Relation between mainshock rupture process and Omori's law for aftershock moment release rate

We compare the source time functions (i.e., moment release rates) of three large California mainshocks with the seismic moment release rates during their aftershock sequences. Aftershock moment release rates, computed by summing aftershock moments in time intervals, follow a power-law time dependence similar to Omori's law from minutes to months after the mainshock; furthermore, in contrast to the previously observed saturation in numbers of aftershocks shortly after the mainshock rupture, no such saturation is seen in the aftershock moment release rates, which are dominated by the largest aftershocks. We argue that the observed saturation in aftershock numbers described by the 'time offset' parameter c in Omori's law is likely an artefact due to the underreporting of small aftershocks, which is related to the difficulty of detecting large numbers of small aftershocks in the mainshock coda. We further propose that it is more natural for c to be negative (i.e. singularity follows the onset of mainshock rupture) than positive (singularity precedes onset of rupture). To make a more general comparison of mainshock rupture process and aftershock moment rates, we then scale mainshock time functions to equalize the effects of the varied seismic moments. For the three California mainshocks, we compare the scaled time functions with similarly scaled aftershock moment rates. Finally, we compare global averages of scaled time functions of many shallow events to the average scaled aftershock moment release rate for six California mainshocks. In each of these comparisons, the extrapolation, using Omori's law, of the aftershock moment rates back in time to the onset of the mainshock rupture indicates that the temporal intensity of the aftershock moment release is about 1.5 orders of magnitude less than the maximum reached by the mainshock rupture. This may be due to the differing amplitudes and relative importance of static and dynamic stresses in aftershock initiation compared to mainshock rupture propagation.