Riemannian wavefield extrapolation of seismic data

1. Riemannian wavefield extrapolationof seismic data J. Shragge, P. Sava, G. Shan, and B. Biondi Stanford Exploration Project S. Fomel UT Austin jeff@sep.stanford.edu

2. Overview • Prelude • Remote sensing/Echo sounding • Seismic wavefield extrapolation • Fugue • Riemannian wavefield extrapolation • Example jeff@sep.stanford.edu

3. Why seismic imaging? • Applied seismology • Hydrocarbon exploration • “Easy” targets already located • remaining large fields located in regions of complex geology • 3-D seismic imaging • Delineate earth structure • property estimation and prediction • improve probability of finding oil jeff@sep.stanford.edu

4. Echo soundings of the earth Transmit sound-waves into earth Record echoes from earth structure Determine earth structure that created echoes jeff@sep.stanford.edu

5. Seismic imaging - Similarities • Related methods • Acoustic wave methods • Ultrasound • Sonar • EM wave methods • Radar • X-ray • Related applications • Medical imaging • Non-destructive testing • Marine navigation • Archaeology site assessment jeff@sep.stanford.edu

6. Seismic imaging - Differences • Complex earth structure • Velocity • V(x,y,z) – 1.5 – 4.5 km/s • Strong gradients • Material properties • heterogeneity • anisotropy • Wave-phenomena • Multi-arrivals, band-limited • Frequency-dependent illumination • Overturning waves • Ray theory cannot capture complexity jeff@sep.stanford.edu

7. Wavefield Extrapolation Monochromatic frequency-domain: Helmholtz equation Recorded wavefield U(x,y,z=0) Want U(x,y,z) Wavefield extrapolation Wave phenomena Wave-equation jeff@sep.stanford.edu

8. One-way wavefield extrapolation Wave-equation dispersion relation Wavefield propagates by advection - with solution Want solution to Helmholtz equation jeff@sep.stanford.edu

9. Migration by wavefield extrapolation • Robust, Accurate, Efficient • Current Limitations • steep dip imaging • no overturning waves jeff@sep.stanford.edu

10. One-way wavefield extrapolation Wave-equation dispersion relation Steep Dip limitation Advection solution on Cartesian grid Overturning wave limitation jeff@sep.stanford.edu

11. Migration by wavefield extrapolation • Robust, Accurate, Efficient • Current Limitations • steep dip imaging • no overturning waves • Our solution • Change coordinate system to be more conformal with wavefield • Riemannian spaces jeff@sep.stanford.edu

12. Riemannian wavefield extrapolation x z jeff@sep.stanford.edu

13. Overview • Prelude • Remote sensing/Echo sounding • Seismic wavefield extrapolation • Fugue • Riemannian wavefield extrapolation • Examples jeff@sep.stanford.edu

14. Helmholtz equation Laplacian Coordinate system (associated) metric tensor jeff@sep.stanford.edu

15. (Semi)orthogonal coordinates jeff@sep.stanford.edu

16. Helmholtz equation 2nd order 1st order 1st order 2nd order jeff@sep.stanford.edu

17. Dispersion relation Riemannian Cartesian jeff@sep.stanford.edu

18. Dispersion relation Riemannian Cartesian jeff@sep.stanford.edu

19. Wavefield extrapolation Riemannian Cartesian jeff@sep.stanford.edu

20. interpolate interpolate jeff@sep.stanford.edu

21. Summary • Riemannian wavefield extrapolation • General coordinate system • Semi-orthogonal (3-D) • Incorporate propagation in coordinates • Applications • Overturning waves • Steeply dipping reflectors jeff@sep.stanford.edu

22. Collaboration? • Numerical development • Wave-based imaging • Ultrasound • Sonar • Radar • Applications • Medical imaging • Non-destructive testing • Marine navigation • Archaeology site assessment jeff@sep.stanford.edu

23. distance depth jeff@sep.stanford.edu

24. RWE vs. time-domain finite differences distance depth jeff@sep.stanford.edu

25. angle time jeff@sep.stanford.edu

26. angle time jeff@sep.stanford.edu

27. distance depth jeff@sep.stanford.edu