Green functions for inhomogeneous weakly anisotropic media

Ivan Psencik


Formulae for the zero-order principal term plus the first-order additional term (0+1A) of the qP and qS wave Green functions in the so-called quasi-isotropic (QI) approximation are derived for an unbounded inhomogeneous weakly anisotropic medium. The basic idea of this approximation is the assumption that the deviation of the tensor of elastic parameters of a weakly anisotropic medium from the tensor of elastic parameters of a nearby "background" isotropic medium is of the order ω-1 for ω-->infinity. Under this assumption, the procedure of constructing the Green functions consists of two steps: (i) calculation of rays, travel times, the geometrical spreading and polarization vectors in the background isotropic medium; (ii) calculation of corrections of travel times, amplitudes and polarization vectors due to the deviation of the weakly anisotropic medium from the isotropic background.

The zero-order QI approximation is especially important for study of qS waves. It removes the well-known problems of the standard ray method for anisotropic media in regions, in which the difference between the phase velocities of qS waves is small. This is the case of weakly anisotropic media as well as of singular regions of qS waves such as vicinities of, for example, kiss and intersection singularities. In such situations, frequency-dependent amplitudes of the qS waves in the zero-order QI approximation are obtained by a numerical solution of two coupled first-order ordinary differential equations along a ray in the background isotropic medium. When variations of phase velocities along a ray are considerably smaller than their differences (the medium is weakly inhomogeneous or homogeneous), approximate closed-form solutions of the two differential equations can be found. The results of the QI approximation describe then two decoupled qS waves, which can be studied separately by the standard ray method for anisotropic media. In the limit of infinitely weak anisotropy, the formulae for the QI approximation smoothly converge to formulae for isotropic media. The formulae for the zero-order QI approximation are regular everywhere with exception of singular regions of the ray method for isotropic media. The accuracy of the QI approximation can be increased by considering the first-order additional terms of the QI approximation.

It is shown that the two coupled equations are equivalent to the equations of the coupling ray theory (CRT) based on a simplification of a coupling volume integral. Use of a different vectorial framework along rays in the background medium in the QI approximation than in the CRT avoids some serious problems of the CRT approach. The QI approximation with the first-order additional terms is expected to yield results of comparable or better quality than the CRT.


Anisotropy, body waves, elastodynamics, Green's function, inhomogeneous media, mode coupling, polarization, ray theory, seismic-wave propagation, shear-wave splitting.

Whole paper

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Revised version

Psencik, I.: Green functions for inhomogeneous weakly anisotropic media. Geophys.J.Int., 135 (1998), 279-288.

In: Seismic Waves in Complex 3-D Structures, Report 6, pp. 263-282, Dep. Geophys., Charles Univ., Prague, 1997.
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