Shaping Regularization

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Shaping Regularization
Best SEG Talk Sergey Fomels

• Yaxun Tang

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Paper Name

• Shaping regularization in geophysical estimation
  problems
• Author Sergey Fomel
• SI 2.4 Wave-Equation and Topology

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Introduction
well-posed
Model
Indirect constraints
Geophysical Estimation
Insufficient Data
ill-posed
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Introduction

• Traveltime tomography
• Migration velocity analysis
• High-resolution Radon transform
• Spectral decomposition

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Regularization theory

• Goal of inversion
• Fit the observed data
• Fit the estimated model to a priorly assumed
  behavior (Tikhonov)
• i.e. to minimize vector

L forward operator m model d data D
regularization operator scaling parameter
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Regularization theory

• The formal solution
• We can also minimize the least-square norm of the
  vector

P model preconditioning operator
mPp
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Regularization Theory

• The formal solution
• Those two solutions are equivalent, if

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Smoothing

• Find a model m that fits the observed data d but
  is in a certain sense smoother
• In this case LI

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Regularized smoothing
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Triangle smoothing
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Shaping regularization

• Take smoothing as a fundamental operation!
• Smoothing implies projection of the input model
  to the space of admissible functions
• Shaping just translates the input into an
  acceptable model

projection operator shaping
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Shaping regularization

• Define
• Then, we get

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Shaping regularization

• Introducing a scaling of L
• When

Suitable for an iterative inversion of the
conjugate-gradient method
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1-D data regularization
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1-D data regularization
Regularization
Model Preconditioning
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1-D data regularization
S is lowpass filtering with a Gaussian
smoother
S is bandpass filtering with a shifted Gaussian
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1-D data regularization
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Velocity estimation
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Conclusion

• The main idea of shaping regularization is to
  take shaping as a fundamental operation
• Shaping regularization is fast convergent
• Compared to the regularization operator, shaping
  operator is often easier to design,