A simple two-dimensional inversion technique for imaging homogeneous targets in stratified media


  • Ilaria Catapano,

    1. Istituto per il Rilevamento Elettromagnetico dell'Ambiente IREA-CNR, Napoli, Italy
    2. Also at Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, Università di Cassino, Cassino, Italy.
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  • Lorenzo Crocco,

    1. Istituto per il Rilevamento Elettromagnetico dell'Ambiente IREA-CNR, Napoli, Italy
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  • Tommaso Isernia

    1. Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Università di Napoli Federico II, Napoli, Italy
    2. Now at Dipartimento di Informatica Matematica Elettronica e Trasporti, Università Mediterranea di Reggio Calabria, Reggio Calabria, Italy.
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[1] The problem of reconstructing in a quantitative fashion dielectric objects embedded in a layered medium with possibly (unknown) rough interfaces is dealt with. In particular, a nonlinear inverse scattering technique is proposed as a suitable tool for giving an accurate imaging of homogenous targets. The proposed approach, which exploits multifrequency data, is not limited to the weak scattering case. Moreover, a simple method is proposed so that simultaneous reconstruction of both unknown permittivity and roughness profile can be performed. In order to assess effectiveness and accuracy of the proposed approach under actual field conditions, a small number of antennas placed on a short measurement line is considered. A large number of numerical simulations confirm the effectiveness of the inversion approach, as well as its robustness with respect to noise on data and approximate knowledge of background parameters.

1. Introduction

[2] In a wide range of applications going from civil engineering to applied geophysics, the most adopted tool for noninvasive surveys is Ground Penetrating Radar (GPR) [Daniels, 1996]. On the other side, commonly adopted techniques for GPR data processing only allow extraction of qualitative information on the investigated region. In fact, they are based on a subjective interpretation of radargrams and on the assumption that velocity of waves in the investigated medium is constant. Hence, the obtained results are subject to the user experience and, as such, may be inaccurate or wrong.

[3] A possible way to overcome these limitations is the adoption of a tomographic approach, which makes it possible to achieve quantitative reconstructions of the domain under test in terms of presence, location, shape and chemical/physical properties of the buried targets. In this paper a tomographic approach is applied to imaging targets embedded in a simple layered structure constituted by three different media with possibly rough interfaces.

[4] Such a configuration is relevant in several applications. As a matter of fact, the considered geometry is mandatory in a number of applications such as diagnostics of concrete walls [Maierhofer and Leipold, 2001] and NDT of materials [Bolomey, 1989]. Moreover, simple extensions of the considered approach are of interest in bridge or asphalt inspection (wherein the scenario is usually composed by three or four layers [Hugenschmidt, 2002; Saarenketo and Scullion, 2000]).

[5] Finally, the considered geometry is also of interest in the subsurface diagnostics of soil. In this latter problem, the soil is usually assumed as a homogeneous half-space [Devaney and Zhang, 1991; Bucci et al., 2001; Galdi et al., 2003; Miller et al., 2000; van den Berg and Abubakar, 2003]. On the other side, because of nonlinearity, solution of inverse scattering problems is usually very sensitive to the availability of adequate information about the reference scenario. As a consequence, any sensible information about the scene, which is available a priori or easily deducible, has to be inserted in the inversion procedure. In this respect, while still being simple, a three-layer medium seems to be a more accurate and general model with respect to the usual assumption. In fact, a 1-D M-layers stratification is a widespread model in the geophysical community [Young and Jol, 2003], and its parameters can be easily derived from soundings in a calibration area. Moreover, because of losses, the deeper interfaces play a progressively decreasing role, so that reflection from the second interface is the main correction one has to take into account (if required) with respect to the more usual half-space assumption. However, such a model still fails to give an accurate modelling of soil because of the fact that interfaces are usually rough (rather than flat). This circumstance is worth to be mentioned since negligence of the rough interface, which is the common situation in the literature but for few cases [Dogaru and Carin, 1998; Galdi et al., 2003], simply does not allow to achieve accurate reconstructions (see Section 4).

[6] In order to give a unified framework for all the above considered problems, a three layer model with possibly rough interfaces is considered in the following and the imaging problem is posed as a properly formulated (and regularized) inverse scattering problem.

[7] In particular, a nonlinear inversion scheme based on those proposed by Isernia et al. [1997] and Bucci et al. [2001] is presented. As the (severely) aspect-limited nature of data implies that monochromatic data are not sufficient to perform effective inversions, a multifrequency approach (requiring the characterization of the environment at a given number of frequencies) is considered throughout. Moreover, taking advantage from the fact that in many cases targets can be represented in a realistic way as homogeneous ones, a suitable regularization scheme [Crocco and Isernia, 2001] is also exploited.

[8] Finally, a suitable extension and generalization of this latter regularization scheme [Crocco and Isernia, 2001] is introduced, which has the remarkable capability of taking into account in a simple and effective way the presence of moderately rough interfaces between the layers. In particular, the same approach is used to simultaneously reconstruct both the roughness profile and the dielectric permittivity distribution of buried homogeneous targets.

[9] Besides the new approach introduced to deal with roughness, the proposed inversion scheme is different from the existing literature for a number of additional reasons. First, although inverse scattering in stratified media is a topic of emerging interest (see above and comments in Xu and Liu [2002]), it seems there are few papers specifically devoted to solution algorithms for such a problem. Second, referring to subsurface diagnostics of soil, it is different from other approaches wherein a rough interface is considered [Galdi et al., 2003] as strong scatterers can be dealt with in our approach. Third, it is different from other approaches wherein a nonhomogeneous soil is considered [Miller et al., 2000] as just smoothly varying backgrounds are considered therein.

[10] In order to assess actual performance of the approach in a relatively simple situation, the canonical yet significant 2-D geometry is considered in this paper. Accuracy, capability to deal with rough interfaces and robustness of the approach with respect to a number of uncertainties suggest that extension to the 3-D case is worth to be pursued. Such an extension, far from obvious from an implementation point of view, can be done in a conceptually easy fashion.

[11] In the following, after the mathematical formulation of the problem, the proposed multifrequency approach is introduced in Section 2. The inversion procedure as well as a simple regularization scheme capable to achieve accurate reconstructions of homogeneous targets and to deal in an effective fashion with moderately rough interfaces are described in Section 3. Finally, in Section 4 numerical examples are given to assess effectiveness and robustness of the proposed approach against noise on data, presence of spurious inclusions in the region under test, and inaccurate a priori information about the reference scenario.

2. Formulation of the Problem

[12] The reference geometry is shown in Figure 1. The three layers are homogeneous with known equivalent dielectric permittivity εieq. The magnetic permeability is everywhere equal to that of the free-space, μ0. The investigated domain Ω is characterized by the unknown equivalent dielectric permittivity εxeq and it is embedded in the second layer, whose extent is d. Let us suppose that the unknown targets are contained within Ω and that their cross sections are invariant with respect to the z-axis.

Figure 1.

Geometry of the problem.

[13] The source of the incident fields is a known distribution of current filaments located in the first layer at a small distance from the interface along the line Γ and whose current is constant along the z-axis. The source probes radiate at different frequencies and also work as receivers for the scattered field. The measurement configuration therefore works in “reflection” mode and the set-up is a multiview-multistatic-multifrequency one in which at each frequency the scattered field is measured in several locations along Γ for each position of the source (moved along Γ).

[14] The considered geometry implies to deal with an aspect limited inverse scattering problem, which is described by the integral relationships [Colton and Krees, 1992]:

equation image
equation image

wherein the time factor exp(jωt) has been omitted. In equations (1) and (2), ρ∈Γ is the position of the measurement probe, ρ′∈Γ is the location of the source probe; Ei, E and Es are the incident field, the total field induced inside Ω and the scattered field in the upper medium (along Γ), respectively; equation image is the complex wave number in the second layer. The quantities gi and ge are the Sommerfeld-Green's functions pertaining the considered geometry. Their expressions, as well as that of the incident field, are reported in Appendix A. It is worth to note that although these functions have a quite cumbersome appearance, their samples can be still calculated in a fast and efficient way by means of FFT codes. Ai, Ae synthetically denote the integral radiation operators, which relate the induced currents (in Ω) to the field inside (in Ω) and outside (on Γ) the scatterer, respectively. The unknown of the problem is the contrast function χ(ω, r) defined as:

equation image

[15] This function relates the complex permittivity one is looking for (in Ω) to that of medium 2 (background). To achieve a quantitative reconstruction of the domain under test in terms of location, shape and electrical properties of the buried targets, we must solve the system given by the integral equations (1) and (2) in terms of χ.

[16] It is easy to verify, by formally inverting (2) in terms of E and substituting into (1), that the problem is nonlinear. Moreover, as thoroughly discussed in Bucci et al. [2001] and Bucci and Isernia [1997] for a similar case, it is ill posed as well because of the compactness of Ae.

[17] Due to the geometry of the problem available data are (strongly) aspect limited. As a consequence, use of monochromatic data, which would allow neglecting the dispersive nature of the layers, cannot be used. As a matter of fact, information related to a single fixed frequency is too poor to guarantee satisfactory reconstructions and a different strategy, able to increase the information content of data, has to be addressed.

[18] By leaving aside Time Domain methods, whose merits and drawbacks are beyond the scope of this paper, a widely adopted solution is the multifrequency tomographic approach, which can indeed improve the reconstruction performances [Bucci et al., 2000]. As multifrequency data are in line with the rising Stepped-frequency trend in the GPR community [Lee, 2002; Sato et al., 2003; van Genderen, 2003] and multifrequency data naturally lend themselves to the use of parallel computing, we focus in the following on such a case.

[19] In setting a multifrequency approach some attention has to be paid in choosing the unknown quantity. Indeed, as it can be noticed from (3), the contrast does not change when changing the illumination, but it changes with frequency. For completely unspecified dispersion relationships, this would imply to solve a number of monochromatic inverse scattering problems, with hardly any advantage over the monochromatic case. However, some definite advantage can be achieved by doing some hypotheses on the nature of the investigated media. In a similar fashion as indicated in [Haak et al., 1996], we suppose that both layers and inclusions are characterized by the complex equivalent relative permittivity having a dependence on frequency given by

equation image

where εi(r) is the relative permittivity and σi(r) is the conductivity.

[20] By means of simple changes we can then rewrite the contrast function as:

equation image

wherein χr (r) = Δε(r) − jΔσ(r), Δε(r) = εx (r) − ε2 and Δσ(r) = [σx (r) − σb]/ωmε0; T is a linear operator which transforms the contrast χr at a reference frequency into the contrast χ evaluated at the current frequency. Therefore, by relying on (5), once χr is known, it is possible to know χ at any frequency, so that χr is a convenient (i.e., frequency independent) unknown for the overall problem. In the following we assume that χr is the contrast at the maximum adopted angular frequency ωm.

[21] Use of equation (5) allows dealing with objects and media which are characterized by the complex equivalent permittivity given in (4). In many applications, this is a reasonable hypothesis for both the background media and the scatterers [Lambert et al., 1998]. However, as constitutive relationships of the background are supposed to be known, the approach can be easily extended to the case of generic backgrounds and the above kind of scatterers. Note this already includes many practically relevant situations, such as occurrence of cavities and land mines, for any kind of soil or stratification. Would this not be the case, a more refined parameterization of the equivalent permittivity of the scatterer (using more degrees of freedom for each complex equivalent permittivity [see, e.g., Budko, 2002]) should be used.

[22] At last, note the increase of the amount of data of a multifrequency approach with respect to the monochromatic case leads to a corresponding increase of both the measurement and computational time. On the other side, as noticed in Bucci et al. [2000], only a finite number of frequencies has to be considered in order to collect all the potential information and only frequencies sufficiently far from each other provide substantially independent data [Leone et al., 1999; Pierri et al., 1999]. In this respect, it proves convenient to consider only data collected at few frequencies in the widest allowable frequency band [Bucci et al., 2000].

3. Inversion Procedure

3.1. Basic Approach

[23] By using (5), the contrast function χ at each frequency can be calculated from the knowledge of χr. Therefore, the problem can be cast as estimating the unknown function χr from the knowledge of a finite number of measured samples of the scattered fields corresponding to a finite collection of known incident fields.

[24] Although using frequency diversity a larger amount of independent data becomes available [Bucci et al., 2000], the nonlinear inverse problem at hand is still ill posed. As a matter of fact, χr belongs to an infinite dimensional space, while the essential dimension of the space of independent data is always finite [Bucci et al., 2001]. Therefore, a stable inversion procedure needs the introduction of a regularized solution [Isernia et al., 1997, 2001].

[25] In this paper, a nonlinear reconstruction method belonging to the class of bilinear approaches [Kleinmann and van den Berg, 1992; Isernia et al., 1997] has been adopted. In this technique, both the contrast χr and the total fields inside the scatterer E are assumed as unknowns. By so doing, the degree of nonlinearity is reduced at expenses of an enlargement of the set of the unknowns. It should also be noted that bilinear approaches do not require approximations, but for that arising from the discretization of the problem.

[26] With respect to this last point, let us assume that the investigated region Ω has been divided into Nc square cells of side Δ, chosen in such a way [Richmond, 1965] that both the contrast and the total field can be assumed to be constant within each of them; moreover, let NM be the number of measurement points equispaced along Γ, NV the number of source points equispaced along Γ and NF the number of considered frequencies.

[27] By so doing it is possible to rewrite equations (1) and (2) in a discrete fashion:

equation image
equation image

wherein m = 1,…, NM, v = 1,…, NV, f = 1,…, NF, p = 1,…,Nc; equation imager(n), equation image(f,n,v) are the contrast and the total field in the n-th cell, respectively; Es(f,m,v) represents the scattered field measured in the location ρm when placing a source in the location ρ′v and operating at the frequency ωf;Ei(f,p,v) is the incident field in the p-th cell when placing a source in the location ρ′v and operating at the frequency ωf.equation image, equation image and equation image are the discrete counterparts of operators T, Ai and Ae, respectively, and equation imagee, equation imagei are the discretized Sommerfeld-Green's functions. Note that the term [ε2eq(ω)]−1 is now included in the kernel of the discrete operators equation image and equation image.

[28] In order to tackle ill-posedness due to the finite amount of data, we look for finite dimensional representations for both the unknowns. In particular, representation of the contrast is given by

equation image

wherein {Ψin}i=1N is a suitable basis defined over the grid and ci is the generic unknown contrast coefficient. In the lack of any a priori information and additional forms of regularization, the number N of terms in (8) has to be lower than the essential dimension of data and can be determined using a criterion like the one proposed in Bucci et al. [2001] for the half-space case.

[29] As far as the choice of the basis {Ψin}i=1N is concerned, a wavelet basis is adopted in the following in order to exploit the multiresolution features of such a class of functions. By so doing, one can possibly accommodate the sought coefficients in a nonuniform fashion within the investigated domain Ω. Such a chance can be useful for at least two different reasons. First, on the basis of a priori information or some kind of step-wise processing [Bucci et al., 2002], one can focus on the regions wherein the scatterers are actually located or a more variable permittivity profile is expected. This possibility allows reconciling reconstruction accuracy with the necessity to have a number of unknown parameters as low as possible [Isernia et al., 2001]. Second, in some cases (such as the half-space case, which can be seen as a subcase of the problem we deal with) a variable degree of resolution is expected with different depths, so that it comes out to be useful to look for a large number of coefficients in the shallow region while using a “coarse” resolution for the deeper part of the region under test [Bucci et al., 2001].

[30] It has to be noted that our use of wavelets is quite different in spirit from the previous works by Miller and Willsky [1996a, 1996b]. In these latter, multiresolution features are adaptively deduced at each step of the iterative scheme, whereas a priori considerations on the mathematical nature of the problem or information arising from simple preprocessing are used herein to select the distribution of wavelets to be used. Such a circumstance allows us to exploit the wavelet capabilities in a relatively simple fashion.

[31] At each frequency and for each view, a finite dimensional representation for the field unknowns is used, given by

equation image

wherein {Φin}i=1Q are the discrete spatial Fourier harmonics defined over the grid and eiv,f is the generic coefficient of the internal field for the view v and the frequency f. The number Q of terms in (9) is given by the discretization of equation (1) performed according to results of Richmond [1965].

[32] The discretized problem is now cast as the global minimization of the cost functional:

equation image

wherein x1 is the vector of the wavelet coefficients of the reference contrast function χr and x2 is the vector containing the Fourier coefficients of the total field for all views and frequencies. Moreover, equation image(equation imager) is the vector containing the point-wise values of the contrast at each processed frequency.

[33] The CG (conjugate gradient) method is the adopted minimization procedure (see Appendix B). High computational efficiency is gained by taking advantage from two different circumstances. First, since (10) is a fourth order polynomial in terms of the unknowns, the line minimization step can be performed in an accurate and computationally effective fashion (see Appendix C). Second, because of the expression of equation imagei (see Appendix A), evaluation of equation image[equation image(equation imager(n))equation image(f,n,v)] can be conveniently reduced to the computation of convolutions and/or correlations. Therefore, by following the same guidelines of Lesselier and Duchêne [1991], operator equation image, as well as its adjoint equation image (which is needed in the overall scheme) can be implemented in a convenient fashion by use of FFT techniques.

[34] The proposed inversion approach simultaneously uses all data related to the processed frequencies as input of the inverse procedure. On the other side, the cost functional to be minimized is a sum of terms of the same kind as those usually exploited in the monochromatic case [Bucci et al., 2000]. Therefore, gradients with respect to field variables are computed one frequency at time (and doing the same set of operations for each frequency), while some summation on all frequencies is necessary in order to compute the gradient with respect to the contrast variables and to update the functional. This is expected, as addition on all frequencies is necessary to make a “weighted” updating of the contrast unknowns. It is worth to note that in the proposed approach the different frequencies can be processed almost in an independent fashion so that the procedure can be implemented on parallel computation facilities in a very simple fashion.

3.2. Homogeneous Targets and Rough Interfaces

[35] In many applications the buried targets are characterized by a constant permittivity. This suggests exploiting such a priori knowledge by introducing an additional form of regularization [Crocco and Isernia, 2001]. By so doing, performance of the reconstruction approach can be improved and the lack of information (due to the aspect limited nature of the data) can be partially compensated. As a matter of fact, while in the lack of additional regularization the number of unknown coefficients in the representation of the contrast has to be less than the degrees of freedom of data, in this case it is possible to take into account a slightly larger number of unknowns.

[36] This further regularization amounts to add to the cost functional a term which takes into account the homogeneous nature of the unknown objects. Thus, the cost functional is redefined and becomes:

equation image

wherein χm is a reference contrast pertaining to the homogeneous scatterer. Equation (11) is the extension to multifrequency case of the cost functional introduced in Crocco and Isernia [2001]. The aim of the additive term is to make the contrast equal to zero or to χm inside each pixel of the investigated domain Ω. The value of χm may be either known or unknown, depending on the available a priori information.

[37] In many practical cases the target is buried under a rough surface. As a consequence, the problem of dealing with scattering phenomena in the presence of rough interfaces arises. The presence of roughness strongly affects the measured field [Bucci et al., 1999; Dogaru and Carin, 1998] so that introduction of appropriate tools to take into account roughness is mandatory.

[38] Referring, by the sake of simplicity, to the first interface (a similar approach can be devised for the second interface), a possible (very simple) way to deal with this problem is to model the roughness of the profile as a volumetric inclusion of air into the background. In such a way, the roughness can be included in the searched permittivity profile, and the overall unknown can assume three possible values corresponding to the permittivity of free space, the permittivity of the background and that pertaining to the inclusion. As long as one is capable to have effective inversion approaches for such a kind of situation, one has two benefic effects. First, the roughness profile and the permittivity distribution of the targets can be imaged simultaneously, thus avoiding possible approximation errors due to a splitting in two separate steps. Second, there is no need to redefine the Green's functions, as done in other approaches [Galdi et al., 2003].

[39] In fact, as the overall problem can be cast as that of imaging a permittivity distribution which can assume three possible values, one can proceed by simply introducing a further regularizing term in the cost functional (9). The new cost functional is given by

equation image

wherein χm is the value of the contrast pertaining to the inclusion, χo is the (known) air/background contrast and M1 and M2 are two masks such that each regularizing term acts on a different part of the investigated domain. In particular, the masks are chosen in such a way that M2 restricts the effect of the pertaining term to the shallow part of the domain under test, while M1 is concerned with the ‘deep’ part. Apart from particular cases, one has generally enough ‘a priori‘ knowledge in order to safely choose M1 and M2. Would this not be the situation, some kind of ‘adaptive’ procedure, which is beyond the scope of this paper, could be devised.

[40] Coefficients α and β are positive weight factors whose choice is a critical point. As a matter of fact, in nonlinear problems such as the one at hand, no simple rule exists to perform such a choice in an “optimal” fashion.

[41] A possible strategy to solve this problem amounts to introducing proper statistical tools [Pascazio and Ferraiuolo, 2003], whose discussion is however beyond the scope of this work. Alternatively, adoption of a multiplicative regularization as devised in Abubakar and van den Berg [2002] can be taken into account, on the other side an additional (“steering”) parameter is required also in this latter.

[42] As far as this paper is concerned, the choice of the weight coefficients is done heuristically and only some empirical considerations can be given. Of course, if the values of α and β are very small, no regularization is enforced, and the minimization would be driven into a false or poorly accurate solution, while, if values of α and β are very high, the contrast simply obeys the regularization terms, thus exhibiting a poor fitting of the measured data (hence loosing the actual solution as well). On the other side, an extensive numerical analysis has shown that the optimal choice of the weight coefficients is influenced by the reference contrast χm, by the intensity of the reflections given by the second interface and by the number of the processed frequencies, which are all known a priori. Note the optimal choice of α and β does not depend on the actual unknowns, id est, the actual shape of the inclusions and their effective positions. Moreover, it is worth to note that frequency diversity makes the choice of α and β less critical. As a final remark, it can be noticed that the additive terms do not modify the computational burden of the approach, as (10) is still a fourth order polynomial in the unknowns.

4. Numerical Examples

[43] In order to prove the effectiveness of the proposed approach, some numerical examples are provided in the following. By sake of clarity the numerical examples are divided in two parts aiming to represent two different possible situations: the former concerning diagnostics of (concrete) walls, and the latter dealing with sensing of objects buried in a stratified soil.

[44] In both cases, and opposite to a large body of literature, particular care has been posed in using realistic values for the background permittivities by using the values in Daniels [1996]. Moreover, as in realistic cases the background permittivity is not a constant, simulated data has been generated assuming that permittivity of the second layer randomly varies around an average value (ε2 = 7 in both cases) within the investigated region. Also, small inclusions of different permittivity have been randomly placed in the investigated domain. Finally, in all the following examples simulated data have been corrupted with a 10% additive noise.

[45] In order to keep things as much realistic as possible, a very small number of equispaced sources and measurement points have been considered. In particular, for each frequency, 5 sources are used and the resulting scattered field is measured by 5 receivers. Both of them are located on a line Γ very close to the first interface.

[46] In all the examples the reference contrast χm is assumed to be known and the contrast function is represented by means of a Haar wavelet basis. Note that such a choice is dictated by the kind of targets we aim to retrieve, since Haar wavelet basis allows a piece-wise constant representation of the unknown permittivity profile.

[47] Finally, in order to evaluate accuracy and performances of the proposed approach, the normalized error given by

equation image

is considered. In (13) χrec is the reconstructed contrast and χideal is given by the reference profile without spurious inclusions and considering the permittivity of the second layer everywhere equal to its average. Moreover, in order to test robustness of the overall approach with respect to false solutions in “worst-case” conditions, we choose as starting guess the background, i.e. χ = 0. Note better starting points or even frequency hopping strategies, such as for example the one in Bucci et al. [2000], could allow still more accurate reconstructions. Finally, the overall approach could be easily extended to the diagnostics of the second of N layers by simply changing R23.

4.1. Diagnostics of a Concrete Wall

[48] As far as diagnostics of walls is concerned, let us consider the case of a wooden structure (εx = 2.70, σx = 0 S/m) embedded in a concrete wall (ε2 = 7, σ2 = 1 × 10−3 S/m). In particular, the concrete layer is considered to be 3.5 λb thick at 400MHz, λb being the wavelength in the second layer. The investigated region is a square 3.5 λb × 3.5 λb wide and it is subdivided into 64 × 64 square pixels. The real part of the reference profile is depicted in Figure 2a. The measurement line Γ is 1.75 λb long and it is placed as shown in Figure 2a.

Figure 2.

(a) Real part of the reference profile and measurement configuration. (b) Reconstructed profile (real part), multifrequency case (εb = 7, α = 0.003). (c) Reconstructed profile (real part), monochromatic case (εb = 7, α = 0.003). (d) Reconstructed profile (real part), multifrequency case with wrong εb = 6 (α = 0.009). (e) Reconstructed profile (real part), multifrequency case with wrong εb = 5 (α = 0.05).

[49] In this case we can safely consider both interfaces being planar, so that the cost functional to minimize is the one given by equation (11), wherein the additive term acts on the whole domain and pushes each pixel to be zero or equal to the (known) value associated to the inclusion. Note that in this case the half space model would be misleading for sure because of the remarkable discontinuity between the second layer and the third one.

[50] By choosing α = 0.003 and selecting the wavelet coefficients in such a way that each basis function has approximately a side of λb/2 in the whole domain (then 64 coefficients are used to describe the unknown contrast), the monochromatic approach (400MHz) and the multifrequency one (f1 = 200MHz, f2 = 300MHz, f3 = 400MHz) give rise to the normalized errors reported in Table 1.

Table 1. Reconstruction Errors
 Reconstruction Error
Monochromatic data1.41
Multifrequency data0.28

[51] Since in (13) we have assumed that χideal is the contrast given in Figure 2a without spurious inclusions, the normalized error actually takes into account the capability of the approach of rejecting presence of spurious objects (usually present in realistic situations). Table 1 and Figures 2b and 2c clearly show that use of multifrequency information dramatically increases performance of the inversion procedure, as it was expected. Note that a satisfying reconstruction is achieved by using just three frequencies.

[52] In order to face conditions as close as possible to the actual ones, we have checked robustness of the proposed approach with respect to uncertainties about the reference background. Accordingly, we have tried to reconstruct the profile of Figure 2a, assuming a wrong value of the average permittivity ε2. In particular, while the actual (mean) value of ε2 is 7, we have solved the inverse problem by supposing ε2 = 6 in a first case and ε2 = 5 in a second one. In Table 2 the obtained reconstruction errors are reported together with the values adopted for the weight coefficient α.

Table 2. Reconstruction Errors and Weight Coefficients With an Incorrect Knowledge of Background Permittivity
εb = 60.379e-3
εb = 50.400.05

[53] It is interesting to observe that the adopted approach is still capable to achieve satisfactory reconstructions, see Figures 2d and 2e, even when the information about the average permittivity of the wall is wrong. Note that a larger value has to be used for the weight factor in these cases, which has to be attributed to the different value of χm and to the need of some “extra” regularization.

[54] As it can be noted, the object shape and extension are correctly estimated, thus allowing to state that the proposed inversion approach is a firm tool able to detect and characterize the target successfully even in the presence of uncertainties on the reference scenario.

4.2. Diagnostics of Soil

[55] In order to discuss the effectiveness of the approach in subsurface diagnostics, let us consider a simply stratified soil constituted by two layers with relative permittivities ε2 = 7, ε3 = 9 and conductivities σ2 = 5e-3S/m, σ3 = 5e-2S/m, respectively. In particular, at 600MHz the second medium is 1.33 λb thick while the rectangular region under test is 2.65 λb × 1.33 λb wide and it is divided into 64 × 32 pixels, λb being the wavelength of the second layer. The measurement line Γ is 2.65 λb long and it is placed as shown in Figure 3a.

Figure 3.

(a) Real part of the reference profile and measurement configuration. (b) Reconstructed profile (real part), multifrequency—three layered model (α = 0.0004). (c) Reconstructed profile (real part), multifrequency—half-space model (α = 0.0004).

[56] As a first example a void (εx = 1, σx = 0S/m) is supposed to be buried under a rough surface in this lossy soil. The real part of the actual profile is depicted in Figure 3a. Due to the presence of an unknown roughness, we should minimize the cost functional given by (12), so that it is now necessary to choose the masks M1 and M2 and both weight factors α and β. However, since the inclusion is an empty cavity, its reference contrasts (χm) is equal to χo so, if we suppose α = β, we can still use the functional given by equation (11).

[57] By using three different frequencies (f1 = 400MHz, f2 = 500MHz, f3 = 600MHz) and choosing α = 4e − 4 we are able to correctly estimate position and extension of the inclusion as it is shown in Figure 3b. Moreover, also the roughness profile is reconstructed in a satisfactory fashion as well. The normalized reconstruction error is err = 0.4. Notice that even in the favourable cases, such an error is quite large due to the presence of noise on data as well as due to the “model” error arising from the (anyway necessary) regularizations.

[58] In this case, the wavelet coefficients have been chosen in such a way that the corresponding pixels have a side roughly equal to λb/10 close to the surface and λb/2 in the bottom part of the domain. By choosing this grid, the contrast unknowns are 104. Note that, thanks to the a priori information introduced by the regularizing terms, we can enlarge the set of the unknowns so that we have enough data to solve the problem correctly even if we use a reduced number of sources, receivers and frequencies.

[59] In the previous reconstruction, the soil has been modeled as a layered structure. In order to remark the usefulness of such a model, let us consider what happens if the more usual half-space model (ε2 = ε3 = 7 and conductivity σ2 = σ3 = 5e-3S/m) is used in the reconstruction. The profile obtained by using the same frequencies and wavelet basis as above and α = 4e − 4 is shown in Figure 3c and corresponds to err = 1.4, which is indeed much larger than that achieved in the previous case. Notwithstanding the fact that the second and third layers are not that different, the half-space model only allows a rather poor estimation of location and shape of the inclusion, so that a layered model is indeed of interest in the subsurface diagnostics of soil. As expected, numerical analysis shows that actual usefulness of a layered model depend upon the thickness of the layer as well as by the amount of discontinuity amongst the second and third layer.

[60] As a second and more difficult example, let us consider the more general case wherein the scatterer and the roughness are volumetric inclusions of different kind, that is volumetric air inclusion and the scatterer correspond to different values of χm. Morever, let us consider the usually neglected case wherein more than one target is buried in the soil. In particular, let us consider the situation wherein two objects with εx = 11 and σx = 0S/m are buried under a rough interface in the lossy soil previously described. As Figure 4a shows, one of the targets is not very deep (about λb/5 from the interface) while the second one is about 4λb/7 far from the interface. The distance among the objects is about λb/2.

Figure 4.

(a) Real part of the reference profile. (b) Reconstructed profile (real part) considering the rough interface (α = β = 0.0004). (c) Reconstructed profile (real part) neglecting the rough interface (α = 0.0004).

[61] In order to retrieve location, shape and electrical property of the targets, multifrequency data are used (f1 = 400MHz, f2 = 500MHz, f3 = 600MHz) and the wavelet coefficients have been chosen in such a way that the corresponding pixels have a side equal to λb/10 close to the surface and λb/2 in the bottom part of the domain, so that the unknown wavelet coefficients are 104.

[62] By minimizing (12), the reconstruction shown in Figure 4b is achieved. In this case, the weight coefficients α and β have been chosen both equal to 1e-4, the reference contrasts χ0 and χm have been assumed known and the masks have been supposed given. As Figure 4b shows, we get a good estimation of number, position, dimensions and electrical property of the targets, clutter is also rejected and reconstruction error is err = 0.26. Notice that also in this case the magnitude of the reconstruction error is affected by noise on data and model error.

[63] In order to better understand the actual need of taking into account roughness, the cost functional given by equation (11) (a single mask on the whole domain) has been also minimized, and the corresponding reconstruction is shown in Figure 4c. Notwithstanding this is the best result we have been capable to achieve (using α = 4e-4), the reconstruction, corresponding to an error err = 0.7, is considerably worse and, in fact, some false targets appear in the investigated domain. Therefore, accuracy in the target retrieval seems to be related to the capability of taking into account and to reconstruct the rough interface. In this respect, it is worth to note that the proposed approach is able to deal with a rough interface in a simple fashion and without increasing the computational burden of the solution procedure. On the other side, choice of the masks M1 and M2 and of α and β has to be properly tuned on the basis of a priori information (M1 and M2) and preliminary simulations (α and β).

5. Conclusions

[64] A nonlinearized, multifrequency and regularized procedure for the quantitative imaging of homogeneous objects embedded in a layered medium with possibly rough interfaces has been presented. This method has been developed by exploiting and extending the bilinear approach [Isernia et al., 1997; Bucci et al., 2001] to the case of multifrequency data, introducing a suitable regularization [Crocco and Isernia, 2001], and introducing a simple method in order to take into account the possible presence of rough interfaces between the layers. The overall procedure can be implemented in a numerically efficient fashion by using FFT codes to compute both the Green's functions and the integral operator equation image.

[65] Frequency diversity can be managed through a suitable formulation capable to tackle the dispersive nature of the scattering problem while considering a frequency independent unknown. However, some a priori information is needed about the reference scenario. In case of scatterers with a simple constitutive relationship (εeq = εr ε0jσ/ω), one can take into account the dispersive nature of the problem while using the same number of unknowns as in the monochromatic case. Note this case includes a number of practically relevant objects such as, for instance, voids (cavities) and plastic landmines. In such a case, a very low number of frequencies allow satisfactory reconstructions.

[66] Possible unknown roughness of the interfaces is accurately taken into account by regarding it as a volumetric inclusion, and using a suitable regularizing term (which does not modify the computational complexity of the problem) in the cost functional whose global minimum defines the solution. Notwithstanding its simplicity, the proposed approach turns out to be effective in taking into account (and recovering) the roughness distribution.

[67] Finally, the overall approach is not limited to the weak scattering case. An extensive numerical analysis shows that the proposed approach is robust against several uncertainties about the reference scenario such as approximate knowledge of the average permittivity of the background, presence of volumetric clutter, nonhomogeneous nature of εb.

Appendix A

[68] In order to underline similarities and differences with respect to the more usual half-space case, in this appendix we report the expressions of the incident field and of the Green's functions. Let us consider a Cartesian coordinate system centered on the first interface (air-background) and with the x-axis parallel to the interfaces (see Figure 1).

[69] By considering a filamentary source located in ρ′ = (x′, y′) (in air) the corresponding incident field in the point r = (x,y) (in the domain Ω) is given by

equation image

wherein κ is the spatial frequency, d is the layer extent, equation image, Imi) < 0

equation image

[70] The function ge represents the field generated in the medium 1 in the location ρ = (xm, ym) by an elementary source located in the point r′ = (x′, y′) in medium 2. Its expression is given by [Chew, 1995]:

equation image

wherein τ21 = τ12.

[71] The function gi represents the field generated in the second layer in the location r = (x, y) by an elementary source located in the point r′ = (x′, y′) in the same layer; its expression is given by [Chew, 1995]:

equation image

wherein H02 is the zero order second kind Hankel function, Δx = xx′, Δ y_ = y − y′ and Δy+ = y + y′.

Appendix B

[72] An efficient manner to minimize functionals (10–12) is to adopt the conjugate gradient procedure, whose scheme is

equation image

where x is a ordered vector containing all the unknown sequences x1, x2, χm, (k) and (k+1) denote the k-th and (k + 1)-th iteration, respectively, ∇Φ is a vector related to the gradient of Φ with respect to x. Moreover, λ is a scalar factor that, step by step, has to be evaluated in order to guarantee the maximum decrease along the direction given by H∇Φ, wherein H is a dyadic depending on the adopted scheme (a Polak-Ribiere scheme [Luenberger, 1991] is adopted in the following). The vector ∇Φ contains, in the same order as x, the sequences equation image.

[73] Let Δequation imager be the variation of equation imager due to an increment of the unknown contrast coefficients Δx1, the corresponding variation of the functional is given by

equation image

wherein * denotes the complex conjugation, + denotes the adjoint operator, 〈.,.〉 is the usual scalar product, c.c. denotes the complex conjugate; note that the operator equation image is self-adjoint.

[74] In a similar way it is possible obtain the variations due to ΔE(v,f) (i.e., Δx2v,f) and Δχm respectively,

equation image
equation image

wherein equation image is the (discrete) identity operator. The expressions of the gradients can be now directly obtained from equations (B2)–(B4) remembering that ΔΦ∣x = 〈∇Φ∣x , Δx〉.

Appendix C

[75] In order to use the conjugate gradient procedure, we must compute, step by step, the scalar factor λ which guarantees the maximum decrease of the functional. By remembering that for a fixed view v and a fixed frequency f the cost functional is:

equation image

it is simple to verify that the behavior of such a functional along an arbitrary line, whose direction is described by [equation imager + λΔequation imager, equation image + λΔequation image, χm + λΔχm], is given by

equation image

[76] Hence, due to the nature of the involved operator, the last expression can be written as:

equation image

The coefficients in (C3) can be obtained considering the three terms present in (C2) one at the time.

[77] The coefficients related to the first term in (C2) are

equation image

Those related to the second term are

equation image

and those related to the third term are

equation image

The overall coefficients are obtained summing the partial ones.

[78] Once the coefficients of the polynomial (C3) have been determined, the stationary points are obtained from

equation image

Hence we can obtain the desired value λ by solving this third degree algebraic equation.