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The surface wave magnitude ( $M_{s}$ ) 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, $M_{s}$ , and one on body waves, $M_{b}$ . 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 $M_{s}$ , using an equation similar to that used for $M_{L}$ .
— 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 $M_{s}$ or $M_{L}$ .

Definition

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

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.^[2]^[4] 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.^[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:

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

where A_N is the north–south displacement in μm,　A_E is the east–west displacement in μm,　T_N is the period corresponding to A_N in s, and T_E is the period corresponding to A_E in s.

Other studies

Vladimír Tobyáš and Reinhard Mittag proposed to relate surface wave magnitude to local magnitude scale M_L, using^[5]

M_{s}=-3.2+1.45M_{L}

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

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

Notes and references

^ William L. Ellsworth (1991). "SURFACE-WAVE MAGNITUDE (M_S) AND BODY-WAVE MAGNITUDE (mb)". United States Geological Survey. Retrieved 2008-09-14.
^ ^a ^b Kanamori, Hiroo (April 1983). "Magnitude scale and quantification of earthquakes". Tectonophysics. 93 (3–4): 185–199. Bibcode:1983Tectp..93..185K. doi:10.1016/0040-1951(83)90273-1.
^ ^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)
^ ^a ^b Bath, M (1966). "Earthquake energy and magnitude". In Ahrens, L. H.; Press, F.; Runcorn, S. (eds.). Physics and Chemistry of the Earth. Pergamon Press. pp. 115–165.
^ Vladimír Tobyáš and Reinhard Mittag (1991-02-06). "Local magnitude, surface wave magnitude and seismic energy". Studia Geophysica et Geodaetica. 35 (4): 354. Bibcode:1991StGG...35..354T. doi:10.1007/BF01613981. S2CID 128567958. Archived from the original on 2013-01-04. Retrieved 2008-09-14.
^ 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)^{[permanent dead link‍]}