The moment magnitude scale (abbreviated as MMS; denoted as MW) is used by seismologists to measure the size of earthquakes in terms of the energy released.[1] The magnitude is based on the seismic moment of the earthquake, which is equal to the rigidity of the Earth multiplied by the average amount of slip on the fault and the size of the area that slipped.[2] The scale was developed in the 1970s to succeed the 1930s-era Richter magnitude scale (ML). Even though the formulae are different, the new scale retains the familiar continuum of magnitude values defined by the older one. The MMS is now the scale used to estimate magnitudes for all modern large earthquakes by the United States Geological Survey.[3]

## Definition

The symbol for the moment magnitude scale is $$M_\mathrm{w}$$, with the subscript $$\mathrm{w}$$ meaning mechanical work accomplished. The moment magnitude $$M_\mathrm{w}$$ is a dimensionless number defined by

$M_\mathrm{w} = {\frac{2}{3}}\log_{10}M_0 - 10.7,$

where $$M_0$$ is the magnitude of the seismic moment in dyne centimeters (10−7 N·m).[1] The constant values in the equation are chosen to achieve consistency with the magnitude values produced by earlier scales, the Local Magnitude and the Surface Wave magnitude, both referred to as the "Richter" scale by reporters.

## Comparative energy released by two earthquakes

As with the Richter scale, an increase of one step on this logarithmic scale corresponds to a 101.5 ≈ 32 times increase in the amount of energy released, and an increase of two steps corresponds to a 103 = 1000 times increase in energy.

The following formula, obtained by solving the previous equation for $$M_0$$, allows one to assess the proportional difference $$f_{\Delta E}$$ in energy release between earthquakes of two different moment magnitudes, say $$m_1$$ and $$m_2$$: $f_{\Delta E} = \frac{10^{(\frac{3}{2}(m_1 + 10.7))}}{10^{(\frac{3}{2}(m_2 + 10.7))}} = 10^{\frac{3}{2}(m_1 - m_2)}.$

Potential energy is stored in the crust in the form of built-up stress. During an earthquake, this stored energy is transformed and results in

• cracks and deformation in rocks
• heat,
• radiated seismic energy $$E_s$$.

The seismic moment $$M_0$$ is a measure of the total amount of energy that is transformed during an earthquake. Only a small fraction of the seismic moment $$M_0$$ is converted into radiated seismic energy $$E_\mathrm{s}$$, which is what seismographs register. Using the estimate

$E_\mathrm{s} = M_0\cdot10^{-4.8}=M_0\cdot1.6\times10^{-5},$

Choy and Boatwright defined in 1995 the energy magnitude [4]

$M_\mathrm{e} = \textstyle{\frac{2}{3}}\log_{10}E_\mathrm{s}-2.9.$

## Nuclear explosions

The energy released by nuclear weapons is traditionally expressed in terms of the energy stored in a kiloton or megaton of the conventional explosive trinitrotoluene (TNT).

A rule of thumb equivalence from seismology used in the study of nuclear proliferation asserts that a one kiloton nuclear explosion creates a seismic signal with a magnitude of approximately 4.0.[5] This in turn leads to the equation [6]

$M_n = \textstyle\frac{2}{3}\displaystyle\log_{10} \frac{m_{\mathrm{TNT}}}{\mbox{Mt}} + 6,$

where $$m_{\mathrm{TNT}}$$ is the mass of the explosive TNT that is quoted for comparison (relative to megatons Mt).

Such comparison figures are not very meaningful. As with earthquakes, during an underground explosion of a nuclear weapon, only a small fraction of the total amount of energy transformed ends up being radiated as seismic waves. Therefore, a seismic efficiency has to be chosen for a bomb that is quoted as a comparison. Using the conventional specific energy of TNT (4.184 MJ/kg), the above formula implies the assumption that about 0.5% of the bomb's energy is converted into radiated seismic energy $$E_s$$.[7] For real underground nuclear tests, the actual seismic efficiency achieved varies significantly and depends on the site and design parameters of the test.

## Comparison with Richter scale

In 1935, Charles Richter and Beno Gutenberg developed the local magnitude ($$M_\mathrm{L}$$) scale (popularly known as the Richter scale) with the goal of quantifying medium-sized earthquakes (between magnitude 3.0 and 7.0) in Southern California. This scale was based on the ground motion measured by a particular type of seismometer at a distance of 100 kilometres (62 mi) from the earthquake's epicenter.[3] Because of this, there is an upper limit on the highest measurable magnitude, and all large earthquakes will tend to have a local magnitude of around 7. The magnitude becomes unreliable for measurements taken at a distance of more than about 600 kilometres (370 mi) from the epicenter.

The moment magnitude ($$M_\mathrm{w}$$) scale was introduced in 1979 by Caltech seismologists Thomas C. Hanks and Hiroo Kanamori to address these shortcomings while maintaining consistency. Thus, for medium-sized earthquakes, the moment magnitude values should be similar to Richter values. That is, a magnitude 5.0 earthquake will be about a 5.0 on both scales. This scale was based on the physical properties of the earthquake, specifically the seismic moment ($$M_0$$). Unlike other scales, the moment magnitude scale does not saturate at the upper end; there is no upper limit to the possible measurable magnitudes. However, this has the side-effect that the scales diverge for smaller earthquakes.[1]

The concept of seismic moment was introduced in 1966,[8] but it took 13 years before the ($$M_w$$) scale was designed. The reason for the delay was that the necessary spectra of seismic signals had to be derived by hand at first, which required personal attention to every event. Faster computers than those available in the 1960s were necessary and seismologists had to develop methods to process earthquake signals automatically. In the mid 1970s Dziewonski[9] started the Harvard Global Centroid Moment Tensor Catalog.[10] After this advance, it was possible to introduce ($$M_w$$) and estimate it for large numbers of earthquakes.

Moment magnitude is now the most common measure for medium to large earthquake magnitudes,[11] but breaks down for smaller quakes. For example, the United States Geological Survey does not use this scale for earthquakes with a magnitude of less than 3.5, which is the great majority of quakes. For these smaller quakes, other magnitude scales are used. All magnitudes are calibrated to the $$M_\mathrm{L}$$ scale of Richter and Gutenberg.

Magnitude scales differ from earthquake intensity, which is the perceptible shaking, and local damage experienced during a quake. The shaking intensity at a given spot depends on many factors, such as soil types, soil sublayers, depth, type of displacement, and range from the epicenter (not counting the complications of building engineering and architectural factors). Rather, magnitude scales are used to estimate with one number the size of the quake.

The following table compares magnitudes towards the upper end of the Richter Scale for major Californian earthquakes.[1]

Date Seismic moment $$M_0\times10^{25}$$ (dyne-cm) Richter scale $$M_\mathrm{L}$$ Moment magnitude $$M_\mathrm{w}$$
1933-03-11 2 6.3 6.2
1940-05-19 30 6.4 7.0
1941-07-01 0.9 5.9 6.0
1942-10-21 9 6.5 6.6
1946-03-15 1 6.3 6.0
1947-04-10 7 6.2 6.5
1948-12-04 1 6.5 6.0
1952-07-21 200 7.2 7.5
1954-03-19 4 6.2 6.4

## Notes

1. Script error
2. "Glossary of Terms on Earthquake Maps". USGS. Retrieved 2009-03-21.
3. USGS Earthquake Magnitude Policy
4. Choy, George L.; Boatwright, John L. (1995), "Global patterns of radiated seismic energy and apparent stress", Journal of Geophysical Research 100 (B9): 18205–28, Error: Bad DOI specified
5. "Nuclear Testing and Nonproliferation", "Chapter 5: Assessing Monitoring Requirements"
6. "What is Richter Magnitude?"
7. Q: How much energy is released in an earthquake?
8. Script error
9. Script error
10. "Global Centroid Moment Tensor Catalog". Globalcmt.org. Retrieved 2011-11-30.
11. Boyle, Alan (May 12, 2008). "Quakes by the numbers". MSNBC. Retrieved 2008-05-12. "That original scale has been tweaked through the decades, and nowadays calling it the "Richter scale" is an anachronism. The most common measure is known simply as the moment magnitude scale."

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