GEOPHYSICAL RESEARCH LETTERS, VOL. 12, NO. 10, PAGES 713-716, OCTOBER 1985 IDENTIFICATION OF AFTERSHOCKS OF DEEP EARTHQUAKES BY A NEW RATIOS METHOD Cliff Frohlich and Scott Davis Institute for Geophysics, University of Texas at Austin, Austin, Texas 78713 Abstract. We describe a new statistical method for identifying pairs or groups of related events in sequences which resemble, but are not identical to, a Poisson process. For a particular earthquake, we form the ratior (No ,Na) of the relativeorigin timesof the Nath subsequent eventand the No th previous event. We then find the probability that such a ratio could occur if the sequence were a Poisson process. Presumably, sequences where the ratio is too small to be probable contain related events, and the subsequent events are aftershocks. Since the method requires knowledge of the origin times of only a few preceding and subsequent events, it is more powerful than methods whichrequire knowledge of the mean activity rate of the Poisson process. Using this ratios method, we searched the ISC catalogfor aftershocks of earthquakes with focal depths exceeding 70 km. From the world as a whole, eventswith at least one aftershock can be found in all depth ranges, including 250 km to 450 km. However, the incidence of aftershocks is significantly lower for events between 100 and 450 km than for shallower or deeper events. Introduction Numerous studies showthat seismic activityis neither a periodic nor a Poisson process [e.g., Utsu, 1969; 1970; 1971; 1972]. Instead some earthquakes cluster in time and space, and many large earthquakes possess "foreshocks" or "aftershocks"[e. g., Vem-Jones, 1970; Kagan and Knopoff, 1976; 1978; Prozorov and Dziewonski,1982; Reasenberg, 1985]. Although there are several methodsfor identifying foreshocks or aftershocks [e.g., Utsu, 1972; Schlienand Toksoz, 1974], no commonly accepted, rigorous definition of aftershock or foreshock is available. In this paperwe propose a new method,the ratios method,to identify aftershocks and foreshocks. This method is superior when the numberof eventsin a sequence is small, or when events often have few or no foreshocks or aftershocks. The ratiosmethod can be applied to sequences with very few events(five or fewer), so one can restrict the sequence to events occurring in a relatively smallgeographic area, and within a relatively short time span. This is a significant advantage of the method, since earthquake catalogs are morelikely to be uniform overshort timeperiods than over long ones[Habermann, 1982], andit usually not necessary to ignore all events smaller than some cutoff magnitude. Copyright 1985 by theAmerican Geophysical Union. Paper number 5L6602 0094-8276/85/005L-6602 $03.00 For all these masons, the ratios method is ideal for identifying aftershocks of deep earthquakes (DEQ). While DEQ seldompossess the well-developed sequences of numerous aftershocks observed for large shallow events [e. g., seePage, 1968] manystudies report DEQ "doublets" or "multiplets" [e.g., Isacks et al., 1967; Oike, 1971], whose incidence may differ depending on focaldepth or on geographic region [Prozorov andDziewonski, 1982; Kagan and Knopoff, 1980]. In this paperwe apply the ratios method to study how focal depthaffects the incidence of DEQ aftershocks and foreshocks.A more comprehensive analysis, including how the incidence of DEQ fore- and aftershocks depends on magnitude and geographic region, is in preparation and will appear elsewhere. The Ratios Method: Comparison of Pm- and Post-Shock Event Times For a particular event, or principal event, the ratios methodevaluates the relative origin times of eventsin a test sequence of events occurring after and before the principal event. The method determines if the observed time differences wouldbe likely to occur by chance alone. When the observed time differences are very improbable, we presume that the testsequence is not generated by a random, or Poisson process, i.e., that some of the earthquakes following theprincipal event areaftershocks. To apply theratios method (Figure 1), consider thetime TNa of theN• th earthquake following the principal event, Ratios Method 1 •--- (% +k-I )! ( i•ok R--O Fig. 1. The ratiosmethod utilizes the ratio r (N, ,N•) of relative origin times for following events (T 1A ,T• .... ) •d preceding events (T • ,T• .... ) relative to theorigin time of the originalevent (T =0) to deterinc if a sequence of events is generated by a Poisson process. 713