THE KIRCHHOFF–HELMHOLTZ INTEGRAL PAIR

The Kirchhoff–Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff–Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff–Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff–Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff–Helmholtz integral.