I compute the impulse response of MZO by considering the process as the combination of full prestack migration followed by zero-offset modeling (Deregowski and Rocca, 1981; Deregowski, 1985). I use a fast algorithm to compute the travel-time map necessary for modeling wave propagation in a general 2-D medium (Van Trier and Symes, 1990). I applied the proposed algorithm for computing the kinematics of the impulse response when velocity is constant and for different velocity functions of depth. The impulse response in varying velocity can differ substantially from the constant velocity one (Popovici, 1990). I use the impulse response in a Kirchoff type algorithm applied to a series of synthetic models. As a result there is a better alignment of prestack data migrated to zero-offset than in the case of conventional algorithms, which imply better stacking for a range of different offsets.

The proposed algorithm for computing the impulse response of MZO using finite-difference travel-time maps is based on the principle that MZO is the combination of two processes: full prestack migration and zero-offset modeling. In a constant velocity medium this definition allows for an analytical formulation of the MZO operator which is identical with the DMO after NMO formulation (Popovici and Biondi, 1989). The algorithm used to investigate the MZO operator in variable velocity media follows the definition of the MZO method and can be divided in two parts:

- Construct the full prestack migration depth model. The depth model is the position in space of all points that can generate a given impulse in a constant-offset section. For a constant velocity medium this is equivalent to constructing the migration ellipse for a constant-offset section. For a variable velocity medium, the loci of points with equal travel-time from source to receiver form a curve resembling an ellipse or a superposition of several ellipses.
- Zero-offset modeling. Given the depth model, raytrace back at 90 degrees from the reflector, to model the zero-offset data. The intersection of the ray with the surface will give the x-coordinate of the MZO operator, while the travel-time along the raypath will provide the zero-offset time-coordinate.

- The two conjugate MZO operators
- MZO and DMO in variable velocity media
- How to get the amplitudes along the MZO operator

12/18/1997