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.
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).
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.