## Quality factor Q in dissipative anisotropic media

**Vlastislav Cerveny** ** & **
**Ivan Psencik**
### Summary

In an isotropic dissipative medium, the attenuation properties
of rocks are usually specified by quality factor *Q*, a positive, dimensionless,
real-valued, scalar quantity, independent of the direction
of wave propagation.We propose a similar, scalar, but direction-
dependent quality *Q*-factor (also called *Q*) for time-harmonic,
homogeneous or inhomogeneous plane waves propagating
in unbounded homogeneous dissipative anisotropic media.
We define the *Q*-factor, 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 the quality factor, denoted
as *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). The quality
factor *Q^* does not depend on the inhomogeneity of the plane wave
under consideration and thus is 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.

### Whole paper

The reprint is available in
PDF (936 kB).

*Geophysics*, **73** (2008), T63-T75.

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