Comparison of response spectrum amplitudes from earthquakes with a lognormally and exponentially distributed return period

Abstract

This paper addresses the significance, for engineering decisions, of replacing the Poissonian by one-step memory models in describing the occurrence of large periodic earthquakes on fault segments along the plate boundaries. A one-step memory model with a lognormally distributed return period was chosen for the analysis. The constraint imposed on both probabilistic models is that, for a given magnitude interval, the median of the lognormally distributed return period is equal to the expected value of the exponentially distributed return period of the Poissonian model. The hypothetical geologic setting chosen for the analysis is in southern California. It consists of a set of faults, one of which exhibits strongly periodic behavior for larger events (a hypothetical segment of the San Andreas fault about 350 km long, for example) whose occurrence is modeled by either of the two probabilistic models, and of other faults for which the earthquakes form a Poissonian sequence of events. PSV spectral amplitudes, PSL LN and PSV EX for the model with a lognormally and exponentially distributed return period, were evaluated for five probabilities of exceedance ( p = 0·01, 0·1, 0·5, 0·9 and 0·99) at a site on the fault and another site away from the fault. The significance of the difference between the PSV amplitudes calculated by the two models, is measured by the ratio of the two amplitudes (by factor f = PSV LN/PSV EX), and by their difference in terms of the overall uncertainty (by the factor f 1 = PSV LN −PSV EX/σ ∗ , where σ ∗ , is a measure of the standard deviation of the distribution of PSV LN). For a realistic geologic setting and plausible simple scenarios of earthquake occurrences, it is determined how the factors f and f 1 depend on the confidence level of the PSV estimate, the exposure time, Y, the time elapsed since the most recent significant event, t e, and on the uncertainty in the one-step memory model (measured by the parameter ζ, equal to the standard deviation of the distribution of the logarithm of the return period for the one-step memory model). The results show that the difference between the predicted PSV amplitudes by the two occurrence models is most significant for high confidence levels (1% or 10% probability of exceedance), during the exposure time following immediately after a larger earthquake on that fault (elapsed time since the most recent event on the segment of the San Andreas fault equal to zero), for the exposure time about 20–25% of the average return period of the events on the hypothetical segment of the San Andreas fault, and for a site on the hypothetical segment of the San Andreas fault. A factor of less than two and a difference within one standard deviation were considered to be of no significance for engineering practice. The results showed that, for a realistic value of the uncertainty of the one-step memory model ( ζ ≥ 0·2), and for a probability of exceedance of 0·1 or higher, even for a site on the fault, the difference between the predicted PSV amplitudes is not significant for engineering purposes. For a probability of exceedance of 0·01, for a site on the fault, the factor f is not much less than one half for the most conservative estimates of the Poissonian model, and barely or not more than two for the most nonconservative estimate. At the same time, the factor f 1 in absolute value can be well beyond one for the most conservative estimates, but, in general, not more than one for the most nonconservative estimates. For sites away from the fault and closer to the Los Angeles metropolitan area, the difference in the predicted spectra by the two occurrence models is insignificant in terms of the defined criterion.