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