Automatic Wave Equation Migration Velocity Analysis

1
TRIP Annual meeting
Differential Semblance Optimization for Common
Azimuth Migration
Alexandre KHOURY
2
Context of the project

• Prestack Wave Equation depth migration
• Wavefield extrapolation method
• Automating the velocity estimation loop
  (time-consuming)

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Motivation of the project

• Encouraging results in 2D for Shot-Record
  migration (Peng Shen, TRIP 2005)

• Efficiency of the Common Azimuth Migration in 3D

enables sparse acquisition in one direction
very economic algorithm

• Goal of the project
• Implement DSO for Common Azimuth Migration in 3D
  after a 2D validation

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Common Azimuth Migration

• Wavefield extrapolation in depth survey
  sinking in the DSR equation

h
M
Subsurface offset

• Variable used for Velocity Analysis Subsurface
  offset

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Subsurface Offset
S
R
S
R
M
M
h
M'
M'
RS
R
S

• For true velocity

• For wrong velocity

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Example two reflectors data set
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True velocity common image gather
Offset gather at x1000 m
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Example two reflectors data set
One gather at midpoint x1000m
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Differential Semblance Optimization

• From

we define the objective function

For

• Criteria for determining the true velocity !

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Differential Semblance Optimization

• Plot of the objective function with respect to
  the velocity

cctrue
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Gradient calculation

• The objective function

• Gradient calculation

• Adjoint-state calculation (Lions, 1971)

code operator
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Migration Structure of the Common Azimuth
Migration

• DSR equation

Wavefield at depth z
in the Fourier domain
Phase-Shift
H1
in the space domain
Lens-Correction
H2
General Screen Propagator or FFD
H3
in the space domain
Imaging condition
Wavefield at depth zDz
Image at depth
zDz
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Algorithm of the gradient calculation
Wavefield pz
Gradient at depth zDz
MIGRATION
H
H,B
H-1
Wavefield pzDz
Gradient at depth z2Dz
H
H,B
H-1
Adjoint variables propagation Dp, Dc
Wavefield pz2Dz
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Algorithm of the gradient calculation
Velocity representation on a B-spline grid
B-Spline transformation
Fine grid
B-Spline grid
LBFGS Optimizer
Adjoint B-Spline transformation
Gradient calculation respect to Fine Grid
Gradient calculation respect to B-Spline grid
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Several critical points
- Avoid wrap-around in the subsurface offset
domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity
in a specified range
-Careful choice of migration parameters for the
accuracy of the gradient (not necessarily for the
migration)
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Several critical points
- Avoid wrap-around in the subsurface offset
domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity
in a specified range
-Careful choice of migration parameters for the
accuracy of the gradient (not necessarily for the
migration)
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• Wrap-around in the subsurface offset domain

h
For wrong velocity
Image
Gather
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Wrap-around in the subsurface offset domain
Effect of padding and split-spread for wrong
velocity
h
Image
Gather
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Several critical points
- Avoid wrap-around in the subsurface offset
domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity
in a specified range
-Careful choice of migration parameters for the
accuracy of the gradient (not necessarily for the
migration)
20
Artifacts propagation
Necessity to taper the data on both offset and
midpoint axes and in time
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Several critical points
- Avoid wrap-around in the subsurface offset
domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity
in a specified range
-Careful choice of migration parameters for the
accuracy of the gradient (not necessarily for the
migration)
22
Several critical points
- Avoid wrap-around in the subsurface offset
domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity
in a specified range
-Careful choice of migration parameters for the
accuracy of the gradient (not necessarily for the
migration)
23
Differential Semblance Optimization

• Tests on different data sets

-Test on flat reflectors with a constant
background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model
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Differential Semblance Optimization

• Tests on different data sets

-Test on flat reflectors with a constant
background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model
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Differential Semblance Optimization
Start of the optimization V2300
Image
Gather
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Differential Semblance Optimization
10 iterations Right position
Gather
Image
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Differential Semblance Optimization

• Tests on different data sets

-Test on flat reflectors with a constant
background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model
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Differential Semblance Optimization
Top of salt image
x5000
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Differential Semblance Optimization
Top of salt one gather
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Differential Semblance Optimization
Plot of localization
of the energy of the objective function
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Differential Semblance Optimization

• Tests on different data sets

-Test on flat reflectors with a constant
background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model
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Differential Semblance Optimization
True velocity
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Differential Semblance Optimization
Starting velocity
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Differential Semblance Optimization
Starting image
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Differential Semblance Optimization
Optimized image
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Differential Semblance Optimization
True image
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Differential Semblance Optimization
Optimized velocity
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Conclusion

• Migration is critical and has to be artifacts
  free.

• Is the DSR Migration precise enough for
  optimization of complex models ?
• Can we deal with complex velocity model ?

• Next test on the Marmousi data set and on a 3D
  data set.

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Acknowledgment

• Prof. William W. Symes
• Total EP
• Dr. Peng Shen, Dr Henri Calandra, Dr Paul
  Williamson