Wide-azimuth angle gathers for wave-equation migration

Discussion

We do not suggest in this paper that the wavefields used for imaging are planar prior to the interaction with the reflector. In complex geology, such an assumption would be unrealistic. However, a wavefield of arbitrary shape can be thought of as a superposition of plane waves propagating in various directions, either because the wavefronts characterizing the wavefields have curvature, or because the wavefields have triplicated during propagation. Each incident plane has a corresponding reflected plane related through Snell's law. Some angle decomposition techniques make use explicitly of a planar decomposition of the wavefields, followed of selection through thresholding of the most energetic plane (Xu et al., 2010). In contrast, we rely on the fact that all planar components of the wavefields have been transformed as planar events in the extended images and rely on slant-stacks or equivalent methods to separate them as a function of azimuth and reflection angles.

As indicated in the preceding sections, we do not need to compute all space-lags at the considered CIP positions. We could compute just two of them, e.g. ${\lambda_x}$ and ${\lambda_y}$ as shown in the examples of this paper, and then reconstruct the third lag using the information given by the reflector normal at the CIP position, equation 18. If the reflector is nearly vertical, it may be more relevant to compute the vertical and one horizontal space-lags. Alternatively, we could avoid computing the reflector normal vector from the conventional image, but instead compute all three components of the space-lag vector ${\boldsymbol {\lambda }}$ . In this case, as indicated by Sava and Vasconcelos (2011), we could estimate the reflector dip from the lag information prior to the angle decomposition.

We have also noted earlier in the paper that the relevant space-lags are constructed in the reflector plane. This fact is a direct consequence of the fact that we have considered equal but with opposite sign time-shift of the source and receiver wavefields. Without this convention, the angle decomposition problem becomes more complex. In our experience to date, we did not find the need to relax this requirement.

The angle-domain CIPs accurately indicate the sampling of the reflector as a function of azimuth and reflection angles. If the shot distribution is sparse, or if the sub-surface geology creates shadow zones, the illumination is also sparse. This is both beneficial, assuming that the angle-domain CIPs are used to evaluate illumination, but it can also be a drawback if the angle-domain CIPs are used for AVA or MVA. However, a sparse sampling of a reflector is not a feature of the angle decomposition, but a feature of the acquisition geometry. Neither our, nor any other angle decomposition, can compensate for the lack of adequate data illuminating the subsurface on a dense angular grid.

Finally, we note that the most likely applications for angle decomposition in complex geology is the study of the reflector illumination itself. Assuming that the sampling is sufficiently dense and that the imaging velocity is accurately known, then we can use the angle decomposition discussed in this paper to evaluate amplitude variation with azimuth and reflection angles. However, we emphasize that this is a relevant exercise only if the reflector illumination is sufficiently dense. Otherwise, AVA effects overlap with illumination effects, rendering the analysis unreliable. Migration velocity analysis in the angle domain may also suffer from the lack of adequate illumination. This partial illumination may deteriorate the moveout which would otherwise be observed in the extended image domain. Furthermore, we do not advocate an implementation of MVA in the angle-domain, but rather in the extended image domain which contains all the relevant information and avoids the additional step of angle decomposition. An extensive discussion of this problem is outside the scope of our paper.

Wide-azimuth angle gathers for wave-equation migration