# Earthquake Occurrence: Statistical Analysis, Stochastic Modeling, Mathematical Challenges

Wednesday, June 12, 2002 - 3:00pm - 4:00pm

Keller 3-180

David Vere-Jones (University of Victoria)

David Vere-Jones (on behalf of Yan Y. Kagan)

Modern earthquake catalogs include origin time, hypocenter, and second-rank seismic moment tensor for each earthquake. The tensor is symmetric, traceless, with zero determinant: hence it has only four degrees of freedom -- one for the norm of the tensor and three for the 3-D orientation of earthquake focal mechanisms. An earthquake occurrence is considered to be a stochastic, tensor-valued, multidimensional, point process.

Earthquake occurrence exhibits scale-invariant, fractal properties: (1) earthquake size distribution is a power-law (Gutenberg-Richter) with an exponential tail. The exponent has a universal value for all earthquakes. (2) Temporal fractal pattern: power-law decay of the rate of the aftershock and foreshock occurrence (Omori's law). (3) Spatial distribution of earthquakes is fractal: the correlation dimension of earthquake hypocenters is 2.2 for shallow earthquakes. (4) Disorientation of earthquake focal mechanisms is approximated by the 3-D rotational Cauchy distribution.

A model of random defect interaction in a critical stress environment explains most of the available empirical results. Omori's law is a consequence of a Brownian motion-like behavior of random stress due to defect dynamics. Evolution and self-organization of defects in the rock medium are responsible for fractal spatial patterns of earthquake faults. The Cauchy and other symmetric stable distributions govern the stress caused by these defects, as well as the random rotation of focal mechanisms.

The major theoretical challenges in describing earthquake occurrence are to create scale-invariant models of stochastic processes, and to describe geometrical/topological and group-theoretical properties of stochastic fractal tensor-valued fields (stress/strain, earthquake focal mechanisms). It needs to be done to connect phenomenological statistical results and attempts of earthquake occurrence modeling with a non-linear elasticity theory appropriate for large deformations.

Modern earthquake catalogs include origin time, hypocenter, and second-rank seismic moment tensor for each earthquake. The tensor is symmetric, traceless, with zero determinant: hence it has only four degrees of freedom -- one for the norm of the tensor and three for the 3-D orientation of earthquake focal mechanisms. An earthquake occurrence is considered to be a stochastic, tensor-valued, multidimensional, point process.

Earthquake occurrence exhibits scale-invariant, fractal properties: (1) earthquake size distribution is a power-law (Gutenberg-Richter) with an exponential tail. The exponent has a universal value for all earthquakes. (2) Temporal fractal pattern: power-law decay of the rate of the aftershock and foreshock occurrence (Omori's law). (3) Spatial distribution of earthquakes is fractal: the correlation dimension of earthquake hypocenters is 2.2 for shallow earthquakes. (4) Disorientation of earthquake focal mechanisms is approximated by the 3-D rotational Cauchy distribution.

A model of random defect interaction in a critical stress environment explains most of the available empirical results. Omori's law is a consequence of a Brownian motion-like behavior of random stress due to defect dynamics. Evolution and self-organization of defects in the rock medium are responsible for fractal spatial patterns of earthquake faults. The Cauchy and other symmetric stable distributions govern the stress caused by these defects, as well as the random rotation of focal mechanisms.

The major theoretical challenges in describing earthquake occurrence are to create scale-invariant models of stochastic processes, and to describe geometrical/topological and group-theoretical properties of stochastic fractal tensor-valued fields (stress/strain, earthquake focal mechanisms). It needs to be done to connect phenomenological statistical results and attempts of earthquake occurrence modeling with a non-linear elasticity theory appropriate for large deformations.