The Relationship between the Gutenberg–Richter b-Value and the Fractal Dimension of Seismicity according to Computer and Laboratory Modeling in Spaces of Various Dimensions

Abstract—The relationship between the Gutenberg–Richter parameters and the fractal dimension of a set of hypocenters is studied on a computer model of the Olami–Feder–Christensen cellular automaton (OFC) in spaces (on grids) of different dimensions. The modeling results are compared to the previous data of laboratory simulations of seismicity by fracturing of rock samples. Computer modeling in spaces of different dimensions has shown that the Gutenberg–Richter parameter (b-value) and the fractal dimension of the event set depend on the dimension of the space within which the fracture process develops, with both parameters increasing as dimensionality increases. In spaces of different dimensionalities, the stored elastic energy is released at rupture from regions that have different dimensionality. In the case of a three-dimensional (3D) space, the energy is released from a region of a certain volume, in the case of a two-dimensional (2D) space, from a region of a certain area. Given the same rupture size and the same critical elastic energy density, more energy is probably be released in the 3D (volumetric) case than in the 2D (areal) case. This can be assumed to be the reason why the power indices of the energy spectrum and fractal geometry of the fracture process differ in spaces of different dimensions. The results of the computer and laboratory modeling of seismicity also support the validity of the Aki formula stating direct proportion between the b-value and the fractal dimension. The substantiation of the validity of the Aki formula for fracture in spaces of different dimensionalities may be useful for the development of methods for a more meaningful and effective transition from seismic statistics to estimates of the physical parameters of the fracture process in regions with different types of fracture in different tectonic conditions.