## Perturbation expansions of complex-valued traveltime
along real-valued reference rays

**Martin Klimes** **&**
**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 traveltime
(complex-valued action function).
A very suitable approximate method
for calculating the complex-valued traveltime
right in real space is represented
by the perturbation from the reference traveltime calculated
along real-valued reference rays
to the complex-valued traveltime defined by
the complex-valued Hamilton-Jacobi equation.

The real-valued reference rays are calculated using
the reference Hamiltonian function.
The perturbation Hamiltonian function is
parametrized by one or more perturbation parameters,
and smoothly connects the reference Hamiltonian function with
the Hamiltonian function corresponding
to a given complex-valued Hamilton-Jacobi equation.
Both the reference Hamiltonian function
and the perturbation Hamiltonian function
may be constructed in different ways,
yielding differently accurate perturbation expansions
of traveltime.
All present perturbation methods use reference rays
calculated in a reference anisotropic non-attenuating medium.

In this paper,
the reference Hamiltonian function is constructed
directly using the Hamiltonian function corresponding
to a given complex-valued Hamilton-Jacobi equation,
and the perturbation Hamiltonian function
is linear with respect to the perturbation parameter.
The direct construction of the reference Hamiltonian function
from the given complex-valued Hamilton-Jacobi equation
is very general and accurate,
especially for homogeneous Hamiltonian functions of degree *N*=-1
with respect to the slowness vector.

### Keywords

Elasticity and anelasticity, body waves, seismic anisotropy,
seismic attenuation, theoretical seismology, wave propagation.

### Whole paper

The reprint is available in
PDF (125 kB).

*Geophys. J. int.*, **186** (2011), 751-759.