Moment magnitude scale
Encyclopedia
The moment magnitude scale (abbreviated as MMS; denoted as MW) is used by seismologists to measure the size of earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s in terms of the energy released. The magnitude is based on the seismic moment
Seismic moment
Seismic moment is a quantity used by earthquake seismologists to measure the size of an earthquake. The scalar seismic moment M_0 is defined by the equationM_0=\mu AD, where*\mu is the shear modulus of the rocks involved in the earthquake...

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
Richter magnitude scale
The expression Richter magnitude scale refers to a number of ways to assign a single number to quantify the energy contained in an earthquake....

(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
United States Geological Survey
The United States Geological Survey is a scientific agency of the United States government. The scientists of the USGS study the landscape of the United States, its natural resources, and the natural hazards that threaten it. The organization has four major science disciplines, concerning biology,...

.

## Definition

The symbol for the moment magnitude scale is , with the subscript meaning mechanical work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

accomplished. The moment magnitude is a dimensionless number defined by

where is the magnitude of the seismic moment in dyne
Dyne
In physics, the dyne is a unit of force specified in the centimetre-gram-second system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN...

centimeters (10−7 N·m). 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
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

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
Equation solving
In mathematics, to solve an equation is to find what values fulfill a condition stated in the form of an equation . These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled...

the previous equation for , allows one to assess the proportional difference in energy release between earthquakes of two different moment magnitudes, say and :

Potential energy is stored in the crust in the form of built-up stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

. During an earthquake, this stored energy is transformed and results in
• cracks and deformation in rocks
• heat,

The seismic moment is a measure of the total amount of energy that is transformed during an earthquake. Only a small fraction of the seismic moment is converted into radiated seismic energy , which is what seismographs register. Using the estimate

Choy and Boatwright defined in 1995 the energy magnitude

## Nuclear explosions

The energy released by nuclear weapon
Nuclear weapon
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission or a combination of fission and fusion. Both reactions release vast quantities of energy from relatively small amounts of matter. The first fission bomb test released the same amount...

s is traditionally expressed in terms of the energy stored in a kiloton or megaton
TNT equivalent
TNT equivalent is a method of quantifying the energy released in explosions. The ton of TNT is a unit of energy equal to 4.184 gigajoules, which is approximately the amount of energy released in the detonation of one ton of TNT...

of the conventional explosive trinitrotoluene (TNT).

A rule of thumb
Rule of thumb
A rule of thumb is a principle with broad application that is not intended to be strictly accurate or reliable for every situation. It is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination...

equivalence from seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

used in the study of nuclear proliferation
Nuclear proliferation
Nuclear proliferation is a term now used to describe the spread of nuclear weapons, fissile material, and weapons-applicable nuclear technology and information, to nations which are not recognized as "Nuclear Weapon States" by the Treaty on the Nonproliferation of Nuclear Weapons, also known as the...

asserts that a one kiloton nuclear explosion
Nuclear explosion
A nuclear explosion occurs as a result of the rapid release of energy from an intentionally high-speed nuclear reaction. The driving reaction may be nuclear fission, nuclear fusion or a multistage cascading combination of the two, though to date all fusion based weapons have used a fission device...

creates a seismic signal with a magnitude of approximately 4.0. This in turn leads to the equation

where 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
Convention (norm)
A convention is a set of agreed, stipulated or generally accepted standards, norms, social norms or criteria, often taking the form of a custom....

specific energy
Specific energy
Specific energy is defined as the energy per unit mass. Common metric units are J/kg. It is an intensive property. Contrast this with energy, which is an extensive property. There are two main types of specific energy: potential energy and specific kinetic energy. Others are the gray and sievert,...

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 . 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
Beno Gutenberg
Beno Gutenberg was a German-American seismologist who made several important contributions to the science...

developed the local magnitude () 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
California
California is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...

. This scale was based on the ground motion measured by a particular type of seismometer
Seismometer
Seismometers are instruments that measure motions of the ground, including those of seismic waves generated by earthquakes, volcanic eruptions, and other seismic sources...

at a distance of 100 kilometres (62.1 mi) from the earthquake's epicenter
Epicenter
The epicenter or epicentre is the point on the Earth's surface that is directly above the hypocenter or focus, the point where an earthquake or underground explosion originates...

. 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 (372.8 mi) from the epicenter.

The moment magnitude () scale was introduced in 1979 by Caltech seismologists Thomas C. Hanks
Tom Hanks (seismologist)
Thomas C. Hanks is an Canadian seismologist. He works for the US Geological Survey in Menlo Park, California. Dr. Hanks is a member of the Seismological Society of America, the American Geophysical Union, the Earthquake Engineering Research Institute, the Geological Society of America, the...

and Hiroo Kanamori
Hiroo Kanamori
is a Japanese American seismologist who has made fundamental contributions to understanding the physics of earthquakes and the tectonic processes that cause them....

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
Seismic moment
Seismic moment is a quantity used by earthquake seismologists to measure the size of an earthquake. The scalar seismic moment M_0 is defined by the equationM_0=\mu AD, where*\mu is the shear modulus of the rocks involved in the earthquake...

(). 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.

The concept of seismic moment was introduced in 1966, but it took 13 years before the () 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 started the Harvard Global Centroid Moment Tensor Catalog. After this advance, it was possible to introduce () and estimate it for large numbers of earthquakes.

Moment magnitude is now the most common measure for medium to large earthquake magnitudes, but breaks down for smaller quakes. For example, the United States Geological Survey
United States Geological Survey
The United States Geological Survey is a scientific agency of the United States government. The scientists of the USGS study the landscape of the United States, its natural resources, and the natural hazards that threaten it. The organization has four major science disciplines, concerning biology,...

does not use this scale for earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s 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 scale of Richter and Gutenberg.

Magnitude scales differ from earthquake intensity
Mercalli intensity scale
The Mercalli intensity scale is a seismic scale used for measuring the intensity of an earthquake. It measures the effects of an earthquake, and is distinct from the moment magnitude M_w usually reported for an earthquake , which is a measure of the energy released...

, 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.

• Earthquake engineering
Earthquake engineering
Earthquake engineering is the scientific field concerned with protecting society, the natural and the man-made environment from earthquakes by limiting the seismic risk to socio-economically acceptable levels...

• Geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...

• List of earthquakes
• Other seismic scales
Seismic scale
A seismic scale is used to calculate and compare the severity of earthquakes....

• Surface wave magnitude
Surface wave magnitude
The surface wave magnitude scale is one of the magnitude scales used in seismology to describe the size of an earthquake. It is based on measurements in Rayleigh surface waves that travel primarily along the uppermost layers of the earth...