## Quality factor *Q* in dissipative anisotropic media

**Vlastislav Cerveny** and
**Ivan Psencik**
### Summary

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 *A*^{in},
*Q*^{-1}=A^{in},
which is a measure of dissipative properties of rocks.
The intrinsic attenuation factor *A*^{in} 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.

### Keywords

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

### Whole paper

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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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