M-OSRP 2006 Annual Meeting, June 7, 2007

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: 160-179

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) 4600 (m/s) 2463.75 (m/s) 3570 (m/s) 3855 (m/s) 4170 (m/s)

25. Shot records x x Source at = 0 m 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-0327778. DOE Basic Energy Sciences award DE-FG02-05ER15697.