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]

Seismic magnitude scales are used to describe the overall strength or "size" of an earthquake. These are distinguished from seismic intensity scales that categorize the intensity or severity of ground shaking (quaking) caused by an earthquake at a given location. Magnitudes are usually determined from measurements of an earthquake's seismic waves as recorded on a seismogram. Magnitude scales vary on what aspect of the seismic waves are measured and how they are measured. Different magnitude scales are necessary because of differences in earthquakes, the information available, and the purposes for which the magnitudes are used.

Earthquake Shaking of the surface of the earth caused by a sudden release of energy in the crust

An earthquake is the shaking of the surface of the Earth, resulting from the sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in size from those that are so weak that they cannot be felt to those violent enough to toss people around and destroy whole cities. The seismicity, or seismic activity, of an area is the frequency, type and size of earthquakes experienced over a period of time. The word tremor is also used for non-earthquake seismic rumbling.

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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 so-called Richter magnitude scale – more accurately, Richter's magnitude scale, or just Richter magnitude – for measuring the strength ("size") of earthquakes refers to the original "magnitude scale" developed by Charles F. Richter and presented in his landmark 1935 paper, and later revised and renamed the Local magnitude scale, denoted as "ML" or "ML". Because of various shortcomings of the ML scale most seismological authorities now use other scales, such as the moment magnitude scale (Mw ), to report earthquake magnitudes, but much of the news media still refers to these as "Richter" magnitudes. All magnitude scales retain the logarithmic character of the original, and are scaled to have roughly comparable numeric values.

Logarithmic scale measurement scale based on orders of magnitude

A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include earthquake strength, sound loudness, light intensity, and pH of solutions.

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]

Seismic moment is a quantity used by seismologists to measure the size of an earthquake. The scalar seismic moment is defined by the equation , where

Shear modulus ratio of shear stress to the shear strain

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain:

In geology, a fault is a planar fracture or discontinuity in a volume of rock, across which there has been significant displacement as a result of rock-mass movement. Large faults within the Earth's crust result from the action of plate tectonic forces, with the largest forming the boundaries between the plates, such as subduction zones or transform faults. Energy release associated with rapid movement on active faults is the cause of most earthquakes.

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]

Beno Gutenberg seismologist

Beno Gutenberg was a German-American seismologist who made several important contributions to the science. He was a colleague and mentor of Charles Francis Richter at the California Institute of Technology and Richter's collaborator in developing the Richter magnitude scale for measuring an earthquake's magnitude.

California State of the United States of America

California is a state in the Pacific Region of the United States. With 39.6 million residents, California is the most populous U.S. state and the third-largest by area. The state capital is Sacramento. The Greater Los Angeles Area and the San Francisco Bay Area are the nation's second- and fifth-most populous urban regions, with 18.7 million and 8.8 million residents respectively. Los Angeles is California's most populous city, and the country's second-most populous, after New York City. California also has the nation's most populous county, Los Angeles County, and its largest county by area, San Bernardino County. The City and County of San Francisco is both the country's second-most densely populated major city after New York City and the fifth-most densely populated county, behind only four of the five New York City boroughs.

Seismometer instrument that records seismic waves (seismograms) by measuring ground motions, caused by earthquakes, volcanic eruptions, and explosions

A seismometer is an instrument that responds to ground motions, such as caused by earthquakes, volcanic eruptions, and explosions. Seismometers are usually combined with a timing device and a recording device to form a seismograph. The output of such a device — formerly recorded on paper or film, now recorded and processed digitally — is a seismogram. Such data is used to locate and characterize earthquakes, and to study the earth's internal structure.

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]

Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructive testing for detecting defects. Rayleigh waves are part of the seismic waves that are produced on the Earth by earthquakes. When guided in layers they are referred to as Lamb waves, Rayleigh–Lamb waves, or generalized Rayleigh waves.

Love wave

In elastodynamics, Love waves, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S–waves) guided by an elastic layer, which is welded to an elastic half space on one side while bordering a vacuum on the other side. In seismology, Love waves are surface seismic waves that cause horizontal shifting of the Earth during an earthquake. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of seismic waves, such as P-waves and S-waves, or Rayleigh waves. Love waves travel with a lower velocity than P- or S- waves, but faster than Rayleigh waves. These waves are observed only when there is a low velocity layer overlying a high velocity layer/ sub–layers.

Wavelength spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

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 ū. (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

(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):

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),

(where E is in Joules and M0 is in Nm), Kanamori approximated Mw by

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)

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

where 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

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 of this stored energy is transformed into

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

where 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 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 is thus poorly known. It could vary highly from one earthquake to another. Two earthquakes with identical but different would have released different .

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

where is radiated efficiency and 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, 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 but different or would have radiated different .

Because and are fundamentally independent properties of an earthquake source, and since 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]

where 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 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 of energy release (potential or radiated) between two earthquakes of different moment magnitudes, and :

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]

See also

Notes

  1. Hanks & Kanamori 1979.
  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. "On Earthquake Magnitudes".
  5. Kanamori 1978.
  6. Kanamori 1978.
  7. Kanamori 1978.
  8. Kanamori 1978.
  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. Kanamori 1977.
  16. Boyle 2008.
  17. Kanamori 1977.
  18. Hanks & Kanamori 1979.
  19. Kostrov 1974; Dahlen 1977.
  20. Choy & Boatwright 1995
  21. Hanks & Kanamori 1979.
  22. USGS Technical Terms used on Event Pages.
  23. Tsuboi et al. 1995.
  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.

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