Quasi-isotropic Approximation of the qS Wave Green Function in Inhomogeneous Weakly Anisotropic Media

Formulae for the leading vectorial term of the qS wave Green function in an unbounded inhomogeneous weakly anisotropic medium obtained 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 approximately of the order (I)-I for (I) -t 00. 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, amplitudes and of the polarization due to the deviation of the weakly anisotropic medium from the isotropic background at 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 except singular regions of the ray method for isotropic media. The formula for the Green function consists of two expressions corresponding to qSl and qS2 waves. When differences between the phase velocities of the two qS waves vary relatively strongly (for example in a vicinity of qS wave singularities), the frequency-dependent amplitudes in the zero-order QI approximation are obtained by a numerical solution of two coupled first-order differential equations along a ray in the background isotropic medium. When the differences of the phase velocities do not vary (the medium is homogeneous) or vary only modestly, the closed-form solutions of the two coupled equations can be found. The qS waves in the latter case are thus decoupled. In the limit of infinitely weak anisotropy, the presented formulae smoothly converge to formulae for isotropic media. For stronger anisotropy, the formulae converge to the formulae for the standard ray method for anisotropic media.