Scaling Relationship between the Number of Aftershocks and the Size of the Main Shock

Abstract

The logarithmic number of aftershocks which occur within one month after a large shallow earthquake in Japan is found to be proportional to the logarithm of the seismic moment of the main shock. The proportional constants for interplate and intraplate earthquakes are different, but it turns out to reflect a similar difference in scaling relations of the fault area to the seismic moment between interplate and intraplate earthquakes. Thus we can derive the fundamental relationship that the aftershock number is proportional to the fault area of the main shock. This is consistent with the hypothesis that aftershocks are generated by unbroken strong patches on the main-shack fault, if the patch density is constant. Combining this new result with the results of previous studies on aftershocks, we propose that a rate of aftershock occurrence is given by
where n(t) indicates the number of aftershocks which occur t days after the main shock, S indicates the fault area of the main shock in km², M_th is the threshold magnitude, b is the b-value of the. Gutenberg-Richter relationship, and c and p are the c- and p-values, respectively, of the modified Omori formula. The constant k amounts to 13.4 for an interplate earthquake and to 31.7-63.5 for an intraplate earthquake depending on the assumption on its fault width. Apparently the areal density of aftershocks on the fault plane is higher for an intraplate earthquake than for an interplate earthquake. Further examination of the results suggests that the areal density of the aftershocks of a continental intraplate earthquake may be systematically higher than that of an oceanic intraplate earthquake. These differences may indicate some intrinsic difference in rupturing process among a continental intraplate, an oceanic intraplate, and an interplate events.