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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book offers an overview of modern advances in Ground Penetrating Radar (GPR) for the reader hoping to understand comprehensive electromagnetic culture, combining instrumental development of radar, signal processing, imaging, and calibration/correction of measured data.
GPR has a multi-disciplinary character that can bring together a diverse and broad community. Of concern are the design and optimization of innovative radars, by virtue of the antennas and associated electronics, imaging algorithms, methodological diversity, calibration procedures, and the development of tools for the interpretation of data in mono-static or multi-static configurations within frequency or transient domains.
This book provides illustrations in civil engineering for the diagnosis of transport infrastructures and buildings, archeological surveys for the appreciation of cultural heritage, detection of underground pipes and cavities, estimation of soil water content for agriculture, and mapping of root trees developing underground, and in planetology, the analysis of the internal structure of planets and other celestial bodies through electromagnetic waves.
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Veröffentlichungsjahr: 2024
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DedicatedtoProfessor Matteo PastorinoAn Inspiring Teacher of Science and Humanity
SCIENCES
Waves, Field Director – Frédérique de Fornel
Instrumentation, Sensors and Measurements,Subject Head – Cyril Lupi
Ground Penetrating Radar
From Theoretical Endeavors to Computational Electromagnetics, Signal Processing, Antenna Design and Field Applications
Coordinated by
Mohammed Serhir
Dominique Lesselier
First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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© ISTE Ltd 2024The rights of Mohammed Serhir and Dominique Lesselier to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023943503
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78945-157-3
ERC code:PE8 Products and Processes Engineering PE8_3 Civil engineering, architecture, maritime/hydraulic engineering, geotechnics, waste treatment
Preface
Mohammed SERHIR1 and Dominique LESSELIER2
1Université Paris-Saclay, CentraleSupélec, SONDRA, Gif-sur-Yvette, France
2Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, Gif-sur-Yvette, France
The present volume, Ground Penetrating Radar: From Theoretical Endeavors to Computational Electromagnetics, Signal Processing, Antenna Design and Field Applications, proposed within the Waves field of the Sciences series, is in the subject “Instrumentation, Sensors and Measurements”, since it covers these three areas of interest. All the above elements are illustrated in this book, emphasis being on how to meet the constraints and how to investigate the ground penetrating radar (GPR) functioning accompanied by a wealth of experiments in properly controlled conditions. Yet, for the sake of completeness and in order to keep in touch with most GPR research, electromagnetic modeling and microwave imaging go far beyond, and they involve complex challenges, as illustrated by the chapter titles of this book.
The material, as proposed in the book, is comprised of eight closely related chapters. The essential aspects of each contribution are highlighted in this preface, however, we must emphasize that each chapter may not do justice to all of the aspects studied at length by the authors. Above all, the information is provided in a pedagogical way, within the framework of the key objectives of the Sciences series.
Chapter 1 is entitled “Electromagnetic Imaging of GPR Data: Theory and Experiments”. Its aim is to propose a careful analysis of tomographic approaches, with a strong focus on qualitative and linearized approaches, since those are quite handy in the field of study, due to their reduced computational burden and the fact that they produce quick yet still quite sound solutions to the end-user. Then, the proposed algorithms are tested on experimental data acquired in controlled conditions, and their pros and cons are cleverly illustrated. In terms of modeling, the approach is based on a Green’s formulation, scalarized according to the commonly made hypothesis of a transverse magnetic mode of polarization. If the analysis starts from a free space model, it quickly goes into the situation of a singly or multiply layered embedding as is necessary for GPR practices. The imaging strategy of choice is, as said above, qualitative. After considering the vast literature on methodologies of that kind, including recent ones by the authors based on discovering the boundaries of the targets and not sampling the whole space, the discussion is on the so-called linear sampling method (LSM), the orthogonality sampling method (OSM) and a more recently introduced method termed boundary inverse source and sparsity (B-IS). In the latter, the quest is of the boundaries of the target support, where equivalent sources are located whatever illuminations are assumed. Then, the sparsity comes into play as only a few non-zero pixels within a possibly vast domain of search are sought. The above is followed by a careful, substantive analysis of the different methodologies, and it is completed by a close look at linearized approaches, based on the Born approximation and the distorted Born approximation. The virtual experiments (VE) approach completes the previous analyses as it particularly enables a more general interpretation beyond LSM here. After a comprehensive discussion of regularization strategies, multistatic tomographic imaging is run on controlled laboratory data, concerning 3D landmines and pipes – those provided by Georgia Tech – and they show that the targets can indeed be identified with accuracy, even if the proposed solutions have been developed in a 2D setting, that is, with slices being superimposed at the end.
Chapter 2 is entitled “Nonlinear and Hybrid Inversion Techniques for Ground Penetrating Radar Imaging”. It focuses on nonlinear and hybrid methods specifically developed within the GPR application framework to map the electromagnetic parameters of buried objects (in the hypothesis of frequency-independent parameters) from multifrequency–multi-view data. The approach is built within Green’s framework. Once the problem is sketched properly into it, with the usual data and state integral operators, a nonlinearized deterministic method is proposed, based on an inexact Newton approach. The algorithm is discussed at length in the chapter, and argues that it is feasible to separate zones wherein an object is detected and nothing is detected. Then, attention is turned to hybrid inversion techniques, in which a qualitative delay-and-sum method is inserted in the above algorithm. In short, the qualitative method yields information on the presence or absence of discontinuities with respect to the background medium. Examples illustrate the algorithmic advances in a quasi-monostatic configuration where results of the qualitative method, the quantitative method and the hybridized method are compared with due detail, confirming the better quality of the latter.
Chapter 3 is entitled “Sensor Deployment in Subsurface GPR Imaging”. The starting point encompasses a variety of approaches dedicated to imaging buried targets (including through-wall cases) to exemplify the challenges faced and the solutions to which those are amenable. Then, the reader is reminded that the so-called linearized imaging solutions lead to a significant reduction of complexity, which in the GPR situation, with very demanding conditions of exploitation on terrain, can be very useful, if run properly, and if accepted that only detection, localization and rough estimates of shape can be achieved. This is the focus of this chapter, which gives careful explanations of the pertinent direct operator in the Green formalism, the inversion of which is then discussed at length and ends up with a so-called regularized reconstruction operator containing everything, and prone to a migration scheme; its theoretical analysis is investigated in-depth and it is illustrated by a simple pedagogical example. Then, the key point is how we deploy the sensors by managing the usual trade-off of having the least data to feed the algorithm with the proper sampling (among other factors) that fits the inverse operator properties (the spectral ones in effect). A careful analysis blending mathematics and numerics follows, considering optimal frequency and space samplings, to devise a proper solution, also carefully compared to solutions in the literature. A host of numerical results follows in various configurations, aiming at depicting the best data in terms of resolution, the cases of free and half-space with sensors away or on the surface of separation being specifically managed. Overall, a detailed mathematical analysis is presented for further reference, aiming at the optimization of sensors’ deployment, and a practitioner can thus, for example, imagine what they can get at best in a given situation, even if idealized for the sake of understanding by attentive linearization.
Chapter 4 is entitled “Ground Penetrating Radar Tomography for Cultural Heritage”. It is about providing a non-invasive assessment of buried/hidden structures in the field of cultural heritage applications. Indeed, in archaeological surveys, underground GPR images enable us to map a given site at some degree in advance in order to properly plan out excavation activities as well as to get precious knowledge about hidden features of the sites that cannot be excavated. There are numerous examples in Italy, as well as in many other places, such as the well-preserved floors of Roman “domus”, churches and a host of historical buildings. Moreover, GPR allows the performance of structural surveys to yield information useful for assessing the state of health of historical assets as well as for characterizing previous restoration actions. In this framework, a further enhancement of the GPR capabilities is by the means of tomographic data processing approaches based on proper models of electromagnetic propagation in complex scenarios, denoted as GPR tomography (GPRT) in the microwave domain. As analyzed previously, this should hopefully provide the end-user with a wealth of high-resolution images, more easily interpretable than standard radargrams. The chapter begins by reviewing the concepts underlying GPR and microwave tomography approaches with an emphasis on linear procedures and discusses effective implementations in detail. This enables the authors to get a sound basis for the practical investigations that follow. The latter are illustrated by several examples of GPRT surveys performed by the authors and colleagues at three famous archaeology sites. In the archaeological park of Paestum in Campania, rather wide areas in the park and close to it had been left unexplored by classical means. GPRT surveys with frequencies around 200 MHz ensure deep and fast surveys, which have allowed us to obtain impressive maps of potential structures, all of which have been subsequently confirmed by proper excavations. For the Knossos Palace in Crete, the main concern was to further examine the possible drastic restoration (by Evans) in the early 20th century of its walls and pillars and how it was deployed, with the need for a sharp resolution now involving the use of the much higher central frequency of 2 GHz. The third case is Paphos’ Tombs of the Kings in Cyprus, for which again earlier restorations were to be verified, especially for cracks and other fractures in the columns, with the potential to lead to further damage. This meant that again, GPRT was run in the 2 GHz range like in the previous case. To conclude, the new or at least additional generation of GPR systems mounted on unmanned aerial vehicles is considered, and their potential in the field of archaeological surveys is deliberated, in particular, how high permittivity of the underground can affect the subsurface probing itself if the antennas are operated in the GHz range, but a contrario enabling a rich surface mapping, as shown with a prototype tested at the archaeological parks of Paestum and Velia (and also in Campania).
Chapter 5 is entitled “The Full-Wave Radar Equation for Wave Propagation in Multilayered Media and Its Applications”. Its primary objective is to present and explain the far- and near-field formulations for full-wave modeling of radar data. Examples are given to illustrate the main features and demonstrate the potential or proven uses in the field. The far-field formulation assumes that the antenna is far enough away and that the field over the antenna aperture is homogeneous. On the other hand, the near-field formulation is a more sophisticated model that applies when the scattered field distribution depends on the medium’s distance and properties. The near-field model generalizes the far-field approach using the superposition principle, decomposing the field distribution into a sum of basic distributions using a set of point sources and receivers. Both formulations are developed within the framework of a planar-stratified lossy dielectric medium, and the spectral and spatial Green’s functions are explained for both. In both the far- and near-field models, the antenna characteristic functions do not depend on the medium distance or its properties. In fact, the far-field model is a particular case of the near-field model, specifically the generalized model with n = 1, where n represents the number of source and receiver points. Retrieving the antenna characteristic functions becomes more complex as n increases. When n = 1, the retrieval is analytical, but for n > 1, it is numerical. An analytical inverse estimation is still being sought. Remote sensing is performed by properly minimizing an objective function. The first application illustrated is soil moisture mapping using quad and drone, with many examples of interest given. The second application is gesture recognition within a given scene or through various materials, with examples provided. The authors also envision using this technology for breathing pattern recognition, which is still in its infancy regarding antenna engineering and processing. The authors conclude that the far-field methodology is well-developed, but further analysis of the near-field approach is necessary, particularly in regards to calibration issues. They propose a wealth of references for further reading on these aspects.
Chapter 6 is entitled “Assessment of Flexible Pavements by GPR: 20 Years of R&D in France”. It is clearly focused on a specific application, yet this application is essential if we wish to maintain roads in good condition at not too high a price, and GPR appears as the perfect tool for this challenge (even if interest is shown in other tools, like gammadensimetry and a host of mechanical devices, as underlined also by the authors). The contribution first proposes a neat and comprehensive analysis of the state of the art of road pavement inspections, with due, but not privileged emphasis on the development of solutions in France where a strong effort has been underway these last two decades and is on a high level nowadays. The first part is devoted to GPR systems and acquisition, as existing, with the peculiar situation of rapid diagnostic being underlined. Then, the reader will go on to find processing and interpretation, proposing a wealth of results to illustrate the main points. Afterward, complements on other non-destructive testing are provided, and a due analysis of their pros and cons is carried out. Then, possibly at the core of this chapter, a wealth of novel approaches is proposed, to enable better evaluations rather than via conventional processing, which may fail unless a stringent array of hypotheses on the pavement is fulfilled. These approaches, discussed in detail, come under the name of high-resolution time-delay algorithms. Yet full-waveform inversion also appears to have high potential, as underlined by the authors, and many investigators, with the promises of artificial intelligence which are specifically mentioned. Tomographic synthetic aperture radar is also shown to be worthwhile to further investigate, as it is delivering fast and highly accurate maps. This is one of the conclusions of the chapter, among many others.
Chapter 7 is entitled “GPR for Tree Roots Reconstruction under Heterogeneous Soil Conditions”. Indeed, the main idea is that we cannot analyze – save exceptions – root tree systems in the way needed for civil engineering to optimize the natural environment and biomass estimation with destructive means, unless a single experiment (e.g. excavation) is attempted for comparison. GPR then appears to be a good and possibly cheap way forward for obtaining further knowledge on those systems with the least amount of hassle. The contribution, after an elaborate discussion of root systems and their environment, develops a processing framework that is valid for complex soil conditions and a variety of root architectures, and it shows how it applies using specifically tailored antennas to map a root system, with both experiments in the laboratory and the field to illustrate the advances and to pinpoint open challenges. This chapter does not emphasize sophisticated solvers, that is, FD-TD which is the (agile powerful) workhorse chosen to obtain the simulated data that are helping the authors, and which uses rather classical processing tools, to remove the clutter, estimate the soil permittivity, and in the end produce well-focused images quickly. This pragmatic yet in-depth investigated approach should speak to many end-users in the GPR community. The key role of tree roots in nature and their climatic evolution should appeal to many readers, particularly since it was Leonardo Da Vinci who was one of the very first to have hypothesized the architecture of tree roots in a mathematical fashion.
Chapter 8, entitled “Sounding Radars and Ground Penetrating Radars Designed for the Exploration of our Solar System – Focus on Planet Mars”, beautifully ends the volume, as it takes us to the fascinating environments of planets and asteroids, and investigates the development and usage of sounding radars and GPRs as essential elements of instrumental payload in almost all planetary missions. To enable the reader to grasp the present advances, first an overview of past, current and future radars developed for planetary exploration is proposed, illustrated by carefully chosen examples showing the breadth of the efforts made so far. Then, because there are evidently many constraints that impact the design and technical performances of a radar consequently, data interpretation is heavily weighed on. After a proper analysis of these issues, the author focuses on the characterization of the Martian subsurface, which has already been envisaged by radars both in orbit and on the ground. The detail of the modes of operation, studies carried out from the collected data, and lessons concerning the diagnoses of subsurface structures are appraised in succession. It is particularly seen that, among other conclusions, a qualitative understanding of the subsurface configuration is rather easily obtained from processed GPR images, yet quantitative analyzes are needed to support the attempted identification of subsurface materials and in-depth understandings of the environment. However, the wide variety of sounders operating within different frequency ranges does provide lots of complementary results and yields significant inputs to tough scientific questions, while knowledge of buried geological structures is indeed essential to understanding the mechanisms of formation and the evolution of the sites studied. Then, the development of two radars to probe the Martian subsurface is discussed: Electromagnetic Investigation of the SubSurface (EISS) and Water Ice Subsurface Deposit Observation on Mars (WISDOM): designed for the purpose of the ExoMars mission. As of today, the first has been “descoped”, so insistence is on the second, as programmed later. However, the work led around them is useful in any case as underlined. In the context of Martian exploration, the WISDOM GPR frequency step 500 MHz–3 GHz radar has been developed to meet the needs for characterizing the subsoil of Mars, as part of the ExoMars Rover Payload Instrument Suite, this again taking place with severe constraints linked to the spatial context (very limited mass, volume and consumption). Thanks to the great results obtained by soundings radars (on Mars, the Moon and Titan) and GPRs (on Mars and the Moon), it is then concluded in the last section about a series of radars that are currently being developed for the future missions that will characterize the subsurface of other bodies of our solar system.
To conclude this preface, we will present a host of questions that still arise today and are certainly not fully answered within the present volume. Yet, in our opinion, as conveners, this book encompasses many tough issues already raised, and this is what makes the analysis a good read both on an educational level as well as on a research level, for theoreticians, numericians, as well as engineers, and practitioners. We can mention: (i) multi-sensors and true three-dimensional modeling, the consistency of Green’s scalar and dyadic formulations of the electromagnetic fields (those may be the basis of solution methods, the mathematical analysis of which can then go quite in-depth and provide further understanding of the imaging itself, since, for example, FD-TD is very powerful, but may seem less amenable to use whenever inverting the signals themselves) and/or related quantities (the well-known S parameters as the most relevant ones); (ii) operating environments since combining dispersive, heterogeneous and in effect heavily uncertain cases; (iii) optimized sparse acquisitions in order to save the precious time of the handler on terrain; (iv) (almost) real-time signal processing and image production with a view to end-user decision-making; and (v), it goes without saying that the fact that any signal is produced/collected by real antenna structures that are also claiming for careful optimizations within the context of the applications envisaged (a set of those applications is discussed in the several chapters as proposed, yet they are certainly not unique), and all of these open questions are recognized as not exhaustive. In addition, deep learning in the present context could have a full volume devoted to it, and is hinted to at least by several contributors, since GPR is a clear case of epistemic and aleatoric uncertainty as is now discussed in the deep learning community under various guises. As for the need to smartly mix possibly sparse and certainly erroneous experimental data with, in principle, perfectly simulated data yet completed in a proper way by cleverly built errors (not reduced to additive white Gaussian noises as too often in truth), while building up the needed networks, it remains, to say the least, a demanding issue.
Finally, as coordinators of the present volume, we would like to say that it would not be without the sophisticated and timely contributions of the authors, nor without the impetus given to us by Dr. Frédérique de Fornel, who is cleverly managing the Waves field of the Sciences series. It is true that a number of other contributors in the field of GPR could have provided their research in this book, but we are certainly not pretending to be exhaustive, mainly in the hope that this volume not only focuses on the right tools in the demanding field of GPR at the highest level of research, but that it also attracts students and young scientists to this field, as it claims many new developments in harmony with concrete and indeed the increasingly urgent demands of society in a wide range of fields.
June 2023
1Electromagnetic Imaging of GPR Data: Theory and Experiments
Michele AMBROSANIO1,3, Martina Teresa BEVACQUA2,3, Tommaso ISERNIA2,3 and Vito PASCAZIO1,3
1 Dipartimento di Ingegneria, Università degli Studi di Napoli “Parthenope”, Naples, Italy
2 Dipartimento di Ingegneria dell’Informazione delle Infrastrutture e dell’Energia Sostenibile, Università Mediterranea di Reggio Calabria, Italy
3 Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT), Parma, Italy
Ground penetrating radar (GPR) represents a powerful technology able to investigate non-accessible scenarios in a non-invasive and non-destructive way, as witnessed by the numerous and different applications, which include demining, lunar explorations, archaeology, geology and civil engineering (Persico 2014; Sehrir et al. 2017; Aboudourib et al. 2020). However, standard GPRs provide a user-dependent map of the underground and subsurface targets, known as “radargram”, whose interpretation requires firm human expertise and does not allow us to infer about the nature of the targets buried below the soil, unless some a priori information is available.
In this respect, tomographic approaches can play a very relevant role (Ambrosanio et al. 2019). These techniques aim at providing images regarding the variations of dielectric permittivity and electrical conductivity profiles of the scene under investigation, which are more reliable and readable than those achieved by using standard GPR data processing. Image formation requires an accurate modeling of the scattering phenomena to properly take into account the complex interactions among sources, receivers, ground and buried objects. As a result, the development of robust and reliable recovery algorithms still has some issues related to both modeling errors as well as the presence of uncertainties in the data. Indeed, the underlying inverse scattering problem (ISP) is both nonlinear and ill-posed (Bertero and Boccacci 1998; Colton and Kress 1998).
Nowadays, most of the tomographic approaches present in the literature can be classified into three main categories (Ambrosanio 2019). The first class includes qualitative methods, which aim at detecting the scatterers hosted in the imaging domain (ID) and retrieving only a limited amount of information, such as scatter location and shape. The second ones, known as approximated methods, adopt some of the approximations of the scattering phenomena, which allow us to simplify the mathematical model and to reduce the computational complexity. Lastly, the nonlinear quantitative methods aim at retrieving both electrical and morphological properties of the ID, by tackling the problem in its full nonlinearity, without involving any approximation. A further class of imaging strategies includes hybrid approaches, which are usually based on a first qualitative step that aims at providing some rough information, that is then exploited in the next quantitative steps to avoid false solutions and to allow for reliable and effective recoveries.
In this respect, this chapter aims at providing to the research community an overview of all the main aspects on tomographic approaches in the framework of the underground exploration of the soil by GPR technologies, with particular attention given to qualitative and linearized approaches. Indeed, they can be relevant in GPR prospecting due to their simplicity and reduced computational burden.
The chapter is organized as follows. In sections 1.1 and 1.2, the equations modeling scattering problem underlying tomographic approaches is reported and extended to the case of planarly layered media. Section 1.3 introduces the basics of qualitative methods, in particular the linear sampling method (LSM), the orthogonality sampling method (OSM) and the boundary retrieval through inverse source and sparsity (B-IS) for the half-space case. In section 1.4, linearized methods are discussed, while section 1.5 is focused on regularization techniques. Finally, in section 1.6 experimental imaging reconstructions are discussed and reported from (Ambrosanio 2021).
1.1. Inverse scattering problem: mathematical formulation
In ISPs, an object-of-interest (OI) is located inside an ID Ω, which is then contained in the problem domain. The problem domain is immersed in a background medium (ε′b, σb, μ0) whose electrical properties are known and usually assumed to be homogeneous. It is worth noting that, for the aim of this chapter, only non-magnetic targets and background will be considered. The ID is surrounded by a boundary Γ of general shape and size depending on the employed imaging setup on which the measuring probes are located. In all of the following applications, a multi-view–multistatic system will be employed, that is, an imaging system made up of more transmitters and receivers in which the samples of scattered fields are collected at receiver locations as per each transmitting antenna.
Figure 1.1.Problem statement. The picture illustrates the incident electric fieldEinc and the scattered electric fieldEscatt collected at receivers locations on the curve Γ. The background is a non-magnetic medium (ε′b, σb, μ0) and the targets are in located the imaging domain Ω
The whole scenario has been sketched in Figure 1.1. In general, it is possible to define the complex permittivity as:
in which ε′ and σ are the relative permittivity and conductivity, respectively, f is the operating frequency, is the position vector identifying each point inside the ID, j refers to the imaginary unit and ε0 = 8.854 × 10−12 F/m is the permittivity value in the void and represents a universal constant. Thus, by adopting the definition in equation [1.1], it is possible to introduce the contrast function “”:
in which .
In this framework, two different electromagnetic scattering problems can be identified: the first problem, also known as forward problem, consists of evaluating the scattered field due to the object of interest inside the ID, assuming as known quantities both the incident field “” and the contrast function “” that encodes the targets properties; the second problem, also referred to as inverse problem, aims at estimating the unknown contrast function “” from the samples of the scattered field “” collected on Γ and from the illuminations “”. The two problems are strictly related to each other; as a matter of fact, a whole class of inversion procedure exploits the forward problem as a fundamental step for obtaining an estimate of the contrast function by means of an iterative strategy.
In order to treat these scattering problems properly, it is convenient to introduce the two equations:
in which ω = 2πf, where “(v)” represents the “V” illuminations of the multi-view–multistatic microwave system whose incident field impinges on the scatterers located in the ID, “” refer to the incident, total and scattered field at the vth illumination, respectively, while “” and “” are the so-called internal and external dyadic Green functions of electric type that describe the relation between the contrast function and the field generated both inside the ID Ω and the scattered field measured along the boundary Γ. These functions define the two operators introduced previously, also known as internal and external integral radiating operators “” and “” or domain and data operators, respectively. It is easy to note that such operators are bilinear related to the pair . These bilinear operators are defined as and , with X ⊂ L∞(Ω) being the subspace of the possible contrast functions, T ⊂ L2(Ω) being a proper subspace for the total electric field inside the object, and Si ⊂ L2(Ω) and Se ⊂ L2(Γ) being two subspaces for the scattered field inside and outside the ID.
Rewriting equation [1.3] by exploiting the operator notation as:
and substituting it in the data equation [1.4], we obtain:
where “I” is the identity operator and the superscript “−1” denotes the inverse operator. Note that although the data and domain operators themselves are linear, the relation between the scattered field and the contrast function is nonlinear. The inversion of the operator in [1.6] is a mathematical representation of the nonlinearity associated with the ISP. Such an issue is handled by casting the ISP as an optimization algorithm over variables representing the unknown electromagnetic properties of the objects to be retrieved, and it occurs because of the information related to the multi-view–multistatic system.
It is worth observing that equations [1.3.] and [1.4] can be rewritten in a different form by introducing the quantity:
which is known as contrast source. In the following, and for the scope of this chapter, the case of transverse magnetic (TM) mode will be considered. Under this assumption, the vectorial equations [1.3] and [1.4] become scalar, since only one component of the electric field is expected.
1.2. Inverse scattering problem in case of planarly-layered media
In section 1.1, a brief introduction on the mathematical statement for the ISP in a homogeneous, unbounded background medium was considered. In this section, a more suitable formulation of the concepts described in the theory of the planarly-stratified media will be dealt with.
Among all the quantities introduced previously, the main actors are still the “incident field” in the ID, the “scattered field” collected at receiver locations and the internal and external Green’s functions.
For the aims proposed in this chapter, the derivation of the two-dimensional (2D) planarly-three-layered medium (known as “slab”) will be considered, whose case also includes the planarly-two-layered one. These formulations belong to the framework of the simplified 2D intra-wall imaging (IWI) and GPR applications, respectively. In the following, the ID will be assumed as buried in medium 2.
To evaluate the electric field generated by a filamentary source located in a simplified 2D planarly-layered medium, the case of a single plane-wave in medium 1 impinging on the planar interface of a slab will be considered. Finally, by exploiting the Sommerfeld’s (1964) identity and integrating on all the spatial frequencies, a closed-form for the incident field generated by a filamentary source in medium 1 and observed in medium 2 will be obtained.
Figure 1.2.Sketch of a planarly-three-layered medium
For a generic 2D TM wave, the scalar electric field for a three-layered medium with source located in (x′, z′) and field observed in (x, z) can be written as:
in which , with i, j = {1,2} and , d2 being the depth of the related interface, and A2 is defined as:
in which and , i, j = {1,2,3}, refers to the transmission and reflection coefficients at interface between media i and j for the TM mode, respectively, A1 is the amplitude of the incident field and d1 is the depth of the first interface (between mediums 1 and 2).
By choosing the reference coordinate system as making the x axis coinciding with the interface between mediums 1 and 2 (i.e. z = −d1 = 0) and assuming d = d2 − d1 and unitary amplitude for the incident field in medium 1 (A1 = 1), then equation [1.8] becomes:
where and .
Finally, by exploiting the inverse Fourier transform for equation [1.10], we get:
Thanks to the reciprocity, it is quite easy to evaluate the external Green function by replacing the observation point (x, z) with the source location and the probe location (x′, z′) with the receivers (Catapano et al. 2004). Finally, the internal Green’s function can be written as (Chew 1995):
which, by adding and subtracting the quantity , making some calculations and exploiting the inverse Fourier Transform along the kx direction results in:
where Δx = x − x′, Δz− = z − z′ and Δz+ = z + z′ (Crocco and Soldovieri 2003). It is easy to derive the previous functions for the case of a two-layered medium (GPR) since it is sufficient to impose R23 = 0.
In particular, for the three-dimensional (3D) half-space case, it is possible to evaluate analytically the expression of the Green’s functions by exploiting similar calculations, which drive into the formulas proposed in Lo Monte et al. (2010) and Lo Monte et al. (2012) that, for the sake of brevity, will not be dealt with in this chapter.
1.3. Solution strategies
The electromagnetic ISP defined by the previous equations is intrinsically nonlinear in the relationship between data and unknowns. However, features that are quite attractive for any inversion approach are the requirement of being fast, computationally effective and reliable from the point of view of reconstructions, trying to exploit as little a priori information as possible. Moreover, such approaches should also avoid the occurrence of false solutions, also known as “local minima” (Isernia et al. 2001).
Nowadays, all the approaches present in the literature for the solution of EIS problems can be classified into three main categories, which are as follows:
qualitative methods
, where only a limited amount of information is provided, such as scatterer location and shape;
linearized methods
, which include famous approximations such as
physical optics
and
Born approximation
(BA), or even high-order approximate models;
nonlinear optimization methods
, in which an approximate solution is obtained by means of an iterative process starting from an initial guess. This class will be not dealt with.
The following sections will be focused on the case of qualitative methods and linear approximations, considering some of the most adopted approaches in practical GPR scenarios.
1.4. Qualitative methods
To overcome the difficulties related to nonlinearity and ill-posedness of the inverse problem underlying GPR imaging as well as the limited number of measurements, many studies have been focused on the introduction of solution methods for the corresponding inverse obstacle problem, whose scope is to recover only the morphological information about the targets, namely their supports, giving up their electromagnetic properties (Cakoni and Colton 2006). Such methods are known as qualitative methods and are different from quantitative methods, which aim at retrieving both electrical and morphological properties of the region of interest (Cakoni and Colton 2006). Indeed, the latter tackle the problem in its full nonlinearity, without involving any approximation and allow to widen the class of retrievable objects considerably. Unfortunately, they are characterized by a higher computational burden and long processing time. On the other hand, qualitative methods are characterized by simplified mathematical models and lower computational burden. Part of the discussion about qualitative methods reported below can be found detailed in the recent paper from the same authors of the chapter (Bevacqua 2019).
Among qualitative methods, it is worth mentioning sampling methods that include the LSM (Colton 2003) (and the related factorization method; Kirsch 2008) as well as the OSM (Potthast 2010). The main idea is that of sampling the investigation domain in an arbitrary grid of points and computing in each point an indicator function whose energy will assume different values depending on whether the sampled point belongs or not to the scatter (see, for instance, Figure 1.3). Although both LSM and OSM belong to the class of sampling methods, they exhibit different features. For instance, the LSM indicator function is computed by solving an auxiliary linear problem, whose solution involves the adoption of a regularization technique. In a different fashion, the OSM does not require the solution of a linear problem, as the indicator function is just related to the evaluation of a scalar product. Moreover, the OSM seems to be more flexible with respect to the measurement configurations.
By taking advantage from recent results in area of compressive sensing and sparsity promoting techniques (Donoho 2006), a new qualitative method has been very recently introduced in literature which, provided some conditions hold true, is able to retrieve the boundary of unknown targets rather than their support (Bevacqua 2017, 2018). Such a method exploits the equivalence theorem in case of dielectric obstacles and takes advantage from the particular distribution assumed by the induced currents in case of perfect electric conductors (Bevacqua 2017). Differently from sampling methods, it does not sample the investigation domain in a grid of points, and it does not require the selection of a fixed threshold to discriminate between points inside and outside the targets. It just retrieves the boundary of the targets (see, for instance, Figure 1.3). However, if compared with LSM and OSM, it shows a heavier computational burden.
Figure 1.3.Normalized indicators against experimental data: the TwinDielTM target. 8GHz and number of measurements M = 36. (a) LSM, (b) OSM and (c) B-IS. Extracted from Bevacqua (2019)
1.4.1. Linear Sampling Method
The LSM is one of the most popular qualitative methods to retrieve objects’ support from the measurements of the corresponding scattered field data (Colton 2003). In detail, it consists of sampling the region under test into an arbitrary grid of points and solving for each point the so-called far-field integral equation (FFE) given by:
wherein ξ represents the unknown function, is the sampling point and Esct∞ is the scattered far-field pattern measured on a closed circle curve Γ in the far zone of the scatterers in direction when a unit amplitude plane wave impinges from direction . Ge∞ is the Green’s function in the far-field zone. Despite the linearity of [1.14], the evaluation of the solution ξ is not straightforward due to the compactness of the operator at the left-hand side, which implies ill-posedness in its inversion (Bertero 1998). Hence, a regularization technique is required. An effective choice is that of solving it by adopting the Tikhonov regularization (Catapano et al. 2007). Then, the estimation of the unknown support is pursued by evaluating the l2 norm (i.e. the “energy”) of the unknown function ξ with respect to given by:
The above defined indicator depends on the sampling point and assumes a finite value when the sampling point belongs to the unknown object, while it blows up elsewhere (Catapano et al. 2007). As such, by selecting a fixed threshold, the indicator [1.15] allows the discrimination between points inside and outside the scatterers and finally retrieve the support of the target (Catapano et al. 2007). It is worth underlining that the computational burden is low, as the solution of [1.14] only involves the evaluation of the SVD of the measured data. The method has been also extended to the case of subsurface imaging (Catapano et al. 2008).
Note that, in case of a lossy background, there is the need of a modified indicator due to the attenuation of the field radiated by a buried electric dipole according to a factor which depends on the amount of losses in the host medium. Then, the energy of the LSM solution is normalized to the energy of the known term of the FFE. As a matter of fact, the field to be matched in the LSM equation becomes progressively less intense when the distance between the sampling point and the measurement domain grows. Interestingly, the presence of losses is not the only case, wherein the modified far-field equation can play a useful role. For example, for a given measurement domain and a given domain under test, in case the directions of the test dipole (whose polarization is chosen equal to the transmitting one) and the measurement dipoles do not match, the different amount of polarization mismatches, which is experienced for different depths implying very different energies for the right-hand term (Catapano 2008).
1.4.2. Orthogonality sampling method
The OSM has been recently introduced in literature in case of free space (Potthast 2010). It consists of computing the reduced scattered field from the far-field pattern . Such a reduced field can be computed by evaluating the scalar product between the far-field and the Green’s function in far-field zone, that is (Potthast 2010):
where 〈·,·〉 denotes the scalar product and γ
