Wave-equation datuming is the process of upward or downward continuing a wavefield between two arbitrarily-shaped surfaces. It is the most precise means of redatuming because it accurately implements solutions to the one-way scalar wave equation to extrapolate seismic data from the true recording surface to some arbitrary output surface. Wave-equation datuming attempts to compute the data that would have been recorded at the output surface.
The concept of wave-equation datuming was first presented by Berryhill (1979) and applied to zero-offset (poststack) data. Berryhill's method is based on an extrapolation 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-receiver gathers. Wiggins (1984) presents the same concept of Kirchhoff integral extrapolation of prestack data but adds an imaging step to the algorithm. With minor modification, migration algorithms based on finite-difference (Claerbout, 1985) and phase-shift (Gazdag, 1978) methods can be recast as datuming operators. Reshef (1991) presents a datuming and migration principle used in conjunction with phase-shift methods to migrate prestack data directly. He performs downward extrapolation from a flat datum and adds data to the extrapolated wavefield each time the topographic surface is intersected. Reshef's method allows direct prestack migration of data recorded on a nonflat topographic surface.
In this chapter, I formulate Kirchhoff, phase-shift, and finite-difference datuming operators using the the principle of adjoint operators. The forward problem is defined as upward continuation and the adjoint process as downward continuation. To point out the similarities and differences of the implementations, I unify the datuming concepts by presenting a matrix formulation where the forward problem is defined as a one-way wave-equation extrapolation from an irregular surface to a flat surface. This formulation is derived in a general context so that any depth extrapolation technique can be used. The adjoint problem, obtained by transposing and conjugating the forward problem, transforms the data from the level surface to the irregular topographic surface.