[email protected] wave-equation migration velocity analysis paul sava* stanford university...
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Wave-equation migration velocity analysis
Paul Sava* Stanford University
Biondo Biondi Stanford University
Sergey Fomel UT Austin
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
In the “big picture”
• Kirchhoff migration
• traveltime tomography
wavefronts
• wave-equation migration
• wave-equation MVA (WEMVA)
wavefields
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
Imaging: Correct velocity
Background velocity
Migrated image
Reflectivity model
What the data tell us...What migration does...
location
depth
location
depth
depthdepth
depth
Imaging: Incorrect velocity
Perturbed velocity
Migrated image
Reflectivity model
What the data tell us...What migration does...
location
depth
location
depth
depthdepth
depth
WEMVA objective
Velocity perturbation
Image perturbation
slownessperturbation(unknown)
WEMVAoperator
imageperturbation
(known)
location
depth
location
depth
sLΔR
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
Double Square-Root Equation
Wikdz
dWz
Δsds
dkkk
0
0
ss
zzz
Fourier Finite DifferenceGeneralized Screen Propagator
Δzikz
Δzzze
W
W
Wavefield extrapolation
βΔsΔzz
0
Δzz
eW
W
βΔsΔzikΔzik0zz
1eWΔW βΔs0
slownessperturbation
backgroundwavefield
wavefieldperturbation
Wavefield perturbation
z
Δzz0s Δss0
ΔW
Δs
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
Linearizations
Unit circle
βΔs2
βΔs2eβΔs
βΔs1
1eβΔs
βΔs1eβΔs 1eWΔW βΔs0
Born approximation
βΔse
Linear WEMVA
slownessperturbation(unknown)
WEMVAoperator
imageperturbation
(known)
sLΔR 0,1ξ
βΔsξΔWWΔW 0 1eWΔW βΔs0
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
What can we do?
• Define another objective function– e.g. DSO
• Construct an image perturbation which obeys the Born approximation
• ...
Analytical image perturbation
0RRΔR
0ρ RfR
Δρdρ
dRΔR
0ρρ
Computed analytically
Picked from data
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Non-linear operator
Linear operator
Image perturbation
WEMVA applications
Other applications
• 4-D seismic monitoring– image perturbations over time– no need to construct
• Focusing MVA– zero offset data
Summary
• Wave-equation MVA• wavefield extrapolation• image space objective• focusing and moveouts • interpretation guided
• Linearization• linear operator• construct image perturbations