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Analytical image perturbations for wave-equation
migration velocity analysis
Paul Sava & Biondo BiondiStanford University
Wave-equation MVA (WEMVA)
• Wavefield-based MVA method
• Closely related to– Wave-equation migration– Wave-equation tomography
• Benefits– Finite-frequency– Multipathing– Hi resolution
A tomography problem
sqs LΔminΔTraveltime
tomography/MVA
Wave-equation tomography
Wave-equation MVA
q t traveltime
d
data
Rimage
L ray field wavefield wavefield
WEMVA: objective function
slowness perturbation
image perturbation
slownessperturbation(unknown)
Linear WEMVAoperator
imageperturbation
(known)
sRs LΔminΔ
Slowness backprojection
slowness perturbation
image perturbation
slowness perturbation
image perturbation
Rs Δ*L
MVA informationTraveltime MVA Wave-equation MVA
• Offset focusing (flat gathers) • Offset focusing (flat gathers)
• Spatial focusing
• Frequency redundancy
z
z
xx
“Data” estimate
Traveltime
MVA
Wave-equation tomography
Wave-equation MVA
t d Rray
tracing
data
modeling
residual
migration
sRs LΔminΔ
Prestack Stolt residual migration
• Background image R1
• Velocity ratio ),( 1 RSR
1RRR • Image perturbation
R
Incorrect velocityCorrect velocity
Zero offset image
Angle gathers
Synthetic model
Residual migration: the problem
Differential image perturbation
1d
dRR
1RRR Image
difference
Image differential
Computed Measured
Field data example
• North Sea– Salt environment
– One non-linear iteration• Migration (background image)
• Residual migration (image perturbation)
• Slowness inversion (slowness perturbation)
• Slowness update (updated slowness)
• Re-migration (updated image)
location
dep
th
location
dep
thde
pth
Zero offset image
Angle gathers
Background slowness
Background image
1
1
1
location
dep
th
Zero offset image
Background image
location
“Ratio” map
1d
dRR
location
dep
th
location
Zero offset image Zero offset image
Background image
Image perturbation
location
dep
th
location
Zero offset image
Image perturbation
Slowness perturbation
sRs LΔminΔ
location
dep
thde
pth
Zero offset image
Angle gathers
Background slowness
Background image
location
dep
thde
pth
Zero offset image
Angle gathers
Updated slowness
Updated image
dep
thde
pth
location
Angle gathers
“Correct” slowness
Zero offset image
“Correct” image
Summary
• Wave-equation MVA– Finite frequency– Multipathing– Hi resolution– Image space objective function
• Image perturbation– From prestack Stolt residual migration– Differential method– Compliant with the Born approximation