## 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 quality factor *Q*, which is a positive, dimensionless,
real-valued, scalar quantity, independent of the direction of wave
propagation. We propose a similar, scalar, but direction-dependent quality
factor for time-harmonic, homogeneous or inhomogeneous plane waves,
propagating in unbounded homogeneous dissipative anisotropic media.
We define the quality factor, similarly as in isotropic viscoelastic
media, as the ratio of the time-averaged complete stored energy and the
dissipated energy, per unit volume. A solution of an algebraic equation
of the sixth degree with complex-valued coefficients is necessary for
the exact determination of *Q*. For weakly inhomogeneous plane waves,
propagating in arbitrarily anisotropic, weakly dissipative media, we
simplify the exact expression for *Q* considerably using the perturbation
method. The solution of the equation of the sixth degree is no longer required.
We obtain a simple explicit perturbation expression for quality factor, denoted
*Q*^{^}. We prove that the direction-dependent *Q*^{^} is
related to the attenuation coefficient *alpha* measured along
a profile in the direction of the energy-velocity vector (ray direction).
*Q*^{^} does not depend on the inhomogeneity of the plane wave under
consideration, and thus may be used as a convenient measure of the intrinsic
dissipative properties of rocks in the ray direction. In all other directions,
the quality factor is influenced by the inhomogeneity of the wave under consideration.
We illustrate the peculiarities in the behavior
of *Q*^{^} and its accuracy on a model of anisotropic, weakly dissipative
sedimentary rock. Examples show interesting inner loops
in polar diagrams of *Q*^{^} in regions of S-wave triplications.

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In: Seismic Waves in Complex 3-D Structures, Report 18,
pp. 163-192, Dep. Geophys., Charles Univ., Prague, 2008.

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