1. Direct horizontal image gathers without velocity or “ironing”
Fang Liu, Arthur B. Weglein, Kristopher A. Innanen, Bogdan G. Nita, Jingfeng Zhang
M-OSRP 2006 Annual Meeting, June 7, 2007
M-OSRP report pages:

2. Key Points
Flattening of the common-image gather without knowing the velocity and waveform distortion
Best-effort plane-wave scenarios for the Zoeppritz equation in AVO analysis
Totally deterministic procedure and rich structures in the common-image gather

3. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

4. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

5. What is common-image gather (CIG)?
One more degree of freedom in the data than in the migration section.
Mapping 3 2
Different ways of mapping D(xg,xs,t) to M(x,z) at the same x location constitute a CIG.
Since there is only one earth, migration with different parameters should achieve the same depth, i.e., flat (horizontal) CIG.

6. CIG : current procedures
Velocity driven: If the velocity is correct, the CIG should be flat (horizontal). Flat CIG is a necessary condition for correct velocity.
Some may produces NMO stretches and other waveform distortions.
If CIG is not flat, “ironing” procedure can damage zero-crossing and other valuable information.
Not totally deterministic.

7. CIG : Inverse series approaches
Driven by the promise of the inverse scattering series: they should automatically give the same depth
Direct formula: flat CIG is no longer a necessary or sufficient condition to strive for
No waveform distortion in the plane-wave world
A natural by-product of the imaging subseries, and a totally deterministic procedure

8. Where does it come from?

9. Where does it come from? Migrated section Input data
Horizontal red lines are drawing to bench-marking the flatness of events

10. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

11. Theory Solving for the wave equation,
with the help of wave propagation in the much simpler reference medium,

12. Solution: the inverse scattering series

13. Solution for the linear term
Solution in the wave-number domain
The triple Fourier transform of the data equals to the double Fourier transform of
We should choose a slice of spectrum in the data to reach an image.
We choose different slices of spectrums such that each slice corresponds to a plane-wave incidence experiment.

14. Essential element 1 : angle θ
It is the incidence angle of the plane-wave. Zoeppritz equation is for many plane waves with different incidence angles.
We don’t have plane wave from the original data. The plane wave can be synthesized by Radon transform (slant stacking, or tau-p transform).
H. Zhang & Weglein 2004

15. Essential element 2 : CMP gather
Image is formed in the CMP gather (i.e., NMO stacking, Clayton & Stolt 1981)
Liu et al. 2005

16. Combining two elements
(1)
(2)
Each slice with a fixed angle θ corresponds to the data from the experiment of a plane-wave with angle θ as the incidence angle.

17. First part of our imaging formula: the linear term
Construct a plane-wave in the CMP gather
Receiver
location
Source
location
Time

18. Second part of our imaging formula: Higher-order imaging subseries (HOIS)
Is the partial capture of the imaging capability of the imaging subseries
More imaging capability than the leading-order subseries to deal with large contrast
Amplitude is left untouched for later AVO analysis
Lightening speed

19. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

20. The promise of the imaging subseries
Accurate image of all reflectors at depth using water-speed, for any angle θ.
Imaging results (locations) from different angle should be the same.
Amplitude is left untouched.

21. The capability captured by current HOIS
Imaging reflectors very close to the actual depth using water speed.
Imaging results (locations) from different angle are much closer than the linear image.
Amplitude is left untouched.

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

23. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

24. Numerical examples x z Big contrast 1500 (m/s) 2328.75 (m/s)

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

26. Linear image : θ=0° D C A B

27. Linear image : θ=9°

28. Higher-order image : θ=0°

29. Higher-order image : θ=9°

30. Case 1: right boundary D

31. D

32. D

33. Case 2: Middle C

34. C

35. C

36. Sum of 11 angles ( )

37. Higher-order imaging :

38. Sum of 11 angles ( )

39. Higher-order imaging :

40. Outline Common image gather Theory
The promise of the imaging subseries and our current capture
Numerical examples
Conclusions and acknowledgements

41. Conclusions
Flattening of the common-image gather without knowing the velocity and waveform distortion
Best-effort plane-wave scenarios for the Zoeppritz equation in AVO analysis
Totally deterministic procedure and rich structures in the common-image gather

42. Acknowledgments M-OSRP members. GX-Technologies for the scholarship.
M-OSRP sponsors.
NSF-CMG award DMS
DOE Basic Energy Sciences award DE-FG02-05ER15697.