Modeling the joint probability distribution of main shock and aftershock spectral accelerations

Seismic risk analysis of deteriorating structures and infrastructure often requires predicting the intensity measures of earthquake ground motions in main shock-aftershock sequences. The uncertainty in the intensity measures of ground motions is typically a dominant contributor to the total uncertainty of the seismic risk analysis. A model for the joint probability distribution of main shock and aftershock intensity measures is thus required to accurately quantify the uncertainty in the seismic risk analysis. The spectral accelerations of ground motions have been identified as significant intensity measures for the seismic risk analysis of structures and infrastructure. The values of spectral accelerations can be affected by many factors representing the characteristics of the seismic source, travel path of seismic waves, and local site conditions. These factors can also introduce statistical dependence among main shock and aftershock spectral accelerations. This paper develops a novel formulation for the joint probability distribution of main shock and aftershock spectral accelerations at multiple periods. We select existing predictive models for the spectral accelerations of main shocks and develop a separate model for the spectral accelerations of aftershocks. The proposed formulation also estimates the correlations between the relevant pairs of model error terms in the two probabilistic predictive models for a wide range of periods. This allows us to separately capture the similarity in source and site and thus present the physical meanings. The increased vulnerability of structures and infrastructure in the aftermath of a damaging mainshock can further highlight the significance of capturing such correlations in the seismic risk analysis.