Constant Q ‐wave propagation and attenuation

A linear model for attenuation of waves is presented, with Q , or the portion of energy lost during each cycle or wavelength, exactly independent of frequency. The wave propagation is completely specified by two parameters, e.g., Q and c 0 , a phase velocity at an arbitrary reference frequency ω 0 . A simple exact derivation leads to an expression for the phase velocity c as a function of frequency: c/c 0 = (ω/ω 0 ) γ , where γ = (1/π) tan −1 (1/ Q ). Scaling relationships for pulse propagation are derived and it is shown that for a material with a given value of Q , the risetime or the width of the pulse is exactly proportional to travel time. The travel time for a pulse resulting from a delta function source at x = 0 is proportional to x β , where β = 1/(1 ‐ γ). On the basis of this relation it is suggested that the velocity dispersion associated with anelasticity may be less ambiguously observed in the time domain than in the frequency domain. A steepest descent approximation derived by Strick gives a good time domain representation for the impulse response. The scaling relations are applied to field observations from the Pierre shale formation in Colorado, published by Ricker, who interpreted his data in terms of a Voigt solid with Q inversely proportional to frequency, and McDonal et al., who interpreted their data in terms of nonlinear friction. The constant Q theory fits both sets of data.