Aftershock Relaxation for Japanese and Sumatra Earthquakes

The modified Omori law provides us with an appropriate representation of the rate of aftershock occurrence r(t) with time t in the form r(t) = τ-1(1+t/c)-p , where τ, c, and p are parameters. No systematic dependence of p was observed on the lower bound of the magnitude m above which aftershocks were counted as well as on the main shock magnitude M while the various values of p were obtained from one aftershock sequence to the other. The c value is influenced by the incomplete detection of small aftershocks in the early stage of sequence. However, careful analyses showed that this value was positive for some sequences. Here we discuss the aftershock relaxation with the emphasis on the parameter c. Our approach is to combine three empirical scaling laws that aftershock sequences approximately satisfy; the Gutenberg-Richter law for the frequency-magnitude relation, the Bath's law for the difference between the magnitude of a main shock and its largest aftershock, and the modified Omori law. This combination results in a so-called generalized Omori law. The main difference from the original form of the modified Omori law is that the parameter c is no longer a constant but it scales with a lower cutoff magnitude m. The proposed law is tested by using the aftershocks of several relatively recent and large earthquakes that occurred in Japan; the 1995 Kobe earthquake (M = 7.2), the 2000 Tottori earthquake (M = 7.3), the 2004 Niigata Chuetsu earthquake (M = 6.8), and the 2005 Fukuoka earthquake (M = 7.0). Moreover, we test the proposed scaling law by using also the aftershock sequence following the 2004 Northern Sumatra earthquake (M = 9.0). Our careful analyses confirm that the proposed law is applicable to the temporal decay of aftershock rates and provide a justification of the dependence of c on m.