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Under the single-scattering (first-order Born) approximation the
scattered field at *r* due to a shot at *s* is given by
| |
(76) |

where *G*(*r*|*s*) is the Green's function response at *r* of the medium
to an impulse at *s*, and represents perturbation in
medium parameters.
Equation () describes the process of calculating the
volume integral in equation () first by downward
continuing the shot wavefield to give *G*(*r*'|*s*), then by upward
continuing the scattered field to the surface, accumulating
contributions from each depth slice on the way back up. The term can be applied as a data-space filter after modeling.
However, for the amplitudes to be correct in
equation (), the extrapolation operators need to
represent true-amplitude Green's functions.

For *v*(*z*) media, the WKBJ Green's function for a smoothly-varying
background velocity field [e.g. Aki and Richards (1980); Stolt and Benson (1986)] is
given by

| |
(77) |

| |
(78) |

The amplitude term in equation () can be
incorporated into the shot-profile modeling procedure;
however, different operators are required for upward versus downward
continuation. Firstly, the true-amplitude downgoing WKBJ
extrapolators should be substituted into
equation (),
and secondly, the upgoing WKBJ extrapolators should be substituted for
the adjoint of in equation ().

** Next:** Towards true-amplitude migration
** Up:** Shot-profile migration and modeling
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Stanford Exploration Project

5/27/2001