Implications of a conservation law for the distribution of earthquake sizes.

Abstract

We investigate the implications that a sum rule for the average velocity of a geological fault has on the distribution of earthquake sizes. Under general conditions the exponent B of the Gutenberg-Richter law for the distribution of small seismic moments is shown to obey B1. This result does not rely on any particular earthquake model and should be equally applicable to the case of friction between two sliding blocks as well as to computer simulations of earthquake dynamics. For the distribution of large earthquakes we specifically consider three different possible models: (a) A single power law exists over the entire range of magnitudes; then B\ensuremath{\approxeq}1. (b) The large events do not fall in the scaling region; in this case we are able to derive a relationship between the exponents relating their frequency and seismic moment to the total size of the system. (c) The large earthquakes also show scaling behavior but with a different exponent B'; then the sum rule implies that B'>1.