SOME AMPLITUDE ISSUES

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),  

$$P_{\rm DMO} = \sum_i \Delta D_i \ P_{D_i},$$ (1)

where P_Di is the wavefield, NMO-corrected for a single dip D_i, and $\Delta D_i$ is the width of the dip-range surrounding D_i. Obviously, if the dip sampling is irregular, the dip-decomposed wavefields stack with a different weight. Beasley 1988 derived an amplitude scheme for Kirchhoff dip moveout from Jakubowicz's weighting by using the following practical approximation:  

$$\Delta D_i \approx \frac{\partial^2 t_0}{\partial x^2} \Delta x.$$ (2)

To the first order, the amplitude along the dip moveout operator is proportional to its time curvature. This approximation is close to the amplitude scheme mathematically derived by Black et al. 1993 from (f,k) dip moveout Blondel (1993). Thus, Jacubowicz's dip moveout by dip-decomposition, Black's Kirchhoff dip moveout, and (f,k) dip moveout have a very similar amplitude distribution.

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 .

 

vel Figure 1 Interval velocity model.

 

imprs
Figure 2 Impulse response of v(z) 2-D DMO for an offset of .8 km and a total two-way traveltime of 1. s. The stars along the operator correspond to a uniform sampling of dips. The denser are the points, the higher is the amplitude of the operator. The dashed line represents the elliptic support of the constant-velocity DMO impulse response for the same offset and traveltime. Both the solid and dashed lines have a similar shape in the region of high amplitudes, differing only by the triplications along the solid line. The unrealistic points beyond the offset show that there is a problem of convergence of the v(z) DMO algorithm for steep dips.

Artley's 1992 v(z) dip moveout method which produced Figure , requires a non-linear inversion of a $4 \times 4$ 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.