1. Multi-dimensional depth imaging without an adequate velocity model
Fang Liu
Arthur B. Weglein
Kristopher A. Innanen
Bogdan G. Nita
University of Houston
Houston, Texas
April 21th, 2005

2. Key Points Migration results True Earth
Data communicate with itself, no need for adequate velocity model
Purposeful perturbation
Physical interpretation: Taylor expansion
Migration results
True Earth

3. Main issue
Success in imaging reflectors in the Earth is closely tied to our ability to find the velocity model.
In complex geological environments, we are often unable to find an adequate velocity model.

4. Objectives Our approach
To improve our ability to accurately locate reflectors, point scatters, especially in areas where the velocity model is difficult to estimate.
Our approach
Inverse scattering series

5. Background of main issue
Current migration and inversion:
FK, Phase-shift, Kirchhoff

6. Background of main issue
Current migration and inversion:
FK, Phase-shift, Kirchhoff

7. Inverse scattering approach
Linear
2nd Order
3rd Order
Free-surface multiple removal
Internal multiple attenuation
Imaging
Inversion

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

9. Equations to be solved:

10. Task separation: Second term

11. Numerical example
Geological model:
300 m
400 m
200 m

12. A sample shot-gather
Modeling algorithm : finite difference

13. X
1000(m) α1 Z
Stolt migration
50(m)

14. X Z 1000(m) Nothing to move the first reflector. 50(m)
Move the second reflector.

15. X
1000(m) Z
50(m)

16. Task separation: Third term
More significant term:
Less significant term:

17. Physical interpretation : Taylor series expansion

18. Possible patterns and subseries
Let’s define:
Can be extended by both:
As:

19. Closed forms

20. Numerical example
versus

21. X
1000(m) Z
50(m)

22. X
1000(m) Z
50(m)

23. Model with three interfaces
1500 m/s
300 m
500 m
1600 m/s
200 m
200 m
1500 m/s
1600 m/s

28. Large contrast model 1500 m/s 2000 m/s 3000 m/s 300 m 500 m 200 m

33. Tentative model

34. Preprocessing requirements
Objective of the research:
2D acoustic medium with 1 parameter (velocity only)
Preprocessing needed in this example:
Wavelet estimation
Deghosting
Free surface multiple removal
Internal multiple attenuation

35. Conclusions All existing tests so far are VERY encouraging.
Depth accurate and velocity independent imaging algorithms.
Closed forms with demonstrated value.

36. Acknowledgments
M-OSRP members
M-OSRP sponsors