Gabor Deconvolution

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Gabor Deconvolution
Extending Wieners method to nonstationarity
Gary Margrave, Linping Dong, Peter Gibson, Jeff
Grossman, Dave Henley, Michael Lamoureux
POTSI
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This is about
The results of research at the University of
Calgary to develop a deconvolution technique that
directly addresses attenuation. The goal higher
resolution, true amplitude reflectivity estimates.
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Limerick
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Limerick
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Limerick
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Limerick
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So

• Who were these guys?
• What was on the other side of the door?
• Why was the door locked?
• How did the window help?
• What does this have to do with deconvolution?

(Thanks to Chris Harrison of CREWES for the
cartoons.)
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Joseph Fourier (1768-1830)
Invented what we now call Fourier analysis to
solve the heat equation. This was done in an
effort to understand the flow of heat when boring
a cannon barrel.
ol Joe Fourier
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Example of Fourier Analysis
48 Hz
38 Hz
Fourier components
28 Hz
19 Hz
9 Hz
signal
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Example of Fourier Analysis
Amplitude spectrum
Phase spectrum
Frequency
Signal
Time
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Fourier Analysis

• Study of the spectrum of signals
• Factorization of convolution (fast computer
  algorithms)
• Simple concept for deconvolution (Wiener)

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Dennis Gabor (1900-1971)

• Nobel Prize for the invention of the hologram.
• Influential 1946 paper in Theory of
  Communication proposed the expansion of a signal
  in Gaussian wave packets.

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Gabors Door
Gabor was standing in the world of stationary
signals that are appropriately decomposed into
sinusoids.
?
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Gabors Door
On the other side of the wall, is the real world
of nonstationary signals. Though Fouriers
analysis could still apply, its meaning was
unclear.
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Fouriers Window
Gabor realized that Fouriers analysis became
much more physically appealing when preceded by a
localizing window.
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Fouriers Window
Gabor realized that Fouriers analysis became
much more physically appealing when preceded by a
localizing window.
Window
X
Nonstationary signal

Localized signal
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Fouriers Window
He proposed that a decomposition into Gaussian
wave packets was therefore possible.
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Problem Physical interpretation of Fourier
Spectra
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60
100
Hz
Total Fourier spectrum
0.2
0.6
1.0
sec
Local Fourier spectra
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The Gabor Idea
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A Partition of Unity
The key to a fast, robust Gabor transform
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The Gabor Idea
A seismic signal
A suite of Gabor slices
The suite of Gabor slices will sum to recreate
the original signal with high fidelity because of
the partition of unity.
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The Gabor Idea
The Gabor transform
Fourier transform
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The Gabor Idea
The inverse Gabor transform done two ways
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Time-Frequency Analysis
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The Utility of the Gabor Transform

• The Gabor transform is a natural extension of
  the Fourier transform into the nonstationary
  realm.
• The Gabor deconvolution approximately
  factorizes a nonstationary convolution.

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Norbert Wiener (1894-1964)
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Enders Robinson (1930-)
The method we call Wiener deconvolution was
really pioneered in seismology by Enders
Robinson. It was based on Wieners theory and
work done with radar in WWII.
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Stationary seismic trace model
Matrix vector multiplication is a superposition
process. In the stationary process, an
unchanging wavelet is scaled, delayed, and
superimposed.
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Stationary seismic trace model
in the Fourier domain
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Wiener deconvolution
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Wiener deconvolution
This algorithm is enabled because the Fourier
transform factorizes the convolution integral.
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Stationarity

• A stationary physical process can be
  characterized by convolution.
• We do not try to characterize a time series as
  stationary or not.

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Nonstationary

• A nonstationary stationary physical process is
  characterized by a superposition process whose
  spectral content evolves.
• Attenuation processes are nonstationary and
  characterized by Q

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Q
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Q
Q is a single number that is a measure of the
quality of a rock.
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Attenuation and minimum phase
Futterman (1962) showed that wave attenuation in
a causal, linear theory is always minimum phase.
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Attenuation Simulation
True Amplitude
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Attenuation Simulation
Normalized
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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.
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Haiku
waveforms evolving phase rotations with decay
nonstationary
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Real Seismic (200 traces)after gain for
spherical spreading
Receiver position
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Gabor spectrum of trace 100after gain
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Stationary seismic trace model
Stationary superposition process.
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Nonstationary seismic trace model
Nonstationary superposition process.
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Gabor Factorizes Nonstationary Trace Model
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Gabor Deconvolution
a)
b)
c)
d)
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Comparison on Synthetic
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Comparison on Synthetic
On a nonstationary synthetic, Gabor beats Wiener
easily. However, in a real processing flow,
Wiener has many helpers
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Real Data Comparison
Stratigraphic line provided by Husky
Energy Dynamite source Data processing provided
by Sensor Geophysical (Peter Cary)
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Real Data Comparison
Standard flow Gain-gtSurface Consistent Wiener -gt
TVSW -gt Stack -gtTVSW (120 Hz) Gabor flow Gabor
-gt Stack -gt Gabor (160 Hz)
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Standard
Gabor
0
200
400
ms
600
800
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Standard
Gabor
0
200
400
ms
600
800
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Gabor
Standard
0
200
ms
400
600
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0
200
ms
400
600
standard
Gabor
Gabor
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Gabor
Standard
0
200
ms
400
600
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Real Data Comparison
Gabor is a viable alternative to standard
high-resolution processing. Comparisons to
Wiener alone were not shown as Gabor easily wins.
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Validation of Gabor wavelets with a VSP
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Comparison of wavelets
0.5 s
0.8 s
1.1 s
Gabor estimates
Wiener estimates
Wiener
Frequency spiking
VSP estimates
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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
nonstationary filters Gabor deconvolution easily
beats Wiener and competes with SCWienerTVSW
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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

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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,
Encana, Sensor Geophysical
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Haiku2
waveforms evolving phase rotations with decay
nonstationary
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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.

Picture unavailable
Sammy Wavelet
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Gabor transform via a partition of unity
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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
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Gabor transform via a partition of unity
The inverse Gabor transform is an inverse Fourier
transform, multiply by the synthesis window, sum
over windows
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Time-Frequency Analysis

• The simultaneous analysis of both time and
  frequency dependence of a signal.
• Fouriers theory allows either time or frequency
  analysis but not both.
• Quantum mechanics provided the first
  applications.

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Early Pioneers in Time-Frequency
Eugene Wigner (1902-1995) Wigner distribution
Hermann Weyl (1885-1955) Pseudodifferential
operators
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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.

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Norbert Wiener
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Dennis Gabor
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Joseph Fourier
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Gabors Door
Gabor was standing in the world of stationary
signals that are appropriately decomposed into
sinusoids.
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Gabors Door
On the other side of the wall, is the real world
of nonstationary signals. Though Fouriers
analysis could still apply, its meaning was
unclear.
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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.
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Compatible windows
Unity and Delta
Leads directly to K-N pseudodifferential operators
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Compatible windows
Gaussians
Extreme Value
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Limerick
There once was a man named Gabor, who was stopped
behind the lock on a door, until one bright day
he saw Joseph Fourier slip through a window to
the fore.
(Thanks to Chris Harrison of CREWES for the
cartoons.)