Simulating earthquake ground motion at a site, for given intensity and uncertain source location

Following a companion article, ground motion acceleration time histories during earthquakes can be described as realizations of non-stationary stochastic processes with evolutionary frequency content and instantaneous intensity. The parameters characterizing those processes can be handled as uncertain variables with probabilistic distributions that depend on the magnitude of each seismic event and the corresponding source-to-site distance. Accordingly, the generation of finite samples of artificial ground motion acceleration time histories for earthquakes of given intensities is formulated as a two-stage Monte Carlo simulation process. The first stage includes the simulation of samples of sets of the parameters of the stochastic process models of 'earthquake ground motion. The second stage includes the simulation of the time histories themselves, given the parameters of the associated stochastic process model. In order to account for the dependence of the probability distribution of the latter parameters on magnitude and source-to-site distance, the joint conditional probability distribution of these variables must be obtained for a given value of the ground motion intensity. This is achieved by resorting to Bayes Theorem about the probabilities of alternate assumptions. Two options for the conditional simulation of ground motion time histories are presented. The more refined option makes use of all the information about the conditional distribution of magnitude and distance for the purpose of simulating values of the statistical parameters of the ground motion stochastic process models. The second option considers all probabilities concentrated at the most likely combination of magnitude and distance for each of the seismic sources that contribute significantly to the seismic hazard at the site of interest.