1. Multi-dimensional depth imaging without an adequate velocity model: towards more challenging geology
Fang Liu, Arthur B. Weglein, Kristopher A. Innanen, Bogdan G. Nita
Annual report:
page
Houston
May 11th, 2006

2. Why ?
The difficulty to get a clear image beneath salt or other complicated overburdens.
All the current migration procedure require :
to get the correct image.
Accurate velocity model
Adequate propagator

3. Current Procedures (first approach)
Velocity
analysis
Adequate velocity
Seismic
migration
Correct Depth Image

4. Our Procedures (second approach)
Velocity
Independent
Imaging
Correct Depth Image

5. How? Inverse scattering series.
The only procedure with the capability of directly achieving all the processing objectives without knowing or determining the actual wave propagation.
Removed multiples without assuming any subsurface information.

6. Key points
New direct imaging method: (a) accommodate inadequate velocity, (b) non-linear in the data.
More imaging terms had being identified and included in subseries with closed-forms.
Encouraging numerical results.
New understanding & more general framework.

7. Background: Perturbation Theory
Actual Medium Reference Medium Perturbation
L – L = V
Reference
Velocity
Perturbation
Actual Velocity
Actual Field - Reference Field = Scattered Field
G G = D

8. Background : Lippmann-Schwinger

9. Background: Born Approximation
FK, Phase-shift, Kirchhoff

10. Background: inverse series
True earth (in perturbation form)
Current pre-stack migration-inversion (constant c0)
Reference velocity is never updated.

11. Background: Imaging subseries
Inversion
Terms with different purposes identified
Subseries identified to capture certain part of the imaging capability.
Closed-form found

12. Assumptions: Remove direct wave and ghosts Known source wavelet
Remove free-surface multiples
Remove internal multiples

13. Physical interpretation : Cascaded Taylor series in multi-D
Express arbitrary function in Taylor expansion :
True Earth
Current migration-inversion
Why cascaded ? The simplest imaging problem is cascaded.

14. Closed forms Explicitly defined
Why should it be cascaded ? Series within another
Implicitly defined
(1) Shift in the z-direction,
(2) amplitude is not modified,
(3) Shift is defined in terms of α1
Final Result
map
Horizontal moving :

15. Large contrast model 1500 m/s 2000 m/s 3000 m/s
Contrast is greatly increased to break the leading imaging subseries

16. Perfect water-bottom
Uneven
pull-up
What will do ?

17. The leading order imaging subseries

18. Shifted leading order imaging subseries
An example of recursive modification

19. The higher order imaging subseries
Water-bottom untouched
Applying the moving term of simultaneous imaging - inversion

20. New understanding Why the moving term implicitly defined?
Innanen K. A , “Reflector location using high-order inverse scattering terms”
M-OSRP Annual report, (2005)

21. From equation (4) in Kris’s last annual report

22. The intuitive leap can be derived naturally as a result of proper integration of the δ–function. And ignoring the amplitude term (Jacobian term).

23. Salt model x z Lateral variation no longer restricted to water bottom
Big contrast
1500 (m/s)
(m/s)
4600 (m/s)
(m/s)
3570 (m/s)
3855 (m/s)
4170 (m/s)

24. Shot records x Source at = 0 m x Source at = 3000 m t t
Conflicting hyperbola

25. Linear imaging result
Strong residual diffractions because we used water to migrate
Bowtie still visible

26. Basic features stay the same as simple models

27. Higher order imaging subseries

28. Linear imaging

29. Higher order imaging subseries

30. Terms dealing with diffractions
First term in without 1D analogy
The linear image

31. Second term ( )
Generalize 1D terms
Terms with no 1D analog

32. Tip of an iceberg
It belongs to at least the following 2 subseries

33. More general framework
Generalize
Issues: missing low frequency

34. Corresponding closed forms
generalize

35. Linear imaging :

36. Linear imaging :

37. Higher-order imaging :

38. Higher-order imaging :

39. Sum of 11 angles ( )

40. Sum of 11 angles ( )

41. Conclusions
New direct imaging method: (a) accommodate inadequate velocity, (b) non-linear in the data.
More imaging terms had being identified and included in subseries with closed-forms.
Encouraging numerical results.
New understanding & more general framework.

42. Acknowledgments M-OSRP members ExxonMobil Upstream Research Company
M-OSRP sponsors
NSF-CMG award DMS
DOE Basic Sciences award DE-FG02-05ER15697