## Earthquake Number Forecasts Testing

Yan Y. Kagan, & David D. Jackson

Submitted August 15, 2016, SCEC Contribution #6943, 2016 SCEC Annual Meeting Poster #301

We study the distributions of earthquake numbers in two global earthquake catalogs: Global Centroid-Moment Tensor (GCMT) and Preliminary Determinations of Epicenters (Monthly Listing) (PDE). The properties of these distributions are especially needed to develop the number test for our forecasts of future seismic activity rate, organized 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. However it is clear 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 occurrence, 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 the time intervals catalog duration is subdivided. 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. Thereafter we investigate whether these distributions fit 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 skewness/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/kurtosis tend both to zero for large earthquake rates: for the Gaussian law these values are identically zero. A calculation of the NBD skewness/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 values of skewness/kurtosis are rising even faster. This means that for small time intervals the earthquake number distribution is even more heavy-tailed than the NBD expects.

Key Words
Forecasts Testing; Probability distributions; Statistical seismology; Poisson and Negative Binomial distributions

Citation
Kagan, Y. Y., & Jackson, D. D. (2016, 08). Earthquake Number Forecasts Testing . Poster Presentation at 2016 SCEC Annual Meeting.

Related Projects & Working Groups
Earthquake Forecasting and Predictability (EFP)