Seismic Moment

Seismic moment is the only measure of earthquake size that is a measurement rather than a scale reading. Keiiti Aki introduced it in 1966, and its logic is almost disappointingly simple: an earthquake is two blocks of rock sliding past each other, so its size is how stiff the rock was, times how much of the fault broke, times how far the two sides moved. $M_0 = \mu A \bar{D}$, and the answer comes out in newton metres because that is what a moment is.

Each of the three factors has a characteristic value and a characteristic mistake. The shear modulus $\mu$ is about 30 GPa in continental crust and 44 GPa is the convention for shallow oceanic settings — using a near-surface sediment value instead can halve the answer. The area is the whole rupture plane, length along strike times width down dip, and the width is capped by the depth at which rock stops failing brittly, usually 15 to 20 km. That cap is why very large earthquakes get longer rather than deeper: the 2004 Sumatra rupture ran roughly 1300 km along strike on a plane only about 150 km wide. And $\bar{D}$ is the AVERAGE slip over that plane, which is typically a third to a half of the peak slip a field survey measures where the fault crosses a road.

The range this quantity covers is what makes a logarithmic magnitude necessary. An earthquake you can just feel is around $10^{12}$ N·m; the 1994 Northridge event was about $1.2 \times 10^{19}$; Tohoku in 2011 was near $5.3 \times 10^{22}$; Valdivia in 1960 was around $2 \times 10^{23}$. That is eleven orders of magnitude, which no linear scale could carry on one axis.

One unit trap dominates all others here. The older literature and the Global CMT catalogue both report moment in dyne-centimetres, and $1\ \text{dyne·cm} = 10^{-7}\ \text{N·m}$. A moment copied from a catalogue without converting is seven decades too large, which the magnitude formula obligingly reports as about 4.7 magnitude units too high — turning a respectable earthquake into an impossible one. If a moment comes back reading Mw 14, this is why.