RECIPROCAL VOLUME AND SURFACE SCATTERING INTEGRALS FOR ANISOTROPIC ELASTIC MEDIA

Linear scattering theory in anisotropic media is useful for describing modelling, inversion and migration algorithms. The single-scattering or Born approximation leads to a volume scattering integral which is f urther simplified by using the geometrical ray approximation (GRA) for Green's functions from the source and receiver to the scattering poin t. Discontinuities of the medium parameters which are confined to smoo th surfaces will reflect and refract the propagating waves. This is of ten described by the Kirchhoff-Helmholtz integral, which uses Green's representation of the reflected field and the Kirchhoff approximation for the field and its normal derivative at the surface. The reflected field and its derivative are often approximated by multiplying the cor responding parts of the incoming field with the plane-wave reflection coefficient computed for the angle between the incoming ray and the su rface normal (Kirchhoff approximation). Besides the inconsistency of i mposing both the field and its normal derivative on the surface to rep resent the field away from it, the Kirchhoff-Helmholtz integral gives rise to a reflected response which is non-reciprocal. The Born and Kir chhoff-Helmholtz integrals have traditionally been treated as complete ly separate formulations in the studies of reflection and transmission of waves due to smooth interfaces. A simple use of the divergence the orem applied to the Born volume integral gives a reciprocal surface sc attering integral, which can be seen as a natural link between the two formulations. This unifying integral has been recently derived in the context of inversion. We call it the Born-Kirchhoff-Helmholtz (BKH) i ntegral. The properties of the BKH integral are investigated by a stat ionary-phase evaluation, and the result is interpreted in ray theoreti cal terms. For isotropic media, explicit expressions are given.