Tutorial on the basics of downward extrapolation operators

John C. Bancroft and Saleh M. Al-Saleh

ABSTRACT

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 (k x , ω) space can be applied in (x, ω) space as a convolution operator to enable a spatially varying velocity. Truncating the spatial operator to an economical size in (x, ω) causes operator instability. This instability occurs when the truncated operator is applied recursively, and any amplitudes in (k x , ω) 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.

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