Tracing real-valued reference rays in anisotropic viscoelastic media

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 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.


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

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Seismic Waves in Complex 3-D Structures, 30 (2020), 77-94 (ISSN 2336-3827, online at