An%20automatic%20wave%20equation%20migration%20velocity%20analysis%20by%20differential%20semblance%20optimization

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An automatic wave equation migration velocity
analysis by differential semblance optimization

• The Rice Inversion Project

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Objective

• Simultaneous optimization for velocity and image
• Shot-record wave-equation migration.

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Theory

• Nonlinear Local Optimization
• Objective function
• Gradient of the objective function
• Remark
• Objective function requires to be smooth .
• Differential semblance objective function is
  smooth.

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Differential semblance criteria
z
x
offset image
angle image
z
z
h
h
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Objective function
I offset domain image c velocity h
offset parameter P differential
semblance operator L2 norm M set of
smooth velocity functions
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(No Transcript)
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Gradient calculation
Downward continuation and upward continuation
Definitions
R0
S0
gradient
derivative cross correlate
down
down
SZ
RZ
DS
DR
cross correlate
up
up
Sz
Rz
image
cross correlate reference field
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Gradient smoothing using spline evaluation

• Vimage I
• gimage

migration
differential migration
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Optimization
BFGS algorithm for nonlinear iteration

• Objective function evaluation

• Gradient calculation

loop

• Update search direction

cout Iout
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Synthetic Examples

• Flat reflector, constant velocity
• Marmousi data set

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Experiment of flat reflector at constant velocity
x
Ccorrect 2km/sec
z
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Initial iterate Image (v0 1.8km/sec) Image
space 401 by 80 Model space 4 by 4
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Iteration 5 Image
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Iterations
v5 Output velocity at iteration 5
vbest - v5
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Marmousi data set
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Marmousi data set
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V
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Initial iterate Image (v01.8km/sec) Image
space 921 by 60 Model space 6 by 6
Offset image
Angle image
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Iterate 5 Image
Offset image
Angle image
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v5 output velocity at iteration 5
vbest best spline interpolated velocity
v5 - vbest
iterations
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Low velocity lense constant velocity background
Vbackground 2 km/sec
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Seismogram
Shot gathers far away from the low velocity lense
Shot gathers near the low velocity lense
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Iteration 1
Start with v0 2km/sec
Iteration 2
Iteration 3
Iteration 4
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1.0 1.5 2.0 2.5 3.0
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Conclusions

• Offset domain DSO is a good substitute for angle
  domain DSO.
• Image domain gradient needs to be properly
  smoothed.
• DSO is sensitive to the quality of the image.
• Differential semblance optimization by wave
  equation migration is promising.