Introduction

Seismic data acquired in areas with irregular topography introduce a numerical problem for migration algorithms that are based on depth extrapolation. Since numerically efficient migration schemes are usually based on depth extrapolation algorithms which allow the extrapolation of a wavefield from a flat surface to another flat surface, datuming is required prior to migration. Datuming is a method of processing to extrapolate a known wavefield at a specified datum of arbitrary shape to another specified datum, also of arbitrary shape. For small differences in elevation and slow velocity variations between the input datum and the output datum, static shifting is a sufficiently accurate datum correction procedure. However, for significant differences in elevations and a more complicated velocity model, the accuracy of the static solution may prove to be insufficient; and a more exact method should be used.

Berryhill presented a wave equation datuming scheme for poststack and prestack data using the Kirchhoff integral method. Following his work, various forms of the Kirchhoff integral solution to the wave equation have been used by different authors for migration Shtivelman and Canning (1988); Wiggins (1984) and layer replacement Berryhill (1986); Yilmaz and Lucas (1986). The Kirchhoff approach is very expensive and cannot be used for a variable velocity medium.

An elegant and simple technique to correct for the error caused by the static time shift was introduced by the "zero-velocity layer" concept Beasley and Lynn (1989). Not only is the static shift required before the migration, but this technique cannot even be applied to the computationally attractive phase-shift algorithms Gazdag (1978), because it includes the nonphysical characteristic of zero velocity.

Recently Reshef 1991 presented a depth migration algorithm from irregular surfaces with depth extrapolation methods. His technique can be applied to prestack depth migration algorithms, with no limitations on surface irregularity, complexity of the subsurface structure, or velocity function.

In a previous SEP report (SEP-75), Ji and Claerbout presented a datuming scheme that uses the phase shift algorithm. Even though the algorithm passed the dot-product test, the computation time for the datuming is much longer than that for the forward modeling. This tells us that the algorithm might not be the optimal conjugate. This paper examines the algorithms again and provide a clearer algebraic representation of the datuming using, various depth extrapolation schemes such as the phase-shiftGazdag (1978), split-step Stoffa et al. (1990), and finite-difference methods Claerbout (1984). It turns out that the datuming algorithm developed in this paper are the same as the extrapolation algorithm used by Reshef 1991 in depth migration.