Surface-wave magnitude

Definition

The formula to calculate surface wave magnitude is:

:M_s = \log_{10}\left(\frac{A}{T}\right)_{\text{max}} + \sigma(\Delta),,

where A is the maximum particle displacement in surface waves (vector sum of the two horizontal displacements) in μm, T is the corresponding period in s (usually 20 2 seconds), Δ is the epicentral distance in °, and

:\sigma(\Delta) = 1.66\cdot\log_{10}(\Delta) + 3.5,.

Several versions of this equation were derived throughout the 20th century, with minor variations in the constant values. Since the original form of M_s was derived for use with teleseismic waves, namely shallow earthquakes at distances 100 km from the seismic receiver, corrections must be added to the computed value to compensate for epicenters deeper than 50 km or less than 20° from the receiver.

For official use by the Chinese government, the two horizontal displacements must be measured at the same time or within 1/8 of a period; if the two displacements have different periods, a weighted sum must be used:

: T = \frac{T_{N}A_{N} + T_{E}A_{E}}{A_{N} + A_{E}},,

where AN is the north–south displacement in μm,　AE is the east–west displacement in μm,　TN is the period corresponding to AN in s, and TE is the period corresponding to AE in s.

Other studies

Vladimír Tobyáš and Reinhard Mittag proposed to relate surface wave magnitude to local magnitude scale ML, using{{cite journal | archive-url=https://archive.today/20130104222201/http://www.springerlink.com/content/lv140444x5m01362/ | url-status=dead | archive-date=2013-01-04 | access-date=2008-09-14 | url-access=subscription

: M_s = -3.2 + 1.45 M_{L}

Other formulas include three revised formulae proposed by CHEN Junjie et al.:

: M_s = \log_{10}\left(\frac{A_\text{max}}{T}\right) + 1.54\cdot \log_{10}(\Delta) + 3.53

: M_s = \log_{10}\left(\frac{A_\text{max}}{T}\right) + 1.73\cdot \log_{10}(\Delta) + 3.27

and

: M_s = \log_{10}\left(\frac{A_\text{max}}{T}\right) - 6.2\cdot \log_{10}(\Delta) + 20.6

References

1. (April 1983). "Magnitude scale and quantification of earthquakes". Tectonophysics.
2. Bath, M. (1966). "Physics and Chemistry of the Earth". Pergamon Press.