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Wave-equation migration velocity analysis
beyond the Born approximation
Paul Sava* Stanford University
Sergey Fomel UT Austin (UC Berkeley)
Imaging=MVA+Migration
• Migration• wavefield based
• Migration velocity analysis (MVA)• traveltime based
• Compatible migration and MVA methods
Imaging: the “big picture”
• Kirchhoff migration
• traveltime tomography
wavefronts
• wave-equation migration
• wave-equation MVA (WEMVA)
wavefields
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
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
Wave-equation MVA: Objective
Velocity perturbation
Image perturbation
slownessperturbation(unknown)
WEMVAoperator
imageperturbation
(known)
sLΔRminΔs
location
depth
location
depth
– migrated images
– moveout and focusing– use amplitudes
– parabolic wave equation– multipathing
– slow
– picked traveltimes
– moveout– ignore amplitudes
– eikonal equation
– fast
Comparison of MVA methods
• Wave-equation MVA • Traveltime tomography
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
What is the image perturbation?
Focusing Flatness Residual process:• moveout• migration• focusing
slownessperturbation(unknown)
WEMVAoperator
imageperturbation
(known)
sLΔRminΔs
location
depth
angle
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
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
ΔW
Δs
Wavefield perturbation
z
Δzz0s Δss0
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
Born approximation
iei 1
ie
Small perturbations!
Born linearization
Non-linear WEMVA
1eWΔW βΔs0
βΔsWΔW 0slowness
perturbation(unknown)
WEMVAoperator
imageperturbation
(known)
sLΔRminΔs
Unit circle
sLΔRminΔs
Does it work?What if the perturbations are not small?
Location [km]
Depth [km
]
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
Wavefield continuation
Wikdz
dWz βΔs
0
eW
W
Bilinear
Implicit
βΔs2
βΔs2
W
W
0
βΔs1
1
W
W
0
Explicit βΔs1W
W
0
(Born approximation)
Exponential approximations
ξβΔs1
βΔsξ11eβΔs
0,1ξ
0ξ
1ξ
0.5ξ
Wikdz
dWz βΔs
0
eW
W
Unit circle
1eWΔW βΔs0
A family of linearizations
ξβΔs1
βΔsξ11eβΔs
0,1ξ
βΔsξΔWWΔW 0Linear WEMVA
slownessperturbation(unknown)
WEMVAoperator
imageperturbation
(known)
sLΔRminΔs
Agenda
Theoretical background
WEMVA methodology
Scattering
Imaging
Image perturbations
Wavefield extrapolation
Born linearization
Alternative linearizations
Summary
• Wave-equation MVA• wavefield-continuation• improved focusing • image space (improve the image)• interpretation guided
• Improved WEMVA• better approximations• no additional cost• further refinement
