Green’s functions for inhomogeneous weakly anisotropic media

Formulae for the zeroth-order principal term plus the ¢rst-order additional term of the qP- and qS-wave Green’s 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 di¡erences 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’s functions is split into two steps: (1) calculation of rays, traveltimes, the geometrical spreading and polarization vectors in the background isotropic medium; (2) calculation of corrections of traveltimes, 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 simpli-¢ed formulae found earlier by the application of the perturbation methods. Application of the QI approximation to qS-wave propagation leads to formulae of basic importance