Theory of differential offset continuation

Conclusions

I have introduced a partial differential equation (1) 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 laws of geometric seismics, 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, an initial value problem is solved for the offset continuation equation (1). 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.

Theory of differential offset continuation