Transcript No Slide Title

Wave-equation
migration velocity analysis
beyond the Born approximation
Paul Sava*
Sergey Fomel
Stanford University
UT Austin (UC Berkeley)
[email protected]
Imaging=MVA+Migration
• Migration
• wavefield based
• Migration velocity analysis (MVA)
• traveltime based
• Compatible migration and MVA methods
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Imaging: 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
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
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Wavefields or traveltimes?
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Wavefields or traveltimes?
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Scattered wavefield
Wavefield
perturbation
Medium
perturbation
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Agenda
Scattering
Theoretical background
Imaging
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
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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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Wave-equation MVA: Objective
location
depth
Velocity
perturbation
WEMVA
operator
min ΔR  L  s
Δs
image
perturbation
(known)
slowness
perturbation
(unknown)
location
depth
Image
perturbation
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Comparison of MVA methods
• Wave-equation MVA
• Traveltime tomography
– migrated images
– picked traveltimes
– moveout and focusing
– use amplitudes
– moveout
– ignore amplitudes
– parabolic wave equation
– multipathing
– eikonal equation
– slow
– fast
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Agenda
Scattering
Theoretical background
Imaging
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
[email protected]
What is the image perturbation?
location
angle
WEMVA
operator
depth
min ΔR  L  s
Δs
image
perturbation
(known)
Focusing
Flatness
slowness
perturbation
(unknown)
Residual process:
• moveout
• migration
• focusing
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Agenda
Scattering
Theoretical background
Imaging
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
[email protected]
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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“Wave-equation” migration
z
s0
z Δz
z  Δz
0
W
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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
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
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Born approximation

ΔW  W0 e
βΔs
WEMVA
operator

1
Non-linear WEMVA
i
e  1  i
e
min ΔR  L  s
ΔWΔs  W0βΔs
image
perturbation
Born
linearization
(known)
slowness
perturbation
(unknown)
i
Small perturbations!
Unit circle
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Does it work?
What if the perturbations are not small?
min ΔR  L  s
Δs
Location [km]
Depth [km]
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Synthetic example
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Born approximation
1%
10%
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Agenda
Scattering
Theoretical background
Imaging
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
[email protected]
Wavefield continuation
W
βΔs
e
W0
dW
  ik z W
dz
Explicit
W
 1  βΔs
W0
Bilinear
W 2  βΔs

W0 2  βΔs
Implicit
W
1

W0 1  βΔs
(Born approximation)
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Exponential approximations
dW
  ik z W
dz
W
βΔs
e
W0
ξ0
ξ  0.5
e
ξ 1
βΔs
1  1  ξ βΔs

1  ξβΔs
ξ  0,1
Unit circle
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A family of linearizations
WEMVA
operator
min ΔR  L  s
Δs

image
perturbation
(known)
ΔW  W0 e
βΔs

slowness
perturbation
(unknown)
1
ΔW  W0  ξΔW βΔs
e
βΔs
1  1  ξ βΔs

1  ξβΔs
Linear WEMVA
ξ  0,1
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Improved linearizations
1%
10%
40%
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Agenda
Scattering
Theoretical background
Imaging
Image perturbations
Wavefield extrapolation
WEMVA methodology
Born linearization
Alternative linearizations
[email protected]
Summary
• Wave-equation MVA
•
•
•
•
wavefield-continuation
improved focusing
image space (improve the image)
interpretation guided
• Improved WEMVA
• better approximations
• no additional cost
• further refinement
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