Earthquake Rupture Area from Moment Magnitude (Wells and Coppersmith)

When an earthquake happens, only part of a fault slips. This relation gives the area of that slipped patch, in square kilometres, from the moment magnitude. It is an empirical regression, not a derived law: Wells and Coppersmith (1994) fitted it to 148 earthquakes. Because it is written for $\log_{10} A$, the area grows by a constant factor for each step in magnitude. Each whole magnitude multiplies the area by about 8, which is 10 to the power 0.91.

Read the results as typical values. A magnitude 6 breaks about 90 km², a magnitude 7 about 760 km², and a magnitude 8 about 6,200 km². The last one is a stretch, because the fit covers moment magnitudes 4.8 to 7.9 and stops near 7.9. The scatter is large: about two thirds of earthquakes fall within a factor of 1.7 either way. So the area is a central estimate, not a prediction for one particular earthquake. The area is that of the fault surface that slipped, not of the ground that shook.

A different published scaling is a useful cross-check. Hanks and Bakun (2002) give $M = \log A + 3.98$ for areas up to 537 km², and $M = (4/3)\log A + 3.07$ above that, with $A$ in km². It comes from a different data set and fit, so it differs from Wells and Coppersmith by less than a fifth, which is small beside the factor of 1.7 scatter. It gives $A = 10^{2.02} = 104.7$ km² for magnitude 6 (the page gives 93 km²) and 886 km² for magnitude 7 (the page gives 759 km²). Run the other way, 104.7 km² returns about magnitude 6.0, where the page gives 6.05.

The inverse form, $M = (\log_{10} A + 3.49)/0.91$, reads a magnitude from a known rupture area. Use either direction inside the range of the fit. The source is Wells, D. L., and Coppersmith, K. J. (1994), Bulletin of the Seismological Society of America, 84(4).