Asymptotic behavior of seismic hazard curves

Hazard curves from probabilistic seismic hazard analysis (PSHA) are plots of the rate of earthquakes exceeding ground motion intensity values vs such threshold values, for a site of interest. In classical PSHA, these curves can be transformed to provide the probability of exceedance of ground motion intensity values in any time interval, utilizing the properties of the homogeneous Poisson process (HPP). In turn, these probability curves can be seen as the plot of the complementary cumulative distribution function of the maximum intensity observed, at the site, in the time interval of interest. One consequence of the HPP framework, within which PSHA is developed, is that, for large time intervals, it can be argued that these curves could asymptotically lead to a probabilistic model for extreme value (EV) random variables. This is discussed, with a simple engineering approach, in this short note, where it is found – via case studies – that exceedance hazard curves seem to converge towards an EV distribution (i.e., the EV type II or Fréchet), with a pace that is impacted by the discontinuity inherent to the curves. It is also seen that other common models, typically used to provide an analytical format to probabilistic curves, do not show the same level of convergence. Besides providing further insights on the results of PSHA, this study can possibly be useful for those cases where a closed-form equation for the hazard curve could be needed, such as reliability-based calibration of building codes, or seismic risk studies involving seismic hazard approximation/extrapolation.