## Basic quantities in homogeneous anisotropic
viscoelastic media using a perturbation approach

**Vaclav Vavrycuk**
### Abstract

Velocity, attenuation and the quality factor of waves propagating in
homogenous media of arbitrary strength of anisotropy and attenuation are
calculated in high-frequency asymptotics using a stationary slowness
vector - the vector evaluated at the stationary point of the slowness
surface. This vector is, in general, complex-valued and inhomogeneous,
meaning that the real and imaginary parts of the vector have different
directions. The slowness vector can be determined by solving three
coupled polynomial equations of the 6th order or by a non–linear
inversion. The procedure is simplified, if perturbation theory is
applied. The elastic medium is viewed as a background medium, and the
attenuation effects are incorporated as perturbations. In the
first-order approximation, the phase and ray velocities and their
directions remain unchanged being the same as those in the background
elastic medium. The perturbation of the slowness vector is calculated
by solving a system of three linear equations. The phase attenuation
and phase quality factor are linear functions of the perturbation of the
slowness vector. Calculating the ray attenuation and ray quality factor
is even simpler than calculating the phase quantities, because they are
expressed just in terms of perturbations of the medium without the
necessity to evaluate the perturbation of the slowness vector.
Numerical modeling indicates that the perturbations are highly accurate,
the errors being less than 0.3% for a medium with a quality factor of 20
or higher. The accuracy can further be enhanced by a simple
modification of the first-order perturbation formulas.

### Keywords

Anisotropy, attenuation, perturbation theory, ray theory, seismic waves,
viscoelasticity.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 18,
pp. 193-221, Dep. Geophys., Charles Univ., Prague, 2008.

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