A major problem with time-domain measurements of seismic surface waves is the significant effect of nondispersed Rayleigh waves and Airy phases, which can occur at both regional and teleseismic distances. This article derives a time-domain method for measuring surface waves with minimum digital processing by using zero-phase Butterworth filters. The method can effectively measure surface- wave magnitudes at both regional and teleseismic distances, at variable periods between 8 and 25 sec, while ensuring that the magnitudes are corrected to accepted formulae at 20-sec reference periods, thus providing historical continuity. For applications over typical continental crusts, the proposed magnitude equation is, for zero- to-peak measurements in millimicrons:

Ms(b) = log(ab) + 1/2 log(sin(Δ)) + 0.0031(20/T)1.8Δ

− 0.66 log (20/T) − log(fc) − 0.43,


fc ≤ 0.6/T√Δ.

To calculate Ms(b), the following steps should be taken:

  1. Determine the epicentral distance in degrees to the event Δ and the period T.

  2. Calculate the corner filter frequency fc using the preceding inequality.

  3. Filter the time series using a zero-phase, third-order Butterworth bandpass filter with corner frequencies 1/Tfc, 1/T + fc.

  4. Calculate the maximum amplitude ab of the filtered signal and calculate Ms(b).

At the reference period of 20 sec, the equation is equivalent to von Seggern’s formula (1977) scaled to Vanĕk (1962) at 50 degrees. For periods 8 ≤ T ≤ 25, the equation is corrected to T = 20 sec, accounting for source effects, attenuation, and dispersion.

Online material: Design and realization of Butterworth filters.

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