Deregowski and Rocca (1981) define migration to zero-offset (DMO+NMO) as the sequence of prestack migration followed by zero-offset modeling. Following this definition an impulse in a prestack constant-offset section is transformed into a depth model that represents all the possible reflector locations (an ellipse in constant velocity). By modeling the reflector back to zero-offset one obtains a curve that represents the impulse response of the MZO operator. The impulse response curve can be applied to constant-offset data as a Kirchhoff integral operator. The authors also define the amplitude along the DMO operator as being proportional to the curvature of the operator. Black et al. (1993) address the problem of the ``true amplitude operator'' by requiring the amplitude along the DMO operator to preserve the reflectivity of a dipping reflector. The DMO amplitudes found in time-space domain coincide with Zhang's (1988) modification of the original Hale Fourier domain DMO.

Hale (1983) has a ray-theoretical definition of DMO. However in the third chapter of his thesis he tries to reconcile the kinematic definition with a wave-equation approach. He starts with the prestack migration equation and isolates fours processing steps: NMO, DMO, stacking and zero-offset migration. The problem can be simplified by dissecting prestack migration into only two processing steps: migration to zero-offset and zero-offset migration. This simplification avoids the subtle flaw that Hale has in his derivation, which leads to an inaccurate analytical expression for DMO.

By including the NMO correction in the Fourier domain Zhang DMO, I obtain an MZO processing step which I compare with the MZO as obtained from the prestack migration dissection. The MZO obtained from prestack migration performs NMO, DMO and stacking in one step. I apply the two different MZO methods to three simple prestack synthetic models and compare the amplitude and phase of the output. The results suggest the two methods are not equivalent.

- First definition: MZO by Fourier transform
- Second definition: MZO extracted from prestack migration
- Integration limits
- Integration order
- Comparing the amplitudes of the methods

11/17/1997