# Moment magnitude scale

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The moment magnitude scale (MMS; denoted explicitly with Mw or Mw, and generally implied with use of a single M for magnitude [1] ) is a measure of an earthquake's magnitude ("size" or strength) based on its seismic moment (a measure of the "work" done by the earthquake [2] ), expressed in terms of the familiar magnitudes of the original "Richter" magnitude scale.

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

Work is the product of force and distance. In physics, a force is said to do work if, when acting, there is a movement of the point of application in the direction of the force.

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.

## Contents

Moment magnitude (Mw) is considered the authoritative magnitude scale for ranking earthquakes by size [3] because it is more directly related to the energy of an earthquake, and does not saturate. (That is, it does not underestimate magnitudes like other scales do in certain conditions. [4] ) It has become the standard scale used by seismological authorities (such as the U.S. Geological Survey [5] ), replacing (when available, typically for M > 4) use of the ML (Local magnitude) and Ms (surface-wave magnitude) scales. Subtypes of the moment magnitude scale (Mww, 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 they can tell us about the earthquake rupture process; the first magnitude scales were therefore empirical. [6] 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. [7] 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. [8] He established a reference point and the now familiar ten-fold (exponential) scaling of each degree of magnitude, and in 1935 published his "magnitude" scale, now called the Local magnitude scale, labeled ML. [9]

Empirical evidence is the information received by means of the senses, particularly by observation and documentation of patterns and behavior through experimentation. The term comes from the Greek word for experience, ἐμπειρία (empeiría).

Kiyoo Wadati was an early seismologist at the Central Meteorological Observatory of Japan, researching deep earthquakes. His name is attached to the Wadati–Benioff zone. It was Wadati's 1928 paper on shallow and deep earthquakes, comparing maximum below surface displacement against distance from the epicentre, which led Charles Richter to develop his earthquake magnitude scale in 1935.

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to baseb is denoted as logb (x) (or, without parentheses, as logbx, or even without explicit base as log x, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so logb (x) for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

The Local magnitude scale was developed on the basis of shallow (~15 km (9 mi) deep), moderate-sized earthquakes at a distance of approximately 100 to 600 km (62 to 373 mi), 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 [10] – a surface-wave magnitude scale (Ms ) by Beno Gutenberg in 1945 [11] , a body-wave magnitude scale (mB ) by Gutenberg and Richter in 1956, [12] and a number of variants [13] – to overcome the deficiencies of the ML scale, but all are subject to saturation. A particular problem was that the Ms scale (which in the 1970s was the preferred magnitude scale) saturates around Ms 8.0, and therefore underestimates the energy release of "great" earthquakes [14] such as the 1960 Chilean and 1964 Alaskan earthquakes. These had Ms 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. [15]

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.

The 1964 Alaskan earthquake, also known as the Great Alaskan earthquake and Good Friday earthquake, occurred at 5:36 PM AKST on Good Friday, March 27. Across south-central Alaska, ground fissures, collapsing structures, and tsunamis resulting from the earthquake caused about 131 deaths.

### 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. [16]

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. [17] A double couple can be viewed as "equivalent to a pressure and tension acting simultaneously at right angles". [18]

Torque, moment, moment of force or "turning effect" is the rotational equivalent of linear force. The concept originated with the studies of Archimedes on the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

Mechanics is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes. During the early modern period, scientists such as Galileo, Kepler, and Newton laid the foundation for what is now known as classical mechanics. It is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of and forces on objects. The field is yet less widely understood in terms of quantum theory.

In mechanics, a couple refers to two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action.

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. [19]

In 1923 Hiroshi Nakano showed that certain aspects of seismic waves could be explained in terms of a double couple model. [20] 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? [21] While Japanese seismologists favored the double couple, most seismologists favored the single couple. [22] 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. [23] 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. [24]

The debate ended when Maruyama (1963), Haskell (1964), and Burridge & Knopoff (1964) 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. [25] 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. [26]

### 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. [27] 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 [28] ), 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. [29]

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. [30] More generally applied to problems of stress in materials, [31] an extension by F. Nabarro in 1951 was recognized by the Russian geophysicist A. V. Vvedenskaya as applicable to earthquake faulting. [32] 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, [33]

In a pair of papers in 1958, J. A. Steketee worked out how to relate dislocation theory to geophysical features. [34] Numerous other researchers worked out other details, [35] 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. [36]

### Seismic moment

Seismic moment – symbol M0 – is a measure of the work accomplished by the faulting of an earthquake. [37] Its magnitude is that of the forces 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 [38] .) Seismic moment is measured in units of Newton meters (N·m) or Joules, or (in the older CGS system) dyne-centimeters (dyn-cm). [39]

The first calculation of an earthquake's seismic moment from its seismic waves was by Keiiti Aki for the 1964 Niigata earthquake. [40] 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. [41] 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). [42] In particular, he derived a now famous equation that relates an earthquake's seismic moment to its physical parameters:

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 ūS with the equivalent D̄A, known as the "geometric moment" or "potency". [43] .) 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. [44]

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. [45] 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. [46] This is typically 10% or less of the total energy, the rest being expended in fracturing rock or overcoming friction (generating heat). [47]

Nonetheless, seismic moment is regarded as the fundamental measure of earthquake size, [48] representing more directly than other parameters the physical size of an earthquake. [49] As early as 1975 it was considered "one of the most reliably determined instrumental earthquake source parameters". [50]

### 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 [51] 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, [52] [ scientific citation needed ] 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,[ citation needed ] which includes the great majority of quakes.

Current practice in official[ who? ] 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[ who? ] 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 [53] 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). [54] 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 1.2×109 N⋅m, while the Great Chilean earthquake of 1960, with an estimated moment magnitude of 9.4–9.6, had a seismic moment between 1.4×1023 N⋅m and 2.8×1023 N⋅m.

## 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. [55] 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 [56]

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

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.

## 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. [57]

• 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 [58] 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. [59]
• 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. [60]

## Notes

1. These are normally not bolded. In the technical literature a single bolded "M" – with or without italicization – is used for several related concepts.
2. Bormann, Wendt & Di Giacomo 2013 , pp. 14, 177.
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 Internet Archive.
4. Miyake 2017 , p. 112.
5. Suzuki 2001 , p. 121. See also Figure 2-22 in Richter 1958 (copy in Bormann, Wendt & Di Giacomo 2013 , p. 60), which replicates Wadati's curves.
6. See Bormann & Saul 2009 for an overview.
7. Kanamori 1977 , p. 2981.
8. ISC-EHB Event 879136 [ IRIS ]; ISC-EHB Event 869809 [ IRIS ].
9. Miyake 2017 , pp. 112-113; Stauder 1962 , p. 39.
10. Miyake 2017 , p. 115.
11. Ben-Menahem 1995 , p. 1210; Maruyama 1963 , p. 484.
12. Shearer 2009 , p. 245.
13. Ben-Menahem 1995 , p. 1210.
14. Miyake 2017 , p. 115.
15. Miyake 2017 , p. 115. See Byerly 1960 for a contemporary account of why many seismologists favored a single couple model.
16. Miyake 2017 , pp. 116, 117.
17. Pujol 2003b , p. 164.
18. Pujol 2003b , p. 165; Miyake 2017 , pp. 117–118.
19. Aki 1966b , p. 84; Pujol 2003b , p. 167.
20. Julian, Miller & Foulger 1998 , §2.2.1.
21. Miyake 2017 , p. 114, 117; Maruyama 1963 , p. 483.
22. Miyake 2017 , p. 112.
23. Miyake 2017 , p. 117.
24. Steketee 1958b , pp. 1168-1169.
25. Stauder 1962 , p. 42; Aki & Richards 2002 , p. 48.
26. Honda 1962 , pp. 32, 65, and see bibliography; Ben-Menahem 1995 , p. 1212; Udías 1991 , p. 90; Maruyama 1963 , p. 467.
27. Miyake 2017 , p. 467; Steketee  1958a , 1958b.
28. Udías 1991 provides a partial overview.
29. Pujol 2003b , pp. 165, 167; Miyake 2017 , p. 118.
30. Aki 1966b , p. 73; Kassaras & Kapetanidis 2018 , p. 410.
31. Beroza & Kanamori 2015 , p. 5.
32. Aki 1966a , pp. 24, 36.
33. Aki 1966a , p. 24.
34. Bormann, Wendt & Di Giacomo 2013 , p. 12, equation 3.1.
35. Aki 1966b , p. 84.
36. Bormann, Wendt & Di Giacomo 2013 , p. 14; Bormann & Di Giacomo 2011 , p. 412.
37. Bormann, Wendt & Di Giacomo 2013 , pp. 39–40.
38. Deichmann 2006 , p. 1268.
39. Abe 1982 , p. 322.
40. Kanamori & Anderson 1975 , p. 1076.
41. Bormann, Wendt & Di Giacomo 2013 , §3.2.8.2, p. 135.
42. Bormann, Wendt & Di Giacomo 2013 , §3.2.8.3, pp. 137–128.

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