Moment magnitude scale

The moment magnitude scale (MMS; denoted explicitly with or Mw, and generally implied with use of a single M for magnitude) is a measure of an earthquake's magnitude ("size" or strength) based on its seismic moment. It was defined in a 1979 paper by Thomas C. Hanks and Hiroo Kanamori. Similar to the local magnitude scale () defined by Charles Francis Richter in 1935, it uses a logarithmic scale; small earthquakes have approximately the same magnitudes on both scales.
Moment magnitude () is considered the authoritative magnitude scale for ranking earthquakes by size. It is more directly related to the energy of an earthquake than other scales, and does not saturate—that is, it does not underestimate magnitudes as other scales do in certain conditions. It has become the standard scale used by seismological authorities like the U.S. Geological Survey for reporting large earthquakes (typically M > 4), replacing the local magnitude () and surface wave magnitude () scales. Subtypes of the moment magnitude scale (, etc.) reflect different ways of estimating the seismic moment.
History.
Richter scale: the original measure of earthquake magnitude.
At the beginning of the twentieth century, very little was known about how earthquakes happen, how seismic waves are generated and propagate through the earth's crust, and what information they carry about the earthquake rupture process; the first magnitude scales were therefore empirical. The initial step in determining earthquake magnitudes empirically came in 1931 when the Japanese seismologist Kiyoo Wadati showed that the maximum amplitude of an earthquake's seismic waves diminished with distance at a certain rate. Charles F. Richter then worked out how to adjust for epicentral distance (and some other factors) so that the logarithm of the amplitude of the seismograph trace could be used as a measure of "magnitude" that was internally consistent and corresponded roughly with estimates of an earthquake's energy. He established a reference point and the now familiar ten-fold (exponential) scaling of each degree of magnitude, and in 1935 published what he called the "magnitude scale", now called the local magnitude scale, labeled . (This scale is also known as the "Richter scale", but news media sometimes use that term indiscriminately to refer to other similar scales.)
The local magnitude scale was developed on the basis of shallow (~ deep), moderate-sized earthquakes at a distance of approximately , conditions where the surface waves are predominant. At greater depths, distances, or magnitudes the surface waves are greatly reduced, and the local magnitude scale underestimates the magnitude, a problem called "saturation". Additional scales were developed – a surface-wave magnitude scale () by Beno Gutenberg in 1945, a body-wave magnitude scale () by Gutenberg and Richter in 1956, and a number of variants – to overcome the deficiencies of the scale, but all are subject to saturation. A particular problem was that the scale (which in the 1970s was the preferred magnitude scale) saturates around and therefore underestimates the energy release of "great" earthquakes such as the 1960 Chilean and 1964 Alaskan earthquakes. These had magnitudes of 8.5 and 8.4 respectively but were notably more powerful than other M 8 earthquakes; their moment magnitudes were closer to 9.6 and 9.3.
Single couple or double couple.
The study of earthquakes is challenging as the source events cannot be observed directly, and it took many years to develop the mathematics for understanding what the seismic waves from an earthquake can tell us about the source event. An early step was to determine how different systems of forces might generate seismic waves equivalent to those observed from earthquakes.
The simplest force system is a single force acting on an object. If it has sufficient strength to overcome any resistance it will cause the object to move ("translate"). A pair of forces, acting on the same "line of action" but in opposite directions, will cancel; if they cancel (balance) exactly there will be no net translation, though the object will experience stress, either tension or compression. If the pair of forces are offset, acting along parallel but separate lines of action, the object experiences a rotational force, or torque. In mechanics (the branch of physics concerned with the interactions of forces) this model is called a "couple", also "simple couple" or "single couple". If a second couple of equal and opposite magnitude is applied their torques cancel; this is called a "double couple". A double couple can be viewed as "equivalent to a pressure and tension acting simultaneously at right angles".
The single couple and double couple models are important in seismology because each can be used to derive how the seismic waves generated by an earthquake event should appear in the "far field" (that is, at distance). Once that relation is understood it can be inverted to use the earthquake's observed seismic waves to determine its other characteristics, including fault geometry and seismic moment.
In 1923 Hiroshi Nakano showed that certain aspects of seismic waves could be explained in terms of a double couple model. This led to a three-decade long controversy over the best way to model the seismic source: as a single couple, or a double couple? While Japanese seismologists favored the double couple, most seismologists favored the single couple. Although the single couple model had some short-comings, it seemed more intuitive, and there was a belief – mistaken, as it turned out – that the elastic rebound theory for explaining why earthquakes happen required a single couple model. In principle these models could be distinguished by differences in the radiation patterns of their S-waves, but the quality of the observational data was inadequate for that.
The debate ended when , Haskell (1964), and showed that if earthquake ruptures are modeled as dislocations the pattern of seismic radiation can always be matched with an equivalent pattern derived from a double couple, but not from a single couple. This was confirmed as better and more plentiful data coming from the World-Wide Standard Seismograph Network (WWSSN) permitted closer analysis of seismic waves. Notably, in 1966 Keiiti Aki showed that the seismic moment of the 1964 Niigata earthquake as calculated from the seismic waves on the basis of a double couple was in reasonable agreement with the seismic moment calculated from the observed physical dislocation.
Dislocation theory.
A double couple model suffices to explain an earthquake's far-field pattern of seismic radiation, but tells us very little about the nature of an earthquake's source mechanism or its physical features. While slippage along a fault was theorized as the cause of earthquakes (other theories included movement of magma, or sudden changes of volume due to phase changes), observing this at depth was not possible, and understanding what could be learned about the source mechanism from the seismic waves requires an understanding of the source mechanism.
Modeling the physical process by which an earthquake generates seismic waves required much theoretical development of dislocation theory, first formulated by the Italian Vito Volterra in 1907, with further developments by E. H. Love in 1927. More generally applied to problems of stress in materials, an extension by F. Nabarro in 1951 was recognized by the Russian geophysicist A. V. Vvedenskaya as applicable to earthquake faulting. In a series of papers starting in 1956 she and other colleagues used dislocation theory to determine part of an earthquake's focal mechanism, and to show that a dislocation – a rupture accompanied by slipping — was indeed equivalent to a double couple.
In a pair of papers in 1958, J. A. Steketee worked out how to relate dislocation theory to geophysical features. Numerous other researchers worked out other details, culminating in a general solution in 1964 by Burridge and Knopoff, which established the relationship between double couples and the theory of elastic rebound, and provided the basis for relating an earthquake's physical features to seismic moment.
Seismic moment.
"Seismic moment" – symbol – is a measure of the fault slip and area involved in the earthquake. Its value is the torque of each of the two force couples that form the earthquake's equivalent double-couple. (More precisely, it is the scalar magnitude of the second-order moment tensor that describes the force components of the double-couple.) Seismic moment is measured in units of Newton meters (N·m) or Joules, or (in the older CGS system) dyne-centimeters (dyn-cm).
The first calculation of an earthquake's seismic moment from its seismic waves was by Keiiti Aki for the 1964 Niigata earthquake. He did this two ways. First, he used data from distant stations of the WWSSN to analyze long-period (200 second) seismic waves (wavelength of about 1,000 kilometers) to determine the magnitude of the earthquake's equivalent double couple. Second, he drew upon the work of Burridge and Knopoff on dislocation to determine the amount of slip, the energy released, and the stress drop (essentially how much of the potential energy was released). In particular, he derived a now famous equation that relates an earthquake's seismic moment to its physical parameters:
with being the rigidity (or resistance) of moving a fault with a surface areas of over an average dislocation (distance) of . (Modern formulations replace with the equivalent , known as the "geometric moment" or "potency".) By this equation the "moment" determined from the double couple of the seismic waves can be related to the moment calculated from knowledge of the surface area of fault slippage and the amount of slip. In the case of the Niigata earthquake the dislocation estimated from the seismic moment reasonably approximated the observed dislocation.
Seismic moment is a measure of the work (more precisely, the torque) that results in inelastic (permanent) displacement or distortion of the earth's crust. It is related to the total energy released by an earthquake. However, the power or potential destructiveness of an earthquake depends (among other factors) on how much of the total energy is converted into seismic waves. This is typically 10% or less of the total energy, the rest being expended in fracturing rock or overcoming friction (generating heat).
Nonetheless, seismic moment is regarded as the fundamental measure of earthquake size, representing more directly than other parameters the physical size of an earthquake. As early as 1975 it was considered "one of the most reliably determined instrumental earthquake source parameters".
Introduction of an energy-motivated magnitude "M"w.
Most earthquake magnitude scales suffered from the fact that they only provided a comparison of the amplitude of waves produced at a standard distance and frequency band; it was difficult to relate these magnitudes to a physical property of the earthquake. Gutenberg and Richter suggested that radiated energy Es could be estimated as
(in Joules). Unfortunately, the duration of many very large earthquakes was longer than 20 seconds, the period of the surface waves used in the measurement of . This meant that giant earthquakes such as the 1960 Chilean earthquake (M 9.5) were only assigned an . Caltech seismologist Hiroo Kanamori recognized this deficiency and took the simple but important step of defining a magnitude based on estimates of radiated energy, , where the "w" stood for work (energy):
Kanamori recognized that measurement of radiated energy is technically difficult since it involves the integration of wave energy over the entire frequency band. To simplify this calculation, he noted that the lowest frequency parts of the spectrum can often be used to estimate the rest of the spectrum. The lowest frequency asymptote of a seismic spectrum is characterized by the seismic moment, . Using an approximate relation between radiated energy and seismic moment (which assumes stress drop is complete and ignores fracture energy),
(where E is in Joules and is in Nformula_4m), Kanamori approximated by
Moment magnitude scale.
The formula above made it much easier to estimate the energy-based magnitude , but it changed the fundamental nature of the scale into a moment magnitude scale. USGS seismologist Thomas C. Hanks noted that Kanamori's scale was very similar to a relationship between and that was reported by 
 combined their work to define a new magnitude scale based on estimates of seismic moment
where formula_8 is defined in newton meters (N·m).
Current use.
Moment magnitude is now the most common measure of earthquake size for medium to large earthquake magnitudes, but in practice, seismic moment (), the seismological parameter it is based on, is not measured routinely 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 includes the great majority of quakes.
Popular press reports most often deal with significant earthquakes larger than . For these events, the preferred magnitude is the moment magnitude , not Richter's local magnitude .
Definition.
The symbol for the moment magnitude scale is , with the subscript "w" meaning mechanical work accomplished. The moment magnitude is a dimensionless value defined by Hiroo Kanamori as
where is the seismic moment in dyne⋅cm (10−7 N⋅m). The constant values in the equation are chosen to achieve consistency with the magnitude values produced by earlier scales, such as the local magnitude and the surface wave magnitude. Thus, a magnitude zero microearthquake has a seismic moment of approximately , while the Great Chilean earthquake of 1960, with an estimated moment magnitude of 9.4–9.6, had a seismic moment between and . 
Relations between seismic moment, potential energy released and radiated energy.
Seismic moment is not a direct measure of energy changes during an earthquake. The relations between seismic moment and the energies involved in an earthquake depend on parameters that have large uncertainties and that may vary between earthquakes. Potential energy is stored in the crust in the form of elastic energy due to built-up stress and gravitational energy. During an earthquake, a portion formula_10 of this stored energy is transformed into
The potential energy drop caused by an earthquake is related approximately to its seismic moment by
where formula_15 is the average of the "absolute" shear stresses on the fault before and after the earthquake (e.g., equation 3 of ) and formula_16 is the average of the shear moduli of the rocks that constitute the fault. Currently, there is no technology to measure absolute stresses at all depths of interest, nor method to estimate it accurately, and formula_15 is thus poorly known. It could vary highly from one earthquake to another. Two earthquakes with identical formula_18 but different formula_15 would have released different formula_10.
The radiated energy caused by an earthquake is approximately related to seismic moment by
where formula_22 is radiated efficiency and formula_23 is the static stress drop, i.e., the difference between shear stresses on the fault before and after the earthquake (e.g., from equation 1 of ). These two quantities are far from being constants. For instance, formula_24 depends on rupture speed; it is close to 1 for regular earthquakes but much smaller for slower earthquakes such as tsunami earthquakes and slow earthquakes. Two earthquakes with identical formula_18 but different formula_24 or formula_23 would have radiated different formula_28.
Because formula_28 and formula_18 are fundamentally independent properties of an earthquake source, and since formula_28 can now be computed more directly and robustly than in the 1970s, introducing a separate magnitude associated to radiated energy was warranted. Choy and Boatwright defined in 1995 the "energy magnitude"
where formula_28 is in J (N·m).
Comparative energy released by two earthquakes.
Assuming the values of are the same for all earthquakes, one can consider as a measure of the potential energy change Δ"W" caused by earthquakes. Similarly, if one assumes formula_34 is the same for all earthquakes, one can consider as a measure of the energy "E"s radiated by earthquakes.
Under these assumptions, the following formula, obtained by solving for the equation defining , allows one to assess the ratio formula_35 of energy release (potential or radiated) between two earthquakes of different moment magnitudes, formula_36 and formula_37:
As with the Richter scale, an increase of one step on the logarithmic scale of moment magnitude 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. Thus, an earthquake of of 7.0 contains 1000 times as much energy as one of 5.0 and about 32 times that of 6.0.
Subtypes of Mw.
Various ways of determining moment magnitude have been developed, and several subtypes of the scale can be used to indicate the basis used.