Average Fault Slip from Moment Magnitude (Wells and Coppersmith)

This is an empirical regression, not a law of physics. Wells and Coppersmith (1994) fitted rupture dimensions and surface displacement against moment magnitude, and the fit for average slip is $\log_{10} D = -4.80 + 0.69\,M$, with $D$ in metres. Because the magnitude sits in an exponent, each step in magnitude multiplies the slip by the same factor, so slip grows steadily rather than by a fixed amount. The page puts a magnitude 6 at about 0.2 m of average slip, a magnitude 7 at about 1.1 m and a magnitude 8 at about 5 m.

Read $D$ as the average distance the two sides of the fault moved past each other over the rupture. The scatter is large. This is the noisiest of their relations: about two thirds of earthquakes fall within a factor of 2.3 either way, so the result is a scale of the slip and not a prediction of it. The fit covers moment magnitudes of 5.6 to 8.1, and a result outside that range is an extrapolation.

A different route shows how far to trust it. The physical identity is the moment balance $M_0 = \mu A D$, with $M_0 = 10^{1.5M + 9.05}$ N·m (Hanks and Kanamori), a rigidity $\mu$ of 30 GPa and a rupture area $A$ from the rupture-area fit. At magnitude 7 that area is 759 km², so $D = 3.548\times10^{19} / (3\times10^{10} \times 7.59\times10^{8}) = 1.56$ m. At magnitude 7.5, with an area of 2,163 km², the balance gives 3.08 m. The page's own values are 1.07 m and 2.37 m, lower than the balance but well inside the fit's scatter of a factor of 2.3. Run backward, 1.07 m of slip returns magnitude 7.0.

The two routes answer different questions. The moment balance is an identity that connects slip to area and moment once two of the three are known, and the seismic-moment page treats it that way. The Wells and Coppersmith relation estimates slip from the magnitude alone, and it carries all the scatter of the data it was fitted to. The rupture-area page gives the area that slipped by this amount, and the moment-magnitude page gives the magnitude this page takes.