Båth's law

Remarks

The law is an observed regularity of earthquake catalogues, not a theorem, and it is named after the Swedish seismologist Markus Båth. Single sequences scatter widely around the value $1.2$. A standard heuristic explanation combines two other laws. The magnitudes of aftershocks follow the Gutenberg–Richter law with the same $b$-value as other earthquakes. If the total number of aftershocks above a fixed magnitude grows like $10^{b M_m}$, then the expected largest of them lies a fixed amount below $M_m$, which gives a constant difference. Example: a mainshock of magnitude $7.0$ is expected to have a largest aftershock of magnitude about $5.8$. Together with Omori's law for the decay of the aftershock rate in time, the law is used to forecast the hazard after a large earthquake.

Going deeper

For a student

Båth's law is one of three classical statistical laws of aftershocks, next to the Gutenberg–Richter law for sizes and Omori's law for times. It is often explained by self-similar statistics. When aftershock productivity and the magnitude distribution both scale exponentially with magnitude, the gap between the mainshock and the largest aftershock no longer depends on the mainshock. In forecasting it gives a quick estimate of how strong the largest aftershock may be.