Mm: A variable‐period mantle magnitude

Abstract

This paper lays the theoretical groundwork for a variable period mantle magnitude, Mm, based on the measurement of the spectral amplitude X(ω) of very long period Rayleigh waves. We retain the concept of magnitude by restricting ourselves to single‐station measurements, ignoring the focal mechanism and the exact depth of the shallow earthquakes considered. Our measurements are made at a series of periods (in all cases greater than 40 s), and the largest value is retained. This procedure effectively avoids the well‐known interference effects leading to saturation of magnitude scales defined at a fixed period. Two corrections are used: a period‐dependent distance correction CD, and a source correction Cs, also period‐dependent, compensating for the variation of the excitation of Rayleigh waves with period. Both of these corrections are fully predictable on the basis of standard surface wave excitation and dispersion theory. The result is a formula of the type Mm = log10 X(ω) + CD + CS + C0 in which all coefficients, including the constant C0 are justifiable on sound theoretical grounds. The analysis of a data set of 256 records from the broadband seismograph at Papeete, Tahiti, the ultra‐long period system at Pasadena, and stations of the GEOSCOPE network, shows that the mean error in the estimation of the seismic moment is on the order of 0.1–0.2 units of magnitude, with the standard deviation at each station also on the order of 0.2 units of magnitude. No significant trend with either distance, period, or station can be identified. The method can also be transposed to the time domain, under some simple assumptions which are justifiable theoretically for typical teleseismic distances across the Pacific Basin. Both versions of the method lend themselves well to automation. Thus, by providing a real‐time estimate of the seismic moment of distant earthquakes, Mm has considerable potential for tsunami warning purposes. Its concept can easily be extended to Love waves and also to intermediate and deep earthquakes.