time reversal techniques in wave imaging and applicationswahab/documents/nust12.pdf · [gallot 11,...

Time Reversal Techniques in Wave Imaging andApplications

Abdul WahabLPMA, University of Paris VII,

&CMAP, Ecole Polytechnique,

France.

CAMP, National University of Science & Technology,Pakistan.

Joint work with H. Ammari (ENS-Paris), E. Bretin (INSA-Lyon), J. Garnier (Paris VII),

S. Gdoura (Supelec-Paris), D. Lesselier (Supelec-Paris)

A. Wahab (CMAP, LPMA) Time Reversal in Wave Imaging NUST, April 18, 2012 1 / 57

Outlines

Principles of Time Reversal

Inverse Source Problems

Time Reversal Cavity

Applications of Time Reversal

Open Questions

Principle of Time reversalIntroduced by M. Fink (Paris VII & ESPCI) in 1989.

The wave equation with compressibility K, density ρ and speed c = 1/√ρK :

W[u](x, t) :=

(∇ ·

ρ∇−

ρc2∂2

)u(x, t) = 0, x ∈ Rd.

is time invariant (under transformation t→ −t).If u(x, t) is solution then so does u(x,−t) !

The fundamental solution, G, to the operator W verifies spatial reciprocity i.e.

G(x− y, t) = G(y − x, t), ∀x, y ∈ Rd, x 6= y.

One can revert a wave from its final state to initial state.

Time reversalIt is a technique that focuses waves onto a source or a scatter by emitting a time reversedversion of the received wave field measured by an array of transducers.(C. Bardos, 09)

W[u](x, t) :=

(∇ ·

ρ∇−

ρc2∂2

)u(x, t) = 0, x ∈ Rd.

G(x− y, t) = G(y − x, t), ∀x, y ∈ Rd, x 6= y.

W[u](x, t) :=

(∇ ·

ρ∇−

ρc2∂2

)u(x, t) = 0, x ∈ Rd.

G(x− y, t) = G(y − x, t), ∀x, y ∈ Rd, x 6= y.

W[u](x, t) :=

(∇ ·

ρ∇−

ρc2∂2

)u(x, t) = 0, x ∈ Rd.

G(x− y, t) = G(y − x, t), ∀x, y ∈ Rd, x 6= y.

W[u](x, t) :=

(∇ ·

ρ∇−

ρc2∂2

)u(x, t) = 0, x ∈ Rd.

G(x− y, t) = G(y − x, t), ∀x, y ∈ Rd, x 6= y.

Principle of Time reversalAcoustic Time-reversal Cavity : Physical Experiment

Time reversal of a scattered wave by an obstacle. [Fink 99]

Principle of Time reversalAcoustic Time-reversal Cavity : Applications

TRCGiven the measurements of wave field scattered by a (point or extended) scatterer at asurface(in far field). Find

- Properties of the medium or scatterer such as refractive index, Young’s modulus,Poisson’s ratio, electric permittivity, magnetic permeability etc.

- Find the shape (for extended scatterers) and the location of the scatterers.

Applications

- Biomedical imaging with diffracting sources.

- Non-destructive evolution.

- Exploration geophysics and seismology.

- Telecommunications.

- Synthetic aperture radar imaging.

- Passive tomography and travel time imaging.

Principle of Time reversalAcoustic Time-reversal Cavity : Applications

TRCGiven the measurements of wave field scattered by a (point or extended) scatterer at asurface(in far field). Find

- Properties of the medium or scatterer such as refractive index, Young’s modulus,Poisson’s ratio, electric permittivity, magnetic permeability etc.

- Find the shape (for extended scatterers) and the location of the scatterers.

Applications

- Biomedical imaging with diffracting sources.

- Non-destructive evolution.

- Exploration geophysics and seismology.

- Telecommunications.

- Synthetic aperture radar imaging.

- Passive tomography and travel time imaging.

Principle of Time reversalInverse Source Problems : Physical Experiment

Time reversal of a secondary (or modulated) wave.

Principle of Time reversalInverse Source Problems : Applications

ISPSuppose a primary incident wave passes through a medium and is modulated or triggers asecondary wave due to the presence of an anomaly. Given the measurements of themodulated or secondary wave field at a surface(in far field). Find the source (i.e. theanomaly) of the modulation or the secondary wave.

Applications

- Biomedical imaging with non-diffracting sources.

- Reservoir localization.

- EM source localization with applications in Robotics.

- Earthquake source localization.

- Identification of the initial state of a dynamical system.

Principle of Time reversalInverse Source Problems : Applications

ISPSuppose a primary incident wave passes through a medium and is modulated or triggers asecondary wave due to the presence of an anomaly. Given the measurements of themodulated or secondary wave field at a surface(in far field). Find the source (i.e. theanomaly) of the modulation or the secondary wave.

Applications

- Biomedical imaging with non-diffracting sources.

- Reservoir localization.

- EM source localization with applications in Robotics.

- Earthquake source localization.

- Identification of the initial state of a dynamical system.

Principle of Time reversalSome Important Results

Physics & Experiments

- [Cassereau & Fink 92, TR Cavity], [Fink 97], [Carminati 07, EM-Waves], [Anderson11, Elastic-ISP], [Johnson, Solids and non-linear elastic media].

Mathematical Analysis

- [Bardos & Fink, TRC], [Blomgren, Papanicolaou, Zhao, Randomness Hypothesis],[Garnier, Papanicolaou, Solna, Wave guides, Layered media, Dispersive media], [Bal,Changing Environment].

Applications

- [Lerosey, Rosny, Tourin, Fink, Science 07], [Larmat 10, Seismology], [Larmat 08 Glacialearthquake], [de Rosny, Telecommunications], [Stojanovic 05, Underwater acoustics],[Gallot 11, Passive elastography].

Biomedical Imaging - [Ammari 11, Elastography], [Burgholzer 08, Photo-acoustics],[Kuchment & Hristova 08, Thermo-acoustics], [Treeby 08, Photo-aocustics].

Inverse Source Problems

Inverse Source ProblemsMotivation : Multi-Physics Imaging

Tissue optical absorption:

p(y,t)

Optical pulsePhoto-acoustic imaging

- Absorbed energy density.

Magneto-acoustic imaging

- Divergence of the Lorentz force.

Acoustic radiation force imaging

- Radiation force.

Inverse Source ProblemsMathematical Formulation

Let f be compactly supported in a bounded smooth domain Ω ⊂ Rd with d = 2, 3 andboundary ∂Ω.

ProblemFind supp

ga(y, t) := pa(y, t) : (y, t) ∈ ∂Ω× [0, T ]

such that :

c2∂2

∂t2−∆− a

∂t∆

)pa(x, t) =

∂tδ0(t)f(x), (x, t) ∈ Rd × R,

pa(x, t) = 0 =∂pa(x, t)

∂t, x ∈ Rd, t 0,

for T sufficiently large.

- a ≥ 0 : Medium dependant attenuation parameter,

- c : Speed of the wave front.

Inverse Source ProblemsMathematical Formulation

Let f be compactly supported in a bounded smooth domain Ω ⊂ Rd with d = 2, 3 andboundary ∂Ω.

ProblemFind supp

ga(y, t) := ua(y, t) : (y, t) ∈ ∂Ω× [0, T ]

such that :

∂t2−

∂tLηλ,ηµ − Lλ,µ

)ua(x, t) =

∂tδ0(t)f(x), (x, t) ∈ Rd × R,

ua(x, t) = 0 =∂

∂tua(x, t), x ∈ Rd, t 0,

for T sufficiently large.

- (λ, µ) : Lame parameters,

- (ηλ, ηµ) : visco-elastic moduli,

- Lα,β [~pa] = (α+ β)∇∇ · ua − β∆ua,

Inverse Source ProblemsAcoustic Time-reversal : Integral formulation

Adjoint WaveLet v be the solution of the wave equation

∂ttv(x, t)−∆v(x, t) = 0, (x, t) ∈ Ω× (0, T )

v(x, 0) = 0, ∂tv(x, 0) = 0, x ∈ Ω

v(x, t) = g0(x, T − t), (x, t) ∈ ∂Ω× [0, T ]

Then,v(x, t) = p0(x, T − t), ∀(x, t) ∈ Ω× [0, T ], and v(x, T ) = f(x)

Exact Integral FormulationGreen’s theorem and integration by parts yield

f(x) = v(x, T ) =

∫∂Ω

∂GD(x, y, t− T )

∂νyg0(y, t− T )dσ(y) ∀x ∈ Ω

where GD is the Dirichlet Green function and v is the adjoint wave.

Inverse Source ProblemsAcoustic Time-reversal : Integral formulation

Adjoint WaveLet v be the solution of the wave equation

∂ttv(x, t)−∆v(x, t) = 0, (x, t) ∈ Ω× (0, T )

v(x, 0) = 0, ∂tv(x, 0) = 0, x ∈ Ω

v(x, t) = g0(x, T − t), (x, t) ∈ ∂Ω× [0, T ]

Then,v(x, t) = p0(x, T − t), ∀(x, t) ∈ Ω× [0, T ], and v(x, T ) = f(x)

Exact Integral FormulationGreen’s theorem and integration by parts yield

f(x) = v(x, T ) =

∫∂Ω

∂GD(x, y, t− T )

∂νyg0(y, t− T )dσ(y) ∀x ∈ Ω

where GD is the Dirichlet Green function and v is the adjoint wave.

Inverse Source ProblemsAcoustic Time-reversal : Experiment

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Simulations carried out by E. Bretin(INSA-Lyon)

Inverse Source ProblemsAcoustic Time-reversal : Experiment

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Simulations carried out by E. Bretin(INSA-Lyon)

Inverse Source ProblemsAcoustic Time-reversal : Integral formulation II

Modified TR-functionalLet G0(x, y, t) be the outgoing fundamental solution and vs(x, t) be such that

∂ttvs(x, t)−∆vs(x, t) = ∂tδs(t)g0(x, T − s)δ∂Ω(x), ∀(x, t) ∈ Rd × R,vs(x, t) = 0, ∂tvs(x, t) = 0 ∀x ∈ Rd, t s.

Then, a modified time-reversal functional is given by

I(x) :=

0vs(x, T )ds

∫∂Ω

∂tG0(x, y, T − s)g0(y, T − s)dσ(y)ds ∀x ∈ Ω.

Remark that g0(y) = −iω∫

ΩG0(z, y)f(z)dz for all y ∈ ∂Ω

Helmholtz-Kirchhoff Identity : For x, z ∈ Ω sufficiently far from y ∈ ∂Ω∫∂Ω

G0(x, y)G0(z, y)dσ(y) '1

G0(x, z)

∫Rω=m

G0(x, z)

dω = δx(z)

Therefore,

I(x) =1

∫Rdf(z)

∫∂Ωω2G0(x, y)G0(z, y)dσ(y)dωdz

∫Rdf(z)

∫Rω=m

G0(x, z)

TheoremFor x far from ∂Ω (w.r.t. wavelength), we have I(x) ' f(x).

G0(x, y)G0(z, y)dσ(y) '1

G0(x, z)

∫Rω=m

G0(x, z)

dω = δx(z)

Therefore,

I(x) =1

∫Rdf(z)

∫Rω=m

G0(x, z)

G0(x, y)G0(z, y)dσ(y) '1

G0(x, z)

∫Rω=m

G0(x, z)

dω = δx(z)

Therefore,

I(x) =1

∫Rdf(z)

∫Rω=m

G0(x, z)

Inverse Source ProblemsAcoustic Time-reversal : Reconstructions

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Left to Right : Initial image, Exact time-reversal, Modified time-reversal

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TR in Attenuating Media

Inverse Source ProblemsAcoustic Time-reversal : Reconstructions

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Left to Right : Initial image, Exact time-reversal, Modified time-reversal

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TR in Attenuating Media

Inverse Source ProblemsAcoustic Time-reversal : TR in attenuating media

Consider the thermo-viscous wave equation∂ttpa(x, t)−∆pa(x, t)− a∂t(∆pa(x, t)) = 0

pa(x, 0) = f(x), and ∂tpa(x, 0) = 0.

Attenuated TR-functionalDefine

Ia(x) =

0vs,a(x, T )ds ∀x ∈ Ω

where vs,a(x, t) is the solution of the adjoint attenuated wave equation [Burgholzer 07], [Treeby 10]

∂ttvs,a(x, t)−∆vs,a(x, t)+a∂t(∆vs,a(x, t)) = ∂tδs(t)ga(x, T − s)δ∂Ω(x).

- Stability,

- Order of correction,

pa(x, 0) = f(x), and ∂tpa(x, 0) = 0.

Ia(x) =

- Stability,

pa(x, 0) = f(x), and ∂tpa(x, 0) = 0.

Ia(x) =

- Stability,

Inverse Source ProblemsAcoustic Time-reversal : Truncated TR functional

LetGa be the fundamental solution of the Lossy Helmholtz equation

ω2 Ga(x, y) + (1 + iaω) ∆yGa(x, y) = −δx(y) in Rd.

Let Ga,ρ(x, y, t) :=1

∫|ω|<ρ

Ga(x, y) exp(−iωt)dω

Consider an approximation vs,a,ρ(x, t) of vs,a(x, t) given by :

vs,a,ρ(x, t) =

∫∂Ω

∂tGa,ρ(x, y, t− s)ga(y, T − s)dσ(y)

We define :

Truncated TR-FunctionalFor all x ∈ Ω :

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds =

∫∂Ω

∂tGa,ρ(x, y, T − s)ga(y, T − s)dσ(y)ds.

∫|ω|<ρ

vs,a,ρ(x, t) =

∫∂Ω

We define :

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds =

∫∂Ω

∫|ω|<ρ

vs,a,ρ(x, t) =

∫∂Ω

We define :

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds =

∫∂Ω

Inverse Source ProblemsAcoustic Time-reversal : Attenuation operators

Remark that p0(x, ω) and pa(x, ω) satisfy(κ2(ω) + ∆

)pa(x, ω) =

iκ2(ω)

ωf(x), and

(ω2 + ∆

)p0(x, ω) = iωf(x).

Therefore, pa(x, ω) =κ(ω)ω

p0(x, κ(ω)) or pa(x, t) = L[p0(x, ·)](t) where

Attenuation Operator

L[φ](t) =1

κ(ω)

∫Rφ(s)eiκ(ω)sds

e−iωtdω.

with κ(ω) =ω

√1− iaω

is the complex wave number.

Remark that p0(x, ω) and pa(x, ω) satisfy(κ2(ω) + ∆

)pa(x, ω) =

iκ2(ω)

ωf(x), and

(ω2 + ∆

)p0(x, ω) = iωf(x).

Therefore, pa(x, ω) =κ(ω)ω

p0(x, κ(ω)) or pa(x, t) = L[p0(x, ·)](t) where

Attenuation Operator

L[φ](t) =1

κ(ω)

∫Rφ(s)eiκ(ω)sds

e−iωtdω.

with κ(ω) =ω

√1− iaω

is the complex wave number.

Moreover, we define operator Lρ associated with κ(ω) = ω√1+iaω

Lρ[φ](t) :=1

∫ ∞0

∫|ω|≤ρ

κ(ω)

ωeiκ(ω)se−iωtdω

We denote its adjoint operator by L∗ρ given by

L∗ρ[φ](t) =1

∫|ω|≤ρ

κ(ω)

ωeiκ(ω)t

∫ ∞0

φ(s)e−iωsds

2caω2s

expiω(s− t)dω =

2π√cas

(t− s)2

for κ(ω) = ω + iaω2

L[φ](t) =1

2√π

)∫ ∞0

φ(s)2π√cas

(t− s)2

Lρ[φ](t) :=1

∫ ∞0

∫|ω|≤ρ

κ(ω)

L∗ρ[φ](t) =1

∫|ω|≤ρ

κ(ω)

ωeiκ(ω)t

∫ ∞0

φ(s)e−iωsds

2caω2s

expiω(s− t)dω =

2π√cas

(t− s)2

L[φ](t) =1

2√π

)∫ ∞0

φ(s)2π√cas

(t− s)2

Lρ[φ](t) :=1

∫ ∞0

∫|ω|≤ρ

κ(ω)

L∗ρ[φ](t) =1

∫|ω|≤ρ

κ(ω)

ωeiκ(ω)t

∫ ∞0

φ(s)e−iωsds

2caω2s

expiω(s− t)dω =

2π√cas

(t− s)2

L[φ](t) =1

2√π

)∫ ∞0

φ(s)2π√cas

(t− s)2

Inverse Source ProblemsAcoustic Time-reversal : Stationary Phase Analysis

Theorem (Stationary Phase Theorem [Hormander 03])

Let K ⊂ [0,∞) be a compact set, X an open neighbourhood of K and k ∈ N. Let

ψ ∈ C2k0 (K) and h ∈ C3k+1

0 (X) be such that :

=mh ≥ 0, =mh(t0) = 0 = h′(t0), h′′(t0) 6= 0, h′ 6= 0 in K\t0

then for ε > 0∣∣∣∣∣∫Kψ(t)eih(t)/εdt− eih(t0)/ε

(h′′(t0)/2iπε

)−1/2∑j<k

εjDj [ψ]

∣∣∣∣∣≤ Cεk

∑α≤2k

supx∈K

∣∣ψ(α)(x)∣∣

Dj [ψ] =∑

ν−µ=j

∑2ν≥3µ

i−j(−1)ν1

2ν ν!µ!h′′(t0)−v

(θµt0ψ

)(2ν)(t0)

θt0 (t) := h(t)− h(t0)−1

2h′′(t0)(t− t0)2.

Inverse Source ProblemsAcoustic Time-reversal : Asymptotic approximation : a ω−1

Proposition

For κ(ω) ' ω + iaω2

2and a→ 0 following results hold :

Let φ(t) ∈ S([0,∞[), then

L[φ](t) = φ(t) +a

(tφ′)′

(t) + o(a).

Let φ(t) ∈ D([0,∞[), then for all ρ > 0

L∗ρ[φ](t) = Sρ[φ](t)−a

2Sρ[(tφ′)′] + o(a).

Let φ(t) ∈ D([0,∞[) and ρ > 0, then

L∗ρ [L[φ]] (t) = Sρ[φ](t) + o(a).

where S is the Schwartz space, D is the space of C∞−functions of compact support and

Sρ[φ](t) =1

∫|ω|≤ρ

e−iωtφ(ω)dω

Inverse Source ProblemsAcoustic Time-reversal : Analysis of truncated functional

Consequently we have

Ia,ρ(x) =

∫∂Ω

∂tGa,ρ(x, y, t)ga(y, t)dσ(y)dt

∫∂ΩLρ [∂tG0(x, y, ·)] (t)L [g(y, ·)] (t)dσ(y)dt

∫∂Ω

∂tG0(x, y, t)L∗ρ [L [g0(y, ·)]] (t)dσ(y)dt

∫∂Ω

∂tG0(x, y, t)Sρ [g0(y, ·)] (t)dσ(y)dt+ o(a)

Finally remark that

δρ,x(z) =1

∫|ω|≤ρ

ω=mG0(x, z)

dω → δx(z) as ρ→ +∞.

Therefore,

Ia,ρ(x) ' δρ,x(y) ∗ f(y) + o(a)

ρ→∞−−−−→

f(x) + o(a).

Ia,ρ(x) =

∫∂Ω

Finally remark that

δρ,x(z) =1

∫|ω|≤ρ

ω=mG0(x, z)

Therefore,

Ia,ρ(x) ' δρ,x(y) ∗ f(y) + o(a)

ρ→∞−−−−→

f(x) + o(a).

Ia,ρ(x) =

∫∂Ω

Finally remark that

δρ,x(z) =1

∫|ω|≤ρ

ω=mG0(x, z)

Therefore,

Ia,ρ(x) ' δρ,x(y) ∗ f(y) + o(a)

ρ→∞−−−−→

f(x) + o(a).

Inverse Source ProblemsAcoustic Time-reversal : Truncated TR-functional Reconstructions

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Test with a = 0.0005. Left to Right : Without correction, with correction & ρ = 15, withcorrection & ρ = 20.

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Test with a = 0.001. Left to Right : Without correction, with correction & ρ = 15, with correction& ρ = 20.

Inverse Source ProblemsAcoustic Time-reversal : Pre-processing TR-scheme

As ga(y, t) = L[g0(y, ·)](t), an alternative strategy is to

- pre-process the measured data ga(y, t) using a pseudo-inverse of L as a filter

- apply the ideal time-reversal functional I(x) to identify source location.

Using higher order asymptotic expansion :

L[φ](t) =

k∑m=0

(tmφ′

)(2m−1)(t) + o(ak)

L−1k [φ](t) =

k∑m=0

amφk,m(t) such that L−1k L[φ](t) = φ(t) + o(ak).

and φk,m verifyφk,0 = φ

φk,m = −m∑l=1

Dl[φk,m−l],and Dmφ(t) =

(tmφ′

)(2m−1)(t).

L[φ](t) =k∑

(tmφ′

)(2m−1)(t) + o(ak)

L−1k [φ](t) =

k∑m=0

φk,m = −m∑l=1

(tmφ′

)(2m−1)(t).

L[φ](t) =k∑

(tmφ′

)(2m−1)(t) + o(ak)

L−1k [φ](t) =

k∑m=0

φk,m = −m∑l=1

(tmφ′

)(2m−1)(t).

Inverse Source ProblemsAcoustic Time-reversal : Pre-processing TR-scheme Reconstructions

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Test with a = 0.001. Left to Right : Without correction, with correction & k = 1, with correction& k = 4.

Inverse Source ProblemsElastic Time-reversal

ProblemFind supp

g0(y, t) := u0(y, t) : (y, t) ∈ ∂Ω× [0, T ]

such that :(

∂tt − Lλ,µ)u0(x, t) = ∂tδ0(t)f(x), (x, t) ∈ Rd × R,

u0(x, t) = 0, ∂tu0(x, t) = 0, x ∈ Rd, t 0,

for T sufficiently large and

Lα,β [u] = (α+ β)∇∇ · u− β∆u.

Elastic TR-functionalConsider

I(x) :=

0vs(x, T )ds,

where vs(x, t) is the adjoint elastic wave :∂ttvs(x, t)− Lλ,µvs(x, t) = ∂tδs(t)g0(x, T − s)δ∂Ω(x), ∀(x, t) ∈ Rd × R,vs(x, t) = 0, ∂tvs(x, t) = 0 ∀x ∈ Rd, t s.

Inverse Source ProblemsElastic Time-reversal

ProblemFind supp

g0(y, t) := u0(y, t) : (y, t) ∈ ∂Ω× [0, T ]

such that :(

∂tt − Lλ,µ)u0(x, t) = ∂tδ0(t)f(x), (x, t) ∈ Rd × R,

u0(x, t) = 0, ∂tu0(x, t) = 0, x ∈ Rd, t 0,

for T sufficiently large and

Lα,β [u] = (α+ β)∇∇ · u− β∆u.

Elastic TR-functionalConsider

I(x) :=

0vs(x, T )ds,

where vs(x, t) is the adjoint elastic wave :∂ttvs(x, t)− Lλ,µvs(x, t) = ∂tδs(t)g0(x, T − s)δ∂Ω(x), ∀(x, t) ∈ Rd × R,vs(x, t) = 0, ∂tvs(x, t) = 0 ∀x ∈ Rd, t s.

Inverse Source ProblemsElastic Time-reversal : Integral formulation and Green’s Tensors

Integral formulation

I(x) := <e

∫Rω2

[∫∂Ω

G(x, y)G(y, z)dσ(y)

]dωf(z) dz

We have defined G(x, y) := G0(x− y) such that G0(x− y) is the fundamental solutionof the Navier equation i.e.(

Lλ,µ + ω2)G0(x) = −δ0(x)I, x ∈ Rd.

It can be expressed as

G0(x) =1

µκ2s

(κ2sG

s0(x)I +∇x∇x

(Gs0 − G

)(x)), x ∈ Rd,

- [∆ + κ2α]Gα0 (x) = −δ(x),

- κ2p = ω2(λ+ 2µ)−1,

- κ2s = ω2µ−1

- α = p, s.

Inverse Source ProblemsElastic Time-reversal : Integral formulation and Green’s Tensors

Integral formulation

I(x) := <e

∫Rω2

[∫∂Ω

]dωf(z) dz

We have defined G(x, y) := G0(x− y) such that G0(x− y) is the fundamental solutionof the Navier equation i.e.(

Lλ,µ + ω2)G0(x) = −δ0(x)I, x ∈ Rd.

It can be expressed as

G0(x) =1

µκ2s

(κ2sG

s0(x)I +∇x∇x

(Gs0 − G

)(x)), x ∈ Rd,

- [∆ + κ2α]Gα0 (x) = −δ(x),

- κ2p = ω2(λ+ 2µ)−1,

- κ2s = ω2µ−1

- α = p, s.

Inverse Source ProblemsElastic Time-reversal : Helmholtz-Kirchhoff identities

Let Gp and Gs be the pressure and shear modes of G such that G(x) = Gp(x) + Gs(x).Then

Proposition (Elastic H.K.-identities)

For all x, z ∈ Ω, we have

∫∂Ω

[∂G(x, y)

∂νG(y, z)− G(x, y)

∂G(y, z)

]dσ(y) = 2i=m

G(x, z)

∫∂Ω

[∂Gα(x, y)

∂νGα(y, z)− Gα(x, y)

∂Gα(y, z)

]dσ(y) = 2i=m

Gα(x, z)

∫∂Ω

[∂Gs(x, y)

∂νGp(y, z)− Gs(x, y)

∂Gp(y, z)

]dσ(y) = 0.

where α = p, s and the co-normal derivative in the outward unit normal direction n isdefined by

∂ν:= λ(∇ · u)n + µ

(∇uT + (∇uT )T

Inverse Source ProblemsElastic Time-reversal : Helmholtz-Kirchhoff identities

Let Gp and Gs be the pressure and shear modes of G such that G(x) = Gp(x) + Gs(x).Then

Proposition (Elastic H.K.-identities)

For all x, z ∈ Ω, we have

∫∂Ω

[∂G(x, y)

∂νG(y, z)− G(x, y)

∂G(y, z)

]dσ(y) = 2i=m

G(x, z)

∫∂Ω

[∂Gα(x, y)

∂νGα(y, z)− Gα(x, y)

∂Gα(y, z)

]dσ(y) = 2i=m

Gα(x, z)

∫∂Ω

[∂Gs(x, y)

∂νGp(y, z)− Gs(x, y)

∂Gp(y, z)

]dσ(y) = 0.

where α = p, s and the co-normal derivative in the outward unit normal direction n isdefined by

∂ν:= λ(∇ · u)n + µ

(∇uT + (∇uT )T

Inverse Source ProblemsElastic Time-reversal : Helmholtz-Kirchhoff identities II

Proposition

If n = y − x and |x− y| 1 then

∂Gα

∂ν(x, y) = iωcαGα(x, y) + o

(|x− y|1−d/2

), α = p, s.

where cs =√µ and cp =

√λ+ 2µ are shear and pressure wave speeds.

LemmaLet Ω ⊂ Rd be a ball with large radius (w.r.t. wavelength). Then, for all x, z ∈ Ω sufficientlyfar from the boundary ∂Ω, we have

1. <e∫

∂ΩGα(x, y)Gα(y, z)dσ(y)

ωcα=m

Gα(x, z)

, α = p, s.

2. <e∫

∂ΩGs(x, y)Gp(y, z)dσ(y)

Inverse Source ProblemsElastic Time-reversal : Helmholtz-Kirchhoff identities II

Proposition

If n = y − x and |x− y| 1 then

∂Gα

∂ν(x, y) = iωcαGα(x, y) + o

(|x− y|1−d/2

), α = p, s.

where cs =√µ and cp =

√λ+ 2µ are shear and pressure wave speeds.

LemmaLet Ω ⊂ Rd be a ball with large radius (w.r.t. wavelength). Then, for all x, z ∈ Ω sufficientlyfar from the boundary ∂Ω, we have

1. <e∫

∂ΩGα(x, y)Gα(y, z)dσ(y)

ωcα=m

Gα(x, z)

, α = p, s.

2. <e∫

∂ΩGs(x, y)Gp(y, z)dσ(y)

Inverse Source ProblemsElastic Time-reversal : Analysis of TR-functional

For x far from ∂Ω,

I(x) = <e

∫Rω2

[∫∂Ω

]dωf(z) dz

cs + cp

∫Rω=m

(Gp + Gs

)(x, z)

dωf(z) dz

+cs − cpcscp

∫Rω=m

(Gp − Gs

)(x, z)

dωf(z) dz

'cs + cp

2cscpf(x) +

cs − cp2cscp

B(x, z)f(z)dz.

The operator B(x, z) :=1

∫Rω=m

(Gp − Gs

)(x, z)

dω, is not diagonal.

The reconstruction mixes the components of f when cs 6= cp.

Inverse Source ProblemsElastic Time-reversal : Analysis of TR-functional

For x far from ∂Ω,

I(x) = <e

∫Rω2

[∫∂Ω

]dωf(z) dz

cs + cp

∫Rω=m

(Gp + Gs

)(x, z)

dωf(z) dz

+cs − cpcscp

∫Rω=m

(Gp − Gs

)(x, z)

dωf(z) dz

'cs + cp

2cscpf(x) +

cs − cp2cscp

B(x, z)f(z)dz.

The operator B(x, z) :=1

∫Rω=m

(Gp − Gs

)(x, z)

dω, is not diagonal.

The reconstruction mixes the components of f when cs 6= cp.

Inverse Source ProblemsElastic Time-reversal : Reconstructions using I

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Left to Right : Initial data, reconstruction with (λ, µ) = (1, 1), with (λ, µ) = (10, 1).

Inverse Source ProblemsElastic Time-reversal : Weighted TR-functional

Let Ψ and Φ be the divergence and the curl free functions respectively such that

I = ∇×Ψ +∇Φ.

Define the weighted time-reversal functional by

I := cs∇×Ψ + cp∇Φ.

∫Rω2

[∫∂Ω

(csGs(x, y) + cpGp(x, y)

)G(y, z)

∫R−iω

[∂G(x, y)

∂νG(y, z)− G(x, y)

∂G(y, z)

]dωf(z) dz

∫Rω=m

G(x, z)

dωf(z) dz

Theorem

Let x ∈ Ω be sufficiently far (w.r.t. wavelength) from the boundary ∂Ω. Then, I(x) ' f(x).

I = ∇×Ψ +∇Φ.

I := cs∇×Ψ + cp∇Φ.

∫Rω2

[∫∂Ω

)G(y, z)

∫R−iω

[∂G(x, y)

∂νG(y, z)− G(x, y)

∂G(y, z)

]dωf(z) dz

∫Rω=m

G(x, z)

dωf(z) dz

Theorem

I = ∇×Ψ +∇Φ.

I := cs∇×Ψ + cp∇Φ.

∫Rω2

[∫∂Ω

)G(y, z)

∫R−iω

[∂G(x, y)

∂νG(y, z)− G(x, y)

∂G(y, z)

]dωf(z) dz

∫Rω=m

G(x, z)

dωf(z) dz

Theorem

Inverse Source ProblemsElastic Time-reversal : Reconstructions using I :

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Reconstruction with (λ, µ) = (1, 1). Left to Right : Initial data, I(x), I(x).

Inverse Source ProblemsElastic Time-reversal : Reconstructions using I

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Reconstruction with (λ, µ) = (10, 1). Left to Right : Initial data, I(x), I(x).

Inverse Source ProblemsElastic Time-reversal : Visco-elastic media

Consider for visco-elastic moduli (ηλ, ηµ)(∂tt − Lλ,µ − ∂tLηλ,ηµ

)ua(x, t) = ∂tδ0(t)f(x), (x, t) ∈ Rd × R,

ua(x, 0) = 0, ∂tua(x, 0) = 0, x ∈ Rd, t s.

Define

vs,a,ρ(x, t) = −1

∫|ω|≤ρ

∫∂Ω

iωG−a(x, y)ga(y, T − s)dσ(y)

e−iω(t−s)dω

where (Lλ,µ ± iωLηλ,ηµ + ω2

)G∓a(x, y) = −δy(x)I, x, y ∈ Rd.

Define

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds

Finally, for Ψ and Φ the divergence and curl free components of Ia,ρ, let

Ia,ρ(x) := cp∇Φ + cs∇×Ψ

)ua(x, t) = ∂tδ0(t)f(x), (x, t) ∈ Rd × R,

ua(x, 0) = 0, ∂tua(x, 0) = 0, x ∈ Rd, t s.

Define

vs,a,ρ(x, t) = −1

∫|ω|≤ρ

∫∂Ω

e−iω(t−s)dω

)G∓a(x, y) = −δy(x)I, x, y ∈ Rd.

Define

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds

)ua(x, t) = ∂tδ0(t)f(x), (x, t) ∈ Rd × R,

ua(x, 0) = 0, ∂tua(x, 0) = 0, x ∈ Rd, t s.

Define

vs,a,ρ(x, t) = −1

∫|ω|≤ρ

∫∂Ω

e−iω(t−s)dω

)G∓a(x, y) = −δy(x)I, x, y ∈ Rd.

Define

Ia,ρ(x) :=

0vs,a,ρ(x, T )ds

Proposition

Let Ω ⊂ Rd be a ball with large radius. Then,

∂ΩGs−a(x, y)Gpa(y, z)dσ(y)

∂ΩGp−a(x, y)Gsa(y, z)dσ(y)

for all x, z ∈ Ω sufficiently far from the boundary ∂Ω w.r.t. wavelength

TheoremFor all x ∈ Ω sufficiently far from the boundary ∂Ω, we have

Ia,ρ(x) = Iρ(x) + o(ν2s/c

2s + ν2

p/c2p)

whereIρ(x)

ρ→∞−→ I(x) ' f(x),

νs and νp are shear and bulk viscosities and

Iρ(x) =

∫∂Ω

0∂t[csGs(x, y, t) + cpGp(x, y, t)

]Sρg0(y, ·)

(t)dt dσ(y)

Proposition

Let Ω ⊂ Rd be a ball with large radius. Then,

∂ΩGs−a(x, y)Gpa(y, z)dσ(y)

∂ΩGp−a(x, y)Gsa(y, z)dσ(y)

for all x, z ∈ Ω sufficiently far from the boundary ∂Ω w.r.t. wavelength

TheoremFor all x ∈ Ω sufficiently far from the boundary ∂Ω, we have

Ia,ρ(x) = Iρ(x) + o(ν2s/c

2s + ν2

p/c2p)

whereIρ(x)

ρ→∞−→ I(x) ' f(x),

νs and νp are shear and bulk viscosities and

Iρ(x) =

∫∂Ω

0∂t[csGs(x, y, t) + cpGp(x, y, t)

]Sρg0(y, ·)

(t)dt dσ(y)

Inverse Source ProblemsElastic Time-reversal : Visco-elastic Weighted TR Reconstructions

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Reconstruction with (λ, µ) = (1, 1) and a = 0.0002. Left to Right : Initial data, without correction

using I(x), correction using Ia,ρ with ρ = 15, with ρ = 20.

Noise Source Localization

Noise Source LocalizationStatement of the Problem

Let p0 satisfy the wave equation1

∂t2p0(x, t)−∆p0(x, t) = n(x, t), (x, t) ∈ Rd × R

p0(x, t) = 0, and∂

∂tp0(x, t) = 0, x ∈ Rd, t 0, d = 2, 3.

n is compactly supported in a bounded smooth domain Ω.

n is a stationary Gaussian process with mean zero and covariance

〈n(x, t)n(y, s)〉 = F (t− s)K(x)δ(x− y).

ProblemFind suppn given

p0(y, t) : (y, t) ∈ ∂Ω× [0, T ]

for sufficiently large T .

- 〈·〉 : Statistical average,

- c : Positive, smooth and boundedfunction,

- F : Time covariance function,

- K : Spatial support of n.

p0(x, t) = 0, and∂

∂tp0(x, t) = 0, x ∈ Rd, t 0, d = 2, 3.

〈n(x, t)n(y, s)〉 = F (t− s)K(x)δ(x− y).

p0(y, t) : (y, t) ∈ ∂Ω× [0, T ]

p0(x, t) = 0, and∂

∂tp0(x, t) = 0, x ∈ Rd, t 0, d = 2, 3.

〈n(x, t)n(y, s)〉 = F (t− s)K(x)δ(x− y).

p0(y, t) : (y, t) ∈ ∂Ω× [0, T ]

Noise Source LocalizationCross-correlation based functional

Imaging functional

I(zS) :=

∫∫∂Ω×∂Ω

G0(x, zS , ω)G0(y, zS , ω)C0(x, y, ω)dσ(x)dσ(y)dω.

c2(x)+ ∆

)G0(x, y, ω) = −δ(x− y), x, y ∈ Rd.

The statistical cross-correlation C0 is defined by

C0(x, y, τ) = 〈p0(x, t)p0(y, t+τ)〉 =1

[ ∫ΩG0(x, z, ω)G0(y, z, ω)K(z)dz

]F (ω)e−iωτ .

TheoremFunctional I gives K up to a smoothing operator, that is

I(zS) '∫

ΩQ(zS , z)K(z)dz, where

Q(zS , z) =

F (ω)

G0(zS , z, ω)

Imaging functional

I(zS) :=

∫∫∂Ω×∂Ω

c2(x)+ ∆

)G0(x, y, ω) = −δ(x− y), x, y ∈ Rd.

C0(x, y, τ) = 〈p0(x, t)p0(y, t+τ)〉 =1

]F (ω)e−iωτ .

I(zS) '∫

Q(zS , z) =

F (ω)

G0(zS , z, ω)

Imaging functional

I(zS) :=

∫∫∂Ω×∂Ω

c2(x)+ ∆

)G0(x, y, ω) = −δ(x− y), x, y ∈ Rd.

C0(x, y, τ) = 〈p0(x, t)p0(y, t+τ)〉 =1

]F (ω)e−iωτ .

I(zS) '∫

Q(zS , z) =

F (ω)

G0(zS , z, ω)

Noise Source LocalizationWeighted imaging functional

Consider the power spectral density F(ω) =

∫∂Ω

C0(x, x, ω)dσ(x).

F(ω) =1

∫ ω+∆ω/2

ω−∆ω/2F(ω′)dω′ ' F (ω)

G0(z, z, ω)

K(z)dz

Moving frequency window ∆ω should be large than 1/T and smaller than noisebandwidth.

Imaging functional

IW (zS) :=

W (ω)

∫∫∂Ω×∂Ω

Theorem

IW (zS) '∫

ΩQW (zS , z)

K0dz, with K0 =

∫ΩK(z)dz and

QW (zS , z) =W (ω)

G0(zS , z, ω)

2dω =

116W (ω)

ω2 J20 (ω|z|)dω, d = 2

116π2

W (ω)

ω2 sinc2(ω|z|)dω, d = 3.

∫∂Ω

C0(x, x, ω)dσ(x).

F(ω) =1

∫ ω+∆ω/2

ω−∆ω/2F(ω′)dω′ ' F (ω)

G0(z, z, ω)

K(z)dz

Imaging functional

IW (zS) :=

W (ω)

∫∫∂Ω×∂Ω

Theorem

IW (zS) '∫

ΩQW (zS , z)

K0dz, with K0 =

∫ΩK(z)dz and

QW (zS , z) =W (ω)

G0(zS , z, ω)

2dω =

116W (ω)

ω2 J20 (ω|z|)dω, d = 2

116π2

W (ω)

∫∂Ω

C0(x, x, ω)dσ(x).

F(ω) =1

∫ ω+∆ω/2

ω−∆ω/2F(ω′)dω′ ' F (ω)

G0(z, z, ω)

K(z)dz

Imaging functional

IW (zS) :=

W (ω)

∫∫∂Ω×∂Ω

Theorem

IW (zS) '∫

ΩQW (zS , z)

K0dz, with K0 =

∫ΩK(z)dz and

QW (zS , z) =W (ω)

G0(zS , z, ω)

2dω =

116W (ω)

ω2 J20 (ω|z|)dω, d = 2

116π2

W (ω)

∫∂Ω

C0(x, x, ω)dσ(x).

F(ω) =1

∫ ω+∆ω/2

ω−∆ω/2F(ω′)dω′ ' F (ω)

G0(z, z, ω)

K(z)dz

Imaging functional

IW (zS) :=

W (ω)

∫∫∂Ω×∂Ω

Theorem

IW (zS) '∫

ΩQW (zS , z)

K0dz, with K0 =

∫ΩK(z)dz and

QW (zS , z) =W (ω)

G0(zS , z, ω)

2dω =

116W (ω)

ω2 J20 (ω|z|)dω, d = 2

116π2

W (ω)

∫∂Ω

C0(x, x, ω)dσ(x).

F(ω) =1

∫ ω+∆ω/2

ω−∆ω/2F(ω′)dω′ ' F (ω)

G0(z, z, ω)

K(z)dz

Imaging functional

IW (zS) :=

W (ω)

∫∫∂Ω×∂Ω

Theorem

IW (zS) '∫

ΩQW (zS , z)

K0dz, with K0 =

∫ΩK(z)dz and

QW (zS , z) =W (ω)

G0(zS , z, ω)

2dω =

116W (ω)

ω2 J20 (ω|z|)dω, d = 2

116π2

W (ω)

Noise Source LocalizationWeighted imaging functional : Remarks

A potential candidate for W should be

W (ω) =

|ω|31|ω|<ωmax

, d = 2

ω21|ω|<ωmax, d = 3.

based on the closure formulae [Abramowitz 65]∫R+

ωJ20 (ω|z|)dω =

|z|δ(z),

and ∫R+

ω2sinc2(ω|z|)dω =1

|z|2δ(z),

where 1 denotes the characteristic function.

IW can seen as an application of I on filtered data p0(x, t) where

p0(x, ω) :=

√W (ω)

F(ω)p0(x, ω).

where (−ωmax, ωmax) is the estimated support of F(ω)

Noise Source LocalizationWeighted imaging functional : Remarks

A potential candidate for W should be

W (ω) =

|ω|31|ω|<ωmax

, d = 2

ω21|ω|<ωmax, d = 3.

based on the closure formulae [Abramowitz 65]∫R+

ωJ20 (ω|z|)dω =

|z|δ(z),

and ∫R+

ω2sinc2(ω|z|)dω =1

|z|2δ(z),

where 1 denotes the characteristic function.

IW can seen as an application of I on filtered data p0(x, t) where

p0(x, ω) :=

√W (ω)

F(ω)p0(x, ω).

where (−ωmax, ωmax) is the estimated support of F(ω)

Noise Source LocalizationAnalogy with time-reversal

I(zS) =

∫∂Ω

G0(x, zS , ω)G0(y, zS , ω)C0(x, y, ω)dσ(x)dσ(y)dω

∫∂Ω

G0(x, zS , ω)G0(y, zS , ω)p0(x, ω)p0(y, ω)dσ(x)dσ(y)dω

∣∣∣ ∫∂Ω

G0(x, zS , ω)p0(x, ω)dσ(x)∣∣∣2dω

0v(zS , t)2dt,

v is the adjoint wave expressed in the form

v(x, t) =

0vs(x, t)ds, with

∂t2vs(x, t)−∆vs(x, t) = δ(t− s)p0(x, T − s)δ∂Ω(x), ∀(x, t) ∈ Rd × (0, T ),

vs(x, t) = 0,∂

∂tvs(x, t) = 0 ∀x ∈ Rd, t < s.

Noise Source LocalizationAnalogy with time-reversal

I(zS) =

∫∂Ω

G0(x, zS , ω)G0(y, zS , ω)C0(x, y, ω)dσ(x)dσ(y)dω

∫∂Ω

G0(x, zS , ω)G0(y, zS , ω)p0(x, ω)p0(y, ω)dσ(x)dσ(y)dω

∣∣∣ ∫∂Ω

G0(x, zS , ω)p0(x, ω)dσ(x)∣∣∣2dω

0v(zS , t)2dt,

v is the adjoint wave expressed in the form

v(x, t) =

0vs(x, t)ds, with

∂t2vs(x, t)−∆vs(x, t) = δ(t− s)p0(x, T − s)δ∂Ω(x), ∀(x, t) ∈ Rd × (0, T ),

vs(x, t) = 0,∂

∂tvs(x, t) = 0 ∀x ∈ Rd, t < s.

Noise Source LocalizationReconstructions

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Top : point sources. Bottom : extended sources.Left to Right : K(x) ; I ; IW withW (ω) = |ω|31|ω|<ωmax .

T = 8, ωmax = 1000, Nx = 28, and Nt = 211.

Noise Source LocalizationEstimation of power spectral density

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F(ω) estimated F(ω) estimated and denoised F(ω) exact

F (ω) = exp

(−π ω2

ω2max

)with ωmax = 1000.

Noise Source LocalizationEstimation of power spectral density

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F(ω) estimated F(ω) estimated and denoised F(ω) exact

F (ω) = 1|ω|≤100 exp

(−π ω2

ω2max

Noise Source LocalizationSpatially correlated sources

Let n be a stationary Gaussian process with mean zero and covariance function

〈n(x, t)n(y, s)〉 = F (t− s)Γ(x, y)

where Γ characterizes spatial support and covariance of the sources.

C0(x, y, τ) =1

[ ∫∫Ω×Ω

G0(x, z, ω)G0(y, z′, ω)Γ(z, z′)dz dz′]F (ω)e−iωτdω.

J(zS , zS′) :=

∫∫∂Ω×∂Ω

G0(x, zS , ω)G0(y, zS′, ω)C0(x, y, ω)dσ(x)dσ(y)dω.

Proposition

J(zS , zS′) :=

∫∫Ω×Ω

F (ω)

G(z, zS , ω)

G(z′, zS

′, ω)dω︸︷︷︸

Ψ(zS ,zS′,z,z′)

Γ(z, z′) dz dz′.

In 3D homogeneous media, Ψ(zS , zS′, z, z′) = ψ(zS − z, zS′ − z′) with

ψ(z, z′) =1

∫RF (ω)sinc(ω|z|)sinc(ω|z′|)dω.

〈n(x, t)n(y, s)〉 = F (t− s)Γ(x, y)

C0(x, y, τ) =1

[ ∫∫Ω×Ω

J(zS , zS′) :=

∫∫∂Ω×∂Ω

Proposition

J(zS , zS′) :=

∫∫Ω×Ω

F (ω)

G(z, zS , ω)

G(z′, zS

′, ω)dω︸︷︷︸

ψ(z, z′) =1

〈n(x, t)n(y, s)〉 = F (t− s)Γ(x, y)

C0(x, y, τ) =1

[ ∫∫Ω×Ω

J(zS , zS′) :=

∫∫∂Ω×∂Ω

Proposition

J(zS , zS′) :=

∫∫Ω×Ω

F (ω)

G(z, zS , ω)

G(z′, zS

′, ω)dω︸︷︷︸

ψ(z, z′) =1

〈n(x, t)n(y, s)〉 = F (t− s)Γ(x, y)

C0(x, y, τ) =1

[ ∫∫Ω×Ω

J(zS , zS′) :=

∫∫∂Ω×∂Ω

Proposition

J(zS , zS′) :=

∫∫Ω×Ω

F (ω)

G(z, zS , ω)

G(z′, zS

′, ω)dω︸︷︷︸

ψ(z, z′) =1

Noise Source LocalizationSpatially correlated sources II

Correlated point sources : Γ(z, z′) =

Ns∑i,j=1

ρijδ(z − zi)δ(z − zj)

- Find zi from I(zS) 'Ns∑i,j=1

F (ω)

G(z, zS , ω)

- Estimate ρij from

J(zi, zj) = ρij

∫R=m

G(zi, zi, ω)

G(zj , zj , ω)

dω ' ρij

∫RF (ω)dω.

√2 1/

√2 0

2 1 0 0

2 0 1 00 0 0 1

1.000 0.733 0.701 0.0610.733 1.000 0.049 0.0610.701 0.049 1.000 0.0300.061 0.061 0.030 1.000

Noise Source LocalizationSpatially correlated sources II

Correlated point sources : Γ(z, z′) =

Ns∑i,j=1

ρijδ(z − zi)δ(z − zj)

- Find zi from I(zS) 'Ns∑i,j=1

F (ω)

G(z, zS , ω)

- Estimate ρij from

J(zi, zj) = ρij

∫R=m

G(zi, zi, ω)

G(zj , zj , ω)

dω ' ρij

∫RF (ω)dω.

√2 1/

√2 0

2 1 0 0

2 0 1 00 0 0 1

1.000 0.733 0.701 0.0610.733 1.000 0.049 0.0610.701 0.049 1.000 0.0300.061 0.061 0.030 1.000

Noise Source LocalizationCorrelated point sources : Reconstruction

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Top : K(z) (left), IW with W (ω) = |ω|31|ω|<ωmax (middle), and z → JW (z1, z) (right).Bottom : z → JW (z2, z) (left),z → JW (z3, z) (middle), and z → JW (z4, z) (right).

Noise Source LocalizationExtended correlated sources : Reconstruction

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Top : K(z) (left), IW with W (ω) = |ω|31|ω|<ωmax (middle), and z → JW (z1, z) (right).Bottom : z → JW (z2, z) (left),z → JW (z3, z) (middle), and z → JW (z4, z) (right).

Time Reversal Cavity

Time Reversal CavitySpherical Dielectric Inclusion

D = z + δB with permittivity ε and permeability µ0, radiated by an electric dipole at y with

direction e.

Let Gee (resp. Gme) be the electric-electric (resp. magnetic-electric) Green’s functionfor the Maxwell’s equations in R3 i.e. for all (x, t) ∈ R3 × R

∇×Gee(x, t) = −µ0∂Gme

∂t(x, t),

∇×Gme(x, t) = ε0∂Gee

∂t(x, t) + Iδ0 (x) δ0 (t) ,

Let Ey(x, t) := −Gee(x− y, t) · e,

direction e.

∂t(x, t),

∂t(x, t) + Iδ0 (x) δ0 (t) ,

direction e.

∂t(x, t),

∂t(x, t) + Iδ0 (x) δ0 (t) ,

Time Reversal CavityElectromagnetic Scattering

Consider the following scattering problem

∇×(

ε0∇×E

)+ µ0

∂t2= −µ0δye

∂δ0

∂t, (R3 \D)× R,

∇×(

ε∇×E

)+ µ0

∂t2= 0, D × R,

ε0(∇×E)+ × ν =

ε(∇×E)− × ν, ∂D × R,

µ0E+ · ν = µ0E− · ν, ∂D × R,

E(x, t) = 0 =∂E

∂t(x, t), t 0,

Lemma (Asymptotic expansion of the scattered field

Let x, y be sufficiently far from z, and ωc = O(δ−α), then

Sωc [E−Ey ](x, t) = δ3

∂tSωc [Gee](x− z, t− τ) ·M(ε, B)Sωc [Ey ](z, τ − t0) dτ +O(δ4−3α)

M =3(ε−ε0)(ε+2ε0)

|B|I is the polarization tensor associated with D [Ammari 07].

Consider the following scattering problem

∇×(

ε0∇×E

)+ µ0

∂t2= −µ0δye

∂δ0

∂t, (R3 \D)× R,

∇×(

ε∇×E

)+ µ0

∂t2= 0, D × R,

ε0(∇×E)+ × ν =

ε(∇×E)− × ν, ∂D × R,

µ0E+ · ν = µ0E− · ν, ∂D × R,

E(x, t) = 0 =∂E

∂t(x, t), t 0,

Lemma (Asymptotic expansion of the scattered field

Let x, y be sufficiently far from z, and ωc = O(δ−α), then

Sωc [E−Ey ](x, t) = δ3

∂tSωc [Gee](x− z, t− τ) ·M(ε, B)Sωc [Ey ](z, τ − t0) dτ +O(δ4−3α)

M =3(ε−ε0)(ε+2ε0)

|B|I is the polarization tensor associated with D [Ammari 07].

ProblemSuppose we are given the tangential component of E on a sphere S for t ∈ [0, t0]. Let Etr bethe field obtained by re-emitting time reversed data. Find Etr and the location z of D.

Theorem

Etr(x, t) ' −δ3

∫RqT (z, τ) ·

[Sωc [Gee(x− z, t0 − τ − t)−Gee(x− z, t− t0 + τ)]

Etr(x, ω) ∝ δ3qT (z, τ) ·(ω2

c2I +∇∇

(ωc|x− z|

q(z, τ) = ε0M(ε, B) ·∂

∂tSωc [Ey ](z, τ).

ProblemSuppose we are given the tangential component of E on a sphere S for t ∈ [0, t0]. Let Etr bethe field obtained by re-emitting time reversed data. Find Etr and the location z of D.

Theorem

Etr(x, t) ' −δ3

∫RqT (z, τ) ·

[Sωc [Gee(x− z, t0 − τ − t)−Gee(x− z, t− t0 + τ)]

Etr(x, ω) ∝ δ3qT (z, τ) ·(ω2

c2I +∇∇

(ωc|x− z|

q(z, τ) = ε0M(ε, B) ·∂

∂tSωc [Ey ](z, τ).

Conclusion, Perspectives and Open Questions

Conclusion, Perspectives andOpen Questions

- Presented time reversal techniques for wave imaging.

- Proposed and analysed, adjoint, pre-processing and weighted TR techniques tocompensate for the attenuation effects.

- Derived Helmholtz-Kirchhoff identities for elastic, visco-elastic and attenuatingacoustic media.

- Variable attenuation correction.

- Time reversal in composite and inhomogeneous media.

- Limited view problem and virtual sources in scattering media.

- Time reversal with a few transducers : lower bound on the number oftransducers for stable reconstructions.

- Coherent interferometric techniques for clutter media and imaging in thepresence of measurements noise (elasticity case).

- Topological derivative based imaging techniques and their analogy with TR.

- Universe of applications.

- Presented time reversal techniques for wave imaging.

- Proposed and analysed, adjoint, pre-processing and weighted TR techniques tocompensate for the attenuation effects.

- Derived Helmholtz-Kirchhoff identities for elastic, visco-elastic and attenuatingacoustic media.

- Variable attenuation correction.

- Time reversal in composite and inhomogeneous media.

- Limited view problem and virtual sources in scattering media.

- Time reversal with a few transducers : lower bound on the number oftransducers for stable reconstructions.

- Coherent interferometric techniques for clutter media and imaging in thepresence of measurements noise (elasticity case).

- Topological derivative based imaging techniques and their analogy with TR.

- Universe of applications.

Related Publications 1

H. Ammari, E. Bretin, J. Garnier, A. Wahab, Noise source localization inan attenuating medium, SIAM Journal on Applied Mathematics,72(1) :(2012), pp. 317-336.

H. Ammari, E. Bretin, J. Garnier, A. Wahab, Time reversal algorithms inviscoelastic media, Submitted to European Journal of Applied Mathematics,(33 pages).

H. Ammari, E. Bretin, J. Garnier, A. Wahab, Time reversal in attenuatingacoustic media, Contemporary Mathematics, vol. 548, pp. 151-163 AmericanMathematical Society 2011.

S. Gdoura, D. Lesselier, A. Wahab, Time reversal and scattering by asmall 3-D dielectric inclusion in lossy media, in preparation

1. Preprints are available at www.cmap.polytechnique.fr/∼wahab

Thank You !

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