Kinematics of 2D specularly-reflected and diffracted multiples in data space and image space

1. Kinematics of 2D specularly-reflected and diffracted multiples in data space and image space Gabriel Alvarez Stanford University

2. Goal Understand how specularly-reflected and diffracted 2D multiples map to subsurface image gathers when migrated with wave equation migration.

3. The Problem Moveout-based multiple attenuation algorithms in image space benefit from the power of migration to handle the complex wave propagation of the primaries. The question remains: What is the moveout of the multiples in image space?

4. Outline • Moveout of 2D diffracted and specularly-reflected multiples • in data space • Mapping of multiples from CMPs to Subsurface Offset Domain • Common Image Gathers (SODCIGs) • Mapping of multiples from SODCIGs to Angle Domain • Common Image Gathers (ADCIGs) • Discussion and conclusions

5. Surface vs. Subsurface Offset hD mD hD

6. mD mξ mξ hD hξ SODCIGs ADCIGs Depth Time Depth Data Space Image Space Image Space Data Space vs. Image Space CMPs

7. ts2 tr2 ZD Moveout of speculary-reflected multiples Flat water-bottom: hD mD hD V ts1 ts2 tr2 tr1 ZD

8. hD hD mD V ZD φ 2φ Moveout of speculary-reflected multiples Dipping water-bottom:

9. ts2 tr2 ZD Moveout of Diffracted Multiples Flat water-bottom: hD mD hD V ts1 ts2 tr2 tr1 ZD Xd

10. ZD αs: takeoff angle of the source ray Moveout of Diffracted Multiple Dipping water-bottom: hD mD hD V Xd φ 2φ

11. Moveout Comparison Dipping water-bottom

12. Moveout of Multiples in Subsurface Offset Domain Common Image Gathers (SODCIGs)

13. V2=ρV1 βr βs mξ hξ hξ Image Coordinates of Non-diffracted Multiple Flat water-bottom: hD mD hD αs αr V1

14. Half-subsurface offset (m) -400 -200 0 200 400 1000 Depth (m) 1200 1400 SODCIG Specularly-reflected multiple. Flat water-bottom

15. Half-subsurface offset (m) -400 -200 0 200 400 1000 Depth (m) 1200 1400 SODCIG Specularly-reflected multiple. Flat water-bottom

16. ~ ts2 ~ tr2 βs+φ βr-φ mξ hξ hξ V2 ~ ~ ~ ~ ~ Image Coordinates of Non-diffracted Multiple hD hD mD αr ts1 tr1 αs+φ αr-φ V1 φ

17. Horizontal position (m) 1200 1400 1600 1800 2000 800 1200 Depth (m) 1600 Constant subsurface-offset section Specularly-reflected multiple from a dipping water-bottom

18. Half-subsurface offset (m) -800 -400 0 400 800 600 Depth (m) 1000 1400 SODCIG Specularly-reflected multiple from a dipping water-bottom

19. ~ ~ ~ ~ ~ Image Coordinates of Diffracted Multiple hD mD hD V1 ts1 tr1 Zdiff αr αs Xdiff βr βs V2 ~ ~ tr2 ts2 mξ hξ hξ

20. Horizontal position (m) Horizontal position (m) 2000 2000 2400 2400 2600 2600 2800 2800 3000 3000 800 800 Depth (m) Depth (m) 1200 1200 1600 1600 Constant subsurface-offset sections Half-subsurface offset 0 m Half-subsurface offset -200 m Diffracted multiple from a flat water-bottom

21. SODCIGs Half-subsurface offset (m) Half-subsurface offset (m) Half-subsurface offset (m) -400 -400 -400 0 0 0 400 400 400 1000 1000 1000 1200 1200 1200 Depth (m) Depth (m) Depth (m) 1400 1400 1400 1600 1600 1600 Diffracted multiple from a flat water-bottom

22. ~ ts2 ~ tr2 ~ ~ ~ ~ ~ Image Coordinates of Diffracted Multiple hD mD hD αr V1 ts1 αs+φ Zdiff tr1 V2 βs+φ αr-φ βr-φ mξ φ hξ hξ

23. Horizontal position (m) Horizontal position (m) 1600 1800 2000 2200 2400 1600 1800 2000 2200 2400 1200 1200 1400 1400 Depth (m) Depth (m) 1600 1600 1800 1800 Constant subsurface offset sections Half-subsurface offset 0 m Half-subsurface offset -200 m Diffracted multiple from a flat water-bottom

24. SODCIGs Half-subsurface offset (m) -800 0 800 1000 1200 Depth (m) 1400 1600 1800 Half-subsurface offset (m) Half-subsurface offset (m) -800 0 800 -800 0 800 1000 1000 1200 1200 Depth (m) Depth (m) 1400 1400 1600 1600 1800 1800 Diffracted multiple from a dipping water-bottom

25. Moveout of Multiples in Angle-Domain Common-Image-Gathers (ADCIGs)

26. ADCIG for specularly-reflected multiple βr βs ~ ~ 2γ tr2 ts2 mξ (xrξ,zrξ) (xsξ,zsξ) hξ hξ (xγξ,zγξ)

27. Half-aperture angle (degrees) 0 10 20 30 40 1200 Depth (m) 1400 1600 ADCIG Specularly-reflected multiple from a flat water-bottom

28. Half-aperture angle (degrees) -40 -20 0 20 40 1000 Depth (m) 1400 1800 ADDCIG Specularly-reflected multiple from a dipping water-bottom

29. ADCIGs Half-aperture angle (degrees) -40 0 40 1200 Depth (m) 1400 1600 Half-aperture angle (degrees) Half-aperture angle (degrees) -40 0 40 -40 0 40 1200 1200 Depth (m) Depth (m) 1400 1400 1600 1600 Diffracted multiple from a flat water-bottom

30. ADCIGs Half-aperture angle (degrees) -40 0 40 1400 Depth (m) 1600 2000 Half-aperture angle (degrees) Half-aperture angle (degrees) -40 0 40 -40 0 40 1400 1400 Depth (m) Depth (m) 1600 1600 2000 2000 Diffracted multiple from a dipping water-bottom

31. Discussion and Conclusions

32. hD mD hD V ZD ZD 2hξ<0 Discussion Water-bottom, specularly-reflected multiples, migrated with sediment velocity migrate to negative subsurface offsets.

33. hD mD hD V ZD ZD 2hξ>0 Discussion On the other hand, primaries, migrated with slower velocities map to positive subsurface offsets.

34. hD mD hD hD mD hD V ZD V ZD ZD 2hξ<0 2hξ>0 Discussion Diffracted multiples may map to positive subsurface offsets in SODCIGs even if migrated with faster velocities.

35. Conclusions Specularly-reflected water-bottom multiples migrate as primaries at twice the water depth and with twice the dip. Diffracted multiples do not migrate like primaries but their moveout in both SODCIGs and ADCIGs can be computed if the location of the diffractor is known.

36. Conclusions Better understanding of the moveout of the multiples in SODCIGs and ADCIGs will help in designing more accurate Radon transforms to attenuate the multiples in image space.

37. Thank you for your attention. I will be happy to entertain your questions.

38. 2γ E F (mξγ,zξγ) D From SODCIGs to ADCIGs βs βr (xsξ,zsξ) mξ (xrξ,zrξ) hξ hξ A C B