1. Some remarks on the leading order imaging series
Depth imaging of reflection data from a layered acoustic medium with an unknown velocity model
Simon A. Shaw
University of Houston
M-OSRP Annual Meeting
April 20th, 2005

2. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

3. Motivation
Current migration theory requires correct velocity model to give correct location
Current methods for deriving the velocity model can be inadequate, especially for complex media
More reservoirs beneath complex geology

4. Objective (of this research)
To improve our ability to accurately locate reflectors, especially in areas where the velocity model is difficult to estimate

5. Objective (of this talk)
To review the 1D constant density variable velocity acoustic leading order imaging series algorithm
To better understand
what it is
what it does
how it does it

6. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

7. Background and review Inverse scattering series-based algorithms:
Do not require any a priori subsurface information
– no velocity model, no event picking, no moveout assumptions …
Require the source wavelet
Allow / require the data to get involved in their own processing. They are data-driven.

8. Background and review Scattering Theory is Perturbation Theory Relates
Actual Field, G
Reference Field, G0
difference between Actual Medium and Reference Medium properties, V

9. Actual Field = Reference Field + Scattered Field
Background and review
Actual Medium Reference Medium Perturbation
L – L = V
Actual Field = Reference Field + Scattered Field
G G S

10. Actual Field = Reference Field + Scattered Field
Background and review
Actual Medium Reference Medium Perturbation
L – L = V
Actual Field = Reference Field + Scattered Field
G G S
Forward Problem

11. Actual Field = Reference Field + Scattered Field
Background and review
Actual Medium Reference Medium Perturbation
L – L = V
Inverse Problem
Actual Field = Reference Field + Scattered Field
G G S

12. Background and review
Inverse Problem
Measured Scattered Field
Perturbation D V
Solution for V is an infinite series in the data, D

13. Reference medium is never updated
Inverse scattering series
V = V1 + V2 + V3 + …
Inverse Series is solution for V in terms of G0 and measured G
Linear
2nd Order
3rd Order
Nonlinear…. Data must multiply itself
Reference medium is never updated

14. Linear inverse scattering
V ≈ V1
Inverse Born approximation for V
Linear
Nonlinear…. Data must multiply itself
Assumes
G ≈ G0

15. Iterative linear G0 towards G Linear Linear Linear
Repeated linear inverse
Linear
Linear
Linear
Updates
G0 towards G
E.g., velocity model updating

16. Reference medium is never updated
Inverse scattering series
Inverse Series is solution for V in terms of G0 and measured G
Reference medium is never updated
V = V1 + V2 + V3 + …
Linear
2nd Order
3rd Order
Multiplication: data events communicate with each other
Nonlinear…. Data must multiply itself

17. Measured Scattered Field
Background and review
Seismic Inverse Problem
Measured Scattered Field
Moses (1956)
Razavy (1975)
Weglein et al. (1981)
Stolt and Jacobs (1980)
Perturbation V D
Free-surface multiple removal
Internal multiple removal
Imaging in space
Target identification

18. Imaging using the inverse series
(Increasing realism)
Production algorithm
Embryonic concepts
Prototype algorithm
Analysis and testing
Generalization
Idea
Non-linear, wavefield at depth
Pattern, isolate subseries
2D, elastic, variable background
Task separation
Taylor series
Risk
time

19. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

20. 1D acoustic inverse problem

21. 1D Earth, 3D wave propagation

22. Inverse scattering series
Linear
2nd Order
3rd Order
Nonlinear…. Data must multiply itself

23. 1D acoustic inverse series

24. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

25. where a1 is the data imaged with the constant reference velocity
Leading order imaging series
1-D with offset
where a1 is the data imaged with the constant reference velocity

26. Leading order imaging series
1-D/1-D

27. Leading order imaging series

28. Leading order imaging series

29. Leading order imaging series

30. Leading order imaging series

31. Leading order imaging series

32. Leading order imaging series

33. Leading order imaging series

34. Leading order imaging series

35. Communication with all deeper events
Depth imaging without the velocity model z

36. + Taylor Series at each mislocated interface
Depth imaging without the velocity model +
First term: image with
the reference velocity
Sum of second and higher terms
Sum of imaging series

37. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

38. How does it work? z A Shift: Taylor Series for: Wrong depth
Correct depth

39. A cascaded series Shift = zb = correct depth zb
zb = wrong depth
zb = correct depth
Shift =
zb zb
Coefficients hold the key to the correct depth
(Depth correction)
(Wrong depth, “time”)
(Data amplitude)
Required information resides in data’s
amplitudes and travel times

40. Higher order imaging contributions (Innanen, 2005)
Leading order shift
4R1
functions of p z
Higher order imaging contributions (Innanen, 2005)

41. Reference/Actual velocity contrast: 10%
Leading order imaging: 3 terms
correction a1
Reference/Actual velocity contrast: 10%

42. Reference/Actual velocity contrast: 10%
Sum 3 leading order terms
correction a1
Reference/Actual velocity contrast: 10%

43. Reference/Actual velocity contrast: 10%
Sum 5 leading order terms
correction a1
Reference/Actual velocity contrast: 10%

44. Leading order imaging series

45. Closed Form
(After R.G. Keys)

46. Convergence properties
Converges for finite kz and
Converges faster for
low kz
small

47. Depth errors ? When will Actual depth of reflector
Depth of reflector predicted by reference (in a1)
Depth of reflector predicted by LOIS

48. Depth errors, condition
when
Actual depth of reflector
Depth of reflector predicted by reference (in a1)
Reference vertical slowness
Actual vertical slowness

49. Analytic example Two interfaces: Transmission coefficient
is satisfied for prestack data independent of
Reflector depths, (za, zb)
Velocity, (z0, z1)

50. Random noise

51. Random noise

52. Coherent noise (internal multiples)
Residual (internal) multiples have two effects:
They will be imaged
They will impact the location at which primaries are imaged

53. Coherent noise (internal multiples)
Data travel times
Image depths
Residual internal multiple
Residual internal multiple

54. Coherent noise (internal multiples)
Data travel times
Image depths
Internal multiple
Internal multiple

55. Coherent noise (internal multiples)
Data travel times
Image depths
Residual internal multiple
Residual internal multiple

56. Coherent noise (internal multiples)
Data travel times
Image depths
Internal multiple
Internal multiple

57. Outline Motivation, objectives Background 1-D acoustic inverse series
A leading order imaging series
Analysis
Conclusions

58. Conclusions
Inverse scattering series has ability to image reflectors in space without requiring or solving for the actual velocity
An imaging subseries has been isolated that is an improvement over conventional imaging with the reference and has good convergence properties
Numerical tests: leading order imaging series algorithm does not require zero frequency to provide benefit

59. Conclusions
To improve imaging series accuracy and lessen low frequency dependence
Keep contrasts smaller (use best background velocity estimation)
Record lower frequencies: industry trend is to recording lower frequencies
Results encouraging therefore 2D, 3D and elastic generalizations progressing

60. Acknowledgments
Drs. Art Weglein, Ken Matson, Doug Foster, Hua-Wei Zhou and Stuart Hall
Craig Cooper, Hugh Rowlett, Rob Habiger
Bob Keys, Dennis Corrigan
M-OSRP colleagues – Kris Innanen, Bogdan Nita, Fang Liu, Haiyan Zhang, Jingfeng Zhang, Einar Otnes, Adriana Ramirez
M-OSRP sponsors, especially BP and ConocoPhillips