A two-asperity fault model with wave radiation

We study the dynamics of a fault containing two asperities having different strengths and subject to a constant strain rate. The fault is modeled by a discrete dynamical system with two degrees of freedom that are the slip deficits of the asperities. The radiation of elastic waves during the slipping modes is taken into account by introducing a term proportional to slip rate in the equations of motion. We give a complete analytical solution of the four dynamic modes of the system. Any seismic event can be expressed as a sequence of modes, for which the moment rate, the spectrum and the total seismic moment can be calculated. We consider the energy budget of the event and calculate the seismic efficiency. The presence of radiation may remarkably change the evolution of the system from a given state, since it moves the boundaries between the different subsets of the sticking region. In addition, the slip amplitude in a seismic event is smaller, while the slip duration is longer in the presence of radiation. The shape of the moment rate function depends on the seismic efficiency and the seismic moment decreases with increasing efficiency, at constant radiated energy. As an application of the model, we consider the 1964 Great Alaska earthquake and calculate the mode durations, the slip distribution, the moment rate function and the seismic moment of the earthquake, obtaining values that are consistent with observations.