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Velocity transform is another form of hyperbolic stacking with the
summation path
| |
(81) |

where *h* corresponds to the offset, *s* is the stacking slowness, and
*t*_{0} is the estimated zero-offset traveltime. Hyperbolic stacking is
routinely applied for scanning velocity analysis in common-midpoint
stacking. Velocity transform inversion has proved to be a powerful
tool for data interpolation and amplitude-preserving multiple
supression Ji (1994a); Lumley et al. (1994); Thorson (1984).
Solving equation (81) for *t*_{0}, we find that the asymptotic
inverse and adjoint operators have the elliptic summation path

| |
(82) |

The weighting functions of the asymptotic pseudo-unitary velocity
transform are found using formulas (34) and (35) to have the
form
| |
(83) |

| (84) |

The factor for pseudo-unitary velocity transform
weighting has been discovered empirically by Claerbout
.

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Stanford Exploration Project

11/12/1997