## 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 the 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 the 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

The reprint is available in
PDF (189 kB).

*Stud. geophys. geod.*, **66** (2022), 124-144.