Evaluating the incompleteness magnitude using an unbiased estimate of the b value

The evaluation of the b value of the Gutenberg-Richter (GR) law, for a sample composed of n earthquakes, presents a systematic positive bias delta b which is proportional to 1/n . In this study, we show how to incorporate in delta b the bias introduced by deviations from the GR law. More precisely we show that delta b is proportional to the square of the variability coefficient CV, defined as the ratio between the standard deviation of the magnitude distribution and its mean value. When the magnitude distribution follows the GR law CV = 1 and this allows us to introduce a new graphical procedure, based on the dependence of b on n, which allows us to identify the incompleteness magnitude mc as the threshold magnitude leading to CV = 1. The method is tested on synthetic catalogues and it is applied to estimate mc in Southern California, Japan and New Zealand.