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

Martin Klimes & Ludek Klimes


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.


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.