inverse problems for stochastic partial differential equations: … · 2016-05-05 · ֒→...
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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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