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Offset continuation is the operator that transforms seismic reflection
data from one offset to another
Bolondi et al. (1982); Salvador and Savelli (1982). If the data are continued
from half-offset *h*_{1} to a larger offset *h*_{2}, the summation path of
the post-NMO integral offset continuation has the following form
Biondi and Chemingui (1994); Fomel (1995b); Stovas and Fomel (1996):
| |
(85) |

where *U* = *h*_{1}^{2} + *h*_{2}^{2} - (*x* - *y*)^{2}, , and *x* and *y* are the midpoint coordinates before and
after the continuation. The summation path of the reverse continuation
is found from inverting (85) to be
| |
(86) |

The Jacobian of the time coordinate transformation in this case is simply
| |
(87) |

Differentiating summation paths (85) and (86), we
can define the product of the weighting functions according to formula
(9), as follows:
| |
(88) |

The weighting functions of the amplitude-preserving offset
continuation have the form^{}
| |
(89) |

| (90) |

It easy to verify that they satisfy relationship (88);
therefore, they appear to be asymptotically inverse to each other.
The weighting functions of the asymptotic pseudo-unitary offset
continuation are defined from formulas (34) and (35), as follows:

| |
(91) |

| (92) |

The most important case of offset continuation is the continuation
to zero offset. This type of continuation is known as *dip moveout
(DMO)*. Setting the initial offset *h*_{1} equal to zero in the general
offset continuation formulas, we deduce that the inverse and forward
DMO operators have the summation paths

| |
(93) |

| (94) |

The weighting functions of the amplitude-preserving inverse and
forward DMO are
| |
(95) |

| (96) |

and the weighting functions of the asymptotic pseudo-unitary DMO are
| |
(97) |

| (98) |

Formulas similar to (95) and (96) have been published
by Fomel 1995b and Stovas and Fomel
1996. Formula (96) differs from the
similar result of Black et al. 1993 by a simple time
multiplication factor. This difference corresponds to the difference
in definition of the amplitude preservation criterion. Formula
(96) agrees asymptotically with the frequency-domain Born DMO
operators Bleistein and Cohen (1995); Bleistein (1990). Likewise, the stacking operator with the
weighting function (95) corresponds to Ronen's inverse DMO
Ronen (1987), as I discussed in an earlier report
Fomel (1995b). Its adjoint, which has the weighting function
| |
(99) |

corresponds to Hale's DMO Hale (1984).

** Next:** NUMERIC TEST
** Up:** EXAMPLES
** Previous:** Velocity Transform
Stanford Exploration Project

11/12/1997