Simplified dynamic and static Green's functions in transversely isotropic media

Abstract

SUMMARY Numerically feasible dynamic Green's function in an unbounded transversely isotropic (TI) medium is obtained in simple dyadic form by evaluating in general an inverse Laplacian operator involved in a previous dynamic Green's function described by Ben-Menahem & Sena (1990). the final dyadic form is close to that of the isotropic dyadic Green's function, therefore, lends itself more easily to analytical and numerical manipulations. It is expressed through three scalar quantities characterizing the propagation of SH, P-SV, and P-SV-SH waves in a transversely isotropic medium. the static Green's function has the same dyadic form as the dynamic Green's function and the three corresponding scalar functions are derived. Using the dynamic Green's function, displacements for three point sources are computed to compare with known numerical results. the singular property of the Green's functions is addressed through the surface integral of the static function in the case of coinciding receiver and source. the singular contribution is shown to be -1/2 of the applied force when the static-stress Green's function is integrated over a half-elliptical surface. Results of this paper are particularly suitable to wavepropagation problems involving the boundary-element method.