Strain energy release by an earthquake rupture with heterogeneous stress-drop distribution

This study theoretically derived the strain energy change in an elastic medium caused by an earthquake faulting and the necessary condition of the initial stress for the earthquake generation. We suppose a heterogeneous slip distribution causing non-uniform stress change on the fault. Employing a stochastic modeling of the slip fluctuation from a reference slip distribution, we represent the ensemble average of the strain energy change by using the power spectral density function of the slip fluctuation. The resultant analytical solutions indicate the followings. (1) When the initial stress is uniform and the non-uniform stress change occurs by the slip fluctuation, the strain energy is less released than the uniform stress change on the fault without the slip fluctuation and the seismic moment is the same. (2) When the initial stress is non-uniform and the final stress is uniform due to the non-uniform stress change on the fault, the strain energy is more released than the uniform stress change on the fault. (3) An effective stress drop in the cases of (1) and (2) becomes larger than the spatially averaged stress drop due to the slip fluctuation in the view point of the strain energy release. This was pointed out in Noda, H. et al. (2013 GJI) and Hirano and Yagi (2017 GJI). This study represents the analytical solution of the effective stress drop using the power spectral density function of the random slip fluctuation. (4) When both the initial stress and the stress change are uniform on the fault, the strain energy is simply proportional to the seismic moment. (5) However, the strain energy change is generally not proportional to the seismic moment when there is slip fluctuation. If the variance of the fractional slip fluctuation over the correlation distance is proportional to the seismic moment to the power of -1/3, the strain energy is proportional to the seismic moment. (5) The energy balance gives a critical value of the initial stress that is required for an earthquake faulting. In order to generate an earthquake, the initial stress needs to be larger than the sum of the half of the effective stress drop and the apparent stress.