We now present a few issues that we think
are important for our Riemannian wavefield extrapolation
In some cases, we discuss ideas which we have not treated yet,
while in other cases we speculate on directions of
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 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.
Our examples of Riemannian wavefield extrapolation are based on
equation (12) which corresponds to the single-square root
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.
Figure 14 Shot-profile migration sketch.
Sources (a) and receivers (b) are extrapolated in the
same ray coordinate system which is appropriate for
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 `` coordinate system'' Alkhalifah et al. (2001); Biondi et al. (1997).
However, our 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
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
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.
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
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.
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-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
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.