# Moment magnitude scale

The moment magnitude scale (M w ) is one of the preferred magnitude scales that are used in seismology to determine the strength of earthquakes . In the case of severe earthquakes in particular, the indication of an earthquake magnitude generally refers to this scale today. The end of the scale is at the value 10.6, based on the assumption that at this value the earth's crust would have to break up completely.

In the symbol M w , W stands for mechanical work , i.e. mechanically implemented work.

## Development of the scale

The first magnitude scales that were developed to quantify earthquakes are based on the measurement of maximum amplitudes of earthquake waves in seismograms . These amplitudes could be brought into a linear relationship with the energy release, making the strength of different earthquakes comparable. In particular, the well-known Richter scale is only valid in a very limited distance range. In addition, most magnitude scales show saturation with very strong earthquakes; This means that the increase in the released energy in the upper range of the scale leads to an ever smaller increase in the magnitude. A comparability of earthquake strengths is no longer guaranteed.

To overcome this limitation, Hiroo Kanamori introduced a new magnitude scale in 1977 based on the seismic moment introduced by Keiiti Aki in 1966 . This is the scalar product of the size of the fracture surface in the subsurface, the mean displacement of the rock blocks and the shear modulus of the rock. Since the seismic moment does not reach saturation, the moment magnitude, in contrast to the other magnitude scales, does not experience any saturation and is therefore also suitable for quantifying earthquakes with a large energy release.

## method

The scalar seismic moment can e.g. B. can be determined from the asymptote of the shift amplitude spectrum at the frequency f = 0 Hz. The moment magnitude is thus linked to the surface wave magnitude scale ( ). According to Gutenberg and Richter, there is the following relationship between the emitted seismic energy ( ) and the magnitude : ${\ displaystyle M _ {\ mathrm {S}}}$${\ displaystyle E _ {\ mathrm {S}}}$${\ displaystyle M _ {\ mathrm {S}}}$

${\ displaystyle \ log _ {10} E _ {\ mathrm {S}} = 1 {,} 5 \ cdot M _ {\ mathrm {S}} +4 {,} 8}$

From this it follows for the seismic moment in the unit Joule : ${\ displaystyle M_ {0}}$

${\ displaystyle \ log _ {10} M_ {0} = 1 {,} 5 \ cdot M _ {\ mathrm {S}} +9 {,} 1}$

If this equation is solved for the magnitude and this is replaced by, the moment magnitude results as a dimensionless characteristic number , which is given by the expression ${\ displaystyle M _ {\ mathrm {w}}}$

${\ displaystyle M _ {\ mathrm {w}} = {\ tfrac {2} {3}} \ left (\ log _ {10} M_ {0} -9 {,} 1 \ right)}$

is defined.

Although with this method the seismic moment is determined from the surface wave magnitude , which, like other scales, reaches saturation, the seismic moment itself is not affected, as it is not derived from the maximum amplitude but from the amplitude spectrum. There are various inversion methods today for determining from the seismogram . The calculated seismic moment depends on the details of the inversion method used, so that the resulting magnitude values ​​can show slight deviations. ${\ displaystyle M_ {0}}$${\ displaystyle M _ {\ mathrm {S}}}$${\ displaystyle M_ {0}}$${\ displaystyle M _ {\ mathrm {w}}}$

## Magnitude value and comparability

### Interscalar comparison

In order to compare two quakes with regard to their strength (i.e. the seismic energy emitted), it should be noted that the scale is logarithmic, so the earthquake strength increases exponentially with the scale value. A quake of magnitude 4 is not twice as strong as a quake of magnitude 2 (see below). An equal difference between two magnitude values ​​always means an equal ratio of the associated intensities (the energies released during the quakes):

${\ displaystyle {\ frac {E_ {2}} {E_ {1}}} = 10 ^ {{\ frac {3} {2}} (M_ {2} -M_ {1})}}$

Examples:

• 0.2 points on the scale correspond to a doubling of the energy
• 0, 6 scale points corresponding to a tenfold
• 1 point on the scale corresponds to the factor ≈31.6
• 2 points on the scale correspond to the factor 1000

### Comparison with TNT equivalents

M w E S
in joules
Quantity of TNT
in tons
Equivalence
Hiroshima
atomic bombs
(12.5 kT TNT)
4th 6.3 · 10 10 000.000.015th 00.000.0012
5 2.0 · 10 12 000,000.475 00.000.0380
6th 6.3 · 10 13 000.015,000 00.001.2000
7th 2.0 · 10 15 000.475,000 00.038, 0000
8th 6.3 · 10 16 015,000,000 01,200, 0000
9 2.0 · 10 18 475,000,000 38,000, 0000

In order to make the meaning of the magnitude value plausible, the seismic energy released in the earthquake is occasionally compared with the effect of the conventional chemical explosive TNT . The seismic energy results from the above formula according to Gutenberg and Richter ${\ displaystyle E _ {\ mathrm {S}}}$

${\ displaystyle E _ {\ mathrm {S}} = 10 ^ {1.5 \ cdot M _ {\ mathrm {S}} +4.8}}$

or converted into Hiroshima bombs:

${\ displaystyle E _ {\ mathrm {S}} = {\ frac {10 ^ {1.5 \ cdot M _ {\ mathrm {S}} +4.8}} {5 {,} 25 \ cdot 10 ^ {13 }}} = 10 ^ {1.5 \ cdot M _ {\ mathrm {S}} -8.92}}$

For the comparison of the seismic energy (in joules) with the corresponding explosion energy, a value of 4.2 · 10 9 joules per ton of TNT applies . The table illustrates the relationship between the seismic energy and the moment magnitude.

### Comparability with other scales

The moment magnitude can only be compared to other magnitude scales to a limited extent, as is already clear from the different determination of the same. The greatest correspondence between the moment magnitude scale (M w ) and the surface wave magnitude scale (M S ) (S stands for surface), which shows only minor deviations in the range of approximately magnitude 5 to 8. Saturation begins above magnitude 8 and is reached at approx. 8.5. The torque magnitude also agrees well with the Richter scale (M L ) in the magnitude range below 6.5. On the other hand, it cannot be compared with the sky wave magnitude scale (m B ), which only corresponds exactly at a magnitude of 7.0 and is saturated at approx. 8.0, as well as with the short-period sky wave magnitude (m b ) , which is only exactly the same at magnitude 5.0 and already saturates at around magnitude 6.8.

## Individual evidence

1. What does the Richter scale say? In: Axel Bach et al .: Earthquake - when the earth strikes back . Quarks & Co (WDR). June 12, 2007. Retrieved June 11, 2015.
2. Peter Bormann: 3.2 Magnitude of seismic events . In: Peter Bormann (Ed.): New Manual of Seismological Observatory Practice NMSOP . revised edition. Deutsches GeoForschungsZentrum GFZ, Potsdam 2009, ISBN 3-9808780-0-7 , p. 36 , doi : 10.2312 / GFZ.NMSOP_r1_ch3 .
3. ^ A b Thorne Lay, Terry C. Wallace: Modern global seismology . Academic Press, San Diego 1995, ISBN 978-0-12-732870-6
4. Thomas C. Hanks, Hiroo Kanamori : A Moment Magnitude Scale . In: Journal of Geophysical Research , Vol. 84, 1979, pp. 2348-2350.
5. Hiroo Kanamori: Magnitude scale and quantification of earthquakes . In: Tectonophysics , Vol. 93 (3-4), 1983, pp. 185-199