The first example is based on a model with two smooth velocity anomalies that generate focusing and defocussing of the coordinate system, without triplication. I construct the coordinate system by ray tracing from an incident horizontal plane-wave at the surface. Figure shows the velocity model with the coordinate system overlayed. Figure shows the coordinate system coefficients defined in equation (9).

RWEimp0.cos
Velocity map and Riemannian coordinate system.
Figure 1 |

The goal of this test model is to illustrate the higher-order extrapolation
kernels in a fairly simple coordinate system
which is close to a Cartesian basis.
The coordinate system is constructed from an
incident plane wave, while the data comes from a point source.
This setting is almost identical to the case of extrapolation
from a point source in Cartesian coordinates,
where high-angle
^{}
propagation requires high-order kernels.
In this case, the Riemannian coordinate system does not match closely the
general direction of wave propagation, so higher order kernels are needed.
In practice, this situation can be addressed better with
synthesized plane-wave data extrapolated in a coordinate system
ray traced from an incident plane, and with point source data
extrapolated in a coordinate system ray traced from a point source.

Figure 2

Figure shows the velocity model and impulse responses for a point source computed with various extrapolators in ray coordinates ( and ). Panel (a) shows the slowness model, panel (b) shows extrapolation with the finite-differences equation, panel (c) shows extrapolation with the finite-differences equation, panel (d) shows extrapolation with the split-step Fourier (SSF) equation, panel (e) shows extrapolation with the pseudo-screen (PSC) equation, and panel (f) shows extrapolation with the Fourier finite-differences (FFD) equation. All plots are displayed in ray coordinates. We can observe that the angular accuracy of the extrapolator improves for the more accurate extrapolators. The finite-differences solutions (panels b and c) show the typical behavior of such solutions for the and equations (e.g. the cardioid for ), but in the more general setting of Riemannian extrapolation. The mixed-domain extrapolators (panels d, e, and f) show increased accuracy. The main differences occur at the highest propagation angles, and the most accurate extrapolators of those compared is the equivalent of Fourier finite-differences (panel f).

Figure shows the corresponding plots in Figure mapped in the physical coordinates, except for panel (a) which in this case shows the impulse response for extrapolation in Cartesian coordinates using a Fourier finite-differences extrapolator with a finite-differences term. The overlay is an outline of the extrapolation coordinate system. After re-mapping to the physical space, the comparison of high-angle accuracy for the various extrapolators is more obvious, since it now has the proper physical meaning.

Figure 3

Figure 4

The second example is based on a model with a large lateral gradient which makes an incident plane wave overturn. A small Gaussian anomaly forces the coordinate system to focus slightly, and another large Gaussian anomaly, not used for the coordinate system, forces the propagating wave to triplicate and move at high angles relative to the extrapolation direction. Figure shows the velocity model with the coordinate system overlayed. Figure shows the coordinate system coefficients defined in equation (9).

RWEimp1.cos
Velocity map and Riemannian coordinate system.
Figure 5 |

The goal of this model is to illustrate Riemannian wavefield extrapolation in a situation which cannot be handled correctly by Cartesian extrapolation, no matter how accurate. In this example, an incident plane wave is overturning, thus becoming evanescent for the solution in Cartesian coordinates. Furthermore, the large Gaussian anomaly, Figure (a), causes serious wavefield triplication, thus requiring high-order kernels in the Riemannian extrapolator.

Figure shows the velocity model and impulse responses for an incident plane wave computed with various extrapolators in ray coordinates ( and ). Panel (a) shows the slowness model, panel (b) shows extrapolation with the finite-differences equation, panel (c) shows extrapolation with the finite-differences equation, panel (d) shows extrapolation with the split-step Fourier (SSF) equation, panel (e) shows extrapolation with the pseudo-screen (PSC) equation, and panel (f) shows extrapolation with the Fourier finite-differences (FFD) equation. All plots are displayed in ray coordinates. As with the preceding example, we can observe increased angular accuracy as we increase the order of the extrapolator. The equivalent FFD is the most accurate.

As in the preceding example, Figure shows the corresponding plots in Figure mapped in the physical coordinates, except for panel (a) which in this case shows the impulse response for extrapolation in Cartesian coordinates using a Fourier finite-differences extrapolator with a finite-differences term. The overlay is an outline of the extrapolation coordinate system.

Figure 6

Panel (a) in Figure clearly shows the failure of the Cartesian extrapolator in propagating waves correctly even up to . All Riemannian extrapolators handle better the overturning waves, including energy that is propagating upward relative to the physical coordinates. As expected, the higher order kernels are more accurate in describing the triplicating wavefields.

Figure 7

Figure 8

10/23/2004