1. Gary Margrave and Michael Lamoureux
Gabor Deconvolution
Gary Margrave and Michael Lamoureux =

2. = Outline The Gabor idea The continuous Gabor transform
The discrete Gabor transform
A nonstationary trace model
Gabor deconvolution
Examples =

3. Gabor Papers in the Research Report
Gabor Deconvolution by Margrave and Lamoureux
Constant-Q wavelet estimation via a nonstationary Gabor spectral model by Grossman, Margrave, Lamoureux, and Aggarwalla
Gabor deconvolution applied to a Blackfoot dataset by Iliescu and Margrave
A ProMAX implementation of
nonstationary deconvolution
by Henley and Margrave =

4. The Gabor Idea
Dennis Gabor proposed (1946) proposed the expansion of a wave in terms of “Gaussian wave packets”.
The mathematics for the continuous Gabor transform emerged quickly.
The discrete Gabor transform
emerged in the 1980s due
to Bastiaans. =

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

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

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

8. Inverse Fourier transform Inverse Fourier transform
The Gabor Idea
The inverse Gabor transform done two ways
Window center time
frequency
Inverse Fourier transform
Sum
time
Window center time
Inverse Fourier transform
Sum

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

10. The Continuous Gabor Transform
Inverse transform
Gaussian
“synthesis” window
Fourier exponential and integration over f and t
The signal
The Gabor transform
The inverse Gabor transform is a 2D integration over the time-frequency plane.

11. A discrete Gabor transform
In the discrete case, Gabor transforms have proven difficult to implement without loss. The issue was finally “solved” in the 1980’s using “frame theory”.
We have implemented a simpler, approximate transform that is based on a convenient summation property of Gaussians.

12. A discrete Gabor transform
The Gaussian summation property.
Summation curve
Gaussian half-width .1 s window increment .05

13. A discrete Gabor transform
The Gaussian summation property.
We show the the sum of a discretely spaced set on Gaussians on an infinite interval is given by
Summation curve
Gaussian half-width .1 s window increment .05
First-order error term
is the Gaussian half-width
is the Gaussian spacing

14. A discrete Gabor transform
The Gaussian summation property.
is the Gaussian half-width
is the Gaussian spacing
Summation curve
Gaussian half-width .1 s window increment .05
First-order error term .5
1.0
1.5
2.0
-21 decibels
-85 decibels
-150 decibels
-340 decibels
The inter-Gaussian spacing should well less than the Gaussian half-width.

15. Gaussian summation test
Gaussian half-width .1 seconds.
0.1
0.25
0.5
1.5 2 1

16. Gabor transform test Unnormalized transform Difference
Seismic signal
After forward and inverse Gabor transform
Difference

17. Gabor transform test Normalized transform Difference
Seismic signal
After forward and inverse Gabor transform
Difference
Normalization is a simple process that reduces end effects.

18. Gabor transform example
Minimum phase wavelet
Reflectivity
Nonstationary synthetic Q = 25

19. Gabor transform example
Wavelet
Q decay curve
Reflectivity
No clipping

20. Gabor transform example
Q decay curve
Clipping

21. Stationary Convolution Model

22. Stationary Convolution Model
in the Fourier domain

23. Nonstationary Convolution Model

24. Nonstationary Convolution Model
We have proven that
Gabor transform of seismic signal
Fourier transform of wavelet
Q-filter
Gabor transform of reflectivity
is approximately given by

25. Proof
therefore

26. Nonstationary Convolution Model example
Gabor transform of seismic signal

27. Nonstationary Convolution Model example
Gabor transform of reflectivity

28. Nonstationary Convolution Model example
Q filter transfer function

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

30. Nonstationary Convolution Model example
wavelet surface

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

32. Nonstationary Convolution Model example
Gabor transform of seismic signal

33. Gabor Deconvolution Gabor transform of the seismic trace
Gabor transform of the reflectivity
Given this
Estimate this
Recall the nonstationary trace model:

34. Gabor Deconvolution

35. Gabor Deconvolution
So, given this:
We must estimate this:

36. Gabor Deconvolution Estimation of “the propagating wavelet”
Can be done by:
Smoothing
Time-variant Burg + smoothing
Least squares modeling of
Lotsa other ways …

37. Gabor Deconvolution Phase calculation
The phase can be locally minimum, implemented by a Hilbert transform over frequency, or locally zero.

38. Gabor deconvolution example
reflectivity
nonstationary seismic signal
nonstationary seismic signal
reflectivity

39. Initial Gabor spectrum

40. Initial Gabor/Burg spectrum

41. Smoothed Gabor spectrum

42. Smoothed Gabor/Burg spectrum

43. Deconvolved Gabor spectrum
Fourier algorithm

44. Deconvolved Gabor spectrum
Burg algorithm

45. Deconvolved Gabor spectrum
Actual reflectivity

46. Gabor deconvolution result
Fourier algorithm
reflectivity
Gabor deconvolution result

47. Gabor deconvolution result
Burg algorithm
Gabor deconvolution result
reflectivity

48. Least squares Q and wavelet estimates
Constant-Q wavelet estimation via a nonstationary Gabor spectral model by Grossman et al.
We fit a constant-Q wavelet model to the Gabor transform of the seismic signal to obtain expressions for a Q estimate and the wavelet amplitude spectrum.

49. Least squares Q and wavelet estimates
The nonstationary convolution model predicts
This, together with the nearly random nature of
suggests that we impose the model
where the equality is taken to hold in the least-squares sense.

50. Least squares Q and wavelet estimates
The Q and wavelet estimates emerge as integrations of the Gabor transform of the signal over a domain in the time-frequency plane.
The integration domain is bounded by hyperbolae that are level-lines of the constant Q operator:

51. Least squares Q and wavelet estimates
This method is still in development but we are hopeful that is will prove superior to simple smoothing as the method of spectral separation in Gabor deconvolution.

52. ProMAX Implementation
A ProMAX implementation of nonstationary deconvolution by Henley and Margrave
This module is in the current CREWES software release. The FORTRAN computation module is self-contained and can be easily ported to other systems.
Both Fourier and Burg methods are included with spectral separation by smoothing. The Q estimator is not there yet.
A word of caution: unwise choice of parameters can mean very long run times.

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

54. Blackfoot testing
Gabor

55. Blackfoot testing
Wiener

56. Blackfoot testing
Typical trace spectra
After Wiener
After Gabor

57. Blackfoot testing
Gabor/Burg stack

58. Blackfoot testing
Wiener stack

59. Gabor/Burg (pre and post stack) and final migration
Blackfoot testing
Gabor/Burg (pre and post stack) and final migration

60. Wiener (pre stack), TVSW and final migration
Blackfoot testing
Wiener (pre stack), TVSW and final migration

61. Conclusions
The discrete Gabor transform can be developed by simple Gaussian slicing.
The nonstationary extension of the convolution model predicts that the Gabor transform of a seismic signal decomposes into the product of the Fourier transform of the wavelet, the symbol of the Q filter, and the Gabor transform of the reflectivity.

62. Conclusions
Gabor deconvolution is implemented by a spectral factorization applied to the Gabor transform of a trace.
Gabor deconvolution generalizes Wiener deconvolution to the nonstationary setting.
Within the Wiener design gate Gabor and Wiener are similar; elsewhere, Gabor seems better.
A new Gabor deconvolution code is released and shows promising results.

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