Green’s functions for inhomogeneous weakly anisotropic media

Abstract

Summary Formulae for the zeroth-order principal term plus the first-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 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’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 simplified 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. The zeroth-order QI approximation removes the well-known problems of the standard ray method for anisotropic media and gives regular solutions for regions in which the differences between the phase velocities of qS waves in the direction of propagation are small. This is the case for weakly anisotropic media as well as for singular regions of qS waves such as in the vicinities of kiss and intersection singularities, for example. In such situations, frequency-dependent amplitudes of the qS waves in the zeroth-order QI approximation are obtained by a νmerical 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 the 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 zeroth-order QI approximation are regular everywhere except for 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 vectorial framework along rays in the background medium that is different in the QI approximation from that used 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 quality to or better quality than those of the CRT. The equivalence of the zeroth-order QI approximation to the CRT promises acceptable results of the QI approximation not only in weakly anisotropic media but also in singular regions of qS waves.