# A Numerical Model for Simulating Ground Motions for the Korean Peninsula
## 3 Ground Motion Simulation Model
### Shapping Window Model Estimation  
In order to generate a ground acceleration in the time domain (\(\ddot{u}_{g}(t)\)) from \(A(f)\), a proper shaping window function should be applied [12]. A shaping window model consists of a ground motion envelope shape and duration time. [PERSON] [12] simulated ground motion recordings using an envelope shape proposed by [PERSON] and [PERSON] [7] and a duration time proposed by [PERSON] and [PERSON] [11]. However, the shaping window model used by [PERSON] [12] may not properly reflect the envelope shape of the ground motions recorded in the Korean Peninsula. In this study, a shaping window model (\(W(t)\)) is proposed based on the ground motions recorded during the 2017 Pohang earthquake (Equation (10)).  
\[\ln(W(t,T_{D}))=c_{0}+c_{1}\ln\left(\frac{t}{T_{D}}\right)+c_{2}\frac{t}{T_{D}} \tag{10}\]  
where \(t\) is the time, and \(T_{D}\) is the duration time. The duration time is calculated from the S-wave arrival time to the time corresponding to the 95% normalized arias intensity energy (\(E(t)\)). \(E(t)\) is estimated using Equation (11).  
\[E(t)=\frac{\int_{0}^{t}\ddot{u}_{g}^{2}(t)dt}{E_{T}} \tag{11}\]  
where \(E_{T}\) is the total energy of \(\ddot{u}_{g}(t)\). Figure 10a-c show the estimated duration time of a ground motion recorded at the BUS2 station, \(E(t)\), and estimated shaping window, respectively. Figure 11 shows the estimated shaping window and duration of ground motions recorded at 111 stations during the 2017 Pohang earthquake. The coefficients \(c_{0}\), \(c_{1}\), and \(c_{2}\) in Equation (10) are estimated to match the median envelope shape denoted with a thick solid line in Figure 7a.  
Figure 10: Estimated shaping window and duration at the BUS2 station for the 2017 Pohang earthquake: (**a**) Ground motion acceleration, (**b**) Normalized arias intensity, (**c**) Shaping window.  
Figure 9: Measured and calculated Fourier amplitude spectrum (FAS) at three sample stations: (**a**) BUS2 station, (**b**) CEA station, (**c**) CEJA station.  
Regression analyses were conducted, by which the values of \(c_{0}\), \(c_{1}\), and \(c_{2}\) were determined to be 1.6546, 0.6227, and \(-\)3.2663, respectively. The envelope shape obtained from the 2017 Pohang earthquake is significantly different from that calculated using the equations proposed by [PERSON] (2003). The envelope shape constructed from the equation proposed by [PERSON] (2003) significantly underestimates the measured envelope shape of ground motions recorded during the 2017 Pohang earthquake.  
Figure 11b shows the estimated duration time (\(T_{D}\)) at each station. Previous studies [10; 11; 30] reported that duration time is mainly affected by corner frequency (\(f_{c}\)) and source-to-site distance (\(R_{H}\)). In this study, the following equation is proposed for \(T_{D}\), according to \(R_{H}\) and \(f_{c}\). Figure 11b shows the duration time calculated using Equation (12). This figure also shows that \(T_{D}\) calculated from Equation (12) distinctively differs from that calculated using the equations proposed by [PERSON] and [PERSON] [11]. The equation of \(T_{D}\) developed for the northeastern United States could underestimate \(T_{D}\) obtained from the 2017 Pohang earthquake. This observation reveals the importance of why numerical models should be developed for low-to-moderate seismic regions, such as the Korean Peninsula.  
\[T_{D}=\frac{1}{f_{c}}+\left\{\begin{array}{ll}3.256&(R_{H}\leq 10\ \mathrm{km})\\ -0.247+0.350R_{hypo}&(10\ \mathrm{km}<R_{H}\leq 50\ \mathrm{km})\\ -19.522-0.045R_{hypo}&(50\ \mathrm{km}<R_{H}\leq 100\ \mathrm{km})\\ 9.005+0.060R_{hypo}&(R_{H}>100\ \mathrm{km})\end{array}\right. \tag{12}\]