INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 20 (2004) S81–S98 PII: S0266-5611(04)79848-2

Nonlinear inversions of immersed objects usinglaboratory-controlled data

B Duchene, A Joisel and M Lambert

Departement de Recherche en Electromagnetisme, Laboratoire des Signaux et Systemes,CNRS-Supelec, 3 rue Joliot-Curie, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France

E-mail: Bernard.Duchene@lss.supelec.fr, Alain.Joisel@lss.supelec.fr andMarc.Lambert@lss.supelec.fr

Received 27 April 2004, in final form 14 October 2004Published 8 November 2004Online at stacks.iop.org/IP/20/S81doi:10.1088/0266-5611/20/6/S06

AbstractThe paper is focused on the characterization of objects immersed in waterusing laboratory-controlled data obtained in the microwave frequency range.Experiments performed at the laboratory represent, at a reduced scale, theelectromagnetic characterization of objects buried at a shallow depth in thesea, with the objects and the emitting and receiving antennas being immersed.Characterization is taken here as an inverse scattering problem whose data,whilst limited in aspect, consist of the values of the time-harmonic scatteredelectric fields measured at several discrete frequencies on a single line of a fewreceivers. The modelling of the wave–object interaction is performed througha domain integral representation of the fields in a two-dimensional transverse-magnetic configuration. The inverse scattering problem is solved by meansof two iterative algorithms tailored for homogeneous objects: the so-calledlevel-set method and a binary specialized contrast source inversion method.Emphasis is put both on the experimental features and their modelling, and theresults obtained for different types of objects are presented.

(Some figures in this article are in colour only in the electronic version.)

1. Introduction

We consider herein an electromagnetic inverse obstacle scattering problem where the goalis to characterize unknown targets illuminated by a known time-harmonic interrogating (orincident) wave in the microwave frequency range (200–900 MHz), from measurements of thescattered electric fields that result from the wave–object interaction in a frequency-diversemultistatic configuration [1–3].

The data of the inverse problem come from a laboratory-controlled experiment [2] wherethe object and the emitting and receiving antennas are immersed in a water tank, the formertwo being fixed and the latter being moved on a limited probing line parallel to the air–waterinterface.

0266-5611/04/060081+18$30.00 © 2004 IOP Publishing Ltd Printed in the UK S81

S82 B Duchene et al

The experiment represents to some extent the low-frequency electromagnetic charact-erization of objects buried at a shallow depth in the sea, at a reduced scale. As a matter of fact,while leaving aside the non-stationary nature of the sea with respect to temperature and salinity(which cannot be easily reproduced in a laboratory-controlled experiment), considering sucha problem at a reduced scale is not straightforward as the ratio wavelength/penetration depthcannot be kept constant while passing from the ELF band to the microwave frequency range,even by varying the salinity of water in the experiment. Nevertheless, water is a dispersive andhighly attenuating medium whose properties depend on the experimental conditions, whichleads to some difficulties in the experimentation as well as in the inversion. Furthermore,as far as the inversion is concerned, the difficulties are enhanced by the aspect-limited dataconfiguration we are faced with, which is kept from the original problem.

Modelling of the wave–object interaction is based on domain integral representations ofthe fields that are suitable for studying diffraction problems in the resonance domain consideredherein. In these representations, the fields appear to be radiated by Huygens-type sources (orcontrast sources) induced within the target by the incident wave, which depend on a contrastfunction representative of the target physical parameters. The inverse problem then consistsin retrieving this contrast function from the measured scattered fields through the inversion oftwo coupled integral equations. This problem is nonlinear and, as is well known, ill-posed.This ill-posedness is inherent in inverse scattering problems; however, it is enhanced here, bythe reduced amount of available information. Thus a strong regularization is required. Thelatter consists in introducing the a priori information that the unknown object is homogeneousin the inversion algorithms, so that, after a suitable normalization, we look for a distributionin space of a binary-valued contrast (1 within the object, 0 elsewhere), i.e. the characteristicfunction of the support of the contrast.

As for the inverse problem, the reader is referred to [4] for an overview of the various classesof inversion methods; especially noteworthy are the investigations of inversion techniquesbased on fast and efficient algorithms [5] or based on gradient methods [6] or dedicated tothe characterization of objects embedded in stratified media [7]. As far as the determinationof the object shape is concerned, let us quote some recent and original contributions [8–10],particularly [11, 12] which concern objects embedded in layered media (see [13] for a reviewof the various methods).

Herein, inversion is carried out by means of two methods tailored to the above-mentionedcase of binary-valued contrast functions, both of them applying to dielectric or metallic(asymptotically at least for the second one) possibly multiply connected objects located infree or layered spaces. They belong to the class of non-linearized iterative techniques.

• The first method is a binary specialized variant of the so-called contrast source inversionmethod introduced by van den Berg and Kleinman [14]. The latter belongs to the class ofalternating direction implicit methods of Kohn and McKenney [15], where two unknownsare iteratively sought for by alternately updating each of the unknowns at each iteration.Hence, the nonlinear nature of the inverse problem is accounted for by looking alternatelyfor the unknown contrast and for the unknown contrast sources. This is done with a gradient-type technique by minimizing a two-term cost functional accounting for the residuals ofboth of the coupled integral equations introduced earlier. These residuals express thediscrepancies between the scattered and incident field data and the fields computed by meansof the iterative solution. The binary nature of the contrast is introduced here as in [16], bylooking for a contrast function that varies continuously between 0 and 1, i.e., in a way thatpreserves the existence of functional derivatives with respect to the contrast needed in thegradient-based inversion algorithm. This idea was first used in the local-shape-function

Nonlinear inversions of immersed objects S83

method developed by Chew and Otto [17] for microwave imaging of strong scatterers(metallic targets) and has also been applied to a controlled laboratory experiment [18]. Letus notice, however, that other methods could be used, for example as in [19, 20], where thisa priori information is introduced as an additive penalty term in the cost functional to beminimized, or in [21], where it is introduced as a multiplicative constraint.

• The second inversion method is the so-called controlled evolution of a level set, pioneeredby Litman et al [22]. It combines two concepts: (i) a level set representation of the shapes,introduced by Santosa [23] in the domain of inverse problems: the homogeneous object isdescribed by its contour taken as the level 0 of the level set whose evolution as a functionof a pseudo-time t is governed by a Hamilton–Jacobi-type equation and involves a velocityfield, and (ii) the speed or velocity method developed by Sokolowski and Zolesio [24] for theshape optimal design and control: it aims at obtaining an optimal domain minimizing a costfunctional (taken here as the error on the observation equation) through an iterative sequenceof deformation of its contour defined from the velocity field.

Let us note that we do not intend, here, to dwell upon the details of the inversion algorithms,as the latter can be found elsewhere in many referenced publications, or to do an exhaustiveexploration of their performances; we prefer to report on current research work carried out at theDepartement de Recherche en Electromagnetisme, laying particular emphasis on experimentalfeatures and their modelling. Indeed, in addition to the inherent difficulties of the inverseproblem, we are faced here with that of collecting experimental data. A careful procedure isneeded in order to overcome the latter problem, which requires, at each stage, a comparisonbetween the measured data and those provided by the model. This is illustrated in the following,and a few examples representative of the behaviour of the inversion algorithms are given.

2. The experimental setup and its modelling

2.1. The geometry

The experimental setup and its modelling are displayed in figure 1. It consists in a watertank whose dimensions (1.8 m × 1.2 m × 1 m) are large enough (compared to the penetrationdepths) to minimize the contribution of the waves reflected on the vertical walls. The tankis filled with tap water. The configuration is modelled as a stratified medium made of threehomogeneous domains separated by horizontal planar interfaces: D1 (the air), D2 (the water)and D3 (the bottom of the tank made of PVC). The three domains are assumed to be unboundedin the horizontal (y, z) plane and, furthermore, D1 and D3 are considered as semi-infinite inthe x direction, delimiting a water layer D2 of depth d (d ≈ 60 cm).

The unknown object is immersed in the tank at a depth of about 25 cm under the watersurface; it is considered as a z-oriented homogeneous infinite cylinder of arbitrary cross-section in the (x, y) plane with contour , and it is supposed to be contained in a test domain D suchthat ⊂ D ⊂ D2 − S − rs.

Two antennas, a source and a receiver, are also immersed in the tank at a depth of about10 cm. They are linked, respectively, to ports 1 and 2 of a network analyser (HP8753)which supplies the transmitter with a power of 10 dBm at a frequency f lying in the range200 MHz < f < 900 MHz, and measures the complex transmission coefficient S21 betweenthe two antennas. The source, fixed and located at rs (rs ∈ D2), radiates a time-harmonicelectromagnetic wave at angular frequency ω (ω = 2πf ) with a time dependence exp(−iωt),and the receiver measures the fields along a probing line S of length L = 39.5 cm parallel tothe y axis and approximately centred above the object, with 80 measurements being carried

S84 B Duchene et al

D3 (PVC)

D2 (water)

sourced

S

Γx

y

D1 (air)

receiver

ΩD

Figure 1. The experimental setup (left) and its geometry (right).

out with a scanning step of 0.5 cm. The scattered field Udif that results from the wave–objectinteraction is obtained through a two-step measurement process (the incident field Uinc in theabsence of the object and the total field U in its presence) by the difference Udif = U − Uinc.

2.2. The propagation medium

All the media are assumed to be homogeneous, stationary, linear, isotropic and non-magnetic,and they are characterized by their propagation constants km (m = 1, 2, 3 or ) so thatk2m = ω2ε0εrmµ0, where ε0 (ε0 = 8.854 × 10−12 F m−1) and µ0 (µ0 = 4π × 10−7 H m−1) are,

respectively, the dielectric permittivity and the magnetic permeability of free space, and εrm isthe relative dielectric permittivity of the medium Dm which is complex when the medium islossy. The media D1 (air) and D3 (PVC) are assumed to be non-dispersive and lossless, andtheir relative permittivities are εr1 = 1 and εr3 = 3, respectively. The object can be dielectric ormetallic, and its permittivity reads εr = ε′

r + iσ/ωε0, where ε′r and σ are the real parts

of the relative permittivity and conductivity of the object, respectively.The medium D2 (tap water) is dispersive and lossy, and its properties depend upon

the experimental conditions. In the microwave frequency range, the dielectric propertiesof pure water can be described as a function of the frequency by means of a model basedupon the Debye relaxation theory [25] and the complex relative permittivity is then givenby εr(ω) = ε∞ + (εs − ε∞) / (1 − iωτ), where τ is the relaxation time and εs and ε∞ arethe dielectric constants at the static and high-frequency limits, respectively. As tap water isconsidered, a term must be added in order to account for losses due to the ionic conductivityof the dissolved salt (σ2); the complex relative permittivity then reads:

εr2(ω) = ε′r2(ω) + iε′′

r2(ω)

ε′r2(ω) = ε∞ + εs − ε∞

1 + ω2τ2, ε′′

r2(ω) = σ2

ωε0+ (εs − ε∞)ωτ

1 + ω2τ2.

(1)

Note that, in the above expression, ε∞ does not depend upon the salinity of water anddepends only slightly upon the temperature, whereas εs, τ and σ strongly depend upon boththe temperature and the salinity. Empirical polynomial approximations of these variables canbe found in [26]. The results obtained by means of these approximations, and by assumingthat the salinity of tap water is about 0.38 g l−1 of NaCl salt, are in good agreement with thoseobtained experimentally by means of a reflectometric technique [27] as shown in figure 2. Withsuch a relative permittivity and a frequency lying in the above-mentioned range, wavelengthλ and penetration depth δ in water are, roughly speaking, such that 16 cm > λ > 4 cm and13.2 cm > δ > 3.3 cm, respectively.

Nonlinear inversions of immersed objects S85

frequency (MHz) frequency (MHz)

ε ’ ε ”

78.5

79

79.5

80

80.5

200 400 600 8004

5

6

7

200 400 600 800

computed

measured

Figure 2. Real (left) and imaginary (right) parts of the relative dielectric permittivity of tap water at22.2 C. The measured permittivity (- - - -) is compared to the permittivity computed with a salinityof 0.38 g l−1 (——).

0

5

10

15

200 400 600 800

frequency (MHz)

SWR

A2: l = 3.1 cm, α = 18˚

A1: l = 2.8 cm, α = 60˚

balun

salinesolution

towardsnetworkanalyzer

αl

Figure 3. A biconical antenna with the associated balun and complementary saline solution device(left) and the standing wave ratio of the two antennas considered herein (right).

2.3. The antennas

Due to their great aptitude to radiate in water in a broad frequency range with a lowcross-polarization and their easy-to-use capabilities, biconical antennas (figure 3, left) havebeen chosen. They have been made at the laboratory [2, 3] and designed so that their impedancedoes not vary too much over their operating frequency range and to avoid secondary lobes in theradiation pattern at high frequencies. The antennas are associated with baluns which ensuretheir symmetry and their matching to the ports of the network analyser. A complementarydevice, consisting of a saline solution confined around the feeding cables, allows reduction ofthe parasitic residual currents that flow along the latter.

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Two types of antennas are used. The first one, denoted A1, is made of cones of lengthl = 2.8 cm and aperture angle α = 60; it can operate over the whole 200–900 MHz frequencyrange but has the drawback of being large. The second type, denoted A2 and made of conesof length l = 3.1 cm and aperture angle α = 18, does not have this drawback; however, itsoperating frequency range is much smaller than the former as shown in figure 3 which displaysthe standing wave ratio (SWR) of both types of antennas as a function of the frequency.

2.4. The incident field

The configuration considered here, i.e. a two-dimensional (2D) structure probed by meansof 3D sources and receivers, is referred to in the literature as a 2.5D configuration. As faras the forward problem is concerned, a lot of work has been published (e.g. [28] for ground-penetrating radar applications, or [29] for transient marine applications) concerning such aconfiguration. However, much less has been published concerning the inverse problem. In [30,31], for example, a 2.5D inverse problem is addressed in the framework of the Born linearizingassumption; another choice is carried out herein, the nonlinear inverse problem being tackledin a simplified configuration: the source is modelled in a 3D framework, whereas the rest ofthe problem is modelled in a 2D-transverse magnetic (TM) configuration (the electric fieldis polarized in the z direction). Indeed, addressing the full 2.5D problem in the inversionwould lead to a rather sophisticated model and a large overload of computations, unwarrantedby the expected improvement of the inversion results. The same considerations hold forprobe compensation techniques, such as SICS (sensor interaction compensation scheme [32]),although they would allow us to take into account the finite 3D nature of the probes and thestrong coupling between the latter and the object evidenced by section 2.6.

It is well known that biconical antennas have a dipole-like behaviour [33]. Thesource is then considered as a Hertzian dipole orientated in the z direction and located atrs (rs = xs = x′, ys = 0, zs = 0).

The incident field observed at r (r = x, y, z = 0) is then given by the z component of theelectric field (Ez) radiated by the dipole in the plane z = 0 (Uinc(r) = Ez(r, rs), r, rs ∈ D2).

The latter is obtained in the spectral domain from the x components of the electric andmagnetic fields Ex and Hx [34]:

Ez(r, r′) = ωµI

4π

∫ +∞

0ky

[−ATE

β2J0(kyy) + β2

kyy

[ATE

β22

− ATM

k22

]J1(kyy)

]dky, (2)

where β2 =√

k22 − k2

y ((β2) 0), J0 and J1 represent the Bessel functions of order 0 and

1, respectively, I is the complex amplitude of the dipole, the superscripts TE and TM standrespectively for electric and magnetic fields transverse to the x axis, and the terms AQ (Q = TEor TM) account for the fact that the field is expressed as the sum of a direct wave and tworeflected waves resulting from reflections on the upper and lower interfaces, and propagatingtowards positive x and negative x, respectively:

ATE = eiβ2|x−x′| + BTE eiβ2x + DTE e−iβ2x,

ATM = sign(x − x′)eiβ2|x−x′| + BTMeiβ2x − DTMe−iβ2x,

with BQ = RQ21[eiβ2|x′| + eiβ2dR

Q23 eiβ2|d−x′|]MQ

DQ = eiβ2dRQ23[eiβ2|d−x′| + eiβ2dR

Q21 eiβ2|x′|]MQ

MQ = [1 − RQ23R

Q21ei2β2d]

−1, Q = TE or TM.

(3)

Nonlinear inversions of immersed objects S87

The coefficients RQ2m (m = 1, 3) are the Fresnel reflection coefficients at the upper (m = 1)

and lower (m = 3) interfaces:

RTM2m = k2

mβ2 − k22βm

k2mβ2 + k2

2βm

, RTE2m = β2 − βm

β2 + βm

, βm =√

k2m − k2

y ((βm) 0).

(4)

The amplitude of the dipole I is determined, at each frequency, by calibrating theexperiment to account for some unknown experimental parameters; its modulus is such thatthe squared norms of the computed and measured incident fields are equal:∫

S|Uinc(r)|2 dr =

∫S

|Ez(r, rs)|2 dr, (5)

and its phase is such that the phases of these fields are equal at the beginning of the probingline.

Once calibration is completed, equation (2) yields the incident field anywhere in D2, andparticularly in D as required for the resolution of the direct and inverse problems, measurementof the field over such a domain being not practically possible in the configuration consideredhere.

2.5. The direct problem

The incident field and the object being known, we now focus on the modelling of theirinteraction. The latter is based upon domain integral representations obtained by applyingGreen’s theorem to the Helmholtz wave equations satisfied by the fields, and by accountingfor continuity and radiation conditions. This leads to two coupled contrast–source integralequations. The first-kind Fredholm integral equation, the so-called observation equation (6),relates the scattered field Udif (Udif = U−Uinc) to the total field U and to the contrast functionχ (χ(r) = k2(r) − k2

2) defined in D and null outside :

Udif (r) =∫

DG22(r, r′)χ(r′)U(r′)dr′, r ∈ S, r′ ∈ D, (6)

where G22(r, r′) is the 2D-TM scalar Green’s function of the stratified medium whichrepresents the field radiated by a line source located at r′ and observed at r in the absence ofobjects, both being located in D2. As in the Hertzian dipole case, G22 can be expressed in thespectral domain as the sum of three terms, i.e., one direct and two reflected waves; however, forcomputational ease, the contribution of the direct wave, with singular behaviour, is processedseparately from that of the reflected waves with regular behaviour [16, 35].

The Lippmann–Schwinger equation, denoted as the coupling equation (7), expresses thefield U in D:

U(r) = Uinc(r) +∫

DG22(r, r′)χ(r′)U(r′)dr′, r ∈ D, r′ ∈ D. (7)

Uinc(r) and χ(r) being known for r ∈ D, the direct problem then consists first in solvingequation (7) for U(r) (r ∈ D), then solving equation (6) for Udif (r) (r ∈ S). In numericalpractice, discrete versions of the above equations are considered. They are obtained by applyingthe method of moments with pulse-basis and point matching, which results in partitioning thetest domain D into elementary pixels, small enough in order to consider the fields and thecontrast as constant over each of them.

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-200

-100

0

100

200

-0.2 -0.1 0 0.1 0.2y (m) y (m)

mod

ulus

(dB

)

phas

e (˚

)

-55

-50

-45

-40

-35

-30

-25

-0.2 -0.1 0 0.1 0.2

Figure 4. Modulus (left) and phase (right) of the measured (- - -) and computed (—) incident fieldson the probing line at a frequency of 434 MHz.

2.6. Validation of the model

As already indicated, one of the key points of the solution of the direct and inverse problemsis the computation of the incident field on the probing line S and on the test domain D. As faras the parameters of the experimentation are accurately known, the dipolar model yields goodresults for both types of antennas at any frequency in their operating range. Figure 4 displays acomparison between the computed and measured incident fields for the A1-type antenna afterthe calibration procedure; the latter are in good agreement except at the minimum of the fieldwhere a few-dB discrepancy is observed. Let us note that the modelling is not very sensitiveto an error in the water depth d (generally much greater than the penetration depth δ), but itis very sensitive to an error in the depths of the source and the receiver, both of them beinglocated near the air–water interface.

As for the scattered field, we are faced with two major difficulties:

• the location and geometry of the receiving antenna, the latter not being accounted for inthe modelling. Figure 5 displays a comparison between the scattered field measured on theprobing line with both types of antennas described in section 2.3 and the field computed asindicated in section 2.5. Due to its large size, theA1-type receiving antenna can produce largeperturbations of the incident field on the object when it is located between the source and theobject, which, in turn, produces large perturbations of the scattered field. The magnitude ofthese perturbations depends upon the frequency and the object, as they result from interactionsbetween the latter and the probes. This problem has been overcome by using the smallerA2-type antenna as evidenced in figure 5, and, in spite of a small discrepancy (≈2 dB), theresults obtained are satisfactory, considering the low level of the scattered field.

• the respective positions of the source, the obstacle and the probing line. Hence, figure 6displays grey level maps of the incident field over a domain that includes the source, theprobing line and the object in a typical case at two different frequencies. It appears that,depending upon the frequency (cf figure 6: map at 250 MHz), the incident field on the probingline can be much higher than that impinging on the object, and in that case the scattered field

Nonlinear inversions of immersed objects S89

-200

-100

0

100

200

-0.2 -0.1 0 0.1 0.2y (m) y (m)

mod

ulus

(dB

)ph

ase

(˚)

-60

-55

-50

-45

-40

-0.2 -0.1 0 0.1 0.2-60

-55

-50

-45

-40

-0.2 -0.1 0 0.1 0.2

-200

-100

0

100

200

-0.2 -0.1 0 0.1 0.2

Figure 5. Modulus (top) and phase (bottom) of the computed (—) scattered field compared to thefields measured (- - -) with A1-type (left) and A2-type (right) antennas in the case of a 5 cm-sidedsquare metallic object at a frequency of 434 MHz.

Figure 6. Grey level maps (scale: 0 to −80 dB) of the incident field at 250 (left) and 840 MHz(right) over a domain (24 cm × 58 cm) including the source, the probing line and the object. Thelocation of the object and of the probing line, in a typical case, are indicated by white lines.

on the probing line is much lower than the former. As the scattered field is obtained in atwo-step procedure from the difference between two quantities much higher than itself, thissituation can lead to large errors in its determination, its amplitude being at the limit of themeasurement capabilities of the experimental apparatus, i.e. at the level of the noise.

S90 B Duchene et al

3. The inverse problem

The inverse problem consists in retrieving the values of the contrast function χ(r) (r ∈ D),at the centres of the pixels partitioning the test domain D, from the measured scattered fieldsUdif (r) (r ∈ S), the incident fields being known on D. Let us note that, as the total fields U inthe test domain are also unknown, this must be done through the inversion of the two coupledintegral equations (6) and (7), resulting in a nonlinear problem.

3.1. The binary contrast source inversion method

The contrast source inversion method (CSI) is based upon rewriting the observation andcoupling integral equations in terms of the contrast sources W induced within the target by theincident wave: W(r) = χ(r)U(r), r ∈ D.

Hence, in an operator notation, the observation equation can be rewritten as:

Udif (r) = (GSW)(r), r ∈ S, (8)

where the integral operator GS acts from L2(D) onto L2(S), and the coupling equation reads:

W(r) = χ(r)Uinc(r) + χ(r)(GDW)(r), r ∈ D, (9)

where the integral operator GD acts from L2(D) onto itself. This allows us to consider W asunknown instead of U.

CSI is an iterative method which consists in looking alternately for the unknown sourcesW and for the unknown contrast χ at each iteration step by minimizing a cost functional Jwhich accounts for the errors ρS and ρD in satisfying both the observation (8) and coupling(9) equations:

J =∑

q

∥∥ρSq

∥∥2

S

/∑q

∥∥Udifq

∥∥2

S +∑

q

∥∥ρDq

∥∥2

D

/∑q

∥∥χqUincq

∥∥2

D,

ρSq = Udif

q − GSq Wq, ρD

q = χqUincq − Wq + χqG

Dq Wq,

(10)

where subscript q stands for the frequency, and || . ||H is the norm associated with the innerproduct 〈., .〉H in L2(H) (〈v, w〉H = ∫

H v(r)w(r) dr, H = S or D, with the overbar denotingthe complex conjugate).

The iterative sequence is constructed as follows: starting from W(n−1)q and χ(n−1)

q , thecontrast sources and the contrast at iteration step (n − 1), we first update the contrast sourcesin the direction Vq of a Polak–Ribiere conjugate-gradient scheme [16]:

W(n)q (r) = W(n−1)

q (r) + α(n)q V (n)

q (r), r ∈ D, (11)

where α(n)q is obtained explicitly by minimizing the cost functional J (equation (10)). Then

the field within D follows immediately from the coupling equation, and we can proceed withthe updating of the contrast.

Let us remember that we look for homogeneous objects so that, after a suitablenormalization by a maximum contrast χmax

q , we look for a distribution in space of binarycontrast values (1 inside the object and 0 elsewhere), and this a priori information can beintroduced here in the inversion algorithm. However, since the contrast is sought by meansof a gradient-based method, as the contrast sources are, it is not looked for directly in abinary form which would prevent the existence of functional derivatives, but in the followingform: χq(r) = χmax

q ψ(τ(r)), where ψ(τ(r)) = [1 + exp(−τ(r)/a)]−1 is a function whichcontinuously depends upon an auxiliary real variable τ, and which runs from 0 to 1 as τ variesfrom −∞ to +∞. The real positive parameter a controls the rate of variation of the function:the lower the parameter a, the closer the latter is to a step function, and τ henceforth stands

Nonlinear inversions of immersed objects S91

for the new unknown contrast function with respect to which the functional derivatives ofinterest are formally derived. Hence, the unknown contrast (more precisely, τ) is updated inthe Polak–Ribiere conjugate-gradient direction ξ:

τ(n)(r) = τ(n−1)(r) + β(n)ξ(n)(r), r ∈ D, (12)

where β(n) is obtained by minimizing the second term of J with constant denominator, i.e.the cost functional JD = ∑

q

∥∥ρD(n)q

∥∥2

D

/∑q

∥∥χ(n−1)q Uinc

q

∥∥2

D.Initialization of the iterative process is as follows: the initial values of the contrast sources

W(0)q are obtained by backpropagating the scattered fields data: W(0)

q = γGS∗q U

difq , where GS∗

is the operator adjoint to GS , acting from L2(S) onto L2(D), and the constant γ is determined

by minimizing∑

q

∥∥∥Udifq − γGS

q GS∗q U

difq

∥∥∥2

S. The initial estimate of the field U(0)

q follows

immediately from the coupling equation. The initial estimate of the contrast is then obtainedby minimizing the error in the contrast–source constitutive equation:

ψ(τ(0)) =(∑

q

[W(0)

q χmaxq U

(0)

q

] /∣∣χmaxq U(0)

q

∣∣2/∑q

|χmaxq U(0)

q |2)1/2

. (13)

As for the update directions Vq and ξ, they are initialized in the directions of the gradients.As is usual with conjugate-gradient-based iterative schemes, after an initial phase of a few

iteration steps where the cost functional shows a rapid decrease, the latter is often stuck in aplateau with a low decrease rate which corresponds to sub-optimal solutions. Applying then aso-called cooling procedure consisting in decreasing the value of parameter a, which controlsthe rate of variation of ψ, can help to overcome this situation by pushing the contrast in thetest domain towards either 0 or 1, once all the quantities of interest have been re-calculatedwith the new a. The efficiency of this procedure in escaping from local minima of the costfunctional has been thoroughly investigated in [16].

Ideally, the strategy used for cooling would be to decrease a each time a plateau is observedin the evolution of the cost functional; however, despite being more advanced, this strategydoes not always lead to a faster convergence of the iterative process when compared to thatobserved with the strategy used herein which consists of applying the cooling process to eachof 32 iteration steps. As for the initial value of the cooling parameter a, it was chosen in [16],where the total field Uq and the contrast τ were updated simultaneously, in order to preconditionthe minimization of the cost functional, by scaling the components of its gradient versus Uq

and τ to the same order of magnitude. This strategy cannot be used here, as the unknownsare updated alternately. So, in the absence of any information, a is initialized to 1, as theoptimization process is not very sensitive to this value. In contrast this process is sensitiveto the rate of decrease of a, the latter being one of the key points of the cooling process: toolarge a decrease would freeze the optimization process, whereas too small a decrease wouldbe ineffective, the 5% decrease chosen here being a good compromise.

Finally, the iterative process is stopped when it no longer succeeds in significantly reducingthe cost functional despite the cooling. Figure 7 displays the evolution of the cost functionalas a function of the iteration step in the case of the circular dielectric cylinder of figure 8. Thecost functional corresponding to an inversion led from experimental data is compared to thatcorresponding to synthetic data, the latter being obtained by solving the direct problem (cfsection 2.5); although the cost functional decreases to a lower level in the latter case than inthe former, it shows, in both cases, the same behaviour. This illustrates, in some sense, therobustness of the algorithm, i.e. its low sensitivity to the perturbations induced on the scatteredfield.

S92 B Duchene et al

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500iteration

JJ0

Figure 7. Evolution of the cost functional as a function of the iteration step for synthetic (——)and experimental (- - - -) data in the case of the circular dielectric cylinder of figure 8.

Figure 8 displays some results obtained with this inversion method after 512 iterationsteps in the following configuration: the objects are either dielectric (ε′

r = 3, σ = 0) ormetallic (σ = 2 × 107 S m−1) with 5 cm circular or square cross-sections. They are locatedat the centre (x = 21 cm, y = 0) of a 10 cm-sided square test domain D partitioned into40 ×40 elementary 2.5 mm-sided square pixels. The source and the receiver, both of A1-type,are located at (xs = 11.5 cm, ys = −37.9 cm) and (xr = 12 cm, −19.9 cm < yr < 19.6 cm),respectively, and operate at six frequencies (250, 450, 540, 630, 730 and 840 MHz), all of thembeing processed simultaneously in the course of the inversion. Figure 8 also displays the initialestimates of the contrast for the two objects with circular cross-sections, which illustrates thelow efficiency of the back-propagation initialization scheme partly due to the fact that theembedding medium is lossy.

Although the various images are very different, retrieval of the objects is not good enoughto distinguish between the various shapes, particularly in the case of a metallic object, whereonly a part of the boundary appears. This is not surprising as the fields do not penetrate insidemetallic objects, so that the only meaningful information that can be inferred from the scatteredfields comes from their boundary. As a general rule, the objects seem to be shifted comparedto their true position; this is due, in part, to the strong coupling between the object and theA1 antennas. Nevertheless, in spite of this coupling, the images do benefit from the broad-band data and from the multi-frequency capabilities of the inversion algorithm, two items thathave been shown to be decisive in the quality of the results (see [36] and many interestingcontributions therein).

3.2. Controlled evolution of level sets

The second method investigated herein is the so-called controlled evolution of a level set,whose extended details can be found in [37–40].

It is based upon a combination of two methods, the level set representation of shapes [41]and the speed method [24], and allows topological identification in the sense that merging,division, emergence and disappearance of domains are enabled. Indeed, the closed contour

Nonlinear inversions of immersed objects S93

0 1 0 1 0 0.5

Figure 8. Grey level maps (scale: 0–1) of the contrast retrieved for dielectric (top) and metallic(bottom) objects with 5 cm square (left) or circular (middle) cross-sections obtained after 512iteration steps with the binary contrast source inversion algorithm. The true positions of the objectsare indicated by the dashed lines. The test domain is a 10 cm-sided square partitioned into 40 × 40elementary square pixels, and the images are built up from six frequencies. The initial estimatesof the contrast (scale: 0–0.5) for the two circular objects are displayed in the right column.

of any object is the zero-level of a level set φ, and this contour simply evolves in space r andpseudo-time t, as a consequence of the evolution of the whole level set, in order to minimizea suitable cost functional JS . Basically, the latter is the mean square norm of the discrepancybetween the scattered field effectively collected on the probing line S and the one associatedwith an object with contour t retrieved at a given t, appropriately normalized with respect tothe mean square norm of the data.

The evolution itself is governed by a Hamilton–Jacobi-type equation (14) which links thepartial derivative of φ with respect to t to the norm of its spatial gradient, through the action ofa velocity field of motion V oriented in the direction of the unit normal n to the contour t:

∂φ

∂t(r, t) + V(r, t)|∇φ(r, t)| = 0, ∀r ∈ D, ∀t 0. (14)

The velocity V is now built so that the derivative of the cost functional JS with respectto the pseudo-time t is negative (pending numerical uncertainties). It has been shownin [37], within a rigorous domain integral framework and using appropriate elements ofcontrol theory, that:

d

dtJS =

∫t

χ U(r)P(r)V(r, t) dr, (15)

where U is the solution of the domain integral equation (7) and P is an adjoint field that isthe solution of domain integral equations similar to those satisfied by U, but corresponding to

S94 B Duchene et al

0.01

0.1

1

0 20 40 60 80 100iteration

JS

Figure 9. Evolution of the cost functional as a function of the iteration step for synthetic (—) andexperimental (- - -) data. The dots indicate the lower levels of the cost functionals correspondingto the best solutions.

a configuration where the sources are located on the probing line, their amplitude being theconjugated error in satisfying the observation equation (6).

Taking V(r, t) = −[g(r, t)], where g, the so-called shape gradient on t extended to allpoints inside D, is proportional to [U(x)P(x)], ensures decreasing of the cost functional JS .

Then the discretization in time (step t) and space (on a fixed grid) of theHamilton–Jacobi-type equation leads to a fast marching algorithm which yields an updatedlevel set at time step (n + 1)t, from a given level set φ at nt and a given velocity. The zerolevel of this updated level set yields a new boundary, and a new velocity follows from the shapegradient calculated by solving the corresponding direct and adjoint problems. This iterativeprocess is stopped either when the objective function J becomes smaller than a given smallvalue or when the number of iterations has reached a given number (here, 100), the ‘final’inversion results being always those associated with the least value of J reached in the courseof the procedure. It should be emphasized that the map of the object t is implicit as it isdefined by the retrieved boundary t , pixels whose centres are located inside the latter beingset to black (corresponding to a non-null contrast), the others being set to white (correspondingto null contrast).

One of the key parameters of the process is the choice of the time step t needed at thediscretization stage of the Hamilton–Jacobi equation; too small a time step leads to a slowdecrease of the cost functional and too high a time step leads to divergence of the algorithm.In [37, 38], t is taken as constant during the inversion and is chosen by pure numericalexperimentation, while in [39], two adaptive schemes have been proposed: (i) the first basedon a Courant–Friedrichs–Levy (CFL) condition [41], where t is a function of the spatialdiscretization and of the maximum amplitude of the speed of motion of the level set, and (ii)the second scheme assumes a maximum time step chosen so that a properly introduced L∞

norm of the level set is decreased from one iteration to the next. It has been shown that allof these approaches lead to similar final results, the difference being the number of iterationsneeded to reach the latter. Herein, the second scheme based on the maximum time step has beenchosen. This scheme may lead to brutal shape deformations linked to too large t variations

Nonlinear inversions of immersed objects S95

Figure 10. Reconstruction of a dielectric object with circular cross-section (diameter, 5 cm) fromsynthetic (middle) and experimental (right) data collected at 434 MHz. The process is initializedwith a small square object (left) located at the centre of the test domain, the latter being constitutedof a 58 cm × 20 cm-sized rectangle partitioned into 90 × 45 elementary rectangular pixels. Thetrue position of the object is indicated by the dashed lines.

but allows a faster convergence of the algorithm. This last point is illustrated in figure 9 whichdisplays the evolution of the cost functional as a function of the iteration step in a typical case.The oscillatory behaviour of the cost functional is induced by the brutal shape deformationswhich correspond to sharp spikes.

This figure also illustrates the difficulty of the inverse problem linked to its ill-posedness.Indeed, the evolution of the cost functional corresponding to an inversion led from experimentaldata is compared to that corresponding to synthetic data. As with the CSI method, in bothcases the cost functional decreases to the same level; furthermore, it shows the same oscillatorybehaviour and the same difficulty in escaping from a plateau with a low decreasing rate at theend of the iterative process.

In this figure, the lower levels of the cost functionals are pointed out. The correspondingsolutions are displayed in figure 10. They have been obtained in the following configuration:the object is a dielectric cylinder (ε′

r = 3, σ = 0) with a circular cross-section (5 cm indiameter), centred at (x = 21 cm, y = 0) and illuminated at a frequency of 434 MHz by anA1 source located at (xs = 11.5 cm, ys = −37.9 cm), the A2 receiving antenna being locatedat (xr = 12 cm, −19.9 cm < yr < 19.6 cm). The inversion process is initialized with a smallobject, made of a few pixels, located at the centre of the test domain D (figure 10, left), a58 cm × 20 cm-sized rectangle partitioned into 90 × 45 elementary rectangular pixels, whichnearly covers the entire water column. The results obtained with synthetic data are displayedin the middle column; the object is retrieved at its right location. The results obtained with

S96 B Duchene et al

-200

-100

0

100

200

-0.2 -0.1 0 0.1 0.2-60

-55

-50

-45

-0.2 -0.1 0 0.1 0.2

y (m) y (m)

mod

ulus

(dB

)

phas

e (˚

)Figure 11. Modulus (left) and phase (right) of the computed (——, black) scattered field comparedto the fields measured (- - -) with the A2-type antenna in the case of a circular dielectric cylinderat a frequency of 434 MHz. The scattered field computed from the best solution (figure 10,right) obtained with the level set method when applied to the measured data is also displayed(——, grey).

experimental data are displayed in the right column; two objects are retrieved here: a smallone located close to the interface and another at the expected location.

Figure 11 displays a comparison (module and phase) between the measured (dashed line)and synthetic (black line) scattered fields; they show an agreement comparable to that obtainedfor the square metallic obstacle considered in figure 5. These fields are compared to thescattered field computed (grey line) by solving the direct problem for a scattering objectcorresponding to the solution given by figure 10 (right). The latter and the measured field arein good agreement which indicates that the solution is acceptable in the sense that it minimizesthe cost functional J ; however, the solution is not satisfactory in the sense that the retrievedobject is rather different from the expected one. This illustrates in some sense the ill-posednature of the inverse problem we are dealing with.

4. Conclusion

Although it does not show the intricacy of a ground probing application where highlyheterogeneous media and rough surfaces are encountered, the electromagnetic characterizationof immersed objects studied herein remains a challenging problem, due to the aspect-limiteddata configuration and the highly attenuating stratified embedding considered.

In this framework, two iterative nonlinear inversion methods have been applied tolaboratory-controlled experimental data. Both of them are based upon the minimization ofa cost functional which expresses the discrepancy between the data and the fields computed bymeans of the iterative solution and are tailored to the case of homogeneous targets, the lattera priori information being introduced in order to regularize the ill-posed inverse problem athand.

However, introducing this a priori information is not always sufficient to ensure theconvergence of the iterative process towards an acceptable solution. An adequate initializationof the iterative schemes could partly overcome this problem, as well as the introduction,in the functionals to be minimized, of additional regularization terms (total variation [42],

Nonlinear inversions of immersed objects S97

edge-preserving potential function [43], constraints of least perimeter or least area [39])accounting for further a priori information, this requiring increased computer resources.

As for the experiment itself, an improved setup is under study through the development ofan active receiving antenna with the same broad-band capabilities as the biconical antennas,but of smaller size, which should limit the perturbations on the measured fields. As a matterof fact, the broad-band capabilities are an important feature to combine the high penetrationdepth of the low frequencies with the better resolution of the high frequencies, knowing thatthe highly attenuating embedding we are working with strongly limits the propagation ofthe latter. Furthermore, multi-frequency data allow not only a good compromise betweenpenetration depth and resolution, but also partly overcome the ill-posedness of the inverseproblem.

Acknowledgments

The authors would like to thank Professor W C Chew and D Lesselier for their invitation tocontribute to this special section. They would like also to acknowledge the contribution of J-MGeffrin (now with Institut Fresnel, Marseille, France) to the realization of the experimentalsetup and the contribution of C Conessa in carrying out the measurements.

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