Do Aftershock Probabilities Decay With Time?

Abstract

Upon hearing about the devastating 22 February 2011 M 6.1 Christchurch earthquake, I remembered the nearby 4 September 2010 M 7.0 event and thought, “It’s a bit late for this to be an aftershock,” followed quickly by, “Don’t be stupid.” The first thought was based on the modified-Omori law (Utsu, 1969) in which the rate of aftershocks, exceeding a particular magnitude, decays as n ( t ) = K ( t + c ) p , where n is the number of earthquakes per unit time as a function of time ( t ) and K, c , and p are constants. The second thought was triggered by remembering that while the probability of this event being an aftershock on 22 February 2011 was small, it was necessary to consider the total probability over all days that I would consider “late.” > We should never think, “It’s a bit late for this to be an aftershock.” So, do aftershock probabilities decay with time? Consider a thought experiment in which we are at the time of the mainshock and ask how many aftershocks will occur a day, week, month, year, or even a century from now. First we must decide how large a window to use around each point in time. Let’s assume that, as we go further into the future, we are asking a less precise question. Perhaps a day from now means 1 day ± 10% of a day, a week from now means 1 week ± 10% of a week, and so on. If we ignore c because it is a small fraction of a day (e.g., Reasenberg and Jones, 1989, hereafter RJ89), and set p = 1 because it is usually close to 1 (its value in the original Omori law), then the rate of earthquakes …