# A Numerical Model for Simulating Ground Motions for the Korean Peninsula
## 3 Ground Motion Simulation Model
### Stochastic Point-Source Model Estimation  
In the stochastic point-source model, the Fourier amplitude spectrum [\(A(f)\)] of a ground motion is calculated using Equation (2). \[A(f)=Source(M_{0},f,f_{c})\times Path(R_{H},f)\times Site(\kappa_{0},f) \tag{2}\]  
where \(Source(M_{0},f,f_{c})\) is the earthquake source effect function, \(Path(R_{H},f)\) is the path effect function, \(Site(\kappa_{0},f)\) is the site effect function, \(M_{0}\) is the seismic moment, \(R_{H}\) is the hypocentral distance, \(f\) is the frequency, and \(f_{c}\) is the corner frequency. A detailed explanation of individual functions is summarized in Table 2.  
The component parameters \(\kappa_{o}\), \(M_{o}\), and \(f_{c}\) to calculate \(A(f)\) were estimated from the ground motions recorded at individual stations during the 2017 Pohang earthquake, whereas the values of other component parameters were adopted from previous studies, as listed in Table 2.  
To effectively determine the values for \(\kappa_{o}\), \(M_{o}\), and \(f_{c}\), the \(A(f)\) of the recorded ground motions were smoothed using a technique proposed by [PERSON] and [PERSON] [21]. Figure 5 shows the original and smoothed \(A(f)\) of ground motions recorded at three sample stations (BUS2, CEA, and CEJA) during the 2017 Pohang earthquake. In this study, the site attenuation coefficient, \(\kappa_{0}\), was determined first. Since the contribution of \(\kappa_{0}\) in Equation (2) is determined without interaction with path effect and site amplifica  
\begin{table}
\begin{tabular}{l l l l} \hline \multicolumn{2}{c}{**Functions**} & \multicolumn{2}{c}{**Parameters**} \\ \hline \multicolumn{2}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \multicolumn{2}{c}{} & & \(\left(R_{0}\right)\) & \(\left(=0.63\right)\) & \(\left(=0.63\right)\) & \(\left(R_{H}\leq 70\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(\sum_{2}\right)\): free surface effect [9] & \(\left(70\text{ km}<R_{H}\leq 100\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi ft_{H}/Q(f)\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(R_{H}>100\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi ft_{H}/Q(f)\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(R_{H}>100\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi ft_{H}/Q(f)\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(R_{H}>100\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi\pi\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) \\ \multicolumn{2}{c}{} & \(\left(R_{H}\right)\times\exp(-\pi\beta_{c})\) & \(\left(70\text{ km}<R_{H}\text{ km}\right)\) & \(\left(7 of path effect and site amplification were removed from Equation (2). The Fourier amplitude spectrum without path effect and site amplification [\(A^{\prime}(f)\)] was calculated using Equation (3). \[A^{\prime}(f)=\frac{A(f)}{Path(R_{H},f)\times AMP(f)}. \tag{3}\]  
[PERSON] and [PERSON] [25] estimated the site attenuation coefficient (\(\kappa_{0}\)) using a residual slope of \(A(f)^{\prime}\) between 10 Hz and 40 Hz, which was also used in this study. Figure 6 shows \(A(f)\) and \(A(f)^{\prime}\). The slope (\(\kappa_{0}\)) of \(A(f)^{\prime}\) is also plotted in Figure 6. The values of \(\kappa_{0}\) estimated for individual stations are summarized in Table 1 of Appendix A. Previous studies [9; 26; 27; 28] estimated \(M_{o}\) and \(f_{c}\) by using the omega-square model proposed by [PERSON] [18; 19] based on the FAS of the recorded ground motions.