Anelastic degradation of acoustic pulses in rock

Measurements on acoustic pulses propagating in massive rock lead to a simple empirical relationship between the pulse rise time, τ and the time of propagation of a pulse, t:

$\text{τ=τ}{}_0\text{+C}\text{∫}\text{)}\text{T}\text{Q}{}^{\mathrm{?1}}\text{d}\text{t}$

where τ₀ is the initial rise time (at t = 0), Q is the anelastic parameter which may be expressed in terms of the fractional loss of energy per cycle of a sinusoidal wave, Q = 2π(ΔE/E)^?1, and is assumed to be essentially independent of frequency, and C is a constant whose value we estimate experimentally to be 0.53 ± 0.04. Of the linear theories of seismic pulse attenuation, model 2 of Azimi et al. (1968) is favoured. Pulse shapes computed from equations of Futterman (1962) also give C = 0.5, but the pulse arrives earlier than in a non-attenuating medium with the same elasticity and density. Pulse shapes calculated using Strick's (1967, 1970, 1971) theory give values of C incompatible with our results. The observations suggest that a method of estimating the Q-structure of the earth from seismic pulse rise times may have a particular advantage over the spectral ratio method.