How to be fooled searching for significant variations of the b-value

An unbiased estimation of the b-value and of its variability is essential to verify empirically its physical contribution to the earthquake generation process, and the capability to improve earthquake forecasting and seismic hazard. Notwithstanding the vast literature on the b-value estimation, we note that some potential sources of bias that may lead to non-physical b-value variations are too often ignored in seismological common practice. The aim of this paper is to discuss some of them in detail, when the b-value is estimated through the popular Aki's formula. Specifically, we describe how a finite data set can lead to biased evaluations of the b-value and its uncertainty, which are caused by the correlation between the b-value and the maximum magnitude of the data set; we quantify analytically the bias on the b-value caused by the magnitude binning; we show how departures from the exponential distribution of the magnitude, caused by a truncated Gutenberg-Richter law and by catalogue incompleteness, can affect the b-value estimation and the search for statistically significant variations; we derive explicitly the statistical distribution of the magnitude affected by random symmetrical error, showing that the magnitude error does not induce any further significant bias, at least for reasonable amplitude of the measurement error. Finally, we provide some recipes to minimize the impact of these potential sources of bias.