Felt Intensity from Magnitude and Distance (Allen, Wald and Worden)

Modified Mercalli intensity describes what people feel at a place, on a scale from I to XII, and it falls with distance from the fault. This relation predicts it from the moment magnitude $M$ and the closest distance $R$ from the site to the fault rupture. It is an empirical attenuation relation, fitted to observed intensities by Allen, Wald and Worden (2012) for shallow earthquakes in active crustal regions, not a derivation from physics. The coefficients (3.950, 0.913, 1.107 and 0.813) are the fitted values of the published Rrup version.

Read it term by term. Intensity rises with magnitude, by 0.913 for each unit. It falls with the natural logarithm of an effective distance $\sqrt{R^2 + h^2}$, weighted by $-1.107$. And $h = 1 + 0.813\,e^{M-5}$ is the near-source term: it stops the intensity growing without limit as $R$ shrinks toward the rupture, and since it grows with magnitude, a bigger earthquake levels off over a wider zone. The distance is measured to the fault, not to the epicentre, because a large rupture is long.

For a magnitude 7 earthquake 50 km away, $h = 1 + 0.813\,e^{2} = 7.0073$, and $I = 3.950 + 6.391 - 1.107\ln\sqrt{2500 + 49.10} = 5.99962$, about VI. At 100 km from a magnitude 6 the relation gives 4.32951, about IV. For a real event, the 1994 Northridge earthquake (M 6.7) was felt at about VIII within a few kilometres of the rupture, and IX in a few places; the page gives 7.85 at 5 km. The relation also runs backwards: 5.99962 from a magnitude 7 puts the site 50 km away, and 4.32951 at 100 km gives magnitude 6.0. Solving for magnitude can have more than one solution, and the page takes the smaller.

Treat the output as a typical felt intensity, not a forecast for a particular site: the scatter is about 0.7 to 0.95 of an intensity unit. It applies to the kind of earthquake it was fitted to, with magnitudes of about 5 to 8.