A New Statistical Perspective on Båth’s Law

Abstract The empirical Båth’s law states that the average magnitude difference (ΔM) between a mainshock and its strongest aftershock is ∼1.2, independent of the size of the mainshock. Although this observation can generally be explained by a scaling of aftershock productivity with mainshock magnitude in combination with a Gutenberg–Richter frequency–magnitude distribution, estimates of ΔM may be preferable because they are directly related to the most interesting information, namely the magnitudes of the main events, without relying on assumptions. However, a major challenge in calculating this value is the bias introduced by missing data points when the strongest aftershock is below the observed cut-off magnitude. Ignoring missing values leads to a systematic error because the data points removed are those with particularly large magnitude differences ΔM. The error can be minimized by restricting the statistics to mainshocks that are at least 2 magnitude units above the cutoff, but then the sample size is strongly reduced. This work provides an innovative approach for modeling ΔM by adapting methods for time-to-event data, which often suffer from incomplete observations (censoring). In doing so, we adequately account for unobserved values and estimate a fully parametric distribution of the magnitude differences ΔM for mainshocks in a global earthquake catalog. Our results suggest that magnitude differences are best modeled by the Gompertz distribution and that larger ΔM are expected at increasing depths and higher heat flows.