Derivation of the complete Gutenberg‐Richter magnitude‐frequency relation using the principle of scale invariance

The purpose of this paper is to show that the various observational parameters characterizing the statistical properties of earthquakes can be related to each other. The fundamental postulate which is used to obtain quantitative results is the idea that the physics of earthquake occurrence scales as a power law, similar to properties one often sees in critical phenomena. When the physics of earthquake occurrence is exactly scale invariant, b = 1, and it can be shown as a consequence that earthquakes in any magnitude band Δm cover the same area in unit time. This result therefore implies the existence of a “universal” covering interval τT, which is here called the “cycle interval.” Using this idea, the complete Gutenberg-Richter relation is derived in terms of the fault area ST, which is available to events of any given size, the average stress drop ΔσT for events occurring on ST, the interval τT for events of stress drop ΔσT to cover an area ST, and the scaling exponent α, which is proportional to the b value. Observationally, the average recurrence time interval for great earthquakes, or perhaps equivalently, the recurrence interval for characteristic earthquakes on a fault segment, is a measure of the cycle interval τT. The exponent α may depend on time, but scale invariance (self similarity) demands that α = 1. It is shown in the appendix that the A value in the Gutenberg-Richter relation can be written in terms of ST, τT, ΔσT, and the parameter α. The b value is either 1 or 1.5 (depending on the geometry of the fault zone) multiplied by α.