Comparing Different Models of Aftershock Rate Decay: The Role of Catalog Incompleteness in the First Times After the Main Shock

We evaluated the efficiency of various models in describing the time decay of aftershock rate of 47 simple sequences occurred in California (37) from 1934 to 2004 and in Italy (10) from 1976 to 2004. We compared the models by the corrected Akaike Information Criterion (AICc) and the Bayesian Information Criterion (BIC). To evaluate the role of catalog incompleteness in the first times after the main shock, we compared the performance of different models by varying the starting time Ts and the minimum magnitude threshold Mmin for each sequence. We found that Omori-type models including parameter c are preferable to those not including it, only for short Ts and low Mmin while the latters generally perform better than the formers for Ts longer than a few hours and Mmin larger than the main shock magnitude M_m minus 3 units. For Ts>1 day or Mmin> M_m-2.5, only about 20% of the sequences still give a preference to models including c. This clearly indicates that a value of parameter c different from zero does not represent a general property of aftershock sequences in California and Italy but it is very likely induced in most cases by catalog incompleteness in the first times after the main shock. We also considered other models of aftershock decay proposed in the literature: the Stretched Exponential Law in two forms (including and not including a time shift) and the band Limited Power Law (LPL). We found that such models perform worse than the Modified Omori Model (MOM) and other Omori-type models for the large majority of sequences, although for LPL, the short duration of the analyzed sequences and the methods used to select them might also contribute to its poor performance. Our analysis demonstrates that the original Omori law and the MOM with c kept fixed to 0 represent the better choices for the modeling (and the forecasting) of simple sequence behavior in California and Italy.