Abstract
The paper considers Poisson temporal occurrence of earthquakes and presents a way to integrate uncertainties of the estimates of mean activity rate and magnitude cumulative distribution function in the interval estimation of the most widely used seismic hazard functions, such as the exceedance probability and the mean return period. The proposed algorithm can be used either when the Gutenberg–Richter model of magnitude distribution is accepted or when the nonparametric estimation is in use. When the Gutenberg–Richter model of magnitude distribution is used the interval estimation of its parameters is based on the asymptotic normality of the maximum likelihood estimator. When the nonparametric kernel estimation of magnitude distribution is used, we propose the iterated bias corrected and accelerated method for interval estimation based on the smoothed bootstrap and second-order bootstrap samples. The changes resulted from the integrated approach in the interval estimation of the seismic hazard functions with respect to the approach, which neglects the uncertainty of the mean activity rate estimates have been studied using Monte Carlo simulations and two real dataset examples. The results indicate that the uncertainty of mean activity rate affects significantly the interval estimates of hazard functions only when the product of activity rate and the time period, for which the hazard is estimated, is no more than 5.0. When this product becomes greater than 5.0, the impact of the uncertainty of cumulative distribution function of magnitude dominates the impact of the uncertainty of mean activity rate in the aggregated uncertainty of the hazard functions. Following, the interval estimates with and without inclusion of the uncertainty of mean activity rate converge. The presented algorithm is generic and can be applied also to capture the propagation of uncertainty of estimates, which are parameters of a multiparameter function, onto this function.
Highlights
The probabilistic seismic hazard is a potential possibility of the occurrence of ground motion caused by seismicity, expressed in the form of likelihoods.This possibility results from probabilistic properties of the seismic source, propagation of seismic waves from the source to a receiver and receiving site
We have presented a way to integrate the uncertainty of mean activity rate and magnitude cumulative distribution function (CDF) estimates in the interval estimation of the most widely used seismic hazard functions, namely the exceedance probability, R(M, D) and the mean return period, T(M)
The proposed algorithm can be used in both situations, either when the parametric model of magnitude distribution is accepted or when the nonparametric estimation is in use
Summary
The probabilistic seismic hazard is a potential possibility of the occurrence of ground motion caused by seismicity, expressed in the form of likelihoods. When the Poisson model for earthquake occurrences is accepted, the exceedance probability, that is the probability that the amplitude parameter of ground motion will exceed a at (x0, y0) in any time intervthe seismic source and pointal of length D time units is:. The other function often used to express probabilistic properties of seismic sources when the Poisson model for occurrence is applied, is the reciprocal of the rate of occurrence of earthquakes of magnitude M or greater referred to as the mean return period The mean return period is the average recurrence time of events of magnitude M or greater Both these hazard functions, R(M, D) and T(M), depend on the mean event rate of the Poisson temporal occurrence of earthquakes and the distribution of magnitude. On synthetic and actual seismicity cases we analyze improvements introduced by such an integrated approach
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