Variability of the b value in the Gutenberg–Richter distribution

The b value of the Gutenberg - Richter distribution is estimated as a function of a threshold magnitude m_th and it is found to depend on m_th for magnitudes larger than the completeness magnitude m_c. We identify a magnitude interval [m_c;m_m] where b is a decreasing function of m_th followed by a regime of increasing b for large magnitudes. This is a common feature of experimental catalogues for different geographic areas. The increase at large m_th 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 b in the intermediate regime to the functional form of the distribution of the b values. We propose two hypotheses: The first is that the spatial and temporal vari- ability of b leads to a b distribution peaked around its average value. The second is that mainshocks and aftershocks are distributed according to the Gutenberg-Richter law with different b values, leading to a bimodal distribution of b. Simulated Epidemic Type Aftershock Sequences (ETAS) catalogues, generated according to this hypothesis, exhibit the same magnitude distribution of experimental ones. In alternative we cannot exclude the b dependence on m caused by magnitudes not homogeneously evaluated in a seismic catalogue. In the latter scenario our results provide the correction terms to the estimated magnitudes.