Phase-encoded Modeling for Efficient Calculation of Green

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Phase-encoded Modeling for EfficientCalculation
of Greens Functions
Samuel Brown February 5, 2009
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Outline

• Phase-encoded Migration
• Phase-encoded Modeling
• Examples
• Signal-to-Noise Ratio
• Applications
• HMMs
• Conclusions

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Phase-encoded Migration
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Phase-encoded Migration
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Phase-encoded Migration

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Phase-encoded Modeling

• Can we extend this idea for modeling?
• Can a few simulations produce as many Greens
  functions as any number of simulations?

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Phase-encoded Modeling

• What could we use these Greens functions for?
  
• Exploration/Earthquake imaging
• Waveform inversion gradients
• Sensitivity kernels
• Angle gathers

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Stochastic Interferometry
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Stochastic Interferometry
Romero et. al. (2000)
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Stochastic Interferometry
Greens functions between any two model points
with two finite-difference simulations!
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Model Extension
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Closed Geometry
G(B A)
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Closed Geometry
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Line Geometry
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Geometry Comparison
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Sigsbee2b Above Salt
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PEM Greens Functions
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Sigsbee2b Below Salt
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Sigsbee2b Below Salt
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Extrapolate Stack

• Place parallel lines of receivers in water
  column.
• Spacing should be greater than the correlation
  length of noise.
• Perform cross-correlations between field
  recorded at imaging point and each line receiver.
• Perform up- and down-going separation of PEM
  Greens functions along each line.
• Extrapolate each line to streamer/source datum
  and stack.

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Signal-to-Noise Enhancement
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Phase-encoded Modeling

• What could we use these Greens functions for?
  
• Exploration/Earthquake imaging
• Waveform inversion gradients
• Sensitivity kernels
• Angle gathers

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Sensitivity Kernels
Rytov Wavepath
Woodward (1992)

• Dynamic ray tracing
• Requires model smoothing
• One-way wave equation
• Requires one simulation for each interrogation
  point
• PEM Greens functions
• Signal-to-noise ratio

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Angle Gathers

• Assume first arrival can be extracted from PEM
  Greens Function and characterized in terms or
  traveltime and amplitude.

• By taking the gradient of traveltimes, we can
  create a ray parameter to determine reflection
  angles.

• This leads naturally to dip or angle gathers
  when imaging with PEM Greens functions.

• Same flexibility as Kirchhoff methods!

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1D HMM First Arrival
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2D HMM First Arrival
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2D HMM First Arrival
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Conclusions

• Phase-encoded modeling can be used to generate
  Greens functions between any two points using
  two finite-difference simulations.
• Cross-talk related noise is a problem
• Extrapolate stack algorithm can improve the
  signal-to-noise ratio.
• Hidden Markov models show promise in extracting
  events from PEM Greens functions.

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Looking Forward
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References
Romero, L. A., D. C. Ghiglia, C. C. Ober, and
S. A. Morton, 2000, Phase encoding of shot
records in prestack migration Geophysics, 65,
426-436. Woodward, M. J., 1992, Wave-equation
tomography Geophysics, 57, 15-26.
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