Seismic magnitudes, entropy and b-value

Abstract

A closed relationship between the Gutenberg–Richter b-value (or \(\beta =b \text{ln}10\)) and the information or Shannon entropy is found and checked through numerical evaluation of the entropy using exact probabilities derived directly from the magnitude exponential distribution. Comparison of the numerical evaluation of the entropy over a finite magnitude range makes it possible to assess the possible contribution to the entropy of real or hypothetical very large magnitudes, and these contributions are found to be quite small. The relationship is also compared with entropies calculated from synthetic data, and Monte Carlo simulations are used to explore the behavior of entropy determinations as a function of sample size. Finally, it is considered how, for the usual case of having data from a single realization, i.e., a single magnitude data set, since estimates of the entropy and of the Aki–Utsu b-value are measured in different ways, they are not redundant and may be complementary and useful in determining when a sample is large enough to give reliable results.