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.