Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media

Vlastislav Cerveny & Ivan Psencik


Perturbation Hamiltonians simplify considerably the solution of various traveltime 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 also depend on one or more dimensionless perturbation parameters, which can be chosen in different ways. General traveltime perturbation procedures, based on the perturbation Hamiltonians, were proposed by Klimes. The evaluation of the perturbation expansions for the traveltime and for its derivatives does not require the computation of perturbed rays in these procedures. It consists in dynamic ray tracing along the reference ray, in the determination of the partial derivatives of the perturbation Hamiltonian with respect to the perturbation parameters, and in additional quadratures along a reference ray, computed in the reference medium. Different forms of the perturbation Hamiltonian yield different perturbation expansions with different accuracy and numerical efficiency of computations. In this paper, the special case of the perturbation Hamiltonian, herein referred to as the linear perturbation Hamiltonian, is used to compute the second-order perturbation expansion of the complex-valued traveltime and the first-order perturbation expansion of the complex-valued traveltime gradient. The advantage of the linear perturbation Hamiltonian is that it is linear with respect to perturbation parameters. 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 perturbation expansions are valid very generally. For the isotropic reference model, the perturbed medium may be isotropic perfectly elastic, isotropic weakly dissipative, weakly anisotropic perfectly elastic or weakly anisotropic weakly dissipative. For the anisotropic reference model, the perturbed medium may be anisotropic perfectly elastic, or anisotropic weakly dissipative. Expressions for the attenuation vector, the dissipation filter, the global absorption factor, the direction-dependent scalar local quality factor and the attenuation coefficient are also derived for seismic body waves propagating in heterogeneous anisotropic weakly dissipative media.


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

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Geophys. J. int., 178 (2009), 939-949.
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