A theoretical discussion of time domain magnitudes: The Prague formula for <i>M_s</i> and the mantle magnitude <i>M_m</i>

Abstract

A combination of surface wave theory and phase‐stationary asymptotics is used to relate the time domain amplitude a ( t ) of a strongly dispersed wave and the moment M 0 of the source. This approximation is valid at sufficiently large distances, over 10° for 20‐s Rayleigh waves. We apply this formalism to justify theoretically the Prague formula for M s . Assuming a Rayleigh Q of 297, we can successfully model the theoretical distance correction as 1.66 log 10 Δ in the range 20–160°. We also predict a relation of the form log 10 M 0 = M s + 19.46, in good agreement with reported empirical values. Finally, we show that the theory requires M s to be described by the product ( aT ); the use of the ratio ( a / T ) is a partial and ad hoc compensation for a large number of frequency‐dependent terms ignored in the Prague formula. The same formalism can be applied to the inversely dispersed branch of mantle Rayleigh waves, between periods of 60 and 230 s. We provide the theoretical justification for the use of time domain measurements to obtain a mantle magnitude, and in particular for the modeling of the ratio a / A of the time domain and spectral amplitudes as 2/ T , at distances ranging from 20 to 120°. At greater distances, and in particular for multiple passages, an additional distance correction must be effected.