Transient Elasticity Imaging and Time Reversal - CiteSeerX

Transient Elasticity Imaging and Time Reversal∗ Habib Ammari†

Lili Guadarrama Bustos‡

Hyeonbae Kang§

Hyundae Lee‡

Abstract In this paper we consider a purely quasi-incompressible elasticity model. We rigorously establish asymptotic expansions of near- and far-field measurements of the transient elastic wave induced by a small elastic anomaly. Our proof uses layer potential techniques for the modified Stokes system. Based on these formulas, we design asymptotic imaging methods leading to a quantitative estimation of elastic and geometrical parameters of the anomaly. Mathematics subject classification (MSC2000): 35R30, 74L15, 92C55 Keywords: transient imaging, asymptotic imaging, time-reversal, quasi-incompressible elasticity

1

Introduction

An interesting approach to assessing elasticity is to use the acoustic radiation force of an ultrasonic focused beam to remotely generate mechanical vibrations in organs [11, 14]. The acoustic force is due to the momentum transfer from the acoustic wave to the medium. The radiation force acts as a dipolar source. A spatio-temporal sequence of the propagation of the induced transient wave can be acquired, leading to a quantitative estimation of the viscoelastic parameters of the studied medium in a source-free region [6, 7]. The Voigt model has been chosen to describe the viscoelastic properties of tissues. Catheline et al. [8] have shown that this model is well adapted to describe the viscoelastic response of tissues to low-frequency excitations. In this paper, we neglect the viscosity effect and only consider a purely quasi-incompressible elasticity model. We derive asymptotic expansions of the perturbations of the elastic wavefield that are due to the presence of a small anomaly in both the near- and far-field regions as the size of the anomaly goes to zero. Then we design an asymptotic imaging method leading to a quantitative estimation of the shear modulus and shape of the anomaly from near-field measurements. Using time-reversal, we show how to reconstruct the location and geometric features of the anomaly from the far-field measurements. We put a particular emphasis on ∗ This work was partially supported by the ANR project EchoScan (AN-06-Blan-0089), the grant NRF 20090085987, the STAR project 190117RD, and KICOS project K20803000001-08B1200-00110. † Department of Mathematics and Applications, Ecole Normale Sup´ erieure, 45 Rue d’Ulm, 75005 Paris, France ([email protected]). ‡ Institut Langevin, CNRS UMR 7587 & ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France ([email protected]). § Department of Mathematics, Inha University, Incheon 402-751, Korea ([email protected], [email protected]).

1

the difference between the acoustic and the elastic cases, namely, the anisotropy of the focal spot and the birth of a near-field like effect by time reversing the perturbation due to an elastic anomaly. The results of this paper extend those in [3] to transient wave propagation in elastic media. The paper is organized as follows. In Section 2 we rigorously derive asymptotic formulas for quasi-incompressible elasticity and estimate the dependence of the remainders in these formulas with respect to the frequency. Based on these estimates, we obtain in Section 3 formulas for the transient wave equation that are valid after truncating the high-frequency components of the elastic fields. These formulas describe the effect of the presence of a small elastic anomaly in both the near- and far-field. We then investigate in Section 4 the use of time-reversal for locating the anomaly and detecting its overall geometric and material parameters via the viscous moment tensor. An optimization problem is also formulated for reconstructing geometric parameters of the anomaly and its shear modulus from near-field measurements.

2

Asymptotic expansions

We suppose that an elastic medium occupies the whole space R3 . Let the constants λ and µ denote the Lam´e coefficients of the medium, that are the elastic parameters in absence of any anomaly. With these constants, Lλ,µ denotes the linear elasticity system, namely Lλ,µ u := µ∆u + (λ + µ)∇∇ · u.

(1)

The traction on a smooth boundary ∂Ω is given by the conormal derivative ∂u/∂ν associated with Lλ,µ , ∂u b := λ(∇ · u)N + µ∇uN, (2) ∂ν b denotes the symmetric gradient, where N denotes the outward unit normal to ∂Ω. Here ∇ i.e., b := ∇u + ∇uT , ∇u (3) where the superscript T denotes the transpose. The time-dependent linear elasticity system is given by ∂t2 u − Lλ,µ u = 0.

(4)

The fundamental solution or the Green function for the system (4) is given by G = (Gij ) where Gij =

1 γi γj − δij 1 3γi γj − δij √µ 1 γi γj r − H√λ+2µ (x, t) + δt= √λ+2µ δt= √rµ . (5) 3 4π r 4π(λ + 2µ) r 4πµ r

Here r √= |x|, γi = xi /r, δij denotes the Kronecker symbol, δ denotes the Dirac delta function, µ and H√λ+2µ (x, t) is defined by √ µ H√λ+2µ (x, t)

  t

r r 2 , the operator − 2(˜ µ−µ) I + (KB ) is invertible as well (see [4]). Thus one can see that T0 is also invertible. In fact, one can readily check that the solution is explicit.

˜ = T −1 (f , g) is given by Lemma 2.1 For (f , g) ∈ H 1 (∂B)3 × L2 (∂B)3 the solution (ϕ, ˜ ψ) 0 0 −1 ϕ˜ = ψ˜ + (SB ) [f ],   −1  1 (˜ µ + µ) 1 0 −1 0 ∗ 0 ∗ ˜ −˜ µ(− I + (KB ) )(SB ) [f ] + g . − I + (KB ) ψ= µ ˜−µ 2(˜ µ − µ) 2

(35) (36)

In view of (33) and (34), one can see that there is 0 > 0 such that T is invertible as long as ω ≤ 0 . Moreover T −1 takes the form T −1 = T0−1 + E,

(37)

kE(f , g)kL2 (∂B)×L2 (∂B) ≤ Cω(kf kH 1 (∂B) + kgkL2 (∂B) ),

(38)

where the operator E satisfies

for some constant C independent of  and ω. Suppose that ω ≤ 0 < 1. Let (ϕ eω , ψeω ) be the solution to (29). Then by (37) we have In view of (30) we have

(ϕ eω , ψeω ) = T0−1 (A, B) + E(A, B). kAkH 1 (∂B) ≤ C(ω 2 + 1).

(39)

On the other hand, according to (31), B can be written as

where B1 satisfies

ˆ 0 (z, ω)N(˜ bU B(˜ x) = (˜ µ − µ)∇ x) + B1 (˜ x), kB1 kL2 (∂B) ≤ C(ω 2 + 1).

(40)

Therefore, we have   ˆ 0 (z, ω)N + T −1 (A, B1 ) + E(A, B). bU (ϕ eω , ψeω ) = (˜ µ − µ)T0−1 0, ∇ 0 8

(41)

Because of (38), (39), and (40), the last two terms in the above equation are error terms satisfying kT0−1 (A, B1 ) + E(A, B)kL2 (∂B)×L2 (∂B) ≤ C(ω 2 + 1). We also need to derive asymptotic expansions for sides of (29) with respect to ω, we obtain ω √ µ ˜

SB

h ∂ϕ eω i ∂ω

ω √ µ

(˜ x) − SB

∂ϕ eω ∂ω

and

eω ∂ψ ∂ω .

By differentiating both

Z ∂A(˜ x) ∂ √ωµ˜ x − y˜)ϕ˜ω (˜ y )dσ(˜ y) (˜ x) = − Γ (˜ ∂ω ∂ω ∂ω ∂B Z ∂ √ωµ Γ (˜ + x − y˜)ψ˜ω (˜ y )dσ(˜ y) ∂ω ∂B

h ∂ ψeω i

(42)

and eω i ∂ √ωµ˜ h ∂ ϕ ∂ √ωµ h ∂ ψeω i ∂B(˜ x) µ ˜ SB (˜ x) − µ SB (˜ x) = ∂n ∂ω − ∂n ∂ω + ∂ω Z Z ω ∂ ∂ ∂ √ωµ˜ ∂ √ Γ (˜ Γ µ (˜ − x − y˜)ϕ˜ω (˜ y )dσ(˜ y) + x − y˜)ψ˜ω (˜ y )dσ(˜ y) ∂n ∂B ∂ω ∂n ∂B ∂ω

(43)

on ∂B. Straightforward computations using (10) and (30) show that the right-hand side of the equality in (42) is of order (ω + 1) in the H 1 (∂B)-norm. We can also show using (31) that ∂G1 2 ∂ω is also of order (ω + 1) in the L (∂B)-norm. Thus, using the same argument as before, we readily obtain ! ˆ0 ∂ϕ eω ∂ ψeω ∂ U −1 b ( , ) = (˜ µ − µ)T0 0, ∇( )(z, ω)N + O((ω + 1)), (44) ∂ω ∂ω ∂ω where the equality holds in L2 (∂B)3 × L2 (∂B)3 . In view of (41) and (44), applying Lemma 2.1 (with f = 0) yields the following result.

Proposition 2.2 Let (ϕ eω , ψeω ) be the solution to (29). There exists 0 > 0 such that if ω < 0 , then the following asymptotic expansions hold: ϕ eω =

and

ψeω =





−(˜ µ + µ) 0 ∗ ) I + (KB 2(˜ µ − µ)

−(˜ µ + µ) 0 ∗ ) I + (KB 2(˜ µ − µ)

−1 −1

ˆ 0 (z, ω)N] + O((ω 2 + 1)), bU [∇ ˆ 0 (z, ω)N] + O((ω 2 + 1)), bU [∇

 −1 ∂ϕ eω −(˜ µ + µ) 0 ∗ ˆ 0 (z, ω)N] + O((ω + 1)), b ∂ U = I + (KB ) [∇ ∂ω 2(˜ µ − µ) ∂ω −1  ∂ ψeω −(˜ µ + µ) 0 ∗ ˆ 0 (z, ω)N] + O((ω + 1)), b ∂ U = I + (KB ) [∇ ∂ω 2(˜ µ − µ) ∂ω

where all the equalities hold in L2 (∂B).

9

(45) (46)

(47) (48)

We are now ready to derive the inner expansion for w. Let Ω be a domain containing D e = 1 Ω − z. After a change of variables, (25) and (26) take the forms: and let Ω  Z  ω 1 √ 2 1  ˆ 0 (˜ Γ µ˜ (˜ x − y˜)(ω 2 U y + z) − ∇ˆ q0 (˜ y + z)) d˜ y   ( − )   µ µ ˜ ω B √ w(˜ x + z, ω) = (49) +SB µ˜ [ϕ˜ω ](˜ x) in B,    ω √  µ x) in R3 \ B, SB [ψ˜ω ](˜

and

Z  1 1  ˆ 0 (˜  ( F(˜ x − y˜) · (ω 2 U y + z) − ∇ˆ q0 (˜ y + z)) d˜ y − )  µ µ ˜ B ω p(˜ x + z, ω) = +QB [ϕ˜ ](˜ x) in B,    ω QB [ψ˜ ](˜ x) in R3 \ B.

Since we have

(50)

√ωµ ω

0

S [ϕ [ϕ eω ] H 1 (∂B) ≤ Cωkϕ e ] − SB eω kL2 (∂B) , B

( 0 ω SB [ϕ e ](˜ x) + O(2 (ω 2 + 1)), x e ∈ B, w(˜ x + z, ω) = 0 eω 2 e \ B. SB [ψ ](˜ x) + O( (ω + 1)), x ˜∈Ω

It then follows from (45) and (46) that

−1  (˜ µ + µ) 0 ∗ 0 ˆ 0 (z, ω)N](˜ bU I + (KB ) [∇ x) + O(2 (ω 2 + 1)) w(˜ x + z, ω) = SB − 2(˜ µ − µ)

(51)

e for x e ∈ Ω. On the other hand, we have  √ω  ω  ∂ϕ e   SB µ˜ (˜ x) + O(2 (ω + 1)), x e ∈ B,   ∂ω ∂w # " (˜ x + z, ω) = ω √  ∂ ψeω ∂ω µ  e \ B.  (˜ x) + O(2 ), x ˜∈Ω SB ∂ω Therefore, from (47) and (48) we obtain that

 −1 (˜ µ + µ) ∂w 0 0 ∗ ˆ 0 (z, ω)N](˜ b ∂ U (˜ x + z, ω) = SB − I + (KB ) x) + O(2 (ω + 1)) (52) [∇ ∂ω 2(˜ µ − µ) ∂ω

e for x e ∈ Ω. Let

 −1 (˜ µ + µ) 0 0 ∗ ˆ 0 (z, ω)N](˜ bU − [∇ x), v(˜ x) := SB ) I + (KB 2(˜ µ − µ) −1  (˜ µ + µ) 0 ∗ ˆ 0 (z, ω)N](˜ bU q(˜ x) := QB − [∇ x). I + (KB ) 2(˜ µ − µ) 10

It is easy to check that (v, q) is the solution to  µ∆v − ∇q = 0 in R3 \ B,      µ ˜∆v − ∇q = 0 in B,      v|− − v|+ = 0 on ∂B,     ∂v ∂v ˆ 0 (z, ω)N bU ) − (qN − µ ) = (˜ (qN − µ ˜ µ − µ)∇ ∂N ∂N  − +      ∇ · v = 0 in R3 ,      v(˜ x) → 0 as |˜ x| → +∞,    q(˜ x) → 0 as |˜ x| → +∞.

on ∂B,

(53)

We finally obtain the following theorem from (51) and (52).

Theorem 2.3 Let Ω be a small region containing D and let   x−z ˆ ˆ 0 (x, ω) − U0 (x, ω) − v R(x, ω) = u , 

x ∈ Ω.

(54)

There exists 0 > 0 such that if ω < 0 , then R(x, ω) = O(2 (ω 2 + 1)),

∇x R(x, ω) = O((ω 2 + 1)),

x ∈ Ω.

(55)

Moreover, ∂R (x, ω) = O(2 (ω + 1)), ∂ω

∇x



 ∂R (x, ω) = O((ω + 1)), ∂ω

x ∈ Ω.

(56)

Note that the estimates for ∇x R in (55) and ∇x ( ∂R ∂ω ) in (56) can be derived using (49). We now derive the outer expansion of u0 . To this end, let us first recall the notion of the viscous moment tensor (VMT) from [4]. Let (vk` , p), for k, ` = 1, 2, 3, be the solution to  µ∆vk` − ∇p = 0 in R3 \ B,      µ ˜∆vk` − ∇p = 0 in B,      0 on ∂B,   vk` |− − vk` |+ =    ∂v ∂vk`  k`  ) − (pN − µ ) = 0 on ∂B, ˜  (pN − µ ∂N − ∂N + (57)   3  ∇ · v = 0 in R , k`     3   δk` X   x ˜j ej = O(|˜ x|−2 ) as |˜ x| → +∞, v (˜ x ) − x ˜ e + k` k `   3   j=1    p(˜ x) = O(|˜ x|−3 ) as |˜ x| → +∞.

Here (e1 , e2 , e3 ) is the standard basis of R3 . The VMT V (˜ µ, µ, B) = (Vijk` )i,j,k,`=1,2,3 is defined by Z b xi ej ) d˜ Vijk` (˜ µ, µ, B) := (˜ µ − µ) ∇vk` (˜ x) : ∇(˜ x, B

11

(58)

P3 where : denotes the contraction of two matrices, i.e., A : B = ij=1 aij bij . ˆ 0 , pˆ0 − qˆ0 ) satisfies Since (ˆ u0 − U  ω2   ˆ 0 ) − 1 ∇(ˆ )(ˆ u0 − U p0 − qˆ0 ) = 0 in R3 \ D,  (∆ +   µ µ         ω2 1 1 1 1  2 ˆ 0 ) − 1 ∇(ˆ  ˆ u − ∇ˆ p0 (∆ + )(ˆ u − U p − q ˆ ) = ω − −  0 0 0 0   µ µ µ µ ˜ µ µ ˜     (ˆ ˆ 0 ) − (ˆ ˆ 0 ) = 0 on ∂D, u0 − U u0 − U + −  1 ∂  ˆ 0 )  p0 − qˆ0 ) + N + (ˆ u0 − U − (ˆ  +  µ ∂N     ˆ 0 ∂ µ ˜ − µ ∂u   = − 1 (ˆ ˆ  p0 − qˆ0 ) − N + (ˆ u0 − U0 ) − + on ∂D,   µ ∂N µ ∂N −     ˆ 0 ) = 0 in R3 , ∇ · (ˆ u0 − U

in D, (59)

together with the radiation condition, the integration of the first equation in (59) against ω √ the Green’s function Γ µ (x, y) over y ∈ R3 \ D and the divergence theorem give us the following representation formula: Z ω ˆ 0 µ ˜ ∂u √ µ ˆ ˆ 0 (x) = U0 (x) + ( − 1) Γ (x, y) u (y)dσ(y) µ ∂N − ∂D Z Z ω ω 1 1 1 1 √ √ Γ µ (x, y)∇ˆ Γ µ (x, y)ˆ p0 (y) dy + ω 2 ( − ) u0 (y) dy. (60) −( − ) µ µ ˜ D µ µ ˜ D It follows from the inner expansion in Theorem 2.3 that, for y ∈ ∂D, ˆ0 ˆ0 ∂U ∂v ∂u (y) = (y) + ∂N ∂N ∂N and, for x ∈ D, µ ˜ ˆ 0 = (4v) ∇ˆ p0 (x) = µ ˜4ˆ u0 + ω u  2

Since µ ˜



x−z 





y−z 



+ O()

1 + O(1) = (∇q) 



(61)

x−z 



+ O(1).

(62)

Z Z ˆ 0 ∂u 2 ˆ 0 (y) dy, (y) dσ(y) − u ∇ˆ p (y) dy = −ω 0 ∂N − D D

Z

∂D

we obtain that for x far away from z, the following outer expansion holds: ˆ 0 (x) −  ˆ 0 (x) ≈ U u

3

(

1 1 − ) µ µ ˜

Z

B

3 X

i,j,`=1

ω √ µ

∂i Γ`j

 Z µ ˜ (x, z) ( − 1) µ ∂B

 ∂j q(ξ)ξi dξ e` ,

ω √ µ

! ˆ0 ∂U ∂v (z) + (ξ) ξi dσ(ξ) ∂N ∂N −

where ∂i Γ`j (x, z) is the differentiation with respect to the x variable and

12

j




∂v ∂N j

is the j-th

component of

∂v ∂N ,

which we may further simplify as follows

ˆ 0 )(x) (ˆ u0 − U

  Z 3 ω X √ µ ˜ µ ˆ ˆ ≈ − ( − 1) ∂i Γ`j (x, z) ∂j vi (ξ) + ∂i vj (ξ) + ∂j U0i (z) + ∂i U0j (z) dξ e` . µ B 3

i,j,`=1

(63)

Here vj denotes the j-th component of v. Since 3 X ˆ 0 (z)p vpq (ξ) − ∇U ˆ 0 (z)ξ, ∂q U v(ξ) =

(64)

p,q=1

we have ˆ 0 )(x) (ˆ u0 − U ≈ −3 (

µ ˜ − 1) µ

3 X

i,j,`,p,q=1



ω √ µ

ˆ 0 (z)p ∂i Γ`j (x, z)∂q U

Z

B

 ∂j (vkl )i (ξ) + ∂i (vkl )j (ξ) dξ e` .

(65)

We have the following theorem for the outer expansion. Theorem 2.4 Let Ω0 be a compact region away from D, namely dist(Ω0 , D) ≥ C > 0 for some constant C, and let 3 ˆ 0 (x, ω) +  ˆ 0 (x, ω) − U R(x, ω) = u µ

3 X

ω √

µ ˆ 0 (z)p e` . Vijkl ∂i Γ`j (x, z)∂q U

(66)

i,j,p,q,`=1

There exists 0 > 0 such that if ω < 0 , then R(x, ω) = O(4 (ω 3 + 1)), Moreover,

3

∂R (x, ω) = O(4 (ω 2 + 1)), ∂ω

x ∈ Ω0 . x ∈ Ω0 .

(67)

(68)

Far- and near-field asymptotic formulas in the transient regime

ˆ 0 satisfies Recall that the inverse Fourier transform, U0 , of U  2 3   (∂t − µ∆)U0 (x, t) − ∇F = δx=¯y δt=0 a in R × R, 3 ∇ · U0 = 0 in R × R,   U0 (x, t) = 0 for x ∈ R3 and t  0. For ρ > 0, we define the operator Pρ on tempered distributions by Z √ ˆ e− −1ωt ψ(ω) Pρ [ψ](t) = dω, |ω|≤ρ

13

(69)

where ψˆ denotes the Fourier transform of ψ. The operator Pρ truncates the high-frequency component of ψ. One can easily show that Pρ [U0 ] satisfies (∂t2 − ∆)Pρ [U0 ](x, t) − ∇Pρ [F ](x − y) = δx=¯y ψρ (t)a ∇ · Pρ [U0 ] = 0 in R3 × R, where ψρ (t) :=

2 sin ρt = t

Z

e−

√

−1ωt

in R3 × R,

(70)

dω.

|ω|≤ρ

The purpose of this section is to derive and asymptotic expansions for Pρ [u0 − U0 ](x, t). For doing so, we observe that Z √ ˆ 0 (x, ω)dω, Pρ [u0 ](x, t) = (71) e− −1ωt u |ω|≤ρ

ˆ 0 is the solution to (19). Therefore, according to Theorem 2.3, we have where u Z 3 √ X e− −1ωt R(x, ω)dω. ∂q Pρ [U0 ](z, t)p [vpq (x) − xp eq ] = Pρ [u0 − U0 ](x, t) −  |ω|≤ρ

p,q=1

Suppose that |t| ≥ c0 for some positive number c0 (c0 is of order the distance between y¯ and z). Then, integrating by parts gives Z Z 1 √ d −√−1ωt − −1ωt e e R(x, ω)dω = R(x, ω)dω |ω|≤ρ t |ω|≤ρ dω Z ∂ 1 (|R(x, ρ)| + |R(x, −ρ)|) + R(x, ω) dω ≤ |t| |ω|≤ρ ∂ω ≤ C2 ρ2 .

Since 

3 X

p,q=1

∂q Pρ [U0 ](z, t)p [vpq (x) − xp eq ] = O(ρ),

we arrive at the following theorem. Theorem 3.1 Suppose that ρ = O(−α ) for some α < 1. Then Pρ [u0 − U0 ](x, t) = 

3 X

p,q=1

∂q Pρ [U0 ](z, t)p [vpq (x) − xp eq ] + O(2(1−α) ).

We now derive a far-field asymptotic expansion for Pρ [u0 − U0 ]. Let G∞ (x, y, t) be the ω √ inverse Fourier transform of Γ µ (x, y). Note that G∞ is the limit of G given by (5) as √ λ + 2µ → +∞. It then follows that Z √ ω √ e− −1ωt Γ µ (x, y)dω Pρ [G∞ ](x, y, t) = |ω|≤ρ

  1 3γi γj − δij 1 γi γj − δij r r = ) − ψρ (t − √ ), φ (t) − φ (t − √ ρ ρ 4π r3 µ 4πµ r µ 14

(72)

Rt where φρ (t) := 0 ψρ (s)ds. From Theorem 2.4, we get Z √ ˆ 0 (x, ω)) dω u0 (x, ω) − U e− −1ωt (ˆ |ω|≤ρ

3 =− µ +

Z

Z

e−

√

−1ωt

|ω|≤ρ

 

3 X

i,j,p,q,`=1

e−

√

−1ωt



ω √ µ

R(x, ω) dω,

ˆ 0 (z)p e`  dω Vijpq ∂i Γ`j (x, z)∂q U

|ω|≤ρ

where the remainder is estimated by Z √ 3 e− −1ωt R(x, ω) dω = O(4(1− 4 α) ). |ω|≤ρ

Since Z

e−

√

−1ωt

|ω|≤ρ

= µ−1



3 X



i,j,p,q,`=1

Z

e−

√

−1ωt

|ω|≤ρ

= µ−1

Z

R

 

3 X

i,j,p,q,k,`=1

 



ω √ µ

ˆ 0 (z)p e`  dω Vijpq ∂i Γ`j (x, z)∂q U 3 X

i,j,p,q,k,`=1

ω √ µ

ω √ µ



Vijpq ∂i Γ`j (x, z)∂q Γpk (z, y¯)ak e`  dω



Vijpq ∂i Pρ [G∞ ]`j (x, z, t − τ )∂q Pρ [G∞ ]pk (z, y¯, τ )ak e`  dτ,

the following theorem holds. ω ˆ 0 (x, ω) := 1 Γ √µ (x − y¯)a. Suppose that ρ = O(−α ) for some α < 1. Theorem 3.2 Let U µ Then for |x − z| ≥ C > 0, the following far-field expansion holds Pρ [u0 − U0 ](x, t)  Z 3 X 3  =− 2 µ R

i,j,p,q,k,`=1 3

+O(4(1− 4 α) ).



Vijpq ∂i Pρ [G∞ ]`j (x, z, t − τ )∂q Pρ [G∞ ]pk (z, y¯, τ )ak e`  dτ

(73)

Note that if we plug (72) in the far-field formula (73) then we can see that, unlike the acoustic case investigated in [3], the perturbation Pρ [u0 − U0 ](x, t) can be seen not only as a polarized wave emitted from the anomaly but it contains, because of the term (1/r3 )φρ (t) in (72), a near-field like term which does not propagate.

4 4.1

Asymptotic imaging Far-field imaging: time-reversal

We present a time-reversal technique for detecting the location z of the anomaly from measurements of the perturbations at x away from the location z. As in the acoustic case, 15

the main idea is to take advantage of the reversibility of the elastic wave equation in a non-viscous medium in order to back-propagate signals to the sources that emitted them [5, 10]. Let S be a sphere englobing the anomaly D. Consider, for simplicity, the harmonic regime, we get Z  S

ω ω  √ √ √ ω ω ω ∂Γ µ ∂Γ µ √ √ √ µ µ (x, z)Γ (x, y) − Γ (x, z) (x, y) dσ(x) = 2 −1=mΓ µ (y, z), ∂n ∂n

ˆ 0 (x, ω), it follows that ˆ 0 (x, ω) − U for y ∈ Ω, and therefore, for w(x) := u Z  S

ω  √ ω ∂w ∂Γ µ √ µ (x, ω)Γ (x, z) − w(x, ω) (x, z) dσ(x) ∂n ∂n 3 √  ω √ ˆ 0 (z, ω)V (˜ = 2 −1 ∇U µ, µ, B)∇z =mΓ µ (y, z) + O(4 ω 3 ), µ

if ω > 1. This shows that the anti-derivative of time-reversal perturbation focuses on the location of the anomaly with an anisotropic focal spot. Because of the structure of the Green ω √ function Γ µ (y, z), time-reversing the perturbation gives birth to a near-field like effect. Moreover, the diffraction limit depends on the direction. It is, unlike the acoustic case, anisotropic. These interesting findings were experimentally observed and first reported in [9]. Our asymptotic formula (73) clearly explains them.

4.2

Near-field imaging: optimization approach

Set Ω to be a window containing the anomaly D. Theorem 3.1 suggests to reconstruct the shape and the shear modulus of the elastic inclusion D by minimizing the following functional: Z

T +∆T

T −∆T

||Pρ [u0 − U0 ](x, t) − 

3 X

p,q=1

∂q Pρ [U0 ](z, t)p [vpq (x) − xp eq ]||2L2 (Ω) ,

√ where T = |¯ y − z|/ µ is the arrival time and ∆T is a window time. One can add a total variation regularization term. The choice of the space and time window sizes are critical. If they are too large, then noisy images are obtained. If they are too small, then resolution is poor. The optimal window sizes are related to the signal-to-noise ratio of the recorded near-field measurements. They express the trade-off between resolution and stability. This will be the subject of a next paper.

5

Conclusion

In this paper we have rigorously establish asymptotic expansions of near- and far-field measurements of the transient elastic wave induced by a small elastic anomaly. We have proved that, after truncation of the high-frequency component, the perturbation due to the anomaly can be seen not only as a polarized wave emitted from the anomaly but it contains unlike 16

the acoustic case a near-field like term which does not propagate. We have also shown that time-reversing this perturbation gives birth to a near-field like effect. Moreover, the diffraction limit is anisotropic. We have then explained the experimental findings reported in [9]. In this paper we have only considered a purely quasi-incompressible elasticity model. In a forthcoming work, we will consider the problem of reconstructing a small anomaly in a viscoelastic medium from wavefield measurements. Expressing the ideal elastic field without any viscous effect in terms of the measured field in a viscous medium, we will generalize the methods described here to recover the viscoelastic and geometric properties of an anomaly from wavefield measurements.

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