Berryhill 1979 gives the following expression for the upward continuation of a scalar wavefield:

(1) |

- The wavefield is two-dimensional and does not vary perpendicular to the seismic line.
- The data are collected on a planar surface.

For a wavefield transformation from one datum *U*(*x*,*z*=*z _{1}* ,

(2) |

datumfig2
Geometry for the continuation of a wavefield between an irregular
topography and a planar datum.
Figure 1 |

Equations (1) and (2) are used for upward continuation. For downward continuation the conjugate process is used. The sum is performed first and then the output traces are crosscorrelated with the filter described in the Appendix.

The program for upward and downward continuation is illustrated by the following pseudo-code:

loop over j { loop over i { shift = time shift between input and output trace if ( conj != 0 ) Q = down [convolve] filter loop over time { if ( conj == 0 ) Q(time-shift,i) = Q(time-shift,i) + up(time,j) else up(time,j) = up(time,j) + Q(time-shift,i) } if ( conj == 0 ) down = up [crosscorrelate] filter }}

The notation in the pseudo-code is consistent with
Figure 1. The data on the upper datum *z*=*z _{2}* corresponds
to index

The actual code used to perform the datuming in this paper is complicated by calculation of geometrical parameters which arise due to non-planar datums and by the incorporation of anti-aliasing. Since the datuming algorithm is based on an integral equation solution it is susceptible to operator aliasing. This is taken care of by calculating triangular smoothing functions along the operator trajectory in a manner analogous to Claerbout 1992.

11/17/1997