A Practical Approach of Probabilistic Seismic Hazard Analysis for Vector IMs Regarding Mainshock with Potentially Largest Aftershock

Seismic hazard and risk may be underestimated if aftershocks are ignored within the traditional probabilistic seismic hazard analysis (PSHA). In this study, a practical approach is proposed to calculate the mean annual rate of the joint exceedance of different ground motion vector intensity measures (IMs) for mainshocks and their potentially largest aftershock. The proposed approach is built based on an indirect vector-valued PSHA framework, which was originally used to compute the joint hazard of multiple IMs for mainshocks. Within the proposed approach, the overall hazard caused by mainshock-aftershock scenarios is calculated by including two key components, namely: 1) the occurrence rate of a given hazard level of mainshock as calculated by the standard PSHA; and 2) the occurrence probability of a given hazard level of aftershock conditioned on the mainshock, which is calculated by establishing the joint distribution of the IMs for both mainshocks and aftershocks using the Copula theory. Based on 662 groups of mainshock-aftershock ground motions from 13 earthquake events in the NGA-West2 database, the empirical joint probability distribution model between the peak ground accelerations of mainshocks and aftershocks (i.e. PGA(MS) and PGA(AS)) is developed. The Clayton Copula function is calibrated as the optimal choice and the marginal distributions of PGA(MS) or PGA(AS) could be properly modelled using the generalized extreme value distributions. The proposed approach is implemented to construct the joint hazard surface regarding both PGA(MS) and PGA(AS) for a hypothetical site. The results indicate that using different types of copula functions and marginal distributions of PGA(MS) and PGA(AS) has a noticeable impact on the final joint hazard level. Moreover, a coarse assumption that PGA(MS) and PGA(AS) follow a joint Gaussian distribution in the logarithm scale would lead to considerable bias in the mainshock-aftershock joint hazard especially at low mean annual rates of exceedance.