## Tracing real-valued reference rays
in anisotropic viscoelastic media

**Ludek Klimes**
### Summary

The eikonal equation in an attenuating medium has the form
of a complex-valued Hamilton-Jacobi equation and must be solved
in terms of the complex-valued travel time. A very suitable
approximate method for calculating the complex-valued travel
time right in real space is represented by the perturbation from
the reference travel time calculated along real-valued reference
rays to the complex-valued travel time defined by the complex-valued
Hamilton-Jacobi equation.

The real-valued reference rays are calculated using the reference
Hamiltonian function. The reference Hamiltonian function is constructed
using the complex-valued Hamiltonian function corresponding to a given
complex-valued Hamilton-Jacobi equation.

The ray tracing equations and the corresponding equations of geodesic
deviation are often formulated in terms of the eigenvectors of
the Christoffel matrix. Unfortunately, a complex-valued Christoffel matrix
need not have all three eigenvectors at an S-wave singularity. We thus
formulate the ray tracing equations and the corresponding equations
of geodesic deviation using the eigenvalues of a complex-valued
Christoffel matrix, without the eigenvectors of the Christoffel matrix.
The resulting equations for the real-valued reference P-wave rays
and real-valued reference common S-wave rays are applicable everywhere,
including S-wave singularities.

### Keywords

Attenuation, anisotropy, heterogeneous media, wave propagation,
ray theory, complex-valued travel time, complex-valued Hamilton-Jacobi
equation, complex-valued eikonal equation, perturbation methods.

### Whole paper
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*Seismic Waves in Complex 3-D Structures*, **30** (2020), 77-94
(ISSN 2336-3827, online at http://sw3d.cz).