High-order kernels for Riemannian Wavefield Extrapolation

Examples

We illustrate the higher-order RWE extrapolators with impulse responses for two synthetic models.

The first example is based on the Marmousi model (Versteeg, 1994). We construct the coordinate system by ray tracing from a point source at the surface in a smooth version of the real velocity model. Figure 1 shows the velocity model with the coordinate system overlaid, and Figures 2(a)-2(b) show the coordinate system coefficients $a$ and $b$ defined in equations 11 and 12. M-coswidth=0.80 Velocity map and Riemannian coordinate system for the Marmousi example.

M-abmRCa,M-abmRCb
Figure 1. Coordinate system coefficients defined in equations 11 and 12. (a) Parameter $a=s {\bf a}$ in ray coordinates. (b) Parameter $b={\bf a}/{\bf j}$ in ray coordinates.

The goal of this test model is to illustrate the higher-order extrapolation kernels in a fairly complex model using a simple coordinate system. In this way, the coordinate system and the real direction of wave propagation depart from one-another, thus accurate extrapolation requires higher order kernels. The coordinate system is constructed from a point at the location of the wave source. This setting is similar to the case of extrapolation from a point source in Cartesian coordinates, where high-angle propagation requires high-order kernels.

Figures 3(a)-3(d) show impulse responses for a point source computed with various extrapolators in ray coordinates ($\tau$ and $\gamma$). Panels (a) and (c) show extrapolation with the $15^\circ$ and $60^\circ$, respectively. Panels (b) and (d) show extrapolation with the pseudo-screen (PSC) equation, and the Fourier finite-differences (FFD) equation, respectively. 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 a and c) show the typical behavior of such solutions for the $15^\circ$ and $60^\circ$ equations (e.g. the cardioid for $60^\circ$), but in the more general setting of Riemannian extrapolation. The mixed-domain extrapolators (panels b and d) are more accurate the finite-differences extrapolators. The main differences occur at the highest propagation angles. As for the case of Cartesian extrapolation, the most accurate kernel of those compared is the equivalent of Fourier finite-differences.

Figures 4(a)-4(d) show the corresponding plots in Figures 3(a)-3(d) mapped in the physical coordinates. 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 apparent, since it now has physical meaning.

Figures 5(a)-5(b) show a side-by-side comparison of equivalent extrapolators in Riemannian and Cartesian coordinates. The impulse response in Figure 5(a) shows the limits of Cartesian extrapolation in propagating waves correctly up to $90^\circ$. The Riemannian extrapolator in Figure 5(b) handles much better waves propagating at high angles, including energy that is propagating upward relative to the physical coordinates.

M-migRC-F15,M-migRC-PSC,M-migRC-F60,M-migRC-FFD
Figure 2. Migration impulse responses in Riemannian coordinates. (a) Extrapolation with the $15^\circ$ finite-differences equation. (c) Extrapolation with the $60^\circ$ finite-differences equation. (b) Extrapolation with the pseudo-screen (PSC) equation. (d) Extrapolation with the Fourier finite-differences (FFD) equation.

M-migCC-F15,M-migCC-SSF,M-migCC-F60,M-migCC-FFD
Figure 3. Migration impulse responses in Riemannian coordinates after mapping to Cartesian coordinates. (a) Extrapolation with the $15^\circ$ finite-differences equation. (c) Extrapolation with the $60^\circ$ finite-differences equation. (b) Extrapolation with the pseudo-screen (PSC) equation. (d) Extrapolation with the Fourier finite-differences (FFD) equation.

M-imgCC,M-migCC-SSF
Figure 4. Comparison of extrapolation in Cartesian and Riemannian coordinates. (a) Split-step Fourier extrapolation in Cartesian coordinates. (b) Split-step Fourier extrapolation in Riemannian coordinates.

The second example is based on a model with a large lateral gradient which makes an incident plane wave overturn. A small Gaussian anomaly, not used in the construction of the coordinate system, forces the propagating wave to triplicate and move at high angles relative to the extrapolation direction. Figure 6 shows the velocity model with the coordinate system overlaid. Figures 7(a)-7(b) show the coordinate system coefficients, $a$ and $b$ defined in equations 11 and 12.

D-cos
Figure 5. Velocity map and Riemannian coordinate system for the large-gradient model experiment.

D-abmRCa,D-abmRCb
Figure 6. Coordinate system coefficients defined in equations 11 and 12. (a) Parameter $a=s {\bf a}$ in ray coordinates. (b) Parameter $b={\bf a}/{\bf j}$ in ray coordinates.

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 an extrapolator we use. In this example, an incident plane wave is overturning, thus becoming evanescent for the solution constructed in Cartesian coordinates. Furthermore, the Gaussian anomaly shown in Figure 7(a) causes wavefield triplication, thus requiring high-order kernels for the Riemannian extrapolator.

Figures 8(a)-8(d) show impulse responses for an incident plane wave computed with various extrapolators in ray coordinates ($\tau$ and $\gamma$). Panels (a) and (c) show extrapolation with the $15^\circ$ and $60^\circ$ finite-differences equations, respectively. Panel (b) and (d) show extrapolation with the pseudo-screen (PSC) equation and the Fourier finite-differences (FFD) equation, respectively. All plots are displayed in ray coordinates. As for the preceding example, we observe higher angular accuracy as we increase the order of the extrapolator. The equivalent FFD extrapolator shows the highest accuracy of all tested extrapolators.

As in the preceding example, Figures 9(a)-9(d) show the corresponding plots in Figures 8(a)-8(d) mapped in the physical coordinates. The overlay is an outline of the extrapolation coordinate system.

Finally, Figures 10(a) and 10(b) show a side-by-side comparison of equivalent extrapolators in Riemannian and Cartesian coordinates. The impulse response in Figure 10(a) clearly shows the failure of the Cartesian extrapolator in propagating waves correctly even up to $90^\circ$. The Riemannian extrapolator in Figure 10(b) handles much better overturning waves, including energy that is propagating upward relative to the vertical direction.

D-migRC-F15,D-migRC-PSC,D-migRC-F60,D-migRC-FFD
Figure 7. Migration impulse responses in Riemannian coordinates. (a) Extrapolation with the $15^\circ$ finite-differences equation. (c) Extrapolation with the $60^\circ$ finite-differences equation. (b) Extrapolation with the pseudo-screen (PSC) equation. (d) Extrapolation with the Fourier finite-differences (FFD) equation.

D-migCC-F15,D-migCC-PSC,D-migCC-F60,D-migCC-FFD
Figure 8. Migration impulse responses in Riemannian coordinates after mapping to Cartesian coordinates. (a) Extrapolation with the $15^\circ$ finite-differences equation. (c) Extrapolation with the $60^\circ$ finite-differences equation. (b) Extrapolation with the pseudo-screen (PSC) equation. (d) Extrapolation with the Fourier finite-differences (FFD) equation.

D-imgCC-bp,D-migCC-SSF
Figure 9. Comparison of extrapolation in Cartesian and Riemannian coordinates. (a) Split-step Fourier extrapolation in Cartesian coordinates. (b) Split-step Fourier extrapolation in Riemannian coordinates.

High-order kernels for Riemannian Wavefield Extrapolation