Gutenberg-Richter b-value maximum likelihood estimation and sample size

The Aki-Utsu maximum likelihood method is widely used for estimation of the Gutenberg-Richter b-value, but not all authors are conscious of the method’s limitations and implicit requirements. The Aki/Utsu method requires a representative estimate of the population mean magnitude; a requirement seldom satisfied in b-value studies, particularly in those that use data from small geographic and/or time windows, such as b-mapping and b-vs-time studies. Monte Carlo simulation methods are used to determine how large a sample is necessary to achieve representativity, particularly for rounded magnitudes. The size of a representative sample weakly depends on the actual b-value. It is shown that, for commonly used precisions, small samples give meaningless estimations of b. Our results give estimates on the probabilities of getting correct estimates of b for a given desired precision for samples of different sizes. We submit that all published studies reporting b-value estimations should include information about the size of the samples used.