Inverse scattering by point-like scatterers in the Foldy regime

Inverse scattering by point-like scatterers in the Foldy regime

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IOP PUBLISHING INVERSE PROBLEMS

Inverse Problems 28 (2012) 125006 (39pp) doi:10.1088/0266-5611/28/12/125006

Inverse scattering by point-like scatterers in theFoldy regime

Durga Prasad Challa1 and Mourad Sini

RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

E-mail: [email protected] and [email protected]

Received 13 July 2012, in final form 13 September 2012Published 15 November 2012Online at stacks.iop.org/IP/28/125006

AbstractThe scattering by point-like scatterers is described in the Born, Foldy andthe intermediate regimes. We explain why the Foldy regime is, rigorously, anatural model for taking into account the multiple scattering. For each regime,we study the inverse problems for detecting these scatterers as well as thescattering strengths. In the first part, we do it for the acoustic case, and in thesecond, we study the corresponding models for the linearized isotropic elasticcase. In this last case, we show how any of the two body waves, namely pressurewaves P or shear waves S, is enough to solve the inverse problem. In the 3Dcase, it is shown that the shear horizontal part (SH) or the shear vertical part(SV) of the shear waves S is also enough for the detection. Finally, we provideextensive numerical tests justifying our findings and discuss the question ofresolution in terms of the distance between the scatterers, the used frequencyand the scattering strengths. In addition, a comparison study between the threementioned regimes is also provided.

(Some figures may appear in colour only in the online journal)

1. Introduction

Scattering by point-like obstacles is well studied in many areas of applied sciences, as inquantum mechanics, acoustic and electromagnetic wave propagation; see [2, 12] and [25] fora review. A commonly used way of modeling the point-like obstacles is by considering thepotentials (resp. the refraction indices) as highly concentrated coefficients on the differentpoints so that they can be naturally considered as approximations of point sources, or Diracimpulses; see [12]. Following this point of view, one considerably simplifies the models.The price to pay is that, these coefficients being singular, the obtained models are no longerregular perturbations of the known operators (as the Laplace operator in the case of acousticpropagation). A way of representing the scattered wave, as a solution of these models, wasfirst given by Foldy [14] who stated formally the fundamental equations of multiple scatteringby finitely many point-like scatterers; see the system of equations (2.8) and (2.9). More details

1 Author to whom any correspondence should be addressed.

0266-5611/12/125006+39$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USA 1

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Inverse Problems 28 (2012) 125006 D P Challa and M Sini

regarding this Foldy model and related works can be found in [12] and [25]. In parallelto this and motivated by applications in the field of quantum mechanics, several rigorousmathematical methods have been proposed to give sense to these singularly perturbed modelsand to solve the scattering by N-particles. The general idea is to take this model as the oneobtained by the limit, in the resolvent sense, of a sequence of operators generated by replacingthe Dirac impulses by smoothed (or less singular) potentials. The arguments are based on theWeinstein–Aronszajn inversion formula or more generally on the Krein inversion formulae forself-adjoint operators; see [2] for a comprehensive study of this issue. Due to the equivalencebetween the forms of the acoustic and the Schrodinger models, we can apply these techniquesto the acoustic case as well. The result is that the represented solution using this obtainedmodel is nothing but the Foldy model where the scattering coefficients (i.e. the scatteringstrengths) should be replaced by the renormalized ones; see section 2.2 for more details on thevalidity of this Foldy model.

Following the ideas in [2], the corresponding model for the scattering by finitely manypoint-like scatterers for the Lame system of equations is derived in [18]. Here, the scatteringis due to high concentrations of the densities on the scatterers which are, then, taken as Diracimpulses.

The purpose of our work here is to study an inverse problem type. Precisely, we areinterested in reconstructing the point-like scatterers and the associated scattering strengthsfrom the far fields corresponding to several incident plane waves. We use as models(1) the Foldy one described above, taking into account the multiple scattering, (2) the Bornapproximation, neglecting the multiple scattering, and (3) the intermediate scattering models,taking into account a finite number of times the interactions between the scatterers. Due tothe quasi-explicit form of the far-field patterns in all the models, we can justify and apply aMUSIC-type algorithm for the reconstruction.

We start with the acoustic case where we describe the scattered fields in each of thesemodels and then provide the inversion algorithm with several numerical tests discussing theresolution of the reconstruction depending on the number of scatterers, their distance, theused wavelength and the scattering strengths. Our focus will be on the target localization eventhough we provide also the corresponding formulae for recovering the scattering strengths.For this last issue, we cite the works [11, 13, 23, 24] for more insight and details on the actualimplementations of those formulae.

As a second step, we describe the three models for the Lame system and provide ajustification of the MUSIC algorithm completing the work [17] where the Born approximationwas used. In this Lame system, we have two body waves, namely pressure waves P or shearwaves S. We show that any of these two waves is enough to solve the inverse problem. Thisobservation was already made in [16] regarding the extended scatterers. In the 3D case, weshow that, in addition, the shear-horizontal part (SH) or the shear vertical part (SV) of theshear waves is also enough for the detection. As for the acoustic case, we provide severalnumerical tests supporting these results and discuss the question of resolution. Note that thisquestion of resolution could not be discussed in [17] since the Born approximation is notappropriate for that. For both the acoustic and the elastic cases, a comparison study betweenthe three mentioned models is provided. Finally, let us mention the following works concerningMUSIC-type algorithms for detecting small inclusions using the near fields in elasticity[4–6, 19].

The rest of the paper is organized as follows. In section 2, we study the acoustic scatteringby point-like scatterers and then the corresponding inverse problems. In section 3, we studythe corresponding problems for the Lame system. Finally, in the appendix, we give the detailedcalculations for justifying the MUSIC algorithm for the Lame system.

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2. Acoustic scattering by point-like scatterers

Let Ui be a solution of the Helmholtz equation (� + κ2)Ui = 0 in Rd, d = 2, 3. Let also Us

be the acoustic field scattered by a medium D ⊂ Rd due to the incident field Ui. The total field

Ut := Ui + Us satisfies the acoustic equation

(� + κ2n)Ut = 0 in Rd, d = 2, 3, (2.1)

and the scattered field Us satisfies the Sommerfield radiation condition:

lim|x|→∞

|x| d−12

(∂Us(x)

∂|x| − iκUs(x)

)= 0 uniformly in all directions x := x

|x| ∈ Sd−1. (2.2)

Here, κ > 0 is the wavenumber and the scattering medium is modeled by the bounded andmeasurable index of refraction n with n = 1 outside of the bounded domain D ⊂ R

d . We canrewrite equation (2.1) as

(� + κ2)Us(y) = −κ2q(y)Ut (y) in D, (2.3)

with q := n−1 as the contrast. Recall that the fundamental solution of the Helmholtz equationis defined as �(x, y) := eiκ|x−y|

4π |x−y| in R3 and �(x, y) := i

4 H10(κ|x − y|) in R

2, where H10 is the

Hankel function of the first kind and order zero. The scattering problem (2.1)–(2.2) is wellposed; see, e.g., [7, 8].

Multiplying equation (2.3) by the fundamental solution and applying integration by parts,we obtain the following Lippmann–Schwinger equation:

Ut (x) = Ui(x) + κ2∫

Dq(y)�(x, y)Ut (y) dy. (2.4)

To describe the scattering by M point-like scatterers y1, y2, . . . , yM, M ∈ N, we follow DeVries–van Coevorden–Lagendijk and take q as q(y) := 1

κ2

∑Mm=1 gmδ(y − ym), gm ∈ R, m =

1, . . . , M; see [25, 12]. Here, δ is the Dirac measure and the gm’s are the scattering strengthsof the point-like scatterers ym’s. Now, we can write equation (2.4) formally as

Ut (x) = Ui(x) +M∑

m=1

gm�(x, ym)Ut (ym), x �= ym, m = 1, 2, . . . , M. (2.5)

To use the formula (2.5), we need to know Ut (ym). However, we cannot calculate Ut (ym), m =1, 2, . . . , M; we cannot evaluate (2.5) at x = ym due to the singularity of �(x, ym) at x = ym.Therefore, few approximations were introduced; see [25] for more information concerningthis issue. In the next sections, we discuss the Born approximation, the Foldy method and thenthe intermediate levels of scattering.

2.1. Approximation methods

2.1.1. Born approximation. In the Born approximation, the total field Ut (ym), at the point-like scatterer ym, is replaced by the incident field Ui(ym) in equation (2.5). It means that theBorn approximation neglects the effect of multiple scattering and deals with weak scattering.We obtain the following representation of the total field:

Ut (x) = Ui(x) +M∑

m=1

gm�(x, ym)Ui(ym), (2.6)

and then, the scattered field is given by

Us(x) =M∑

m=1

gm�(x, ym)Ui(ym).

3


Using plane waves, Ui(x, θ ) = eiκx·θ with the direction of incidence θ ∈ Sd−1, and knowing

the asymptotic expansion of �(x, y) as |x| → ∞, we obtain the far-field pattern related to theBorn approximation as

U∞(x, θ ) =M∑

m=1

gm eiκym·(θ−x), x, θ ∈ Sd−1. (2.7)

2.1.2. Foldy’s method. In contrast to the Born approximation, and following the methodof Foldy, used also for the scattering by many small obstacles, see [14, 25], we replaceequation (2.5) by

Ut (x) = Ui(x) +M∑

m=1

gm�(x, ym)Um, (2.8)

where the terms Um are calculated from the Foldy algebraic system given by

Um = Ui(ym) +M∑

j=1j �=m

gj�(ym, y j)Uj ∀m = 1, . . . , M. (2.9)

In [14], (2.8)–(2.9) are called the fundamental equations of multiple scattering. From thesystem (2.9), we see how the Foldy method takes into account the multiple scattering effectbetween the scatterers.

From (2.8)–(2.9), we obtain the scattered field Us(x) as

Us(x) =M∑

m=1

gm�(x, ym)Um. (2.10)

In particular, for plane incident waves Ui(x, θ ) = eiκx·θ , we obtain the far-field pattern relatedto the Foldy method as

U∞(x, θ ) =M∑

m=1

gm e−iκ x·ymUm, x, θ ∈ Sd−1. (2.11)

2.1.3. Intermediate levels of scattering. Here, we give a common platform which deals withintermediate levels of scattering between the Born and the Foldy models. For any non-negativeinteger k, let Ut

k and Usk denote the total and the scattered fields, respectively, in the kth level

scattering. We set Um,0 := Ui(ym),∀m = 1, . . . , M, to be the incident wave. Then, the totalfield in the kth level scattering is calculated by

Utk(x) = Ui(x) +

M∑m=1

gm�(x, ym)Um,k, (2.12)

where the terms Um,k’s, indicating the exciting fields, are defined recursively by

Um,k+1 := Ui(ym) +M∑

j=1j �=m

gj�(ym, y j)Uj,k for m = 1, . . . , M. (2.13)

From (2.12)–(2.13), the scattered field Usk (x) in the kth level scattering is given by

Usk (x) =

M∑m=1

gm�(x, ym)Um,k. (2.14)

4


From the above equations, we observe that k = 0 and ∞ deal with the Born approximationand the Foldy model, respectively. The system (2.13) is nothing but the (k + 1)th iteration ofthe Foldy algebraic system (2.9). In particular, for plane incident waves Ui(x, θ ) = eiκx·θ , weobtain the far-field pattern related to the kth level scattering as

U∞k (x, θ ) =

M∑m=1

gm e−iκ x·ymUm,k. (2.15)

For each of these models, we study the following inverse problem.

Remark. As we can see, all the above models are derived by approximating the total field atym, m = 1, . . . , M, i.e. Ut (ym) in (2.5).

Inverse problem. Given the far-field pattern U∞(x, θ ) for several incident and observationdirections θ and x, locate the point-like scatterers y1, y2, . . . , yM and reconstruct their scatteringstrengths g1, g2, . . . , gM .

2.2. The validity of the Born, Foldy and intermediate models

• Foldy’s model. First, we observe that the Foldy algebraic system (2.9) is obtained from (2.5)by taking x tending to ym, m = 1, . . . , M, and deleting the singular term in the sum. Thereare several ways to justify and give sense to this step. The first is related to the regularizationof the model (2.1), or (2.3). We mention the work [2, chapter 2] where this is studied in theframework of interactions of point-like particles in quantum mechanics. We highlight themain idea behind this method. Replacing the scattering coefficients gm by the parameter-dependent coefficients gm(ς ) := (g−1

m + ς

2π2 )−1, ς ∈ R+, and the Fourier transform ofthe delta distribution by its truncated part, up to ς , they obtain a parameter family of self-adjoint operators, with ς being a parameter, in the Fourier variable. These operators arefinite-rank perturbations of the multiplication operator (which is the Fourier transform ofthe Laplacian). Based on the Weinstein–Aronszajn formula, they show that the resolvent ofthis family of operators converges, as ς → ∞, to the resolvent of a closed and self-adjointoperator which they define as the Fourier transform of the operator modeling the scatteringby finitely many point-like obstacles. As a result, the scattering fields, computed via theresolvent of this operator, is represented by nothing but (2.10), where gm is replaced by(g−1

m − iκ4π

)−1, i.e. exactly the Foldy representation. We can deduce then that the Foldymodel is a natural model to describe the multiple interactions of point-like obstacles. Letus also mention that approximating models of the form (2.3) replacing q by less singularpotentials than the delta-type potentials, i.e. the compactly supported Rollnik potentials,are provided in [2, chapter 2]. It is proved that the corresponding family of self-adjointoperators converge in the norm-resolvent sense to the operator modeling the scattering byfinitely many point-like obstacles described above. The scattering strengths gm are relatedto the limits of those Rollnik potentials.

A second way to justify the Foldy model is demonstrated in details in [15], see also[2], where the Krein formula of the resolvent of the extensions of self-adjoint operatorsis used, instead of the Weinstein–Aronszajn determinant formula. This provides a moregeneral representation of the scattered field due to point-like scatterers where the Foldymodel is a particular one.

• Born’s approximation. Assuming that the point-like scatterers are far away from each other,i.e. |ym − y j| � 1, m �= j, then a good approximate solution of the linear system (2.9) isindeed the vector (Ui(y j)) j=1,...,M . This implies that the Foldy model reduces to the Bornmodel.

5


• Intermediate levels of scattering. Let k = 1, then

Um,1 = Ui(ym) +∑j �=m

gj�(ym, y j)Ui(y j), (2.16)

which means that the total field on the point scatterer ym is given by the incident waveUi(ym) plus the scattered field by each of the other scatterers, y j, j �= m, taken separately.This model takes into account one time interaction between the scatterers.

Let k = 2, then

Um,2 = Ui(ym) +∑j �=m

gj�(ym, y j)Uj,1,

which we can write, using (2.16), as

Um,2 = Ui(ym) +∑j �=m

gj�(ym, y j)Ui(y j) +

∑j �=m

gj�(ym, y j)

⎡⎣∑

s�= j

gs�(y j, ys)Ui(ys)

⎤⎦ .

This means that the total field on the point scatterer ym is given by the incident field Ui(ym)

plus the scattered field, due to the incident field Ui, by each of the other scatterers, y j,j �= m, taken separately and plus the scattered field of each scatterer y j, j �= m, due tothe incident field given by the scattered wave by the other scatterers ys, s �= j. This modeltakes into account the two-level interaction between the scatterers.

Iterating this process, we can see how the kth level of scattering takes into account thek-level interactions between the scatterers.

2.3. The inverse problems for the Born and Foldy models

2.3.1. Localization of ym’s via the MUSIC algorithm. The MUSIC algorithm is a methodto determine the locations ym, m = 1, 2, . . . , M, of the scatterers from the measuredfar-field pattern U∞(x, θ ) for a finite set of incidence and observation directions, i.e.x, θ ∈ {θ j, j = 1, . . . , N} ⊂ S

d−1. We refer the reader to the monographs [3] and [20] formore information about this algorithm. We follow the way presented in [20]. We assume thatthe number of scatterers is not larger than the number of incident and observation directions,i.e. N � M. We define the response matrix F ∈ C

N×N by

Fjl := U∞(θ j, θl ). (2.17)

In order to determine the locations ym, we consider a grid of sampling points z ∈ Rd in a region

containing the scatterers y1, y2, . . . , yM . For each point z, we define the vector φz ∈ CN by

φz := (e−iκθ1·z, e−iκθ2·z, . . . , e−iκθN ·z). (2.18)

MUSIC characterization of the scatterers. The MUSIC algorithm is based on the property thatφz is in the range R(F ) of F if and only if z is at one of locations of the scatterers. Precisely,let P be the projection onto the null space N (F∗) = R(F )⊥ of the adjoint matrix F∗ of F ,and then,

z ∈ {y1, y2, . . . , yM} ⇐⇒ Pφz = 0.

This property can be proved based on the factorization F = H∗T H of F ∈ CN×N ,

where the matrix T ∈ CM×M is invertible and H ∈ C

M×N defined in terms of the vectorsφy1 , φy2 , . . . , φyM has a maximal rank.

For the Born approximation, this factorization is clear from (2.7) with T :=Diag(g1, g2, . . . , gM ) ∈ C

M×M , the diagonal matrix with diagonal entries as gm, and

6


H ∈ CM×N , defined by Hpq := eiκθq.yp, 1 � p � M, 1 � q � N. The maximal rank property

of H is justified, e.g., in [20, chapter 4].1

In the case of Foldy, we can write the Foldy algebraic system (2.9) in a compact form as

AUI = UI, (2.19)

where the matrix A ∈ CM×M and the vectors UI, UI ∈ C

M×1 are given by

A :=

⎛⎜⎜⎝

1 −g2�(y1, y2) −g3�(y1, y3) . . . −gM�(y1, yM )

−g1�(y2, y1) 1 −g3�(y2, y3) . . . −gM�(y2, yM )

. . . . . . . . . . . . . . .

−g1�(yM, y1) −g2�(yM, y2) −g3�(yM, y3) . . . 1

⎞⎟⎟⎠ ,

UI := [U1,U2, . . . ,UM] and UI := [Ui(y1),Ui(y2), . . . ,UM(yM )].

We suppose that A is non-singular and denote its inverse by B := (bi j) ∈ CM×M. From (2.19),

we obtain

UI = BUI .

Then, using (2.11), the response matrix F ∈ CN×N can be factorized as

F = H∗T H, (2.20)

where, in this case, T := Diag(g1, g2, . . . , gM )B, while H is the same matrix we introducedbefore.

In the case of Born approximation, it is clear that T is invertible. Let us deal with the Foldycase. We observe that the matrix A ∈ C

M×M can be factorized as A = Ag with A ∈ CM×M

defined by

A :=

⎛⎜⎜⎜⎜⎝

1g1

−�(y1, y2) −�(y1, y3) . . . −�(y1, yM )

−�(y2, y1)1g2

−�(y2, y3) . . . −�(y2, yM )

. . . . . . . . . . . . . . .

−�(yM, y1) −�(yM, y2) −�(yM, y3) . . .1

gM

⎞⎟⎟⎟⎟⎠

and g := Diag(g1, g2, . . . , gM ).

Then, T = gB = gA−1 = g(Ag)−1 = A−1. Hence, it is enough to consider the invertibilityof A.

2.3.2. Invertibility of the matrix A. We discuss here the conditions under which the matrixA ∈ C

M×M is invertible.Case 1 (Diagonally dominant condition). As the matrix A is symmetric, row-wise and

the column-wise diagonally dominant conditions match. The diagonally dominant conditionfor A is

M∑m=1m�= j

|�(ym, y j)| <1

|g j| , ∀ j = 1, 2, . . . , M. (2.21)

We have, �(x, y) = �(y, x) := eik|x−y|4π |x−y| in R

3 and �(x, y) = �(y, x) := i4 H1

0(κ|x − y|) inR

2. In R3, (2.21) can be written as

M∑m=1m�= j

|�(ym, y j)| <1

|g j| ⇐⇒M∑

m=1m�= j

1

|ym − y j| <4π

|g j| , ∀ j = 1, 2, . . . , M. (2.22)

1 We show the idea and the details of the proof in the appendix in the framework of the inverse elastic scattering.

7


Relation (2.22) tells us that if the scattering strength of every point-like scatterer y j, j =1, 2, . . . , M, is less than 4π over the sum of the reciprocals of the distance of the scatterery j, j = 1, 2, . . . , M, from the other scatterers, then A is invertible.

In particular, the condition that the left part of (2.21) is much smaller than its right partfor every j (i.e. when the scatterers are relatively far away from each other compared to thescattering strengths) leads to weak scattering, e.g., the Born approximation.

Case 2 (Non-diagonally dominant condition). The necessary and sufficient condition forthe invertibility of A in the case of two scatterers is �2(y1, y2) �= 1

g1g2. Fixing the wavenumber

κ and the scattering strengths g1 and g2, this condition holds almost every time except forthe distributions of the scatterers satisfying |y1 − y2|2 = 1

16π2 |g1||g2| in the 3D case forinstance. These exceptions where A is singular are called resonances; see [2, chapter 2.1].This observation can be generalized to the case of finitely many point-like scatterers asfollows. Fix the wavenumber κ and the scattering strengths g1, g2, . . . , gM and look at detA asa function of the M(M−1)

2 real variables ξm j := |ym − y j| for m, j = 1, . . . , M with m < j andset ξ := (ξ1,2, ξ1,3, . . . , ξ1,M, ξ2,3, . . . , ξM−1,M ). Then, due to the explicit form of A, we see

that ξ → detA(ξ ) is a real analytic function in RM(M−1)

2+ . Hence, it has locally a finite number ofzeros. These zeros are related to the possible distributions of the scatterers for which Foldy’smethod does not apply.

2.3.3. Recovering the scattering strengths gm’s. Once we locate the scatterers from the givenfar-field patterns using the MUSIC algorithm, we can recover the scattering strengths fromthe factorization of F ∈ C

N×N . Indeed, from theorem 4.1 of [20], we know that the matrixH has a maximal rank, see also the appendix. So, the matrix HH∗ ∈ C

M×M is invertible.Let us denote its inverse by IH . Once we locate the scatterers y1, y2, . . . , yM by using theMUSIC algorithm for the given far-field patterns, we can recover IH , and hence, the matrixT ∈ CM×M given by T = IHHFH∗IH , where IHH is the pseudo-inverse of H∗. As we knowthe structure of T ∈ C

M×M in both Born (T = g) and Foldy (T = A−1) approximations, wecan recover the scattering strengths g1, . . . , gM from the diagonal entries of T or of T −1 in theBorn approximation and the Foldy model, respectively.

2.3.4. Numerical results and discussions. In this section, we illustrate the performance ofthe MUSIC algorithm for this acoustic case and present results for locating the scatterers usingthe Foldy method. We also present the results for comparing weak (Born) and multiple (Foldy)scatterings.

For the convenience of visualization, we only show the results for two-dimensionalproblems. However, we should mention that the algorithm in two- and three-dimensionalspaces are the same. Denote by Nd the number of incident directions used in a quarter of a unitcircle, which are the same for the observation directions. We consider the following angles(figure 1(a)):

θ j = x j = ( j − 1)π

2Nd, j = 1, 2, . . . , 4Nd .

In the following examples, we consider Nd = 4 and the point-like scatterers of thesame scattering strength located at the points y1 = (0, 0), y2 = (0, 0.5), y3 = (0.5, 0), y4 =(0.5, 0.5), y5 = (1, 1), y6 = (1,−1), y7 = (−1,−1), y8 = (−1, 1), y9 = (1,−1.5), y10 =(1.5, 0.5), y11 = (−1.5, 1), y12 = (0, 0.4), y13 = (0,−1), y14 = (1.5, 1.5) and y15 =(0.6, 0.6).

Since the MUSIC algorithm is an exact method, the reconstruction is very accurate in theabsence of noise in measured data, for both Born and Foldy models. It can be observed in

8


(a) (b) (c)

Figure 1. (a) Incidence and observation directions with Nd = 4, (b) Born-based and (c) Foldy-basedreconstructions with 0% noise, gi = 1 and κ = π for three scatterers.

(a) (b) (c) (d)

Figure 2. Born- and Foldy-based reconstructions, respectively, from left to right with 2% noise,gi = 1 and κ = 2π for eight scatterers.

figures 1(b) and (c), from the pseudo-spectrum of the scatterers located at the points y1, y2, y5

having scattering strength 1 for each with the wavenumber κ = π (i.e. the minimum distancebetween the scatterers is quarter of the wavelength) with respect to the Born approximationand the Foldy model.

To analyze the effect of the noise level on the resolution of the algorithm, different noiselevels are used. To distinguish the differences between the Born approximation and the Foldymodel, we used different scattering strengths, noise levels and distance between the scatterers.

Figures 2 and 3 are related to the eight scatterers located at the pointsy1, y2, y5, y6, y7, y8, y10 and y11 having scattering strength 1 for each with 5% random noisein the measured far-field pattern. Figure 2 shows the pseudo-spectrum of the eight mentionedscatterers for the wavenumber κ = 2π, whereas figure 3 shows the pseudo-spectrum for thewavenumber κ = π . We can observe that the scatterers satisfy largely the condition (2.21) andthe reconstruction looks similar in both the Born approximation and the Foldy model. Hence,we observe that if the scatterers are well separated with low scattering strengths, there is notmuch difference in the reconstruction between the Born approximation and the Foldy model.

Now, we look for some examples where scatterers failed to satisfy the condition (2.21).Figure 4 shows the pseudo-spectrum of the three scatterers located at y2, y3 and y4, each havingthe scattering strength 10 for κ = π with 5% random noise in the measured far-field patternswith respect to the Born approximation and the Foldy method. Figure 5 shows the pseudo-spectrum of the nine scatterers located at y1, y4, y6, y7, y8, y9, y10, y11 and y12, each having the

9


(a) (b) (c) (d)

Figure 3. Born- and Foldy-based reconstructions, respectively, from left to right with 2% noise,gi = 1 and κ = π for eight scatterers.

(a) (b) (c) (d)

Figure 4. Born- and Foldy-based reconstructions, respectively, from left to right with 5% noise,gi = 10 and κ = π for three scatterers.

(a) (b) (c) (d)

Figure 5. Born- and Foldy-based reconstructions, respectively, from left to right with 1% noise,gi = 10 and κ = π for nine scatterers.

scattering strength 10 for κ = π with 1% random noise in the measured far-field patterns withrespect to the Born approximation and the Foldy model.

Compared to figures 2 and 3, we see in figures 4 and 5 how the reconstruction deterioratesdue to the effect of multiple scattering created by the close obstacles. In this case, we can seethe differences between the Born approximation and the Foldy model.

As a conclusion, we have seen that if the condition (2.21) is satisfied largely, then the effectof the multiple scattering is quite low and the reconstruction is similar in both Born and Foldybut above the condition (2.21) the use of the Born approximation gives better reconstruction

10


than the use of the Foldy method. However, in the latter case, the Born approximation is notvalid as the scatterers are relatively close. It is observed that, in general, the increase of thenoise level, the decrease of the distance between the scatterers and the increase in the numberof scatterers make the reconstruction worse in both the approximations in the presence ofnoise. It is also observed that when the scatterers have different scattering strengths and if theyare not well separated, the visibility of the scatterer is proportional to the scattering strengthof the respective scatterer.

2.4. The inverse problem for the intermediate levels of scattering

In this section, we deal with the intermediate levels of scattering. Recall that Usk , the scattered

fields in the kth level scattering, have the form

Usk (x) =

M∑m=1

gm�(x, ym)Um,k

with the terms, the exciting field and Um,k that are defined recursively by

Um,k+1 := Ui(ym) +M∑

j=1j �=m

gj�(ym, y j)Uj,k for m = 1, . . . , M. (2.23)

Define the vector UI,k ∈ CM with components Um,k. Now, recall the definition of UI ∈ C

M

in section 2.3.1; then, the exciting fields in different levels of scattering can be calculated asfollows:

UI,k =k∑

l=0

(−M)lUI for k = 0, 1, . . . , (2.24)

where the matrix M ∈ CM×M is defined by Mpq := −gq�(yp, yq) for p �= q and Mpp := 0.2

Then, using (2.15), the response matrix in the kth level scattering can be factorized as

F = H∗T H, (2.25)

with T := g∑k

l=0(−M)l , where g and H are defined in section 2.3.1. To apply the MUSICalgorithm, the invertibility of the matrix T is needed and the norm of M less than half is thesufficient condition for that in every level of scattering. In this case, the reconstruction looksquite similar in all levels of scattering when the scatterers are far enough from each other.We can observe the similar kind of differences which we mentioned between weak (Born)and multiple (Foldy) scatterings, as the level k of the scattering increases, with respect to thecondition (2.21). We can observe this in figure 6, for the same data as in figure 4.

The more the condition (2.21) is satisfied (i.e. L.H.S � R.H.S) by the scatterers, themore multiple scattering can be neglected which leads to weak scattering, i.e. reconstructionlooks similar in various levels of scattering. But once this condition is violated, we see a cleardifference between the reconstruction in different levels of scattering.

As discussed in section 2.3.3, we can recover the matrix T ∈ CM×M as T = IHHFH∗IH ,

where F is the given far-field pattern in the kth level scattering and H and IH are as mentionedearlier in sections 2.3.1 and 2.3.3, respectively. By comparing this evaluated T with its explicitform, i.e. g

∑kl=0(−M)l , in the kth level scattering, we can recover the scattering strengths gm.

2 Observe that (I + M) = A, which we mentioned in (2.19) of section 2.3.1 for the Foldy case. Its inverse can beapproximated by the truncated Neumann series in the case that the norm of M is less than 1 and, in the case of 1-normof M, it is equal to the invertibility condition of A in case 1 of section 2.3.2.

11


(a) (b) (c) (d)

Figure 6. Figures are related to three scatterers with 5% noise, gi = 10 and κ = π : (a), (b) firstlevel scattering and (c), (d) third level scattering.

In Born and Foldy models, it is clear as mentioned in section 2.3.3 for M scatterers. In the casek = 1, we have T = g − gM. As we know that g is a diagonal matrix and the diagonal entriesof M are zero, the diagonal entries of T are equal to the scattering strengths g1, g2, . . . , gM

of the M scatterers. But, for intermediate level scattering k > 1, it is difficult to recover thescattering strengths due to the complicated structure of the matrices (−M)l , for l = 2, . . . ,

and hence of T . For this reason, we restrict ourselves to the special case of two point-likeobstacles y1, y2 with the corresponding scattering strengths g1, g2. In this case, we have theexplicit form of (−M)l for l = 0, 1, 2, . . . as follows:

(−M)l =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

[g

l21 g

l22 �l(y1, y2) 0

0 gl21 g

l22 �l(y1, y2)

], l ∈ 2N ∪ {0},

[0 g

l−12

1 gl+1

22 �l(y1, y2)

gl+1

21 g

l−12

2 �l(y1, y2) 0

], l ∈ 2N − 1.

The matrix (−M)l is either diagonal or anti-diagonal for every l ∈ N ∪ {0}. This structure isno longer valid for the case of more than two scatterers. From this structure, we obtain theexplicit form of T = g

∑kl=0(−M)l in the kth-order scattering as follows:

T =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[g1 00 g2

], k = 0,

⎡⎢⎢⎢⎣

g1

k2∑

l=0gl

1gl2�

2l(y1, y2)

k2∑

l=1gl

1gl2�

2l−1(y1, y2)

k2∑

l=1gl

1gl2�

2l−1(y1, y2) g2

k2∑

l=0gl

1gl2�

2l(y1, y2)

⎤⎥⎥⎥⎦ , k ∈ 2N,

⎡⎢⎢⎢⎣

g1

k−12∑

l=0gl

1gl2�

2l(y1, y2)

k−12∑

l=0gl+1

1 gl+12 �2l+1(y1, y2)

k−12∑

l=1gl+1

1 gl+12 �2l+1(y1, y2) g2

k−12∑

l=0gl

1gl2�

2l(y1, y2)

⎤⎥⎥⎥⎦ , k ∈ 2N − 1,

[1g1

−�(y1, y2)

−�(y1, y2)1g2

]−1

, k = ∞.

12


From the above explicit form of T , we observe the following points.

• The diagonal entries of T give the scattering strengths in the Born approximation, i.e.k = 0.

• Substituting the non-diagonal entries in the diagonal entries give the scattering strengths

in every even level scattering k, i.e. k ∈ 2N. Indeed, define a := ∑ k2l=1 gl

1gl2�

2l−1(y1, y2),

and then, the non-diagonal entries of T are equal to a. Also the diagonal entries T11 andT22 of T are equal to g1(1 + �(y1, y2)a) and g2(1 + �(y1, y2)a), respectively. Now, withthe knowledge of the scatterers y1 and y2 from the MUSIC algorithm and by substitutingthe value of a in the diagonal entries, we can evaluate the scattering strengths g1 and g2.

• Substituting the diagonal entries in the non-diagonal entries give the scattering strengths in

every odd level scattering k, i.e. k ∈ 2N− 1. Indeed, define b1 := g1∑ k−1

2l=0 gl

1gl2�

2l(y1, y2)

and b2 := g2∑ k−1

2l=0 gl

1gl2�

2l(y1, y2), and then, the diagonal entries T11 and T22 of T areequal to b1 and b2, respectively. Also the non-diagonal entries T12 and T21 of T are thesame and are equal to g1b2�(y1, y2) = g2b1�(y1, y2). Now again with the knowledge ofthe scatterers y1 and y2 from the MUSIC algorithm and by substituting the diagonal entriesin the non-diagonal entries of T , we can evaluate the scattering strengths g1 and g2.

• The diagonal entries of T −1 give the scattering strengths in the method of Foldy, i.e.k = ∞.

3. Elastic scattering by point-like scatterers

Assume that the Lame coefficients λ and μ are constants satisfying μ > 0 and dλ + 2μ >

0, d = 2, 3. We denote by ρ a bounded and measurable density function such that ρ = 1outside of the bounded domain D ⊂ R

d . The inhomogeneous problem associated with theLame system reads as follows:

(�e + ω2ρ)ut = 0 in Rd, [�e := μ� + (λ + μ)∇div] (3.1)

with the Kupradze radiation conditions

lim|x|→∞

|x| d−12

(∂up

∂|x| − iκpup

)= 0, and lim

|x|→∞|x| d−1

2

(∂us

∂|x| − iκsus

)= 0, (3.2)

where the two limits are uniform in all the directions x := x|x| ∈ S

d−1. Here, we denoted thetotal field by ut := ui +u, the incident field by ui and the scattered field by u. Also, we denotedby up := −κ−2

p ∇(∇ · u), the longitudinal (or the pressure or P) part of the field u and byus := −κ−2

s ∇ × (∇ × u), the transversal (or the shear or S) part of the field u correspondingto the Helmholtz decomposition u = up + us. The constants κp := ω√

λ+2μand κs := ω√

μare

known as the longitudinal and transversal wavenumbers, respectively, and ω is the frequency.The scattering problem (3.1)–(3.2) is well posed; see, e.g., [21, 22]. In addition, the scatteredfield u satisfies the following asymptotic expansion at infinity:

u(x) := eiκp|x|

|x| d−12

u∞p (x) + eiκs|x|

|x| d−12

u∞s (x) + O

(1

|x| d+12

), |x| → ∞ (3.3)

uniformly in all directions x ∈ Sd−1. The longitudinal part of the far field, i.e. u∞

p (x), is normalto S

d−1, while the transversal part u∞s (x) is tangential to S

d−1. Due to this property, they canbe measured separately. Note that it is not necessarily true for near-field measurements. Asusual in scattering problems, we use plane waves as incident waves in this work. For the Lamesystem, they have the following analytic forms:

ui,p(x, θ ) := θ eiκpθ ·x and ui,s(x, θ ) := θ⊥ eiκsθ ·x, (3.4)

13


where θ⊥ is any direction in Sd−1 perpendicular to θ . Pressure incident waves are in the

direction of θ , whereas shear incident waves are in the direction of θ⊥. In the two-dimensionalcase, the shear waves have only one direction. But in the three-dimensional case, they havetwo orthogonal components called vertical and horizontal shear directions denoted by θ⊥v

and θ⊥h , respectively. So, θ⊥ = αθ⊥h + βθ⊥v /|αθ⊥h + βθ⊥v | for arbitrary constants α andβ. To give the explicit forms of θ⊥h and θ⊥v , we recall the Euclidean basis {e1, e2, e3},where e1 := (1, 0, 0), e2 := (0, 1, 0) and e3 := (0, 0, 1), write θ := (θ1, θ2, θ3)

andset r2 := θ2

1 + θ22 . Let R3 be the rotation map transforming θ to e3. Then, in the basis

{e1, e2, e3},R3 is given by the matrix

R3(θ ) = 1

r2

⎡⎣ θ2

2 + θ21 θ3 −θ1θ2(1 − θ3) −θ1r2

−θ1θ2(1 − θ3) θ21 + θ2

2 θ3 −θ2r2

θ1r2 θ2r2 θ3r2

⎤⎦ . (3.5)

It satisfies R3 R3 = I and R3(θ ) = e3. Correspondingly, we write θ⊥h := R

3 (e1) andθ⊥v := R

3 (e2). These two directions represent the horizontal and the vertical directions ofthe shear wave and they are given by

θ⊥h = 1

r2(θ2

2 + θ21 θ3, θ1θ2(θ3 − 1),−r2θ1)

, θ⊥v = 1

r2(θ1θ2(θ3 − 1), θ2

1 + θ22 θ3,−r2θ2)

.

(3.6)

As in the acoustic case, by setting q := ρ − 1, multiplying equation (3.1) with thefundamental tensor G(x, y) of the Navier equation and performing integration using Betti’sthird identity, see [22], we obtain the following Lippmann–Schwinger equation:

ut (x) = ui(x) + ω2∫

Dq(y)G(x, y)ut (y) dy. (3.7)

Here, we recall the form of G(x, y),

G(x, y) := − 1

μ

∇∇κ2

s

�κp (x, y) + 1

μ

(I + ∇∇

κ2s

)�κs (x, y), (3.8)

where the first and the second parts represents the P-part and the S-part of G(x, y), respectively,and �κp (x, y) and �κs (x, y) are given by

�κp (x, y) :=

⎧⎪⎪⎨⎪⎪⎩

i

4H1

0(κp|x − y|) in 2D

eiκp|x−y|

4π |x − y| in 3D, �κs (x, y) :=

⎧⎪⎨⎪⎩

i

4H1

0(κs|x − y|) in 2D

eiκs|x−y|

4π |x − y| in 3D.

To describe the elastic scattering by M point-like scatterers y1, y2, . . . , yM, M ∈ N, as in theacoustic case, we take q as q(y) := 1

ω2

∑Mm=1 gmδ(y − ym), gm ∈ R, m = 1, . . . , M. Here,

again δ is the Dirac measure and gm are the scattering strengths of the point-like scatterers ym.Now, we can write equation (3.7) formally as

ut (x) = ui(x) +M∑

m=1

gmG(x, ym)ut (ym), x �= ym, m = 1, 2, . . . , M. (3.9)

As in the acoustic case, we cannot calculate ut (ym), m = 1, 2, . . . , M, since we cannotevaluate (3.9) at x = ym due to the singularity of G(x, ym) at x = ym. We discuss the Bornapproximation, the Foldy method and then the intermediate levels of scattering.

14


3.1. The Born approximation

In the Born approximation, the total field ut (ym) is replaced by the incident field ui(ym) at thepoint-like scatterers ym, m = 1, . . . , M, in equation (3.9). So, the total field can be written as

ut (x) = ui(x) +M∑

m=1

gmG(x, ym)ui(ym), (3.10)

and then, the scattered field can be written in the following form:

u(x) =M∑

m=1

gmG(x, ym)ui(ym). (3.11)

The asymptotic behavior of Green’s tensor at infinity is given as follows:

G(x, ym) = apx ⊗ xeiκp|x|

|x| n−12

e−iκpx.ym + as(I − x ⊗ x)eiκs|x|

|x| n−12

e−iκsx.ym + O(|x|− n+12 ), (3.12)

with x = x|x| ∈ S

d−1 and I being the identity matrix in Rd , ap = κ2

p

4πω2 and as = κ2s

4πω2 ; see, e.g.,[1].

It follows from (3.3), (3.11) and (3.12) that the P-parts and the S-parts of the far-fieldpattern associated with the incident wave ui,p are given by

u∞,pp (x, θ ) = ap

M∑m=1

gm(x ⊗ x) · θ eiκpym·(θ−x), (3.13)

u∞,ps (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) · θ eiκpym·θ e−iκsym·x. (3.14)

Similarly, the P-parts and the S-parts of the parts of the far-field pattern associated with Sincident wave ui,s can be written as

u∞,sp (x, θ ) = ap

M∑m=1

gm(x ⊗ x) · θ⊥ eiκsym·θ e−iκpym·x, (3.15)

u∞,ss (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) · θ⊥ eiκsym·(θ−x). (3.16)

3.2. Foldy’s method

Now, consider the multiple elastic scattering between the point-like obstacles. Similarly, as inthe acoustic case, in the method of Foldy, the total field ut (x) has the form

ut (x) = ui(x) +M∑

m=1

gmG(x, ym)um, (3.17)

where the terms um can be calculated from the Foldy algebraic system given by

um = ui(ym) +M∑

j=1j �=m

gjG(ym, y j)u j, ∀m = 1, . . . , M. (3.18)

We can write the Foldy algebraic system (3.18) in a compact form as

AuI = uI, (3.19)

15


where the matrix A ∈ CdM×dM and the vectors uI, uI ∈ C

dM are given by

A :=

⎛⎜⎜⎝

I −g2G(y1, y2) . . . −gMG(y1, yM )

−g1G(y2, y1) I . . . −gMG(y2, yM )

. . . . . . . . . . . .

−g1G(yM, y1) −g2G(yM, y2) . . . I

⎞⎟⎟⎠ ,

uI :=

⎛⎜⎜⎜⎝

u1

u2...

uM

⎞⎟⎟⎟⎠ and uI :=

⎛⎜⎜⎜⎝

ui(y1)

ui(y2)...

ui(yM )

⎞⎟⎟⎟⎠ .

We denote uI by uI,p for P incident waves and by uI,s for S incident waves.Here, we inserted the fundamental matrix G, the identity matrix I, the incident vectors ui

and the Foldy terms um element wise. We suppose that A is non-singular and denote its inverseby B. From (3.19), we obtain the following representation:

uI = BuI .

For each m = 1, 2, . . . , M, set Bm ∈ Cd×dM as a submatrix of B formed by the

rows related to the Foldy term um, i.e. Bm is formed by d consecutive rows, from((m − 1)d + 1)th row to mdth row of B. With this setting, we obtain the scattered field from(3.18) and (3.17) as

u(x) =M∑

m=1

gmG(x, ym)um =M∑

m=1

gmG(x, ym)BmuI . (3.20)

It follows from (3.3), (3.12) and (3.20) that the P-parts and the S-parts of the far-field patternassociated with the P incident wave ui,p are given by

u∞,pp (x, θ ) = ap

M∑m=1

gm(x ⊗ x) e−iκpx·ym · Bm · uI,p(θ ), (3.21)

u∞,ps (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) e−iκsx·ym · Bm · uI,p(θ ). (3.22)

Similarly, it follows from (3.3), (3.12) and (3.20) that the P-parts and the S-parts of the far-fieldpattern associated with the S incident wave ui,s can be written as

u∞,sp (x, θ ) = ap

M∑m=1

gm(x ⊗ x) e−iκpx·ym · Bm · uI,s(θ ), (3.23)

u∞,ss (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) e−iκsx·ym · Bm · uI,s(θ ). (3.24)

As we mentioned earlier, the P-parts of the far fields are normal to Sd−1 and the S-parts are

tangential. Using these properties, we define the following scalar far fields which will be usefulin the statement and the justification of the MUSIC algorithm.

The scalar far-field pattern associated with PP scattering (P incident wave and P part ofthe far field) is

u∞(x, θ ) := x · u∞,pp (x, θ )

ap=

M∑m=1

e−iκpx·ym gm(x · Bm · uI,p(θ )). (3.25)

16


The scalar far-field pattern associated with SP scattering (S incident wave and P part of the farfield) is

u∞(x, θ ) := x · u∞,sp (x, θ )

ap=

M∑m=1

e−iκpx·ym gm(x · Bm · uI,s(θ )). (3.26)

The scalar far-field pattern associated with PS scattering (P incident wave and S-part of thefar field) is

u∞(x, θ ) := x⊥ · u∞,ps (x, θ )

as=

M∑m=1

e−iκsx·ym gm(x⊥ · Bm · uI,p(θ )). (3.27)

The scalar far-field pattern associated with SS scattering (S incident wave and S-part of thefar field) is

u∞(x, θ ) := x⊥ · u∞,ss (x, θ )

as=

M∑m=1

e−iκsx·ym gm(x⊥ · Bm · uI,s(θ )). (3.28)

3.3. Intermediate levels of scattering

For any non-negative integer k, let utk and us

k denote the total and the scattered fields,respectively, in the kth level scattering. We set um,0 := ui(ym),∀m = 1, . . . , M, to be theincident wave. Then, the total field in the kth level scattering is calculated by

utk(x) = ui(x) +

M∑m=1

gmG(x, ym)um,k, (3.29)

where the terms um,k, indicating the exciting fields, are defined recursively by

um,k+1 := ui(ym) +M∑

j=1j �=m

gjG(ym, y j)u j,k for m = 1, . . . , M. (3.30)

The system (3.30) is nothing but the (k + 1)th iteration of the Foldy algebraic system (3.18).Define the vector uI,k ∈ C

dM with components um,k arranged element wise as in the patternof uI in section 3.2.3 Recall the definitions of A ∈ C

dM×dM , uI ∈ CdM in section 3.2 and set

M := A − I,4 and then, (3.30) can be written in a compact form as

uI,k =k∑

l=0

(−M)luI for k = 0, 1, . . . . (3.31)

Define the matrix Ck ∈ CdM×dM by Ck := ∑k

l=0(−M)l for k = 0, 1, . . .. For eachm = 1, 2, . . . , M, set Cm,k ∈ C

d×dM as a submatrix of Ck formed by the rows related tothe exciting field term um,k, i.e. Cm,k is formed by d consecutive rows, from ((m − 1)d +1)th row to mdth row of Ck. With this setting and from (3.30) and (3.29), we obtain thescattered field us

k(x) in the kth level scattering as

usk(x) =

M∑m=1

gmG(x, ym)um,k =M∑

m=1

gmG(x, ym)Cm,kuI . (3.32)

3 From (3.29) and (3.30), we can observe that k = 0 and ∞ deals with the Born approximation and the Foldy model,respectively.4 In the case that the norm of M is less than 1, the inverse of A can be approximated by the truncated Neumann series.

17


It follows from (3.3), (3.12) and (3.32) that the P-parts and the S-parts of the far-field patternin the kth level scattering associated with the P incident wave ui,p are given by

u∞,pp,k (x, θ ) = ap

M∑m=1

gm(x ⊗ x) e−iκpx·ym · Cm,k · uI,p(θ ), (3.33)

u∞,ps,k (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) e−iκsx·ym · Cm,k · uI,p(θ ). (3.34)

Similarly, it follows from (3.3), (3.12) and (3.32) that the P-parts and the S-parts of the far-fieldpattern in the kth level scattering associated with the S incident wave ui,s can be written as

u∞,sp,k (x, θ ) = ap

M∑m=1

gm(x ⊗ x) e−iκpx·ym · Cm,k · uI,s(θ ), (3.35)

u∞,ss,k (x, θ ) = as

M∑m=1

gm(I − x ⊗ x) e−iκsx·ym · Cm,k · uI,s(θ ). (3.36)

As in the Foldy model, we define the following scalar versions of the far-field patterns.The scalar far-field pattern in the kth level scattering associated with PP scattering

(P incident wave and P part of the far field) is

u∞k (x, θ ) := x · u∞,p

p,k (x, θ )

ap=

M∑m=1

e−iκpx·ym gm(x · Cm,k · uI,p(θ )). (3.37)

The scalar far-field pattern in the kth level scattering associated with SP scattering (S incidentwave and P part of the far field) is

u∞k (x, θ ) := x · u∞,s

p,k (x, θ )

ap=

M∑m=1

e−iκpx·ym gm(x · Cm,k · uI,s(θ )). (3.38)

The scalar far-field pattern in the kth level scattering associated with PS scattering (P incidentwave and S-part of the far field) is

u∞k (x, θ ) := x⊥ · u∞,p

s,k (x, θ )

as=

M∑m=1

e−iκsx·ym gm(x⊥ · Cm,k · uI,p(θ )). (3.39)

The scalar far-field pattern in the kth level scattering associated with SS scattering (S incidentwave and S-part of the far field) is

u∞k (x, θ ) := x⊥ · u∞,s

s,k (x, θ )

as=

M∑m=1

e−iκsx·ym gm(x⊥ · Cm,k · uI,s(θ )). (3.40)

From the scalar far-field patterns related to the Born approximation, the Foldy model andintermediate level scatterings, we observe that there is not scattered field in the perpendiculardirections (i.e. x ⊥ θ ) for PP and SS scatterings. Similarly, we observe that there is noscattered field in the parallel and anti-parallel directions (i.e. x ‖ θ or −x ‖ θ ) for PS and SPscatterings.

Remark. As in the acoustic case, all the above models are derived by approximating the totalfield at ym, m = 1, . . . , M, i.e. ut (ym) in (3.9).

Inverse problem. Given the far-field pattern u∞(x, θ ), corresponding to each of the foursituations described above for several incident and observation directions θ and x, locate thepoint-like scatterers y1, y2, . . . , yM and reconstruct their scattering strengths g1, g2, . . . , gM .

18


3.4. The validity of the Born, Foldy and intermediate models

Regarding the Born and the intermediate models, similar comments as in the acoustic case canbe made. The Foldy model (3.17)–(3.18) is justified in [18] where the scattering strengths gm

are replaced by[g−1

m − iω2λ + 5μ

12πμ(λ + 2μ)

]−1

for the 3D case and by[g−1

m + 1

4π

[λ + 3μ

μ(λ + 2μ)CE + λ + μ

μ(λ + 2μ)− 1

2

(ln μ

μ+ ln(λ + 2μ)

λ + 2μ

)]]−1

in the 2D case, where CE is Euler’s constant. This extends the corresponding results in [2],known for the acoustic case, to the linearized isotropic elastic case.

3.5. The inverse problems for the Foldy model

Regarding the inverse problem for the Born model, we refer to [17] for the details. Let usmention that, there, it is shown that indeed only one type of elastic wave is enough fordetecting the point-like scatterers, confirming the earlier results shown in [16] concerningextended scatterers. In the next sections, we deal with the Foldy and the intermediate modelsgeneralizing those results and provide a detailed study on the resolution of the reconstructiondepending on the distance between the scatterers, the frequency used, the scattering strengthsand the type of incident wave. This study could not be made in [17] since the Born model isnot appropriate to analyze the resolution.

3.5.1. MUSIC algorithm for elastic waves. Here, again we assume that the number ofscatterers is not larger than the number of incident and observation directions, preciselyN � dM. We define the response matrix F ∈ C

N×N by

Fjl := U∞(θ j, θl ). (3.41)

In order to determine the locations ym, we consider a grid of sampling points z ∈ Rd . For each

point z, we define the vectors φjz,p and φ

jz,s in C

N by

φ jz,p := ((θ1 · e j) e−iκpθ1·z, (θ2 · e j) e−iκpθ2·z, . . . , (θN · e j) e−iκpθN ·z), (3.42)

φ jz,s := ((θ⊥

1 · e j) e−iκsθ1·z, (θ⊥2 · e j) e−iκsθ2·z, . . . , (θ⊥

N · e j) e−iκsθN ·z),∀ j = 1, . . . , d.

(3.43)

In PP, PS, SS and SP scatterings, denote the response matrix F by F pp , F p

s , Fss and Fs

p ,respectively, and these can be factorized as

F pp = H p∗gBH p, (3.44)

F ps = Hs∗gBH p, (3.45)

Fss = Hs∗gBHs, (3.46)

and Fsp = H p∗gBHs. (3.47)

19


Here, g := Diag(g1I, g2I, . . . , gMI) ∈ CdM×dM is the diagonal matrix with diagonal entries as

gm (each gm appear d times in g) and the matrices H p ∈ CdM×N and Hs ∈ C

dM×N are definedas

H p :=

⎛⎜⎜⎝

θ1 eiκpθ1·y1 θ2 eiκpθ2·y1 . . . θN eiκpθN ·y1

θ1 eiκpθ1·y2 θ2 eiκpθ2·y2 . . . θN eiκpθN ·y2

. . . . . . . . . . . .

θ1 eiκpθ1·yM θ2 eiκpθ2·yM . . . θN eiκpθN ·yM

⎞⎟⎟⎠

and

Hs :=

⎛⎜⎜⎝

θ⊥1 eiκsθ1·y1 θ⊥

2 eiκsθ2·y1 . . . θ⊥N eiκsθN ·y1

θ⊥1 eiκsθ1·y2 θ⊥

2 eiκsθ2·y2 . . . θ⊥N eiκsθN ·y2

. . . . . . . . . . . .

θ⊥1 eiκsθ1·yM θ⊥

2 eiκsθ2·yM . . . θ⊥N eiκsθN ·yM

⎞⎟⎟⎠ .

We already assumed that A ∈ CdM×dM is invertible and its inverse is B ∈ C

dM×dM . We canobserve that A can be factorized as A = Ag with A ∈ C

dM×dM defined by

A :=

⎛⎜⎜⎜⎝

1g1

I −G(y1, y2) . . . −G(y1, yM )

−G(y2, y1)1g2

I . . . −G(y2, yM )

. . . . . . . . . . . .

−G(yM, y1) −G(yM, y2) . . . 1gM

I

⎞⎟⎟⎟⎠ .

So, the matrix A is invertible if and only if A is invertible. As we assumed that A is invertible,the matrix A is invertible and we obtain gB = gA−1 = g(Ag)−1 = A−1. This gives us thefactorization of the response matrices in PP, PS, SS and SP scatterings as

F pp = H p∗A−1H p, (3.48)

F ps = Hs∗A−1H p, (3.49)

Fss = Hs∗A−1Hs, (3.50)

and Fsp = H p∗A−1Hs. (3.51)

Under the assumption that A is invertible, we have the following theorem to justify the MUSICalgorithm for elastic wave scattering.

Theorem 3.1. Let {θ j : j ∈ N} ⊂ Sd−1 be a countable set of directions such that any analytic

function on Sd−1 that vanishes in θ j for all j ∈ N vanishes identically. Let K be a compact

subset of Rd containing {ym : m = 1, . . . , M}. Then, there exists N0 ∈ N such that for any

N � N0 the following characterization holds for every z ∈ K:

z ∈ {y1, . . . , yM} ⇐⇒ φ jz,p ∈ R(H p∗

)

�φ j

z,s ∈ R(Hs∗), for some j = 1, . . . , d. (3.52)

Furthermore, we have the following.

• The ranges of H p∗ and Fp (where Fp := F pp or Fp := Fs

p ) coincide and thus

z ∈ {y1, . . . , yM} ⇐⇒ φ jz,p ∈ R(Fp) ⇐⇒ Ppφ

jz,p = 0, for some j = 1, . . . , d, (3.53)

where Pp : CN → R(Fp)

⊥ = N (Fp∗) is the orthogonal projection onto the null space

N (Fp∗) of Fp

∗.

20


• The ranges of Hs∗ and Fs (where Fs := F ps or Fs := Fs

s ) coincide, and thus,

z ∈ {y1, . . . , yM} ⇐⇒ φ jz,s ∈ R(Fs) ⇐⇒ Psφ

jz,s = 0, for some j = 1, . . . , d, (3.54)

where Ps : CN → R(Fs)

⊥ = N (Fs∗) is the orthogonal projection onto the null space

N (Fs∗) of Fs

∗.

Proof. The idea of the proof is essentially the same as the one of theorem 4.1 in [20] concerningthe acoustic case. The main task in proving this theorem is to show that the matrices Hs andH p have the maximal rank. For convenience of the reader, we give it in the appendix sincesome technical difficulties have to be taken care of. �

We can prove this theorem also for the different set of incident and the observationaldirections given by the assumption that ‘{θ j : j ∈ N} ⊂ S

d−1 and {x j : j ∈ N} ⊂ Sd−1 are

countable set of incident and observational directions such that any analytic function on Sd−1

that vanishes on one of these sets will vanish identically’.From theorem 3.1, the MUSIC algorithm holds for the response matrices corresponding

to the PP, PS, SS and SP scatterings using the Foldy method, under the assumption of theinvertibility of the matrix A. To make the best use of the singular value decompositionin SP and PS scatterings, we apply the MUSIC algorithm to Fs

pFsp∗(resp. Fs

p∗Fs

p ) andF p

s∗F p

s (resp. F ps F p

s∗) with the help of the test vectors φ

jz,p(resp. φ

jz,s), respectively.

The point-like scatterers can then be located in the PP, PS, SS and SP scatterings of elasticwaves in the Foldy regime. In addition, in the three-dimensional case, while dealing with Sincident wave or S-part of the far-field pattern, it is enough to use one of its horizontal (Sh) orvertical (Sv) parts. Hence, it is enough to study the far-field pattern of any of the PP, PSh, PSv ,ShSh, ShSv , SvSh, SvSv , ShP, SvP elastic scatterings to locate the scatterers. In other words, inthe three-dimensional case, instead of using the full incident wave and the full far-field pattern,it is enough to study one combination of pressure (P), horizontal shear (Sh) or vertical shear(Sv) parts of the elastic incident wave and a corresponding part of the elastic far-field patterns.

Indeed, define the vectors φjz,sh , φ

jz,sv ∈ C

N and the matrices Hsh, Hsv ∈ C

3M×N exactly

as φjz,s and Hs replacing θ⊥

i for i = 1, . . . , N by θ⊥hi and θ

⊥v

i , respectively; see (3.6). Wedenote the response matrices by F p

sh , Fsh

p , F psv , Fsv

p , Fsh

sh , Fsh

sv , Fsv

sh and Fsv

sv in the elastic PSh, ShP,PSv , SvP, ShSh, ShSv , SvSh, SvSv scatterings, respectively, and then, we can factorize them asfollows:

F psh = Hsh ∗

A−1H p, Fsh

p = H p∗A−1Hsh, (3.55)

F psv = Hsv ∗A−1H p, Fsv

p = H p∗A−1Hsv

, (3.56)

Fsh

sh = Hsh ∗A−1Hsh

, Fsh

sv = Hsv ∗A−1Hsh, (3.57)

and Fsv

sh = Hsh ∗A−1Hsv

, Fsv

sv = Hsv ∗A−1Hsv

. (3.58)

Hence, in the 3D case, we can state the following theorem related to the MUSIC algorithm,

Theorem 3.2. Let {θ j : j ∈ N} ⊂ S2 be a countable set of directions such that any analytic

function on S2 that vanishes in θ j for all j ∈ N vanishes identically. Let K be a compact subset

of R3 containing {ym : m = 1, . . . , M}. Then, there exists N0 ∈ N such that for any N � N0

the following characterization holds for every z ∈ K :

z ∈ {y1, . . . , yM} ⇐⇒ φjz,t ∈ R(Ht∗), for some j = 1, 2, 3 and for all t ∈ {p, sh, sv}. (3.59)

21


Furthermore, the ranges of Ht∗ and Frt coincide, and thus,

z ∈ {y1, . . . , yM} ⇐⇒ φjz,t ∈ R(Fr

t ) ⇐⇒ Ptφjz,t = 0,

for some j = 1, 2, 3 and for all r, t ∈ {p, sh, sv}, (3.60)

wherePt : CN → R(Fr

t )⊥ = N (Frt

∗) is the orthogonal projection onto the null spaceN (Frt

∗)of Fr

t∗.

Proof. The proof of this theorem is the same as the one of theorem 3.1, by proving the maximal-rank property of the matrices Hsh

, Hsv

and H p and by using the test vectors φjz,p, φ

jz,sh and φ

jz,sv ;

see the appendix for more details. �

3.5.2. Invertibility of the matrix A. As in the acoustic case, when the scatterers are relativelyfar away from each other comparing to the scattering strengths, the invertibility condition ofA is the diagonally dominant condition and it is given by

M∑m=1m�= j

||G(ym, y j)||∞ <1

|g j| , ∀ j = 1, 2, . . . , M. (3.61)

Here, || · ||∞ is the infinite norm and it is defined for a matrix, L = [Lmn] ∈ CM×N , as

||L||∞ := max1�m�M

∑Nn=1 |Lmn|. Recall the matrix G(x, y) from (3.8). It can be written explicitly

in 3D as

G(x, y) = 1

μk2s

�p(x, y)

r2

[k2

pRR + (1 − ikpr)(I − 3RR)]

− 1

μk2s

�s(x, y)

r2

[k2

s RR + (1 − iksr)(I − 3RR)] + �s(x, y)

μI, (3.62)

where R = x − y, r = |x − y| and R = Rr ; see [10]. Similarly, by writing the explicit form

of G(x, y) in 2D, we observe that it is expressed explicitly in terms of x − y. In (3.62), weremark that the entries of G(x, y) are analytic in terms of the variables ηm jl = (ym − y j)l ,m, j = 1, . . . , M and l = 1, . . . , d for ηm jl ∈ R\{0}. Remark also that detA is given by theproducts and sums of g−1

m and the entries of G(ym, y j) for m, j = 1, . . . , M. From the abovediscussion, using the analyticity of the determinant of A in terms of the d M(M−1)

2 real variablesηm jl for m, j = 1, . . . , M with m < j, l = 1, . . . , d, fixing the frequency ω and the scatteringstrengths gm, for m = 1, . . . , M, we can show that except for few distributions of the scatterers,y1, . . . , yM , the matrix A is invertible.

3.5.3. Recovering the scattering strengths gm. We have the factorization of the responsematrix in PP, PS, SS and SP elastic scatterings using the Foldy method as

Frt = Ht∗A−1Hr, ∀r, t ∈ {p, s}.

As in the acoustic case, once we locate the scatterers for the given scalar far-field patterns,we can recover the scattering strengths from the factorization of Fr

t ∈ CN×N . Indeed, from

theorem 3.1, we know that the matrices H p and Hs have the maximal rank. So, the matricesH pH p∗ ∈ C

dM×dM and HsHs∗ ∈ CdM×dM are invertible. Let us denote these inverses by

IH p and IHs , respectively. Once we locate the scatterers y1, y2, . . . , yM by using the MUSICalgorithm for the given far-field patterns, we can recover the matrix A−1 ∈ CdM×dM asA−1 = IHt HtFr

t Hr∗IHr , IHt Ht (resp. Hr∗IHr ) is the pseudo-inverse of Ht∗ (resp. Hr) . Then,

22


(a) (b)

Figure 7. Incidence and observation directions with Nd = 4: (a) PP and SS cases (the incidenceand observations coincide) and (b) PS and SP cases (‘*’ incidence directions and ‘o’ observationdirections).

we can recover the scattering strengths g1, . . . , gM from the diagonal entries of A. In a similarway, also in the 3D case, we can recover the scattering strengths in the case of PSh, ShP, PSv ,SvP, ShSh, ShSv , SvSh and SvSv elastic scatterings for r, t ∈ {p, sh, sv}.

3.5.4. Numerical results and discussions. In this section, we illustrate the performance ofthe MUSIC algorithm for the elastic waves for locating the scatterers using the Foldy method.We present the results comparing the weak (Born) and multiple (Foldy) scatterings using onetype of wave. In addition, we compare the results for the case of S and P incident plane waves.

Here also, we only show results for two-dimensional problems. From the previous sections,we know that one type of transverse wave is sufficient to locate the scatterers in the three-dimensional case. Denote by Nd , the number of incident directions used in a quarter of a unitcircle which are the same for the observational directions. There are no restrictions on incidentand observational directions but there are some points one should consider.

(1) It is better to avoid perpendicular directions for P incident waves and P part of far-fieldpatterns (PP case), or S incident waves and S-part of far-field patterns (SS case) as they do notprovide any useful information due to no scattered far field in these directions. To avoid theperpendicular directions, in the first and the third quarters, we use the incidence angles (seefigure 7(a))

θ j = ( j − 1)π

2Nd,

θ2Nd+ j = π + ( j − 1)π

2Nd, j = 1, . . . , Nd,

and in the second and the fourth quarters, we make use of the incidence angles

θNd+ j = π

2+ π

4Nd+ ( j − 1)

π

2Nd,

θ3Nd+ j = 3π

2+ π

4Nd+ ( j − 1)

π

2Nd, j = 1, . . . , Nd .

The observation directions are taken to be the same as the incidence one. In this setup, wehave |x · θ | � sin( π

4Nd) for all x, θ ∈ {θ j, j = 1, . . . , 4Nd}.

23


(a) (b)

(c) (d)

Figure 8. Multiple (Foldy) PP and SP scatterings, respectively, from left to right with 2% noise,gi = 1 and κ = π for three scatterers.

(2) It is better to avoid parallel or anti-parallel directions in the case of P incident wavesand the S-part of far-field patterns (PS case) or S incident waves and the P part of far-fieldpatterns (SP case) as they do not provide any useful information due to no scattered far fieldin these directions. To avoid this, we choose the incident and observation angles as follows(figure 7(b)):

θ j = ( j − 1)π

2Nd, j = 1, . . . , 4Nd,

x j = θ j + π

4Nd, j = 1, . . . , 4Nd .

With this choice, the minimum angle between the incidence and the observation angles is π4Nd

.To show the differences between the Born approximation and the Foldy model, we used

different scattering strengths, noise levels and distance between the scatterers. We observedthe similar kind of variations between the Born approximation and the Foldy model which arementioned in the case of acoustic scattering in section 2.3.4. It is important to mention that,by converting the vector far-field patterns to the scalar ones as in equation (3.25), the noisein the measured far-field patterns is amplified in the modified multi-static response matrixresulting worse results than the acoustic case. In the following examples, the parameters arechosen as λ = 1 and μ = 1 resulting in κp = κ/

√3 and κs = κ . Let us consider the points

y2, y3, y4, y5, y6, y7, y8, y13 and y14 which we mentioned in section 2.3.4. We have chosenNd = 4, which gives us the total number of incident directions as 4Nd = 16.

Figures 8 and 10 are related to three scatterers located at the points y2, y5 and y13. Infigures 8 and 10, each scatterer is of scattering strength 1 for the wavenumber κ = π with 2%random noise in the measured far-field patterns. Figure 9 is related to six scatterers located

24


(a) (b)

(c) (d)

Figure 9. Multiple (Foldy) PP and SP scatterings, respectively, from left to right with 2% randomnoise, gi = 1 and κ = 1.5π for six scatterers.

at the points y2, y5, y6, y7, y8 and y13 of each having scattering strength 1 for the wavenumberκ = 1.5π with 2% random noise in the measured far-field pattern.

Figures 8 and 9 shows the results for PP and SP far-field patterns in the case of the Foldymodel. Figure 8 shows good reconstruction for all the scatterers in both PP and SP cases eventhough in the SP case the peaks are sharper at locations y2, y5 and y13. Here, we can observethat the reconstruction looks better in the SP case than in the PP case. This is more clear infigure 9 with a greater number of scatterers.

Figure 10 shows the results for PP far-field patterns in the cases of the Born approximationand the Foldy model. In this case, 2% random noise in the measured far-field pattern is used.Here, the scatterers satisfy the diagonally dominant condition (3.61) and the reconstruction isgood and similar for both the Born approximation and the Foldy model.

Figures 11–14 are related to the three scatterers located at the points y2, y3 and y14.Figure 11 shows the pseudo-graphs for PP scattering of the Born approximation and the Foldymodel for the wavenumber κ = 1.5π with 5% random noise in the measured far-field pattern.Here, scatterers violates the diagonally dominant condition (3.61) and we can see the effect ofmultiple scattering in the case of Foldy.

Figures 12 and 13 show the reconstruction results for SP scattering of the Bornapproximation and the Foldy method for the wavenumber κ = π with 2% and 10% randomnoises, respectively, in the measured far-field pattern. In both figures, we can see the effect ofmultiple scattering.

In all these figures, we showed the differences between the Born approximation andFoldy’s method. Figure 10 is the one for which scatterers satisfy the diagonally dominantcondition (3.61) and in the rest of the figures, this condition is violated by the scatterers. In

25


(a) (b)

(c) (d)

Figure 10. Born- and Foldy-based PP scatterings, respectively, from left to right with 2% randomnoise, gi = 1 and κ = π for three scatterers.

(a) (b)

(c) (d)

Figure 11. Born- and Foldy-based PP scatterings, respectively, from left to right with 5% randomnoise, gi = 10 and κ = 1.5π for three scatterers.

all the cases, we study the effect of noise level, the relative distance between the obstaclescompared to their scattering strengths. It is clear that the effect of multiple scattering ismore in the reconstruction related to the Foldy model (multiple scattering) than in the

26


(a) (b)

(c) (d)

Figure 12. Born- and Foldy-based SP scatterings, respectively, from left to right with 2% randomnoise, gi = 10 and κ = π for three scatterers.

Born approximation (weak scattering). Similarly, we see that S incident waves give betterreconstruction than P incident waves since the former have shorter wavelengths than the latter,see (3.4), and note that κs > κp.

To finish this section, let us mention that the reconstruction depends on the choice of thesignal and noise subspaces of the multi-scale response matrix. For small measurement noise,these two subspaces are easy to choose due to the clear cut in the distribution of the singularvalues of the multi-scale response matrix. However, for large noise, the distribution of thesingular values are smooth and it becomes more difficult to separate the singular values of thenoise and signal subspaces, for example, see figure 14 for the SP case with three scatterers.We can observe this more clearly in elastic scattering than in acoustic scattering.

3.6. The inverse problems for the intermediate levels of scattering

We recall, see section 3.3, that the scattered field in the kth level scattering is calculated by

usk(x) =

M∑m=1

gmG(x, ym)um,k, (3.63)

with um,k defined recursively by

um,k+1 := ui(ym) +M∑

j=1j �=m

gjG(ym, y j)u j,k for m = 1, . . . , M. (3.64)

27


(a) (b)

(c) (d)

Figure 13. Weak and multiple SP scatterings, respectively, from left to right with 10% noise,gi = 10 and κ = π for three scatterers.

(a) (b)

Figure 14. Singular values in the Foldy-based SP scattering in the presence of three scatterers with2% and 5% random noises, respectively, from left to right (gi = 10 and κ = π ).

Also recall that we can summarize the different models by the formula

uI,k =k∑

l=0

(−M)luI for k = 0, 1, . . . , (3.65)

where uI,k, uI and M are as mentioned earlier in section 3.3. Then, by a similar approach as insection 3.5.1, we obtain the factorization of the response matrix in the kth level scattering as

Frt = Ht∗T Hr, (3.66)

28


(a) (b)

Figure 15. PP scattering for three scatterers with 5% noise, gi = 10 and κ = 1.5π : (a) third levelscattering and (b) eighth level scattering.

where T = g∑k

l=0(−M)l , r (resp. t) is either p or s or sh or sv based on the type of the far-fieldpattern (resp. incident wave). Here, g, H p, Hs, Hsh

and Hsv

are defined as in section 3.5.1.To apply the MUSIC algorithm, the invertibility of the matrix T is needed and the norm ofM less than half is the sufficient condition for that in every level of scattering. In this case,the reconstruction looks quite similar in all the levels of scattering when the scatterers arefar enough from each other. However, we can observe the similar kind of differences whichwe mentioned between weak (Born) and multiple (Foldy) scatterings, as the level k of thescattering increases, with respect to the condition (3.61). We can observe this in figure 15, forthe same data as in figure 11.

As discussed in section 3.5.3, we can recover the matrix T ∈ CdM×dM as T =

IHt HtFrt Hr∗IHr , where Fr

t is the given far-field pattern in the kth level scattering. By comparingthis evaluated T with its explicit form, g

∑kl=0(−M)l , in the kth level scattering, we can recover

the scattering strengths gm. In the Foldy model, it is clear as mentioned in section 3.5.3 for Mscatterers, where as in the Born model T = g and so the diagonal entries of T produces thecorresponding scattering strengths of the scatterers. In the case k = 1, we have T = g − gM.As we know that g is a diagonal matrix and the diagonal blocks of the size d × d of M arezero, the diagonal entries of T are equal to the scattering strengths of the M scatterers. But forintermediate level scattering k > 1, it is difficult to recover the scattering strengths due to thecomplicated structure of the matrices (−M)l , for l = 2, . . . , and hence of T . For this reason,as in the acoustic case, we restrict ourselves to the special case of two point-like obstaclesy1, y2 with the corresponding scattering strengths g1 and g2. In this case using the reciprocityrelation of the fundamental matrix G(x, y), i.e. G(x, y) = [G(y, x)], we have the explicitform of (−M)l for l = 0, 1, 2, . . . as follows:

(−M)l =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

[g

l21 g

l22 Gl(y1, y2) 0

0 gl21 g

l22 Gl(y1, y2)

], l ∈ 2N ∪ {0},

[0 g

l−12

1 gl+1

22 Gl(y1, y2)

gl+1

21 g

l−12

2 Gl(y1, y2) 0

], l ∈ 2N − 1.

Here, 0 is the zero matrix of order d. The matrix (−M)l is either diagonal or anti-diagonalby blocks of the size d × d. This structure is no longer valid for the case of more thantwo scatterers. From this structure, we obtain the explicit form of T = g

∑kl=0(−M)l in the

29


kth-order scattering as follows:

T =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[g1I 00 g2I

], k = 0,

⎡⎢⎢⎢⎣

g1

k2∑

l=0gl

1gl2G2l(y1, y2)

k2∑

l=1gl

1gl2G2l−1(y1, y2)

k2∑

l=1gl

1gl2G2l−1(y1, y2) g2

k2∑

l=0gl

1gl2G2l(y1, y2)

⎤⎥⎥⎥⎦ , k ∈ 2N,

⎡⎢⎢⎢⎣

g1

k−12∑

l=0gl

1gl2G2l(y1, y2)

k−12∑

l=0gl+1

1 gl+12 G2l+1(y1, y2)

k−12∑

l=1gl+1

1 gl+12 G2l+1(y1, y2) g2

k−12∑

l=0gl

1gl2G2l(y1, y2)

⎤⎥⎥⎥⎦ , k ∈ 2N − 1,

⎡⎢⎢⎣

1

g1I −G(y1, y2)

−G(y1, y2)1

g2I

⎤⎥⎥⎦

−1

, k = ∞.

From the above explicit form of T , we can deduce the scattering strengths g1 and g2 byfollowing the similar steps that we mentioned in section 2.4 of the acoustic scattering.

4. Conclusion

We used the Foldy method to model point-like scatterers and we defined the intermediatelevels of scattering between Born and Foldy. Using MUSIC-type algorithms, we can locate thescatterers and then recover the scattering strengths from far fields corresponding to incidentplane waves. We have shown that the accuracy of reconstruction is proportional to the distancebetween the scatterers but inversely proportional to the wavelength, the noise in measuredfar-field patterns, the scattering strengths appearing and to the number of point-like scatterers.In the elastic wave scattering, S incident waves provide more accurate reconstruction of thelocations of the scatterers compared to P incident waves. This is true for Born, Foldy and forany intermediate level of scattering. In particular, the larger the Lame parameter λ, the betterthe reconstruction with the S incident waves compared to the P incident waves. We have alsoproved that one type of incident wave (P or S) and one part of the far field (P or S) is sufficientto locate the point-like scatterers. In addition, one part of the S wave (SH wave or SV wave)is sufficient in the three-dimensional case.

Acknowledgments

DPC was supported by the Austrian Science Fund (FWF) P22341-N18. MS was partiallysupported by the Austrian Science Fund (FWF) P22341-N18.

Appendix. Proofs of theorems 3.1 and 3.2

We prove the results in the 3D case, i.e. theorem 3.2. The same proof can be applied in the 2Dcase as well.

30


A.1. Part 1: the H p case

First we note that φjz,p ∈ R(H p∗) ∀ j = 1, 2, 3 if z ∈ {y1, . . . , yM} because φ

jym,p, m =

1, . . . , M, j = 1, 2, 3, are the columns of the matrix H p∗ ∈ CN×3M .

We show that there exists N0 ∈ N such that for every point z ∈ K\{y1, y2, . . . , yM},the elements of {φ1

y1,p, φ2y1,p, φ

3y1,p, φ

1y2,p, φ

2y2,p, φ

3y2,p, ..., φ1

yM ,p, φ2yM ,p, φ

3yM ,p, φ

jz,p} are linearly

independent for every j = 1, 2, 3 and for all N � N0. In particular, this would imply thatH p∗ has the maximal rank 3M and that φ

jz,p /∈ R(H p∗) for every j = 1, 2, 3 and for all

z ∈ K\{y1, y2, . . . , yM}. Let us deal with j = 1 first.Assume on the contrary that this is not the case. Then, there exist sequences Nl → ∞,

{λ(l)mt

} ⊂ CM for t = 1, 2, 3, {z(l)} ⊂ K\{y1, y2, . . . , yM} and {μ(l)} ⊂ C, such that

|μl| +

m=Mt=3∑m=1t=1

|λlmt

| = 1

and

μ(l) e−iκpz(l)·θs (θs · e1) +

m=Mt=3∑m=1t=1

λ(l)mt

e−iκpym·θs (θs · et ) = 0 ∀s = 1, 2, . . . , Nl . (A.1)

Since all the sequences are bounded, there exist converging subsequences z(l) → z ∈ K,{λ(l)

mt} → λ ∈ C

3M and μ(l) → μ ∈ C as l tends to infinity. We fix s ∈ N and let l tend toinfinity. Then,

|μ| +

m=Mt=3∑m=1t=1

|λmt | = 1 and μ e−iκpz·θs (θs · e1) +

m=Mt=3∑m=1t=1

λmt e−iκpym·θs (θs · et ) = 0. (A.2)

Since it holds for every s ∈ N we conclude from the assumption on the ‘richness’ of the set{θs : s ∈ N} that

μ e−iκpz·θ (θ · e1) +

m=Mt=3∑m=1t=1

λmt e−iκpym·θ (θ · et ) = 0 ∀θ ∈ S2.

The left-hand side is the far-field pattern of the function

x �−→ μ(∇z�κp (x, z) · e1) +

m=Mt=3∑m=1t=1

λmt (∇y�κp (x, ym) · et ).

31


Here, �κp is the fundamental solution of the Helmholtz equation with wavenumber κp.Therefore, by Rellich’s lemma and unique continuation,

μ(∇z�κp (x, z) · e1

) +

m=Mt=3∑m=1t=1

λmt

(∇y�κp (x, ym) · et) = 0 ∀x /∈ {z, y1, y2, . . . , ym}. (A.3)

Now, we distinguish between two cases.

(A) Let z /∈ {y1, y2, . . . , ym}. By letting x tends to z and then to ym for m = 1, . . . , M, weconclude that all the coefficients μ and λmt for t = 1, 2, 3 and m = 1, . . . , M have tovanish. Indeed,

(1) By letting x tends to z, ∇z�κp (x, z) · e1 �−→ ∞ and from (A.3), μ has to be zero.(2) We write

3∑t=1

λmt (∇y�κp (x, ym) · et ) = eiκp|ym−x|

4π |ym − x|2(

iκp − 1

|ym − x|)

(λm1 (ym1 − x1) + λm2 (ym2 − x2) + λm3 (ym3 − x3)).

By taking x = (ym1 , ym2 , x3) and x3 tending to ym3 , we observe that λm3 has to be zero.Similarly, by considering various directions of x, we deduce that λm1 and λm2 also vanish.This contradicts the first equation of (A.2).

(B) Let now z ∈ {y1, y2, . . . , ym}. Without loss of generality, we assume that z = y1. By thesame arguments as in part (A) we conclude that

μ + λ11 = 0, λ12 = 0, λ13 = 0 and λmt = 0

for m = 2, . . . , M and for t = 1, 2, 3. (A.4)

Now, we write (A.1) in the following form:[μ(l) + λ

(l)11

]e−iκpy1·θs (θs · e1) + μ(l)

[e−iκpz(l)·θs − e−iκpy1·θs

](θs · e1)

+3∑

p=2

λ(l)1p

e−iκpy1·θs (θs · ep) +

m=Mt=3∑m=2t=1

λ(l)mt

e−iκpym·θs (θs · et ) = 0 (A.5)

for all s = 1, 2, . . . , Nl . The quantity

ρl = |μ(l) + λ(l)11

| +3∑

p=2

|λ(l)1p

| +

m=Mt=3∑m=2t=1

|λ(l)mt

| + |z(l) − y1|

converges to zero as l tends to infinity. By Taylor’s formula, we have that

e−iκpz(l)·θs − e−iκpy1·θs = −iκpθs.(z(l) − y1) e−iκpy1·θs + O(|z(l) − y1|2) (A.6)

32


as l tends to infinity. Replacing (A.6) in (A.5) and dividing by ρl yields

3∑p=2

λ(l)1p

e−iκpy1·θs (θs · ep) +

m=Mt=3∑m=2t=1

λ(l)mt

e−iκpym·θs (θs · et )

+ [λ

(l)11

− iκpμ(l)(θs · a(l))

]e−iκpy1·θs (θs · e1) = O(|z(l) − y1|),

for all s = 1, 2, . . . , Nl , where

λ(l)11

= μ(l) + λ(l)11

ρl, λ

(l)1p

=λ

(l)1p

ρl, p = 2, 3, λ(l)

mt= λ(l)

mt

ρl,

m = 2, 3, . . . , M, t = 1, 2, 3, and a(l) = z(l) − y1

ρl.

All these sequences are bounded; hence, we can extract further subsequences λ(l)mt

→ λmt

for m = 1, 2, . . . , M, t = 1, 2, 3 and a(l) → a ∈ R3 as l tends to infinity. We have that

m=Mt=3∑m=1t=1

|λ(l)mt

| + |a| = 1 (A.7)

and

[λ11 − iκpμ(θs · a)] e−iκpy1·θs (θs · e1) +3∑

p=2

λ1p e−iκpy1·θs (θs · ep)

+

m=Mt=3∑m=2t=1

λmt e−iκpym·θs (θs · et ) = 0

for all s ∈ N. Again, by the assumption on the set θs : s ∈ N, we conclude that this equationholds for all θ ∈ S

2. The left-hand side is now the far-field pattern of the function

x �−→ λ11 (∇y�κp (x, y1) · e1) + μa · (∇y∇y�κp (x, y1)) · e1

+3∑

p=2

λ1p

(∇y�κp (x, y1) · ep) +

m=Mt=3∑m=2t=1

λmt

(∇y�κp (x, ym) · et).

So, by Rellich’s lemma and unique continuation again,

λ11 (∇y�κp (x, y1) · e1) + μa · (∇y∇y�κp (x, y1)) · e1

+3∑

p=2

λ1p (∇y�κp (x, y1) · ep) +

m=Mt=3∑m=2t=1

λmt (∇y�κp (x, ym) · et ) = 0, (A.8)

33


for all x /∈ {y1, . . . , yM}. By letting x tends to ym for m = 2, . . . , M, we conclude that allthe coefficients λmt for m = 2, . . . , M and for t = 1, 2, 3 have to vanish.

By letting x tends to y1, the most singular part of the term

λ11 (∇y�κp (x, y1) · e1) + μa · (∇y∇y�κp (x, y1)) · e1 +3∑

p=2

λ1p (∇y�κp (x, y1) · ep)

is μa · (∇y∇y�κp (x, y1)) · e1 and because of (A.8), μa should vanish. Then, coming to theremaining term by taking different directions of x as in part (A) and because of (A.8), wededuce that λ1t for t = 1, 2, 3 have to vanish. From (A.2) and (A.4), we obtain |μ| = 1/2,

and thus, a = 0.This finally contradicts (A.7).

We have performed the converse part for j = 1. In the same manner, we can show it forj = 2 and 3 as well. These arguments prove that H p∗ and so H p have maximal ranks.

Hence, φjz,p ∈ R(H p∗) iff z ∈ {y1, . . . , yM} for some j = 1, 2, 3.

A.2. Part 2: the Hs case

Following a similar way as part 1, we can prove that φjz,s ∈ R(Hs∗) iff z ∈ {y1, . . . , yM} for

some j = 1, 2, 3 by proving the maximal rank property of the matrix Hs∗ and so for Hs aswell. Due to some technical differences compared to the H p case, we provide here the detailsas well.

Indeed, we show that there exists N1 ∈ N such that for all points z ∈ K\{y1, y2, . . . , yM},the vectors of {φ1

y1,s, φ2y1,s, φ

3y1,s, φ

1y2,s, φ

2y2,s, φ

3y2,s, ..., φ

1yM ,s, φ

2yM ,s, φ

3yM ,z, φ

jz,s} are linearly

independent for every j and for all N � N1. In particular, this would imply that Hs∗ hasthe maximal rank 3M and that φ

jz,s /∈ R(Hs∗) for every j and for all z ∈ K\{y1, y2, . . . , yM}.

Let us deal with j = 1 first.Assume on the contrary that this is not the case. Then, there exist sequences Nl → ∞,

{λ(l)mt

} ⊂ CM for t = 1, 2, 3, {z(l)} ⊂ K\{y1, y2, . . . , yM} and {μ(l)} ⊂ C such that

|μl| +

m=Mt=3∑m=1t=1

|λlmt

| = 1

and

μ(l) e−iκsz(l)·θs (θ⊥s · e1) +

m=Mt=3∑m=1t=1

λ(l)mt

e−iκsym·θs (θ⊥s · et ) = 0 ∀s = 1, 2, . . . , Nl . (A.9)

Since all of the sequences are bounded, there exist converging subsequence z(l) → z ∈ K and{λ(l)

mt} → λ ∈ C

3M and μ(l) → μ ∈ C as l tends to infinity. We fix s ∈ N and let l tend toinfinity. Then,

|μ| +

m=Mt=3∑m=1t=1

|λmt | = 1 and μ e−iκsz·θs (θ⊥s · e1) +

m=Mt=3∑m=1t=1

λmt e−iκsym·θs (θ⊥s · et ) = 0. (A.10)

34


Since it holds for every s ∈ N, we conclude from the assumption on the ‘richness’ of the set{θs : s ∈ N} that

μ e−iκsz·θ (θ⊥ · e1) +

m=Mt=3∑m=1t=1

λmt e−iκsym·θ (θ⊥ · et ) = 0 ∀θ ∈ S2.

We know that, θ⊥ = αθ⊥h + βθ⊥v /|αθ⊥h + βθ⊥y | with α and β being arbitrary constants. Bytaking α = 1, β = 0, we have

Hs = Hshand θ⊥ = θ⊥h = 1

θ21 + θ2

2

(θ2

2 + θ21 θ3, θ1θ2(θ3 − 1),−θ1

(θ2

1 + θ22

)).

(If we take α = 0, β = 1, then θ⊥ = θ⊥v and it can be used to show the maximal rank propertyof Hs = Hsv

.)Now, the left-hand side of (A.10) is the far-field pattern of the function

x �−→ μ

(−iκs

∂2

∂z22

�κs (x, z) + ∂3

∂z21∂z3

�κs (x, z)

)

+m=M∑m=1

λm1

(−iκs

∂2

∂ym22

�κs (x, ym) + ∂3

∂ym21∂ym3

�κs (x, ym)

)

+m=M∑m=1

λm2

(∂3

∂ym1∂ym2∂ym3�κs (x, ym) + iκs

∂2

∂ym1∂ym2�κs (x, ym)

)

−m=M∑m=1

λm3

(∂3

∂ym31

�κs (x, ym) + ∂3

∂ym1∂ym22

�κs (x, ym)

).

Here, �κs is the fundamental solution of the Helmholtz equation with the wavenumber κs.Therefore, by Rellich’s lemma and unique continuation,

μ

(−iκs

∂2

∂z22

�κs (x, z) + ∂3

∂z21∂z3

�κs (x, z)

)

+m=M∑m=1

λm1

(−iκs

∂2

∂ym22

�κs (x, ym) + ∂3

∂ym21∂ym3

�κs (x, ym)

)

+m=M∑m=1

λm2

(∂3


∂2


)

−m=M∑m=1

λm3

(∂3

∂ym31

�κs (x, ym) + ∂3

∂ym1∂ym22

�κs (x, ym)

)= 0, (A.11)

for all x /∈ {z, y1, y2, . . . , ym}. Again, we distinguish between two cases.

(A) Let z /∈ {y1, y2, . . . , ym}. By letting x tend to z and to ym for m = 1, . . . , M, we concludethat all the coefficients μ and λmt for t = 1, 2, 3 and m = 1, . . . , M have to vanish. Indeed,

(1) Taking x = (z1, x2, z3) with x2 tending to z2, then due to the singularity of ∂2

∂z22�κs (x, z)

in (A.11), μ has to be zero.

35


(2) Consider the term

λm1

(−iκs

∂2

∂ym22

�κs (x, ym) + ∂3

∂ym21∂ym3

�κs (x, ym)

)

− λm3

(∂3

∂ym31

�κs (x, ym) + ∂3

∂ym1∂ym22

�κs (x, ym)

)

+ λm2

(∂3


∂2


).

Taking x = (x1, ym2, ym3) and tending x1 to ym1 , we observe that λm3 has to bezero due to the singularity of ∂3

∂ym31�κs (x, ym) in (A.11). Similarly, by considering the

other directions, we can show that λm1 and λm2 also vanish.

This contradicts the first equation of (A.10).

(B) Let now z ∈ {y1, y2, . . . , ym}. Without loss of generality, we also assume that z = y1.By the same arguments as part (A) we conclude thatμ + λ11 = 0, λ12 = 0, λ13 = 0 and λmt = 0 for m = 2, . . . , M and for t = 1, 2, 3.

(A.12)Now, we write (A.10) in the following form:[μ(l) + λ

(l)11

]e−iκsy1·θs

(θ⊥

s · e1) + μ(l)[e−iκsz(l)·θs − e−iκsy1·θs ]

(θ⊥

s · e1)

+3∑

p=2

λ(l)1p

e−iκsy1·θs(θ⊥

s · ep) +

m=Mt=3∑m=2t=1

λ(l)mt

e−iκsym·θs(θ⊥

s · et) = 0

(A.13)for all s = 1, 2, . . . , Nl . The quantity

ρl = ∣∣μ(l) + λ(l)11

∣∣ +3∑

p=2

|λ(l)1p

| +

m=Mt=3∑m=2t=1

∣∣λ(l)mt

∣∣ + |z(l) − y1|

converges to zero as l tends to infinity. By Taylor’s formula, we have that

e−iκsz(l)·θs − e−iκsy1·θs = −iκsθs.(z(l) − y1) e−iκsy1·θs + O(|z(l) − y1|2) (A.14)

as l tends to infinity. Replacing (A.14) in (A.13) and dividing by ρl yields

3∑p=2

λ(l)1p

e−iκsy1·θs(θ⊥

s · ep) +

m=Mt=3∑m=2t=1

λ(l)mt

e−iκsym·θs(θ⊥

s · et)

+ [λ

(l)11

− iκsμ(l)(θs · a(l))

]e−iκsy1·θs

(θ⊥

s · e1)

= O(|z(l) − y1|)for all s = 1, 2, . . . , Nl , where

λ(l)11

= μ(l) + λ(l)11

ρl, λ

(l)1p

=λ

(l)1p

ρl, p = 2, 3,

λ(l)mt

= λ(l)mt

ρl, m = 2, 3, . . . , M, t = 1, 2, 3 and a(l) = z(l) − y1

ρl.

36


All these sequences are bounded as well, i.e. we can extract further subsequencesλ(l)

mt→ λmt for m = 1, 2, . . . , M, t = 1, 2, 3, and a(l) → a ∈ R

3 as l tends to infinity.We have that

m=Mt=3∑m=1t=1

∣∣λ(l)mt

∣∣ + |a| = 1 (A.15)

and

[λ11 − iκsμ(θs · a)] e−iκsy1·θs(θ⊥

s · e1) +

3∑p=2

λ1p e−iκsy1·θs(θ⊥

s · ep)

+

m=Mt=3∑m=2t=1

λmt e−iκsym·θs(θ⊥

s · et) = 0,

for all s ∈ N. Again, by the assumption on the set θs : s ∈ N, we conclude that thisequation holds for all θ ∈ S

2. The left-hand side is now the far-field pattern of thefunction

x �−→ μa · ∇y1

(−iκs

∂2

∂y122

�κs (x, y1) + ∂3

∂y121∂y13

�κs (x, y1)

)

+m=M∑m=1

λm1

(−iκs

∂2

∂ym22

�κs (x, ym) + ∂3

∂ym21∂ym3

�κs (x, ym)

)

+m=M∑m=1

λm2

(∂3


∂2


)

−m=M∑m=1

λm3

(∂3

∂ym31

�κs (x, ym) + ∂3

∂ym1∂ym22

�κs (x, ym)

).

So, by Rellich’s lemma and unique continuation again,

μa · ∇y1

(−iκs

∂2

∂y122

�κs (x, y1) + ∂3

∂y121∂y13

�κs (x, y1)

)

+m=M∑m=1

λm1

(−iκs

∂2

∂ym22

�κs (x, ym) + ∂3

∂ym21∂ym3

�κs (x, ym)

)

+m=M∑m=1

λm2

(∂3


∂2


)

−m=M∑m=1

λm3

(∂3

∂ym31

�κs (x, ym) + ∂3

∂ym1∂ym22

�κs (x, ym)

)= 0, (A.16)

for all x /∈ {y1, . . . , yM}. By letting x tend to ym for m = 2, . . . , M, we conclude thatall the coefficients λmt for m = 2, . . . , M and for t = 1, 2, 3 have to vanish.

37


By letting x tend to y1, the most singular part of the term

μa · ∇y1

(−iκs

∂2

∂y122

�κs (x, y1) + ∂3

∂y121∂y13

�κs (x, y1)

)

+λ11

(−iκs

∂2

∂y122

�κs (x, y1) + ∂3

∂y121∂y13

�κs (x, y1)

)

+λ12

(∂3

∂y11∂y12∂y13�κs (x, y1) + iκs

∂2

∂y11∂y12�κs (x, y1)

)

−λ13

(∂3

∂y131

�κs (x, y1) + ∂3

∂y11∂y122

�κs (x, y1)

)

is μa ·∇y1

(−iκs

∂2

∂y122�κs (x, y1) + ∂3

∂y121∂y13

�κs (x, y1))

and because of (A.16), μa should vanish.Then, coming to the remaining term by taking different directions of x as in part (A) andbecause of (A.16), we deduce that λ1t for t = 1, 2, 3 have to vanish. Now, from (A.10)and (A.12), we obtain |μ| = 1/2 and thus a = 0.

This finally contradicts (A.15).We have performed the converse part for j = 1. In the same manner, we can show it

for j = 2 and 3 as well. From these arguments, we obtain the maximal rank property of thematrix Hs∗ and so for Hs.

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Inverse scattering by point-like scatterers in the Foldy regime