paul sava


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© 2007 PC Sava

Page updated on
April 7, 2018

Riemannian wavefield extrapolation


Riemannian spaces are described by non-orthogonal curvilinear coordinates. One-way wavefield extrapolation can be generalized to semi-orthogonal Riemannian coordinate systems, which include, but are not limited to, ray coordinate systems. The result are one-way wavefield extrapolation methods which are not limited to any particular extrapolation direction. Such extrapolation techniques can be used for both wavefield modeling and migration.


The pictures below depict wavefronts emerging from a point source in the middle of the model: one frame is computed with one-way Riemannian extrapolation in the frequency domain, and the other frame is computed with time-domain finite differences. The wavefronts are overlain over the velocity model which is smoothed to avoid backscattering at sharp discontinuities. Aside from a slight phase difference, the two wavefields are almost identical. Which frame corresponds to which method?


The Marmousi velocity model depicted in Cartesian coordinates and in Riemannian coordinates, defined by rays from a point source.
Impulse responses for one-way wavefield extrapolation in Cartesian and Riemannian coordinates using the 15 degrees equation. In Cartesian coordinates, the extrapolation is only accurate within 15 degrees from the extrapolation direction, which is straight down. In Riemannian coordinates, the extrapolation is accurate to much larger angles, including for overturning waves.



This picture shows a comparison of wavefield extrapolation in Cartesian and Riemannian coordinates. Overturning waves cannont be propagated using downward continuation, but they are well propagated by Riemannian extrapolation.

References

Sava, P.C., Fomel, S., 2005, Riemannian wavefield extrapolation, Geophysics, v. 70, no. 3, pp. T45-T56. (PDF)

Figures can be used freely for non-profit educational purposes by acknowledging their source:
© Paul Sava, Center for Wave Phenomena, Colorado School of Mines
http://www.mines.edu/~psava