## 3 Ground Motion
### Conditional Spectrum  
The CS was computed based on the dominant earthquake magnitude and epicentral distance determined by the deaggregation of the PSHA at the control frequency and the corresponding spectral acceleration shared by the reference UHRS. The CS is significantly affected by the control frequency. This study adopted a total of three control frequencies. The first control frequency, \(f_{c1}\), is half the first modal frequency, the second control frequency, \(f_{c2}\), is the first modal frequency, and the third control frequency, \(f_{c3}\), is the second modal frequency. The first control frequency, \(f_{c1}=1.85\) Hz, is an approximately half of \(f_{c2}\) and reflects a decrease in stiffness owing to the inelastic behavior of the containment building subjected to ultimate deformation, although it was not determined in a quantitative manner. Hereafter, the CS derived for the three control frequencies, \(f_{c1}\), \(f_{c2}\), and \(f_{c3}\) are denoted as CS1, CS2, and CS3, respectively. Table 4 summarizes the control frequencies and corresponding spectral accelerations. In a CS, the correlation between the spectral accelerations, of which one is at the control frequency and other frequencies, is defined using the epsilon coefficient, which indicates how much higher the spectral acceleration at the control frequency is compared to the median spectral acceleration predicted by the GMPEs used in the PSHA ([PERSON] et al., 2018). Therefore, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Response Spectrum** & **Control Frequency** & **Spectral** & **\(\mathbf{r_{r}}\)** \\
**for Seismic Input** & **(Hz)** & **Acceleration (g)** & **Number of PGA in** \\  & & & **UHRS 1 ** \\ \hline CS1 & 1.85 & 0.153 & 0.648 \\ CS2 & 3.47 & 0.311 & 1.318 \\ CS3 & 9.96 & 0.517 & 2.191 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Control frequencies and corresponding spectral accelerations for conditional spectra. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **Median Allowable Drift Ratio** & \begin{tabular}{c} **Logarithmic** \\ **Standard Deviation** \\ \end{tabular} \\ \hline \begin{tabular}{c} Bending \\ controlled walls \\ \end{tabular} & \(V_{ave}>0.50\sqrt{f_{c}^{\prime}}\)(MPa) & 0.007 & \\ \
\begin{tabular}{c} \(V_{ave}<0.25\sqrt{f_{c}^{\prime}}\)(MPa) \\ \end{tabular} & 0.009 & 0.34 \\
\begin{tabular}{c} Shear controlled walls \\ \end{tabular} & 0.007 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Median and logarithmic standard deviation of allowable drift ratio for limit state [1]. the construction of a CS requires information on the magnitude and epicentral distance characterizing the dominant earthquake scenario identified by the deaggregation of the seismic hazard at the control frequency. Contribution of the magnitude-epicentral distance combination obtained by the deaggregation of the seismic hazard at the control frequency for each CS is represented in Figure 4. It was observed that the contribution of the scenario with a low magnitude and close distance increased as the control frequency increased. Although the scenario with magnitude 6.2 to 6.4 and epicentral distance 0 to 20 km contributes the most significantly, the proportion of the total is only 13.3 to 13.6%, which is difficult to regard as the single dominant scenario. This is because the seismic source model used in the PSHA was a type of area seismic source with a constant annual rate of exceedance for the earthquake magnitude over the entire area. Therefore, it is necessary to consider various scenarios besides the one with the maximum contribution to the calculation of the CS. In this study, the median and standard deviation of the CS, assumed to be lognormally distributed, were calculated by summing the contributions of all the scenarios shown in Figure 4, based on the methodology proposed by [PERSON] et al. (2013) [30]. Figure 5 shows the median spectra of CS1, CS2, and CS3 along with the UHRS. It was observed that each CS was in contact with the UHRS at the control frequency.