Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media

Vlastislav Cerveny and Ivan Psencik


Perturbation Hamiltonians simplify considerably the solution of various travel-time perturbation problems of seismic body waves propagating in heterogeneous, isotropic or anisotropic, weakly dissipative media. In addition to the phase-space coordinates, representing the Cartesian coordinates and Cartesian components of the slowness vector, the perturbation Hamiltonians depend also on one or more perturbation parameters, which can be chosen in different ways. General travel-time perturbation procedures, based on the perturbation Hamiltonians, were proposed by Klimes (2002). Evaluation of travel-time perturbations of any order does not require the computation of perturbed rays in these procedures, and consists fully in the quadratures along a reference ray, computed in the reference medium. Different forms of perturbation Hamiltonian yield different accuracy and numerical efficiency of computations. In this paper, the special case of perturbation Hamiltonian, called here the difference perturbation Hamiltonian, is used to compute the first- and second-order perturbations of complex-valued travel time and the first-order perturbation of the complex-valued travel-time gradient. Heterogeneous perfectly elastic, isotropic or anisotropic, reference media are considered, in which reference rays can be calculated without problems by well-known methods. The derived perturbed equations are valid very generally. For isotropic reference model, the perturbed medium may be isotropic perfectly elastic, isotropic weakly dissipative, weakly anisotropic perfectly elastic, or weakly anisotropic weakly dissipative. Similarly, for reference anisotropic model, the perturbed medium may be anisotropic perfectly elastic, or anisotropic weakly dissipative. For perturbed weakly dissipative anisotropic media, also expressions for the attenuation vector, the global absorption factor, and the direction-dependent scalar local quality factor are derived. It is shown that the dissipative perturbation influences also the real-valued travel times. Simple expressions are given for such travel-time perturbation corrections, caused fully by dissipation.


Seismic anisotropy, seismic attenuation, wave propagation, theoretical seismology.

Whole paper

The image of the paper in GIF 150dpi (1 160 kB !) is designed for an instant screen preview.

The paper is available in PostScript (263 kB) and GZIPped PostScript (97 kB).

In: Seismic Waves in Complex 3-D Structures, Report 18, pp. 223-244, Dep. Geophys., Charles Univ., Prague, 2008.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .