κ-generalised Gutenberg–Richter law and the self-similarity of earthquakes

Abstract

• Generalised Gutenberg–Richter law appropriately describes the earthquakes in a wide range of magnitudes. • Self-organised criticality of earthquakes and the fractal nature of the geological faults are introduced in the context of Kaniadakis statistics. • The entropic index is a universal parameter in the analysis of more than 150,000 earthquakes. • The earthquakes are well described by the same physics, for all scales of magnitude. The earthquakes statistical analysis are essential for understanding the seismic activity of a region and consequently, in seismic hazard studies. Nowadays, the statistics that describe the relationship between the earthquake magnitude and the total number of quakes in a given region has been dominated by the Gutenberg–Richter (GR) power-law. However, the GR law is only valid from a threshold magnitude (local scale invariance). To mitigate this limitation, we introduce in this work a statistical mechanics approach to appropriately describe the earthquakes in a wide range of magnitudes. In this way, we formulate a generalisation of the GR law based on a κ -probability function, which is a distribution linked to Kaniadakis statistics (or κ -statistics). We have tested the viability of our proposal using real data sets recorded in 6 different regions of the Earth by considering more than 150,000 earthquakes. The results show that the κ -generalised GR law describes appropriately enough the energy distribution in an ample range of magnitudes, especially for low magnitude earthquakes where the standard GR law fails. Besides, the entropic index of the Kaniadakis entropy is a universal parameter with a value close to 1. Furthermore, the results reveal the fractal nature of the fragments filling the space between tectonic plates. They also show the self-similarity of earthquakes regardless of their magnitude.