The Kirchhoff-Helmholtz integral for anisotropic elastic media

The Kirchhoff-Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth inter-face in acoustic or isotropic elastic media d ue to a given incident wavefield and observation points sufficiently far aw ay from the interface. This integral makes use of the Kirchhoff approximati on of the unknown scattered wavefield and its normal derivative at the inte rface in terms of the Corresponding quantities of the known incident field. An attractive property of the Kirchhoff-Helmholtz integral is that its asy mptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. H ere, we extend the Kirchhoff-Helmholtz modeling integral to general anisotr opic elastic media. It uses the natural extension of the Kirchhoff approxim ation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff-Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff-He lmholtz integral, we also derive an extension to anisotropic media of a use ful decomposition formula of the geometrical spreading of a primary reflect ion ray. (C) 2001 Elsevier Science B.V. All rights reserved.