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Nonlinear and Hybrid Inversion Techniques for Ground Penetrating Radar Imaging
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A DNN architecture referred to as GPRInvNet was proposed to tackle the challenges of mapping the ground-penetrating radar (GPR) B-Scan data to complex permittivity maps of subsurface structures. The GPRInvNet consisted of a trace-to-trace encoder and a decoder. It was specially designed to take into account the characteristics of GPR inversion when faced with complex GPR B-Scan data, as well as addressing the spatial alignment issues between time-series B-Scan data and spatial permittivity maps. It displayed the ability to fuse features from several adjacent traces on the B-Scan data to enhance each trace, and then further condense the features of each trace separately. As a result, the sensitive zones on the permittivity maps spatially aligned to the enhanced trace could be reconstructed accurately. The GPRInvNet has been utilized to reconstruct the permittivity map of tunnel linings. A diverse range of dielectric models of tunnel linings containing complex defects has been reconstructed using GPRInvNet. The results have demonstrated that the GPRInvNet is capable of effectively reconstructing complex tunnel lining defects with clear boundaries. Comparative results with existing baseline methods also demonstrated the superiority of the GPRInvNet. For the purpose of generalizing the GPRInvNet to real GPR data, some background noise patches recorded from practical model testing were integrated into the synthetic GPR data to retrain the GPRInvNet. The model testing has been conducted for validation, and experimental results revealed that the GPRInvNet had also achieved satisfactory results with regard to the real data.
The quantitative inspection of unknown targets or bodies by means of microwave tomography requires a proper modeling of the field scattered by the structures under test, which in turn depends on several factors related to the adopted antennas and measurement configuration. In this paper, a multifrequency tomographic approach in nonconstant-exponent Lebesgue spaces is enhanced by a preliminary step that processes the measured scattered field with a neural network based on long short-term memory cells. In the considered cases, this approach allows dealing with measurements in three-dimensional settings obtained with non-ideal antennas and measurement points, while retaining a canonical two-dimensional formulation of the inverse problem. The adopted data-driven model is trained with a set of simulations of cylindrical targets performed with a finite-difference time domain method, considering a simplified bistatic measurement configuration as an initial case study. The inversion procedure is then validated with numerical simulations involving cylindrical and spherical structures.
Ground penetrating radar (GPR) technology for underground exploration consists in the transmission of an electromagnetic signal in the ground for sensing the presence of buried objects. While monostatic or bistatic configurations are usually adopted, a limited number of multistatic GPR systems have been proposed in the scientific literature. In this manuscript, we investigate the recovery performance of a specific and unconventional contactless multistatic GPR system, designed at the Georgia Institute of Technology for the subsurface imaging of anti-tank and anti-personnel plastic mines. In particular, for the first time, tomographic approaches are tested against this experimental multistatic GPR system, while most GPR processing in the scientific literature processes multi-monostatic experimental data sets. Firstly, by mimicking the system at hand, an accurate theoretical as well as numerical analysis is performed in order to estimate the data information content and the performance achievable. Two different tomographic linear approaches are adopted, i.e. the linear sampling method (LSM) and the Born approximation (BA) method, this latter enhanced by means of the compressive sensing (CS) theoretical framework. Then, the experimental data provided by the Georgia Institute of Technology are processed by means of a multi-frequency CS and BA-based method, thus generating very accurate 3D maps of the investigated underground scenario.
Ground Penetrating Radar (GPR) systems are nowadays standard inspection tools in several application areas, such as subsurface prospecting, civil engineering and cultural heritage monitoring. Usually, the raw output of GPR is provided as a B-scan, which has to be further processed in order to extract the needed information about the inspected scene. In this framework, inverse-scattering-based approaches are gaining an ever-increasing interest, thanks to their capabilities of directly providing images of the physical and dielectric properties of the investigated areas. In this paper, some advances in the development of such inversion techniques in the GPR field are revised and discussed.
gprMax is open source software that simulates electromagnetic wave propagation, using the Finite-Difference Time-Domain (FDTD) method, for the numerical modelling of Ground Penetrating Radar (GPR). gprMax was originally developed in 1996 when numerical modelling using the FDTD method and, in general, the numerical modelling of GPR were in their infancy. Current computing resources offer the opportunity to build detailed and complex FDTD models of GPR to an extent that was not previously possible. To enable these types of simulations to be more easily realised, and also to facilitate the addition of more advanced features, gprMax has been redeveloped and significantly modernised. The original C-based code has been completely rewritten using a combination of Python and Cython programming languages. Standard and robust file formats have been chosen for geometry and field output files. New advanced modelling features have been added including: an unsplit implementation of higher order Perfectly Matched Layers (PMLs) using a recursive integration approach; diagonally anisotropic materials; dispersive media using multi-pole Debye, Drude or Lorenz expressions; soil modelling using a semi-empirical formulation for dielectric properties and fractals for geometric characteristics; rough surface generation; and the ability to embed complex transducers and targets.
Ultra wideband (UWB) Microwave imaging is one of the most promising emerging imaging technologies for breast cancer detection, and is based on the dielectric contrast between normal and cancerous tissues at microwave frequencies. UWB radar imaging involves illuminating the breast with a microwave pulse and reflected signals are used to determine the presence and location of significant dielectric scatterers, which may be representative of cancerous tissue within the breast. Beamformers are used to spatially focus the reflected signals and to compensate for path dependent attenuation and phase effects. While these beamforming algorithms have often been evaluated in isolation, variations in experimental conditions and metrics prompts the assessment of the beamformers on common anatomically and dielectrically representative breast models in order to effectively compare the performance of each. This paper seeks to investigate the following beamforming algorithms: Monostatic and Multistatic Delay-And-Sum (DAS), Delay-Multiply-And-Sum (DMAS) and Improved Delay-And-Sum (IDAS). The performance of each beamformer is evaluated across a range of appropriate metrics.
The physical basis for breast tumor detection with microwave imaging is the contrast in dielectric properties of normal and malignant breast tissues. Confocal microwave imaging involves illuminating the breast with an ultra-wideband pulse from a number of antenna locations, then synthetically focusing reflections from the breast. The detection of malignant tumors is achieved by the coherent addition of returns from these strongly scattering objects. In this paper, we demonstrate the feasibility of detecting and localizing small (<1 cm) tumors in three dimensions with numerical models of two system configurations involving synthetic cylindrical and planar antenna arrays. Image formation algorithms are developed to enhance tumor responses and reduce early- and late-time clutter. The early-time clutter consists of the incident pulse and reflections from the skin, while the late-time clutter is primarily due to the heterogeneity of breast tissue. Successful detection of 6-mm-diameter spherical tumors is achieved with both planar and cylindrical systems, and similar performance measures are obtained. The influences of the synthetic array size and position relative to the tumor are also explored.
A hybrid inversion procedure for subsoil prospecting by ground penetrating radar (GPR) measurements is proposed and experimentally validated in this letter. The approach integrates the benefits of qualitative beamforming in an adaptive multifrequency inexact-Newton scheme formulated in Lebesgue spaces with variable exponents. After a first preprocessing step aimed at estimating the incident field, a delay-and-sum method provides an initial image of the region of interest. Then, the inexact-Newton algorithm exploits the qualitative indicator for both setting the exponent function and selectively weighting the update of the unknown during iterations. The proposed approach is tested against experimental GPR data.
Ground penetrating radar (GPR) enables infrastructure inspection using consecutive and long survey lines. However, the existing GPR data processing methods may lead to distortions or dislocations in the reconstructed shapes of the detected objects or even inconsistencies of the inverted dielectric values when performing inversion or object identification directly using GPR data obtained from the consecutive and long survey lines. To overcome these issues, this study proposed a novel deep neural network (DNN) architecture named GPRI2Net to simultaneously reconstruct the permittivity maps and categorize the object class labels from the GPR data of consecutive and long survey lines. GPRI2Net combined a convolutional neural network (CNN) based on DenseUnet and a recurrent neural network (RNN) based on bidirectional convolutional long short-term memory (Bi-ConvLSTM) to exploit the contextual information in and between the B-Scan segments extracted from the GPR data of a consecutive and long survey line. In addition, GPRI2Net performed inversion and object identification simultaneously using one network, which highly shared features between the two tasks and greatly reduced the computational complexity. Validation experiments were performed at two levels: first, using synthetic data based on the tunnel liner defects model and then using a sandbox model test in a realistic scenario. The results demonstrated that GPRI2Net can reconstruct consecutive permittivity maps and categorize the object class labels from GPR data with different dominant frequencies and survey line lengths and achieved a superior performance using the synthetic data compared to several other methods. Moreover, GPRI2Net also achieved satisfactory results using real-world GPR data.
An iterative multiscaling approach for solving the electromagnetic inverse scattering problem related to the imaging of shallow subsurface targets with the ground-penetrating radar (GPR) is proposed. The approach combines the zooming properties of the multiscaling technique with the reconstruction capabilities of an Inexact-Newton (IN) method developed in Lp spaces. It is based on multifrequency processing that allows one to face the ill-posedness of the inverse scattering problem by exploiting the regularization properties of a truncated Landweber (LW) method. Experimental data, extracted from radargrams obtained by the GPR in a real situation, are used for validation. The reconstruction results are also compared with those from competitive alternatives, such as a standard IN method or a state-of-the-art multifrequency Conjugate Gradient (CG)-based approach.
In recent years, deep learning (DL) is becoming an increasingly important tool for solving inverse scattering problems (ISPs). This paper reviews methods, promises, and pitfalls of deep learning as applied to ISPs. More specifically, we review several state-of-the-art methods of solving ISPs with DL, and we also offer some insights on how to combine neural networks with the knowledge of the underlying physics as well as traditional non-learning techniques. Despite the successes, DL also has its own challenges and limitations in solving ISPs. These fundamental questions are discussed, and possible suitable future research directions and countermeasures will be suggested.
An efficient multiresolution inverse scattering approach is presented for profiling high-contrast buried targets in large investigation domains (IDs). The proposed technique is based on an iterative multiscale approach (IMSA) that starts with coarse meshes and successively zooms in and marches toward the detected target location. Hence, by tightening the ID, the resolution could be enhanced without any increase in the number of meshes of the search domain. Here, the global evolutionary programming (EP) optimization algorithm is used in each step of IMSA, to guarantee the success of inversion process. In particular, an efficient variation in EP with Cauchy mutation is implemented for enhanced convergence. Moreover, a hybrid combination of multifrequencies (MFs) and frequency hopping (FH) schemes is combined with the proposed technique to better deal with the problem's nonlinearity and also simultaneously make a consistent trade-off between mesh length and frequencies. The proposed FH-MF-IMSA is evaluated by implementing it to various large-scale subsurface problems. Furthermore, the superior performance of this technique is shown by comparing its results with the standard IMSA, as well as with other well-known global optimization techniques.
Ground-penetrating radar inspection of vertical structures, such as columns or pillars, is relevant in several applicative contexts. Unlike conventional subsurface prospecting, where the medium is accessible only from one side, the columns can be probed from various sides with measurement domains possibly encircling the structure. This makes it possible to retrieve more information about the scene, thanks to an increased view and data collection diversity. This article proposes an imaging approach for structures probed all around via vertical scans. The approach faces the imaging as a full 3-D electromagnetic inverse scattering problem and accounts for the vectorial nature of the scattering phenomenon. Moreover, the imaging approach is based on an approximate model of scattering and the inversion is regularized by means of the truncated singular value decomposition to produce stable and accurate results. The reconstruction capabilities of the proposed imaging approach are evaluated in terms of the achievable spatial resolution. To this end, a numerical analysis exploiting synthetic data allows investigating how the imaging quality depends on the number of vertical scans. Reconstruction results referred to data gathered in controlled conditions provide an experimental assessment of the achievable imaging capabilities.
This paper is devoted to solving a full-wave inverse scattering problem (ISP), which is aimed at retrieving permittivities of dielectric scatterers from the knowledge of measured scattering data. ISPs are highly nonlinear due to multiple scattering, and iterative algorithms with regularizations are often used to solve such problems. However, they are associated with heavy computational cost, and consequently, they are often time-consuming. This paper proposes the convolutional neural network (CNN) technique to solve full-wave ISPs. We introduce and compare three training schemes based on U-Net CNN, including direct inversion, backpropagation, and dominant current schemes (DCS). Several representative tests are carried out, including both synthetic and experimental data, to evaluate the performances of the proposed methods. It is demonstrated that the proposed DCS outperforms the other two schemes in terms of accuracy and is able to solve typical ISPs quickly within 1 s. The proposed deep-learning inversion scheme is promising in providing quantitative images in real time.
An innovative nonlinear inverse scattering approach in variable exponent Lebesgue spaces is proposed for microwave imaging purposes. The main objective of the approach is to overcome one of the main problems associated to reconstruction procedures developed in Lebesgue spaces with constant exponent, i.e., the need for the selection of the optimum Lebesgue-space norm parameter, which is critical for obtaining accurate reconstructions, and no exact rules exist for this choice. The approach proposed in this paper is oriented to tomographic imaging applications at microwave frequencies, for which the adopted inversion procedure is based on a Gauss-Newton method, which has been reformulated in the unconventional variable exponent Lebesgue spaces. The capabilities of the approach are demonstrated by a set of numerical simulations, in which cylindrical dielectric targets are inspected. Moreover, an experimental result is reported. IEEE
Antenna effects alter the detection of buried objects during ground-penetrating radar (GPR) surveys. In this paper, we propose a novel approach based on full-wave inversion to filter out antenna effects from GPR data. The approach, which is exact for locally planar layered media, resorts to a recently developed electromagnetic model that takes advantage of an intrinsic, closed-form solution of Maxwell's equations to describe the antenna-medium system. As any multilayered medium can be reduced to a half-space medium with effective, frequency-dependent, global reflection coefficients, the method consists in inverting the radar data to retrieve a frequency-dependent, half-space complex conductivity. Converted into the time domain, this quantity represents the filtered radar image. We validated the approach through numerical simulations and laboratory experiments with pipes buried in a sandbox. The results demonstrated the validity of the concept and showed that the filtered radar images include only medium reflections, which means an easier interpretation in terms of medium structures. Antenna radiation pattern effects are, however, not removed. This physically based approach including the full-wave antenna model appears to be very promising for improved subsurface imaging and provides the basis for multifrequency GPR data fusion as the source is inherently normalized. IEEE
An iterative subsurface inverse scattering algorithm has been proposed to profile high-contrast 2-D dielectric objects buried under a lossy ground. The proposed iterative modified diffraction tomography (IMDT) method is based on the combination of the traditional iterative Born method and DT technique which is well known for its simplicity and robustness. In effect, IMDT is an iterative Born algorithm that utilizes the spectral domain concept of DT for solving the inversion problem. The proposed iterative approach results in removing the Born approximation's limitation of DT technique in dealing with high-contrast scatterers. To this end, the total field inside the reconstruction domain is renewed and then expanded into upgoing and downgoing plane waves in each iteration. Consequently, by exponential expansion of the fields and deriving a modified DT formulation, high-contrast targets are also efficiently reconstructed. To assess the proposed IMDT method, various high-contrast objects and noise conditions are studied. It is shown that the IMDT algorithm significantly outperforms the DT technique and is capable of reconstructing high-contrast objects efficiently and accurately even in noisy environments.
This letter addresses the problem of reconstructing the geometrical features, i.e., location, shape, and size of targets embedded into a nonaccessible region. In particular, an approach recently introduced for the case of free space and full aspect measurements is extended, discussed, and validated for the more realistic and challenging case of data collected under "aspect limited" measurement configurations, which include subsurface sensing, cross-borehole imaging, and many other cases. Results obtained by considering 2-D scenario and noise affected simulated data confirm the potentialities of the proposed method in dealing with realistic applications.
One of the main computational drawbacks in the application of 3-D iterative inversion techniques is the requirement of solving the field quantities for the updated contrast in every iteration. In this paper, the 3-D electromagnetic inverse scattering problem is put into a discretized finite-difference frequency-domain scheme and linearized into a cascade of two linear functionals. To deal with the nonuniqueness effectively, the joint structure of the contrast sources is exploited using a sum-of-ℓ₁-norm optimization scheme. A cross-validation technique is used to check whether the optimization process is accurate enough. The total fields are, then, calculated and used to reconstruct the contrast by minimizing a cost functional defined as the sum of the data error and the state error. In this procedure, the total fields in the inversion domain are computed only once, while the quality and the accuracy of the obtained reconstructions are maintained. The novel method is applied to ground-penetrating radar imaging and through-the-wall imaging, in which the validity and the efficiency of the method are demonstrated.
An innovative inverse scattering (IS) technique for the simultaneous processing of multifrequency (MF) ground-penetrating radar (GPR) measurements is proposed. The nonlinear IS problem is solved by profitably integrating a customized MF version of the particle swarm optimizer (PSO) within the iterative multiscaling approach (IMSA) to jointly exploit the reduction of the ratio between unknowns and uncorrelated data with a pervasive exploration of the multidimensional search space for minimizing the probability that the solution is trapped into local minima corresponding to false solutions of the problem at hand. Both numerical and experimental test cases are reported to assess the reliability of the MF-IMSA-PSO method toward accurate GPR tomography as well as improvements with respect to the competitive state-of-the-art inversion approaches.
Ground Penetrating Radar (GPR) is a non-destructive imaging system able to provide high-resolution images of the subsurface. From a theoretical point of view, it requires to solve an inverse scattering problem, where a set of parameters describing the underground scenario must be retrieved starting from samples of the measured electromagnetic field. In this chapter, an overview of different methods/algorithms for quantitative and qualitative buried scatterer reconstruction widespread in literature is provided.
An introduction to the most relevant theoretical and algorithmic aspects of modern microwave imaging approaches. Microwave imaging-a technique used in sensing a given scene by means of interrogating microwaves-has recently proven its usefulness in providing excellent diagnostic capabilities in several areas, including civil and industrial engineering, nondestructive testing and evaluation, geophysical prospecting, and biomedical engineering. Microwave Imaging offers comprehensive descriptions of the most important techniques so far proposed for short-range microwave imaging-including reconstruction procedures and imaging systems and apparatus-enabling the reader to use microwaves for diagnostic purposes in a wide range of applications. This hands-on resource features: A review of the electromagnetic inverse scattering problem formulation, written from an engineering perspective and with notations. The most effective reconstruction techniques based on diffracted waves, including time- and frequency-domain methods, as well as deterministic and stochastic space-domain procedures. Currently proposed imaging apparatus, aimed at fast and accurate measurements of the scattered field data. Insight on near field probes, microwave axial tomographs, and microwave cameras and scanners. A discussion of practical applications with detailed descriptions and discussions of several specific examples (e.g., materials evaluation, crack detection, inspection of civil and industrial structures, subsurface detection, and medical applications). A look at emerging techniques and future trends. Microwave Imaging is a practical resource for engineers, scientists, researchers, and professors in the fields of civil and industrial engineering, nondestructive testing and evaluation, geophysical prospecting, and biomedical engineering.
An electromagnetic inverse scattering approach for imaging of shallow subsurface objects is reported. It extends to multifrequency processing, an efficient method previously developed for single frequency imaging. The considered approach is an iterative procedure based on an inexact-Newton method developed in Banach spaces, which exhibits effective regularization capabilities and reduced over-smoothing effects. The approach is validated using numerical simulations in which cylindrical scatterers are reconstructed in a homogeneous lossy medium. Results are compared with those obtained using standard single frequency operating conditions.
In this paper, we present a new model-based linear inversion approach for cross-borehole ground penetrating radar quantitative imaging. The approach is reliable and computationally effective as it consists of the cascade solution of two linear inverse problems. The first problem yields a qualitative image of the targets (i.e., their location and approximate shape) and the information needed to cast a set of virtual experiments wherein a linear scattering model that implicitly depends on unknown targets holds true. By relying on such a model, it is possible to achieve, via linear inversion, a quantitative estimate of the targets' electric permittivity and conductivity in a much broader range of cases as compared with traditional approximations, such as the Born approximation. The quantitative imaging capabilities of the proposed method are enhanced by means of an original strategy, in which the features of virtual experiments are exploited to counteract the data reduction caused by the aspect limitation of the measurement configuration. Results against simulated data are reported to show the capability to successfully image nonweak scatters.
A nonlinear tomographic approach for microwave imaging of dielectrics buried under a rough surface is presented. It has been made possible to efficiently apply the contrast-source-inversion method, which is proven to be one of the most successful nonlinear inversion techniques when the Green's function of the background medium is available, to the given imaging problem. This has been achieved through the application of the buried object approach (BOA) which enables the calculation of the Green's function of layered media with rough interfaces by considering the roughness as a series of objects located alternately on both sides of a planar interface between two half spaces. Furthermore, the calculation of the Green's function of the two-layered medium with a planar interface required in the BOA has been accelerated through an adaptation of the two-level discrete complex image method. By making use of the strength of nonlinear inversion and fast and accurate computation of the Green's function of the layered media with rough interface, superior results have been achieved in a feasible computational time for dielectrics having constitutive parameters in a considerably wide range even if they are inhomogeneous or buried under substantially large rough surfaces.
Electromagnetic inspection techniques are becoming powerful tools for buried object detection and subsurface prospection in several applicative fields, such as civil engineering and archeology. However, the nonlinearity and ill-posedness of the underlying inverse problem make the development of efficient imaging techniques a very challenging task. In the present paper, an algorithm based on a regularizing approach in Lp Banach spaces is proposed for tackling such problems. The effectiveness of the approach is verified by means of numerical simulations in a noisy environment aimed at evaluating the reconstruction capabilities with respect to the choice of several model parameters. The reported results show that, for small targets, the use of Lp Banach spaces with 1 < p < 2 allows to obtain a better localization of different buried scatterers.
In GPR prospecting, strong reflections due to the background material interface can hinder the detection of localized buried scatterers, especially when the targets are close (in terms of probing wavelength) to the interface. Moreover, signals due to objects located outside the investigated domain and occurring in the observation time window may dramatically affect the reliability of the results. In order to mitigate such kind of clutter, an entropy based approach has been recently proposed in the frame of intra-wall diagnostic. In this paper, we assess the performance of such an approach by processing experimental data collected in laboratory controlled conditions and referred for the challenging situation of shallower dielectric and metallic targets, whose back-scattered fields overlap in time with the air-soil interface signal. In addition, a performance comparison of the proposed method is performed with other two approaches, i.e., the mean subtraction method and the subspace projection procedure.
In this contribution, an adaptive multiscale approach for the localization and characterization of buried objects in a half-space is proposed. The main goal of the approach is to reduce the number of elements to be estimated and so the degrees of freedom in the unknown profile. This leads to improvement of the robustness of the inversion and to an increase in the quality of reconstruction. The proposed inversion scheme is based on an adaptive, coarse-to-fine iterative strategy using spline pyramids. The global procedure consists of sequences of non-linear inversions separated by refinement steps, which overall produces an accurate, low-order representation of the sought object.
The characterization of conductive obstacles in subsoil is investigated in the induction regime. Validated by numerical experimentation, a simple model is proposed to calculate the main electromagnetic quantities of interest, the interaction between the obstacles being taken into account. The model is based on the extended Born approximation, the Lax-Foldy multiple diffraction theory, and introduction of equivalent spherical obstacles. Those are retrieved via a hybrid algorithm of differential evolution using a communication strategy between groups, which in particular enables separation of coupled obstacles close to one another.
This paper addresses the problem of reconstructing geometrical features of 3-D targets embedded into a nonaccessible region from multiview multistatic scattered field data. Sampling methods (SM) are simple and computationally effective approaches to pursue this task. However, their implementation requires a large number of multipolarization sources and probes. Moreover, their performances are often unsatisfactory for aspect-limited measurement configurations and lossy media. In order to tackle these drawbacks, usually faced in subsurface imaging, we propose a simplified and improved formulation based on the physical interpretation of SM. In particular, such a formulation relies on a small number of single polarization probes and exploits multifrequency data, for the first time in the framework of SM. The performances of the resulting approach are verified by monitoring 3-D regions of large extent.
In this paper, we consider a solution method of the inverse problem of imaging two-dimensional (2D) objects buried underground by cross-hole radar data in the time domain. In addition to less information on the targets due to restriction on the arrangement of transmitters and receivers than for full-view cases such as imaging of objects in free space, the large search region between boreholes makes solving the inverse problem difficult. Although iterative optimization approaches take long computing time, these approaches give much better image qualities for high-contrast objects than linear inversions such as a diffraction tomography. However, the reconstruction in a large search region with limited-view measurements often fails trapped in a local minimum. To overcome this difficulty, we propose a two-step iterative approach: the first step is to reduce the search region to a smaller one and the second step is the accurate reconstruction of the targets in the small region. Both steps are based on an iterative optimization approach, i.e., the forward-backward time-stepping method previously proposed. This two-step approach is tested for detection of tunnel-like objects surrounded by a heterogeneous background medium to evaluate its performance. Numerical results indicate the efficiency of the approach and its ability of circumventing local minima.
In inverse scattering problems, only a limited amount of
independent data is actually available whenever the finite accuracy of
the measurement set up is taken into account. The authors deal with the
problem of quantifying such an amount in the subsurface sensing case. In
particular, an alternative formulation of the problem is given which
also allows one to understand how to dimensionate the measurement setup
in an optimal fashion. Analytical results are reported for the case of a
lossless soil, while a numerical study is carried out in the general
case. By relying on the same formulation and tools, the authors also
discuss the kind of unknown profiles that can actually be retrieved. In
particular, it is shown that the class of retrievable functions exhibits
intrinsic multiresolution features. This suggests that adoption of
wavelet expansions to represent the unknown function may enhance the
reconstruction capabilities. Numerical examples support this conclusion
The aim of this paper is twofold. First, why and how exploitation
of multifrequency information is of great usefulness in inverse
scattering problems is discussed. Second, three different solution
strategies, all based on a recently introduced approach, are presented,
discussed, and compared in the actual case of noise affected data in the
two-dimensional (2D) scalar case. The first one is a (nonlinear)
frequency-hopping technique, which favorably compares with approaches of
the same kind that use linear inversion steps. As a second approach, the
contemporary use of the different frequencies data is considered.
Contrary to common assumptions, it is shown that such an approach may
perform better than the previous one, the different performances between
the two being dictated by the spatial frequency content of the unknown
contrast. Finally, a hybrid and, to the best of the authors' knowledge,
novel strategy, which exploits the advantages of the the previous ones,
is introduced and discussed
Multifrequency microwave tomography in Lebesgue spaces with nonconstant exponents
- Fedeli A.
Variable exponent Lebesgue space inversion for cross‐borehole subsurface imaging
- Estatico C.