# Surface wave magnitude

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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 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 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, $M_s$, 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 during that time, commonly attributed to Richter, could be either $M_s$ or $M_L$.

## Definition

The formula to calculate surface wave magnitude is:[1][3]

$M = \log_{10}\left(\frac{A}{T}\right)_{\text{max}} + \sigma(\Delta)$

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, Δ is the epicentral distance in °, and

$\sigma(\Delta) = 1.66\cdot\log_{10}(\Delta) + 3.5$

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:

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

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[4]

$M_s = -3.2 + 1.45 M_{L}$

Other formulas include three revised formulae proposed by CHEN Junjie et al.:[5]

$M_s = \log_{10}\left(\frac{A_{max}}{T}\right) + 1.54\cdot \log_{10}(\Delta) + 3.53$
$M_s = \log_{10}\left(\frac{A_{max}}{T}\right) + 1.73\cdot \log_{10}(\Delta) + 3.27$

and

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

## Notes and references

1. ^ a b 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. Retrieved 2008-09-14.
2. ^ William L. Ellsworth (1991). "SURFACE-WAVE MAGNITUDE (MS) AND BODY-WAVE MAGNITUDE (mb)". USGS. Retrieved 2008-09-14.
3. ^ It is obvious that the entire formula cannot stand dimensional analysis without additional qualifications. References here provide no such qualification.
4. ^ Vladimír Tobyáš and Reinhard Mittag (1991-02-06). "Local magnitude, surface wave magnitude and seismic energy". Studia Geophysica et Geodaetica. Retrieved 2008-09-14.
5. ^ 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.