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

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∗($\tau$) 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∗($\tau$) 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.