M_m: a variable-period mantle magnitude

This paper lays the theoretical groundwork for a variable period mantle magnitude, M_m, 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 C_p, and a source correction C_s, also period-dependent. 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. Both versions of the method lend themselves well to automation. -from Authors