Moment Magnitude (Mw)

Moment magnitude exists because the older scales lied about the biggest earthquakes. Charles Richter's ML, defined in 1935, read the peak amplitude on a Wood–Anderson torsion seismometer and corrected it for distance using a table calibrated in Southern California. It worked beautifully for the moderate local events it was built for, and it SATURATED near 7 — beyond that, every great earthquake produced roughly the same instrument reading, because the instrument simply could not follow the long periods where a great earthquake keeps its energy. Surface-wave magnitude Ms pushed the ceiling up to about 8.3 and then saturated too.

Hiroo Kanamori and Thomas Hanks fixed this in 1979 by refusing to measure amplitude at all. They defined a magnitude directly from seismic moment: $M_w = \frac{2}{3}(\log_{10} M_0 - 9.05)$ with $M_0$ in newton metres. Because moment has no ceiling, neither does Mw. The two constants were chosen so that the new scale would agree with ML and Ms in the middle of their working ranges, which is the whole reason a 6.0 still means what it always meant.

The $\frac{2}{3}$ is not arbitrary. It comes from the older energy–magnitude relation, $\log_{10} E = 4.8 + 1.5 M_s$: inverting that slope of 1.5 gives $2/3$, so Mw is deliberately constructed to keep the familiar "one magnitude unit is about 32 times the energy" property. The scale was built to slot into the arithmetic people already had rather than to replace it.

Three things to carry away. First, the constant belongs to SI — in dyne-centimetres the same relation reads $M_w = \frac{2}{3}(\log_{10} M_0 - 10.7)$, and mixing the forms is the standard error. Second, "the Richter scale" as used in news reports almost never means Richter's ML any more; agencies report Mw and the name simply stuck. Third, and most important: a magnitude is a scale reading, not a quantity. You cannot average magnitudes meaningfully, you cannot add them, and the difference between 6 and 7 is not one of anything — it is a factor of 31.6.