An axiomatic method for studying the truth or falsity of the Hirano-Utsu law describing aftershocks

TL;DR

This paper introduces an axiomatic framework to evaluate the validity of the Hirano-Utsu law in aftershock modeling, revealing a two-stage relaxation process and challenging the assumption that it is an improved version of Omori's law.

Contribution

It proposes an axiomatic method and a phenomenological deactivation coefficient to test the Hirano-Utsu law, uncovering a two-stage relaxation mode in aftershock evolution.

Findings

Discovered a two-stage source relaxation process.

Established a method to experimentally test the Hirano-Utsu law.

Identified a bifurcation point ending the Omori epoch.

Abstract

The power law of aftershock evolution was proposed by Hirano in 1924 and introduced by Utsu into seismology in the second half of the last century. The Hirano-Utsu law is widely used in studying the relaxation of earthquake source after the main shock of an earthquake. The prevailing view in the literature is that the Hirano-Utsu law is an improved version of Omori's hyperbolic law, formulated in 1894. The author disagrees with this notion. The paper proposes an axiomatic approach to the study of aftershocks. A phenomenological parameter of the source, called the deactivation coefficient, was introduced. The theory is based on axioms that do not contain any a priori statements regarding the form of the law of aftershock evolution. Formulas for the deactivation coefficient are derived from the axioms, allowing one to experimentally establish the truth or falsity of the Hirano-Utsu and Omori laws. A two-stage mode of source relaxation was discovered. In the first stage, called the Omori epoch, the Omori law is strictly followed. The Omori epoch ends with a bifurcation, after which aftershock activity becomes unpredictable. Omori's law is not fulfilled at the second stage of evolution. The Hirano-Utsu law is not fulfilled either at the first or second stage. Keywords: earthquake source, main shock, relaxation, deactivation coefficient, evolution equation, inverse problem, Omori epoch, bifurcation, two-stage relaxation mode.