Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 137–152. ISSN 2079-6641

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MSC 60G22, 37M10, 60J80, 33E12

Research Article

Approximation of the waiting times distribution laws for foreshocks based on a fractional model of deformation activity

O. V. Sheremetyeva, B. M. Shevtsov

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Paratunka, Mirnaya str., 7, Russia

The article discusses two algorithms for constructing sequences of foreshocks associated with the main event of a given energy, based on the statistical model of the deformation process previously developed by the authors. Catalog of the Kamchatka Branch of the Geophysical Survey of Russia Academy of Sciences for the period from 1 January 1962 to 31 December 2002 for the Kuril-Kamchatka island arc subduction zone is used for research (area 46◦–62◦ N, 158◦–174◦ E) [28]. The method of «epochs» is applied to the sequences of foreshocks to obtain an empirical cumulative distribution function (eCDF) P∗(τ) of relative frequency of foreshocks occurrence depending on the time before the mainshock. Based on the fractional model of the deformation process developed by the authors, the empirical cumulative distribution function P∗(τ) of foreshocks waiting time are approximated by the Mittag-Leffler function and the exponential function. It is shown that the accuracy of the approximation by the Mittag-Leffler function is higher than the exponential one. A comparative analysis of three parameters of approximating functions for the empirical distributions obtained from the results of two algorithms for constructing sequences of foreshocks is carried out. Based on the obtained values of the parameters of the Mittag-Leffler function, the deformation process in the considered region can be considered non-stationary and close to the standard Poisson process.

Key words: foreshocks, approximation, fractional Poisson process, Mittag-Leffler function, non-local effect, non-stationarity, statistical model, fractional model

DOI: 10.26117/2079-6641-2022-40-3-137-152

Original article submitted: 12.10.2022

Revision submitted: 29.10.2022

For citation. Sheremetyeva O. V., Shevtsov B. M. Approximation of the waiting times distribution laws for foreshocks based on a fractional model of deformation activity. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 137-152. DOI: 10.26117/2079-6641-2022-40-3-137-152

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

Funding. The work was carried out within the framework of the state assignment on the topic «Physical processes in the system of near space and geospheres under solar and lithospheric influences» (No. AAAA-A21-121011290003-0).

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Sheremetyeva O. V., Shevtsov B. M., 2022


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Sheremetyeva Olga Vladimirovna – Ph.D. (Tech.), Research
Scientist, Laboratory of Physical Process Modeling, Institute of
Cosmophysical Research and Radio Wave Propagation FEB RAS,
Paratunka, Kamchatka, Russia, ORCID 0000-0001-9417-9731.

Shevtsov Boris Mikhaylovich – D. Sci. (Phys. & Math.), Chief Scientific Officer, Laboratory of Electromagnetic Radiation, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Kamchatka, Russia, ORCID 0000-0003-0625-0361.