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:

- 1.
- Downward extrapolate data from a datum to a flat surface.
- 2.
- Continue to downward extrapolate and image for migration.

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.

11/17/1997