# Surface wave magnitude

The surface wave magnitude (${\displaystyle M_{s}}$) scale is one of the magnitude scales used in seismology to describe the size of an earthquake. It is based on measurements in Rayleigh surface waves that travel primarily along the uppermost layers of the Earth. It is currently used in People's Republic of China as a national standard (GB 17740-1999) for categorising earthquakes.[1]

Surface wave magnitude was initially developed in the 1950s by the same researchers who developed the local magnitude scale ML in order to improve resolution on larger earthquakes:[2]

The successful development of the local-magnitude scale encouraged Gutenberg and Richter to develop magnitude scales based on teleseismic observations of earthquakes. Two scales were developed, one based on surface waves, ${\displaystyle M_{s}}$, and one on body waves, ${\displaystyle M_{b}}$. Surface waves with a period near 20 s generally produce the largest amplitudes on a standard long-period seismograph, and so the amplitude of these waves is used to determine ${\displaystyle M_{s}}$, using an equation similar to that used for ${\displaystyle M_{L}}$.

— William L. Ellsworth, The San Andreas Fault System, California (USGS Professional Paper 1515), 1990–1991

Recorded magnitudes of earthquakes during that time, commonly attributed to Richter, could be either ${\displaystyle M_{s}}$ or ${\displaystyle M_{L}}$.

## Definition

The formula to calculate surface wave magnitude is:[1]

${\displaystyle M=\lg \left({\frac {A}{T}}\right)_{\text{max}}+\sigma (\Delta )\,,}$

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, Δ is the epicentral distance in °, and

${\displaystyle \sigma (\Delta )=1.66\cdot \log _{10}(\Delta )+3.5\,.}$

According to GB 17740-1999, 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, weighed sum must be used:

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

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[3]

${\displaystyle M_{s}=-3.2+1.45M_{L}}$

Other formulas include three revised formulae proposed by CHEN Junjie et al.:[4]

${\displaystyle M_{s}=\log _{10}\left({\frac {A_{max}}{T}}\right)+1.54\cdot \log _{10}(\Delta )+3.53}$
${\displaystyle M_{s}=\log _{10}\left({\frac {A_{max}}{T}}\right)+1.73\cdot \log _{10}(\Delta )+3.27}$

and

${\displaystyle M_{s}=\log _{10}\left({\frac {A_{max}}{T}}\right)-6.2\cdot \log _{10}(\Delta )+20.6}$