Wavefield extrapolation in phase-ray coordinates

This section examines the use of phase-rays as a coordinate system for wavefield extrapolation. The velocity model examined here (shown in figure a) is characterized by a slow, Gaussian-shaped velocity anomaly (-600 m/s) in a medium of otherwise constant velocity (1200 m/s).

 

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Figure 4 Gaussian-shaped velocity anomaly model. a) Gaussian-shaped anomaly of -600 m/s maximum velocity perturbation superposed on a constant 1200 m/s velocity field; b) 5 Hz wavefield computed for a shot located at 8000 m using a single-velocity split step Fourier operator in Cartesian coordinates; and c) phase-rays traced through the wavefield of b).

This model was chosen to ensure that extrapolated wavefields triplicate, as illustrated by figure b. This wavefield was generated for a shot point located at 8000 m using a split-step Fourier operator in a Cartesian coordinate system. Figure c shows phase-rays traced through the wavefield of figure b. Phase-rays in the upper portions of the model have fairly smooth coverage. In areas of wavefield triplication, though, significant coverage gaps are noticeable.

The phase-rays shown in figure c were subsequently used as a coordinate system for the generalized coordinate wavefield extrapolation approach Sava and Fomel (2003). Figure a shows the velocity model of figure a in phase-ray coordinates.

 

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Figure 5 A comparison of wavefield extrapolation results computed in phase-ray and Cartesian coordinates. a) Velocity model of figure a mapped to phase-ray coordinates; b) wavefield extrapolated in phase-ray coordinates using a split-step Fourier method; c) the map of wavefield in b) to Cartesian coordinates; d) wavefield solved in Cartesian coordinates using a split-step Fourier operator.

Figure b presents the results of wavefield extrapolation in phase-ray coordinates using a split-step Fourier operator and the velocity model shown in figure a. Figure c shows this result mapped to Cartesian coordinates. The wavefield presented in figure d was computed in Cartesian coordinates using a split-step Fourier operator. In areas where wavefields are present in significant amplitude, the phase-ray and Cartesian results are similar. Areas of low wavefield amplitude beneath the Gaussian velocity anomaly (e.g. [z,x]=[4200 m,6800 m]), though, are markedly different. This difference is related to the inability to map the results from ray coordinates to Cartesian in areas of minimal or non-existent ray coverage.

This experiment highlights a consequence of using phase-ray coordinates for wavefield extrapolation. Monochromatic wavefield triplication is generally identified by interference patterns created by converging wavefield components. (See, for example, the checkerboard pattern beneath the Gaussian velocity anomaly.) Because phase-ray direction is dependent on the total wavefield gradient, it is similarly dependent on the gradients of each converging wavefield component. The gradient vector, being unable to unwrap individual convergent phases, chooses a weighted average of individual gradients. Accordingly, phase-rays are usually steered in the direction of the convergent component with the largest individual gradient magnitude, but they will never triplicate since the weighted gradient is uniquely defined at each wavefield point. This fact suggests that phase-ray coordinates represent a trade off between introducing inaccuracy associated with triplicating coordinates and inaccuracy of wavefield extrapolation at greater angles to the phase-ray direction.

Figure presents a comparison between wavefields extrapolated in phase-ray and conventional ray coordinates Sava and Fomel (2003).

 

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Figure 6 A comparison of wavefield extrapolation results computed in phase-ray and conventional ray coordinates. a) wavefield extrapolated in phase-ray coordinates; b) wavefield of a) interpolated into Cartesian coordinates; c) wavefield extrapolated in conventional ray coordinates Sava and Fomel (2003); and d) wavefield of c) interpolated in Cartesian coordinates.

Figures a and b present wavefields extrapolated in phase-ray coordinates and after interpolation into Cartesian coordinates, respectively. Figures c and d present similar results, but with conventionally traced rays. Of the two ray coordinate systems tested, the phase-ray coordinate extrapolated wavefield (figure b) better resembles the wavefield calculated in Cartesian coordinates (figure c). However, the sampling of phase-ray extrapolated wavefields, and their Cartesian maps, must be greatly improved before a definitive comparison is possible.

 

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