan earthquake, while the simulation results show stronger sensitivity to the subfault division number at backward stations, such as HOW station in the Northridge earthquake and 51 BXZ and 51 BXM stations in the Lushan earthquake. It needs to be verified with more earthquake events in the future. Figure 14: Simulation results of the 51 YAM station.