Abstract.A fault containing two asperities with different strengths is considered. The fault is embedded in a shear zone subject to a constant strain rate by the motions of adjacent tectonic plates. The fault is modelled as a discrete dynamical system where the average values of stress, friction and slip on each asperity are considered. The state of the fault is described by three variables: the slip deficits of the asperities and the viscoelastic deformation. The system has four dynamic modes, for which analytical solutions are calculated. The relationship between the state of the fault before a seismic event and the sequence of slipping modes in the event is enlightened. Since the moment rate depends on the number and sequence of slipping modes, the knowledge of the source function of an earthquake constrains the orbit of the system in the phase space. If the source functions of a larger number of consecutive earthquakes were known, the orbit could be constrained more and more and its evolution could be predicted with a smaller uncertainty. The model is applied to the 1964 Alaska earthquake, which was the effect of the failure of two asperities and for which a remarkable post-seismic relaxation has been observed in the subsequent decades. The evolution of the system after the 1964 event depends on the state from which the event was originated, that is constrained by the observed moment rate. The possible durations of the interseismic interval and the possible moment rates of the next earthquake are calculated as functions of the initial state.

We consider a fault containing two regions with different mechanical behaviours: a strong, velocity-weakening region (asperity) and a weak, velocity-strengthening region. The fault is embedded in a shear zone subject to a constant strain rate by the motions of adjacent tectonic plates. The fault is modelled as a discrete dynamical system whose state is described by two variables expressing the slip deficits of the two regions. Because of plate motion, the asperity accumulates stress and eventually releases it, producing an earthquake, when a frictional threshold is exceeded. The weak region is subject to a very slow creep during interseismic intervals and may slip at a higher rate (afterslip) as a consequence of coseismic stress imposed by the asperity failure. The evolution equations of the system are solved analytically for the interseismic intervals, the asperity slip and the afterslip in the weak region. It is found that the amount of afterslip is proportional to the seismic slip of the asperity, in agreement with observations. The model shows that afterslip is a natural consequence of seismic slip in a fault containing a velocity-strengthening region. Afterslip may have any duration, according to the intensity of velocity strengthening, thus accounting for the wide range of observed durations. The model is applied to the fault of the 2011 Tohoku-Oki earthquake. The results suggest that the first four months after the event were dominated by afterslip, while the subsequent postseismic deformation was probably due to viscoelastic relaxation in the asthenosphere.

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Abstract. 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.

Abstract. A fault containing two asperities with different strengths is considered. The fault is embedded in a viscoelastic shear zone, subject to a constant strain rate by the motions of adjacent tectonic plates. The fault is modelled as a discrete dynamical system where the average values of stress, friction and slip on each asperity are considered. The state of the fault is described by three variables: the slip deficits of the asperities and the viscoelastic deformation. The system has four dynamic modes, for which the analytical solutions are calculated. The relationship between the state of the fault before a seismic event and the sequence of slipping modes in the event is enlightened. Since the moment rate depends on the number and sequence of slipping modes, the knowledge of the source function of an earthquake constrains the orbit of the system in the phase space. If the source functions of a larger number of consecutive earthquakes were known, the orbit could be constrained more and more and its evolution could be predicted with a smaller uncertainty. The model is applied to the 1964 Alaska earthquake, which was the effect of the failure of two asperities and for which a remarkable postseismic relaxation has been observed in the subsequent decades. The evolution of the system after the 1964 event depends on the state from which the event was originated, that is constrained by the observed moment rate. The possible durations of the interseismic interval and the possible moment rates of the next earthquake are calculated as functions of the initial state.