Earthquake Rupture Length from Moment Magnitude (Wells and Coppersmith)

The bigger the earthquake, the longer the stretch of fault that slips, and this relation turns a moment magnitude $M$ into a rupture length $L$ along strike, in kilometres. It is an empirical regression, not a physical law: Wells and Coppersmith (1994) fitted a straight line, in the logarithm of the length, to the subsurface rupture of 167 earthquakes. Nothing about fault mechanics is derived; the line summarises what those earthquakes did.

The logarithmic form makes the scaling easy to read. Each whole magnitude lengthens the rupture by a factor of about 3.9, which is 10 to the power 0.59, so the lengths climb steeply: about 12 km at magnitude 6, 49 km at magnitude 7 and 190 km at magnitude 8. A step of one in magnitude multiplies the length instead of adding to it.

A real event shows how to use it. The 1992 Landers earthquake (M 7.3) is quoted with a rupture 70 to 85 km long, with slip models near 70 km and the surface break near 85 km. The relation gives 73.6 km, inside that range. It also runs backward, $M = \frac{\log_{10} L + 2.44}{0.59}$, so a rupture of 77.5 km, the middle of the quoted range, corresponds to about magnitude 7.3.

A regression has scatter. About two thirds of earthquakes fall within a factor of 1.45 of the line, so the answer is a typical length for that magnitude, not a prediction for a particular quake, and the fit covers magnitudes from 4.8 to 8.1. The length is that of the fault that slipped, which is why the distance to a big earthquake is not one number: the waves come from along the whole length. The area of the same rupture and the displacement along it are on neighbouring pages.