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by Seismology TechnologyAugust 1st, 2024

**Authors:**

(1) Anatol Guglielmi, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences;

(2) Boris Klain, Borok Geophysical Observatory of Schmidt Institute of Physics of the Earth, Russian Academy of Sciences;

(3) Alexey Zavyalov, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences;

(4) Oleg Zotov, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences and Borok Geophysical Observatory of Schmidt Institute of Physics of the Earth, Russian Academy of Sciences.

100 years ago, an event occurred that determined for a long time the style of studying the evolution of aftershocks: Hirano published an article [1], in which he proposed replacing Omori’s law (6) with a power law of the form

The years went by. In 1938, the famous mathematician and geophysicist Jeffreys drew attention to Hirano's formula [19]. Another couple of decades passed. And so, thanks to the efforts of Utsu [20–22], formula (8) finally attracted the close attention of seismologists. Since then, formula (8) has become firmly established in the practice of aftershock research. It was found that the exponent is on average p =1.1. , but varies from case to case within wide limits (approximately from 0.7 to 1.5).

Formula (8), like Omori formula (6), is one-parameter. In fact, we have already spoken about the parameter c in connection with formula (6). As for the parameter k , it does not have any physical meaning for Hirano, since it does not have a fixed dimension. The dimension k depends on the value p , and this is not accepted in physics. Thus, the Hirano-Utsu formula (8) does not have the status of a geophysical law, but nothing prevents us from considering it as a completely acceptable fitting formula for approximating observational data.

We would like, with the greatest respect to the discoverer, to express directly our opinion that Hirano's search for a correct aftershock formula, different from the Omori formula, was justified, since the Omori formula (6) is not universal. Another thing is that the choice of Hiroano's formula (8) turned out to be unsuccessful. However, it would be a mistake to think that we do not appreciate Hirano's creative efforts. After all, we always take risks, since our main method of searching for scientific truth is trial and error. One way or another, the long search in this direction, begun by Milne and Omori in the century before last, continued by Hirano and Utsu in the last century, has finally been successfully completed. The universal one-parameter formula for the evolution of aftershocks (7) has been found. It differs from the Omori formula (6) in that it takes into account the difference between the proper time τ and the world time t .

And finally, one more historical observation. It is quite clear that formulas (6)–(8) are not only pictograms that schematically reflect one or another seismologist’s idea of the direction of the evolution of aftershocks. These formulas are primarily and mainly tools for processing and analyzing observations in order to calculate the phenomenological parameters of the earthquake source. Omori formula (6) allows you to calculate the parameter k . The Hirano-Utsu formula (8) allows you to calculate parameter p . Formula (7), by which we calculate the source deactivation factor σ , turned out to be logically more justified and much more effective.

This paper is available on arxiv under CC BY 4.0 DEED license.

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