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The moment magnitude scale (abbreviated as MMS; denoted as M_\mathrm{w}, where w indicates work accomplished) is used by seismologists to measure the size of earthquakes in terms of the energy released. The magnitude is based on the 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. The scale was developed in the 1970s to succeed the 1930s-era Richter magnitude scale, M_\mathrm{L}. 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.

Like the Richter scale, the MMS is logarithmic; on the scale, an earthquake one number higher is approximately thirty-one times (the square root of 1,000) more powerful (for example, 7.0 is about thirty-one times the power of 6.0).

Comparison to Richter scale

In 1935, Charles Richter developed the local magnitude (M_\mathrm{L}) scale (also known as the Richter scale) with the goal of quantifying medium-sized earthquakes (between magnitude 3.0 and 7.0) in Southern Californiamarker. This scale was based on the responses of seismographs and their distance from the epicentre. Because of this, there is an upper limit on the highest measurable magnitude; all large earthquakes will have a local magnitude of around 7. The local magnitude's estimate of earthquake size is also unreliable for measurements taken at a distance of more than about 350 miles (600 km) from the earthquake's epicenter.

The moment magnitude (M_\mathrm{w}) scale was introduced in 1979 by Caltechmarker 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 of low-energy earthquakes clustering together.

Moment magnitude is now the most common estimate of both medium and large earthquake magnitudes, but is rarely used 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 similar to the Richter scale are used.

Neither scale measures the earthquake intensity, which is the perceptible moving, shaking, and local damages 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, they are used to estimate only the total energy released by the quake.

The following table compares magnitudes towards the upper end of the Richter Scale for major Californian earthquakes.{{Table
type=class="wikitable sortable"
hdrs=Date M_0\times10^{25} M_\mathrm{L} M_\mathrm{w}
row1=1933-03-11marker 2 6.3 6.2
row2=1940-05-19 30 6.4 7.0
row3=1941-07-01 0.9 5.9 6.0
row4=1942-10-21 9 6.5 6.6
row5=1946-03-15 1 6.3 6.0
row6=1947-04-10 7 6.2 6.5
row7=1948-12-04 1 6.5 6.0
row8=1952-07-21marker 200 7.2 7.5
row9=1954-03-19 4 6.2 6.4}}


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

M_\mathrm{w} = {2 \over 3}\log_{10}\left(M_0\right) - 10.7

where M_0 is the seismic moment in dyne centimetres (10-7 Nm). The constant values in the equation are chosen to achieve consistency with the magnitude values produced by earlier scales, most importantly the Local Moment (or "Richter") scale.

As with the Richter scale, an increase of 1 step on this logarithmic scale corresponds to a 101.5 = 31.6 times increase in the amount of energy released, and an increase of 2 steps corresponds to a 103 = 1000 times increase in energy.

Radiated seismic energy

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_\mathrm{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

M_\mathrm{e} = {2\over 3}\log_{10}\left(E_\mathrm{s}\right)-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).

Many academics refer to a 1 kt TNT explosion being roughly equivalent to a magnitude 4 earthquake (an often quoted rule of thumb in seismology), which in turn leads to the equation

M_\mathrm{n} = {2 \over 3}\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.

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_\mathrm{s}. For real underground nuclear tests, the actual seismic efficiency achieved varies significantly and depends on the site and design parameters of the test.

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