## 4 Conclusions  
In order to improve the accuracy of the stochastic Green's function method in simulating near-fault broadband ground motion, based on the numerical simulation results of Day on the earthquake slip function, an improved approximate expression for the rise time on a rectangular fault considering that the rupture starting point may be at any position and the aspect ratio may be arbitrary is proposed in this study. On this basis, considering the contributions of P wave, SV wave, and SH wave to the 3D ground motion, respectively, an improved stochastic Green's function method for simulating 3D broadband ground motion is proposed. Taking the 1994 Northridge earthquake in the United States and the 2013 Lushan earthquake in China as examples, under different subfault division numbers, the source spectra synthesized using the improved approximate expression for the rise time are compared with the theoretical \(\omega^{-2}\) source spectra of the large earthquake, then the ground motions at the observation points are simulated using the improved method, and the simulation results are compared with the observed records. After that, the effectiveness of the improved stochastic Green's function method proposed in this study is verified. The following conclusions can be drawn:  
Figure 19: The simulation misfit of the ground motion of the Lushan earthquake. (The yellow line represents the simulation misfit with the constant rise time when the subfault division number is 11 \(\times\) 7, the green line represents the simulation misfit with the modified rise time expression when the subfault division number is 11 \(\times\) 7, the brown line represents the simulation misfit with the constant rise time when the subfault division number is 44 \(\times\) 28, and the blue line represents the simulation misfit with the modified rise time expression when the subfault division number is 44 \(\times\) 28). If it is assumed that the rise time is uniformly distributed on the fault plane, as the subfault division number gradually increases, the sag phenomenon of the source spectrum of the large earthquake synthesized by superimposing small earthquakes using the stochastic Green's function method in the intermediate frequency band will become more and more significant. After improving the expression for the rise time, as the subfault division number increases, the amplitude change in the synthesized source spectrum of the large earthquake is not significant. 2. If the rise time is assumed to be constant, the amplitude of the response spectrum and amplitude spectrum of the simulated ground motion at the observation point will decrease in the intermediate frequency range. As the subfault division number increases, the amplitude decrease becomes more pronounced. After using the improved rise time expression, the sag phenomenon that appeared in the intermediate frequency range is alleviated, and the simulation results are closer to the observed records. As the subfault division number increases, the decrease in amplitude in the intermediate frequency range is not as significant as the decrease when the rise time is constant. 3. After using the improved rise time expression proposed in this study, when the subfault division number of the Northridge earthquake fault is 9 \(\times\) 10, in the period range of 0.01 s to 6 s, the simulation misfit values of the NS, EW, and UD components are within \(\pm\)0.35, \(\pm\)0.3, and \(\pm\)0.25, respectively; when the subfault division number of the Lushan earthquake fault is 11 \(\times\) 7, in the period range of 0.01 s to 6 s, the simulation misfit values of the NS and EW components are within \(\pm\)0.2, and the simulation misfit of the UD component is within \(\pm\)0.3.