## Exact and asymptotic Green functions in homogeneous
anisotropic viscoelastic media

**Vaclav Vavrycuk**
### Summary

An asymptotic Green function in homogeneous anisotropic viscoelastic
media is derived. The Green function in viscoelastic media is formally
similar to that in elastic media, but its computation is more involved.
The stationary slowness vector is, in general, complex-valued and
inhomogeneous. Its computation involves finding two independent
real-valued unit vectors which specify the directions of its real and
imaginary parts and can be done either by iterations or by solving a
system of coupled polynomial equations. When the stationary slowness
direction is found, all quantities standing in the Green function such
as the slowness vector, polarization vector, phase and energy
velocities, and principal curvatures of the slowness surface can
readily be calculated.

The formulas for the exact and asymptotic Green functions are
numerically checked against closed-form solutions for isotropic and
simple anisotropic, elastic and viscoelastic models. The calculations
confirm that the formulas and developed numerical codes are correct.
The computation of the P-wave Green function in two realistic materials
with a rather strong anisotropy and absorption indicates that the
asymptotic Green function is accurate at distances greater than several
wavelengths from the source. The error in the modulus reaches at most
4% at distances greater than 15 wavelengths from the source.

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In: Seismic Waves in Complex 3-D Structures, Report 17,
pp. 213-241, Dep. Geophys., Charles Univ., Prague, 2007.

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