Velocity dispersion due to anelasticity; implications for seismology and mantle composition

Abstract

The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann aftereffect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the 'standard linear solid', and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms.The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.

Absorption band Q model for the Earth

“…In this paper we apply the absorption band concept [Liu et al, 1976;Kanamori and Anderson, 1977;Anderson et al, 1977;Anderson and Minster, 1980;Minster and Anderson, 1981] to the attenuation of body waves, surface waves, and free oscillations. The basic method is described by Anderson and Hart [1978a, b] except that we replace the frequencyindependent Q assumption with the absorption band assumption.…”

Optimized viscoelastic wave propagation for weakly dissipative media

“…1: zero for negative time and decaying to a constant positive value for positive times (Christensen 1982). This can be approximated with a discrete relaxation spectrum (Liu et al 1976)…”

“…A first method to transform the stress-strain relation into a differential form using Padé approximations is introduced by Day & Minster (1984). Later, Emmerich & Korn (1987) and Carcione et al (1988) suggested to improve this by approximating the medium properties with a discrete relaxation spectrum (Liu et al 1976) and fitting the parameters used to describe this spectrum to the observed behaviour numerically. These methods are still common today (e.g.…”

Global Imaging of the Earth's Deep Interior: Seismic Constraints on (An)isotropy, Density and Attenuation

“…The constant-Q model is closely related to the notion of a continuous absorption band, i.e. a continuous distribution of relaxation mechanisms that lead to nearly frequency-independent absorption (Liu et al, 1976), as shown in Figure 11.6. This is in contrast to the appearance of isolated absorption peaks that are, for instance, commonly found in metals (e.g.…”

“…the frequency-dependence of phase velocity (Figure 11.6). In the constant-Q model (Liu et al, 1976), the phase velocity dispersion relation is given by 19) where ω ref is a reference frequency. Equation (11.19) holds, provided that Q >> 1, and that ω lies within the absorption band.…”