Thomsen-type parameters and attenuation coefficients for constant-Q transverse isotropy

TL;DR

This paper derives analytic formulas for Thomsen-type attenuation parameters in constant-Q transverse isotropic media, revealing frequency dependence in some parameters and emphasizing the importance of accounting for this in attenuation analysis.

Contribution

It provides the first explicit analytic expressions for Thomsen-type attenuation parameters in Kjartansson's constant-Q TI model, including the frequency-dependent parameter δ_Q.

Findings

The parameter δ_Q varies with frequency in constant-Q TI models.

Attenuation coefficients depend on frequency for oblique wave propagation.

Simplified expressions are derived for the special case of viscoacoustic TI media.

Abstract

Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called ``constant-$Q$'' transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks. The attenuation coefficients in constant-$Q$ TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy parameters. Recent research indicates that not all those parameters for such constant-$Q$ media are frequency-independent. Here, we present concise analytic formulae for the Thomsen-type attenuation parameters for Kjartansson's constant-$Q$ TI model and show that one of them ($\delta_{Q}$) varies with frequency. The analytic expression for $\delta_{Q}$ helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by introducing the newly defined ``dispersion factors''. Viscoacoustic constant-$Q$ transverse isotropy is also discussed as a special case, for which the elliptical condition and simplified expressions for the parameters $\delta$ and $\delta_{Q}$ are derived. Our results show that in the presence of significant absorption the attenuation coefficients of the ``constant-$Q$'' model vary with frequency for oblique propagation with respect to the symmetry axis. This variation needs to be taken into account when applying the spectral-ratio method and other attenuation-analysis techniques.