In Earthquake Magnitude you learned that each step on the magnitude scale means much stronger shaking. Seismologists today use moment magnitude, $M_w$, because it measures the earthquake at its source, the fault slip itself, and it keeps working for the largest quakes. It runs on logarithms.
Why it works
The size of an earthquake is how much rock broke and how far it moved. Seismic moment captures that: $M_0 = \mu A D$, where $A$ is the area of fault that slipped, $D$ the average slip, and $\mu \approx 3 \times 10^{10}\,\text{Pa}$ the rock's stiffness, giving newton-meters. Moments range over many powers of ten, so the magnitude takes a log: $M_w = \tfrac{2}{3}(\log_{10} M_0 - 9.1)$. Because of the $\tfrac23$, multiplying $M_0$ by 1,000 adds 2 to the magnitude, and each step of 1 means about $10^{1.5} \approx 32$ times more energy (and seismic waves about 10 times larger).
Key idea
$M_w = \tfrac{2}{3}(\log_{10} M_0 - 9.1)$ with $M_0 = \mu A D$. Each magnitude step means about 32 times more energy; two steps mean about 1,000 times.
Worked examples
Worked example: Magnitude from a fault
A fault ruptures over an area 100 km long and 20 km deep, with 2 m of average slip ($\mu = 3 \times 10^{10}\,\text{Pa}$). What is $M_w$?
- $A = 100 \times 20 = 2,000\,\text{km}^2 = 2 \times 10^{9}\,\text{m}^2$.
- $M_0 = (3 \times 10^{10})(2 \times 10^{9})(2) = 1.2 \times 10^{20}\,\text{N·m}$.
- $\log_{10} M_0 = 20.08$, so $M_w = \tfrac23(20.08 - 9.1) \approx 7.3$.
Answer: About $M_w = 7.3$.
Worked example: Energy ratio
How much more energy does a magnitude 8 release than a magnitude 6?
- $10^{1.5 \times 2} = 10^{3} = 1,000$ times.
Answer: About 1,000 times.
The original Richter scale (1935) used the height of waves on one kind of seismograph. For very large earthquakes those waves stop growing, so the old scale saturates near magnitude 7 to 8. Moment magnitude is built from the fault itself, so it keeps working for the largest quakes.
Common mistake
Many people think magnitude 10 is the top of a 1-to-10 scale. The scale has no top: magnitude grows with fault area times slip. But size is limited by real faults. The largest recorded earthquake, Chile 1960, was about $M_w$ 9.5, on a rupture roughly 1,000 km long; a magnitude 10 would need a rupture far longer than any known fault. Check with $M_0 = \mu A D$: what area would it take?