Reply to “Comment on ‘Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function’ by A. Mignan” by S. Hainzl and A. Christophersen

Abstract

ABSTRACT Hainzl and Christophersen (2016) recently suggested that the approach used by Mignan (2016) to visualize the Omori law in a log–log plot using the complementary cumulative density function (CCDF) was misleading. They showed that one should use a temporal upper bound t max =max( t obs ), with t obs the occurrence time of a set of observed aftershocks, instead of t max →∞, as used by Mignan. They found that both the Omori law and stretched exponential function (SEF) are undistinguishable on a corrected CCDF log–log plot but that the Omori law is preferred based on maximum‐likelihood estimations (MLEs). I first clarify the rationale for using the CCDF log–log plot for comparison of the SEF with a power law but verify that the comparison indeed becomes misleading when the data sequence is incomplete. However, I then show that MLEs obtained for the Omori law are ambiguous, because this function may win versus the SEF, even if the true relaxation process follows an SEF (subject to early data incompleteness). This debate highlights that statistics alone may not be enough to choose between the stretched exponential and the Omori law and that physical considerations should be added to the discussion.