## 2 Stochastic Green's Function Method for 3D Ground Motion Simulation and Its Improvement
### Fourier Amplitude Spectra of Small Earthquakes in Subfaults  
Based on the \(\omega^{-2}\) model, the acceleration Fourier spectrum \(A_{ij}^{K}(f)\) excluding the local site amplification effect of the small earthquake at the observation point caused by the _ij_ subfault can be expressed as [7; 9; 10]:  
\[A_{ij}^{K}(f)=\frac{F_{s}^{K}R_{\Phi\Phi ij}^{K}}{4\pi\rho\sigma^{3}}\frac{(2 \pi f)^{2}M_{0 ij}}{1+\left(\frac{f}{f_{\rm maxij}}\right)^{2}}\left[1+\left( \frac{f}{f_{\rm maxij}}\right)^{2}\right]^{-\frac{1}{2}}\left(\frac{\rho v}{ \rho_{x}v_{\rm z}}\right)^{\frac{1}{2}}N_{ij}^{K}(f)\frac{1}{r_{ij}}\exp\left[ \frac{-\pi fr_{ij}}{Q_{K}(f)v}\right] \tag{1}\]  
Figure 1: Stochastic Green’s function method for 3D ground motion simulation. (\(OXYZ\) 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, \(\alpha y\) 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, red arrows indicate the vibration direction of waves). In which \(f\) is the frequency; the superscript \(K\) represents the P wave, SV wave, or SH wave; and the subscripts \(i\), \(j\) indicate the serial number of the subfault along the length and the width. \(F_{\text{s}}^{K}\) is the amplification factor of the free surface. As seismic waves near the ground surface propagate in a nearly vertical direction, \(F_{\text{s}}^{K}\) is taken as 2.0 for all types of waves. \(R_{\Phi\phi ij}^{K}\) is the radiation pattern considering the frequency and distance dependence [18]. \(M_{0 ij}\) and \(r_{ij}\) represent the seismic moment and the source distance of the small earthquake, respectively. \(f_{cij}^{K}\) and \(f_{\text{max}ij}\) are the corner frequency [21; 22] and high-frequency cutoff frequency [23] of the small earthquake, respectively. \(\rho\), \(v\) represent the medium density and the P-wave or S-wave velocity near the source, respectively. \(\rho_{\text{z}}\), \(v_{\text{z}}\) represent the medium density and P-wave or S-wave velocity at the seismic bedrock, respectively. \(N_{ij}^{K}(f)\) is the correction term including the influence of the near-field and intermediate-field terms of the _ij_ subfault [9; 10; 24]. \(Q_{K}(f)\) is the quality factor, which represents the anelastic attenuation characteristics of the propagation path. In this study, in order to improve the accuracy of low-frequency ground motion simulation using the stochastic Green's function method, according to the study of [PERSON] et al. [25] and [PERSON] et al. [26] on the frequency and distance dependence of radiation pattern, \(R_{\Phi\phi ij}^{K}\) in Equation (1) is calculated using the formula that considers the effects of both frequency and distance [18]:  
\[R_{\Phi\phi ij}^{K}(f,r_{ij})=\left\{\frac{R_{\Phi\phi ij}^{K}(f)}{\frac{[\log (r_{2})-\log(r_{ij})]R_{\Phi\phi ij}^{K}(f)+[\log(r_{ij})-\log(r_{1})]R_{\Phi \phi\phi}^{K}}{\log(r_{2})-\log(r_{1})},r_{1}<r_{ij}\leq r_{2}}{R_{\Phi\phi \phi}^{K}},r_{ij}>r_{2}\right. \tag{2}\]  
where  
\[R_{\Phi\phi\phi ij}^{K}(f)=\left\{\frac{[\log(f_{2})-\log(f)]R_{\Phi\phi ij}^{ K}+[\log(f)-\log(f_{1})]R_{\Phi\phi\phi}^{K}}{\log(f_{2})-\log(f_{1})},f_{1}<f \leq f_{2}}{R_{\Phi\phi\phi}^{K}},f_{1}<f>f_{2}\right. \tag{3}\]  
In Equations (2) and (3), \(r_{ij}\) represents the source distance of the _ij_ subfault, \(r_{1}\) and \(r_{2}\) represent the distance range where the radiation coefficient transitions linearly with the logarithm of the source distance, with \(r_{1}=40\) km and \(r_{2}=100\) km, respectively;\(f_{1}\) and \(f_{2}\) represent the frequency range in which the radiation coefficient transitions linearly with the logarithm of the frequency. In this study, it is assumed that \(f_{1}=0.5\) Hz and \(f_{2}=5\) Hz.