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# Introduction

Migration to zero-offset (MZO) transforms prestack data to zero-offset data. In constant velocity, MZO is equivalent to the processing sequence comprised of dip moveout correction (DMO), normal moveout correction (NMO) and stacking. The output of MZO processing is the input to the zero-offset migration (ZOM) algorithm.

Another definition of MZO given by Deregowski and Rocca (1981) is based on the decomposition of MZO into 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.

Hale (1983) tries to derive the ray-theoretical DMO from wave-equation principles by decomposing the prestack migration equation (double-square root in )in four major processing steps:

1.
DMO,
2.
NMO,
3.
Stacking,
4.
Zero-offset migration.
I define MZO as the sequence of steps 1,2 and 3 therefore I need to isolate only two steps:
1.
Migration to zero-offset (MZO).
2.
Zero-offset migration (ZOM).
Examining the MZO processing sequence, I observe that it is also the solution to a partial differential equation which can be used further to generalize MZO to a depth variable velocity.

I apply the variable velocity MZO to three synthetic models and obtain very good coincidence between the initial zero-offset section and the output of the zero-offset migration. The drawback of the method is that summing over the offset is performed as part of the MZO. Further work is needed to isolate the stacking sequence.

Next: MZO from prestack migration Up: Popovici : PDE for Previous: Popovici : PDE for
Stanford Exploration Project
11/17/1997