Instanton Representation of Foreshoсk–Aftershock Sequences

Abstract—The description of foreshock and aftershock activation processes is of great significance in seismology both from practical and theoretical standpoints. An analogy between the mathematical expressions describing the seismicity patterns in the direct and inverse Omori–Utsu laws has been empirically established. Studies of the generalized vicinity of a large earthquake (LEGV, also abbreviated as GVLE) revealed an even closer analogy between the properties of foreshocks and aftershocks. This analogy also holds for the characteristics of the activation process, in particular, for the anomalous changes in the Gutenberg–Richter b-value. We propose a unifying model for the entire foreshock–aftershock process, which is described by kinetic equations with solutions in the form of strongly temporally localized maxima, called instantons by analogy with solitons for lozalized waves. A demonstrating example of an instanton solution is a graph of the time derivative of the logistic dependence describing a transient process. The rate of this process initially increases significantly, reaches a peak, and then asymptotically decreases to zero. The aim of this paper is to demonstrate effectiveness of the instanton model which generalizes the self-developing process (SDP) model but does not involve the formation of a physically unrealizable singularity typically simulating the explosive growth in the number of foreshocks and aftershocks in the vicinity of the main event. The new model is compared with empirical data from the earthquakes that occurred between 2003 and 2023 in the southern part of Sakhalin, an area with the best capabilities for seismicity recording. A reasonable consistency between theoretical and empirical time dependences is obtained both for the LEGV constructed for a region within 44.5°–50.5° N and 141.5°–143.5° E and for separate strong Sakhalin earthquakes.