Application of Transform Spaces for Multichannel Processing and Imaging

1
Application of Transform Spaces for Multichannel
Processing and Imaging

• William Burnett

2
Outline

• Review of The Fourier Transform
• Useful Filtering Properties
• Nonstationary Filtering
• Data Processing Transforms
• Transforms by Matrix-Vector Multiplication
• NMO Example
• Conclusions

3
The Fourier Transform

• Stationary filtering properties
• Inversion Theorem (Exactly Reversible)
• Faltung Theorem (Convolution)
• Shifting Property

4
Nonstationary Filtering

• Nonstationary Convolution
• Nonstationary Shift Theorem

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Nonstationary Filtering

• Many seismic data processing steps can be viewed
  as nonstationary shifts of the form
  t(t) t, or more
  generally
  input coordinate as a function of
  output coordinate output coordinate

• f-k migration
• ? (? 0) ? 0
• Wavefield extrapolation
• ? (? z) ? z

• NMO
• tx(t0 ) t0
• DMO
• tn(t0) t0

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Data Processing Transforms

• Forward Mixed-Domain Processing Transform
• With a similar derivation, and the introduction
  of
• Inverse Mixed-Domain Processing Transform

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Matrix-Vector Multiplication

• Integrals can be discretely performed by
  matrix-vector multiplication.
• Fourier Transform example
• where,

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Matrix-Vector Multiplication

• Integrals can be discretely performed by
  matrix-vector multiplication.
• Fourier Transform example
• where,

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Matrix-Vector Multiplication
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NMO Example

• Processes that require interpolation are not
  reversible.
• Conventional NMO example
• Conventional time-mapping algorithms loop over
  data for values of t0, and then solve for t.
  
• The data at the calculated t value is then
  mapped to t0 in the output image.

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NMO Example

• Conventional NMO example (contd)
• The calculated tx values do not care that the
  data is actually discrete.
• If tx falls between samples, the value that gets
  mapped is an interpolated estimate of surrounding
  values.
• This process cannot be reversed exactly, as there
  are infinite ways to redistribute the amplitude
  over surrounding points.

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NMO Transform

• Two parameters are needed to implement NMO as a
  transform
• aNMO describes NMO stretch exactly.

x
t
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NMO Transform

• Substituting these two values in to the
  mixed-domain transforms gives
• The Forward NMO Transform
• The Inverse NMO Transform

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NMO Transform Matrices
Forward NMO
Inverse NMO
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NMO Applied then Removed
Velocity model varied linearly from 2000-3000
m/s from top to bottom.
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Conclusions

• Why bother with transforms?
• Processing by reversible transform opens up a
  variety of new, but familiar, processing domains.
  
• Dip filters, noise reduction, and multiple
  attenuation could benefit from being applied in
  say an NMO Domain or Migrated Domain.
• Implementation by matrix-vector multiplication
  reveals matrix symmetry and could make use of a
  variety of fast matrix-vector algorithms.
• Error can be controlled while efficiency
  increases.

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References

• Barnes, A. E. 1992 Another look at NMO stretch.
  Geophysics, 57, 749-751.
• Claerbout, J. F. 1992 Earth Soundings Analysis
  Processing versus Inversion. Blackwell Science
• Ferguson, R. J., and Margrave, G. F. 2002
  Prestack depth migration by symmetric
  nonstationary phase shift. Geophysics, 67,
  594-603.
• Horn, R.A. and Johnson, C.R. 1992 Topics in
  Matrix Analysis. Cambridge University Press
• Karl, J. H. 1989 An Introduction to Digital
  Signal Processing. Academic Press, San Diego.
• Margrave, G. F. 1998 Theory of nonstationary
  linear filtering in the Fourier domain with
  application to time-variant filtering.
  Geophysics, 63, 244-259.
• Yilmaz, O. 2001 Seismic Data Analysis. Society
  of Exploration Geophysics, Tulsa.

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Acknowledgements

• Rob Ferguson, Mrinal Sen, and Bob Tatham, my
  advisor and committee
• You, the sponsors of the EDGER forum!