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I have introduced a partial differential
equation () and proved that the process described
by it provides for a kinematically and dynamically equivalent offset
continuation transform. Kinematic equivalence means that in constant
velocity media the reflection traveltimes are transformed to their
true locations on different offsets. Dynamic equivalence means that,
in the OC process, the geometric spreading term in the amplitudes of
reflected waves transforms in accordance with the geometric seismics
laws, while the angle-dependent reflection coefficient stays the same.
The offset continuation equation can be applied directly to design OC
operators of the finite-difference type. To construct integral OC
operators, I have posed and solved an initial value problem for the
offset continuation equation (). For the special
cases of continuation to zero offset (DMO) and continuation from zero
offset (inverse DMO), the OC operators are related to the known forms
of DMO operators: Hale's Fourier DMO, Born DMO, and Liner's ``exact
log DMO.'' The discovery of these relations sheds additional light on
the problem of amplitude preservation in DMO.

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Stanford Exploration Project

12/28/2000