Inverse problems for stochastic partial differential equations:

application to correlation-based imaging

Josselin Garnier (Universite Paris Diderot)

http://www.josselin-garnier.org/

• Consider a classical inverse problem, sensor array imaging:

- probe an unknown medium with waves,

- record the waves transmitted through or reflected by the medium,

- process the recorded data to extract relevant information.

→ The data set can be huge.

→ Only part of the unknown medium is of interest.

→ Least squares imaging is appropriate in the presence of measurement noise. What

about medium noise or source noise ?

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Conventional reflector imaging through a homogeneous medium

~xs ~xr

~yref

• Sensor array imaging of a reflector located at ~yref . ~xs is a source, ~xr is a receiver.

Measured data: u(t, ~xr; ~xs), r = 1, . . . , Nr, s = 1, . . . , Ns.

• Mathematical model:

( 1

c20

+1

c2ref

1Bref(~x− ~yref)

)∂2u

∂t2(t, ~x; ~xs)−∆~xu(t, ~x; ~xs) = f(t)δ(~x− ~xs)

• Purpose of imaging: using the measured data, build an imaging function I(~yS) that

would ideally look like 1

c2ref

1Bref(~yS − ~yref), in order to extract the relevant

information (~yref , Bref , cref) about the reflector.

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• Classical imaging functions:

1) Least-Squares imaging: minimize the quadratic misfit between measured data and

synthetic data obtained by solving the wave equation with a candidate (~y, B, c)test.

2) Linearized Least-Squares imaging: simplify Least-Squares imaging by

“linearization” of the forward problem (Born).

3) Reverse Time imaging: simplify Linearized Least-Squares imaging by forgetting

the normal operator.

4) Kirchhoff Migration: simplify Reverse Time imaging by substituting travel time

migration for full wave equation.

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• Classical imaging functions:

1) Least-Squares imaging: minimize the quadratic misfit between measured data and

synthetic data obtained by solving the wave equation with a candidate (~y, B, c)test.

2) Linearized Least-Squares imaging: simplify Least-Squares imaging by

“linearization” of the forward problem (Born).

3) Reverse Time imaging: simplify Linearized Least-Squares imaging by forgetting

the normal operator.

4) Kirchhoff Migration: simplify Reverse Time imaging by substituting travel time

migration for full wave equation.

• Kirchhoff Migration function:

IKM(~yS) =

Nr∑

r=1

Ns∑

s=1

u( |~xs − ~yS |

c0+

|~yS − ~xr|

c0, ~xr; ~xs

)

It forms the image with the superposition of the backpropagated traces.

|~yS − ~x|/c0 is the travel time from ~x to ~yS .

- Very robust with respect to (additive) measurement noise.

- Sensitive to medium noise: If the medium is scattering, then Kirchhoff Migration

(usually) does not work.

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Conventional reflector imaging through a scattering medium

~xs ~xr

~yref

• Sensor array imaging of a reflector located at ~yref . ~xs is a source, ~xr is a receiver.

Data: u(t, ~xr; ~xs), r = 1, . . . , Nr, s = 1, . . . , Ns.

( 1

c2(~x)+

1

c2ref

1Bref(~x− ~yref)

)∂2u

∂t2(t, ~x; ~xs)−∆~xu(t, ~x; ~xs) = f(t)δ(~x− ~xs)

• Random medium model:

1

c2(~x)=

1

c20

(

1 + µ(~x))

c0 is a reference speed,

µ(~x) is a zero-mean random process.

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Strategy: Stochastic and multiscale analysis

• The medium noise u− u0 (where u0 is the data that would be obtained in a

homogeneous medium) is very different from an additive measurement noise !

• A detailed multiscale analysis is possible in different regimes of separation of scales

(small wavelength, large propagation distance, small correlation length, . . .).

→ Stochastic partial differential equations (driven by Brownian fields).

→ Analysis of the distribution (moments) of u.

• General results obtained by a stochastic multiscale analysis:

• The mean (coherent) wave is small.

=⇒ The Kirchhoff Migration function (or Reverse Time imaging function, or Least

Squares imaging function) is unstable in randomly scattering media:

E[

IKM(~yS)]

Var(

IKM(~yS))1/2

≪ 1

• The wave fluctuations at nearby points and nearby frequencies are correlated.

The wave correlations carry information about the medium.

=⇒ One should use local correlations to solve the inverse problem.

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Imaging below an “overburden”

From van der Neut and Bakulin (2009)

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Imaging below an overburden

~xs

~xr

~yref

Array imaging of a reflector at ~yref . ~xs is a source, ~xr is a receiver located below the

scattering medium. Data: u(t, ~xr; ~xs), r = 1, . . . , Nr, s = 1, . . . , Ns.

If the overburden is scattering, then Kirchhoff Migration does not work:

IKM(~yS) =

Nr∑

r=1

Ns∑

s=1

u( |~xs − ~yS |

c0+

|~yS − ~xr|

c0, ~xr; ~xs

)

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Numerical simulations

Computational setup Kirchhoff Migration

(simulations carried out by Chrysoula Tsogka, University of Crete)

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Imaging below an overburden

~xs

~xr

~yref

~xs is a source, ~xr is a receiver. Data: u(t, ~xr; ~xs), r = 1, . . . , Nr, s = 1, . . . , Ns.

Image with migration of the special cross correlation matrix:

I(~yS) =

Nr∑

r,r′=1

C( |~xr − ~yS |

c0+

|~yS − ~xr′ |

c0, ~xr, ~xr′

)

,

with

C(τ, ~xr, ~xr′) =

Ns∑

s=1

∫

u(t, ~xr; ~xs)u(t+ τ, ~xr′ ; ~xs)dt , r, r′ = 1, . . . , Nr

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Remark: General CINT function (with u(ω, ~xr; ~xs) =∫

u(t, ~xr; ~xs)e−iωtdt:

ICINT(~yS) =

Ns∑

s,s′=1

|~xs−~xs′ |≤Xd

Nr∑

r,r′=1

|~xr−~xr′ |≤X′

d

∫∫

|ω−ω′|≤Ωd

dωdω′ u(ω, ~xr; ~xs)u(ω′, ~xr′ , ~xs′)

× exp

− iω[ |~xr − ~yS |

c0+

|~xs − ~yS |

c0

]

+ iω′[ |~xr′ − ~yS |

c0+

|~xs′ − ~yS |

c0

]

• If Xd = X ′d = Ωd = ∞, then ICINT(~y

S) =∣

∣IKM(~yS)∣

∣

2.

• If Xd = 0, X ′d = ∞, Ωd = 0, then ICINT(~y

S) is the previous imaging function.

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Numerical simulations

Kirchhoff Migration Correlation-based Migration

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Application: Ultrasound echography in concrete

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Ultrasound echography in concrete

Reciprocity: u(t,xr;xs) = u(t,xs;xr) for (almost) all r, s.

→ The data set is good !

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Ultrasound echography in concrete

350.0 400.0 450.0 500.0 550.0 600.0 650.0 700.0 750.0

Y axis

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

450.0

500.0

550.0

600.0

650.0

Z a

xis

x=400.0mm

0.00

0.15

0.30

0.45

0.60

0.75

0.90

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Extension: Ambient noise imaging

• Observation: Sources are needed, but they do not need to be known/controlled.

→ One can apply correlation-based imaging techniques to signals emitted by ambient

noise sources.

• The emission of ambient noise sources is modeled by stationary (in time) random

processes with mean zero.

→ This is another situation where the measured data have mean zero and the

information is carried by the correlations.

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Reflector imaging with a passive receiver array

• Ambient noise sources () emit stationary random signals.

• The signals (u(t, ~xr))r=1,...,Nrare recorded by the receivers (~xr)r=1,...,Nr

(N).

• The reflector () is imaged by migration of the cross correlation matrix [1]:

I(~yS) =

Nr∑

r,r′=1

CT

( |~xr′ − ~yS |

c0+

|~xr − ~yS |

c0, ~xr, ~xr′

)

with CT (τ, ~xr, ~xr′) =1

T

∫ T

0

u(t+ τ, ~xr′)u(t, ~xr)dt

0 50 100−50

0

50

z

x

x1

x5

−150 −100 −50 0 50 100 150−0.5

00.5

1

τ

coda correlation x1−x

5

0 100 200 300 400−0.5

00.5

1

t

signal recorded at x5

0 100 200 300 400−0.5

00.5

1

t

signal recorded at x1

Good image provided the ambient noise illumination is long (in time) and diversified

(in angle) [1].

[1] J. Garnier and G. Papanicolaou, SIAM J. Imaging Sciences 2, 396 (2009).

Conclusions

• In order to solve inverse problems in complex environments (beyond the additive

noise situation), one should use and process well chosen cross correlations of data, not

data themselves.

• Correlation-based imaging can be applied with ambient noise sources instead of

controlled sources.

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Conclusions

• In order to solve inverse problems in complex environments (beyond the additive

noise situation), one should use and process well chosen cross correlations of data, not

data themselves.

• Correlation-based imaging can be applied with ambient noise sources instead of

controlled sources.

See recent book (Cambridge University Press, April 29, 2016):

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