Båth’s Law Derived from Gutenberg-Richter’s Law: a Simple Deduction with Implications for Earthquake Sequence Analysis

We provide the derivation of Båth’s law from Gutenberg-Richter’s law with a simple deduction through Zipf distribution for a set of data exhibiting power law scaling. It turns out that D, the difference between the magnitude of the mainshock and that of the largest aftershock (or the second largest event in the mainshock-aftershock sequence) has positive correlation with the magnitude of the mainshock. The parameters of the aftershock sequence subject to analysis, including number of samples, cutoff magnitude, and b‑value, also contribute to D and its uncertainty. The uncertainty of D is even larger associated with the ‘dragon king’ events which are the statistical outlier of the power law scaling.