The Gutenberg–Richter Law and Entropy of Earthquakes: Two Case Studies in Central Italy

Abstract

A cumulative frequency-magnitude relation, the Gutenberg–Richter law, dominates the statistics of the occurrence of earthquakes. Although it is an empirical law, some authors have tried to give some physical meaning to its a and b parameters. Here, we recall some theoretical expressions for the probability of occurrence of an earthquake with magnitude M in terms of a and b values. A direct consequence of the maximum likelihood estimation (MLE) and the maximum entropy principle (MEP) is that a and b values can be expressed as a function of the mean magnitude of a seismic sequence over a certain area. We then introduce the definition of the Shannon entropy of earthquakes and show how it is related to the b value. In this way, we also give a physical interpretation to the b value: the negative logarithm of b is the entropy of the magnitude frequency of earthquake occurrence. An application of these concepts to two case studies, in particular to the recent seismic sequence in Abruzzi (central Italy; mainshock Mw 6.3, 6 April 2009 in L’Aquila) and to an older 1997 sequence (Umbria-Marche, central Italy; mainshock Mw 6.0, 26 September 1997 in Colfiorito), confirms their potential to help in understanding the physics of earthquakes. In particular, from the comparison of the two cases, a simple scheme of different regimes in succession is proposed in order to describe the dynamics of both sequences.