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# Integral inverse DMO

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 J2 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 t2, 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