Conditions for the occurrence of seismic sequences in a fault system

We consider a fault system producing a sequence of seismic events of similar magnitudes. If the system is made up of n faults, there are n! possible sequences, differing from each other for the order of fault activation. Therefore the order of events in a sequence can be expressed as a permutation of the first n integers. We investigate the conditions for the occurrence of a seismic sequence and how the order of events is related to the initial stress state of the fault system. To this aim, we consider n coplanar faults placed in an elastic half-space and subject to a constant and uniform strain rate by tectonic motions. We describe the state of the system by n variables that are the Coulomb stresses of the faults. If we order the faults according to the magnitude of their Coulomb stresses, a permutation of the first n integers can be associated with each state of the system. This permutation changes whenever a fault produces a seismic event, so that the evolution of the system can be described as a sequence of permutations. A crucial role is played by the differences between Coulomb stresses of the faults. The order of events implicit in the initial state is modified due to changes in the differences between Coulomb stresses and to different stress drops of the events. We find that the order of events is determined by the initial stress state, the stress drops and the stress transfers associated with each event. Therefore the model allows the retrieval of the stress states of a fault system from the observation of the order of fault activation in 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.