# Moment magnitude scale

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The moment magnitude scale (MMS; denoted as Mw or M) is one of many seismic magnitude scales used to measure the size of earthquakes.[1]

The scale was developed in the 1970s to succeed the 1930s-era Richter magnitude scale (ML). Even though the formulas are different, the new scale was designed to produce magnitude values for a given earthquake similar to those produced by the older one. Under suitable assumptions, as with the Richter magnitude scale, an increase of one step on this logarithmic scale corresponds to a 101.5 (about 32) times increase in the amount of energy released, and an increase of two steps corresponds to a 103 (1,000) times increase in energy. Thus, an earthquake of Mw of 7.0 releases about 32 times as much energy as one of 6.0 and nearly 1,000 times one of 5.0.

The moment magnitude is based on the seismic moment of the earthquake, which is equal to the shear modulus of the rock near the fault multiplied by the average amount of slip on the fault and the size of the area that slipped.[2]

While earthquake magnitudes are usually calculated on several scales, the moment magnitude value, being more directly related to the energy of an earthquake, is generally preferred. The generic "M" magnitude the U.S. Geological Survey uses for reporting magnitudes to the public is (for M > 4 events) the moment magnitude, which the press often calls the "Richter magnitude".[3]

## History

### Richter scale: the original measure of earthquake magnitude

In 1935, Charles Richter and Beno Gutenberg developed the local magnitude (ML ) 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 (a Wood-Anderson seismograph) at a distance of 100 kilometres (62 mi) from the earthquake's epicenter. 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.[4] Further, the magnitude becomes unreliable for measurements taken at a distance of more than about 600 kilometres (370 mi) from the epicenter. Since this ML  scale was simple to use and corresponded well with the damage which was observed, it was extremely useful for engineering earthquake-resistant structures, and gained common acceptance.[5]

### Modified Richter scale

The Richter scale was not effective for characterizing some classes of quakes. As a result, Beno Gutenberg expanded Richter's work to consider earthquakes detected at distant locations. For such large distances the higher frequency vibrations are attenuated and seismic surface waves (Rayleigh and Love waves) are dominated by waves with a period of 20 seconds, corresponding to a wavelength of about 60 km. Their magnitude was assigned a surface wave magnitude scale (Ms ). Gutenberg also combined compressional P-waves and the transverse S-waves (which he termed "body waves") to create a body-wave magnitude scale (mb ), measured for periods between 1 and 10 seconds. Ultimately Gutenberg and Richter collaborated to produce a combined scale which was able to estimate the energy released by an earthquake in terms of Gutenberg's surface wave magnitude scale (Ms ).[6]

### Correcting weaknesses of the modified Richter scale

The Richter scale, as modified, was successfully applied to characterize localities. This enabled local building codes to establish standards for buildings which were earthquake resistant. However a series of quakes were poorly handled by the modified Richter scale. This series of "great earthquakes" included faults that broke along a line of up to 1000 km. Examples include the 1957 Andreanof Islands earthquake and the 1960 Chilean quake, both of which broke faults approaching 1000 km. The Ms  scale was unable to characterize these "great earthquakes" accurately.[7]

The difficulties with use of Ms  in characterizing the quake resulted from the size of these earthquakes. Great quakes produced 20 s waves such that Ms  was comparable to normal quakes, but also produced very long period waves (more than 200 s) which carried large amounts of energy. As a result, use of the modified Richter scale methodology to estimate earthquake energy was deficient at high energies.[8]

### Seismic moment

The concept of seismic moment was introduced in 1966 by Keiiti Aki, a professor of geophysics at the Massachusetts Institute of Technology. Using detailed field studies of the 1964 Niigata earthquake and data from a new generation of seimographs in the World-Wide Standardized Seismograph Network (WWSSN), he first confirmed that an earthquake is "a release of accumulated strain energy by a rupture",[9] and that this can be modeled by a "double couple".[10] With further analysis he showed how the energy radiated by seismic waves can be used to estimate the energy released by the earthquake.[11] This was done using seismic moment, defined[12] as

M0 = μūS

with μ being the rigidity (or resistance) of moving a fault with a surface areas of S over an average dislocation (distance) of . (Modern formulations replace μūS with the equivalent D̄A, known as the "geometric moment" or "potency".[13].) 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.[14]

### Introduction of an energy-motivated magnitude Mw

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

${\displaystyle \log E_{s}\approx 4.8+1.5M_{S},}$

(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 Ms . This meant that giant earthquakes such as the 1960 Chilean earthquake (M 9.5) were only assigned an Ms   8.2. Caltech seismologist Hiroo Kanamori[15] recognized this deficiency and he took the simple but important step of defining a magnitude based on estimates of radiated energy, Mw , where the "w" stood for work (energy):

${\displaystyle M_{w}=2/3\log E_{s}-3.2}$

Kanamori recognized that measurement of radiated energy is technically difficult since it involves 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, M0 . Using an approximate relation between radiated energy and seismic moment (which assumes stress drop is complete and ignores fracture energy),

${\displaystyle E_{s}\approx M_{0}/(2\times 10^{4})}$

(where E is in Joules and M0  is in N${\displaystyle \cdot }$m), Kanamori approximated Mw  by

${\displaystyle M_{w}=(\log M_{0}-9.1)/1.5}$

### Moment magnitude scale

The formula above made it much easier to estimate the energy-based magnitude Mw , but it changed the fundamental nature of the scale into a moment magnitude scale. Caltech seismologist Thomas C. Hanks noted that Kanamori's Mw  scale was very similar to a relationship between ML  and M0  that was reported by Thatcher & Hanks (1973)

${\displaystyle M_{L}\approx (\log M_{0}-9.0)/1.5}$

Hanks & Kanamori (1979) combined their work to define a new magnitude scale based on estimates of seismic moment

${\displaystyle M=(\log M_{0}-9.05)/1.5}$

where ${\displaystyle M_{0}}$ is defined in newton meters (N·m).

Although the formal definition of moment magnitude is given by this paper and is designated by M, it has been common for many authors to refer to Mw  as moment magnitude. In most of these cases, they are actually referring to moment magnitude M as defined above.

### Current use

Moment magnitude is now the most common measure of earthquake size for medium to large earthquake magnitudes,[16] 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.

Current practice in official earthquake reports is to adopt moment magnitude as the preferred magnitude, i.e., Mw  is the official magnitude reported whenever it can be computed. Because seismic moment (M0 , the quantity needed to compute Mw ) is not measured if the earthquake is too small, the reported magnitude for earthquakes smaller than M  4 is often Richter's ML .

Popular press reports most often deal with significant earthquakes larger than M  ~ 4. For these events, the official magnitude is the moment magnitude Mw , not Richter's local magnitude ML .

## Definition

The symbol for the moment magnitude scale is Mw , with the subscript "w" meaning mechanical work accomplished. The moment magnitude Mw  is a dimensionless value defined by Hiroo Kanamori[17] as

${\displaystyle M_{\mathrm {w} }={\frac {2}{3}}\log _{10}(M_{0})-10.7,}$

where M0  is the seismic moment in dyne⋅cm (10−7 N⋅m).[18] 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.

## 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.[19] During an earthquake, a portion ${\displaystyle \Delta W}$ of this stored energy is transformed into

• energy dissipated ${\displaystyle E_{f}}$in frictional weakening and inelastic deformation in rocks by processes such as the creation of cracks
• heat ${\displaystyle E_{h}}$
• radiated seismic energy ${\displaystyle E_{s}}$.

The potential energy drop caused by an earthquake is related approximately to its seismic moment by

${\displaystyle \Delta W\approx {\frac {\overline {\sigma }}{\mu }}M_{0}}$

where ${\displaystyle {\overline {\sigma }}}$ is the average of the absolute shear stresses on the fault before and after the earthquake (e.g., equation 3 of Venkataraman & Kanamori 2004) and ${\displaystyle \mu }$ 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 ${\displaystyle {\overline {\sigma }}}$ is thus poorly known. It could vary highly from one earthquake to another. Two earthquakes with identical ${\displaystyle M_{0}}$ but different ${\displaystyle {\overline {\sigma }}}$ would have released different ${\displaystyle \Delta W}$.

The radiated energy caused by an earthquake is approximately related to seismic moment by

${\displaystyle E_{\mathrm {s} }\approx \eta _{R}{\frac {\Delta \sigma _{s}}{2\mu }}M_{0}}$

where ${\displaystyle \eta _{R}=E_{s}/(E_{s}+E_{f})}$ is radiated efficiency and ${\displaystyle \Delta \sigma _{s}}$ 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 Venkataraman & Kanamori 2004). These two quantities are far from being constants. For instance, ${\displaystyle \eta _{R}}$ 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 ${\displaystyle M_{0}}$ but different ${\displaystyle \eta _{R}}$ or ${\displaystyle \Delta \sigma _{s}}$ would have radiated different ${\displaystyle E_{\mathrm {s} }}$.

Because ${\displaystyle E_{\mathrm {s} }}$ and ${\displaystyle M_{0}}$ are fundamentally independent properties of an earthquake source, and since ${\displaystyle E_{\mathrm {s} }}$ 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[20]

${\displaystyle M_{\mathrm {E} }=\textstyle {\frac {2}{3}}\log _{10}E_{\mathrm {s} }-3.2}$

where ${\displaystyle E_{\mathrm {s} }}$ is in J (N·m).

## Comparative energy released by two earthquakes

Assuming the values of σ̄/μ are the same for all earthquakes, one can consider Mw  as a measure of the potential energy change ΔW caused by earthquakes. Similarly, if one assumes ${\displaystyle \eta _{R}\Delta \sigma _{s}/2\mu }$ is the same for all earthquakes, one can consider Mw  as a measure of the energy Es radiated by earthquakes.

Under these assumptions, the following formula, obtained by solving for M0  the equation defining Mw , allows one to assess the ratio ${\displaystyle E_{1}/E_{2}}$ of energy release (potential or radiated) between two earthquakes of different moment magnitudes, ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$:

${\displaystyle E_{1}/E_{2}\approx 10^{{\frac {3}{2}}(m_{1}-m_{2})}.}$

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 Mw  of 7.0 contains 1000 times as much energy as one of 5.0 and about 32 times that of 6.0.

## Comparison with Richter scale

The moment magnitude (Mw >) scale was introduced to address the shortcomings of the Richter scale (detailed above) 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. 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.[21]

## Subtypes of Mw

Various ways of determining moment magnitude have been developed, and several subtypes of the Mw  scale can be used to indicate the basis used.[22]

• Mwb – Based on moment tensor inversion of long-period (~10 – 100 s) body-waves.
• Mwr – From a moment tensor inversion of complete waveforms at regional distances (~ 1,000 miles). Sometimes called RMT.
• Mwc – Derived from a centroid moment tensor inversion of intermediate- and long-period body- and surface-waves.
• Mww – Derived from a centroid moment tensor inversion of the W-phase.
• Mwp (Mi) – Developed by Seiji Tsuboi[23] for quick estimation of the tsunami potential of large near-coastal earthquakes from measurements of the P-waves, and later extended to teleseismic earthquakes in general.[24]
• Mwpd – A duration-amplitude procedure which takes into account the duration of the rupture, providing a fuller picture of the energy released by longer lasting ("slow") ruptures than seen with Mw .[25]

## Notes

1. ^
2. ^ "Glossary of Terms on Earthquake Maps". USGS. Archived from the original on 2009-02-27. Retrieved 2009-03-21.
3. ^ The "USGS Earthquake Magnitude Policy" for reporting earthquake magnitudes to the public as formulated by the USGS Earthquake Magnitude Working Group was implemented January 18, 2002, and posted at https://earthquake.usgs.gov/aboutus/docs/020204mag_policy.php. That page was removed following a web redesign; a copy is archived at the Wayback Machine.
4. ^
5. ^
6. ^
7. ^
8. ^
9. ^ Aki 1966a, p. 25.
10. ^ Aki 1966a, p. 36.
11. ^ Aki 1966b, p. 87.
12. ^ Aki 1966b, p. 84, equation 12.
13. ^ Bormann, Wendt & Di Giacomo 2013, p. 12, equation 3.1.
14. ^ Aki 1966b, p. 84.
15. ^
16. ^
17. ^
18. ^
19. ^
20. ^ Choy & Boatwright 1995
21. ^
22. ^
23. ^
24. ^ Bormann, Wendt & Di Giacomo 2013, §3.2.8.2, p. 135.
25. ^ Bormann, Wendt & Di Giacomo 2013, §3.2.8.3, pp. 137–128.

## Sources

• 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..
• Kostrov, B. V. (1974), "Seismic moment and energy of earthquakes, and seismic flow of rock [in Russian]", Izvestiya, Akademi Nauk, USSR, Physics of the solid earth [Earth Physics], 1: 23–44 (English Trans. 12–21).
• Tsuboi, S.; Abe, K.; Takano, K.; Yamanaka, Y. (April 1995), "Rapid Determination of Mw from Broadband P Waveforms", Bulletin of the Seismological Society of America, 85 (2): 606–613
• Utsu, T. (2002), Lee, W.H.K.; Kanamori, H.; Jennings, P.C.; Kisslinger, C., eds., "Relationships between magnitude scales", International Handbook of Earthquake and Engineering Seismology, International Geophysics, Academic Press, A (81), pp. 733–46.