Quality factor Q in dissipative anisotropic media

Vlastislav Cerveny and Ivan Psencik


In an isotropic dissipative medium, the attenuation properties of rocks are usually specified by the quality factor Q, which is a dimensionless, direction-independent, scalar quantity. We propose a similar, but direction-dependent, dimensionless quality factor Q for anisotropic dissipative media.

We present an algorithm for the computation of quality factor Q of a homogeneous or an inhomogeneous time-harmonic plane wave propagating in a homogeneous viscoelastic anisotropic medium. The quality factor is defined as the ratio of the time-averaged complete stored energy and the dissipated energy, per unit volume (Buchen, 1971). The algorithm for the computation of Q is exact, and may be applied to homogeneous or inhomogeneous plane waves propagating in media of unrestricted anisotropy (including isotropy) and unrestricted viscoelasticity. Generally, the algorithm requires a solution of an algebraic equation of the sixth degree with complex-valued coefficients.

For weakly inhomogeneous plane waves, propagating in arbitrarily anisotropic, weakly dissipative media, the expression for the quality factor simplifies considerably. The solution of the algebraic equation of the sixth degree is no more required; simple explicit analytical expression for the quality factor Q is obtained. The approximate formula for the quality factor Q does not depend on the inhomogeneity of the considered plane wave; the concept of homogeneous plane waves is not needed and it is not used in the derivation of Q. The quality factor Q is inversely proportional to the intrinsic attenuation factor Ain, Q-1=Ain, which is a measure of dissipative properties of rocks. The intrinsic attenuation factor Ain is proportional to the scalar product of the attenuation vector with the energy-flux vector. The proposed algorithm for calculation of Q is generalized to waves generated by point sources, and to high-frequency body waves, propagating in heterogeneous anisotropic viscoelastic media. The expression for Q simplifies further if weak anisotropy concept is used or if Q is specified for higher-symmetry anisotropy.

Numerical examples show that the quality factors (or intrinsic attenuation factors) of plane waves propagating in anisotropic viscoelastic media may be strongly directionally dependent, even for weakly anisotropic media.


Anisotropy, weak attenuation, inhomogeneous plane waves, quality factor, viscoelasticity.

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In: Seismic Waves in Complex 3-D Structures, Report 17, pp. 173-194, Dep. Geophys., Charles Univ., Prague, 2007.
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