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

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α = V 2p

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β = V 2s

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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

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Marmousi

β

δα

δβ

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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