elastic wave eld tomography with physical...
TRANSCRIPT
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Elastic wavefield tomography withphysical constraints
Yuting Duan∗ and Paul Sava
Center for Wave PhenomenaColorado School of Mines
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Vp∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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Vs∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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Vp∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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Vs∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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Vp
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Vs
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isotropic wave equation
ρü = (λ + 2µ)∇(∇ · u)− µ∇× (∇× u)
I ρ : density
I λ,µ: Lamé parameters
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isotropic wave equation
ü = α∇(∇ · u)− β∇× (∇× u)
I α = λ+2µρI β = µρ
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objective function J= JD + JM + JC
J = JD + JM + JC
I JD : data misfitI JM : geometrical constraintI JC : physical constraint
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objective function J = JD+JM + JC
JD =1
2‖dp − do‖2
I dp: predicted dataI do: observed data
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ASM gradient
∂αJD∂βJD
= ∑e
−[∇(∇ · u)]T ? a[∇× (∇× u)]T ? a
I u: state variableI a: adjoint variable
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ASM gradient
∂αJD∂βJD
= ∑e
−[∇(∇ · u)]T ? a[∇× (∇× u)]T ? a
I u: state variableI a: adjoint variable
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ASM gradient
∂αJD∂βJD
= ∑e
−[∇(∇ · u)]T ? a[∇× (∇× u)]T ? a
I u: state variableI a: adjoint variable
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α
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β
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∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣RS
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do
dp
dp − do
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do
dp
dp − do
P
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do
dp
dp − do
S
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@@RP
@@RS
@@RP
@@RS
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@@RP
@@RS
@@RP @@R
S
@@RP
@@RS
@@RP @@R
S
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@@RP
@@RS
@@RP
@@RS@@R
P
@@RS
@@RP
@@RS
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@@RP
@@RS
@@RP
@@RS@@R
P
@@RS
@@RP
@@RS
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∂αJD ∂βJD
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improved model update
I illumination compensation
I parameter rebalancing
I geometrical constraint
I physical constraint
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gradient
∂αJD∂βJD
= ∑e
−[∇(∇ · u)]T ? a[∇× (∇× u)]T ? a
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gradient with illumination compensation
∂αJD∂βJD
= ∑e
− [∇(∇ · u)]
T ? a
‖∇(∇ · u)‖2 ‖a‖2
[∇× (∇× u)]T ? a‖∇ × (∇× u)‖2 ‖a‖2
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∂αJD ∂βJD
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improved model update
I illumination compensation
I parameter rebalancing
I geometrical constraint
I physical constraint
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isotropic wave equation
ü = α∇(∇ · u)− β∇× (∇× u)
I α = λ+2µρI β = µρ
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isotropic wave equation
ü = α∇(∇ · u)− βc∇× (∇× u)
I α = λ+2µρ
Iβc =
µρ
I c : scaling factor
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∂αJD ∂βJD
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improved model update
I illumination compensation
I parameter rebalancing
I geometrical constraint
I physical constraint
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objective function J = JD + JM+JC
JM =1
2‖Wα (α− ᾱ) ‖2 +
1
2‖Wβ
(β − β̄
)‖2
I Wα,Wβ : inverse model covariance
I ᾱ,β̄ : prior models
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improved model update
I illumination compensation
I parameter rebalancing
I geometrical constraint
I physical constraint
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hl
hu
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hl > 0
hu < 0
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objective function J = JD + JM + JC
JC = −η∑
x
[log (−hu) + log (hl)]
η : weighting parameter
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hl > 0
hu < 0
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∂αJC ∂βJC
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α
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α w/ constraints
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β
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β w/ constraints
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∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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cross-well example
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α
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β
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x z
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x z
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α
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β
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α
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β
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∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
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T
S
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conclusions
multi-parameter inversion
improved model update
I illumination compensation
I parameter rebalancing
I geometrical constraint
I physical constraint
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α
β
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α
β
α
β