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Entropy
Volume 27
Issue 2
10.3390/e27020130
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Article

Mathematical Theory of Seismic Activity and Its Specific Cases: Gutenberg–Richter Law, Omori Law, Roll-Off Effect, and Negative Binomial Distribution

by
Roumen Borisov
and
Nikolay K. Vitanov
*
Institute of Mechanics, Bulgarian Academy of Sciences, Academician Georgi Bonchev Street, Block 4, 1113 Sofia, Bulgaria
*
Author to whom correspondence should be addressed.
Entropy 2025, 27(2), 130; https://doi.org/10.3390/e27020130
Submission received: 29 August 2024 / Revised: 9 January 2025 / Accepted: 24 January 2025 / Published: 26 January 2025
(This article belongs to the Special Issue Time Series Analysis in Earthquake Complex Networks)
Versions Notes

Abstract

We discuss a model of seismic activity that is based on the concept of energy in a cluster of sources of seismic activity. We show that specific cases of the studied model lead to the Gutenberg–Richter relationship and the Omori law. These laws are valid for earthquakes that happen in a single cluster of sources of seismic activity. Further, we discuss the distribution of earthquakes for several clusters containing sources of seismic activity. This distribution contains, as a specific case, a version of the negative binomial distribution. We show that at least a part of the roll-off effect connected to the parameter b of the Gutenberg– Richter law occurs because one records earthquakes that happen in more than one cluster of sources of seismic activity.
Keywords: seismic activity; earthquakes; energy-based model; Gutenberg–Richter law; Omori law; negative binomial distribution; roll-off effect seismic activity; earthquakes; energy-based model; Gutenberg–Richter law; Omori law; negative binomial distribution; roll-off effect

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MDPI and ACS Style

Borisov, R.; Vitanov, N.K. Mathematical Theory of Seismic Activity and Its Specific Cases: Gutenberg–Richter Law, Omori Law, Roll-Off Effect, and Negative Binomial Distribution. Entropy 2025, 27, 130. https://doi.org/10.3390/e27020130

AMA Style

Borisov R, Vitanov NK. Mathematical Theory of Seismic Activity and Its Specific Cases: Gutenberg–Richter Law, Omori Law, Roll-Off Effect, and Negative Binomial Distribution. Entropy. 2025; 27(2):130. https://doi.org/10.3390/e27020130

Chicago/Turabian Style

Borisov, Roumen, and Nikolay K. Vitanov. 2025. "Mathematical Theory of Seismic Activity and Its Specific Cases: Gutenberg–Richter Law, Omori Law, Roll-Off Effect, and Negative Binomial Distribution" Entropy 27, no. 2: 130. https://doi.org/10.3390/e27020130

APA Style

Borisov, R., & Vitanov, N. K. (2025). Mathematical Theory of Seismic Activity and Its Specific Cases: Gutenberg–Richter Law, Omori Law, Roll-Off Effect, and Negative Binomial Distribution. Entropy, 27(2), 130. https://doi.org/10.3390/e27020130

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