the adjoint-state method - colorado school of...

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