Hereditarian Properties of Foreshock Distributions in the Framework of a Fractional Model of the Deformation Process

The study of the relationships between seismic events based on a large number of statistical models has shown correlations between seismic events in the catalogs under consideration based on selected criteria. Correlations between events lead to the appearance of nonlocality properties in time (hereditarity) and in space in these sequences. Then the representation of the seismic process as a stream of independent random changes of dislocations and its description by the standard Poisson process using an exponential function becomes incorrect. A logical generalization of this approach is the use of a fractional Poisson process to describe the seismic deformational process, taking into account these properties. The fractional model considers the deformation process from a probabilistic point of view as a transition from one regime (or state) to another. To describe the probability of preserving the deformation process in a certain regime, the fractional Mittag-Leffler function as a generalization of the exponential function is used. This function takes into account the properties of the hereditary (i.e., the history of the process) and of the non-stationary of the event stream, and its fractional parameters are determined by the parameters of the medium. The study of the activation regime in the foreshock phase made it possible to construct foreshock sequences (clusters) based on criteria related to the characteristics of the earthquake energy and medium of the earthquake preparation area. Using the method of superpose epochs, empirical functions of the waiting time distribution of the foreshocks depending on the time before the mainshock are obtained. Varying the fractional parameters and the scale factor of the Mittag-Leffler function made it possible to approximate the empirical functions with greater accuracy than by the exponential function. Fractional parameters take values less than one, which allows us to conclude that there are hereditary and non-stationary properties in the foreshock sequences. The parameter values become closer to one with an increase in the energy class of the mainshock. Such trends in the behavior of foreshock sequences for stronger mainshocks can be interpreted as a difference of the hereditary properties for mainshocks of different energies. Also, the reason for the weakening of hereditary properties in the foreshock sequences for the high-energy mainshocks may be the inclusion of events that are not related to this mainshock, but fall into its large spetial-temporal area. It should be noted that for a reason of the small amounts of data, seismology is statistically insufficient to strictly solve the problem of choosing a deformation process model. However, the fractional Poisson process model is preferable because it has a more universal nature.