The adjoint-state method Francesco Perrone * and Paul Sava Center for Wave Phenomena Colorado School of Mines
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The adjoint-state method
Francesco Perrone∗ and Paul Sava
Center for Wave PhenomenaColorado School of Mines
Forward problem
m (x)
f (xs , t)
Forward problem
m (x)
f (xs , t)
F
Forward problem
m (x)
f (xs , t)
F dobs (xr , t)
Forward problem
m (x)
f (xs , t)
F dobs (xr , t)
Forward problem
m (x)
f (xs , t)
F dobs (xr , t)
Inverse problem
m (x)
f (xs , t)
F dobs (xr , t)
scattering
inverse
theory
inverse
problem
optimization
f (xs , t) dobs (xr , t)
geometry of seismic experiment
f (xs , t) dobs (xr , t) dcal (xr , t)
forward propagation
compare wavefields at receivers
f (xs , t) dobs (xr , t)
forward and backward propagation
compare wavefields everywhere
Objective function
H (m) = 12‖d
obs − d cal (m) ‖2
I dobs : observed data
I d cal (m): computed data
Fréchet
derivatives method
adjoint−state
gradient of a
function
data model
Frechet derivatives model
objective function model
gradient model
The adjoint-state method
efficient method for computing thegradient
The adjoint-state method
I state variables
I adjoint sources
I adjoint variables
I gradient
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
objective function
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
wave equation
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
model parameter
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
state variable
Initial model
u (x, t): state variable
u (x, t): state variable
u (x, t): state variable
u (x, t): state variable
u (x, t): state variable
u (x, t): state variable
u (x, t): state variable
Constrained optimization
A = H (u,m)−F∗ (u,m) · a
adjoint variable
Physical solutions
A = H (u,m)−F∗ (u,m) · a
Physical solutions
A = H (u,m)−F∗ (u,m) · a
F (u,m) = 0
Physical solutions
A = H (u,m)
F (u,m) = 0
Optimization of A (u, a,m)
∂A∂a = 0
∂A∂u = 0
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0
∂A∂u = 0
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = ∂H
∂u
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
∂F∂u : Wave-equation derivative
F (u,m) = m ∂2
∂t2u −∇2u − f
∂F∂u : Wave-equation derivative
F (u,m) = m ∂2
∂t2u −∇2u − f
∂F∂u : Wave-equation operator
F (u,m) = m ∂2
∂t2u −∇2u − f
∂F∂u = m ∂2
∂t2−∇2
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = ∂H
∂u
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = dcal (m)− dobs
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
d cal (m): computed data
dobs: observed data
d cal (m)− dobs: adjoint source
a (x, t): adjoint variable
a (x, t): adjoint variable
a (x, t): adjoint variable
a (x, t): adjoint variable
a (x, t): adjoint variable
a (x, t): adjoint variable
a (x, t): adjoint variable
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = dcal (m)− dobs
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = dcal (m)− dobs
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = dcal (m)− dobs
∂A∂m = ∂H
∂m −[∂F∂m
]∗· a
Optimization of A (u, a,m)
F (u,m) = 0[∂F∂u
]∗a = dcal (m)− dobs
∂A∂m = −
[∂F∂m
]∗· a
∂F∂m: wave-equation derivative
F (u,m) = m ∂2
∂t2u −∇2u − f
∂F∂m: wave-equation derivatives
F (u,m) = m ∂2
∂t2u −∇2u − f
∂F∂m = ∂2
∂t2u
−[∂F∂m
]∗ · a: gradient
−[∂F∂m
]∗ · a: gradient
Anomaly
The adjoint-state method
I state variables: Fu = f
I adjoint sources: g = ∂H∂u
I adjoint variables: F∗a = g
I gradient: ∂A∂m = −
[∂F∂m
]∗ · a + ∂H∂m
The adjoint-state method
I state variables: Fu = f
I adjoint sources: g = ∂H∂u
I adjoint variables: F∗a = g
I gradient: ∂A∂m = −
[∂F∂m
]∗ · a + ∂H∂m
Take-home message
I general method
I simple implementation
I no error analysis
References
Fichtner, A., J. Trampert, 2011, Hessian kernels of seismic datafunctionals based upon adjoint techniques: Geophys. J. Int., 185 ,775 - 798
Plessix, R.-E., 2006, A review of the adjoint-state method forcomputing the gradient of a functional with geophysical applications:Geophys. J. Int., 167 , 495–503.
Tromp, J., C. Tape, and Q. Liu, 2005, Seismic tomography, adjointmethods, time reversal and banana-doughnut kernels: Geophys. J.Int., 160, 195-216
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