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 (complex-valued action function). The solution of the complex-valued Hamilton-Jacobi equation for complex-valued travel time by Hamilton's equations of rays would require complex-valued rays (complex-valued geodesics). Since the material properties are known in real space only, we cannot calculate complex-valued rays. 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.

For this perturbation from the reference travel time to the complex-valued travel time, we need a complex-valued perturbation Hamiltonian function, i.e., a family of complex-valued Hamiltonian functions smoothly parametrized by one or more perturbation parameters. The perturbation Hamiltonian function must smoothly connect the reference Hamiltonian function with the Hamiltonian function corresponding to a given complex-valued Hamilton-Jacobi equation, and Hamilton's equations corresponding to the reference Hamiltonian function must yield real-valued reference rays. In order to be able to perform the perturbation from the reference travel time to the complex-valued travel time, we need the perturbation Hamiltonian function to be a holomorphic function of the complex slowness vector. This paper is devoted to the construction of the reference Hamiltonian function for a given complex-valued Hamilton-Jacobi equation, and to the construction of the corresponding complex-valued perturbation Hamiltonian function.

The perturbation Hamiltonian function
may be constructed in different ways,
yielding differently accurate perturbation expansions
of travel time. Unlike in previous papers,
we construct the reference Hamiltonian function
directly using the Hamiltonian function corresponding
to a given complex-valued Hamilton-Jacobi equation.
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.

Ray theory, complex-valued travel time (complex-valued action function), complex-valued Hamilton-Jacobi equation, complex-valued eikonal equation, perturbation methods, attenuation, anisotropy, heterogeneous media, wave propagation.

The paper is available in PDF (122 kB).

In: Seismic Waves in Complex 3-D Structures, Report 20, pp. 193-205, Dep. Geophys., Charles Univ., Prague, 2010.