Recommendations for determining p values for aftershock sequences and catalogs

Abstract

Abstract Using synthetic aftershock sequences, we investigate the factors that influence the maximum likelihood estimate (MLE) of the Omori-law p value (pMLE) 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 pMLE, called σMLE, 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 pMLE itself; this dependence on pMLE leads to over- or underestimating σMLE. Moreover, σMLE depends on the aftershock times themselves only as they affect pMLE 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 Wn2, 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, pMLE is approximately equal to the weighted mean of the individual sequence p values.