## Quasi-isotropic approximation of the qS wave Green
function in inhomogeneous weakly anisotropic media

**Ivan Psencik**
### Abstract

Formulae for the leading vectorial term of the *qS* wave Green function
in an unbounded inhomogeneous weakly anisotropic medium obtained by using
the so-called quasi-isotropic (QI) approximation are presented. 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 *omega*^{-1} for *omega --> infinity*. Under this
assumption, the procedure of constructing the Green function 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 and of the polarization vectors due to the
deviation of the weakly anisotropic medium from the isotropic background
at the termination points of rays.

The QI approximation 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 *qS* wave singular regions such as
vicinities of, for example, kiss and intersection singularities. The
formulae for the leading vectorial term of the *qS* wave Green function
in the QI approximation are thus regular everywhere with exception of
singular regions of the ray method for isotropic media. The formula for
the Green function consists of two independent expressions corresponding
to *qS1* and *qS2* waves, which are given by simple closed-form formulae.
The *qS* waves are thus decoupled in the QI approximation. The formulae
are applicable to weakly anisotropic as well as isotropic media. Their
results smoothly converge to results for isotropic media in the limit of
infinitely weak anisotropy.

Exp. Abstracts of the 5th SBGf, Vol.1, Sao Paulo, 259-262, 1997.

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