Conditions for the occurrence of seismic sequences in a fault system

We consider a system made of n coplanar faults and study the conditions under which it may originate a sequence of n seismic events of similar magnitudes. The system is placed in an elastic half-space and is subject to a constant and uniform strain rate by tectonic motions. The state of the system is described by n variables that are the Coulomb stresses of the faults. Coulomb stresses increase steadily during the interseismic intervals and change suddenly in connection with seismic events. A permutation of the first n integers can be associated with each state of the system, expressing the order of Coulomb stresses according to their magnitudes. The permutation controls the evolution of the system. The state space can be divided into n! subsets corresponding to each of the n! permutations. The representative point of the system moves continuously in one of the subsets during the interseismic intervals and switches to a different subset in connection with seismic events. The order of events in a sequence can be also expressed as a permutation of the first n integers, so that the number of possible sequences is equal to n!. The order of events is determined by the initial stress state, the stress drops and the stress transfers associated with each event. If the sequence is made of n similar events and the differences between Coulomb stresses are always greater than the mutual stress transfers, the order is given by the initial permutation. However, the order implicit in the initial permutation is generally modified during the sequence, due to changes in the differences between Coulomb stresses and to different stress drops of the events. The model allows the retrieval of the stress states of a fault system from the observation of a seismic sequence. As an example, the model is applied to the 2012 Emilia (Italy) seismic sequence and enlightens the complex interplay between the fault dislocations that produced the observed order of events.