1. The FOCI™ method of depth migration
Gary Margrave
Saleh Al-Saleh
Hugh Geiger
Michael Lamoureux
POTSI

2. Wavefield extrapolators and stability A stabilizing Wiener filter
Outline
Wavefield extrapolators and stability
A stabilizing Wiener filter
Dual operator tables
Spatial resampling
Post stack testing
Pre stack testing
FOCI

3. Wavefield Extrapolators
The phase-shift extrapolation expression
output wavefield
Fourier transform of input wavefield

4. Wavefield Extrapolators
The phase-shift operator
wavelike
evanescent

5. Wavefield Extrapolators
The PSPI extension

6. Wavefield Extrapolators
wavenumber

7. Wavefield Extrapolators
wavenumber

8. Wavefield Extrapolators
In the space-frequency domain
where

9. Wavefield Extrapolators
In the space-frequency domain
imaginary
real
meters

10. Wavefield Extrapolators
Back to the wavenumber domain
wavenumber

11. Wavefield Extrapolators
Back to the wavenumber domain
wavenumber

12. Stabilization by Wiener Filter
Two useful properties
Product of two half-steps make a whole step.
The inverse is equal to the conjugate in the wavelike region.

13. Stabilization by Wiener Filter
A windowed forward operator for a half-step
Solve by least squares

14. Stabilization by Wiener Filter
is a band-limited inverse for
Both have compact support
Form the FOCI™ approximate operator by
FOCI™ is an acronym for
Forward Operator with Conjugate Inverse.

15. Properties of FOCI operator
Let
Then

16. Properties of FOCI operator
determines phase accuracy.
determines stability.
Empirical observation:

17. Properties of FOCI operator
Amount of evanescent filtering is inversely related to stability

18. Operator tables
Since the operator is purely numerical, migration proceeds by construction of operator tables.

19. Operator tables
We construct two operator tables for small and large h. The small h table is used most of the time, with the large h being invoked only every nth step.

20. Operator Design Example

21. Improved Operator Design

22. Spatial Resampling
Wavenumber
Frequency
Wavelike

23. In red are the wavenumbers of a 7 point filter
Spatial Resampling
Frequency
In red are the wavenumbers of a 7 point filter
Wavenumber

24. Spatial Resampling Frequency
Downsampling for the lower frequencies uses the filter more effectively

25. Spatial resampling is done in frequency “chunks”.

26. Run Time Experiment

27. Run Times log-log Scale
PS slope 1.07
FOCI slope 1.03
FOCI slope 1.05
Slopes for last three points

28. Phase Shift Impulse Response

29. FOCI Impulse Response
nfor=7, ninv=15, (21 pt), no spatial resampling

30. FOCI Impulse Response
nfor=7, ninv=15, (21 pt), with spatial resampling

31. Marmousi Velocity Model

32. Exploding Reflector Seismogram

33. FOCI Post-Stack Migration
nfor=21, ninv=31, nwin=0

34. FOCI Post-Stack Migration
nfor=21, ninv=31, nwin=51

35. FOCI Post-Stack Migration
nfor=21, ninv=31, nwin=21

36. FOCI Post-Stack Migration
nfor=7, ninv=15, nwin=7

37. Marmousi Velocity Model

38. FOCI Pre-Stack Migration
nfor=21, ninv=31, nwin=0, deconvolution imaging condition

39. FOCI Pre-Stack Migration
nfor=21, ninv=31, nwin=0, deconvolution imaging condition

40. FOCI Pre-Stack Migration
Shot 30

41. FOCI Pre-Stack Migration
Shot 30

42. FOCI Pre-Stack Migration
Stack +50*Shot 30

43. Detail of Pre-Stack Migration

44. Marmousi Reflectivity Detail

45. Marmousi Velocity Model

46. FOCI Pre-Stack Migration
Velocity model convolved with 200m smoother

47. FOCI Pre-Stack Migration
Velocity model convolved with 400m smoother

48. FOCI Pre-Stack Migration
Exact Velocities

49. FOCI Pre-Stack Migration
Velocities smoothed over 200m

50. FOCI Pre-Stack Migration
Velocities smoothed over 400m

51. Detail of Pre-Stack Migration
Exact Velocities

52. Detail of Pre-Stack Migration
Velocities smoothed over 200m

53. Detail of Pre-Stack Migration
Velocities smoothed over 400m

54. Run times
Full prestack depth migrations of Marmousi on a single 2.5GHz PC using Matlab code.
20 hours for the best result
1 hour for a usable result

55. Conclusions
Explicit wavefield extrapolators can be made local and stable using Wiener filter theory.
The FOCI method designs an unstable forward operator that captures the phase accuracy and stabilizes this with a band-limited inverse operator.
Reducing evanescent filtering increases stability.
Spatial resampling increases stability, improves operator accuracy, and reduces runtime.
The final method appears to be ~O(NlogN).
Very good images of Marmousi have been obtained.

56. Research Directions Better phase accuracy. Extension to 3D.
Extension to more accurate wavefield extrapolation schemes.
Migration velocity analysis.
Development of C/Fortran code on imaging engine.

57. A US patent application has been made for the FOCI process.
Acknowledgements
We thank:
Sponsors of CREWES
Sponsors of POTSI
NSERC
MITACS
PIMS
A US patent application has been made for the FOCI process.

58. Detail of Pre-Stack Migration

59. FOCI Pre-Stack Migration
nfor=21, ninv=31, nwin=0, deconvolution imaging condition

60. FOCI Pre-Stack Migration
nfor=21, ninv=31, nwin=0, deconvolution imaging condition