# 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 of Rayleigh surface waves that travel along the uppermost layers of the Earth. This magnitude scale is related to the local magnitude scale proposed by Charles Francis Richter in 1935, with modifications from both Richter and Beno Gutenberg throughout the 1940s and 1950s.[1][2] It is currently used in People's Republic of China as a national standard (GB 17740-1999) for categorising earthquakes.[3]

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 through the mid 20th century, commonly attributed to Richter, could be either ${\displaystyle M_{s}}$ or ${\displaystyle M_{L}}$.

## Definition

The formula to calculate surface wave magnitude is:[3]

${\displaystyle M_{s}=\log _{10}\left({\frac {A}{T}}\right)_{\text{max}}+\sigma (\Delta )\,,}$

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

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

Several versions of this equation were derived throughout the 20th century, with minor variations in the constant values.[2][4] Since the original form of ${\displaystyle M_{s}}$ 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.[4]

For official use by the Chinese government,[3] 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:

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

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[5]

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

Other formulas include three revised formulae proposed by CHEN Junjie et al.:[6]

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

and

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

3. ^ a b c XU Shaokui, LU Yuanzhong, GUO Lucan, CHEN Shanpei, XU Zhonghuai, XIAO Chengye, FENG Yijun (许绍燮、陆远忠、郭履灿、陈培善、许忠淮、肖承邺、冯义钧) (1999-04-26). "Specifications on Seismic Magnitudes (地震震级的规定)" (in Chinese). General Administration of Quality Supervision, Inspection, and Quarantine of P.R.C. Archived from the original on 2009-04-24. Retrieved 2008-09-14.{{cite web}}: CS1 maint: multiple names: authors list (link)
6. ^ CHEN Junjie, CHI Tianfeng, WANG Junliang, CHI Zhencai (陈俊杰, 迟天峰, 王军亮, 迟振才) (2002-01-01). "Study of Surface Wave Magnitude in China (中国面波震级研究)" (in Chinese). Journal of Seismological Research (《地震研究》). Retrieved 2008-09-14.{{cite web}}: CS1 maint: multiple names: authors list (link)