1. POTSI Gaboring Advances
Gary Margrave, Linping Dong, Peter Gibson, Jeff Grossman, Michael Lamoureux
POTSI

2. This is about … The Utility of the Gabor transform
Theoretical advances in Gabor theory
Field testing of Gabor deconvolution

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

4. The key to a fast, robust Gabor transform
A Partition of Unity
The key to a fast, robust Gabor transform

5. Gabor transform via a partition of unity
Given a partition
Define analysis window
and synthesis window
so that

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

7. Gabor transform via a partition of unity
The inverse Gabor transform is an inverse Fourier transform, multiply by the synthesis window, sum over windows

8. The Utility of the Gabor Transform
The Gabor transform is a natural extension of the Fourier transform into the nonstationary realm.
Gabor deconvolution is enabled because the Gabor transform approximately “factorizes” the nonstationary seismic trace (Margrave et al, this report)
The product of two Gabor spectra is a nonstationary convolution. A nonstationary convolution is a pseudodifferential operator.
Based on a “partition of unity”, a numerical Gabor transform can easily accommodate an irregular sampling lattice (Lamoureux et al, Grossman et al, this report)
Whatever you did with the Fourier transform you can “sorta” do “nonstationarily” with the Gabor transform.

9. What about the Wavelet Transform?
Applications are largely restricted to data compression and de-noising.
There is no convolution theorem for the Wavelet transform.
The product of two Wavelet transforms is not anything useful.

10. Advancements in Gabor Theory
Gabor transforms can exactly factorize certain pseudodifferential operators if “compatible” window pairs are used (Gibson et al)
We have identified a small number of compatible window pairs.
The use of “non-compatible” windows merely means that any factorization will be approximate.

11. Compatible windows
The Gabor analysis and synthesis windows are called compatible if
This is precisely the condition needed to solve unambiguously for the “Gabor multiplier” given the pseudodifferential operator.

12. Compatible windows Unity and Delta:
Leads directly to K-N pseudodifferential operators

13. Compatible windows
Gaussians:
Extreme Value:

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

15. Gabor spectrum of trace 100 after gain

16. Attenuation and nonstationarity
each reflected arrival is minimum phase
source is minimum phase
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.

17. Gabor Deconvolution a) b) c) d)

18. Comparison on Synthetic

19. Real Data Comparison Stratigraphic line provided by Husky Energy
Dynamite source
Data processing provided by Sensor Geophysical (Peter Cary)

20. Gabor -> Stack -> Gabor (160 Hz)
Real Data Comparison
Standard flow:
Gain->Surface Consistent Wiener -> TVSW -> Stack ->TVSW (120 Hz)
Gabor flow:
Gabor -> Stack -> Gabor (160 Hz)

21. Standard Processing
200
400 ms
600
800

22. Pre and post stack Gabor
200
400 ms
600
800

23. Standard Processing
200
400 ms
600
800

24. Pre and post stack Gabor
200
400 ms
600
800

25. Standard Processing
200 ms
400
600

26. Pre and post stack Gabor
200 ms
400
600

27. 200 ms
400
600
Gabor
standard
Gabor

28. Standard Processing
200 ms
400
600

29. Pre and post stack Gabor
200 ms
400
600

30. Summary
The Gabor transform is a very promising tool for exploration seismology
The Gabor transform extends Fourier concepts to the nonstationary realm
Gabor succeeds because it can factorize pseudodifferential operators
Gabor deconvolution easily beats Wiener and edges out Wiener+TVSW

31. Research Goals Incorporate well information as a constraint
Better phase estimation (remove more delay)
Develop theory on nonstationary minimum phase filters
Extend to multiple attenuation
Gabor wavefield extrapolator

32. = 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
GEDCO, Husky Energy, Encanna =