On the theory of earthquakes: Paradoxical contradiction of Omori's law to the law of energy conservation

TL;DR

This paper examines the paradox where Omori's law of aftershock decay conflicts with energy conservation, proposing theoretical solutions to reconcile the phenomenological law with fundamental physics.

Contribution

It introduces a theoretical framework that resolves the contradiction between Omori's law and energy conservation by modeling source deactivation and evolution equations.

Findings

Identified the paradoxical contradiction between Omori's law and energy conservation.

Proposed two methods to eliminate the paradox within the theoretical model.

Enhanced understanding of aftershock phenomenology through source deactivation concepts.

Abstract

After the main shock of an earthquake the aftershocks are observed. According to Omori's law, the frequency of aftershocks decreases hyperbolically over time. We noticed that, strictly speaking, Omori's law paradoxically contradicts the law of energy conservation. The contradiction is that the excitation of each aftershock consumes a finite portion of the source's energy, so that the total energy released by the source tends to infinity over time. The paradox is formally theoretical, but its analysis has proved useful. Eliminating the contradiction between Omori's law and the fundamental law of conservation of energy allowed us to further understand the nature of the phenomenological theory of aftershocks. We used the concept of deactivation of a source after the formation of a main rupture in it. We have based the theory on the original aftershock evolution equation, which has the form of a first-order linear differential equation. Two ways to eliminate the paradoxical situation are indicated. Key words: earthquake source, aftershock, evolution equation, deactivation coefficient, inverse problem, Omori epoch, source bifurcation, logistic equation, Hirano-Utsu formula.