1. The recovery of seismic reflectivity in an attenuating medium Gary Margrave, Linping Dong, Peter Gibson, Jeff Grossman, Michael Lamoureux University of Calgary, POTSI project

2. The Earth is not perfectly elastic Gary Margrave, Linping Dong, Peter Gibson, Jeff Grossman, Michael Lamoureux University of Calgary, POTSI project

3. Gabor Deconvolution Gary Margrave, Linping Dong, Peter Gibson, Jeff Grossman, Michael Lamoureux University of Calgary, POTSI project

4. This talk is about … The design and application of nonstationary inverse filters as Gabor multipliers … which are a generalization of the familiar Fourier multipliers … to remove the effects of seismic attenuation and source signature … which can be modelled as a pseudodifferential operator applied to the reflectivity function of the earth … which is an extension of the concept of convolution … the resulting nonstationary deconvolution extends Wiener’s stationary algorithm … an understanding of the physics of seismic waves is crucial.

5. Why this matters As seismic imaging moves into a new era where the direct estimation of lithology and pore fluids is possible, it is important to estimate the reflectivity with the highest accuracy and resolution possible. An estimate that is correct “modulo a smooth function” is not sufficient.

6. Purely elastic finite-difference simulation Distance from source Depth

7. Depth Purely elastic finite-difference simulation

8. Surface recording of vertical motion Receiver position First breaks Ground roll Reflection

9. Real Seismic (200 traces) after gain for spherical spreading Receiver position

10. Trace 100

11. Gabor spectrum of trace 100 after gain

12. The Gabor Idea A seismic signal A Gaussian slice or wave packet. A shifted Gaussian Multiply

13. The Gabor Idea A suite of Gaussian slices A seismic signal Remarkably, the suite of Gaussian slices can be designed such that they sum to recreate the original signal with high fidelity.

14. The Gabor Idea Fourier transform time Window center time Suite of Gabor slices Window center time frequency Gabor spectrum or Gabor transform The Gabor transform

15. The Gabor Idea The inverse Gabor transform done two ways Inverse Fourier transform Sum Inverse Fourier transform (Window) & Sum time Window center time frequency

16. The Continuous Gabor Transform The signal Gaussian “analysis” window Fourier exponential and integration over t The Gabor transform The Gabor transform is a 2D time-frequency decomposition of a 1D signal. Forward transform

17. Gabor transform via a partition of unity Given a partition Define analysis window and synthesis window

18. Gabor transform via a partition of unity Form a Gabor slice

19. Gabor transform via a partition of unity The forward Fourier transform over the set of Gabor slices gives the Gabor transform

20. Gabor transform via a partition of unity The inverse transform

21. Gabor spectrum of trace 100 after gain

22. Minimum Phase A causal signal with a causal inverse is minimum phase. Many physical systems have minimum phase impulse responses.

23. Minimum Phase The causality of a signal, s(t), and its inverse imply a symmetry of its Fourier transform: The phase spectrum is the Hilbert transform of the logarithm of the amplitude spectrum.

24. Minimum phase and attenuation Futterman (1962) showed that wave attenuation in a causal, linear theory is always minimum phase. The imposed symmetry in the Fourier domain was called a “Kramers-Kroenig” relation.

25. Constant Q model of attentuation Kjartanssen (1978) and others x is distance travelled, v is velocity Q is the rock “Quality factor” H is the Hilbert transform

26. Attenuation Simulation True Amplitude

27. Attenuation Simulation Normalized

28. Attenuation and nonstationarity Attenuation depends on path length. Therefore the seismic recording is inherently nonstationary, being a linear superposition of many different minimum-phase arrivals with differing degrees of attenuation. source is minimum phase each reflected arrival is minimum phase

29. nonstationary Nonstationary seismic trace model reflectivity source signature seismogram with attenuation

30. Nonstationary seismic trace model 0 0.5 1.0 seconds The nonstationary trace model is a pseudodifferential operator. Here is its depiction as a singular integral operator that, in the discrete case, is a matrix operator.

31. Stationary seismic trace model 1.00.50 seconds 0 0.5 1.0 seconds This more common trace model, which ignores attenuation, is expressed as a Toeplitz matrix

32. Stationary seismic trace model

33. in the Fourier domain

34. Wiener’s deconvolution measure the reflectivity and calculate its Fourier amplitude spectrum estimate the amplitude spectrum of the source signature. Assume minimum phase. calculate the reflectivity spectrum by spectral division

35. Wiener’s deconvolution This algorithm is enabled because the Fourier transform factorizes the convolution integral. An extension of this algorithm to the nonstationary case would be possible with a transform that factorizes a pseudodifferential operator. We have shown that the Gabor transform induces a factorization of a pseudodifferential operator that, while usually approximate, can sometimes be exact.

36. Stationary seismic trace model 1.00.50 seconds 0 0.5 1.0 seconds This more common trace model, which ignores attenuation, is expressed as a Toeplitz matrix

37. Nonstationary seismic trace model 0 0.5 1.0 seconds The nonstationary trace model is a pseudodifferential operator. Here is its depiction as a singular integral operator that, in the discrete case, is a matrix operator.

38. stationary Nonstationary seismic trace model nonstationary

39. Nonstationary seismic trace model We have proven that Gabor transform of seismic signal Projection Stuff we want to get rid of Gabor transform of reflectivity

40. Proof

41. Nonstationary Convolution Model example Gabor transform of seismic signal

42. Nonstationary Convolution Model example Gabor transform of reflectivity

43. Nonstationary Convolution Model example Q filter transfer function

44. Nonstationary Convolution Model example Q filter times Gabor transform of reflectivity

45. Nonstationary Convolution Model example wavelet surface

46. Nonstationary Convolution Model example wavelet times Q filter times Gabor transform of reflectivity

47. Nonstationary Convolution Model example Gabor transform of seismic signal

48. Gabor Transform Input signal (reflectivity) Attenuated signal (seismic record) seconds

49. Gabor Transform Time (sec) Each row is a Fourier transform for a different window position

50. Gabor Deconvolution Given thisEstimate this Recall the nonstationary trace model: Gabor transform of the seismic trace Gabor transform of the reflectivity

51. Gabor Deconvolution Reflectivity Data Model of attenuation and source sig.

52. Gabor decon step 1 Signal Forward Gabor transform Window and Fourier transform Gabor spectrum of signal

53. Input Gabor spectrum Gabor decon step 2 Spectral smoothing

54. Smoothed Gabor spectrum Gabor decon step 2 Spectral smoothing

55. Input Gabor spectrum Smoothed Gabor spectrum Gabor decon step 3 Spectral division

56. Gabor decon step 4 Bandlimiting

57. Gabor decon step 4 Bandlimiting

58. Gabor decon step 5 Inverse Gabor transform Reflectivity estimate

59. Gabor deconvolution result Trace comparison Hyperbolic smoothing

60. Gabor deconvolution result Fourier spectra comparison

61. Blackfoot testing Gabor deconvolution applied to a Blackfoot dataset by Iliescu and Margrave Gabor/Burg deconvolutionWiener spiking deconvolution

62. Gabor Blackfoot testing

63. Wiener Blackfoot testing

64. Gabor Deconvolution 0.5 1.0 0.0

65. State-of-the-art alternative 0.5 1.0 0.0

66. Gabor Deconvolution

67. State-of-the-art alternative

68. Validation of Gabor wavelets with a VSP 1000 m500 m 75 3C receivers in well at 20 m spacing 78 3C receivers on surface at 30 m spacing 500 m 1000 m 1500 m 5 shots

69. Comparison of wavelets Wiener Wiener estimates Frequency spiking VSP estimates Gabor estimates 0.5 s 0.8 s 1.1 s

70. Surface wavelet estimates (time domain) Wiener decon estimates Fourier spiking decon estimates Gabor decon estimates Wiener decon estimates Fourier spiking decon estimates Gabor decon estimates 0.5 s 0.8 s 1.1 s

71. Compare wavelets by crosscorrelation Wiener estimates Frequency domain spiking estimates Gabor estimates

72. Summary We have developed a new seismic deconvolution algorithm that corrects for both source waveform and (nonstationary) attenuation. Our method generalizes that of Wiener to nonstationary seismic records. Our method is based on the Gabor transform and succeeds in part because the Gabor transform approximately factorizes a pseudodifferential operator.

73. Acknowledgements MITACS: Mathematics of Information Technology and Complex Systems Imperial Oil Charitable Foundation PIMS: Pacific Institute of the Mathematical Sciences All of the following provided support NSERC: Natural Sciences and Engineering Research Council of Canada CREWES: Consortium for Research in Elastic Wave Exploration seismology =