Discussion

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 rough velocity, similarly to the method used by (15). An alternative method of creating ray coordinate systems is discussed by (98). The coordinate system can be initiated from a point source or an arbitrary surface in 3D (or a line in 2D) which positions it optimally relative to the imaging target. Furthermore, with Riemannian wavefield extrapolation, we can address a particular target in the image and thus we do not need to construct a coordinate system which is appropriate for the entire image.

Coordinate system regularization:

The coordinate system coefficients for Riemannian wavefield extrapolation given by coefs.3d have singularities at caustics, i.e. when the geometrical spreading term J, defining a cross-sectional area of a ray tube, is zero. We can address this problem through a simple numerical regularization, by adding a small non-zero quantity to the denominators to avoid division by zero. This strategy worked reasonably well for the current examples, although better strategies are needed.

In principle, it is best if coordinate system triplications are avoided. However, for velocity models with large contrasts (e.g. salt), avoiding such triplications may require large smoothing prior to ray tracing. In these situations, there is a strong possibility that the waves do not propagate close to the extrapolation axis, thus requiring higher-order terms in the extrapolator at increased cost.

Prestack data:

My current examples of Riemannian wavefield extrapolation are based on oneway.3d 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. spmig 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 a particular set of overturning waves. This is also an illustration of how Riemannian wavefield extrapolation can be used for target-oriented wave-equation migration. In general, source and receiver wavefields can be migrated in different coordinate systems, with the imaging condition applied after interpolation to common (Cartesian) coordinates.

 

spmig
Figure 13 Shot-profile migration sketch. Sources (a) and receivers (b) are both extrapolated in a ray coordinate system appropriate for overturning waves.

Interpolation:

The images created with wavefield extrapolation in Riemannian coordinates require interpolation to a Cartesian coordinate system. This is a shared difficulty of all methods operating on non-Cartesian grids. In the current implementation, I use simple sinc-type interpolation based on the explicit mapping of the Cartesian coordinates $(\xx,\zz)$function of the ray coordinates $(\qt,\qg)$ given by ray tracing.

Cost:

The main cost of an implicit finite-difference solution to the one-way equation in Riemannian coordinates is related to solving a tridiagonal system in 2D (24) or a pentadiagonal system in 3D (77). In this respect, the cost of Riemannian wavefield extrapolation is identical to the cost of Cartesian downward continuation for the same number of samples. However, computing the coefficients of the tridiagonal system adds modestly to the cost, since they can be precomputed ahead of time.

A second consideration is that we are comparing extrapolation in different domains (space for downward continuation and shooting angle for Riemannian extrapolation). Since in Riemannian coordinates we extrapolate at small angles, we can sample the wavefronts less and achieve same or better angular accuracy than in Cartesian coordinates at lower cost.