Variability of the b value in the Gutenberg–Richter distribution

Abstract

The <it>b</it> value of the Gutenberg–Richter (GR) distribution is estimated as a function of a threshold magnitude <it>m</it><inf>th</inf> and it is found to depend on <it>m</it><inf>th</inf> for magnitudes larger than the completeness magnitude <it>m</it><inf><it>c</it></inf>. We identify a magnitude interval [<it>m</it><inf><it>c</it></inf>, <it>m</it><inf><it>m</it></inf>] where <it>b</it> is a decreasing function of <it>m</it><inf>th</inf> followed by a regime of increasing <it>b</it> for large magnitudes. This is a common feature of experimental catalogues for different geographic areas. The increase at large <it>m</it><inf>th</inf> is explained in terms of an upper magnitude cut-off in experimental catalogues due to finite size effects. We develop a rigorous mathematical framework to relate the decrease of <it>b</it> in the intermediate regime to the functional form of the distribution of the <it>b</it> values. We propose two hypotheses: The first is that the spatial and temporal variability of <it>b</it> leads to a <it>b</it> distribution peaked around its average value. The second is that main shocks and aftershocks are distributed according to the GR law with different <it>b</it> values, leading to a bimodal distribution of <it>b</it>. Simulated Epidemic Type Aftershock Sequences catalogues, generated according to this hypothesis, exhibit the same magnitude distribution of experimental ones. In alternative, we cannot exclude the <it>b</it> dependence on <it>m</it> caused by magnitudes not homogeneously evaluated in a seismic catalogue. In the latter scenario our results provide the correction terms to the estimated magnitudes.