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Elastic least-squares migration
Yuting Duan∗, Antoine Guitton, and Paul SavaCenter for Wave Phenomena
Colorado School of Mines∗presently at Shell International Exploration and Production Inc.
α = V 2p
α = V 2p
β = V 2s
β = V 2s
���
α = V 2p
���
β = V 2s
���
higher resolutiond
fewer artifactsd
improved amplitudes
objective function
J =∑e
1
2‖Fm− d‖2
I F: demigration
I FT: migration
isotropic wave equation
u− α∇(∇ · u) + β∇× (∇× u) = f
I α = λ+2µρ
I β = µρ
I u (e, x, t): background wavefield
I f (e, x, t): source
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
Born
I δα: α perturbation
I δβ: β perturbation
I u: background wavefield
I δu (e, x, t): perturbed wavefield
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
Born
I δα: α perturbation
I δβ: β perturbation
I u: background wavefield
I δu (e, x, t): perturbed wavefield
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
Born
I δα: α perturbation
I δβ: β perturbation
I u: background wavefield
I δu (e, x, t): perturbed wavefield
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
= [∇(∇ · u) d −∇× (∇× u)]
[δα
δβ
]= Bm
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
= [∇(∇ · u) d −∇× (∇× u)]
[δα
δβ
]= Bm
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
= [∇(∇ · u) d −∇× (∇× u)]
[δα
δβ
]= Bm
δα, δβ → δu
δu− α∇(∇ · δu) + β∇× (∇× δu)
= δα∇(∇ · u)− δβ∇× (∇× u)
= [∇(∇ · u) d −∇× (∇× u)]
[δα
δβ
]= Bm
forward operator (demigration)
dK = KKKKKKKKKKPKKKKKBKKKm
data forward Bornextraction modeling source
I m: model perturbation
[δα
δβ
]I d: data
forward operator (demigration)
dK = KKKKKKKKKKPKKKKKBKKKm
data forward Bornextraction modeling source
I m: model perturbation
[δα
δβ
]I d: data
forward operator (demigration)
dK = KKKKKKKKKKPKKKKKBKKKm
data forward Bornextraction modeling source
I m: model perturbation
[δα
δβ
]I d: data
forward operator (demigration)
dK = KKKKKKKKKKPKKKKKBKKKm
data forward Bornextraction modeling source
I m: model perturbation
[δα
δβ
]I d: data
adjoint operator (migration)
mK = KKKBTKKKKKPTKKKKKKTKKKd
imaging backward datacondition modeling injection
I m: model perturbation
[δα
δβ
]I d: data
adjoint operator (migration)
mK = KKKBTKKKKKPTKKKKKKTKKKd
imaging backward datacondition modeling injection
I m: model perturbation
[δα
δβ
]I d: data
adjoint operator (migration)
mK = KKKBTKKKKKPTKKKKKKTKKKd
imaging backward datacondition modeling injection
I m: model perturbation
[δα
δβ
]I d: data
imaging condition
δα =∑e,t
[∇(∇ · us)] · ur
δβ =∑e,t
[−∇× (∇× us)] · ur
I us : source wavefield
I ur : receiver wavefield
objective function
J =∑e
1
2‖W (KBFm− d) ‖2
I W (e, x, t): data weighting
objective function
J =∑e
1
2‖W (KPBm− d) ‖2
I W (e, x, t): data weighting
objective function
J =∑e
1
2‖W (KPBm− d) ‖2
I W (e, x, t): data weighting
example
Marmousi
α
Marmousi
β
δα
δβ
δα���
���
δβ���
���
recorded dz
recorded dz
δα���
���
δβ���
���
δβ���
���
α β
Volve data
I 240 receivers
I 242 shots
I Models: Vp, Vs , δ, ε
I frequency: 0− 15Hz
Volve data
I 240 receivers
I 242 shots
I Models: Vp, Vs , δ, ε
I frequency: 0− 15Hz
Volve data
I 240 receivers
I 242 shots
I Models: α, β, δ, ε
I frequency: 0− 15Hz
α model
β model
bandpass
3D to 2D compensation
data weighting
data
@@@R
@@@R
α elastic RTM image
@@@R
@@@R
α elastic LSRTM image
@@@R
@@@R
β elastic RTM image
β elastic LSRTM image
vertical component: observation
vertical component: prediction with α & β image
@@@R
vertical component: data misfit
@@@R
summary
elastic imaging conditionelastic model perturbationno polarity reversal
elastic LSM methodhigher resolutionimproved amplitudes
acknowledgement
d
Statoil ASA and the Volve license partnersExxonMobil E&P Norway ASBayerngas Norge AS
d
dd • Disclaimer: The views expressed in this presentation are the views of the authors anddo not necessarily reflect the views of Statoil ASA and the Volve field license partners.
vertical component: prediction with α image
vertical component: prediction with β image