Surface-wave magnitude

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.

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:

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:


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

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


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

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


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

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


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


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

and


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