How can one compare the magnitude-9.0 earthquake in Japan with the magnitude-6.3 quake that struck New Zealand?

News stories about the disaster in Japan bandy around two sorts of figures when comparing earthquakes: magnitude and energy. It's not hard to keep them straight and not much harder to calculate the comparisons yourself.

Magnitude measures shaking. The old Richter magnitudes, which aren't used anymore, were calculated from ground movements (the amplitudes of seismic waves). So-called moment magnitudes, used since the 1970s, are based on a more complicated formula but were designed to be comparable for most earthquakes.

The magnitude scale is logarithmic. That just means that if you add 1 to an earthquake's magnitude, you multiply the shaking by 10. An earthquake of magnitude 5 shakes 10 times as violently as an earthquake of magnitude 4; a magnitude-6 quake shakes 10 times as hard as a magnitude-5 quake; and so on.

To compare two earthquakes in terms of shaking, you subtract one magnitude from the other and raise 10 to that power: 10^(M1-M2).

For example, if the magnitude of one quake is 6 and another is 4, than the difference in magnitudes is 2, so the stronger earthquake shakes 10^2 or 100 times as hard as the milder one.

Fractional differences work the same way. Increase the magnitude by 0.1, and you multiply the shaking by 10^(0.1), or about 1.259—an increase of 26%. Increase the magnitude by 0.3, and the shaking almost exactly doubles. That's a handy rule of thumb to keep in mind.

Shaking isn't the only way to compare earthquakes. Another common approach is to talk about the relative amounts of energy they release. To do that, just insert one simple extra step: subtract the magnitudes and then add 50% before using the result as a power of 10. In mathspeak, the formula is 10^((M1-M2)*1.5).

(This works because the energy scales with 3/2 of magnitude, for reasons that don't matter.)

Final example: comparing the magnitude-9.0 earthquake in Japan with the magnitude-6.3 quake that struck New Zealand in February. The difference in magnitudes is 2.7, so the difference in *shaking* is 10^2.7, or just over 500 times as big—a figure you've probably seen. The difference in energy, however, is 10^(2.7*1.5) = 10^4.05, or about 11,220 times.

You can learn a lot more at the U.S. Geological Survey's earthquake site [1].