Derivation of the Gutenberg-Richter Empirical Formula from the Solution of the Generalized Logistic Equation

TL;DR

This paper derives the Gutenberg-Richter earthquake magnitude distribution formula from the generalized logistic equation, providing a theoretical basis and demonstrating its fit to observed seismic data across multiple regions.

Contribution

It introduces a new generalized logistic equation approach to derive the Gutenberg-Richter formula, linking empirical observations with theoretical modeling.

Findings

Excellent fit of theoretical curves to observed earthquake data

Validates the generalized logistic equation as a model for earthquake statistics

Derives the Gutenberg-Richter formula as an asymptotic case

Abstract

We have written a new equation to study the statistics of earthquake distributions. We call this equation "the generalized logistic equation". The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case in approximation of large magnitudes. To illustrate how the found solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N-25N, 5W-35W), Canary Islands, Magellan Mountains (20N-9S, 148E-170E), and the Sea of Japan. This approximation showed the excellent fit between the theoretical curves and observed data for earthquake magnitudes 1<m<9.