## Description of weak anisotropy and weak attenuation
using the first-order perturbation theory

**Vaclav Vavrycuk**
### Summary

Velocity anisotropy and attenuation in weakly anisotropic and weakly attenuating
structures can be treated uniformly using the weak anisotropy-attenuation (WAA)
parameters. The WAA parameters are constructed in a very analogous way to weak
anisotropy (WA) parameters designed for weak elastic anisotropy. The WAA parameters
generalize the WA parameters by incorporating the attenuation effects. The WAA
parameters can be represented alternatively by one set of complex values or by two sets of
real values. Assuming high-frequency waves and using the first-order perturbation theory,
all basic wave quantities such as the slowness vector, polarization vector, propagation
velocity, attenuation and quality factor are linear functions of the WAA parameters.

Numerical modeling shows that the perturbation formulas have different accuracy
for different wave quantities. The propagation velocity is usually calculated with high
accuracy. However, the attenuation and quality factor may be reproduced with appreciably
lower accuracy. This happens mostly when strength of velocity anisotropy is higher than
10% and attenuation is moderate or weak (*Q*-factor > 20). In this case, the errors of the
attenuation or quality factor can attain values comparable with strength of anisotropy or can
be even higher. It is shown that a simple modification of the formulas by including some
higher-order perturbations improves the accuracy three to four times.

### Keywords

Anisotropy, attenuation, perturbation theory, theory of wave propagation.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 19,
pp. 191-218, Dep. Geophys., Charles Univ., Prague, 2009.

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