Acceleration time histories have a significant impact on the safety evaluation of key structures because earthquake duration and loading process contribute significantly to uncertainty in structural analysis. Thus, determining acceleration time histories for time history response analysis is a significant practical problem, particularly for non-rock sites. Epsilon (ε) is the number of standard deviations by which the ground motion is above or below the median-predicted motion for the attenuation relationship. This study clarified the role of ε in determining a scenario earthquake to obtain acceleration time histories. As an example, the seismic hazard at a real site is disaggregated into its contributions from discrete variables (M, R, ε) to determine a scenario earthquake. M is the surface wave magnitude and R is the projected epicentral distance along the minor axis of the equivalent ellipse. The target peak ground acceleration (PGA) and a consistent spectrum for the rock site with a 2% probability of exceedance in 50 years using probabilistic seismic hazard analysis were obtained. As the earthquake ground motion at the site increased, the number of potential seismic sources contributing to the probability of exceedance decreased. The fifth potential seismic source dominated the seismic hazard at the real site, given that the target probability of exceedance in 50 years is 2%, so the scenario earthquake was located in this potential seismic source. A sample space formed of M, R, and ε that may generate a PGA greater than or equal to the target PGA at the site was constructed. Thus, the normalized probability of the exceedance of the target PGA is the joint distribution of M, R, and ε. The mean and mode of M, R, and ε are the expected and the most likely event in the sample space, respectively. As the site is located near the geometrical center of the fifth potential seismic source, high-magnitude, near-field seismic events are a major contribution to the seismic hazard at the site. The predictive PGA of the mean and mode of M, R, and ε were computed using an attenuation relationship:the values are significantly larger than those of the target PGA. The difference between the mean/mode response spectra of M, R, and the target spectrum is obvious, especially for the acceleration response at low natural frequencies. For the computed response spectrum of the scenario earthquake fitted to the target PGA and consistent spectrum, the sample space was adapted so that the PGA of M, R, and ε computed from the attenuation relationship was approximately the same as the target PGA. Strong ground motion records were obtained from the NGA database based on the scenario earthquake for use in simulating aleatory uncertainty in rock ground motion. Stochastically generated soil profiles were used to investigate the uncertainty of the dynamic characteristics of soil and shear-wave velocity testing results. The strong ground motion records were combined with the soil profiles to create input files that were used to perform an equivalent linear site response analysis, which included an assessment of uncertainty in the amplification factor. Here, the amplification factor is the ratio between the response spectrum of soil surface acceleration and that of rock acceleration time histories. Thus, the distribution of the amplification factor of the spectrum was obtained. The response spectrum of the scenario earthquake was multiplied by the estimated amplification factor to act as the soil surface acceleration response spectrum. Although the scenario earthquake is neither the expected nor most probable event, its seismic influence field at the site exceeds the target PGA. By taking into account both the target PGA and consistent spectrum, in which all of the events in the fifth potential seismic source will generate ground motion at the site, the safety of important structures can be achieved.