Tutorial on the basics of downward extrapolation operators

John C. Bancroft, Saleh M. Al-Saleh

There are many algorithms for downward extrapolating a seismic time section for migration. This tutorial demonstrates the downward extrapolation process, and focuses on the phase-shift extrapolator. A phase-shift extrapolator that is defined in (kx, ) space can be applied in (x, ) space as a convolution operator to enable a spatially varying velocity. Truncating the spatial operator to an economical size in (x, ) causes operator instability. This instability occurs when the truncated operator is applied recursively, and any amplitudes in (kx, ) space that exceed unity will tend to a very large number.

These concepts are illustrated with numerous diagrams, and a simple descriptive operator is presented to illustrate the effect of operator size and the amplitude of the sinc ringing.