Magnitude dependence of radiated energy spectra: Far-field expressions of slip pulses in earthquake models

We examine the radiated waves emitted by events on a model fault. The model deterministically produces a complex sequence of events, with a wide range of sizes, from a uniform frictional instability. The spontaneous rupture events emit a rich spectrum of radiated waves as they nucleate, propagate, and decelerate within the complex stress field left by previous events. Two model innovations, a new driving boundary condition on the fault and a new radiating boundary condition which allows a spatially varying prestress away from the fault, allow us to directly measure the radiation without problems from boundary reflections in our two-dimensional model. We quantify the radiation by first measuring the energy spectral density and then averaging over events of a similar size to examine the magnitude dependence. Assuming only a physics of the tractions on the fault, we obtain a full spectra of radiated waves for a complex population of events with a wide range of sizes. To quantify the resulting spectra, we consider two different spectral measures. One, the peak amplitude of the spectral energy density, occurs at a period which scales with the rupture length and corresponds with the classical corner frequency measurement. The other, the peak amplitude of the spectral average acceleration or the low-frequency corner in the case of a flat acceleration spectrum, occurs at a period that scales with the duration of slip of points on the fault. The period of the peak spectral acceleration saturates for large events. Looking at the rupture motions on the fault, we find that this spectral behavior corresponds with the behavior of slip pulses in the model. Intense narrow pulses of slip develop for very long rupture events. We quantify this by measuring the mean slip duration as a function of rupture length and show that it is has the same behavior as the peak period of spectral acceleration. Thus the duration of the slip pulses in these ruptures is directly expressed in their radiated spectra. Moreover, we find that these corner periods exhibit a nontrivial dependence on event magnitude for the different frictional instabilities that we have examined, suggesting that any observed dependence of these corner periods on earthquake magnitude might provide insight into the frictional instability of earthquakes