Our first example is designed to illustrate our method
in a fairly simple model. We use a 2-D model with
horizontal and vertical gradients
m/s which gives waves
propagating from a point source a pronounced
tendency to overturn
(Figure 2).
The model also contains a diffractor located
around *x*=3800 m and *z*=3000 m.

We use ray tracing to create an orthogonal
ray coordinate system corresponding to a point
source on the surface at *x*=6000 m.
Figure 2(a) shows the velocity
model and the rays in the original Cartesian
coordinate system (*x*,*z*).
Figure 2(c) shows the velocity
model mapped into the ray coordinate system ().
The diffractor is mapped to s and
measured from the vertical.
The synthetic data we use is represented by
impulses at the source location at every 0.25 s.
In ray coordinates, this source is represented by a
plane-wave evenly distributed over all
shooting angles .Ideally, an image obtained by migrating such a dataset
is a representation of the acoustic wavefield
produced by a source which pulsates periodically.

Figure 2(b) shows the image obtained by downward continuation in Cartesian coordinates using the standard equation. Figure 2(d) shows the image obtained by wavefield extrapolation in ray coordinates using the equivalent equation. The overlays in panels (b) and (d) are wavefronts at every 0.25 s and rays shot at every to facilitate one-to-one comparisons between the images in ray and Cartesian coordinates.

Figure 3 is a direct comparison of the results obtained by extrapolation in the two coordinate systems. The image created by extrapolation in Cartesian coordinates (a) is mapped to ray coordinates (b). The image created by extrapolation in ray coordinates (c) is mapped to Cartesian coordinates (d). Since we use the same velocity for ray tracing and for wavefield extrapolation, we expect the wavefields and the overlain wavefronts to be in agreement. The most obvious mismatch occurs in regions where the equation fails to extrapolate correctly at steep dips (around .This is not surprising since, as its name indicates, this equation is only accurate up to .However, this limitation is eliminated in ray coordinates, because the coordinate system brings the extrapolator in a reasonable position and at a good angle, although the extrapolator uses an equation of a similar order of accuracy.

Another interesting observation in Figures 3 (a) and (d) concerns the diffractor we introduced in the velocity model. When we extrapolate in Cartesian coordinates, the diffraction is only accurate to a small angle relative to the extrapolation direction (vertical). In contrast, the diffraction develops relative to the propagation direction when computed in ray coordinates, thus being more accurate after mapping to Cartesian coordinates.

Figure 2

Figure 3

We can also observe that the diffractions created by the anomaly in the velocity model are not at all limited in the ray coordinates domain. In a beam-type approach, such diffraction would not develop beyond the extent of any particular beam which interacts with it. Neighboring beams would be completely insensitive to the presence of the velocity anomaly.

The second example concerns a smooth velocity with a negative Gaussian anomaly which creates a triplication of the ray coordinate system (Figure 4). Everything other than the velocity model is identical to its counterpart in the preceding example. Similarly to Figure 2, panels (a) and (b) correspond to Cartesian coordinates, and panels (c) and (d) correspond to ray coordinates. Using regularization of the ray coordinates parameters, we are able to extrapolate through the triplication. The small discrepancy between the wavefields and the corresponding wavefronts indicate that our method of ray tracing is not perfectly accurate in the triplicating region, and the wavefield extrapolation is correcting for the kinematic differences.

Figure 4

Figure 5

Figure 5 is similar to Figure 3. The ``butterfly'' in panel (b) is another indication that the ray coordinate system is triplicating, and different shooting directions pick up the same energy from the wavefield extrapolated in Cartesian coordinates (a). None of this happens when we extrapolate in ray coordinates (c) and interpolate to Cartesian coordinates (d).

Our next example concerns the more complicated Marmousi
model. Figure 6 shows
the velocity models mapped into the two
different domains, and the wavefields obtained
by extrapolation in each one of them.
We create the ray coordinate system by
ray tracing in a smooth version of the model,
and extrapolate in the rough version.
The source is located on the surface at *x*=5000 m.

In this example, the wavefields triplicate in both domains (Figure 7). Since we are using a equation, extrapolation in Cartesian coordinates is only accurate for the small incidence angles, as can be seen in panels (a) and (b). In contrast, extrapolating in ray coordinates (c) does not have the same angle limitation, which can also be seen after mapping back to Cartesian coordinates (d).

Figure 6

Figure 7

Figure 8

Figure 8 is a close-up comparison of the wavefields obtained by extrapolation with different methods in different domains. Panel (a) is a window of the velocity model for reference. Panels (b) and (c) are obtained by extrapolation in ray coordinates using the and split-step equations, respectively. Panels (d), (e) and (f) are obtained by downward continuation in Cartesian coordinates using the , and split-step equations, respectively. The ray coordinates extrapolation results are similar to the Cartesian coordinates results in the regions where the wavefields propagate mostly vertically, but are much better in the regions where the wavefields propagate almost horizontally.

Next, we present another example in the
Sigsbee 2A model. We consider two types of
ray coordinates: one initiated by a
plane wave at the surface
(Figure 9), and one
initiated by a point source at the surface
at *x*=16000 m.
(Figure 10).
Similarly to the preceding examples,
we observe complicated wavefield
propagation, with many triplications,
of the extrapolated wavefields.

Figure 9

Figure 10

Figure 11

Figure 11 is a close-up comparison of the wavefields obtained by extrapolation with different methods in different domains. Panel (a) is a window of the velocity model for reference. Panel (b) is the wavefield obtained by finite-difference modeling using the two-way acoustic wave equation. This panel contains not only waves propagating forward, but also reflections which are not going to be modeled in the one-way extrapolation results. Panel (c) is obtained by downward continuation in Cartesian coordinates using the equation, and panel (d) is obtained by extrapolation in ray coordinates using the equation. We can observe good match of the forward propagating wavefields in (b) and (d), in contrast to the poor match with panel (c) in the regions of nearly horizontal propagation.

Finally, we present an example of zero-offset migration of overturning reflections using Riemannian wavefield extrapolation. Figure 12 depicts the velocity model (a), the recorded data (b), and the migrated image (c). The overlay is a sketch of the ray coordinate system used for migration. The first event in the data corresponds to the overturning reflection from the ball and is imaged correctly, and the later events are multiple reverberations inside the ball which are not imaged with our method.

Figure 12

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