A visual relationship between Kirchhoff migration and seismic inversion
John C. Bancroft
The exploding reflector model and the finite-difference methods automatically take care of the amplitudes when using the downward continuation method. Similarly, the FK method of migration applies a defined amplitude scaling when moving the data in the FK space. Estimation of the diffraction stack amplitudes proved more of a challenge until the Kirchhoff integral solution to the wave equation provided a theoretical foundation.
Assumptions used in the design of geological models are reviewed in preparation for evaluating the design of migration programs that are derived from the wave-equation. A review of Kirchhoff migration is then presented that begins as a diffraction stack process, and then proceeds to matched-filtering concepts and the integral solution to the wave-equation. One-dimensional (1D) convolution modelling and deconvolution are then used to introduce inversion concepts that lead to "transpose" processes and matched filtering. These concepts are then expanded for two-dimensional (2D) data to illustrate that Kirchhoff migration is a "transpose" process or matched filter that approximates seismic inversion.