Outline

1. Simultaneous Signal and Noise Modelling via the Radon TransformM.D.Sacchi, C. Moldoveanu-Constantinescu and D. Trad (Veritas DGC)EAGE Paris 2004 Signal Analysis and Imaging Group Department of Physics & Institute for Geophysical ResearchUniversity of Alberta, Edmonton, AB, Canada

2. Outline • Motivation • Simultaneous Signal and Noise Modeling • Radon operators • Operator Classes • Radon Transform • Hybrid Transform • Radon Transform via Generalized convolution • Summary

3. Motivation • Build a general framework for the design and implementation of transformations for SNR enhancement, blind signal separation, and cross-talk attenuation.

4. Operator Classes • Brief description of 3 operator classes utilized to attenuate noise • Class I • Class II • Class III

5. Transform Methods - Class I Signal + Noise Random noise Same signal template over the full aperture Global Signal Focusing Operator Focused Signal + Cross-talk Unfocused Noise + Cross-talk Cross-talk can be minimized using sparse inversion

6. Transform Methods - Class II Signal + Noise Coherent noise Two or more signal templates over the full aperture Global Signal and Noise Focusing Operators Focused Signal + Cross-talk Focused Noise + Cross-talk

7. Transform Methods - Class III Signal + Noise Coherent and incoherent noise Template can be defined over a small aperture Local Signal and Noise Local Focusing Operators Modes capturing the signal Modes capturing the noise Focused Signal + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Synthesized Noise + Cross-talk Synthesized Signal + Cross-talk

8. Radon Transforms (Class I) Templates

9. Radon Transform (Class I) Only one integration path (single template) Focusing cannot be simultaneously achieved when more that one type of waveform is contained in the data

10. Linear Radon Transform Hyperbolic Radon Transform Data

11. Hybrid Radon Transform (Class II) Trad, Sacchi and Ulrych, 2001, JSE

12. Augmented Radon Operator L augmented Radon operator; m combined vector of Linear and Hyperbolic Radon domain parameters

13. Cost Function of the Problem Cn noise covariance matrix; Cm model covariance matrix

14. Solution: Minimizer of J In LS form: In Minimum Norm form:

15. General form of the covariance matrix of theaugmented problem If the linear and hyperbolic Radon panels are uncorrelated (Clh=Chl=0):

16. Estimator of the linear noise

17. Estimator of the linear noise

18. Estimator of the linear noise and the Wiener filter

19. Hybrid Radon Transform (Class II) p v v p m d Inverted m Sparse Inversion

20. Hybrid Radon Transform Recovered model of hyperbolic events Recovered Model of linear events

21. Field Shot Gather

23. Hybrid Radon Observations Prediction Residuals Linear Noise Signal

24. Transform Methods - Class III Class III methods can collapse to Class I and II depending on the selection of the template and template aperture Class III methods involve the concept of generalized convolution (convolvers) and generalized deconvolution. Simultaneous modeling of noise and signal is achieved by working with localized operator

25. Transform Methods - Class III Signal + Noise Coherent and incoherent noise Template define over a small aperture Local Signal and Noise Local Focusing Operators Modes capturing the signal Modes capturing the noise Focused Signal + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Focused Noise + Cross-talk Focused Noise + Cross-talk Focused Signal + Cross-talk Synthesized Noise + Cross-talk Synthesized Signal + Cross-talk

26. Transform Methods - Class III Local Wavefield Operator (LWO)

27. One Local Wavefield Operator (LWO) t (s) (nt=69,nx=17) h (m) We could have also used local parabolas, hyperbolas etc etc

28. One shifted and scaled operator:

29. One shifted and scaled operator: Linear superposition of one operator:

30. One shifted and scaled operator: Linear superposition of one operator: Superposition of many operator:

31. Generalized convolution In matrix form

32. Generalized convolution Modal Decomposition Data k mode

33. Generalized cross-correlation

34. Generalized Filtering

35. LS Inverse Transform

36. Example 1: Local Wavefield Operators 69 17 LWO

37. Example 1: Transform Invertivility One operator sliding on the data

38. Example 1: Modal decomposition

39. Example 1: Event Synthesis ??

40. Example 2: LWO

41. Example 2: Modal Decomposition FR: Full Reconstruction, PR: Partial Reconstruction

42. Example 3: Dealing with Alias Operator is designed in an unaliased grid Sampling operator is used to match the decomposition to the data grid

43. Example 3: FR and Reconstruction Error a) Original data b) Decimated data c) Reconstructed data d) Reconstruction error

44. Example 3: FK Spectra a) Original data b) Decimated data c) Reconstructed data d) Reconstruction error

45. Example 4: Noise enhanced by CMP stacking Data PR k=24:26 Noise Estimate An example where class I and II operators will not work Linear noise events do not span the complete aperture K=1,49 (LWOs) Coherent noise in marine seismic data, Larner et. al, 1983, Geophysics, 48, No. 7

46. Data PR k=24:26 Noise Estimate K=1,49 (LWOs)

47. Data PR k=24:26 True Data

48. Data PR k=21:29 Noise Estimate

49. Data PR k=21:29 True data

50. Summary Two parametric alternative to design transformations for SNR enhancement has been discussed Hybrid operators (Class II) can be used to model noise and signal when both components are well represented by two or more Radon templates. Data domain estimators of the noise can be constructed. All research so far has focused on covariance matrices with diagonal form. Localized and non-localized noise can be attenuated with Radon operators of Class III. These operators can be efficiently implemented via generalized deconvolution.