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Recommendations for determining p values for aftershock sequences and catalogs

Department of Geological Sciences and Institute for Geophysics University of Texas at Austin, 4412 Spicewood Springs, # 600, Austin, Texas, 78759-8500
University of Texas Institute for Geophysics, 4412 Spicewood Springs, # 600, Austin, Texas 78759-8500

Abstract

Using synthetic aftershock sequences, we investigate the factors that influence the maximum likelihood estimate (MLE) of the Omori-law p value (p_MLE) determined for aftershock sequences of individual earthquakes, as well as for "stacked" sequences constructed using aftershocks from two or more mainshock events. The estimated uncertainty in p_MLE, called _MLE, depends explicitly on the aftershock sequence population N, the ratio of the begin and end times of the time interval of interest T/S, and on p_MLE itself; this dependence on p_MLE leads to over- or underestimating _MLE. Moreover, _MLE depends on the aftershock times themselves only as they affect p_MLE and is unaffected by whether the sequence decay resembles the Omori decay model. Thus, we recommend that reported p values be accompanied by a goodness-of-fit test, such as the Anderson-Darling statistic W_n², to assess how well the sequence resembles Omori's law (OL), as well as by values for N, S, and T. Stacking aftershock sequences permits the study of p values for entire catalogs as well as sequences possessing too few events to allow individual analysis. However, stacking sequences with differing begin and end times introduces artifacts into the aftershock time series that need to be avoided when determining p values. One artifact arising at small times after the mainshock mimics the effect of the c parameter in the modified Omori's law (MOL). We can avoid the artifacts by excluding time spans containing changes in the slope of the log rate of aftershock activity from the analysis. For most situations where sequences with varying p, N, and begin and end times are stacked, p_MLE is approximately equal to the weighted mean of the individual sequence p values.