Surface-wave magnitude

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 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.

Vladimír Tobyáš and Reinhard Mittag proposed to relate surface wave magnitude to local magnitude scale M_L, 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

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

• Seismic magnitude scales

• Robert E. Wallace, ed. (1991). "The San Andreas Fault System, California (Professional Paper 1515)". USGS. Retrieved 2008-09-14.