## 4 Probabilistic Seismic Hazard Analysis and Evaluation of Example Scenario Earthquake
### Target Spectrum for Selecting Seed Ground Motions in Risk Assessment  
Based on the preceding steps, once the risk assessment spectrum has been evaluated and the scenario earthquake has been identified, the next step is to select appropriate ground motion records that match both the spectrum shape and seismic characteristics. The risk assessment spectrum, UHS, assumes a uniform annual frequency of exceedance at all  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Magnitude bin (Mw) & 6.4–6.6 \\ \hline Distance\({}^{*}\)bin (km) & 0–10 \\ \hline \(\varepsilon\) bin & 0.5–1.0 \\ \hline Contribution to Hazard (\%) & 3.93 \\ \hline \end{tabular} \({}^{*}\) Closest distance to the conseismic fault rupture plane.  
\end{table}
Table 2: Parameters for scenario earthquake based on the PSHA de-aggregation results.  
Figure 4: The PSHA de-aggregation result for magnitude, distance, and epsilon at a spectral period of 0.2 s based on the UHS with a 0.2 \(\times\) 10\({}^{-4}\) annual frequency of exceedance.  
periods; however, this does not represent a single ground motion because the hazard curves of each period are obtained independently. This means that selecting a ground motion compatible with the UHS conservatively implies large amplitude spectral acceleration values for most spectral periods for a single ground motion. Due to the conservatism in the UHS, the scenario earthquake defined by magnitude-distance-epsilon pairs may have lower spectral acceleration values except at the target period specified in the de-aggregation procedure. When selecting ground motions compatible with the UHS, those with spectral acceleration values lower than the target were selected. To exclude the conservatism of the UHS, a conditional mean spectrum (CMS) was developed that considered both the natural period of the structure and scenario earthquake [36]. CMS provides the mean response spectrum conditioned on the occurrence of the UHS spectral acceleration value at the natural period of the structure. In other words, the specific natural period of the CMS represents the UHS values, and other period bands of the CMS are determined based on the values between the median of the GMPE and the UHS, considering the correlation between periods [36]. Therefore, when ground motions are selected based on the CMS, they can preserve the scenario earthquake conditions estimated by hazard de-aggregation. In this study, the CMS was calculated based on the de-aggregation results in Table 2, and the selection of ground motion time histories based on the CMS was presented as a procedure for generating the acceleration ground motion for seismic risk assessment. The CMS calculation equation is given by Equation (4):  
\[\ln Sa^{CMS}(T_{i})\big{|}\ln Sa^{CMS}(T^{*})=\overline{\ln Sa}(M,R,T_{i})+ \rho(T_{i},T^{*})\times\sigma_{\ln Sa(T_{i})}\times\varepsilon(T^{*}) \tag{4}\]  
where \(\ln Sa^{CMS}(T_{i})\big{|}\ln Sa^{CMS}(T^{*})\) represents the CMS at all periods (\(T_{i}\)) and target period (\(T^{*}\)) and \(\overline{\ln Sa}(M,R,T_{i})\) is the logarithmic mean spectral acceleration value for magnitude (\(M\)) and distance (\(R\)) at all periods, and \(\rho(T_{i},T^{*})\) is the correlation coefficient between spectral accelerations at all periods and target period. \(\sigma_{\ln Sa(T_{i})}\) is the standard deviation of mean spectral acceleration from GMPE at all periods, and \(\varepsilon(T^{*})\) is the number of standard deviations by which given target spectral acceleration differs from the logarithmic mean spectral acceleration value. The risk assessment spectrum, UHS, and calculated CMS for the scenario earthquakes in Table 2 are shown in Figure 5. As shown in Figure 5, at the target period of 0.2 s, the spectral acceleration of the CMS matches that of the UHS. In other periods, the spectral acceleration appeared as a conditional mean relative to the GMPE median value, depending on the ground motion correlation coefficient.  
Figure 5: Comparison of the CMS for the structure’s natural period and UHS.