Jakubowicz 1990 gave an elegant formulation of (f,k) dip moveout where a finite number of dips are processed separately and stacked according to equation (1),
In the general case of depth-variable velocity, there is no analytic expression of the amplitude along the operator but the dip-decomposition method gives us a qualitative idea. Indeed, the operator of variable-velocity dip moveout may be build point by point for a range of dips regularly sampled. Then, a region where the points are close to each other allows a significant stack of the dipping segments of the operator, and thus corresponds to an area of high amplitude. Figure shows an impulse response of dip moveout in a time-variable velocity model whose profile is represented in Figure .
Figure 1 Interval velocity model.
Artley's 1992 v(z) dip moveout method which produced Figure , requires a non-linear inversion of a system of equations for each time, offset, and dip we consider. Though it is a time consuming process, it provides us with amplitude information. First, the triplication of the impulse response have a low energy for this velocity model (a troff in the velocity profile would show a high amplitude triplication). Secondly, the amplitude varies a lot along the operator, starting high at gentle dips, decreasing, and increasing again just before a triplication.
The shape of a v(z) DMO operator differs from the constant-velocity DMO operator essentially by a number of triplications along its branches (Figure ). However, since these triplications are generally low-amplitude, the shape is not the most discriminative feature of the operator. The amplitude variations along the operator strongly depend on the velocity model, and therefore play the important role in the dip moveout correction.