## 4 Probabilistic Seismic Hazard Analysis and Evaluation of Example Scenario Earthquake
### Development of a Uniform Hazard Spectrum as the Risk Assessment Spectrum  
PSHA calculates the probability of exceeding a specific ground motion intensity within a specific spectral period and uses one of the approaches to efficiently quantify large uncertainties in earthquakes [23]. The uncertainties in earthquake ground motions to be quantified were associated with the seismic characteristics of the source, path, and site effects. These uncertainties can be quantified using the following equation with the magnitude, distance, and epsilon as parameters [24]:  
\[\
u(Sa>y)=\sum_{k}\sum_{j}\
u_{j}\iiint f_{M,\ R,\ E}(m,r,\varepsilon)P(Sa>y|m,r, \varepsilon,GMPE_{k})\ dmdr\&P(GMPE_{k}) \tag{1}\]  
Figure 1: Schematic of the procedure for generating ground motions using recorded ground motions in low to moderate seismicity regions for seismic risk assessment. The frequency of earthquake occurrences was calculated through a statistical analysis of the recorded historical earthquake data. In Equation (1), \(\
u_{j}\) represents the annual rate of occurrence associated with the potential earthquake sources [24]. This rate represents the occurrence of earthquakes with magnitudes greater than the minimum magnitude defined by the Gutenberg-Richter law [25]. The function \(f_{M,\ R,\ E}(m,r,\varepsilon)\) is a joint probability density function for magnitude (\(m\)), distance (\(r\)), and epsilon (\(\varepsilon\)). As path and site effects, to predict the probability of ground motion intensity at the site, the distribution of ground motion intensity according to the magnitude and distance from the earthquake source to the site must be considered. Therefore, the ground motion prediction equation (GMPE) calculated for a specific region of seismic activity considers the attenuation characteristics of ground motion intensity based on different magnitudes and distances. In GMPE, the uncertainty of ground motions is modeled as a log-normally distributed random variable. Therefore, epsilon (\(\varepsilon\)) is defined as the number of logarithmic standard deviations by which ground motions deviate from the median values for a given magnitude (\(m\)) and distance (\(r\)) scenario. \(P(Sa>y|m,r,\varepsilon,GMPE_{k})\) is the probability that the ground motion intensity measure (\(Sa\)) exceeds intensity level (\(y\)) given magnitude, distance, epsilon, and \(GMPE_{k}\). \(P(GMPE_{k})\) denotes the weight of each GMPE. Finally, \(\
u(Sa>y)\) is the result of the PSHA, indicating the annual frequency of exceedance for the ground motion intensity measure exceeding a specified intensity level. By integrating these steps, the PSHA result is expressed as a seismic hazard curve, which is defined as the relationship between the ground motion intensity and the annual frequency of exceedance for a site of interest. In addition, the calculation of seismic hazard across a wide range of natural periods allows for the determination of the UHS based on the value of the ground motion intensity at which all spectral periods exhibit a uniform annual frequency of exceedance. The UHS is a seismic design or assessment spectrum used to select ground motions for the seismic response analysis of structures at a site of interest. According to regulatory guideline 1.208 [13], the 1 \(\times\) 10\({}^{-4}\) and 1 \(\times\) 10\({}^{-5}\) UHS of free-field motions are site-specific earthquake ground motions. The design-based earthquake (DBE) ground motion is now defined using modern probabilistic techniques and is referred to as the SSE ground motion. Appendix A of the ASCE 4-16 guidelines [11] provides an overview of the methodologies used in seismic margin assessment and seismic probabilistic risk assessment with respect to plant facility integrity for ground motions greater than the design level ground motion.