Bounds on the distribution of amplitudes in ground motion prediction models

Abstract

In a probabilistic seismic hazard study, the imprecise prediction of ground motion parameters by empirical attenuation models is usually taken into account by assuming a lognormal distribution for the prediction imprecision. For some important engineering structures, such as hydro-power stations in New Zealand and nuclear power plants and nuclear waste storage overseas, their critical importance requires ground-motion estimates that have very low annual probability (very long return period). For such a level of ground motions, an assumption of lognormal distribution leads to an almost monotonic increase in the estimated ground motion parameters with increasing return period, without a limit. These properties of the estimated ground motions cause a major difficulty in the ground-motion assessment – are these estimates realistic? If not, what would be the upper limits?

In the present study, we examine the root of the problem – to estimate the upper limit of the range within which the prediction imprecision has a lognormal distribution. We use a sub-dataset from a very large dataset used for developing Japanese attenuation models. Because of data ownership and the quality of data from analogue and early versions of digital instruments, the sub-dataset consists of records from the K-net and Kik-net arrays only. We employ two methods to tackle the problem.

The first method uses graphical inspection of the probability plots and formal statistical tests. We plot the theoretical values derived from a lognormal distribution against the actual values computed from data and the attenuation models. Using visual inspection, we can identify where the upper tails of the distribution depart from the lognormal distribution for a number of spectral periods. Using formal statistical tests, we can also identify the upper tail departures from the lognormal distribution for a number of spectral periods, but they do not all correspond to those periods identified by the probability plots. This method produces mixed results.

The second method is to compare two important parameters of the data and the attenuation models. The first parameter is the expected number of records in a dataset that have a value larger than a specified spectral acceleration. The second parameter is the actual number of records exceeding this specified value. We find that the actual number of exceedances at moderately strong and strong ground shaking is much smaller than the expected number of exceedances for all spectral periods. At very high spectral accelerations (the level of design ground motion for important structures such as hydro-power stations), the actual numbers of exceedances are only 5-10% of the expected numbers of exceedances. Although we cannot put an upper limit to the ground motion parameters using this method, our results strongly suggest that there are some physical constraints that limit the maximum spectral ground accelerations in the sub-dataset used in the present study.

If these results are considered in a probabilistic seismic hazard study, the continuing increase in the estimated ground-motion parameters with increasing return period may not occur.

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Technical Abstract

In current ground-motion models, the uncertainty in predicted ground motion is modelled with a lognormal distribution. One consequence of this is that predicted ground motions do not have an upper limit. In reality, there probably exist physical conditions that limit the ground motion estimates that may be unrealistically large, especially at the low annual probabilities considered for important structures, such as dams or nuclear reactors. 

Attempts to estimate the upper limits have been made by others by using ground-motion records from a single event, but it is not clear if the conclusions derived are applicable to attenuation models which are derived from a large number of records generated by a large number of earthquakes.

We have analysed very large strong-motion data sets from the K-net and Kik-net strong-motion networks in Japan and determined the total residuals from the ground-motion model developed for Japan. These residuals are then used to construct normal probability lots, and the departures of the residuals from lognormal distributions are quantified by visual inspection and statistical tests. For some periods, departure from a lognormal distribution at about 2.5-3 standard deviations can be identified, with the departure suggesting a shortening of the upper tail. For other periods, departure from a lognormal distribution can be identified if the largest one or two residuals are disregarded. At a few spectral periods, the distribution of the upper tail suggests long tails. Statistical tests suggest that, at a few periods, the distribution at the upper tail differs from lognormal distribution at a significance level of 5%.

We have also used a statistical procedure to examine the actual and expected numbers of predicted spectral accelerations exceeding a given spectral acceleration. Our results show that, for moderate, large and very large spectral accelerations, the actual number of exceedances is much less than the expected number of exceedances. Our results from the statistical procedure do not put any limits on the upper tail, but suggest that physical constraints may limit the maximum spectral accelerations.