https://doi.org/10.1140/epjb/e2006-00075-3
Power law distribution of seismic rates: theory and data analysis
1
Mathematical Department, Nizhny Novgorod State University, Gagarin Prosp. 23, Nizhny Novgorod, 603950, Russia
2
Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, 90095, USA
3
Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los Angeles, CA, 90095, USA
4
Laboratoire de Physique de la Matière Condensée, CNRS UMR 6622 and Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
Corresponding author: a sornette@moho.ess.ucla.edu
Received:
28
September
2005
Revised:
1
January
2006
Published online:
10
March
2006
We report an empirical determination of the
probability density functions Pdata(r) (and
its cumulative version)
of the number r of earthquakes in finite space-time windows
for the California catalog, over fixed spatial boxes 5 ×5 km2,
20 ×20 km2 and 50 ×50 km2
and time intervals and 1000 days. The data can
be represented by asymptotic power law tails together with several
cross-overs. These observations are
explained by a simple stochastic branching process previously studied by
many authors, the ETAS (epidemic-type aftershock sequence) model which
assumes that each earthquake can trigger other earthquakes
(“aftershocks”). An aftershock sequence results in this model from the
cascade of aftershocks of each past earthquake. We develop the full
theory in terms of generating functions for describing the space-time
organization of earthquake sequences and develop several approximations
to solve the equations.
The calibration of the theory to the empirical observations
shows that it is essential to augment the ETAS model by taking account
of the pre-existing frozen heterogeneity of spontaneous earthquake
sources. This seems natural in view of the complex multi-scale nature of
fault networks, on which earthquakes nucleate.
Our extended theory is able to account for the empirical
observation but some discrepancies, especially for the shorter time windows,
point to limits of both our theoretical approach and of the ETAS model.
PACS: 64.60.Ak – Renormalization-group, fractal, and percolation studies of phase transitions (see also 61.43.Hv Fractals; macroscopic aggregates) / 02.50.Ey – Stochastic processes / 91.30.Dk – Seismicity (see also 91.45.gd–in geophysics appendix)
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2006