Instanton representation of foreshoсk—aftershock sequences

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Abstract

Descriptions of the processes of foreshock and aftershock activations are of great significance in seismology, both for practical and theoretical reasons. An analogy of mathematical expressions describing the origin of the direct and inverse Omori–Utsu laws has been established empirically. Investigations of the generalized vicinity of a large earthquake (GVLE) have revealed an even closer analogy between the properties of foreshocks and aftershocks. This similarity also applies to the characteristics of the activation process, in particular, anomalous changes in the slope of the repeatability plot. It is proposed in this paper to use kinetic equations for the unifying model of the entire foreshock-aftershock process, the equations having solutions in the form of dependencies with explicit maxima, localized in time and called instantons (by analogy with solitons–localized waves). A clear pattern of an instanton solution is a plot of the time derivative of the logistic dependence describing the transition process. The speed of such a process first increases significantly, reaches a maximum, and then asymptotically decreases to zero.

The aim of the work is to demonstrate the efficiency of using the instanton model, which generalizes the model of self–developing processes (SDP), but does not provide for the development of physically unrealistic singularity which is a pattern usually simulating an explosive growth in the number of foreshocks and aftershocks in the vicinity of the main event. A comparison of the new model with empirical data is performed by the example of earthquakes in the southern part of Sakhalin Island in 2003–2023.This zone is the most equipped with facilities for seismic events registration. The satisfactory correspondence between theoretical and empirical temporal dependences is shown both for the GVLE built for the territory within (44.5°–50.5° N., 141.5°–143.5° E.) and for individual strong earthquakes on Sakhalin.

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About the authors

L. M. Bogomolov

Institute of Marine Geology and Geophysics, FEB RAS

Author for correspondence.
Email: bleom@mail.ru
Russian Federation, Yuzhno-Sakhalinsk

M. V. Rodkin

Institute of Marine Geology and Geophysics, FEB RAS; Institute of earthquake prediction theory and mathematical geophysics, RAS

Email: rodkin@mitp.ru
Russian Federation, Yuzhno-Sakhalinsk; Moscow

V. N. Sychev

Institute of Marine Geology and Geophysics, FEB RAS

Email: bleom@mail.ru
Russian Federation, Yuzhno-Sakhalinsk

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Supplementary files

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2. Fig. 1. Accumulation rate of the number of events per month n, before the Simushir earthquake of November 15, 2006, Ms = 7.9–8.2 (a), and the Gornozavodskoye earthquake of August 17, 2006, M = 5.6 (b), based on the materials of [Tikhonov et al., 2008; Tikhonov, Kim, 2010]. Regression lines with singularity are drawn according to the authors’ works.

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3. Fig. 2. Time dependence of the daily number of foreshocks and aftershocks in the EOZ for the ISC catalog based on the materials of [Rodkin, 2020]. The points correspond to the average values ​​for successive groups of earthquakes, explanations in the text.

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4. Fig. 3. Characteristic time dependences of solutions of kinetic equation (13) for s = 2 with normalization to the maximum activity value: (a) f = a = 1, Af = Aa = 1; (b) f = 0.8, a = 0.2, Af = 1, Aa = 4; (c) f = 6, a = 0.15, Af = 1, Aa = 40; (d) f = 0.4, a = 1.6, Af = 4, Aa = 1; (e) f = 1, a = 0.01, Af = 1, Aa = 100. In cases (a) and (b), marker h indicates the graphs of the dependence n(t) = K/(c + t)p, the values ​​of K, c, p are given in the text. The intervals of overlapping graphs are marked with a strip.

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5. Fig. 4. Characteristic time dependences of solutions of kinetic equation (19) with normalization to the maximum activity value: (a) p = 0.6, tm = 8; (b) p = 0.6, tm = 2; (c) p = 1.5, tm = 8. Graphs in panel (a): marker ptm is solution (19), marker fa is the compared dependence (15) with parameters: f = 0.72, a = 0.12, Af = 1, Aa = 6.

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6. Fig. 5. Location of seismic stations of the Sakhalin branch of the Federal Research Center of Geological Surveys of the Russian Academy of Sciences and earthquake epicenters in the southern part of Sakhalin Island for the period 2003–2023. Black lines show the largest fault zones: 1 — West Sakhalin, 2 — Central Sakhalin, and regional faults: 3 — Susunai, 4 — East Sakhalin (Hokkaido–Sakhalin), according to [Voyeikova et al., 2007; Zelenin et al., 2022].

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7. Fig. 6. Time dependence of the daily number of foreshocks and aftershocks in the EOE for the catalogue of the south of Sakhalin Island and its approximation (solid line) using the instanton solution (15) with parameters (a) f = 0.4, a = 0.4, nm = 300, Af = Aa = 1; (b) the same dependence and its approximation on a logarithmic scale on an interval of ± 10 days from the main event.

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8. Fig. 7. Seismic activity before and after the earthquakes: (a) Nevelskoy, 02.08.2007, Mw = 6.2; (b) Gornozavodskoy, 17.08.2006, Mw = 5.9 and their approximations using instanton solutions (15). Approximation parameters: f = 0.4, a = 0.1, nm = 87, Af = Aa = 1 (a); f = 0.37, a = 0.19, nm = 86, Af = Aa = 1 (b).

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