A theoretical discussion of time domain magnitudes: the Prague formula for M_s and the mantle magnitude M_m

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_o 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 log₁₀ Δ in the range 20-160°. We also predict a relation of the form log₁₀ M₀=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. -from Author