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Earthquake Number Forecasts Testing

Yan Y. Kagan

Published September 23, 2016, SCEC Contribution #7071

We study the distributions of earthquake numbers in two global earthquake catalogs: Global Centroid-Moment Tensor (GCMT) and Preliminary Determinations of Epicenters (PDE). The properties of these distributions are especially required to develop the number test for our forecasts of future seismic activity rate, tested by the Collaboratory for Study of Earthquake Predictability (CSEP). A common assumption, as used in the CSEP tests, is that the numbers are
described by the Poisson distribution. It is clear, however, that the Poisson assumption for the earthquake number distribution is incorrect, especially for the catalogs with a lower magnitude threshold. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrences, the negative-binomial distribution (NBD) has two parameters. The second parameter can be used to characterize the clustering or over-dispersion of a process. We investigate the dependence of parameters for both distributions on the catalog magnitude threshold and on temporal subdivision of catalog duration. Firstly, we study whether the Poisson law can be statistically rejected for various catalog subdivisions. We find that for most cases of interest the Poisson distribution can be shown to be rejected statistically at a high significance level in favor of the NBD. Therefore we investigate whether these distributions fit the observed distributions of seismicity. For this purpose we study upper statistical moments of earthquake numbers (skewness and kurtosis) and compare them to the theoretical values for both distributions.

Empirical values for the skewness and the kurtosis increase for the smaller magnitude threshold and increase with even greater intensity for small temporal subdivision of catalogs. As is known, the Poisson distribution for large rate values approaches the Gaussian law, therefore its skewness and kurtosis both tend to zero for large earthquake rates: for the Gaussian law these values are identically zero. A calculation of the NBD skewness and kurtosis levels based on the values of the first two statistical moments of the distribution, shows rapid increase of these upper moments levels. However, the observed catalog values of skewness and kurtosis are rising even faster. This means that for small time intervals the earthquake number distribution is even more heavy-tailed than the NBD predicts. Therefore for small time intervals we propose using empirical number distributions appropriately smoothed for testing forecasted earthquake numbers.

Key Words
Probability distributions, Seismicity and tectonics, Statistical seismology, Dynamics, seismotectonics

Kagan, Y. Y. (2016). Earthquake Number Forecasts Testing. arXiv,. https://arxiv.org/abs/1609.08217