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DMO is a method of transformation of finite-offset data
to zero-offset data. Let the normal moveout corrected input data be
denoted and the zero-offset desired
output denoted . Assume known relationships between
the coordinates of the general form
| |
(14) |

The DMO operator can be
defined in the zero-offset frequency and midpoint
wavenumber as Liner (1988)
| |
(15) |

| (16) |

whereas its inverse can be defined as
| |
(17) |

where
| |
(18) |

A detailed derivation of *J*_{2} is given by Liner 1988.
The method
is based on a general formalism Beylkin (1985); Cohen and Hagin (1985)
for inverting integral equations
such as dmo.eq.
It involves inserting dmo.eq into dmoinv.eq and expanding
the resulting amplitude and phase as a Taylor series and making a
change of variables according to Beylkin 1985. The solution
provides an asymptotic inverse for dmo.eq, where the weights are given by

| |
(19) |

In this expression, is the Jacobian of the change of variables in the forward DMO given by
| |
(20) |

which reduces to ,
assuming the general coordinate relationships coord.relat where
is independent of *t*_{2}, leading to a zero lower left element in
the determinant matrix above.
The quantity
is the inverse of the Beylkin determinant, *H*, and is given by

| |
(21) |

If we recognize that is independent of , then the lower
element of *H*^{-1} is zero and Beylkin_inv reduces to
| |
(22) |

where and are, respectively,
| |
(23) |

| |
(24) |

Notice that and depend on the coordinate
relationships coord.relat. Therefore, the Beylkin determinant, *H*,
varies according to the DMO operator but is constant
for kinematically equivalent operators.

** Next:** Hale DMO and its
** Up:** Amplitude-preserving AMO
** Previous:** Chaining DMO and inverse
Stanford Exploration Project

1/18/2001