Discussion

We now present a few issues that we think are important for our Riemannian wavefield extrapolation method. In some cases, we discuss ideas which we have not treated yet, while in other cases we speculate on directions of future research.

3-D Riemannian extrapolation:

All our examples of Riemannian extrapolation in 2-D using finite-differences use implicit methods, since they are more stable and there is no reason not to use them. In 3-D, however, implicit solutions to the one-way wave-equation become much more difficult, even in Cartesian coordinates Fomel and Claerbout (1997); Rickett et al. (1998). This problem seems even more complicated for Riemannian wavefield extrapolation, since the extrapolation equation also contains mixed $k_\xi k_\eta$ terms. However, we speculate that for wavefield extrapolation in ray coordinates, this problem is not as difficult as it seems at first sight. The reason is that energy propagates roughly in the forward direction of the coordinate system and, therefore, we do not need extrapolators accurate at high dips. Thus, we could either use explicit methods with reasonably small stencils, or we could use low order mixed-domain methods from the split-step family which are easy to implement even in 3-D.

Prestack data:

Our examples of Riemannian wavefield extrapolation are based on equation (12) which corresponds to the single-square root (SSR) equation of standard Cartesian wavefield extrapolation. Riemannian wavefield extrapolation can be extended to prestack data either for shot-profile, plane-wave or S-G migration by appropriate definitions of the underlying ray coordinate system. Figure 14 is a schematic representation of shot-profile migration in ray coordinates, where both source and receivers are extrapolated in the same ray coordinate system appropriate for overturning waves. However, the sources and receivers do not necessarily have to be migrated in the same coordinate system. We could extrapolate both sides differently and apply the imaging condition after interpolation to the Cartesian grid.

 

spmig Figure 14 Shot-profile migration sketch. Sources (a) and receivers (b) are extrapolated in the same ray coordinate system which is appropriate for overturning waves.

Time wave-equation migration:

Our Riemannian wavefield extrapolation allows the output image to be presented either in one-way traveltime, which is the extrapolation coordinate, or in depth, after interpolation to Cartesian coordinates. A ray coordinate system initiated by a plane wave propagating vertically is related to what has been known in the literature as a ``$\tau$ coordinate system'' Alkhalifah et al. (2001); Biondi et al. (1997). However, our $\tau$ ray coordinate system is different, because it allows energy to move laterally, in contrast to the vertical traveltime coordinate system which does not allow such movement. Thus, another application of Riemannian wavefield extrapolation is time wave-equation migration (Figure 9), which has interesting properties, e.g., for migration velocity analysis Clapp (2001). Furthermore, wave-equation MVA Biondi and Sava (1999); Sava and Fomel (2002) could also be reformulated as a function of one-way propagation time.

Regularization at caustics:

The coordinate system coefficients for Riemannian wavefield extrapolation given by equations (8) have singularities at caustics, e.g., when the geometrical spreading term J, defining a cross-sectional area of a ray tube, goes to zero. In our current examples, we have used simple numerical regularization, by adding a small non-zero quantity to the denominators to avoid division by zero. This strategy worked reasonably well for our current examples.

Adaptive grid:

A potential difficulty of our method is represented by the uneven sampling of the wavefronts caused by focusing and defocussing of the rays defining the coordinate system. One solution to this problem is to use adaptive gridding, by increasing or decreasing sampling along the wavefronts, similarly to the techniques employed by the wavefront construction method Qian and Symes (2002); Vinje et al. (1993). Furthermore, each frequency in the data can be extrapolated on its own grid, sparser at lower frequencies and denser at higher frequency, thus reducing cost and increasing accuracy.

Interpolation:

The images created with wavefield extrapolation in ray coordinates require interpolation to a Cartesian coordinate system. This is a shared difficulty of all methods that do not operate on a Cartesian grid. In our examples, we have successfully used a simple sinc-type interpolation method. In principle, we could use better interpolation methods using prediction-error filters at higher cost, although we have not seen the need for this in our current examples.

Coordinate system construction:

The ray coordinate systems do not need to be created using the same velocity model as the one used for extrapolation. We can use a smooth velocity model to create the coordinate system by ray tracing, and then interpolate the unsmoothed velocity, similarly to the method used by Brandsberg-Dahl and Etgen (2003). Such a strategy opens up the possibility of defining coordinate systems using arbitrary velocity models which favor selected parts of the image. For example, we could use for ray tracing a velocity model optimized to reduce, in a least-squares sense, the angle between the extrapolation grid and the dips in the image. An alternative method of creating ray coordinate systems is discussed by Shragge and Biondi (2003).

Amplitude preservation:

Amplitude-preserving imaging using one-way wavefield extrapolation operators is difficult. Recent research has advanced our knowledge on this subject Sava et al. (2001); Shan and Biondi (2003); Zhang et al. (2001), but the goal of true-amplitude wave-equation migration is still unachieved. The biggest practical difficulty is associated with amplitude preservation at high scattering angles relative to the extrapolation direction. Since we are normally using low angle operators relative to the wave propagation direction, we speculate that Riemannian extrapolation can also improve the amplitude characteristics of wave-equation migration.