The concept of wave-equation datuming was first presented by Berryhill (1979) and applied to zero-offset (poststack) data. The method employed by Berryhill is based on an extrapolating scheme using the Kirchhoff integral solution to the scalar wave-equation. Berryhill (1984) generalizes his method to prestack sections by applying the same extrapolation algorithm to common-source gathers and then to common-receivers gathers. Wiggins (1984) presents the same concept of Kirchhoff integral extrapolation of prestack data but adds an imaging step to the algorithm. We can use the same algorithm to:
Yilmaz and Lucas (1986) demonstrate how datuming can be used twice to replace a layer of arbitrary shape and known velocity to another layer with a different velocity in order to eliminate raypath bending in the area with severe velocity contrast. They call the technique prestack layer replacement. The method uses wave-equation datuming twice; first to downward continue the wavefield from an initial surface to a datum, using a known velocity, and second to upward continue the wavefield from the datum to the initial surface using a different velocity. For prestack data the sources and the receivers are downward continued independently to the datum and subsequently upward continued.
Reshef (1991) presents a datuming and migration principle to be used in conjunction with phase-shift methods to directly migrate prestack data. He performs downward extrapolation from a flat datum and adds data to the extrapolated wavefield each time the topographic surface is intersected. This method allows direct prestack migration of data recorded on a nonflat topographic surface.
Ji and Claerbout (1992) also present an improved datuming and migration algorithm for depth varying v(z) velocity media. Bevc (1992) improves the datuming and migration algorithm via an antialiasing Kirchhoff method. Both papers use the conjugate transpose concept to examine the algorithm.
In this paper I present a method to perform zero-offset datuming and migration for any velocity model and any topographic surface. The algorithm is based on the phase-shift extrapolation of the wavefield. For laterally varying velocity I use a Phase-Shift Plus Interpolation (PSPI) and a Split-Step method. Datuming is performed in both directions, from an uneven topographic datum to a flat surface and from a flat surface up to a topographic datum. The two algorithms are conjugate transpose to each other. To complement the two datuming algorithms I use another pair of conjugate operators to perform migration and modeling via a PSPI or Split-Step algorithm. I show results on different velocity models for the two datuming algorithms and the accompanying migration and modeling algorithms.