A review and new insights on the estimation of the b-value and its uncertainty

The estimation of the b-value of the Guteuberg-Richter Law and its uncertainty is crucial in seismic hazard studies, as well as in verifying theoretical assertions, such as, for example, the universality of the Gutenberg-Richter Law. In spite of the importance of this issue, many scientific papers still adopt formulas that lead to different estimations. The aim of this paper is to review the main concepts relative to the estimation of the b-value and its uncertainty, and to provide some new analytical and numerical insights on the biases introduced by the unavoidable use of binned magnitudes, and by the measurement errors on the magnitude. It is remarked that, although corrections for binned magnitudes were suggested in the past, they are still very often neglected in the estimation of the b-value, implicitly by assuming that the magnitude is a continuous random variable. In particular, we show that: i) the assumption of continuous magnitude can lead to strong bias in the b-value estimation, and to a significant underestimation of its uncertainty, also for binning of ΔM = 0.1; ii) a simple correction applied to the continuous formula causes a drastic reduction of both biases; iii) very simple formulas, until now mostly ignored, provide estimations without significant biases; iv) the effect on the bias due to the measurement errors is negligible compared to the use of binned magnitudes.