## 2 Methodology
### Stochastic Point-Source Model  
#### 2.2.1 Parameters of Stochastic Point-Source Model  
The Fourier Amplitude Spectrum (FAS) of ground motion can be determined using the stochastic point-source model [6]:  
\[A(M_{0},R,f)=E(M_{0},f)P(R,f)G(f)I(f) \tag{1}\]  
where \(M_{0}\) is the seismic moment, \(R\) is the source-to-site distance, and \(f\) is the frequency. This equation consists of four components: the source effect, \(E(M_{0},f)\), which characterizes the earthquake source mechanism; the path effect, \(P(R,f)\), which describes the attenuation of seismic waves along their propagation path; the site effect, \(G(f)\), which accounts for local site amplification and attenuation; and the type of motion, \(I(f)\), which facilitates expressing the spectrum as acceleration \((=-2\pi f)\). The site effect, \(G(f)\), is divided into two distinct factors: the site amplification factor, denoted by \(A^{*}(f)\), and the diminution factor, denoted by \(D(f)\):  
\[G(f)=A^{*}(f)D(f) \tag{2}\]  
The expression for \(D(f)\) involves the high-frequency spectral decay factor, \(\kappa_{0}\). \[D(f)=\exp\{-\pi f\kappa_{0}\} \tag{3}\]  
To determine \(\kappa_{0}\), the slope of the Fourier amplitude is calculated for each station that recorded seismic data during the 2017 Pohang Earthquake [25]. For Fourier spectra with a negative slope, the decrease in amplitude beyond a specific frequency is estimated [26]. In this study, a \(\kappa_{0}\) value of 0.014 s is employed, where it represents the zero-distance intercept of the values obtained from each station, as originally defined in [25]. The site amplification factor is determined using the horizontal-to-vertical spectral ratio, with a damping ratio of 5% applied at each station [27]. \[A^{*}(f)=\frac{RSA_{H}(f)}{RSA_{V}(f)} \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Information** & **Details** \\ \hline UTC time & 15 November 2017 05:29:31 \\ KST time (local time) & 15 November 2017 14:29:31 \\ Magnitude & \(M_{L}\) 5.4, \(M_{w}\) 5.4 \\ Depth (km) & 7 km \\ PGA (g) & 0.58 g (\(R=7.46\) km) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed information on the 2017 Pohang Earthquake. where \(RSA_{H}(f)\) and \(RSA_{V}(f)\) denote the response spectra for the horizontal and vertical components of ground motion, respectively. The path effect, \(P(R,f)\), consists of geometrical spreading, \(R^{n}\), and the residual term, which is the anelastic attenuation function:  
\[P(R,f)=R^{n}\text{exp}\bigg{\{}\frac{-\pi fR}{\beta Q(f)}\bigg{\}} \tag{5}\]  
where the geometrical spreading function determined for low-to-moderate seismicity regions of Eastern North America (ENA) is uniformly applied in this study as follows [28]:  
\[R^{n}=\left\{ \begin{array}{ccc}R^{-1.3}&(R<70\;\;\text{km})\\ 70^{-1.3}(R/70)^{0.2}&(70\;\;\text{km}\leq R<130\;\;\text{km})\\ 70^{-1.3}(130/70)^{0.2}(R/70)^{-0.5}\;(130\;\;\text{km}\leq R)\end{array} \right. \tag{6}\]  
In Equation (5), \(\beta\), representing the shear wave velocity near the source, is 3.5 km/s according to [29]. The quality factor, \(Q(f)\), is derived by separating observed earthquake ground motions into unknown terms associated with the source, path, and site effects. Using this methodology, the quality factor for frequencies above 1 Hz is obtained through regression analysis, resulting in the formula: \(Q(f)=163f^{0.745}\). The source effect, \(E(M_{0},f)\) in Equation (1), is expressed as:  
\[E(M_{0},f)=CM_{0}S(f)=\frac{CM_{0}}{1+\left(f/f_{c}\right)^{2}} \tag{7}\]  
Here, \(f_{c}\) is the corner frequency and the constant \(C\) is given by:  
\[C=\frac{R_{p}VF}{4\pi\rho\beta^{3}R_{0}} \tag{8}\]  
where \(R_{p}\) (the wave radiation pattern), \(V\) (the partition of total shear-wave energy into horizontal components), and \(F\) (a parameter indicating the effect of the free surface) are assumed to be 0.55, 0.707, and 2, respectively. The value of 0.55 is widely accepted as the average of body-wave radiation patterns over various azimuths and take-off angles [30; 31].