Hong Life CalculatorsUnderstanding the moment magnitude (Mw) calculator — concept and principle
Moment magnitude (Mw) measures the size of an earthquake from the physics of the fault itself: how stiff the rock is, how large an area ruptured and how far the two sides slipped. It is the scale seismologists use for moderate and large earthquakes.
How it is calculated
M₀ = μ × A × D
Mw = (2/3) × (log₁₀M₀ − 9.1)
- μ is the shear modulus (rigidity) of the rock in GPa; the calculator multiplies by 10⁹ to get Pa. About 30 GPa is a common value for crustal rock.
- A is the rupture area in km², multiplied by 10⁶ to get m².
- D is the average slip on the fault in metres.
- M₀ is the seismic moment in newton-metres (N·m).
The principle
A fault slip is equivalent to a pair of opposing force couples acting on the fault. Their strength, the seismic moment, equals rigidity times area times slip, a quantity that can also be measured from long-period seismic waves, so it is grounded in mechanics rather than in the response of one instrument. Hanks and Kanamori defined Mw in 1979 so that it matches the older magnitude scales over the range where they work, giving the factor 2/3 and the constant 9.1 for M₀ in N·m (equivalent to 16.1 when M₀ is in dyne-cm). Because the moment keeps growing with fault size, Mw does not saturate the way the local and surface-wave scales do above about magnitude 8.
Worked example
- Inputs: μ = 30 GPa, A = 1,000 km², D = 2 m.
- Convert units: μ = 3 × 10¹⁰ Pa, A = 1 × 10⁹ m².
- M₀ = 3 × 10¹⁰ × 1 × 10⁹ × 2 = 6 × 10¹⁹ N·m.
- log₁₀(6 × 10¹⁹) = 19.77815.
- Mw = (2/3) × (19.77815 − 9.1) = (2/3) × 10.67815 = 7.1187675, about Mw 7.1.
| Shear modulus (GPa) | Seismic moment (N·m) |
|---|---|
| 7.5 | 1.5e+19 |
| 15 | 3e+19 |
| 22.5 | 4.5e+19 |
| 30 | 6e+19 |
| 37.5 | 7.5e+19 |
| 45 | 9e+19 |
| 60 | 1.2e+20 |
How to read the result
Each unit of Mw corresponds to about 31.6 times more seismic moment, and 0.2 units is roughly a doubling. A fault about 50 km long and 20 km wide slipping 2 m, as in the example, gives a large earthquake near Mw 7.
Common mistakes
- Forgetting the unit conversions: entering μ in Pa or area in m² inflates M₀ enormously.
- Using maximum slip instead of average slip, which overstates the magnitude.
- Mixing up the constants: 9.1 is for N·m, 16.1 for dyne-cm.
Frequently asked questions
Why is Mw preferred over the Richter scale?
The Richter (local) scale saturates for big earthquakes because short-period waves stop growing, while seismic moment keeps increasing with fault size.
How much slip did the largest earthquakes have?
Great subduction earthquakes have produced average slip of several metres to over ten metres across areas of tens of thousands of square kilometres.
Does a larger shear modulus mean a stronger earthquake?
For the same area and slip, yes, because moment is proportional to μ. In reality stiffer rock also tends to slip less.
Sources: T. C. Hanks and H. Kanamori (1979), "A moment magnitude scale", Journal of Geophysical Research 84(B5), 2348–2350 · H. Kanamori (1977), "The energy release in great earthquakes", Journal of Geophysical Research 82(20), 2981–2987