1. Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California, Santa Cruz Sanya Symposium, 2011

2. Outline Introduction: Physical wavelet Time-slicing and depth-slicing of 4-D data Physical wavelet defined on observation planes: Dreamlet Dreamlet and beamlet propagator and imaging Applications Conclusion

3. Introduction Wavefield or seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways. Wave solutions can only exist on the light cone (hyper-surface) in the 4D Fourier space defined by dispersion relation. Physical wavelet is a localized wave solution by extending the light cone into complex causal tube. dreamlet can be considered as a type of physical wavelet defined on an observation plane (data plane on the earth surface or extrapolation planes at depth z in the migration/imaging process).

4. Physical wavelet Physical wavelet: localized wave field defined in the 4-dimensional time-space, satisfies the wave equation: – Globally for homogeneous media; – Locally for inhomogeneous media Localized by analytic extension to the complex 4-D time-space Only exit on the causal tube (nature of wave solution)

5. Special features of seismic data Seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways. The time-space distributions must observe causality which is dictated by the wave equation. Wave solutions can only exist on the light cone (hyper-surface) in the 4D Fourier space. Often the data are only available on the surface of the earth (the observation plane)

6. 4D Fourier domain 4D space-time domain Wavefield data are solutions from the wave equation:

7. where space-time four-vector wavenumber-frequency four-vector by the wave equation absolute value of frequency Lorentz-invariant scalar product Measure on the light-cone (Minkowski measure)

8. Light cone in the Fourier space ( )

9. light cone Lorentz-invariant measure on C Light cone Wave equation solutions satisfy the dispersion relation (causality) And therefore can only exist on the “light cone”

10. Space-time light cone (from Wikipedia)

11. Light cone: skeleton of the causal tube 

12. Construction of localized wave solutions Kaiser 1994 (Analytic signal transform) Kiselev and Perel, 2000; Perel and Sidorenko, 2007 (Continuous wavelet transform)

13. To construct the wavelets (localized wave solution) (Physical wavelet ), extend from the real space-time to the causal tube in complex space-time, by applying the analytic-signal transform where is the unit step function and Is an acoustic wavelet (physical wavelet)

14. Analytic Signal Transform and Windowing in the Fourier domain is an acoustic wavelet of order  in the Fourier domain. the AST can be looked as a windowing in the Fourier domain (windowed Fourier transform)

15. Space localization at t=0 r

16. Time localization at r=0 (real part- solid; imag- dashed) t

17. Wavefield data on planes:  Data acquisition plane on surface  Extrapolation planes during migration/imaging

18. Surface Extrapolated planes Data acquisition on the surface Wave field downward continuation Depth migration by downward-continuation or Survey sinking + Imaging condition

19. Two different decomposition schemes For Time-slices: All the space-axis are symmetric Depth-slices: Time-axis and space axes are different and need to be treated differently

20. Time-slicing in 4-D A time-slice

21. Depth-slicing in 4D Depth (x)(x) A depth slice

22. Two different decomposition schemes For time-slices: All the space-axis are symmetric: e.g. Curvelet Depth-slices: Time-axis and space axes are different and need to be treated differently: e.g. Pulsed-beam; wavepacket; Dreamlet (Drumbeat-beamlet)

23. Dreamlet: A type of physical wavelet defined on observation planes (data planes) Wu et al., 2008; 2009; 2011 (SEG abstracts)

24. Dreamlet (localized time-space solution of wave equation) Dreamlet: Physical wavelet on a plane x=(x,y) Time-space wavelet (directional wavepacket, “pulsed beam”) through dispersion relation: :Drumbeat;:Beamlet

25. Construction of dreamlet atoms: Drumbeat (t-f atom) beamlet (x-k atom) Windowing in frequency and horizontal wavenumber domains  Windowing on the light-cone (through the dispersion relation)

26. Dreamlet = Wavepacket Windowing on the light cone

27. Integration on the light-cone On the light cone we have ( and k as variables) New measure on the light-cone The integration on the light cone for wave solution:

28. Discrete wavelet atoms obtained by windowing on the light-cone  Dreamlets Discrete wavelet transform (Orthogonal or sparse frame) vs. Continuous wavelet transform

29. The window defined on the observation plane (red segment) and window for the whole space (green disk).

30. Dreamlet = Wavepacket Windowing on the light cone

31. Examples of dreamlet decomposition on seismic data

32. The poststack data of SEG 2D salt model

33. Dreamlet decomposition of the SEG salt data by local exponential frames t-f

34. Dreamlet decomposition of the SEG salt data using different window widths of drumbeat: 32 points t-f

35. Dreamlet decomposition of the SEG salt data using different window widths of drumbeat: 16 points

36. Dreamlet decomposition of the SEG salt data using different window widths of drumbeat: 8 points

37. Dreamlet decomposition of the SEG salt data using different thresholds: 1% f - t

38. Dreamlet decomposition of the SEG salt data using different thresholds: 2%

39. Dreamlet decomposition of the SEG salt data using different thresholds: 3%

40. Dreamlet decomposition of the SEG salt data using different thresholds: 4%

41. Compression Ratio (CR) for Dreamlet decomposition of seismic data Figure 1: Comparison Ratios of different decomposition methods (SEG/EAGE salt model poststack data). Dreamlets Curvelets Beamlets (Local-cosine basis)

42. Curvelet (a type of directional wavelet frame) Curvelet tiling in the wavenumber domain (on the left) & the corresponding spatial localization

43. Curvelet coefficients of the SEG 2D salt poststack data

44. Curvelet coefficients: finest scale

45. Features of Dreamlet and Beamlet Different levels of localization Wave data decomposition and compression Wave propagation, scattering and imaging Imaging in compressed domain Other applications: Illumination, resolution, velocity analysis and tomography, demultiples

46. Beamlet Localization (space-direction) Figure 4: Spreading of beamlet ( )propagation. Top is the beamlet of, and bottom.

47. Dreamlet localization (t-f-x-k) Figure 3: Snapshots of a single dreamlet propagation. On the left is the dreamlet of,and on the right,.

48. Beamlet localization (Space-direction localization) Space localization  Local perturbation theory: Beamlet propagator – Efficient migration algorithm in strongly heterogeneous media Direction localization  Local angle domain analysis: – Local imaging matrix and angle gathers – Energy-flux Green’s function – Directional illumination analysis (DIA) – Local resolution analysis – Local inversion

49. SEG 2D Salt model

50. Local perturbation vs. global perturbation Global references and global perturbationsLocal references and local perturbations

51. Illumination analysis and True-reflection imaging Directional illumination analysis Acquisition-aperture correction in the local dip-angle domain with beamlet migration

52. image by common-shot prestack G-D migration Total Acquisition-Dip-Response intensity from all the 325 shots Total illumination intensity from all the 325 shots

53. Acquisition-Dip-Response (horizontal) from all the 325 shots Acquisition-Dip-Response (45  down from horizontal) from all the 325 shots image by common-shot prestack G-D migration

54. With z-dependent gain After dip angle domain acquisition –aperture correction) ZOOM-IN COMPARISON for the 2D SEG-EAGE salt model

55. 速度模型 (Velocity model on slice C of the SEG 3D salt model) Example of 3D true-reflection beamlet migration (see Mao and Wu)

56. True-reflection image (right) vs. standard migration (left) 普通成像 ( 左 ) 和真反射成像 ( 右 ) 的对比

57. Dreamlet localization (Full phase-space localization) Efficient seismic data decomposition (Ideal decomposition) Dreamlet propagator and migration – Link to fast asymptotic wave-packet propagation – Imaging in the compressed domain

58. Changes of dreamlet coefficients with depth during Shot-domain prestack migration Scattered field (data) Source field Scattered field (high-compression) CR=5.6 CR=15.2

59. Coefficient changes during dreamlet survey-sinking prestack depth migration Variation of dreamlet coefficient amount during migration. The black line is for the survey sinking dreamlet coefficients using sunk data. Full data Sunk data

60. Conclusion Wave solutions can only exist on the light cone in the 4D Fourier space defined by the dispersion relation Physical wavelet defined by Kaiser is a localized wave solution by extending the light cone into complex causal tube. The effect is windowing on the light-cone. Dreamlet can be considered as a type of discrete physical wavelet defined on an observation plane

61. Conclusion-continued Curvelet is good for decomposition of time- slice 4-D data; while dreamlet is good for depth-slice 4-D data. Causality (or dispersion relation) built into the wavelet (dreamlet) and propagator is a distinctive feature of physical wavelet which is advantageous for applications in wave data decomposition, propagation and imaging.

62. Conclusion-continued The applications in illumination, true- reflection imaging, local angle domain analysis, imaging in compressed domain are only in the beginning.

63. Acknowledgments This is a Group effort mainly conducted in the Modeling and Imaging Lab at UCSC. I thank all my colleagues and students. Bangyu Wu, Yu Geng and Jian Mao directly involved in the work of this talk. I am grateful to Chuck Mosher for initiating the study of wavelet transform on wave propagation and the continuous interaction with our group. I thank Jinghuai Gao for the collaboration, Dr. Howard Haber and Dr. Gerald Kaiser for their discussions and comments. This work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz.