Formulae for the zero-order principal term plus the first-order additional
term 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 to seek
the asymptotic solution of the elastodynamic equation as an expansion with
respect to two small parameters of the same order: the small parameter used
in the standard ray method and a parameter characterizing differences of
tensors of elastic parameters of a weakly anisotropic medium and of a
nearby "background" isotropic medium. As a result, the procedure of
constructing the Green functions splits into 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.

Application of the QI approximation to *qP* wave propagation leads to
useful simplified formulae found earlier by the application of the
perturbation methods. Application of the QI approximation to the *qS*
wave propagation leads to formulae of basic importance. The zero-order
QI approximation removes the well-known problems of the standard ray
method for anisotropic media and gives regular solutions in regions, in
which the difference between the phase velocities of *qS* waves in the
direction of propagation 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 a medium is strongly anisotropic and/or
high frequencies are considered, approximate closed-form solutions of
the two coupled differential equations have a form of the ray solutions
describing two decoupled *qS* waves. The standard ray method for anisotropic
media can substitute the QI approximation in such regions. On the other
hand, in the limit of infinitely weak anisotropy, the formulae for the QI
approximation smoothly converge to formulae for isotropic media. Thus the
QI approximation represents a link between ray formulae for anisotropic and
isotropic media. The formulae for the zero-order QI approximation are
regular everywhere except 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.

The two coupled differential 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
problems of the CRT approach. The QI approximation including the
first-order additional terms is expected to yield results of comparable
or better quality than the CRT. The equivalence of the zero-order QI
approximation with the CRT promises acceptable results of the QI
approximation not only in weakly anisotropic media but also in singular
regions of *qS* waves.

Geophys.J.Int.,

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