## Statistical distributions of earthquake numbers: consequence of branching process

Yan Y. Kagan

Published 2010, SCEC Contribution #1304

We discuss various statistical distributions of earthquake numbers.
Previously we derived several discrete distributions to describe
earthquake numbers for the branching model of earthquake occurrence:
these distributions are the Poisson, geometric, logarithmic, and the
negative binomial (NBD). The theoretical model is the `birth and
immigration' population process. The first three distributions above
can be considered special cases of the NBD. In particular, a point
branching process along the magnitude (or log seismic moment) axis with
independent events (immigrants) explains the magnitude/moment-frequency
relation and the NBD of earthquake counts in large time/space windows,
as well as the dependence of the NBD parameters on the magnitude
threshold. We discuss applying these distributions, especially the NBD,
to approximate event numbers in earthquake catalogs. There are many
different representations of the NBD. Most can be traced either to the
Pascal distribution or to the mixture of the Poisson distribution with
the gamma law. We discuss advantages and drawbacks of both
representations for statistical analysis of earthquake catalogs. We
also consider applying the NBD to earthquake forecasts and describe the
limits of the application for the given equations. In contrast to the
one-parameter Poisson distribution so widely used to describe
earthquake occurrence, the NBD has two parameters. The second parameter
can be used to characterize clustering or over-dispersion of a process.
We determine the parameter values and their uncertainties for several
local and global catalogs, and their subdivisions in various time
intervals, magnitude thresholds, spatial windows, and tectonic
categories. The theoretical model of how the clustering parameter
depends on the corner (maximum) magnitude can be used to predict future
earthquake number distribution in regions where very large earthquakes
have not yet occurred.

Citation
Kagan, Y. Y. (2010). Statistical distributions of earthquake numbers: consequence of branching process. Geophysical Journal International, 180(3), 1313-1328. doi: 10.1111/j.1365-246X.2009.04487.x.