## 3 Validation of Effectiveness  
### Ground Motion Simulation of the 1994 Northridge Earthquake  
The 1994 Northridge earthquake in the United States is taken as an example for analysis. The moment magnitude of the earthquake is Mw 6.7. The epicenter was located at 34.211\({}^{\circ}\) N, 118.546\({}^{\circ}\) W, and the focal depth was 17.5 km. Referring to the inversion source model of [PERSON] et al. [44] and [PERSON] [45], the fault parameters used in this study are listed in Table 1. The quality factor \(Q_{K}(f)\) is expressed as follows [46]:  
\[Q_{K}(f)=150f^{0.5} \tag{16}\]  
The fault location and station location of the 1994 Northridge earthquake are shown in Figure 2. In the figure, \(OXYZ\) is the overall coordinate system, \(O\) is the projection of the upper left corner of the fault plane on the surface, \(X\), \(Y\), and \(Z\) are the north, east, and vertical downward direction, respectively, \(oxy\) is the local coordinate system of the fault plane, \(o\) is the upper left corner of the fault plane, \(x\) and \(y\) are the direction of fault length and width, respectively, \(Q\) is the rupture starting point on the fault plane, and \(d_{\text{t}}\) is the top depth of the fault plane. To test the effectiveness of the improved method proposed in this study, firstly, the source spectra of the 1994 Northridge earthquake are synthesized using the following two methods with subfault division numbers of 9 \(\times\) 10, 18 \(\times\) 20, and 36 \(\times\) 40, respectively:  
1. Assuming that the rise time is a constant on the fault plane, the stochastic Green's function method where the commonly used Equation (11) is used to calculate the rise time;
2. Considering the non-uniform distribution of rise time on the fault plane, the stochastic Green's function method where the proposed expression (12) is applied to calculate the rise time. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Fault Parameter** & **Parameter Value** \\ \hline \hline  & Strike, dip, rake & 122\({}^{\circ}\), 40\({}^{\circ}\), 101\({}^{\circ}\) \\  & Fault length and width & 18 km, 21 km \\  & Rupture area & 378 km\({}^{2}\) \\  & Depth at the top of the fault & 5.0 km \\ Macroscopic fault & Stress drop & 5.0 MPa \\ parameter & Seismic moment & \(1.3\times 10^{19}\) N\(\cdot\)m \\  & Average slip & 1.0185 m \\  & Medium density near the source & 2.8 g/cm\({}^{3}\) \\  & S wave and P wave velocity near the source & 3.7 km/s, 6.4 km/s \\  & Rupture propagation velocity & 3.0 km/s \\ \hline  & Number of asperities & 2 \\  & Area of the first asperity & 58.8 km\({}^{2}\) \\ Microscopic fault & Slip of the first asperity & 2.016 m \\ parameter & Area of the second asperity & 8.4 km\({}^{2}\) \\  & Slip of the second asperity & 1.779 m \\  & Area of the background & 310.8 km\({}^{2}\) \\  & Slip of the background & 0.6618 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fault parameters of 1994 Northridge earthquake. Figure 2: Fault location and station location of 1994 Northridge earthquake. (\(OXYZ\) is the overall coordinate system, \(O\) is the projection of the upper left corner of the fault plane on the surface, \(X\), \(Y\), and \(Z\) are the north, east, and vertical downward direction, respectively, \(oxy\) is the local coordinate system of the fault plane, \(o\) is the upper left corner of the fault plane, \(x\) and \(y\) are the direction of fault length and width, respectively, \(Q\) is the rupture starting point on the fault plane, and \(d_{\rm t}\) is the top depth of the fault plane. The black star marks the hypocenter, and the black triangles mark the locations of the stations.). The source spectrum here refers to the Fourier amplitude spectrum of the moment-rate function (without considering the influence of high-frequency cutoff frequency) [36].