1. Jeffrey.c.shragge@exxonmobil.com Riemannian Wavefield Migration: Wave-equation migration from Topography Jeff Shragge, Guojian Shan, Biondo Biondi Stanford University Paul Sava, Sergey Fomel Bureau of Economic Geology UT Austin

2. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Incomplete Data Computational Power Physical Inaccuracy

3. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Incomplete Data Computational Power Physical Inaccuracy

4. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Propagator inaccuracy (illumination) Topographic surface limitations Coordinate system not conformal to propagation direction or acquisition surface Migration physics decoupled from geometry

5. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Propagator Inaccuracy (Illumination) Migration Physics decoupled from Geometry Difficult Steep Dip Imaging Coordinate system not conformal to propagation direction or acquisition surface

6. Jeffrey.c.shragge@exxonmobil.com Steep Dip Imaging Accuracy of wavefield extrapolation decreases as propagating waves tend to horizontal Extrapolation Direction

7. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Propagator Inaccuracy (Illumination) Coordinate system not conformal to propagation direction or acquisition surface Migration Physics decoupled from Geometry Difficult Steep Dip Imaging No use of Overturning waves

8. Jeffrey.c.shragge@exxonmobil.com Using Overturning Waves Currently do not use potentially useful information provided by overturning waves Extrapolation Direction

9. Jeffrey.c.shragge@exxonmobil.com Proposed Solution Coordinate system conformal with wavefield propagation Extrapolation Direction

10. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Topographic Surface Limitations Coordinate system not conformal to propagation direction or acquisition surface Migration Physics decoupled from Geometry Difficult Steep Dip Imaging No use of Overturning waves Extrapolation from complex free-surface Propagator Inaccuracy (Illumination)

11. Jeffrey.c.shragge@exxonmobil.com Free Surface Topography How to define extrapolation surface orthogonal to free surface?

12. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Topographic Surface Limitations Coordinate system not conformal to propagation direction or acquisition surface Migration Physics decoupled from Geometry Difficult Steep Dip Imaging No use of Overturning waves Extrapolation from complex free-surface Propagator Inaccuracy (Illumination) Extrapolation from deviated boreholes

13. Jeffrey.c.shragge@exxonmobil.com VSP Deviated Well Topography Receiver wavefield acquired in well deviated in 3-D How to define wavefield extrapolation from borehole surface?

14. Jeffrey.c.shragge@exxonmobil.com Proposed Solution Coordinate system conformal with borehole geometry

15. Jeffrey.c.shragge@exxonmobil.com The Dossier: WE Imaging Limitations Topographic Surface Limitations Difficult Steep Dip Imaging No use of Overturning waves Extrapolation from complex free-surface Propagator Inaccuracy (Illumination) Extrapolation from deviated boreholes Coordinate system not conformal to propagation direction or acquisition surface Migration Physics decoupled from Geometry

16. Jeffrey.c.shragge@exxonmobil.com A Solution… Summerized Reduce Propagator Inaccuracy (Illumination) Enable W.E. imaging directly from Topographic Surfaces Perform Migration on Coordinate systems conformal to propagation direction/acquisition surface Couple Migration Physics with Geometry Improve Steep Dip Imaging Use Overturning waves No datuming at the free-surface W.E. Imaging for massive 3-D VSP data

17. Jeffrey.c.shragge@exxonmobil.com Migration from Topography – I Make wave-equation imaging conformal with acquisition geometry

18. Jeffrey.c.shragge@exxonmobil.com Migration from Topography – II Make wave-equation imaging conformal with acquisition geometry

19. Jeffrey.c.shragge@exxonmobil.com Migration from Topography – III Make wave-equation imaging conformal with acquisition geometry Two requirements: i) Propagating wavefields in generalized coordinate systems ii) Creating the specific topographic coordinate system

20. Jeffrey.c.shragge@exxonmobil.com Agenda Riemannian Wavefield Extrapolation Coordinate System Generation Example: Migration from Topography –Imaging Husky

21. Jeffrey.c.shragge@exxonmobil.com Agenda Riemannian Wavefield Extrapolation Coordinate System Generation Example: Migration from Topography –Imaging Husky

22. Jeffrey.c.shragge@exxonmobil.com RWE in 2-D ray-coordinates RiemannianCartesian Extrapolation Direction Orthogonal Direction

23. Jeffrey.c.shragge@exxonmobil.com RWE: Helmholtz equation (associated) metric tensor Laplacian Coordinate system Sava and Fomel (2005)

24. Jeffrey.c.shragge@exxonmobil.com RWE: (Semi)orthogonal coordinates

25. Jeffrey.c.shragge@exxonmobil.com 1 st order2 nd order 1 st order RWE: Helmholtz equation Ray-coordinate Interpretation α = velocity function J = geometric spreading (i.e. Jacobian) Sava and Fomel (2005)

26. Jeffrey.c.shragge@exxonmobil.com RWE: Dispersion relation Riemannian Cartesian

27. Jeffrey.c.shragge@exxonmobil.com RWE: Dispersion relation Riemannian Cartesian

28. Jeffrey.c.shragge@exxonmobil.com RWE: Kinematic dispersion relation Riemannian Cartesian

29. Jeffrey.c.shragge@exxonmobil.com RWE: Kinematic dispersion relation Riemannian Cartesian

30. Jeffrey.c.shragge@exxonmobil.com RWE: Wavefield extrapolation Riemannian Cartesian

31. Jeffrey.c.shragge@exxonmobil.com interpolate Extrapolation Work Flow PHYSICAL DOMAIN CANONICAL DOMAIN MODEL DOMAIN WAVEFIELD DOMAIN

32. Jeffrey.c.shragge@exxonmobil.com Agenda Riemannian Wavefield Extrapolation Coordinate System Generation Example: Migration from Topography –Imaging Husky

33. Jeffrey.c.shragge@exxonmobil.com Conformal Mapping Technique PHYSICAL DOMAIN O O O O O O O O O O

34. Jeffrey.c.shragge@exxonmobil.com Conformal Mapping Technique PHYSICAL DOMAINCANONICAL DOMAIN O O O O O O O O O OOOOOOOO O O O g f -1

35. Jeffrey.c.shragge@exxonmobil.com Conformal Mapping Technique PHYSICAL DOMAINCANONICAL DOMAIN O O O O O O O O O OOOOOOOO O O O OOOOOOO O O O g f -1

36. Jeffrey.c.shragge@exxonmobil.com Conformal Mapping Technique PHYSICAL DOMAINCANONICAL DOMAIN O O O O O O O O O O O O O O O O O O O OOOOOOOO O O O OOOOOOO O O O g f -1 g -1 f

37. Jeffrey.c.shragge@exxonmobil.com Conformal Mapping Technique O O O O O O O O O OOOOOOOO O O O Shot Profile Migration –use same grid for both source and receiver wavefields

38. Jeffrey.c.shragge@exxonmobil.com RWE: Shot-profile migration Riemannian Cartesian

39. Jeffrey.c.shragge@exxonmobil.com Single shot Imaging Work Flow interpolate PHYSICAL DOMAIN CANONICAL DOMAIN MODEL DOMAIN IMAGE DOMAIN interpolate

40. Jeffrey.c.shragge@exxonmobil.com Agenda Riemannian Wavefield Extrapolation Coordinate System Generation Example: Migration from Topography –Imaging Husky

41. Jeffrey.c.shragge@exxonmobil.com Husky data set – Topography 10x Surface Topography Exaggeration

42. Jeffrey.c.shragge@exxonmobil.com Husky data set – Time Migration + Geology

43. Jeffrey.c.shragge@exxonmobil.com TopoWEM TopoWEM Image

44. Jeffrey.c.shragge@exxonmobil.com TopoWEM TopoWEM Image

45. Jeffrey.c.shragge@exxonmobil.com TopoWEM + Angle Gathers TopoWEM Image

46. Jeffrey.c.shragge@exxonmobil.com TopoWEM Angle Gathers

47. Jeffrey.c.shragge@exxonmobil.com Kirchhoff from Flat Datum Kirchhoff Image

48. Jeffrey.c.shragge@exxonmobil.com Kirchhoff from Flat Datum Kirchhoff Image

49. Jeffrey.c.shragge@exxonmobil.com Kirchhoff + surface offset gathers Kirchhoff Image

50. Jeffrey.c.shragge@exxonmobil.com TopoWEM vs. Statics+Datum Kirchhoff TopoWEM Image Kirchhoff Image

51. Jeffrey.c.shragge@exxonmobil.com TopoWEM vs. Statics+Datum Kirchhoff TopoWEM Image Kirchhoff Image

52. Jeffrey.c.shragge@exxonmobil.com Summary TopoWEM produced good image –Resolved near-surface structures and dips Kirchhoff migration from flat datum produced good image –Better at mid-to-basement depths but could not resolve near-surface Bottom line: statics + migration from datum is less accurate –Kinematic errors in static assumptions TopoWEM is more expensive –Requires Taylor series expansion about 2 coefficient parameters Imaging from acquisition coordinate works

53. Jeffrey.c.shragge@exxonmobil.com Acknowledgements Anatoly Baumstein Tom Dickens Ernestine Dixon Dave Hinkley Matt Hong Linda Price Mike Rainwater Peter Traynin Joe Vanderslice Kim Wilmott Husky Oil Talisman Oil Friday Lunch Folks John Sumner No doubt many others