On the relation between seismic moment and stress drop in the presence of stress and strength heterogeneity

The seismic moment is related by definition to the average slip on the fault plane of an earthquake. Here we derive an exact expression for the seismic moment in terms of a general heterogeneous stress drop distribution and the geometry of the fault of a complex event. We find that the seismic moment is proportional to a weighted integral of the stress drop on the fault. The weight in this linear relationship is the slip for a hypothetical event with the same source geometry but uniform stress drop. This relationship between seismic moment and stress drop depends on geometry. In particular, for multiple sources the weight is reduced by factors of the order of ρ/ R , where ρ is the radius of a typical subfault and R is the radius of the total source area. As a consequence of these results we find that for a given stress drop, a simple fault generates a larger seismic moment than a multiple fault of the same total surface. Conversely, for a given moment and source area, a complex event would need higher stress drops on the subfaults than a simple smooth fault. We test these results with three rectangular models of faulting. The first is a simple, smooth fault with uniform stress drop. The second model is a simple fault with zero stress drop in the central section of the fault. The last model is a complex event where the central section of the fault remains unbroken. We show that the last two models are difficult to distinguish from their far‐field radiation.