## 2 Methodology
### Seismic Source Characterization and PGA Distribution Modeling  
#### 2.3.1 Statistical Modeling of Source Parameters for the 2017 Pohang Earthquake  
Two critical parameters affecting the source effect, \(f_{c}\) and \(M_{0}\), were analyzed for the 2017 Pohang Earthquake. These parameters were obtained from 111 out of 130 stations, all located within a 500 km radius of the epicenter, with the source-to-site distances ranging from 11.58 to 413.18 km. According to Equation (1), the Fourier amplitude is influenced by the source-to-site distance, \(R\), and assumes stochastic variations in \(f_{c}\) and \(M_{0}\). The randomness of these parameters introduces uncertainty into the Fourier amplitude, leading to variability in the predicted PGAs for artificial earthquakes. Spearman correlation analysis explored relationships between source parameters and source-to-site distance, and among the source parameters themselves. Despite appearing random within a valid range of distance, as depicted in Figure 1a,b, a dependency existsamong the source parameters, as illustrated in Figure 1c. The correlation coefficient between source-to-site distance and \(f_{c}\) was 0.0537, indicating virtually no correlation. Similarly, the correlation coefficient between source-to-site distance and \(M_{0}\) was 0.0349, suggesting a negligible correlation. In contrast, the correlation between \(f_{c}\) and \(M_{0}\) was estimated at -0.4570, indicating a weak negative correlation. This finding underscores the importance of generating sampling points that account for this correlation. Copula functions are widely used to estimate joint probability density functions because they effectively capture the dependence structure between two variables while allowing for the coupling and separation of each probability distribution [39; 40; 41]. The application of copulas is also well-established in earthquake research [42; 43]. A copula is a multivariate cumulative distribution function that uses the complex dependency structures between marginal probability distributions and random variables. According to [PERSON]'s  
Figure 1: Relationship between (**a**) the source-to-site distance and seismic moment, (**b**) the source-to-site distance and corner frequency, and (**c**) corner frequency and seismic moment. theorem, the joint cumulative distribution function of the source parameters \(y_{1}\) and \(y_{2}\) is defined as follows [44]:  
\[F(y_{1},y_{2})=C(F(y_{1}),F(y_{2})|\theta) \tag{18}\]  
where \(C(F(y_{1}),F(y_{2})|\theta)\) represents the copula, \(F(y_{1})\) and \(F(y_{2})\) are the marginal cumulative distribution functions of the source parameters, and \(\theta\) is a parameter indicating the dependency. In the comparative evaluation of five copula functions (Clayton, Frank, Gaussian, Gumbel, and t), the Frank copula was selected as the superior option for calculating the joint probability distribution and generating sampling points for \(f_{c}\) and \(M_{0}\) supported by its highest log-likelihood value as detailed in Table 2.  
This selection was further reinforced by the Frank copula's precise alignment with the target Spearman correlation coefficient of -0.4570, derived from the observed source parameters. The Frank copula function is expressed as follows:  
\[C(u,v)=-\frac{1}{\theta}\text{log}\bigg{[}1+\frac{(\text{exp}\{-\theta u\}-1) (\text{exp}\{-\theta v\}-1)-1}{\text{exp}\{-\theta\}-1}\bigg{]} \tag{19}\]  
where \(u\) and \(v\) are the uniformly distributed random variables and \(\theta\) is a parameter that characterizes the dependence structure between these variables. It is observed that recorded values of \(f_{c}\) and \(M_{0}\) from the 111 data points are more suitably represented by distinct probability distributions. The \(f_{c}\) values follow a lognormal distribution, as expressed in:  
\[f_{1}(x|\mu,\sigma)=\frac{1}{x\sigma\sqrt{2\pi}}\text{exp}\Bigg{\{}\frac{-( \text{ln}\,x-\mu)^{2}}{2\sigma^{2}}\Bigg{\}} \tag{20}\]  
where the mean \(\mu=-0.507\), and the variance \(\sigma^{2}=0.228\).