1. Interferometric Interpolation of 3D OBS Data Weiping Cao, University of Utah Oct. 29 2009

2. Outline Problems: Missing and sparse traces Methodology: Interferometric interpolation Numerical results: –3D layered model –Anti-aliasing condition for interferometric redatuming Conclusions

3. Problems: Missing and sparse traces Methodology : Interferometric interpolation Numerical results: –3D layered model –Anti-aliasing condition for interferometric redatuming Conclusions Outline

4. Motivations Problem: Receiver interval of OBS data is (sometimes) too large Solution: Interferometric interpolation Water Benefits of Interferometric Interpolation: Accuracy (wave-equation based scheme)Accuracy (wave-equation based scheme) No sedimentary velocity neededNo sedimentary velocity needed

5. Outline Problems: Missing and sparse traces Methodology: Interferometric interpolation Numerical results: –3D layered velocity model –Anti-aliasing condition for interferometric redatuming Conclusions

6. Interferometric Interpolation of OBS Data G(B|A) Interpolated OBS Data Seabed Reflectors Ocean Surface x B A G(x|A) Natural OBS Green’s Function Seabed Reflectors Ocean Surface x A G o (x|B)* Model based Green’s Function B Seabed Ocean Surface x A Dong S. and G. T. Schuster, 2008, Interferometric interpolation and extrapolation of sparse OBS and SSP data: UTAM 2007 annual meeting, 39 – 48.

7. Interferometric Interpolation of OBS Data 2-state reciprocity equation: Up-down separation, far-field approximation Water-layer reflection OBS reflection Artifacts? (up-down separation, far-field approx., limited aperture, wavelet, sampling… Matching filter!

8. Workflow Input Field Data Water Layer Model Generate GF for Water Multiples Interpolate M issing D ata Max. Itr (MF ) Get Virtual CSG Max Itr Intr/Extr Final CSG N Matching Filter N Y Y Time (s) 03.0 X (km) 04.5 Seabed Ocean Surface x Time (s) 03.0 X (km) 04.5 Time (s) 03.0 X (km) 04.5 Input Data Unfiltered Virtual Filtered Virtual

9. Outline Problems: Missing and sparse traces Methodology: Interferometric interpolation Numerical results: –3D layered velocity model –Anti-aliasing condition for interferometric redatuming Conclusions

10. Numerical Results 3D velocity model size: 3000 x 3000 x 1400 m 3 Source located at (10 m,10 m, 30 m) 300 by 300 receivers dx = dy = 10 m Sea bed is flat at a depth of 750 m 3 km 1.4 km Source

11. Layered Velocity Model Velocity (m/s)15002400 Sea bed Reflector 1 Reflector 2

12. Synthetic Data Line y=1000m 0 5 Time (s) 03000 X (m) CSG in the x direction: y=1000 m, dx = 10 m 0 5 Time (s) 03000 Y (m) CSG in the y direction: x=1000 m, dx = 10 m

13. 3D Interpolation Goal: dense OBS data Recording interval: 10 m × 10 m Total number of receivers: 300 × 300 = 90, 000 Input: sparse OBS data Recording interval 50 m × 50 m ( =104 m ) Total number of receivers: 60 × 60 = 3, 600

14. Sparse Data 0 5 Time (s) 03000 X (m) Line y=1000m Decimated CSG in the X direction: Y = 1000 m, dx = 50 m 0 5 Time (s) 03000 Y (m) Decimated CSG in the Y direction: X = 1000 m, dx = 50 m

15. Interpolation Results: X direction 0 5 Time (s) 03000 X (m) Line y=1000m Decimated CSG in the X direction: Y =1000 m, dx = 50 m 0 5 Time (s) 03000 Y (m) Virtual dense data, dx = 10 m

16. Time (s) 03.0 X (km) 04.5 Time (s) 03.0 X (km) 04.5 Local Matching Filter

17. 0 5 Time (s) 03000 X (m) 0 5 Time (s) 03000 X (m) Line y=1000m Filtered virtual data, dx = 10 m Decimated CSG in the X direction: Y =1000 m, dx = 50 m Interpolation Results: X direction

18. 0 5 Time (s) 03000 X (m) 0 5 Time (s) 03000 X (m) Line y=1000m Real dense data, dx = 10 m Decimated CSG in the X direction: Y =1000 m, dx = 50 m Interpolation Results: X direction

19. 0 5 Time (s) 03000 Y (m) Line y=1000m 0 5 Time (s) 03000 Y (m) Virtual dense data, dx = 10 m Decimated CSG in the Y direction: X =1000 m, dx = 50 m Interpolation Results: Y direction

20. 0 5 Time (s) 03000 Y (m) 0 5 Time (s) 03000 Y (m) Line y=1000m Virtual data after filtering, dx = 10 m Decimated CSG in the Y direction: X =1000 m, dx = 50 m Interpolation Results: Y direction

21. 0 5 Time (s) 03000 Y (m) 0 5 Time (s) 03000 Y (m) Line y=1000m Real dense data, dx = 10 m Decimated CSG in the Y direction: X =1000 m, dx = 50 m Interpolation Results: Y direction

22. 1.0 3.5 Time (s) 102710 X offset (m) True vs. Virtual traces before Filtering True Virtual Interpolation Results: Trace Comparison

23. 1.0 3.5 Time (s) True vs. the Virtual Traces after Filtering True Virtual 102710 X offset (m) Interpolation Results: Trace Comparison

24. 1.0 3.5 Time (s) True vs. the Virtual Traces after Filtering 102710 X offset Interpolation Results: Trace Comparison

25. Different Recording Spacings 0.5 1.2 Normalized error 0.2 0.05 Interpolation error vs. recording spacing Recording spacing of input data (λ x min ) The normalized error =

26. Outline Problems: Missing and sparse traces Methodology: Interferometric interpolation Numerical results: –3D layered velocity model –Anti-aliasing condition for interferometric redatuming Conclusions

27. Interferometric redatuming equation: Anti-aliasing Condition for Interferometric Redatuming Phase difference between and less than G(A|x) G(B|x) Anti-aliasing condition

28. 0 3 Time (s) 0.6 4 X (km) Remove Interf. Artifacts with the Anti-aliasing Condition Anti-aliased Interf. Result Regular Interf. Result Recording interval 0.49 λ 0 3 Time (s) 0.6 4 X (km)

29. 0 3 Time (s) 0.6 4 X (km) Remove Interf. Artifacts with the Anti-aliasing Condition Anti-aliased Interf. Result with Up-down Separation Regular Interf. Result with Up-down Separation Recording interval 0.97λ 0 3 Time (s) 0.6 4 X (km)

30. Outline Problems: Missing and sparse traces Methodology: Interferometric interpolation Numerical results: –3D layered velocity model –Anti-aliasing condition for interferometric redatuming Conclusions

31. Encouraging results obtained for interpolating sparse OBS data (recording spacing: ) Degraded interpolation results when the recording spacing of the input sparse data increases Remaining artifacts: up-down separation, anti-aliasing condition up-down separation, anti-aliasing condition

32. Acknowledgments  Thank UTAM 2008 sponsors for the support of the research.  Thank you all for your attention.