A forest-fire analogy to explain the b-value of the Gutenberg-Richter law for earthquakes

TL;DR

This paper uses a forest-fire model analogy to explain the origin of the Gutenberg-Richter law's b-value in earthquakes, showing how secondary fires (aftershocks) influence earthquake statistics.

Contribution

Introducing a second class of trees in the forest-fire model to generate aftershock-like events, resulting in earthquake statistics that match the Gutenberg-Richter law with realistic parameters.

Findings

Secondary fires produce a Gutenberg-Richter-like distribution.

The b-value change is analytically demonstrated in a simplified model.

The mechanism has testable experimental implications.

Abstract

The Drossel-Schwabl model of forest fires can be interpreted in a coarse grained sense as a model for the stress distribution in a single planar fault. Fires in the model are then translated to earthquakes. I show that when a second class of trees that propagate fire only after some finite time is introduced in the model, secondary fires (analogous to aftershocks) are generated, and the statistics of events becomes quantitatively compatible with the Gutenberg Richter law for earthquakes, with a realistic value of the $b$ exponent. The change in exponent is analytically demonstrated in a simplified percolation scenario. Experimental consequences of the proposed mechanism are indicated.