Conditional mean spectrum of aftershocks

This study proposes a pragmatic method to generate the conditional mean spectrum of aftershocks (CMS_A), which reasonably considers the dependence of the spectral shapes (epsilons) between mainshocks and aftershocks using the copula technique. To obtain the CMS_A, the conditional mean epsilons of aftershocks are computed, conditioned on the corresponding values of mainshocks, and then these conditional mean epsilons are further used to modify the predicted response spectra of aftershocks estimated by a specific ground-motion prediction equation. To validate the presented method, a suite of 662 real mainshock–aftershock ground-motion sequences from 13 earthquake sequences are selected to form the database for this study. The Abrahamson et al. (2014, hereafter, ASK14) model is selected to predict the response spectra of aftershocks. This study builds the joint probability distributions between mainshock epsilons and aftershock epsilons with copula functions. The CMS_A values are calculated for all the selected mainshock ground motions according to the procedures of generating CMS_A. The results show that, compared with the predicted response spectra estimated by the ASK14 model, the CMS_A matches the as-recorded response spectrum of aftershocks better, especially for long periods. This result validates the presented method of generating the CMS_A.