Evaluation of Vector Hazard for Conditional Mean Spectrum with Different Definitions of Multivariate Exceedance Rate

The conditional mean spectrum (CMS) and its generalized forms, like the generalized CMS (GCMS), and the CMS conditioned at the logarithm-average spectral acceleration over a period range (CMS-AvgSa), are anchored on a conditional period (or periods) with a target hazard level. This study evaluated the vector hazard of multiple spectral accelerations (Sa) at different periods for CMS, GCMS, and CMS-AvgSa. By invoking the invariance property of scalar disaggregation and Copula function, we developed the computational methods for three types of multivariate exceedance rate (MER): multivariate simultaneous exceedance rate (MERs), multivariate arbitrary exceedance rate (MERa), and Kendall-function-based MER (MERk). Taking the cities of Xi'an and Kunming (China) as examples, scalar probabilistic seismic-hazard analysis (PSHA), and disaggregation were performed and CMS was built at different conditional periods (T* = 0.05, 0.2, 1.0, and 2.0 s, respectively). The MERk computation results lies between the other two extreme cases (MERa and MERs), which are either too strict or too tolerant compared with the anchored hazard level. It is found that when the earthquake disaggregation characteristics not differ markedly between T* and other periods in CMS, the calculated MERk results would be roughly consistent with the anchored hazard level at Sa(T*). The conclusion holds true when different Sa(T) were considered in the calculation of MERk . On the other hand, the calculated MERa and MERs would be significantly influenced by the choice of Sa(T) and the results are not comparable with the anchored hazard level. We also evaluated the MERk for GCMS and CMS-AvgSa, which were constructed using mean scenario earthquake from vector-valued PSHA and disaggregation. Comparing with the UHS, the MERk of GCMS and CMS-AvgSa are more close to the anchored hazard level within a certain period range. Although no benchmark exists for vector hazard level, the MERk is likely to be a promising approach for measuring the hazard over a vector of intensity measures in target spectrum.