Productivity within the epidemic-type seismicity model

The productivity of a magnitude m event can be characterized in term of triggered events of magnitude above m-d: it is the number of direct descendants v(d) and the number of all descendants V(d). There is evidence in favour of the discrete exponential distribution for both v(d) and V(d) with a dominant initial magnitude m (the case of aftershock cluster). We consider the general Epidemic Type Aftershock Sequence model adapted to any distribution of v(d). Our first result shows that models with branching aftershock structure do not allow for the coincidence of distribution types of v(d) and V(d) (say, the discrete exponential, as in the scientific literature). The second problem is related to the tail behaviour of the V(d) distribution. We show the fundamental difference in tail behaviour of the V(d)-distributions for general-type clusters and clusters with a dominant initial magnitude: the tail is heavy in the former case and light in the latter. The real data demonstrate the possibilities of this kind. This result provides theoretical and practical constraints for distributional analysis of V(d).