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Wave-equation
migration velocity analysis
Paul Sava*
Biondo Biondi
Sergey Fomel
Stanford University
Stanford University
UT Austin
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The problem
• Depth imaging
– image: migration
– velocity: migration velocity analysis
• Migration and MVA are inseparable
• “Everyhing depends on v(x,y,z)”
» JF Claerbout, 1999
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An approximation
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A better approximation
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In the “big picture”
wavefronts
wavefields
• Kirchhoff migration
• wave-equation migration
• traveltime tomography
• wave-equation MVA
(WEMVA)
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Wavefield scattering
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Wavefield scattering
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Scattered wavefield
Wavefield
perturbation
Medium
perturbation
ΔW  f  s 
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Imaging: Correct velocity
location
depth
depth
Background
velocity
location
Reflectivity
model
depth
depth
What migration does...
What the data tell us...
depth
Migrated
image
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Imaging: Incorrect velocity
location
depth
depth
Perturbed
velocity
location
Reflectivity
model
depth
depth
What migration does...
What the data tell us...
depth
Migrated
image
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WEMVA objective
location
depth
Velocity
perturbation
ΔR  L  s
location
image
perturbation
(known)
WEMVA
operator
slowness
perturbation
(unknown)
depth
Image
perturbation
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Wavefield extrapolation
dW
  ik z W
dz
Double Square-Root Equation
Fourier Finite Difference
Generalized Screen Propagator
z  Δz
W
ik z Δz
e
z
W
W
z  Δz
z  Δz
0
W
dk z
 kz 0ik z0 Δz  βΔs
Δs
 ikk z Δz
ds s s0
e
βΔs
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Slowness perturbation
z
s0
s 0  Δs
z Δz
z  Δz
0
W
z  Δz βΔs
0
W
e
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Wavefield perturbation
z
s 0  Δs
s0
z Δz
background
wavefield

ΔW
ΔW Δs
 W0 e
wavefield
perturbation
βΔs

1
slowness
perturbation
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Linearizations

ΔW  W0 e
e
βΔs

1
e
βΔs
 1  βΔs
Born approximation
e
βΔs
e
βΔs
βΔs
2  βΔs

2  βΔs
1

1  βΔs
Unit circle
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Linearizations

ΔW  W0 e
βΔs

1
e
βΔs
1  1  ξ βΔs

1  ξβΔs
ξ  0,1
e
ξ0
βΔs
ξ  0.5
ξ 1
Unit circle
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Linearizations

ΔW  W0 e
βΔs

ΔW  W0  ξΔW βΔs
1
ξ  0,1
e
ξ0
βΔs
ξ  0.5
ξ 1
Unit circle
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Linear WEMVA

ΔW  W0 e
βΔs

1
ΔW  W0  ξΔW βΔs
ξ  0,1
ΔR  L  s
image
perturbation
(known)
WEMVA
operator
slowness
perturbation
(unknown)
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Correct velocity
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Incorrect velocity
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Image perturbation
R0
R
ΔR  R  R 0
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Failure!
ΔR  R  R 0
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Small phase limitation
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What can we do?
• Define another objective function
– e.g. DSO
• Construct an image perturbation which
obeys the Born approximation
• ...
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Residual migration
ρ
R  f ρ R 0 
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Analytical image perturbation
R  f ρ R 0 
ΔR  R  R 0
Picked from data
dR
ΔR 
Δρ
dρ ρρ0
Computed analytically
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Analytical image perturbation
R0
dR
dρ
ρ ρ 0
Δρ
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Image perturbations comparison
ΔR  L  s
ΔR  R  R 0
ΔR 
dR
Δρ
dρ ρρ0
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Slowness perturbations
ΔR  L  s
ΔR  R  R 0
ΔR 
dR
Δρ
dρ ρρ0
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Migrated images
ΔR  L  s
ΔR  R  R 0
ΔR 
dR
Δρ
dρ ρρ0
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Migrated images: angle gathers
ΔR  L  s
ΔR  R  R 0
ΔR 
dR
Δρ
dρ ρρ0
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Agenda
Scattering
Theoretical background
Imaging
Non-linear operator
WEMVA methodology
Linear operator
Image perturbation
WEMVA applications
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Other applications
• 4-D seismic monitoring
– image perturbations over time
– no need to construct
• Focusing MVA
– zero offset data
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4D seismic monitoring
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4D seismic monitoring
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4D seismic monitoring
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Focusing MVA
Correct image
Incorrect image
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Focusing MVA
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Focusing MVA
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Focusing MVA
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Focusing MVA
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Focusing MVA
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Focusing MVA
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Summary
• Wave-equation MVA
•
•
•
•
wavefield extrapolation
image space objective
focusing and moveouts
interpretation guided
• Linearization
• linear operator
• construct image perturbations
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