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In general, residual migration improves the quality of an image
without re-migration of the original data; instead, a transformation
is applied to the current migrated image.
In prestack Stolt residual migration, we attempt to correct
the effects of migrating with an inaccurate reference velocity by
applying a transformation to images transformed to the
Fourier domain.
Supposing that the initial
migration was done with the velocity *v*_{0}, and that the correct
velocity is *v*, we can use dsr-sg
to derive , the vertical wavenumber for the reference
velocity, and , the vertical wavenumber for the correct velocity.

Mathematically, the goal of prestack Stolt residual migration
is to obtain from .
If we elliminate from and ,and make the notation ,we obtain the residual migration equation for full 3D
prestack seismic images:

| |
(33) |

which can also be represented in midpoint-offset coordinates using the
change of variables in changevar.
If we make the change of variables
| |
(34) |

we obtain a simplified version of myresmig-3d-pr:
| |
(35) |

In the 3D post-stack case, when ,
myresmig-3d-pr becomes:

| |
(36) |

In the 2D prestack case (*k*_{m}_{y}=0 and *k*_{m}_{x}=0),
we can write myresmig-3d-pr as:

| |
(37) |

For 2D post-stack data, myresmig-3d-po and myresmig-2d-pr become

| |
(38) |

which can also be written in the familiar form (102):
| |
(39) |

where, by definition, , and .

** Next:** Common-azimuth Stolt residual migration
** Up:** Prestack residual migration
** Previous:** Stolt migration
Stanford Exploration Project

11/4/2004