Acoustic and elastodynamic 3D Green’s functions for isotropic media with a weak velocity gradient

Abstract

Approximate analytical formulae for complete acoustic and elastodynamic 3D Green’s functions in isotropic media with a weak and constant velocity gradient are presented. The formulae were derived by analytical calculation of higher-order approximations of the ray series. The ray series of the acoustic Green’s function consists of only one non-zero term, the ray series of the elastodynamic Green’s function consists of three non-zero terms including the zeroth-order term. Since the ray series is finite for both the Green’s functions, the formulae are complete and valid in the whole frequency range. The formulae are approximate because we assumed a weak velocity gradient and used the first-order perturbation theory. Moreover, the formulae are valid only in a limited region around a point source. A wavefield generated by an explosive point source, and the elastostatic Green’s function have also been derived.