Development of a Time-Domain, Variable-Period Surface-Wave Magnitude Measurement Procedure for Application at Regional and Teleseismic Distances, Part I: Theory

Abstract

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: M s(b) = log( a b ) + 1/2 log(sin(Δ)) + 0.0031(20/ T ) 1.8 Δ − 0.66 log (20/ T ) − log( f c ) − 0.43, where: f c ≤ 0.6/ T √Δ. To calculate M s(b) , the following steps should be taken: Determine the epicentral distance in degrees to the event Δ and the period T . Calculate the corner filter frequency f c using the preceding inequality. Filter the time series using a zero-phase, third-order Butterworth bandpass filter with corner frequencies 1/ T − f c , 1/ T + f c . Calculate the maximum amplitude a b of the filtered signal and calculate M s(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.