Elastodynamic green's functions for a smoothly heterogeneous half-space

In this paper, the response of a vertically heterogeneous elastic half-space with a smooth modulus variation under a set of time-harmonic ring- and point-sources is derived analytically. A method of evaluation via asymptotic decomposition for the singular Green's functions is presented. In the technique, the Green's functions are decomposed into an analytical part and a residual component. Capturing the corresponding singular behavior, the analytical parts of the ring- and point-load Green's functions are expressible in terms of the elliptic integrals and algebraic functions, respectively. The residual integrals which are regular can be evaluated by numerical contour integration. To obtain correct results, one must note and take into account the existence of multiple poles along the formal path of the inversion integrals, the details of which are discussed in the paper. To highlight the various aspects of the physical problem, a set of illustrative numerical results is included.