Transcript Automatic Wave Equation Migration Velocity Analysis

TRIP Annual meeting
Differential Semblance Optimization
for Common Azimuth Migration
Alexandre KHOURY
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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
d (m, h,  )
I (m, h)
p(m, h,  )
Subsurface offset
•Variable used for Velocity Analysis : Subsurface offset
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Subsurface Offset
S
M
R
S
R
M
h’
M'
M'
R’=S’
h'  0
R’
S’
h'  0
•For true velocity
P(M , h' , z, t  0)  0
P(M ' ,0, z, t  0)  0
•For wrong velocity
P(M , h' , z, t  0)  0
P(M ' ,0, z, t  0)  0
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Example: two reflectors data set
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True velocity common image gather
Offset gather at x=1000 m
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Example: two reflectors data set
One gather at midpoint x=1000m
c  ctrue
c  ctrue
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Differential Semblance Optimization
•From I (m, h) we define the objective function :
1
J (c)  h.I
2
For
c  ctrue
c  ctrue
2
1
  | h.I (m, h) |2
2 h,m
h  0  I (m, h)  0
J 0

h  0  I (m, h)  0
h  0  I (m, h)  0
J 0

h  0  I (m, h)  0
•Criteria for determining the true velocity !
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Differential Semblance Optimization
•Plot of the objective function with respect to the velocity
c=ctrue
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Gradient calculation
•The objective function :
•Gradient calculation :
J (c ) 
1
h.I
2
2
 J (c),c  h.I , h.I 
 c, (
I * 2
) h I 
c
I * 2
J (c )  ( ) h I
c
•Adjoint-state calculation (Lions, 1971):
code operator (
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I *
)
c
Migration: Structure of the Common Azimuth Migration
• DSR equation:
1
2
2

[(
k

k
)

(
k

k
)
]
2
m
h
m
h
x
x
y
y
cs
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Wavefield at depth z
kz 
2
H1
Phase-Shift
.eikz 0z
cr 2
1
 [(kmx  khx ) 2  (kmy  khy ) 2 ]
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in the Fourier domain
i(
.e
2
1 1 2
  ) iz
cs cr c0
H2
Lens-Correction
H3
General Screen Propagator or FFD
in the space domain
in the space domain
Imaging condition
Wavefield at depth z+z
Image at depth z+z
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Algorithm of the gradient calculation
B
Gradient at depth
z+z
Wavefield pz
MIGRATION
H
p
c
H
H-1
H*,B*
Gradient at depth
z+2z
Wavefield pz+z
H
H-1
H*,B*
Adjoint variables
propagation Dp, Dc
Wavefield pz+2z
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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 B-Spline grid
Gradient calculation
respect to Fine 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)
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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)
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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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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:
V=2300
Image
Gather
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Differential Semblance Optimization
10 iterations: Right position
Image
Gather
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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
x=5000
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Differential Semblance Optimization
Top of salt : one gather
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Differential Semblance Optimization
Plot of  | h.I ( x, h) |
function h
2
: localization of the energy of the objective
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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 E&P
• Dr. Peng Shen, Dr Henri Calandra, Dr Paul Williamson
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