User manual | Accurate Ground Penetrating Radar Numerical

Accurate Ground Penetrating Radar Numerical
Modeling for Automatic Detection and Recognition
of Antipersonnel Landmines
DISSERTATION
zur
Erlangung des Doktorgrades (Dr. rer. Nat.)
der
Mathematisch-Naturwissenschaftlichen Fakultät
der
Rheinischen Friedrich-Wilhelms-Universität Bonn
vorgelegt von
Maria Antonia Gonzalez Huici
aus
Ordizia (Spanien)
Bonn, Dezember 2012
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen
Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn
1. Gutachter: Prof. Dr. Andreas Hördt
2. Gutachter: Prof. Dr. Andreas Kemna
Tag der Promotion: 14.10.2013
Erscheinungsjahr: 2013
A mi familia
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Acknowledgments
This research project was funded by the Federal Office of Defense Technology and Procurement
(BWB), an agency of the German Ministry of Defense, and carried out in the Fraunhofer
Institute for High Frequency Physics and Radar Techniques (former FGAN e.V.), Wachtberg,
Germany.
This dissertation could have never been completed without the help, support and efforts of a
lot of people. Firstly, I would like to express my deepest gratitude to my initial mentor and
co-supervisor of this thesis, Prof. Andreas Hördt, for introducing me the methods of applied
Geophysics and for his crucial support and assistance, in particular, throughout my first stage
at the University of Bonn. Similarly, my most sincere thanks go to Prof. Andreas Kemna, for
accepting the co-supervision of this thesis and for all his valuable comments and recommendations.
I also want to profoundly acknowledge my project director in the Fraunhofer Institute FHR, Dr.
Udo Uschkerat, for giving me the opportunity of working with him in this project and for his
fundamental guidance and insight during all these years. I feel in depth too with the Passive
Sensor Systems and Classification department director in FHR, Dr. Joachim Schiller, for taking
care every time there was a problem and for his constant disposal to help. Also my gratitude to
our secretary, Ms. Winandy, always cheerful and ready to make the things more simple.
I am be very grateful to my friends in the Fraunhofer, specially, to my dear Macarena, who gave
more colour to my routine inside and outside the institute, to Alfred, for making my life easier
several times and for countless funny moments in the climbing wall, to Angel, for his honest
friendship and so many personal and technical advices, to my friend Mariano, for our endless
conversations, his help and the productive discussions about antennas and hardware during the
coffee breaks; also a mention to Fernando, for his encouragement and last review reading of this
thesis, to Carlos, Iole, Giulia, Diego, Robert, Mario, Chris, Omar, Jens, Christian, Thomas and
all the colleagues who along these years have contributed to make me feel ‘at home’.
In addition, gratitude words to some GPR and microwave colleagues who I had the luck to meet
in annual conferences: to my friend Merchi Solla, who I appreciate as person and as a researcher,
and with whom I feel always happy to share a post-conference trip, to Alex Novo, for your
humor and sympathy and for the nice days spent traveling in China! You know I am really in
depth with you...I dedicate also some words to Clemente Cobos, who I met in the best GPR
workshop ever...the one in Granada!, I am very thankful for the beautiful days spent together in
Andalucı́a and in Bonn. Sincere thanks as well to José Luis Gómez, who gave me new strength and
motivation, and to my friend Sebas, for his brilliant theoretical suggestions and all the fantastic
shared moments, specially during your research stay in the FHR and my visit to Murcia.
I would also like to convey sincere thanks to the colleagues in the Leibniz Institute for Applied
Geophysics (LIAG) in Hannover, Dr. Jan Igel and Dr. Holger Preetz, for providing the test field
for the measurement campaign and their friendly aid and assistance.
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This thesis is also dedicated to my dearest friend Naroa whose continuous support and emphasis
has decisively contributed to the completion of this work, eskerrik asko!, to Sandra, for your close
friendship, our great girl dinners and conversations and the beautiful time spent together, to
Julia, for your loyal friendship and your great help with the German language, to my dear Anne
Drew, another mountain lover...thank you for the fantastic months enjoyed together practicing
our favorite sport and for that nice trip to the Alps; to Alejandro Garcia, for your big encourage at
the very beginning, to Peter who has shared with me many funny moments since my early times in
the student dormitory, to Ricardo for several meals, parties and thrilling theoretical discussions
during my first years in Bonn, to Nesrin, for the interesting conversations and pleasant walks
along the Rhein, to Suni, my very first friend in Bonn, and to Jenny, Andrés, Daniel, Felix, Bram
...and all the friends who were with me throughout this long journey.
A mis companeros de carrera, Fabri, Antonio, Santi, Juan Francisco..., y en particular a Raquel,
quien desde la distancia y a pesar de nuestros largos silencios, todavı́a mantiene un lugar muy
grande en mi corazón. A mi prima Maribel, por tu constante apoyo, por tantos viajes y aventuras
compartidas y porque eres como una hermana para mı́, y a mi prima Noelia, que con su optimismo
y vitalidad siempre fue capaz de transmitirme energı́a positiva.
Very warm thanks to my beloved Fabio, who has permanently motivated and accompanied me
throughout the latest phase of this work. Thank you for making me smile even in the moments
of more stress. Ti amo.
Finalmente, quiero expresar mi gratitud más profunda a mi familia, a los que quiero con locura.
A mi hermano, que siempre ha sido un modelo y referente para mı́. Especialmente, a mis padres,
que desde la distancia han estado constantemente presentes apoyándome de manera incondicional
y transmitiéndome todo su cariño y fuerza. Gracias por creer en mı́, sin vosotros nunca habrı́a
llegado hasta donde estoy. Y finalmente a mis abuelos, allá donde estén, en particular a mi yaya,
que siempre permanererá de manera especial en mi recuerdo.
Bonn, November 2012.
Marı́a A. Gónzalez Huici
Zussamenfassung
Der Bodenradar (Ground penetrating radar, GPR), der flach vergrabene Objekte mit niedrigem
dielektrischem Kontrast durch non-invasive Messung des Untergrunds aufspüren kann, gilt als viel
versprechende Technologie für die Abbildung wenig oder kein Metall enthaltender Landminen.
Aufgrund der schwachen Radarreflexion dieser Minen und des Auftretens unerwünschter Effekte
wie Antennenkopplung, System Ringing und Reflektionen der Oberflche und des Bodens, stellt
sich dies als besondere Herausforderung dar. Diese Effekte können die Antwort des Zielobjektes
verdunkeln und die numerische Modellierung bietet die Möglichkeit ein solch komplexes Problem
Rückstreuung (Backscattering) zu analysieren.
Die Aufgabe, die sich im Rahmen der Detektierung von Landminen stellt, ist demnach nicht
nur, diese zu finden, sondern auch die Rate des falschen Alarms aufgrund von weiterer Stördaten
(Clutter) zu verringern, d.h., die Reflektoren zu identifizieren und eine präzise Modellierung als
notwendige Grundlage für die korrekte Interpretation des durch den GPR gewonnenen Outputs
zu erstellen.
In der vorliegenden Arbeit beschreiben wir den GPR Modellierungsprozess sorgfältig hinsichtlich
Frequenz- und Zeitbereich und entwickeln ein vollständiges Modell eines realistischen GPR
Szenarios, welches validiert und angepasst wurde, bis eine zufriedenstellende Übereinstimmung
zwischen Freiraum-Messungen und Simulationen erreicht wurde. Dieses Modell beinhaltet eine
genaue Aufführung des aktuellen GPR Impuls-Systems, der Schnittstelle, des Bodens und der
Zielobjekte und wurde mittels der Methode der finiten Elemente (FEM) berechnet.
Das Antennenmodell wurde den Dimensionen und bekannten Charakteristika unseres GPR Systems entsprechend erstellt und zunächst dahingehend optimiert, Impedanz und Antennencharakteristik mehrerer Konfigurationen im Frequenzbereich zu analysieren, um eine passende Antenneneffizienz und -richtcharakteristik sicher zu stellen. Es vergleicht weiterhin das simulierte
Übersprechen der Antenne und Signaturen einfacher Zielobjekte mit den Messungen im Zeitbereich. Bei dem Boden wird von einem nicht-streuenden verlustbehafteten Medium ausgegangen,
welcher heterogen und dessen Oberfläche uneben sein kann. Die statistische Verteilung, die der
Beschreibung von Topografie und Bodeninhomogenität zu Grunde liegt, wird im Detail erläutert,
und kann leicht über nicht konstante elektrische Parameter und variable oberflächenhöhe in das
Modell eingegliedert werden. Die Zielobjekte werden exakt durch entsprechende CAD-Modelle
beschrieben.
Das oben beschriebene Modell konnte nun genutzt werden, Zeitbereichsignaturen für unterschiedliche Testminen und kleine Objekte unter mehreren Oberflächen- und Bodenbedingungen
zu erhalten. Die simulierten Antworten geben uns ein weitreichendes Verständnis über die Faktoren, die die elektromagnetischen Streuungen durch kleine vergrabene Objekte kontrollieren. Sie
dienen der Interpretation der Signaturcharakteristika bezogen auf Zielobjekt und Hintergrundparameter. Die daraus gezogenen Schlussfolgerungen werden zuletzt zusammengefasst und erste
Betrachtungen zur Erstellung einer repräsentativen Datenbank für die existierenden Landminen
werden kurz erörtert.
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Schlielich wurde bewiesen, dass die berechneten Zielsignaturen zufriedenstellend als Referenzsignale für eine effiziente Clutter-Unterdrückung und verbesserte Landminendetektion/-erkennung
eingesetzt werden knnen. Um dies zu tun, definieren wir eine kombinierte Strategie aus
einem Energie-basierten Detektionsalgorithmus und einer Kreuzkorrelation-basierten Identifikationstechnik. Letztere kann vor dem Aufbringen der Detektion als zusätzlicher Filterschritt
in Form einer Ähnlichkeitszwangsbedingung zwischen gemessenen und synthetischen Signalen
umgesetzt werden. Die vorgeschlagene Methodik wird mit experimentellen Daten in einem inhomogenen Testfeld am Leibniz-Institut für Angewandte Geowissenschaften LIAG in Hannover
(Deutschland) validiert, wo mehrere Minensimulanten und Testziele vergraben waren. Im Besonderen ergibt die Anwendung des kombinierten Verfahrens mit experimentellen Daten eine deutliche Verbesserung der Detektionsrate, insbesondere für die Minen, die mit alleiniger Betrachtung
von rückgestreuter Energie sehr schwer zu erkennen sind. Das Potential der Methode für Zielunterscheidung wurde ebenfalls belegt.
Abstract
Ground penetrating radar (GPR) is a promising non-invasive technology for imaging shallowly
buried low-metal or non-metallic antipersonnel (AP) landmines. However, the application of
GPR to the landmine problem remains nowadays a complex scientific and technical task due
to the weak echoes produced by the dielectric landmines and the presence of undesirable effects
from antenna coupling, system ringing and interface/soil contributions (clutter). In this context,
accurate simulations, which are of great help in prediction and correct interpretation of GPR
output, may become crucial for an efficient detection, clutter removal and eventual classification
of the mines.
This work presents a full forward model of a realistic GPR scenario which includes targets, soil,
ground surface and an accurate representation and radiation characteristic analysis of the considered ultra-wideband (UWB) impulse GPR system. The modeling procedure is comprehensively
described and the GPR model optimized until a good agreement between measurements and
simulations is achieved. The problem is solved numerically in time and frequency domains via
the Finite Element Method (FEM) and using COMSOL Multiphysics Simulation Tool.
The final model is then used to perform a parametric study of the scattering signatures (onedimensional synthetic responses) by several buried landmine-like targets and a series of configurations (depth, soil conditions, target size and shape, etc.). The extracted conclusions are
summarized together with some guidelines to build a representative target signature database.
Finally, this research demonstrates that the computed signatures can be satisfactorily employed as reference waveforms for efficient clutter suppression and enhanced landmine detection/recognition. This is done through a combined strategy consisting of an energy based detection algorithm and a cross-correlation based identification technique. The latter is implemented
before conducting the detection as an additional filtering step in the form of a similarity constraint
between measured and synthetic reference signals. The proposed methodology is validated using
experimental data acquired in a prepared inhomogeneous test field at the Leibniz Institute for
Applied Geosciences LIAG in Hannover (Germany) where diverse mine simulants were buried at
different depths. In particular, the application of the combined strategy to field data yields a
clear improvement in the detection sensitivity, especially for those mines which are most difficult to detect through backscattered energy considerations alone. The potential of the method
for target discrimination is also evidenced and quantified via Receiver Operating Characteristic
(ROC) curves.
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Contents
1 Introduction
1
1.1
Motivation of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Objectives and Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Demining Problem and the Ground Penetrating Radar
7
2.1
Standard Methods of Demining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2
Ground Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1
GPR Performance and Operating Principles . . . . . . . . . . . . . . . . . . 14
2.2.2
GPR Design
2.2.3
Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4
Data Visualization: A, B and C Scans . . . . . . . . . . . . . . . . . . . . . 18
2.2.5
The GPR System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Numerical Methods
25
3.1
State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2
Integral Equation Method, the Method of Moments . . . . . . . . . . . . . . . . . . 26
3.3
Finite Difference Time Domain Method . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4
Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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Contents
3.5
3.4.1
Rayleigh-Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.2
Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3
COMSOL Simulation Tool
. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) . . . 33
4 Physical and Geophysical Background
4.1
4.2
4.3
35
Theory of Electromagnetic Wave Propagation . . . . . . . . . . . . . . . . . . . . . 35
4.1.1
Fundamental equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.2
Dispersion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.3
Reflection and Transmission of Electromagnetic Waves . . . . . . . . . . . . 39
4.1.4
GPR Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Analytical Methods of determining Electromagnetic Scattering . . . . . . . . . . . 44
4.2.1
Rayleigh Scattering (RS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2
Mie Scattering (MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3
Geometrical Optics (GO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Antenna Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1
Infinitesimal Dipole (Hertzian Dipole) . . . . . . . . . . . . . . . . . . . . . 48
4.3.2
Half-wave Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.3
Bow-Tie Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4
Electrical Properties of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5
Spatial Variability of Soils: Fluctuations in Electromagnetic Parameters . . . . . . 55
4.5.1
Correlation Length and Statistical Considerations . . . . . . . . . . . . . . 56
4.5.2
Rough Air-Ground Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 A 2D Parametric Study of the Scattering by Small Objects
59
5.1
COMSOL Electromagnetic Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2
PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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Contents
5.3
The Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4
Scattering by Circular and Rectangular Cylinders in Frequency Domain . . . . . . 63
5.5
5.4.1
Free Space
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.2
Wet and Dry Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Signatures of Circular and Rectangular Cylinders in Time Domain . . . . . . . . . 75
5.5.1
Synthetic Radargrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 GPR Antenna Modeling in Frequency Domain
83
6.1
Bow-Tie Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2
Antenna Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3
Antenna Radiation Pattern and Impedance . . . . . . . . . . . . . . . . . . . . . . 85
6.4
Antenna Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5
6.4.1
The Antenna Flare Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.2
The Antenna Shielding
6.4.3
The Absorbing Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4.4
The Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Soil Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.1
Soil Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.2
Antenna Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5.3
Interface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 GPR Antenna and Target Responses in Time Domain
7.1
7.2
107
Time domain Characteristics of GPR antennas . . . . . . . . . . . . . . . . . . . . 107
7.1.1
Definition of source pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.2
Optimization of the GPR Model . . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.3
Field Distributions for Different Antenna Configurations . . . . . . . . . . . 118
Target Scattering Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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Contents
7.2.1
Source Pulse Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.2
Frequency Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.3
Target Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.4
Soil Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.5
Summary and Some Guidelines to Create a Representative Signature
Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8 Experimental Analysis and Validation
143
8.1
Test objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.2
Test site description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.4
8.3.1
Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.3.2
Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A GUI for automatic landmine detection and recognition
9 Conclusions
9.1
. . . . . . . . . . . . . . 169
171
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Appendices
A Boundary Conditions in COMSOL
177
A.1 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.1.1 Perfectly Matched Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.1.2 Scattering Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.2 Interface Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.2.1 Perfect Electric Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.2.2 Continuity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B Plane Wave Scattering by Simple Canonical Objects
181
Contents
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B.1 Scattering by Circular Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
B.2 Scattering by a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Definitions
185
Acronyms
189
Bibliography
191
List of Figures
199
List of Tables
209
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Contents
1
Introduction
Whatever you can do, or dream you can do, do it. Boldness has genius, power and magic
in it. Begin it now
Goethe
1.1
Motivation of the Study
The landmines were used for the first time during the World War I and since then, they were
massively employed in warfare along the last century. They constitute one of the worst types of
global pollution as they may remain active for more than 50 years after placement. Since 1975,
antipersonnel (AP) landmines have killed or maimed more than 1-million people, being many of
them civilians and children.
Together with the direct tragic personal consequences, the widespread use of landmines and the
huge cost of demining labors, have a terrible impact on the economy of a country, in particular in
underdeveloped regions. The presence of mines affects enormously the social infrastructures and
the return to normal life after a conflict results seriously hindered. They make reconstruction of
rail and road networks, power lines, and waterways almost impossible, prevent fertile land from
being cultivated and restrict animal grazing. Commercial activities are also interrupted since
farmers and other people are unable to move along mined trails and roads to transport their
products to the market. Such consequences, ruin people’s capacity to meet their basic needs
increasing hunger and poverty.
All this has led to a worldwide effort to ban further landmine production and use and clear away
existing landmines. This international effort to completely ban all AP mines was formalized in
1997 through the Ottawa Convention or the Mine Ban Treaty, formally the Convention on the
Prohibition of the Use, Stockpiling, Production and Transfer of Anti-Personnel Mines and on
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1: Introduction
their Destruction. Nowadays, in 2012, there are 160 states parties to the treaty. Two states
have signed but not yet ratified while 36 UN countries are non-signatories to the Convention,
making a total of 39 states out of the treaty including USA, China and Russia. But even if this
treaty prevents new mines to be laid, the challenge of safely detecting and removing all these
devices is overwhelming. This is known as humanitarian demining, and in this context the ideal
clearance program must present a very high efficiency and minimum number of casualties. The
UN criterion is that nearly all mines should be cleared (99.6% is required), recognizing that in
reality no detection system achieves 100%.
Conventional hand-held metal detectors (MD) employed in humanitarian clearance procedures
find every piece of metal, giving rise to a large number of false alarms. Moreover, demining
operations with standard MD may become particularly slow, dangerous and cost expensive, since
many modern AP mines contain little or no metal.
Therefore, intensive research is being conducted to develop and improve new demining methods
based on other parameters or characteristics different from the metal content. There are infrared
(IR) cameras which produce images of the thermal contrast associated with the disturbed soil
layer surrounding the mine or the alteration of the heat flow due to the presence of a mine. They
are, however, strongly dependent on environmental conditions and they present limited penetration depth and low sensitivity to non recent mines. Other methods are for example artificial vapor
sensors, which detect the odor from the explosive material within the mine, but in general they
lack sensitivity, speed and portability and don’t result very well suited for demining applications.
The interest is also growing in techniques for detecting bulk explosives such as nuclear methods
and nuclear quadrupole resonance but the operator security and system complexity are some of
their drawbacks where more research need to be done. Only a few of these alternative methodologies are currently employed in real mine affected areas. Probably the most advanced of them
is Ground Penetrating Radar (GPR), which is a promising non-invasive technique able to detect
both metallic and dielectric buried objects in different soils. Nevertheless, to be successfully used
in landmine detection, the resolution needed implies ultra wideband (UWB) signals of frequencies
up to some GHz, which decreases soil penetration and raises the image clutter, making necessary
further and often complex signal processing. GPR systems are also quite expensive and the challenge is to translate the potential of this technology into practical and affordable systems which
should include robust signal processing algorithms for automatic detection and identification of
landmines.
1.2
Objectives and Scientific Contributions
Typical environmental conditions for a GPR survey are generally neither ideal nor uniform, so
that GPR antennas have to cope with changing soil parameters and topographic fluctuations.
Analytical solutions are only available for unrealistic idealized scenarios, such as infinitesimal
dipole sources located on or above homogeneous half-spaces.
Numerical modeling permits to analyze the scattering problem under realistic conditions and
provides an important understanding about the phenomena which control the electromagnetic
response.
1.2: Objectives and Scientific Contributions
3
There are several works dealing with the forward modeling of GPR problems. Many of them focus
only in one element of the model, as for example the GPR antennas [Uduwawala et al., 2005],
[Lampe & Holliger, 2001], [Warren & Giannopoulos, 2009] or the inhomogeneous dispersive soil
[Texeira et al., 1998]. Others consider the whole system (antenna, interface, soil, targets) but
most of them include approximations in one or more aspects. One of the first trials to model
a complete GPR scenario in a realistic way was carried out by [Bourgeois & Smith, 1996] and
afterwards by Gürel et. al in [Gürel, 2001], [Gürel, 2001]. However, even if they perform full 3D
modeling, they still assume canonical targets without internal structure, flat ground surface and
simplified models to represent soil inhomogeneity as a collection of random inclusions. Moreover,
the numerical method massively employed for full GPR modeling is the Finite Difference Time
Domain, which is relatively fast and easy to implement but it shows clear limitations to model
irregular geometries and complex materials; hence, certain simplifications need to be done.
Sometimes these approximations are very convenient to handle the numerical problem and to
reduce its computational weight without introducing significant errors in the calculations; but
some other times, on the contrary, they generate inaccurate results which may not be valid to
interpret real measurements.
When we tackle the problem of GPR modeling for detection and classification of small buried
dielectric landmines, high precision and spatial resolution are desirable in order to find all the
mines and reduce the high number of false alarms produced by unwanted reflections (clutter).
Then, in such a context, it becomes essential to understand and take properly into account the
role played by all the elements and parameters introduced in the model. Only then, it will
be possible to reach an adequate trade-off between the assumptions acquired and the accuracy
required for achieving satisfactory results/interpretations.
In this thesis we accomplish satisfactorily the following major goals:
1. To present a detailed description and analysis of the full modeling process of a realistic GPR
scenario in order to obtain a broad understanding about the factors which control and affect
the electromagnetic scattering by small targets as well as the GPR antenna performance.
Validate the results with analytical solutions when possible.
2. Carry out laboratory experiments to adjust all the modeling parameters, until good correlation between measurements and simulations is achieved: set the proper model to accurately
represent the GPR scenario including the antenna, interface, soil and targets.
3. Use the obtained target responses to interpret the main features of the scattering signatures
according to the object and background parameters, and develop a representative database
for some typical landmines and canonical objects buried in different soil types.
4. Define a detection/recognition methodology which incorporates a minimum distance or
similarity constraint between measured and synthetic reference target signatures. Test the
potential of this method to eliminate the clutter and recognize the landmines using field
data, i.e., study its ability to reduce the amount of false alarms in real conditions.
4
1.3
1: Introduction
Thesis Outline
The following paragraphs are intended to provide a brief overview of the structure and content
of this thesis.
Chapter 2 presents the demining problem and the standard methods for demining available,
describing thoroughly the GPR types, operating principles and components, as well as the
data acquisition methodology.
Chapter 3 concentrates on the numerical modeling of the GPR problem describing the integral
and differential techniques applied in GPR simulations. The discretization scheme, the
boundary conditions and the advantages and disadvantages of each method are discussed.
In particular, the Finite Element Method is explained in detail and the COMSOL simulation
tool is introduced.
Chapter 4 is a review of the physical and geophysical background related to the GPR modeling
problem. We start from the Maxwell equations to describe the propagation of EM waves
through different materials and interfaces, then we present the analytical description of
the light scattering, we show the principles of electromagnetic (EM) radiation by physical
antennas and introduce the bow-tie dipole, and finally we describe some methods to model
the electrical parameters of soils and their spatial variability.
Chapter 5 is a compilation of frequency and time domain simulations in 2D. As a first approach
we work in the far field of the source, i.e., we assume plane wave excitation. We study the
scattering produced by several small objects of different shapes and materials in free space
and buried in homogeneous and inhomogeneous soils.
Chapter 6 describes the development of the model for our GPR antenna system in frequency
domain. The process to create the model, and how the geometry and the different elements
affect to the radiation pattern, directivity and impedance of the antenna are studied comprehensively. These antenna characteristics are analyzed in both free space and material
half-space.
Chapter 7 contains the description of the transient simulations and the optimization and validation of the antenna model in time domain. The antenna direct coupling signal behaviour
is first analyzed for several antenna configurations, and these results are afterwards summarized and employed to adapt the antenna model until good correlation with the measured
signal is obtained. Additionally, the scattering responses by different test objects are compared with measurements and the final model is so selected and validated. Finally we
present some time domain snapshots for the considered models and study the behaviour
of mine-like target signatures when modifications on object size, structure, depth and soil
parameters are introduced.
Chapter 8 presents the experimental application of the obtained model. First, we summarize
the preprocessing methodology for measured data and some postprocessing algorithms for
1.3: Thesis Outline
5
target focusing and energy based detection. Then, after describing the measurement campaign carried out in a testfield where different mine surrogates were buried, we use our
GPR antenna model along with realistic CAD models of the testmines to simulate the target responses under different conditions. With these signatures, together with some other
obtained from geometrically simple clutter objects, a database is created. In the end, we describe a correlation based filtering algorithm that applied before the energy based detection
algorithm to the measured data, results in a clear reduction of the clutter, an improvement
of the true detection rate of those mines most difficult to detect, and a potential way to
classify the targets. A graphical user interface for visualization, detection and recognition
has been written to automatize the process and make it user friendly.
Chapter 9 summarizes the most important conclusions and provides some recommendations for
further research.
6
1: Introduction
2
Demining Problem and the Ground
Penetrating Radar
It has become appallingly obvious that our technology has exceeded our humanity
A. Einstein
The mines are cheap and easy to produce explosive devices which are designed to be activated
through contact or pressure. When used by military forces they are intended to disable any
person or vehicle either by the explosion or by the fragments launched at high speeds.
They are often laid in groups, called mine fields, and are usually strategically placed to slow the
enemy or to prevent them from passing through a certain area. The landmines are usually buried
shallow or surface laid, in regular or irregular distributions. They are deployed over large areas
either manually, by aircraft or by mechanical minelayers and the variety of environments where
they can be found is huge. Currently, it is estimated that around 60-100 million landmines remain
uncleared in at least 62 countries around the world killing and injuring a big number of civilians
every year (see Table 2.1 and Fig. 2.1).
The worst mine-affected areas are located in poor regions with few resources to face the consequences, being Afghanistan one of the countries most seriously affected by mines. According to
the Mine Clearance Planing Agency, over a 15 years period an estimated 20000 civilians have
been killed and 400000 wounded by landmines in this country. Africa is probably the most contaminated continent with more than 40 million of landmines spread over at least 19 countries.
There are an estimated 30 millions of landmines only in Angola, Cambodia and Mozambique
with a correspondingly high number of casualties. Former Yugoslavia also suffers this problem,
since more than 5 million landmines were placed during the Balkan conflict.
In general, demining consists of detecting and neutralizing the landmines. Nowadays, the equipment available to mine detection teams is very similar to that used during World War II. Gener7
2: Demining Problem and the Ground Penetrating Radar
Global Contamination from Mines and Cluster Munition Remnants
DENMARK
SERBIA
RUSSIA
UNITED KINGDOM
RUSSIA
CROATIA
BIH
MOLDOVA GEORGIA UZBEKISTAN
ABKHAZIA
KYRGYZSTAN
MONTENEGRO
KOSOVO
GREECE
CYPRUS
LEBANON
PALESTINE
MOROCCO
Atlantic Ocean
WESTERN
SAHARA
Tropic of Cancer
ARMENIA
AZERBAIJAN
TURKEY
ALBANIA
ALGERIA
SYRIA
IRAQ
ISRAEL
JORDAN
LIBYA
TAJIKISTAN
NAGORNOKARABAKH
AFGHANISTAN
IRAN
KUWAIT PAKISTAN
NEPAL
TAIWAN
OMAN
SENEGAL
LAO
PDR
PHILIPPINES
GUINEA-BISSAU
SOMALIA
UGANDA
COLOMBIA
CONGO,
REPUBLIC OF THE
CONGO, DR
PERU
THAILAND
SOMALILAND
ETHIOPIA
Pacific Ocean
ECUADOR
Pacific Ocean
YEMEN
ERITREA
SUDAN
VENEZUELA
Equator
INDIA
NIGER
MALI
CHAD
NICARAGUA
MYANMAR
EGYPT
CUBA
MAURITANIA
NORTH
KOREA
SOUTH
KOREA
CHINA
ANGOLA
ZAMBIA
VIETNAM
CAMBODIA
SRI LANKA
RWANDA
BURUNDI
Indian Ocean
MOZAMBIQUE
ZIMBABWE
NAMIBIA
Tropic of Capricorn
CHILE
ARGENTINA *
No contamination
Mines
FALKLAND ISLANDS/MALVINAS
Cluster munition remnants
Mines and cluster munition remnants
* Argentina has declared that it is mine-affected by virtue of its claim of sovereignty over the Falkland Islands/Malvinas.
8
vb2
Figure 2.1 – Distribution of the AP mines and UXO in the world. Source ICBL [ICBL, 2009].
© ICBL 2009
9
Country
Afghanistan
Angola
Bosnia and Herzegovina
Cambodia
China
Colombia
Croatia
Egypt
Eritrea
Ethiopia
China
Iran
Irak
Mozambique
Myanmar
Russia
Somalia
Sri Lanka
Sudan
Ukraine
Vietnam
Yugoslavia
Estimated number Reported casualties
of landmines
(1999-2008)
10,000,000
15,000,000
1,000,000
6,000,000
10,000,000
3,000,000
23,000,000
1,000,000
500,000
16,000,000
10,000,000
1,000,000
1,000,000
1,000,000
1,000,000
3,500,000
500,000
12,069
2,664
7,300
6,696
1,947
2,931
5,184
2,325
2,795
2,354
1,272
1,748
1,545
-
Table 2.1 – Landmines around the world; - indicates insufficient data. Source ICBL.
10
2: Demining Problem and the Ground Penetrating Radar
ally, demining operations are conducted with a handheld MD and a prodding device, such as a
pointed stick or screwdriver. The mined area is previously cleared of vegetation and divided into
small lanes. Then, a deminer slowly progress down each lane while swinging the MD close to the
ground. When the detector produces an acoustic alarm, the deminer probes the suspected area
to determine whether a buried mine is present. Sometimes trained dogs and mechanical demining
equipment are also part of the procedure.
The major limitation of the standard process is that the MD cannot differentiate a mine or any
unexploded ordnance (UXO) from other metallic items (shrapnel, metal scraps, cartridge cases,
etc.), which are abundant in most battlefields. This leads to many false alarms: 100-1000 false
alarms for each mine detected. Tuning the sensitivity of a conventional MD to decrease the false
alarm rate reduces simultaneously the probability of detection, which means that more mines
remain uncleared when the demining labor is completed and under the humanitarian demining
scope this is unacceptable. On the other hand, if the detector is tuned to signal even the small
metal amount present in some landmines, it becomes extremely sensitive to other metallic debris
present in the area, making the mine clearance a very slow and dangerous process.
The operator must achieve the best compromise between both competing goals of minimizing
the false alarm and maximizing the number of mines detected. This balance can be quantified
by what is known as a receiver operating characteristic (ROC) curve. A ROC curve plots the
probability of finding a buried mine (the probability of detection, or PD) against the probability
that a detected item is a false alarm (the probability of false alarm, or PFA). Both probabilities
are represented in a curve as a function of the threshold used to make a declaration (e.g., the
intensity of the tone produced by a MD). Figure 2.2 illustrates some theoretical examples of ROC
curves [MacDonald & R., 2003]. The ROC curve for a perfect detector will be that approaching
100% detection at 0% false alarm, while the curve associated to a random guessing would be a
diagonal line.
Figure 2.2 – ROC curves (Photo RAND).
There are more than 350 types of mines which can be classified into two main categories: anti-
11
Figure 2.3 – Various AP blast landmines (Photo GICHD).
personnel (AP) mines and anti-tank (AT) mines. The basic function of both of these varieties of
landmines is the same, but there are a couple of important differences between them.
Anti-tank mines are typically larger, in order of 20-40cm and contain several times more explosive
material than anti-personnel mines. They are commonly buried between 5cm and 40cm deep in
soil and they contain enough explosive to destroy a tank or a truck, as well as to kill people in
or around the vehicle. In addition, more pressure is usually required to detonate an anti-tank
mine. Many of this type of mines are located on bridges, roads and large areas where tanks may
travel. Although some classes with low metal content exist, they are in general easier to detect.
They do not concern humanitarian demining, since they are usually not triggered by the weight
of humans.
The AP mines can be separated into two main groups: blast and fragmentation landmines.
Blast mines are typically cylindrical shaped, 6 to 12cm in diameter and 4 to 7cm thick, and
they can weight only 30gr. The casing is made of wood, plastic or metal and they usually have
a small amount of explosive. They are triggered by pressure and usually just a small weight
(∼20kg) can activate the mechanism, causing the affected object (normally the foot/leg) to blast
into fragments. As it was mentioned before, in many cases they have little or no metal content
what makes these mines extremely difficult to locate using traditional MDs. Fig. 2.3 shows
some examples of typical AP blast landmines and in the present work we will concentrate on the
detection and recognition of this sort of mines.
The fragmentation mines throw multiple fragments upwards at high speed. They can cause
several casualties at distances of up to 50m. Since all of the modern fragmentation mines employ
steel, they are in principle easily found by MDs. However, they are often activated by tripwires
which means that the mine can be triggered by movement at distances up to 20m before being
located by the MD. These landmines, which are out of the focus of this thesis, come in diverse
sizes and shapes and they normally contain more explosive than a blast mine.
12
2.1
2: Demining Problem and the Ground Penetrating Radar
Standard Methods of Demining
For 15 years several research groups have been actively investigating new detection methods for
humanitarian demining applications. One aim of these methods is to lower the false alarm rate
while maintaining a high probability of detection, thus saving time and reducing the chance of
injury to the deminer. Table 2.2 is a compilation of most of the methods currently available.
The second column depicts the detection principle on which each technology is based while the
remaining columns outline the strengths and limitations of the different methods. An exhaustive
description can be also found in [Acheroy, 2007]. Looking at the table and considering the different
methods, we can notice that no single sensor technology is effective finding all types of mines in all
conditions. Given the limitations of individual sensor technologies, only an integrated multisensor
system may bring significant improvement in mine detection capability. Latest developments on
GPR-based systems are hand-held dual-sensors (e.g. HSTAMIDS, MINEHOUND, ALIS), which
are a combined sensor platform of a MD and a GPR; however, this kind of sensors do not
make optimal use of the totality of information available since the operator receives two separate
outputs and the GPR is just employed as discrimination sensor over the targets detected by the
MD. Accordingly, the integration of technologies at the design level of the multisensor, and the
development of algorithms for advanced signal processing and data fusion from individual sensors,
is likely to yield an important gain in terms of detection rate and false alarm reduction. And even
though the dual sensors are out the scope of this thesis, the investigation of the performance of
the GPR alone in target detection and recognition is fundamental for its successful incorporation
in such a dual sensor or its future standalone application in demining labors.
2.2
Ground Penetrating Radar
The terms ground penetrating radar (GPR), ground probing radar, subsurface radar, or surface
penetrating radar (SPR) refer to a non-destructive technique which employs radio waves to probe
the underground (or man-made visually opaque structures). The correct interpretation of the reflected electromagnetic (EM) field yields information on subsurface structural variations as well
as changes in material properties.
GPR is used in a wide spectrum of applications in fields such as geophysics, civil engineering or
archeology. Some well-developed applications are the geotechnical studies for contaminated land
assessment and bedrock profiling, concrete, road and railway inspection, buried pipe and utility mapping, geological applications like ice profiling and glaciology, groundwater mapping and
mineral exploration, archaeological studies for structural mapping, exploration and excavation
planning, and finally military and security applications, like landmine and UXO detection. The
kind of objects and features investigated can vary from a couple of centimeters to hundreds or
even kilometers of meters deep. A few GPR systems have been mounted on aircrafts and satellites to sense geological structures buried beneath the Saharan desert as well as to measure the
depth of the Moon or features on Mars. The range of the GPR in the ground is limited because
of the absorption suffered by the signal while it travels on its two-way path through the ground
Technology
Limitations
Induces eddy currents in metallic
components through alternating
magnetic fields
Emits and measures reflected radio waves from soil variations
Determines electrical conductivity distribution
Assesses thermal, light reflectance contrasts
Cheap; low complexity; performs
well in a variety of environments
Metal clutter; low-metal and
plastic mines; slow
Detects both dielectric and
metallic anomalies
Detects both dielectric and
metallic anomalies
Operates from safe standoff distances and scans wide areas
quickly
Natural clutter; very moist or
dry environments; cost expensive
Dry environments; can detonate
mine
Low spatial resolution and penetration depth; presence of foliage; heavily dependent on environmental conditions
Emits and measures reflected
sound/seismic waves
Low false alarm rate; not reliant
on EM properties
Deep mines; presence of foliage;
frozen ground
Induce radiation emissions from
the atomic nuclei in explosives
Identify elemental content of
bulk explosives
Not specific to explosives
molecule; soil topography; shallow penetration; high complexity
Induces radio frequency pulse
that causes the chemical bonds
in explosives to resonate
Identifies bulk explosives
TNT; liquid explosives; radio frequency interference; quartz bearing and magnetic soils
Advanced Prodders/ Probes
Provide feedback about nature
of probed object and amount of
force applied by probe
Could deploy almost any type of
detection method
Hard ground, roots, rocks; requires physical contact with
mine; slow and dangerous
Explosive Vapor: Biological
(dogs, rodents, bacteria), Fluorescent, Piezoelectric, Electrochemical, Spectroscopic
Detect explosive vapors or measure changes upon exposure to
explosive vapors
Confirms presence of explosives
Dry soils; lack of speed, sensitivity and portability
Electromagnetic
Metal Detector
GPR
Electrical impedance tomography
Infrared/ hyperspectral
Acoustic/ Seismic
Bulk explosive detection
Nuclear Methods:
thermal
neutron activation,
neutron
backscatter, and X-ray backscatter
Nuclear Quadrupole Resonance
Table 2.2 – Summary of Detection Technologies [MacDonald & R., 2003].
13
Strengths
2.2: Ground Penetrating Radar
Operating principle
14
2: Demining Problem and the Ground Penetrating Radar
material.
In humanitarian demining applications, the GPR technology may provide useful information on the location and geometrical properties of buried metallic and non-metallic
landmines [Brunzell, 1999], [Bruschini et al., 1998], [Scheers, 2001], [Savelyev et al., 2007],
[van den Bosch, 2006]. Therefore, GPR may permit the detection of landmines which are not
detectable by the widely used MD and could contribute to reduce the false alarm rate in mine
clearing operations.
The most important requirements for a GPR system dedicated to landmine detection are substantial antenna elevation above the ground, large dynamic range, time stability, high accuracy
of the measured data, efficient clutter reduction and high down-range and cross-range resolution
[Groenenboom & Yarovoy, 2002]. Moreover, the need for immediate output and the significant
risk to the life of the operator make it a severe technical challenge. Some of these topics will be
further developed in the following sections.
2.2.1
GPR Performance and Operating Principles
Several references discuss and review the principles of GPR in general and present different case
studies [Beres & Haeni, 1991], [Daniels, 1996], [Davis & Annan, 1976], [Davis & Annan, 1989].
The operational principle resembles that of a conventional radar system but applied to subsurface
features. Basically, a transmitter is connected to a transmitting antenna through a waveguiding
structure (usually a coaxial waveguide). The transmit antenna, which is directed towards the
ground, radiates electromagnetic (EM) waves that follow different paths before being recorded
by a receiver. If only one single antenna is employed, the radar is called monostatic. However,
most GPR systems use separate transmit and receive antennas in what is termed bistatic mode.
When the system consists of an array of more than two antennas the radar is called multistatic.
At frequencies below 1kHz, the EM behaviour is inductive in the nature and is typically described
by the diffusion equation. This physical character is dominant for the EM fields used in metal detectors. At frequencies above 1MHz, the EM fields become “wave-like” and electromagnetic propagation is described by the wave equation. Then, GPR starts to be applicable [Annan, 2003]. At
higher frequencies, above 1Ghz, depth penetration decreases drastically [Daniels, 1996], whereas
resolution increases. When the frequencies approach 10 GHz, the relaxation of the dipolar water
molecule causes very high attenuation making the use of GPR impractical in wet soils.
GPR systems typically operate in the VHF/UHF range (i.e. 30MHz-3GHz) of the radio spectrum. The wavelengths of the illuminating radiation are similar in dimension to the target, which
makes its GPR image very different from its optical one: it is highly affected by the propagation
characteristics of the ground and presents a lower definition. Moreover, the beam pattern of the
antenna is widely spread in the soil degrading the spatial resolution of the image.
At GPR frequencies the following relationships are valid:
the phase velocity,
c0
1
vph = √ ≈ √
ǫµ
ǫr
1
c0 = √
µ 0 ǫ0
(2.1)
(2.2)
15
2.2: Ground Penetrating Radar
the attenuation coefficient,
α=
r
µ σ
σ
· ≈ Z0 · √
ǫ 2
2 ǫr
(2.3)
and the EM impedance,
Z=
r
Z0
µ
≈√ ,
ǫ
ǫr
(2.4)
with the relative permittivity ǫr defined as
ǫr =
ǫ
ǫ0
(2.5)
where ǫ is the dielectric permittivity of the material, ǫ0 is the free space permittivity (8.854x10−12
farad/metre), µ is the magnetic permeability of the material, µ0 the permeability of the free space
(12.57x10−7 henrys/metre), σ is the electric conductivity and c0 is the velocity of light in free
space. The above approximations are accurate when the magnetic properties of the medium are
close to the values in free space (which is true for most Earth materials), i.e., µ = µ0 µr ≈ µ0 .
As we can see, the physical properties relevant for radio wave propagation are dielectric permittivity, electrical conductivity and magnetic permeability. More specifically, dielectric permittivity
(or dielectric constant) controls the wave velocity and conductivity affects to the signal absorption
by the medium, hence determining the signal attenuation.
In general, GPR performance is satisfactory through materials with a low conductivity such as
granite, dry sand, snow, ice, and fresh water, but will not penetrate certain clays that are high
in salt content or salt water because of the high absorption of electromagnetic energy of such
materials [Daniels, 2004]. The radar reflections will occur when there is a change in the EM
properties, especially in the aforementioned dielectric permittivity, since this parameter plays the
main role on the variation of the characteristic impedance of the medium at the frequencies of
interest. At low microwave frequencies, the relative permittivity of dry geologic materials ranges
from 3 to 6 while the relative permittivity of water is about 80. Soils have properties between
these two extremes depending mostly on the water content [Topp et al., 1980]. On the other
hand, the relative permittivity for materials employed to fabricate AP mines is 3 or less, which
often produces little dielectric contrast with the background, in particular for dry soils.
In most practical cases, part of the GPR signal will be scattered when it encounters a plastic
landmine in soil, but the backscattered signal will be often difficult to isolate and identify from
the whole response. This response contains not only the reflection from a potential target, but
also undesirable effects from antenna coupling, system ringing and interface/soil contributions,
which can mask the target response [Annan, 2003], [Daniels, 1996]. Different techniques are used
to reduce such clutter contributions and retrieve the target reflection.
The principal parameters which influence the characteristics and magnitude of the signal reflected
by the targets are: the contrast of the electromagnetic parameters between the host material and
the target (in particular, the dielectric permittivity), the target size, the system operating frequency and resolution (i.e. the bandwidth), and the signal attenuation in the host material.
The impact of all these factors on the recorded target signatures and the GPR antenna performance will be explored later in this thesis.
16
2.2.2
2: Demining Problem and the Ground Penetrating Radar
GPR Design
The design of a GPR system can be classified into different categories according to its hardware
implementation. There are two main classes: time domain radars and frequency domain radars
and each type can be further subdivided depending on the modulation of the signal. Systems
that transmit a short time pulse (or impulse) and receive and process the backscattered signal by
means of a sampling receiver can be considered to operate in time domain. Systems that transmit
individual frequencies in a sequential manner or as a swept frequency and receive the reflected
signal using a frequency conversion receiver can be considered to operate in frequency domain.
[Daniels, 2004].
Time Domain Radar
The time domain GPRs (also called impulse radars), constitute the majority of commercially
available radar systems. They transmit a sequence of pulses (typically of Gaussian-like shape)
of amplitudes lying between 20V to 200V and pulse widths within the range 200ps to 50ns at
a pulse repetition interval between several hundred microseconds and one microsecond. The
impulse generator is generally based on the technique of rapid discharge of the stored energy in a
short transmission line. It is quite feasible to generate pulses of several hundred kV but at long
repetition intervals. The output from the receiver antenna is applied to a flash A/D converter or
a sequential sampling receiver.
The central frequency fc of the pulse (which is the same as the carrier frequency) can vary from
some MHz up to GHz, being its half-power (-3dB) bandwidth, almost equal to fc . For example,
a 1ns monocycle has a fc and a 3dB bandwidth both equal to 1Ghz.
These GPRs are called Ultra-Wide Band (UWB), because of the very large relative bandwidth
involved. The block diagram of a time domain UWB GPR is illustrated in Fig. 2.4. In general,
the choice of the bandwidth for a UWB system depends on the desired spatial resolution and
signal penetration. Unfortunately, attenuation increases with frequency in most environments.
The resolution will be mainly dictated by the time duration of the pulse τ or, equivalently, its
frequency bandwidth B:
1
τ= .
(2.6)
B
The equivalent spatial dimension ∆x is obtained multiplying the time duration by the pulse travel
velocity v in the medium
v
∆x = v · T =
(2.7)
B
For traditional radar systems it is accepted that two identical targets can be separated in range
if they are at least the half of pulse width apart in time. Increasing the frequency bandwidth,
results in shorter time duration signals, which has the ability to image or resolve closely spaced
points in the ground. For instance, a receiver bandwidth of typically 1Ghz is required to provide
a resolution between 5 and 20cm, depending on the relative permittivity of the material.
2.2: Ground Penetrating Radar
17
Figure 2.4 – Block diagram of a time domain UWB GPR.
Frequency Domain Radar
The main potential advantages of the frequency domain radar are the wider dynamic range,
lower noise figure, and higher mean powers that can be radiated. There are two main types of
frequency domain radar, Frequency Modulated Carrier Wave (FMCW) and Stepped Frequency
Carrier Wave (SFCW). FMCW radar transmits a continuously changing frequency over a chosen
frequency range on a repetitive basis. The received signal is mixed with a sample of the transmitted waveform and results in a difference in frequency, which, although fundamentally related to
the phase of the received signal, is a measure of its time delay and hence the range of the target.
The SFCW radar transmits a series of incremental frequencies and stores the received signal to
afterwards carry out a Fourier transform reconstruction of the time-domain equivalent waveform.
The SFCW has found many applications in GPR because the requirements on scan rate are relatively modest.
However, we are not going to give here any further detail about frequency domain radars since
the commercial GPR system employed for this work is a pulsed radar.
2.2.3
Data Collection
Many GPR systems use separate, man-portable, transmit and receive antennas, which are placed
and moved over the surface of the ground or material under investigation. By systematically
surveying the area in a regular grid pattern, a radar image of the ground can be built up on a
display in real time [Daniels, 2004].
The recorded data can be presented in the time or frequency domains, being possible to go from
one domain to another via Fourier transform. Time domain data representation is useful for
discriminating reflections from different objects and interfaces, which may help to interpret the
18
2: Demining Problem and the Ground Penetrating Radar
Figure 2.5 – Common Offset acquisition mode.
underground structure. Frequency domain representation provides information about the spectral
dependence of the radar cross section (RCS) for a given target.
Acquisition Modes
The transmitter and receiver antennas can be used in different orientations: parallel (end-fire),
perpendicular (broadside) and cross polarized orientation to the survey direction. A particular
orientation of the antennas can significantly enhance responses from certain type of targets (for
example, pipes or objects with symmetry in one direction) because the radiation pattern of GPR
antennas is not omnidirectional, which means the antennas radiate or receive more efficiently in
some directions than in others [Daniels, 2004].
There are four main modes of radar data acquisition: common offset, common source point,
common receiver point and common midpoint. We describe below only the first of them, which
is the one of interest for our case.
In common offset (CO) operation mode, the transmitter and receiver antennas are kept at a
constant distance and moved along the survey track simultaneously. The measured travel times
of the recorded signals are displayed on the vertical axis, while the antenna position is plotted
along the horizontal axis. Most of GPR surveys use a common offset survey mode. Figure 2.5
shows a typical CO data acquisition configuration.
2.2.4
Data Visualization: A, B and C Scans
The GPR data visualization and display are crucial issues for the correct interpretation of subsurface structures. There are three standard ways of displaying the recorded data: a one-dimensional
trace (A-scan), a two dimensional slice (B-scan or radargram), and a three dimensional cut (Cscan) [Daniels, 2007].
19
2.2: Ground Penetrating Radar
i. A-Scan
Time or frequency domain data acquired by a GPR antenna for one spatial localization is termed
A-scan. When the velocity of propagation in the soil is known, the time vector can be transformed
into distance/depth. Examples of time and frequency domain A-scans resulting from laboratory
measurements in free space are shown at Fig. 2.6. The antenna crosstalk, which is the main
contribution to the received voltage, is shown in black, while the echoes coming from both a
metallic and a plastic sphere (after crosstalk subtraction), located approximately 8cm far from
the antennas, are displayed in blue and red respectively. If the targets are buried in soil, the surface
contribution (which is often mixed up with the crosstalk for small antenna heights) occurs earlier
and is usually much stronger than the target reflection.
Time Domain − Measurement 51
4
2
x 10
Cross−talk in air
Metallic Sphere
Plastic Sphere
Voltage
1
0
−1
−2
−3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
t(s)
20*log10(abs(fft(Voltage)))
Frequency domain − Measurement 51
120
Cross−talk in air
Metallic Sphere
Plastic Sphere
100
80
60
40
20
0
0.2
0.4
0.6
0.8
1
f (Hz)
1.2
1.4
1.6
1.8
2
10
x 10
Figure 2.6 – Preprocessed A-scans in time (top) and frequency domain (bottom) for a metallic and
a plastic sphere of radius r=5cm.
ii. B-scan
A B-scan is the denomination of a set of A-scans gathered alongside a line that forms a two
dimensional data set. Fig. 2.7 schematically illustrates how GPR signal evolves with the position
of the antennas with respect to the target. Reflections from targets appear as hyperbolic curves
in the recorded data due to the difference in round-trip travel time between the target and the
antenna system as the latter is moved along the measurement line. The apex of the hyperbola
corresponds to the antenna system located just above the target. Its shape depends on the depth
of the reflector as well as the wave propagation velocity (i.e. dielectric permittivity) in soil.
20
2: Demining Problem and the Ground Penetrating Radar
Figure 2.7 – B-scan and hyperbola formation.
Additionally, the response of the target decreases as the antennas are moved away from the target,
due to the travel path lengthening and the reduction of the antenna gain in the direction of the
target. Examples of measured time domain B-scans are given in Fig. 2.8.
Figure 2.8 – Measured B-scans, received amplitude with a metallic sphere (left), with a metallic
sphere after average background removal (middle) and with a plastic sphere after background removal (right).
21
2.2: Ground Penetrating Radar
In particular, the two-way wave travel time from the transmitter antenna Tx to the receiver
antenna Rx is determined according to the following formula:
tT x2Rx =
dT x2P + dP 2Rx
v
(2.8)
where v is the propagation velocity in the medium (soil) and dT x2P and dP 2Rx are respectively the
distances between the transmitter antenna and a point target and vice versa. More specifically,
if x is the scanning position along the measurement line respect to the scatterer, and d is the
scatterer depth, the aforementioned travel paths can be calculated by:
p
dT x2P = (x − xT x )2 + d2 ),
(2.9)
p
2
2
(2.10)
dRx2P = (x + xRx ) + d )
with xT x and xRx the distances between the scanning point and the transmitter and receiver
antenna respectively. These values will become zero if we have or we assume we have a monostatic
GPR system without any offset between the transmit and receive antennas. Moreover, generally
dT x2P = dRx2P and the expressions above get simplified. In particular, assuming the latter
simplifications and after some mathematical manipulation we get:
t2T x2Rx
x2
−
,
(2d/v)2
d2
(2.11)
which represents a hyperbola with the vertex in the origin and an apex in 2d/v.
On the other hand, when we assume a target of a certain radius r instead of an ideal punctual
scatterer, the travel paths are now given by:
dT x2P = dT x2P c − r,
dP 2Rx = dP 2Rxc − r
(2.12)
(2.13)
where dT x2P c and dRx2P c are the corresponding distances to the center of the targets.
iii. C-Scan
A C-scan is represented by a horizontal slice of a number of stacked B-scans, measured by repeated
line scans along the measurement plane. Three dimensional displays are basically block views
of GPR traces recorded at different positions on the surface. Obtaining good three-dimensional
images is of great help for interpreting images and identifying targets, which are usually easier to
isolate and identify on three dimensional datasets than on conventional two dimensional profiles.
Figure 2.9 illustrates a sequence of C-scans for a metallic and a plastic sphere in air at a distance
of 10cm to the antenna head. No signal processing has been applied and the targets show up as
a “wave-like” structure.
2.2.5
The GPR System
The radar system chosen for our experiments is the SPRScan, a commercial impulse UWB GPR
manufactured by ERA Technology (see Fig. 2.10 and 2.11). Its sampling head is able to acquire
22
2: Demining Problem and the Ground Penetrating Radar
73
Metallic
Sphere
y
y
y
Plastic
Sphere
x
91
82
x
x
Figure 2.9 – Measured C-scans (raw data) at different time instants. Recorded amplitude with a
plastic and a metallic sphere situated at the same distance from antenna head.
a maximum of 195 A-scans, of 512 points each, per second (or 390 A-scans of 256 points in
coarse model). Before the A/D conversion, the signal is analogically averaged (10 or 20 samples)
to improve the signal to noise ratio (S/N) and a time varying gain correction can be applied to
partially compensate the soil attenuation and geometrical spreading, increasing the overall system
dynamic range.
A pair of parallel bow-tie antennas enclosed in a sealed shielding box, are used as transmitter
and receiver. The pulse generator (pulse width: 0.5ns, pulse repetition frequency: 1MHz) is
integrated into the antenna case to minimize losses and transmission reflections. This antenna
has a central operating frequency and bandwidth of 2GHz, which leads to an expected resolution
of less than 5cm. No additional information is available regarding the SPRScan antennas.
Figure 2.10 – View of GPR Transmitter/Receiver Head.
2.2: Ground Penetrating Radar
Figure 2.11 – Test field and SPRScan Radar in the LIAG (Hannover).
23
24
2: Demining Problem and the Ground Penetrating Radar
3
Numerical Methods
I do not fear computers. I fear the lack of them
Isaac Asimov
3.1
State of the Art
The numerical modeling, and specifically the forward models, has been an active area of research
since early 1990s, primarily due to a demand for a complete understanding of the fundamental
GPR phenomena. In particular, accurate simulations may help to conceive a more effective
sensor by testing different designs on the same scene without having to manufacture them;
it can set an upper limit on the performance of a GPR system given the soil and target EM
and geometrical properties; it can also be used for testing signal processing algorithms on
GPR signals either free from or containing controlled measurement noise; finally, the predicted
responses can be helpful to identify clutter and targets. In addition, the sophistication, size,
and accuracy of GPR models have accelerated over the last years as computational resources
have improved and become more accessible. All this, has made numerical modeling a useful and
widely appealed approach to the GPR problem.
In computational electromagnetics, there are several methods for solving the Maxwell equations
or equations related to Maxwell’s theory. Many of these procedures can be classified as either
boundary or domain techniques. Both classes involve a series expansion of a unknown function
f , where f is typically a vector field. For a domain technique the computational domain Ω is
discretized and the solution of the field equations in Ω has to be approximated numerically. Along
a given boundary ∂Ω, the series expansion have to fulfill analytically the corresponding boundary
conditions. For a boundary technique, in the other hand, the boundary ∂Ω of the domain Ω must
25
26
3: Numerical Methods
be discretized and the boundary conditions are solved numerically, whereas the expansion of f
has to fulfill analytically the given field equations in Ω. Some well-known examples of domain
techniques are the Finite Element Method (FEM), the Method of Moments (MoMs), and the
Finite Difference Method (FDM). An example of boundary technique is the Boundary Element
Method (BEM).
All the computational techniques present compromises between computational efficiency, stability
and the ability to model complex geometries. Our first task to carry out this work was to search
and select a suitable method to simulate realistic GPR scenarios in the context of landmine
detection.
3.2
Integral Equation Method, the Method of Moments
The integral equations methods, commonly referred as the Method of Moments (MoM), which
was firstly introduced by Harrigton in 1968, are derived from solutions of the integral form of
Maxwell equations and are between the most commonly used numerical techniques for solving
electromagnetic problems [Balanis, 2005], [Harrigton, 1968], [Mittra, 1973], [Wang, 1990].
The key to solve any antenna or scattering problem is getting the physical or equivalent current
density distributions on the volume or surface of the antenna or scatterer. Once these are known,
the radiated or scattered fields can be found using the standard radiation integrals. The prediction of the current densities over the antenna or scatterer is accomplished by the Integral Equation
Method (IEM), whose objective is to obtain the induced current density in the form of an integral equation where the unknown current is part of the integrand. The integral equation is then
solved using numerical techniques such as the commonly applied MoM. For time-harmonic electromagnetics, two of the most popular integral equations are the Electric Field Integral Equation
(EFIE) and the Magnetic Field Integral Equation (MFIE) [Balanis, 2005]. The EFIE enforces the
boundary condition on the tangential components of the electric field while the MFIE enforces
the boundary condition on the tangential magnetic field. These integral equations can be used for
both radiation and scattering problems. And since MoM involves expanding the currents, which
are restricted to a finite domain, instead of the fields, that may extend to infinite, it is convenient
for open domains.
The MoM technique essentially transforms a general operator into a matrix equation which can
be solved easily on a computer. The procedure is called matrix method because it reduces the
original functions to matrix equations. A brief mathematical description of this method will be
given in the following paragraphs.
The linear boundary-value problem is defined by a governing differential equation in the operational form
L̂f = g
(3.1)
on the domain Ω, where L̂ is a known linear operator (i.e., the EFIE), g is a known function (the
excitation or the source), and f is the unknown field or response function to be determined. The
objective here consists of determining f once L and g are specified. The MoM requires that the
27
3.3: Finite Difference Time Domain Method
unknown response function can be expanded as a linear combination and be written as
X
f=
cj φj ,
(3.2)
j
where the cj are the scalars to be determined, i.e., the complex current amplitudes, and φj are
usually called basis or expansion functions. A certain number of terms in equation (3.2) leads to
an approximation of the current distribution. The substitution of equation (3.2) into (3.1) and
application of linearity gives
X
cj L̂φj = g.
(3.3)
j
The basis functions φj are chosen so that each L̂φj in equation (3.3) can be evaluated conveniently.
The only task remaining is to find the unknown constants cj . Next, it is necessary to define a
set of testing or weighting functions (ω1 , ω2 , ω3 , ...) in the L domain. These are essentially basis
functions used to approximate the right hand side of equation (3.1). Thus they should be linearly
independent and capable of approximating the excitation field. With every ωi an inner product
is taken, which results in:
X
cj hωi , L̂φj i = hωi , gi
i = 1, 2, 3...
(3.4)
j
Equation (3.4) corresponds to a set of equations and can be written in a matrix form as
[l]~c = ~g ,
(3.5)
[l] = [hωi , L̂φj i],
(3.6)
where [l] is the matrix
and ~c = [cj ] and ~g = [hωi , gi] are column vectors. Then, if L̂ is non-singular, the unknowns ~c will
then be given by
~c = [l]−1~g .
(3.7)
The MoM has the advantages that it is conceptually simple, from an application viewpoint devoid
of complicated mathematics and it is suitable for open domains. As a frequency domain technique,
it can also solve problems very quickly if only one frequency is required.
Probably, one of the most important drawbacks of the use of MoM for the GPR problem is the
difficulty of its setup for stratified media (since it involves the use of dyadic Green functions)
and for complex inhomogeneous bodies, mainly due to the complexity of the associated surface
integral equation [van den Bosch, 2006].
3.3
Finite Difference Time Domain Method
The FDM is one of the oldest and most popular numerical techniques. Richard W. Southwell
[Southwell, 1946] used such method in his book published in the mid 1940’s. Originally, the FD
method was predominantly implemented in the frequency domain but is now widely applied in the
time domain because of the increasing capacity of modern computers. The Finite Differences Time
28
3: Numerical Methods
Domain (FDTD) algorithm was presented in 1966 by Yee [Yee, 1966] and since then, continuously
extended and improved [Bergman et al., 1998], , [Texeira et al., 1998], [Taflove & Hagness, 2005],
[Giannopoulos, 2002]. By means of this technique, the differential operators in Maxwell equations
or the derived differential equations are discretized in a staggered grid, where the electric and
magnetic field components are offset both in time and space by half discretization intervals. The
detailed formulation of the discretized fields can be easily found in the literature.
The building block of this discretized FDTD grid is the Yee cell, which is illustrated in Fig. 3.1.
Figure 3.1 – Discretization scheme of the Yee cell. The six components of the EM field are discretized
in a staggered grid and referenced by the spatial coordinates x, y and z directions,
respectively. In addition to the spatial staggering the components of the magnetic field
are also offset in time from those of the electric field by a half-time step.
Since the Maxwell equations, which are applied in each Yee cell, are discretized in both space and
time, the solution is obtained in an iterative manner. In each iteration, the electromagnetic fields
advance (propagate) in the FDTD grid and each iteration corresponds to an evolved simulated
time of one ∆t. Therefore, one can command the FDTD solver to simulate the fields for a given
time window by setting the number of iterations. The values ∆t, ∆x, ∆y, and ∆z cannot be
assigned independently. In order to ensure numerical stability, ∆t must be bounded, and for the
three dimensional case involving all six coupled electric and magnetic field vector components, it
will be given by the Courant stability condition,
∆t ≤ q
1
c (∆x)
2 +
1
1
(∆y)2
+
1
(∆z)2
(3.8)
where c is the velocity of light.
Complex shaped objects can be also included in the models by assigning appropriate constitutive
29
3.4: Finite Element Method
parameters to the locations of the electromagnetic field components. However, objects with
curved boundaries are usually represented using a staircase approximation.
The main variables when applying FDTD are the problem space size in cells required to model
the scenario, and the number of steps needed. These determine the computer run time and
computational cost. The cell size must be small enough to provide accurate results at the highest
frequency of interest, and still be large enough to keep the resource requirements viable. Cell
size is directly affected by the materials present. The greater the permittivity, the shorter the
wavelength and the smaller the cell size needed. Due to the approximation inherent in FDTD,
waves of different frequencies will propagate at slightly different speeds through the grid. This
difference in speed depends on the direction of propagation relative to the grid. For accurate
and stable results, this grid dispersion error must be minimized by reducing the cell size. The
fundamental constraint is that the cell size must be much less than the smallest wavelength. In
particular, it is generally accepted that the discretization step should be at least ten times smaller
than the smallest wavelength of the propagating electromagnetic fields:
∆l =
λ
.
10
(3.9)
Another consideration, and one of the most relevant issues when modeling open boundary problems is the truncation of the computational space at a finite distance from sources and targets.
An approximate condition known as absorbing boundary condition (ABC) needs to be defined
in order to terminate the numerical domain without introducing additional reflections from the
borders. The role of such an ABC is to absorb any waves impinging on it, in this way simulating
an unbounded space. Therefore, the only reflected waves will be ideally the ones scattered by the
target. A more detailed description of different ABC and a brief analysis of their influence on the
simulation results is given in the Appendix A and the Section 5.3 respectively.
3.4
Finite Element Method
The Finite Element Method (FEM) is a numerical method used to approximate solutions of Ordinary and Partial Differential Equations (PDE). The procedure to reach a solution consists either
of eliminating the differential equation completely (steady state problems), or translating the
PDE into an approximate system of ordinary differential equations, which are then numerically
integrated employing standard techniques. The FEM originated from the need of solving complex elasticity and structural analysis problems in civil, mechanical and aeronautical engineering.
Its development can be traced back to the works by Alexander Hrennikoff (1941) and Richard
Courant [Courant, 1943] in the early 40s. While the approaches proposed by these pioneers were
totally different, they had one crucial characteristic in common: mesh discretization of a continuous domain into a set of discrete subdomains. The actual term ‘finite element’ appeared in
a paper by R.W. Clough in 1960 [Clough, 1960] and since these early days, the technique has
experimented a rapid growth in usage due to its versatility and underlying rich and robust mathematical basis. Nowadays, FEM is a well-developed numerical method [Huebner et al., 2001],
[Jin, 2002] to solve boundary-value problems in a large variety of non-structural areas and new
applications show up regularly in literature. Like other pure domain techniques, the FEM does
30
3: Numerical Methods
not explicitly account for open infinite domains and it is required the use of absorbing boundary
conditions (ABC) at the model edges to terminate the computational domain.
It is interesting to compare this technique with the above presented FDM. Both methods discretize first the solution domain to approximate the solution of a given boundary-value problem,
the former giving a point-wise approximation to the governing equations, and the latter a piecewise approximation. Specifically, the FEM creates a collection of arbitrarily shaped elements
(normally triangular in 2D and tetrahedral in 3D) assigning a solution everywhere in the solution domain while FDM just on a set of orthogonal grid of points. This feature makes FEM
much better suited for unstructured meshes and irregular geometries, such as arbitrary volumes
or complicated curved faces, which are straightforward to deal with. Likewise, FEM is more
flexible in handling complex boundary conditions when these are needed. All this makes FEM a
suitable method to accurately model a near field GPR scenario, where the real antenna geometry/parameters along with surface roughness and soil heterogeneity should be accommodated. It
is also a crucial issue to consider the amount of computational power needed to formulate and
solve the problem, which usually supersedes FD method for problems in two or more dimensions.
Then, special care needs to be taken in order to optimize the problem definition and select the
most proper solver. In this context, the question of adaptivity involving either remeshing or increased interpolation order during the solution process (see Definitions), becomes very important.
Within the FEM, the most widely used methods to approximate the governing differential equation (3.1) are the Rayleigh-Ritz and Galerkin methods. These techniques form the basis for the
FEM and their fundamental principles will be introduced in the next section.
3.4.1
Rayleigh-Ritz Method
The underlying mathematical foundation of the FEM comes from the classical Rayleigh-Ritz
method. The Rayleigh-Ritz procedure reformulates the original differential equation boundaryvalue problem, already given in (3.1), as a variational problem. So defined, the calculus of
variations is applied to find the minimum or maximum of a given functional and this value corresponds to the solution of the differential equation. An approximate solution to this variational
problem can be found by setting up a solution with respect to a number of variable parameters;
hence, the minimization of the functional with respect to these variables gives the best approximation. This kind of approach provides the reasons why the finite element method worked well
for the class of problems in which variational statements could be obtained.
The problem can be illustrated for the simplest case defining an inner product by
Z
hu | vi ≡
u∗ vdΩ
(3.10)
Ω
and assuming the operator L̂ to be self-adjoint and positive definite, and is referred to as the
standard variational principle. The requirement of L̂ to be self-adjoint, limits its application to
lossless media. In such a case, the solution to Eq. (3.1) can be found by minimizing the functional
given by
1
1
1
I(f ) = hL̂f | f i − hf | gi − hg | f i.
(3.11)
2
2
2
31
3.4: Finite Element Method
Proof of this statement follows in two steps: first, it must be shown that the differential equation
(3.1) is a necessary consequence when I(f ) is stationary (either at a maximum or minimum) and
second, that the stationary point is a minimum.
Once the functional given in Eq. (3.11) is determined, an approximate solution is developed using
a finite basis trial expansion
N
X
~=φ
~ · ~c,
ft =
cj φj = ~c · φ
(3.12)
j=1
where φj are a finite set of expansion functions defined over Ω, and cj are the coefficients to be
determined. Then, Eq. (3.12) becomes
1
~ | φi
~ · ~c − 1 ~c · hφ
~ | gi − 1 hg | φi
~ · ~c.
I(ft ) = ~c · hL̂φ
2
2
2
(3.13)
The next step is the minimization of I(ft ) with respect to the coefficients cj . When the problem
is real, the partial derivatives are forced to be zero and the following set of linear equations are
constructed
∂
1
~ · ~c + 1 ~c · hL̂φ
~ | φi i − hφi | gi
I(ft ) = hL̂φi | φi
∂ci
2
2
1
~ + hL̂φ
~ | φi i) · ~c − hφi | gi
= (hL̂φi | φi
2
~ · ~c − hφi | gi = 0
= hφi | L̂φi
for i= 1,2, ...,N , and where the last step follows because L̂ is self-adjoint and the problem is real.
In case the operator involved is complex (as it happens in lossy media), Eq. (3.11) is still valid
but the inner product needs to be redefined.
The former result can be written as a matrix equation
Ax = b
(3.14)
where
Aij = hφi | L̂φj i,
bi = hφi | gi,
and where x=c are the coefficients to be determined. By the self-adjoint property of L̂ it is also
seen that A is a symmetric matrix (Aij = Aji ).
If the differential equation operator L̂ is not capable of being formulated as self-adjoint, the
Galerkin method is often used.
3.4.2
Galerkin Method
For the application of FEM to more sorts of problems, the classical theory cannot be applied
(e.g. fluid related problems). And the extension of the mathematical basis to non-linear and
non-structural problems is achieved through the method of weighted residuals (MWR), originally
conceived by Galerkin. The MWR was found to provide the ideal theoretical basis for a much
wider spectrum of problems as opposed to the Rayleigh-Ritz method. Basically, the method is
based on weighting the residual of the differential equation and the resulting product integrated
32
3: Numerical Methods
over space. Technically, Galerkin’s method is a special case of the general MWR procedure, since
various types of weights can be employed.
If we have an approximate solution, ft , the residual is defined as follows
r ≡ L̂ft − g,
(3.15)
which is not equal to zero for the approximate solution ft . By forcing the weighted residuals,
defined below, to be zero on Ω it gives the best approximation
Z
wi rdΩ = 0,
(3.16)
Ri ≡
Ω
where wi are a given set of weighting functions. Galerkin’s method employs as weighting functions
the same as those used for the expansion of the trial solution in Eq. (3.12), i.e., wi = φi . The
weighted residual integrals become
Z
~ · ~c − φi g)dΩ.
Ri = (φi L̂φ
(3.17)
Ω
Similar to Eq. (3.14), this ends up in a matrix system for the coefficients ~c. When the operator L̂
is self-adjoint, the matrix system produced by Galerkin’s method will reduce to the same matrix
system produced by the Rayleigh-Ritz method. It should be added that a variety of other residual
methods exist that employ different sets of weighting functions.
The Rayleigh-Ritz and Galerkin methods described above expand approximate solutions of Eq.
(3.1) using a finite basis set of functions defined all over Ω (Eq.3.12). For problems in more
than one dimension, it is usually a nontrivial matter defining these basis functions. The FEM
approaches this problem by dividing the domain Ω into many subdomains; by making the subdomains small enough such that the solution does not vary in any complicated way, a trial function
can be built from a linear combination of simple approximate solutions on each subdomain. Once
these functions have been defined, either the Rayleigh-Ritz or Garlekin method may be used to
solve the problem but most practitioners of the FEM now use Galerkin’s method.
Hence, the FEM consists of dividing the domain into subdomains and constructing a trial solution composed of a linear combination of basis functions defined over each respective subdomain,
which differs from the classical Rayleigh-Ritz and Galerkin methods since they construct the trial
solution composed of a linear combination of basis functions defined over whole domain.
3.4.3
COMSOL Simulation Tool
The COMSOL Multiphysics is a powerful commercial tool for modeling and solving all kind of
physical problems based on PDEs via the FEM. This simulation environment facilitates all steps
in the modeling process defining your geometry, specifying your physics, meshing, solving and
then post-processing the results.
Model set up is fast, thanks to a number of predefined modeling interfaces for applications ranging
from fluid flow and heat transfer to structural mechanics and electromagnetic analyses. Material
properties, source terms and boundary conditions can all be arbitrary functions of the dependent
variables which is of special interest in our case, since we pretend to model the soil in a realistic
3.5: Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML)
33
manner.
For a 2D geometry, COMSOL uses a meshing algorithm to generate an unstructured mesh consisting of triangular elements or a mapped mesh consisting of quadrilateral elements. A 3D mesh
is either generated as an unstructured mesh containing tetrahedral elements or by extruding or
revolving a 2D mesh. Figure 3.2 shows a pair of examples of typical domain discretization in
both 2D and 3D.
Figure 3.2 – 2D and 3D FEM Meshes.
In fact, one major advantage of the FEM is that it allows the user to arbitrarily fine tune a mesh,
such that there is more resolution in areas of the geometry where there may be abrupt variations
in the solution, e.g., antenna elements. Specifying mesh size manually to minimize the error in
the desired output, is often not easy and the mesh refinement and adaptive mesh generation
(see Definitions) available in COMSOL identify the regions where high resolution is needed and
produce an adequate mesh automatically. This factor was critical in the simulations presented
in this thesis, since the resolution needed in such areas would have made a uniform mesh of the
entire scenario computationally unviable.
In selecting the interpolation functions, COMSOL provides default cases optimized for each application mode. These include, first, second and higher-order polynomials in some 2-dimensional
cases and linear vector elements for 2- and 3-dimensional cases. Linear vector elements are necessary in certain electromagnetic application modes in order to make the boundary conditions
between subdomains self-consistent with Maxwell equations. The final steps of the FEM, formulating and solving the systems of equations, are both handled by numerous subalgorithms within
COMSOL.
3.5
Absorbing Boundary Conditions (ABC) and Perfectly
Matched Layers (PML)
One of the first issues to consider when dealing with radiation problems and one of the great
challenges in finite element modeling is how to treat open boundaries. Due to the non-continuous
nature of the discretized model space, some boundary reflections are always present. This reflected energy can, in general, be minimized by a suitable choice of the boundary conditions.
The Electromagnetics Module offers two closely related types of absorbing boundary conditions,
34
3: Numerical Methods
the scattering boundary condition and the matched boundary condition. The former is perfectly
absorbing for a plane wave at normal incidence. The matched boundary condition is also perfectly absorbing for guided modes, provided that the correct value of the propagation constant
is supplied. They are mainly intended to be used at boundaries that do not represent a physical
boundary. However, in many scattering and antenna modeling problems, the incident radiation
cannot be described as a plane wave with a well-known direction of propagation. In those situations, the use of Perfectly Matched Layers (PMLs) may be considered. A PML is strictly speaking
not a boundary condition but an additional domain that absorbs the incident radiation without
producing reflections. It provides good performance for a wide range of incidence angles and is
not very sensitive to the shape of the wave fronts. The PML formulation can be deduced from
Maxwell equations by introducing a complex-valued coordinate transformation under the additional requirement that the wave impedance should remain unaffected as explained in [Jin, 2002].
From the implementation viewpoint, it is more practical to describe the PML as an anisotropic
material with losses. This formulation is used by COMSOL and also covered by [Jin, 2002]. In
Appendix A we address how to implement planar, cylindrical and spherical PMLs in COMSOL
along with the definition of the other absorbing boundary conditions utilized.
4
Physical and Geophysical Background
The eternal mystery of the world is its comprehensibility
A. Einstein
The basis for quantitatively describing GPR signals may be found by combining the physics of
electromagnetic (EM) wave propagation with the material properties of the media.
4.1
Theory of Electromagnetic Wave Propagation
The EM analysis is basically a problem of solving a set of fundamental equations subject to given
boundary conditions. In the next sections we review some basic concepts and equations of EM
theory. Our emphasis will be on the presentation of the differential equations and boundary
conditions that define the GPR problem.
4.1.1
Fundamental equations
In general, electric and magnetic fields are described as coupled, three-dimensional polarized vector quantities that have both magnitude and direction. The relations and variations of the electric
and magnetic fields, charges and currents associated with electromagnetic waves are governed by
the Maxwell equations. These equations can be written either in differential or in integral form.
The differential form of Maxwell equations is the most widely employed representation to solve
boundary-value electromagnetic problems. It is used to describe and relate the field vectors,
current densities, and charge densities at any point in space and time. These expressions are
valid if it is assumed that the field vectors are single-valued, bounded, continuous functions of
35
36
4: Physical and Geophysical Background
position and time and have continuous derivatives. Field vectors associated with electromagnetic
waves possess these characteristics except where abrupt changes in charge and current densities
are present. These discontinuities usually happen at interfaces between media where there are
discrete changes in the electrical parameters across the interface. The variations of the field vectors across such boundaries (interfaces) are related to the discontinuous distributions of charges
and currents by what are usually referred to as the boundary conditions. Hence, a complete
description of the fields vectors at any point (including discontinuities) at any time requires not
only the Maxwell equations but also the associated boundary conditions [Balanis, 2005].
In differential form, the Maxwell equations are:
~
~ × E(~
~ r , t) = − ∂ B(~r, t) ,
∇
∂t
~
~ × H(~
~ r , t) = J(~
~ r , t) + ∂ D(~r, t) ,
∇
∂t
~
~
∇ · D(~r, t) = ρe ,
~ · B(~
~ r , t) = 0,
∇
(4.1)
(4.2)
(4.3)
(4.4)
~ in [V/m] is the vector representing the electric field intensity, D
~ in [C/m2 ] is the electric
where E
~ in [A/m] is the magnetic field intensity, B
~ in [T] is the magnetic flux density, J~
flux density, H
2
3
~ is the
in [A/m ] is the electric current density, ρe in [C/m ] is the electric charge density and ∇
vector differential operator.
All the above electromagnetic field variables depend on the spatial position with respect to some
coordinate system, ~r in [m], and the elapsed time, t in [s].
The first equation is known as Faraday’s law of induction, the second is Ampere’s law as amended
~
by Maxwell to include the displacement current ∂∂tD and the third and fourth are Gauss’ laws for
the electric and magnetic fields.
The charge and current densities ρe , J~ may be thought as the sources of the electromagnetic
fields. For wave propagation problems, these densities are localized in space; for example, they
are restricted to flow on an antenna. The generated electric and magnetic fields are radiated away
from these sources and can propagate to large distances to the receiving antennas. Far from the
sources, that is, in source-free regions of space, these terms become zero.
~ and B
~ as well as D
~ and H
~ respectively
In those four Maxwell equations only the vector E
are linked. Assuming isotropic media and field vectors with not too high intensities all four
aforementioned vectors can be related. Therefore three constitutive relations are introduced.
~ r , ω) = ǫ0 ǫr (~r, ω)E(~
~ r , ω),
D(~
~ r , ω) = µ0 µr (~r, ω)H(~
~ r , ω),
B(~
(4.5)
(4.6)
where ǫ0 and µ0 are the electric and magnetic field constants and ǫr and µr are the relative
dielectric permittivity and the relative magnetic permeability respectively. The parameter ω is
the angular frequency which is related to the frequency f by,
ω
.
(4.7)
f=
2π
The Ohm’s law complete the previous constitutive equations:
~ r , ω) = σ(~r, ω)E(~
~ r , ω),
J(~
(4.8)
37
4.1: Theory of Electromagnetic Wave Propagation
~ and H
~ or E
~ and D
~
with σ the electrical conductivity. Although a relationship between B
respectively is established, a direct transformation cannot be performed. If anisotropy is present,
the three electrical properties ǫr , µr and σ become tensors. Generally, the four Maxwell equations
(4.1) through (4.4) are valid in the time domain, whereas the constitutive equations (4.5), (4.6)
and (4.8) are valid in the frequency domain. Only by assuming non-dispersive media, i.e. the
relative permittivity, relative magnetic permeability and electric conductivity are not frequency
dependent, the equations acquire the same form in time domain. Otherwise, the multiplication
in frequency domain has to be substituted by a convolution in time domain. In many real cases,
the assumption of non-dispersive media cannot be made. To allow the application of the three
aforementioned constitutive equations, the four Maxwell equations have to be transformed into
the frequency domain. This can be achieved by applying the Fourier transform to the Maxwell
equations. Through the inverse Fourier transform, any time-varying field can be expressed as a
linear combination of single-frequency solutions:
Z ∞
1
~
~ r , ω)eiωt dω,
A(~r, t) =
A(~
(4.9)
2π −∞
~ r , t) and A(~
~ r , w) is the vector field to be transformed from time to frequency domain.
where A(~
We assume that all fields are time-harmonic, i.e., harmonically oscillating functions with a single
frequency that can be represented by eiωt :
~ r , t) = E(~
~ r )eiωt ,
E(~
~ r , t) = H(~
~ r )eiωt ,
H(~
~ r), H(~
~ r ) are complex-valued.
where the phasor amplitudes E(~
Applying the transformation rule (4.9) to the Maxwell equations and substituting the constitutive
relations (4.5), (4.6) and (4.8), we may rewrite equations (4.1) to (4.4) in the form:
~ × E(~
~ r , ω) = −iω B(~
~ r, ω),
∇
~ × H(~
~ r , ω) = (σ(~r, ω) + iωǫ0 ǫr (~r, ω))E(~
~ r, ω),
∇
~ · ǫ0 ǫr (~r, ω)E(~
~ r , ω) = 0,
∇
~ · B(~
~ r, ω) = 0,
∇
(4.10)
(4.11)
(4.12)
(4.13)
where we have assumed a space absent of electrical charges (ρ), which is a correct assumption for
most georadar applications since normally no free charges are present in the field.
To complement all fundamental equations to describe all phenomena in electromagnetism the
Lorentz force has to be mentioned. The Lorentz force will not be considered here, since it
describes effects of moving charges.
4.1.2
Dispersion Equations
For the following derivations an additional assumption is necessary. The investigated region is
homogeneous, i.e. the relative permittivity, the relative magnetic permeability and the electrical
conductivity do not show dependence on the position. With these assumptions equation (4.12)
can be written as:
~ · ǫ0 ǫr (ω)E(~
~ r , ω) = 0.
∇
(4.14)
38
4: Physical and Geophysical Background
~ · (aA)
~ = a(∇
~ · A)
~ + (∇a)
~ · A,
~ for a homogeneous medium (4.14) results in
Using the relation ∇
~ · E(~
~ r , ω) = 0.
∇
(4.15)
~ ×∇
~ ×A
~ = ∇(
~ ∇
~ · A)
~ −∇
~ · (∇A)
~
Then, we apply the curl to (4.10) and employing the identity ∇
and introducing (4.15), this leads to
~ ·∇
~ E(~
~ r, ω) = iω ∇
~ × B(~
~ r , ω).
∇
(4.16)
~ r , w) can be expressed by the constitutive equation (4.6). If we use the rule ∇
~ × (aA)
~ =
B(~
~ × A)
~ + (∇a)
~ ×A
~ and assuming µr to be constant inside the considered area, (4.16) results in
a(∇
~ r , ω) = iωµ0 µr (ω)∇
~ × H(~
~ r, ω),
∆E(~
(4.17)
~ ∇
~ is the Laplace operator. If we express H(~
~ r , ω) according to the Maxwell equation
where ∆ = ∇·
(4.11), the above equation finally yields
σ(ω) ~
2
~
∆E(~r, ω) = −ω µ0 µr (ω)ǫ0 ǫr (ω) − i
E(~r, ω),
(4.18)
ǫ0 ω
which is known as Helmholtz equation. Generally all three parameters, i.e. relative magnetic permeability, relative permittivity and electrical conductivity, are complex and frequency dependent
parameters:
ǫr (ω) = ǫ′r (ω) − iǫ′′r (ω),
µr (ω) = µ′r (ω) − iµ′′r (ω),
′
′′
σ(ω) = σ (ω) − iσ (ω),
(4.19)
(4.20)
(4.21)
where ǫ′r (ω) is the dielectric polarization term, ǫ′′r (ω) represents energy loss due to polarization
lag, σ ′ (ω) refers to ohmic conduction and σ ′′ (ω) is related to faradaic diffusion.
Considering typical GPR scenarios, a few constrictions can be assumed. In most soils the relative
magnetic permeability is equal to one and will thereby be neglected. In the frequency range of
common georadar applications from 10 MHz to few GHz, the imaginary part of the electrical
conductivity can be ignored and the real part is assumed to be frequency independent and equal
to the DC conductivity [Knight & Endres, 2005]. Hence, the expression inside the bracket in
equation (4.18) is often merged into one parameter called the effective relative permittivity, ǫeff ,
or,
σ ′ (ω)
′
′′
′
′′
ǫeff = ǫeff (ω) − iǫeff (ω) = ǫr (ω) − i ǫr (ω) +
,
(4.22)
ǫ0 ω
with ǫ′eff and ǫ′′eff the real and imaginary part of the effective relative permittivity. The parameter
ǫ′r is associated with the electric permittivity, which may also be expressed in terms of relative
permittivity. The parameter ǫ′′r is related to losses due to both conductivity and frequency. In
the frequency range of GPR, displacement currents are usually higher than conduction currents
and for practical purposes, at frequencies up to 1GHz and conductivities below 0.1S/m, the effect
of the ǫ′′r term will be small and is usually disregarded.
~ field by a plane wave ‘ansatz’:
Equation (4.18) can be easily solved substituting the E
~ r , ω) = E0 ei(~k·~r−ωt) ,
E(~
(4.23)
4.1: Theory of Electromagnetic Wave Propagation
with ~k the wave vector. The solution is then the dispersion relation
σ ′ (ω)
2
2
′
k (ω) = ω µ0 ǫ0 ǫr (ω) − i
= ω 2 µ0 ǫ0 ǫeff .
ǫ0 ω
39
(4.24)
The parameter k can be complex depending on the effective relative permittivity. It may be
separated into real and imaginary parts:
√
k = β + iα = ω µ0 ǫ0 ǫeff ,
(4.25)
where α corresponds to the attenuation factor and β is the phase constant.
From the solution (4.25) the following relationships can be derived:
1
q
2
µ0 ǫ0 ǫ′eff
2
α=ω
1 + tan (δ) − 1
,
2
1
q
2
µ0 ǫ0 ǫ′eff
2
β=ω
1 + tan (δ) + 1
,
2
′′ ǫeff
.
tan(δ) =
ǫ′eff
(4.26)
(4.27)
(4.28)
The dimensionless parameter tan(δ) is more commonly called the material loss tangent
[Daniels, 2004] and can be interpreted as the ratio between the conduction current density to
the displacement current density. Then, the phase velocity vph can be determined by:
vph (ω) =
ω
c0
c0
p
,
=
1 ≈
2
β
µr (ω)ǫ′r (ω)
ǫ′′eff p
1 + tan2 (δ) + 1
2
(4.29)
with c0 the speed of light in free space. In low-loss media where ǫ′′eff << ǫ′eff and where wave
propagation will occur, tan(δ) approaches zero. In this case and assuming non-magnetic materials,
the phase velocity of an EM wave can be simplified to the well-known relation:
c0
vph = p ′ .
ǫeff
(4.30)
For most georadar applications (4.30) is sufficiently accurate since media with significant loss
tangents will supply inferior data quality caused by the higher attenuation (4.26). For media
with insignificant dispersion, i.e. ǫeff (ω) ≈ ǫeff (ω + δω), vph is equal to the EM propagation
velocity v.
4.1.3
Reflection and Transmission of Electromagnetic Waves
In 4.1.2 the various phenomena of EM wave propagation through a homogeneous medium, i.e.
attenuation and propagation velocity, were described. For typical georadar applications on the
surface such condition is not given. In this section we will show what happens if a EM wave
reaches an interface. Basically, a part of the incident energy will be reflected, while the remaining
energy will be transmitted to the lower medium. For this analysis we choose a coordinate system as
displayed in Fig. 4.1. In this system the position vector ~r is necessarily contained in the separation
40
4: Physical and Geophysical Background
Figure 4.1 – Incidence, reflection and refraction angles of an electromagnetic plane wave at the interface between two dielectric media.
interface between the two media which corresponds to the XY-plane; the XZ-plane coincides with
the incidence plane, which contains the incident ~ki vector. Let’s assume a plane wave (Eq. 4.23)
′A
approaching an interface from medium A with a relative permittivity ǫA
eff = ǫeff to a medium B
′B
′′B
with a relative permittivity ǫB
eff = ǫeff − iǫeff and tan(δ) << 1. The boundary conditions at a flat
~ and H
~
interface between two homogeneous media require that the tangential components of E
~ and B
~ are temporally and spatially continuous [Jackson, 1999].
and the normal components of D
Consequently, the phase factors of the incident, reflected, and transmitted wave must be identical
along the interface for all times: ~ki · ~r − ωi t = ~kr · ~r − ωr t = ~kt · ~r − ωt t.
To fulfill these conditions the frequencies of the three waves need to be equal: ωi = ωr = ωt . And
similarly, the tangential components of the ~k vectors have to remain constant, implying that the
~kr and ~kt must lie in the plane defined by the incident wave vector ~ki and perpendicular to the
interface.
Since the incidence plane is chosen as the XZ-plane of the coordinate system (kyi = 0) and
considering that the z component of ~r is also zero, they can be simplified as follows:
kxi = kxr = kxt ,
(4.31)
kyr
(4.32)
0=
=
kyt .
Introducing now the angles of incidence ϕi , reflection ϕr and transmission ϕt we have
|~ki | sin ϕi = |~kr | sin ϕr = |~kt | sin ϕt .
(4.33)
Thus, as |~ki | = |~kr |, the following relationships can be deduced
|~ki | sin ϕi = |~kr | sin ϕr =⇒ ϕi = ϕr ,
q
i
ǫB
sin ϕ
eff
|~ki | sin ϕi = |~kt | sin ϕt =⇒
=q .
sin ϕt
ǫA
eff
(4.34)
(4.35)
41
4.1: Theory of Electromagnetic Wave Propagation
Equation (4.34) is the so called reflection law and equation (4.35) is better known as Snells
law. If the relative permittivity, which usually depends on the frequency, is of complex value the
resulting angles and ~k vectors become also complex.
In the second part of this section we will obtain the amplitudes of the reflected and transmitted
waves for a given incident wave taking into account the EM parameters of the media. From Eq.
~ and B
~ to be plane waves we get:
4.1 and assuming again the fields E
~k × E~0 (~r, ω) = −ω B~0 (~r, ω).
(4.36)
We define now a vector ~ez normal to the interface. Then, using Eq. 4.36, the continuity conditions
of the fields at the interface take the form:
~ 0i (~r, ω) + ǫA E
~ 0r (~r, ω)) = ~ez · ǫB E
~ 0t (~r, ω),
~ez · (ǫA E
~ez · (~k × E~0i (~r, ω) + ~k × E~0r (~r, ω)) = ~ez · ~k × E~0t (~r, ω),
~ i (~r, ω) + E
~ r (~r, ω)) = ~ez × E
~ t (~r, ω),
~ez × (E
0
~ez × (
0
0
1
1
1 ~
k × E~0i (~r, ω) + A ~k × E~0r (~r, ω)) = ~ez × B ~k × E~0t (~r, ω).
µA
µ
µ
(4.37)
(4.38)
(4.39)
(4.40)
~ 0 can be decomposed
To describe the whole behavior of the EM waves at an interface the vector E
into a component parallel and one perpendicular to the plane of incidence (parallel and perpendicular polarizations). If the incident wave has another polarization, it can be expressed as a
combination of both elementary polarizations. Moreover, due to the symmetry of the interface
respect to the incidence plane, the polarization does not change when the wave is reflected or
transmitted. Hence, both directions will be treated separately.
After applying some vectorial algebra to the continuity of the fields and using the Snell law, the
coefficients of transmission τ and reflection ρ can be derived:
q
i
t
ǫA
2
~
E0⊥
sin(ϕt − ϕi )
eff cos(ϕ )
q
=
τ⊥ =
=q
,
~i
sin(ϕt + ϕi )
B cos(ϕt )
i) +
E
cos(ϕ
ǫA
ǫ
0⊥
q eff
q eff
(4.41)
i
A
B cos(ϕt )
r
t
i
ǫ
cos(ϕ
)
−
ǫ
~
E
2 sin(ϕ ) cos(ϕ )
eff
eff
q
=
ρ⊥ = 0⊥ = q
,
i
~
sin(ϕi + ϕt )
E0⊥
ǫA cos(ϕi ) + ǫB cos(ϕt )
eff
τk =
ρk =
~t
E
0k
~i
E
0k
~r
E
0k
~i
E
0k
eff
q
i
2 ǫA
tan(ϕi − ϕt )
eff cos(ϕ )
q
=
=q
,
tan(ϕi + ϕt )
B cos(ϕi )
t) +
cos(ϕ
ǫA
ǫ
q eff
q eff
i
t
B
ǫeff cos(ϕ ) − ǫA
2sin(ϕt ) cos(ϕi )
eff cos(ϕ )
q
q
=
=
.
sin(ϕi + ϕt ) cos(ϕi − ϕt )
B cos(ϕi )
t) +
cos(ϕ
ǫA
ǫ
eff
eff
(4.42)
Equations (4.41) and (4.42) are also known as Fresnel equations and describe the behavior of a
plane wave when reaching an interface. As we can see, the coefficients of reflection and transmission depend on the EM parameters of the media on both sides of the interface, the frequency,
the angle of incidence, and the polarization of the incident wave. If we assume normal incidence
(ϕi = 0), which is often the case for GPR applications, the polarization is of no significance anymore and (4.41) and (4.42) get simplified. We have to note that these relationships are valid for
42
4: Physical and Geophysical Background
incoming plane waves and this condition is only assured for sufficient distance between the EM
source and the interface. Otherwise, the wave vector ~k will be a variable of the spatial position.
Hence, to investigate shallow objects or interfaces in the close proximity of the illuminating antenna, the plane wave assumption and the Fresnel equations can only be used for a preliminary
analysis.
4.1.4
GPR Resolution
The radar resolution is defined as the system capacity to discriminate between individual scatterers. When dealing with GPR technology, the resolution concept is essentially divided into two
classes: vertical (downrange, depth or longitudinal) resolution RV and horizontal (cross-range,
angular, lateral, or plan) resolution RH [Daniels, 1996].
In general, resolution will be the highest when the antenna is placed over the surface of the
medium, and it decreases as the antenna is elevated above the surface [van der Kruk, 2004]. It is
also demonstrated that spatial resolution improves when the attenuation rises [Daniels, 1996].
Vertical Resolution
Vertical resolution provides knowledge about the equipment’s ability to differentiate in time two
adjacent reflections as different events. For the type of system considered in this thesis (pulsed
radar) the vertical resolution mainly depends on the duration of the radar pulse, which is related
to the center frequency of the antenna. The shorter the pulse duration, the better its resolution
will be. It is popularly accepted that two close events can be distinguished if the targets are
separated in time by a difference of half the effective The shorter the pulse duration, the better
its resolution will be. It is popularly accepted that two close events can be distinguished if the
targets are separated in time by a difference of half the effective pulse duration τP , which is
obtained from the width of the signal envelope at its -3 dB level. Therefore the expected spatial
vertical resolution can be calculated from τP and the wave propagation velocity v in the medium
as follows [Annan, 2003]:
τP c
τP v
= √ .
RV =
(4.43)
2
2 ǫr
Fig. 4.2 displays graphically the resulting vertical resolution for different pulse lengths and media.
This theoretical approach does not take into account that the characteristics of the transmitted
radar signal will change in the media it is traveling through. In most natural materials, the
attenuation of the electromagnetic waves increases with frequency, widely known as the dispersion
effect. This low-pass filtering effect produced by the media causes an increase in the length of
the pulse (the effective bandwidth is reduced) and, therefore, worsens the resolution. The above
formula will be also affected by the spreading losses of the signal, which means that targets
that are far away from the source have different vertical resolution than those closer. Another
important issue to consider when trying to estimate the vertical resolution is the composition
of the targets to resolve. Materials with a high dielectric contrast are more likely to mask the
objects that are close to them since they produce strong reflections.
43
4.1: Theory of Electromagnetic Wave Propagation
Vertical resolution (m)
0.4
epr
=5
epr
=10
epr
=15
epr
=20
soil
0.3
soil
soil
0.2
soil
0.1
0
0
0.5
1
1.5
2
2.5
3
Pulse length (ns)
3.5
4
4.5
5
Figure 4.2 – Vertical resolution against pulse length for different media.
Horizontal Resolution and First Fresnel Zone
Horizontal resolution indicates the minimum distance that should exist between two reflectors
located next to the other at the same depth (parallel to the analyzed medium surface) so that
the radar detects them as separate events [Daniels, 1996]. To simplify the problem, we consider
that the wave is emitted and recorded at the same point (zero offset).
The horizontal resolution of any antenna depends on the trace interval, the beam width, the
radar cross section of the reflector and the depth of the target. The beam geometry depends on
the propagation medium, the antenna characteristics and its height above surface. In general,
a narrower beam results in a better horizontal resolution, which also means that closer to the
source (shallow targets) the resolution will be in general higher since the beam is narrower. The
beam can be approximately considered as the cone of energy that intersects with the reflector
surface, illuminating an area that is called antenna footprint. An estimation of the footprint size
can be obtained by various formulas proposed in specialized bibliography. In fact, there is much
controversy with respect to the horizontal resolution since there are many factors influencing it
and different criteria to set a definition. A common approximation identifies the antenna footprint
with the diameter d1 of the first Fresnel Zone [Igel, 2007], [Rial et al., 2009], which determines
the maximal horizontal resolution of the radar system: RH = d1 , i.e, objects which are less than
d1 apart cannot be resolved. The zone construction was first proposed by Fresnel in 1818 in an
attempt to explain diffraction phenomena using Huygens principle [Born & Wolf, 1975].
To illustrate the importance of the notion of a Fresnel zone we consider a spherical wave incident
on a reflecting circular target. Each element of the target’s surface originates a reflected spherical
wave; the net disturbance at the location of a detector consists of the coherent sum of all these
reflected waves. If the two-way path difference ∆L is less than λ/2, then all the reflected waves
constructively interfere. On the other hand, waves originating from the annular region for which
λ/2 < ∆ < λ contribute to the detected signal with opposite sign, resulting in partial destructive
interference. Successively larger annular regions contribute with alternating signs, resulting in a
well-defined progression of reflecting zones. The first Fresnel zone is traditionally defined as the
largest reflecting disk for which all reflected waves reach the detector with phase shifts ∆φ ≤ π.
The diameter of the first Fresnel zone is given by the following expression,
r
λ2
d1 = hλ +
(4.44)
4
44
4: Physical and Geophysical Background
Horizontal resolution with epr soil=10
0.4
epr
soil
Horizontal resolution (m)
Horizontal resolution (m)
Horizontal resolution for distance=15cm
=5
eprsoil=10
0.3
epr
soil
0.2
=15
0.1
0
0
1
2
Frequency (Hz)
3
4
dist.=10cm
dist.=15cm
dist.=20cm
0.4
0.3
0.2
0.1
0
0
9
x 10
1
2
Frequency (Hz)
3
4
9
x 10
Figure 4.3 – Horizontal resolution versus pulse length for different media.
where h is the distance between the antenna and the reflector surface and λ is the center wavelength of the emitted radar pulse. According to this relationship and as it is shown in Fig. 4.3,
the horizontal resolution worsens with decreasing soil permittivity and with object depth (i.e.
larger distances to the antenna) and improves for higher frequencies. Another formula for the
radius r1 of the antenna footprint that has been found to fit the results well in real conditions, is
given by the following expression [Conyers & Goodman, 1997]:
r1 =
4.2
λ
h
+p
4
(ǫr + 1)
d1 = 2r1
(4.45)
Analytical Methods of determining Electromagnetic Scattering
The physical models which are used to predict the propagation and scattering of electromagnetic
waves in dielectric materials have two main sources: the already described electromagnetic wave
theory and the geometrical optics (GO). The latter method is only relevant when the wavelength of
the electromagnetic radiation employed is considerably shorter than the dimensions of the object
or medium being illuminated and when the materials involved can be considered to be electrical
insulators. Optical theory is therefore most relevant for dry materials. Materials containing
significant amounts of moisture will behave as conducting dielectrics, especially if the water
contains ions.
4.2.1
Rayleigh Scattering (RS)
Rayleigh scattering (named after Lord Rayleigh) takes place when the particle is much smaller
than the wavelength (scattering from particles up to about tenth of the wavelength of light),
i.e., the particle is electrically small [Bohren & Huffman, 1983]. It happens when light travels in
transparent solids and liquids, but is most prominently seen in gases. RS can be considered to
be elastic scattering since the energy of the scattered photos does not change. The amount of RS
that occurs to a beam of light is dependent upon the size of the particles and the wavelength of the
4.2: Analytical Methods of determining Electromagnetic Scattering
45
light; in particular, the scattering coefficient, and thus the intensity of the scattered light, varies
as the sixth power of the particle size and inversely with the fourth power of the wavelength, a
relation known as the Rayleigh law. The angular intensity polarization relationships for this type
of scattering are conveniently simple. For particles not larger than the Rayleigh limit, there is
complete symmetry of scattering about a plane normal to the incident direction of the radiation,
so that the forward scatter equals backward scatter. This is due to the symmetry of the angular
distribution of Rayleigh scattering, governed by the (1 + cos2 θ) term as shown in Eq. 4.46.
In general, if the incident field is a beam of unpolarized light of wavelength λ and intensity Ii ,
the intensity Is of the scattered light is [Bohren & Huffman, 1983]:
Is =
Ii 8π 4 N r 6 |n2 − 1|2
(1 + cos2 θ),
λ4 d2 |n2 + 2|2
(4.46)
where r is the radius of the scattering sphere, d is the distance from scatterer, n is the refractive
index, and N is the number of scatterers. Integrating over the sphere surrounding the particle
gives the Rayleigh scattering cross section for a single particle,
σs =
2π 5 d6 |n2 − 1|2
.
3 λ4 |n2 + 2|2
(4.47)
Rayleigh scattering of sunlight from particles in the atmosphere is the reason why the light from
the sky is blue. This means that blue light (shorter wavelength) is scattered much more than red
light.
4.2.2
Mie Scattering (MS)
Perhaps the most important exactly soluble problem in the theory of absorption and scattering
by small particles is that for a sphere of arbitrary radius and refractive index. Although the
formal solution to this problem has been available for many years, only since the advent of large
digital computers, it has been a practical means for detailed computation. In 1908 Gustave Mie
[Born & Wolf, 1975] developed the theory based on the Maxwell equations for a plane monochromatic wave incident upon a homogeneous sphere in a nonconducting medium in an effort to
understand the varied colors in absorption and scattering exhibited by small colloidal particles
of gold suspended in water.
The Mie theory allows the calculation of the electric and magnetic fields inside and outside a
spherical object and the formalism is applicable to spheres of all sizes, refractive indexes and for
radiation at all wavelengths. MS is then more general than RS.
Briefly, Maxwell equations are solved in spherical coordinates through separation of variables.
The incident plane wave is expanded in Legendre polynomials so that the solution inside and
outside the sphere can be matched at the boundary. The solution sought is at a distance much
larger than the wavelength, d ≫ λ, in the so called far field zone. Some remarkable features of
these results are the Mie resonances, sizes that scatter particularly strongly or weakly.
When the particle size becomes larger than around 10% of the wavelength of the incident radiation, the RS starts to break down and MS model can be used to determine the intensity of
the scattered radiation. This scattered light pattern is then like an antenna radiation lobe, with
46
4: Physical and Geophysical Background
sharper and more intense forward lobe for larger particles. This effect can be observed in Fig.
4.4 where an incoming plane wave of 50cm wavelength is scattered by a circular target of different diameters. The transition between RS and MS is clear, as the scattering goes from more
symmetric scattering by the smallest infinite cylinder to increasing forward scattering as its size
becomes of the order of the wavelength.
MS is roughly independent of wavelength and produces almost white glare around the sun when
the density of particles in air is high. It is also responsible of the white light from mist and fog.
The available analytical solutions for the scattering by an infinite circular cylinder in 2D and a
sphere in 3D can be found in the Appendix B.
4.2.3
Geometrical Optics (GO)
Since the wavelength of visible light is only of order a micron, it is very easy to find situations in which the wavelength of light is much smaller than the dimensions of the radiating or
scattering objects. In this case, high-frequency asymptotic techniques are applied. Three of
such techniques widely used are the Geometrical Optics (GO), Geometrical Theory of Diffraction
(GTD) [Keller, 1962], [Borovikov & Kinber, 1994] and the Physical Theory of Diffraction (PTD)
[Ufimtsev, 2007].
GO is the oldest and most popular theory of light propagation, and it is an approximate highfrequency method where the scattered light is obtained as a superposition of reflected, refracted
and diffracted rays. Because it uses ray concepts, it is often referred to as ray optics. In GO
light is treated as a set of rays emanating from a source, which propagate through homogeneous
media according to a set of three simple laws. The first law is the law of rectilinear propagation,
which states that light rays propagating through a homogeneous transparent medium propagate
in straight lines. The second law is the law of reflection, which dictates the interaction of light
rays with conducting surfaces. The third law is the law of refraction, which dictates the behaviour
of light rays as they traverse a sharp boundary between two different transparent media (e.g. air
and glass). Originally, GO was developed to analyze the propagation of light at sufficiently high
frequencies where it was not necessary to consider the wave nature of light. Instead, the transport
of energy from one point to another in an isotropic lossless medium is accomplished by Snell’s law
of reflection: the angle of reflection is equal to the angle of incidence. The scattering intensity in
the geometrical optics theory is given by [van de Hulst, 1957] as
Is(j) (p, m, θi ) =
ǫ2j I0 a2 sin θi cos θi dθi φ
,
r 2 sin θdθdφ
(4.48)
where θi is the incidence angle, θ is the scattering angle, a corresponds to the radius of the spherical
particle and m refers to the relative refractive index. The subscript j=1 is for the perpendicular
and j=2 for the parallel polarization component of light with respect to the scattering plane. The
fraction ǫj of the scattering intensity due to the reflection and/or the refraction for the emergent
ray is given in terms of the Fresnel coefficients already defined in Section 4.1.3.
For sufficiently high frequencies GO fields may dominate the scattering phenomena and may not
require any corrections. This is more evident for backscattering from smooth curved surfaces
whose curvature is large compared to the wavelength.
4.2: Analytical Methods of determining Electromagnetic Scattering
47
Figure 4.4 – Modeled amplitude and phase of the scattered electric field by a dielectric circular
cylinder of 5 cm (top), 10cm (middle) and 25cm (bottom) radius applying ABC at the
borders, f=0.6Ghz (λ = 50cm).
48
4: Physical and Geophysical Background
GTD is a generalization of GO which accounts for diffractions, introducing diffracted rays in
addition to the usual rays of GO. These rays are produced by incident rays which hit edges,
corners or vertices of boundary surfaces, or which graze these surfaces. Various laws of diffraction,
analogous to the laws of reflection and refraction, are employed to characterize diffracted rays.
PTD is another high-frequency asymptotic technique which consists of using the ray optics to
estimate the field induced on the objects and then integrating that field over the scattering
surface to calculate the radiated field. This theory is the natural extension of the physical
optics approximation for nonuniform sources, i.e, sources that concentrate near edges, being
a satisfactory approach specially for objects of complicate shape.
4.3
Antenna Structures
Antennas are probably the main components on a radar system. Most commercially available
GPR antennas are either dipole type (or element) or horn (or aperture) type antennas and the
majority of GPR equipments employ resistively loaded dipoles for ground-coupled applications,
and unloaded horns for air-launched applications. These antennas are designed to achieve a large
bandwidth, stable transient response and efficient impedance matching to the ground.
Typically, for ground-coupled applications, independent and identical transmitter and receiver
antennas are used in close proximity. They are housed in the same enclosure and they operate in
direct contact or a few centimeters above the ground. Most of them are shielded to minimize the
aboveground clutter and radio-frequency interference as well as to focus the radiation downwards.
An inherent problem to ground-coupled antennas is that their radiation characteristic changes
greatly with varying soil conditions due to the modification of the surface current distribution
along the radiating element. As we will see in Chapter 6 and Chapter 7, numerical modeling can
provide valuable information to quantify and determine this change on the antenna performance
under different conditions.
Dipole antennas can be further subdivided into linear dipoles or bow-tie dipoles. In the next
sections we briefly describe and compare the infinitesimal Hertzian dipole and the half-wave
dipole with the commonly used bow-tie dipole.
4.3.1
Infinitesimal Dipole (Hertzian Dipole)
A type of antenna regularly considered when studying the basic characteristics of an electromagnetic radiator is a short dipole, also called Hertzian dipole. Both the length dl and radius a of
this antenna are very small relative to the wavelength λ, i.e., dl ≪ λ (dl ≪ λ/50), so that the
electric current used to excite the antenna is spatially constant along its length. This type of
idealized short dipole cannot be physically constructed but can be utilized as building block to
simulate real antennas.
The corresponding radiation pattern shape, which is not a function of the radial distance r, is a
circular section toroid shaped and symmetrical about the axis of the dipole. Then, emission is
maximal in the plane perpendicular to the dipole and zero in the direction of the wire, that is, the
4.3: Antenna Structures
49
current direction. The maximum theoretical antenna gain for this type of dipole (a parameter
closely connected with its maximum directivity) has a typical value of 1.5, which corresponds
to 1.76 dBi, where dBi means decibels gain relative to an isotropic antenna (definitions of some
basic antenna concepts are found in the Definitions section at the end of this thesis).
Within the scope of this work the Hertzian dipole is used to simulate a punctual current source
of intensity I=1A.
4.3.2
Half-wave Dipole
Antennas whose lengths are much less than that of the emitted radiation tend to be extremely
inefficient. In fact, it is necessary to have l ∼ λ in order to obtain an efficient antenna where l is
the total length of the dipole. Probably the most common practical antennas are the half-wave
antenna and the full-wave antenna.
A half-wave dipole is typically formed by two quarter wavelength conductors or elements placed
back to back for a total length of λ/2. The two conducting wires are fed at the center of the dipole.
A standing wave on an element of a length λ/4 produces the maximum voltage differential, since
one end of the element is at a node while the other is at an antinode of the wave. The larger the
differential voltage, the greater the current flow between the elements. The half-wave antenna
radiation pattern is very similar to the characteristic pattern of a Hertzian dipole but the former
provides a more efficient radiator. The radiation resistance of a half-wave antenna is Rr ≈ 73Ω,
a value substantially larger than that for a Hertzian dipole. On the other hand, the full-wave
antenna radiation pattern is considerably sharper (i.e., it is more concentrated in the transverse
directions θ = ±π/2). The emission diagram results accordingly in a slightly flattened torus. In
other words, a half-wave antenna is a significantly more efficient electromagnetic radiator than a
Hertzian dipole; more specifically its maximum theoretical gain is 2.15dBi.
According to standard transmission line theory, if a transmission line is terminated by a resistor
whose resistance matches the characteristic impedance of the line, then all of the power transmitted down the line is dissipated in the resistor. On the contrary, if the resistance does not match
the impedance of the line then some of the power is reflected and returned to the generator.
We can think of a half-wave antenna, center-fed by a transmission line, as a resistor terminating
the line. The only difference is that the power absorbed from the line is radiated rather than
dissipated as heat. The resistance, however, is not enough to characterize the dipole impedance,
as there is also an imaginary part.
The gain of a linear half-wave dipole antenna is ∼ 2.15. To simulate the λ/2-dipole, we assume a
dipole radius r=1.6µm and length l= 7.5cm which corresponds to 2Ghz resonance frequency. A
voltage of 1V is applied.
4.3.3
Bow-Tie Dipole
The most interesting case to model a realistic scenario is the use of the actual antenna in the
GPR system, namely, a triangular (bow-tie) dipole.
Bow-tie antennas are very popular within the GPR community because they are easy and cheap
50
4: Physical and Geophysical Background
90
1
120
0.8
60
0.6
150
30
0.4
0.2
180
0
210
330
240
300
270
Hertzian
Linear
Bow−tie
Figure 4.5 – Radiation pattern of the three dipoles in the far field (left) and 3D radiation pattern of
the bow-tie dipole (right).
Hertzian dipole
Linear dipole
90
90
1
120
60
120
0.6
60
0.8
0.6
150
30
0.6
150
30
0.4
0.2
0
330
210
0.2
180
0
300
180
330
210
240
270
300
2 Ghz
0
330
210
240
270
1.5 Ghz
30
0.4
0.2
180
1
120
0.8
0.4
240
90
60
0.8
150
Bow−tie dipole
1
300
270
2.5 Ghz
Figure 4.6 – Radiation patterns of the dipole antennas at different frequencies: 1.5, 2 and 2.5GHz.
to design and manufacture and reasonably ultra-wideband [Daniels, 1996].
The antenna structure consists of two triangular metal sheets and provides a 3dB gain over a
simple dipole. They are usually connected to a symmetric line (twin line), which is matched to
the feed point impedance. This input impedance will depend on the antenna length and flare
angle. Figure 4.5 (left) shows the far field (FF) E-plane pattern in air of the a Hertzian, a linear
half-wave and a bow-tie dipole (with same length and port excitation as the λ/2-dipole and 53o
flare angle). It can be clearly appreciated that the patterns of the half-wave and bow-tie dipole
antennas are narrower (i.e. they show more directivity) than that of an ideal infinitesimal dipole.
Figure 4.6 displays the corresponding near field (NF) (at a distance d=25cm) radiation patterns
of the three dipoles we have introduced so far assuming 3 different operating frequencies. As
expected, the radiated power by the bow-tie stays almost constant for the considered frequency
range while for both dipoles it decays drastically. This happens due to the broadband performance
of this sort of antenna, what makes it a very popular one in GPR commercial systems.
Several models with bow-tie illumination are presented in Chapter 6, where we analyze in detail
the radiation characteristic of a GPR antenna under different configurations and scenarios.
51
4.4: Electrical Properties of Soils
MATERIAL
Air
Distilled Water
Fresh Water
Sea Water
Dry Sand
Saturated Sand
Limestone
Shales
Silts
Clays
Granite
Dry Salt
Ice
ǫr
1
80
80
80
3-5
20-30
4-8
5-15
5-30
5-40
4-6
5-6
3-4
σ (mS/m)
0
0.01
0.5
3e3
0.01
0.1-1
0.5-2
1-100
1-100
2-1000
0.01-1
0.01-1
0.01
v (m/ns)
0.3
0.033
0.033
0.01
0.15
0.06
0.12
0.09
0.07
0.06
0.13
0.13
0.16
α (dB/m)
0
2e-3
0.1
103
0.01
0.03-0.3
0.4-1
1-100
1-100
1-300
0.01-1
0.01-1
0.01
Table 4.1 – Relative permittivity (ǫr ), conductivity (σ), velocity (v) and attenuation (α) from
[Davis & Annan, 1989].
4.4
Electrical Properties of Soils
The ability of the GPR system to locate buried objects depends strongly on the EM properties of
the soil, namely, relative dielectric permittivity, ǫ, conductivity, σ and magnetic permeability, µ.
As mentioned above, magnetic permeability will not be considered here, since for nonmagnetic
materials µr equals 1 in the GPR frequency range and it can be assumed that the only factor
influencing the speed and attenuation of the radar wave is the complex dielectric constant of the
media.
In Table 4.1 we display typical values of these parameters for different materials and soils. As
explained in section 4.1, the real part of the dielectric permittivity ǫ′ (the relative permittivity)
of a material is related to its capability to store energy when an alternating electrical field is
applied whereas the imaginary (loss) part ǫ′′ is mainly associated with the energy dissipation.
The electrical conductivity, which is included within the loss part, controls the detection range of
the system: the radar signals travel with least attenuation through insulating materials (materials
having low electrical conductivity) and, on the contrary, they cannot penetrate through conductive
materials. Thus, materials like air, sand and gravel soils and fresh water present low attenuation
for GPR signals while clay and silt soils and salt water are conductive and cause high attenuation.
The determination of the dielectric properties of earth materials remains largely experimental.
Rocks, soils, and concrete are complex materials composed of many different minerals in widely
varying proportions, and their dielectric parameters may differ notably even within materials
that are nominally similar. A big number of researchers have investigated the relationships
between the physical, chemical, and mechanical properties of materials and their electrical and,
in particular, microwave properties. In general, they have tried to develop suitable models to link
the properties of the material to its electromagnetic parameters. Such models provide a basis for
52
4: Physical and Geophysical Background
understanding the behavior of electromagnetic waves within the media and some of them will be
shortly described in the next sections.
Frequency Dependence of the Electrical Parameters
An important process contributing to the frequency dependence of permittivity is the polarization
arising from the orientation with the imposed electric field of molecules that have permanent
dipole moments. The mathematical formulation of Debye [Debye, 1929] describes this process for
pure polar materials:
ǫs − ǫ∞
ǫ′ (f ) = ǫ∞ +
,
(4.49)
1 + (2πf τ )2
σdc
ǫs − ǫ∞
2πf τ +
,
(4.50)
ǫ′′ (f ) =
1 + (2πf τ )2
2πf ǫ0
where ǫ∞ represents the permittivity at so high frequencies that molecular orientation does not
have time to contribute to the polarization, ǫs represents the static permittivity (i.e., the value
at zero frequency), σdc is the DC conductivity (Sm−1 ) and τ is the relaxation time.
Water in its liquid state is a prime example of polar dielectric. The Debye parameters of water
are: ǫs = 80.1, ǫ∞ = 4.2, and τ = 9.3x10−12 s at 25◦ C [Hasted, 1973]. In sandy soils, most
water is in its free liquid state while in high clay content soils, pore water is not necessarily in
its free liquid state. Sometimes it is physically absorbed in capillarities, limited in motion by
electrostatic interaction with clay particles. Dielectric relaxation of absorbed water takes place
at lower frequencies than the relaxation of free water.
In the case of GPR measurements, which commonly have a frequency band from 10 MHz to 3GHz,
ǫ′′ (f ) is often small compared with ǫ′ (f ). Furthermore, many soils do not exhibit relaxation of
permittivity in this frequency range.
Water Content-Permittivity Relationships
The dielectric properties of a soil depend on a number of factors, the volumetric water content,
the frequencies of interest, the texture of the soil particles (sand, silt, or clay), the bulk density,
and the temperature [Hoekstra & Delaney, 1974].
The water content is the component that plays the main role in the electrical permittivity
of the soil (due to its high permittivity value compared with that of the background soil);
hence, many empirical and semi-empirical relationships between volumetric moisture content
Θ and the apparent permittivity of a soil have been proposed in literature [Topp et al., 1980],
[Dobson et al., 1985], [Peplinski et al., 1995], [Mironov et al., 2004], [Wang & Schmugge, 1980],
[Wobschall, 1977]. One of the simplest and most popular empirical models is probably Topp’s
equation [Topp et al., 1980], which fits a third order polynomial function to the empirically determined permittivity response of mineral soils:
ǫ′eff = 3.03 + 9.3Θ + 146Θ2 − 76.7Θ3 ,
(4.51)
with ǫ′eff the real part of the bulk effective permittivity. This model is appropriate for frequencies
in the 10MHz-1GHz range and it agrees quite well with the experimental data for a wide range of
53
4.4: Electrical Properties of Soils
water contents (5-50%). However, for getting accurate results, this formula needs to be adjusted
for each type of soil.
In contrast to empirical relationships, dielectric mixing models seek to determine the resulting
relative permittivity of a mixture on the basis of the relative permittivity and volume fractions
of its constituents. These semi-empirical models basically assume that the material is a matrix
with a multi-phase mixture of geometrically simple inclusions. From them, the most general
is the Complex Refracted Index Method (CRIM) [Mavko et al., 1998], which is a volumetric
model that requires only knowledge about the permittivity of the materials and their fractional
volume percentages. It can be used on both the real and imaginary components of the complex
permittivity and its general form is written as follows:
ǫmix
eff =
l−1
X
i=N
√
f i ǫi
!2
,
(4.52)
where ǫmix
eff is the complex bulk effective permittivity of the mixture, fi is the volume fraction
of the ith component and ǫi is the complex permittivity of the ith component. Although any
number of phases can be incorporated, in most cases, it is assumed a three-phase soil consisting
of mineral solids, air and water. It is easy to apply and it produces accurate results for various
soils in the range 1-10GHz. For such a mixture, the CRIM formula becomes:
√
√
√
√
ǫ = (1 − Φ) ǫP + Θ ǫw + (Φ − Θ) ǫa ,
(4.53)
with ǫw , ǫP and ǫa , the relative dielectric permittivities of the water, the soil matrix and the air
respectively, Θ the volumetric water content and Φ the porosity.
Another example of a mixing model is the semi-empirical power-law presented by
[Peplinski et al., 1995]. This model includes the textural composition of the soil and provides
frequency dependent expressions for the complex relative dielectric constant in terms of the sand
and clay fractions, the volumetric water content and the bulk density of the soil. The deduced
model is valid for a frequency range between 0.3 and 1.3GHz and it is based on an earlier model
for dielectric constants in the 1.4-18GHz frequency band developed by [Dobson et al., 1985]. The
real part of the complex relative dielectric permittivity for the bulk soil is approximated as
ǫ′eff
σs ϕ
′
= 1.15 1 +
ǫP − 1 + Θβ ǫ′ϕ
fw − Θ
σP
The imaginary part is derived as
h ′′
iϕ
ǫ′′eff = Θβ ǫ′′ϕ
,
fw
1
ϕ
− 0.68.
(4.54)
(4.55)
where: σs is the bulk density of the soil and typically varies between 1.1 gr/cm for clay and
1.6 gr/cm for sandy soil, σP = 2.66 gr/cm 3 is a typical value for the specific density of the solid
soil particles for sand textural class, ǫP = (1.01+0.44σP )2 is the empirical model for the dielectric
permittivity of the soil particles, ϕ = 0.65 is empirically obtained in [Dobson et al., 1985] and
[Peplinski et al., 1995], and finally, β ′ = 1.2748 − 0.519S − 0.152C and β ′′ = 1.33797 − 0.603S −
0.166C are two frequency independent constants which join the soil type into the model, being S
the mass fraction of the sand constituent and C the mass fraction of the clay constituent.
Furthermore, the frequency dependent variables ǫ′f w and ǫ′′f w are the real and imaginary parts,
54
4: Physical and Geophysical Background
Figure 4.7 – Comparison between measured and CRIM modelled complex, frequency-dependent permittivity of sandy soil with 20% water content and < 2% clay content [Cassidy, 2009].
respectively, of the dielectric constant for free water given by a modified Debye model. The latter
′ , which is derived in [Peplinski et al., 1995] as:
depends on the effective conductivity σeff
′
= 0.0467 + 0.2204σS − 0.4111S + 0.6614C.
σeff
(4.56)
In the next illustrations we show the practical use of CRIM and Peplinski models. In Fig. 4.7
[Cassidy, 2009], the CRIM-model based effective permittivity spectrum of a damp sandy soil with
approximately 20% water content and < 2% clay content is compared to its measured values. In
general, the mixing model performs well over the GPR frequency range.
In Fig. 4.8 (left) different values of the real and imaginary parts of ǫeff are shown for frequencies
below 1.4 GHz and 15% moisture assuming the Peplinski formula and sandy soil with 60% sand
content. We have selected such a soil to match the values obtained from the CRIM model. If
the sand and water contents are increased, ǫ′eff grows up to 20 for pure sand and 20% moisture.
Then, it seems that Peplinski model is overestimating ǫ′eff for a given moisture, the more for
higher water contents. A similar conclusion is achieved in [Sabouroux & Ba, 2011], where the
performance of the above presented empirical and semi-empirical models are compared with
measured data for sandy soils. The above results, according to the results reported also by other
authors, indicate that the real part of the dielectric permittivity stays almost constant over the
frequency band 300MHz-1.3GHz (and even higher) whereas the imaginary part slightly decreases
when the frequency increases. The former observation is of particular relevance when working
with an ultra-wideband system, since we wish to assume a constant soil permittivity value when
carrying out the GPR simulations.
Nevertheless, more investigation is required about soil hydrological processes to investigate the
role played by the different parameters. In principle, the values obtained employing both models
may be a satisfactory approximation to estimate the soil effective permittivity. For certain soils,
it may be important to take into account soil porosity and the CRIM model performs better than
a model purely based on soil texture as in the Peplinski approach, but for other soil types, this
model may work very well. On the other hand, if we apply the Topp’s formula for Θ = 20%,
55
4.5: Spatial Variability of Soils: Fluctuations in Electromagnetic Parameters
Sandy soil (60% Sand, 1% Clay), θ=15%
Sandy soil (60% Sand, 1% Clay), Freq.=1GHz
50
Real
Imaginary
20
Real
Imaginary
40
15
eff
ε
ε
eff
30
10
20
5
10
0
2
4
6
8
10
Frequency (Hz)
12
0
0
14
8
x 10
0.1
0.2
θ
0.3
0.4
0.5
Figure 4.8 – Real and imaginary parts of the relative effective permittivity for sandy soil against
frequency (left) and water content fraction (right).
a value of ǫ′eff = 10.11 is obtained, which is not a bad estimate. However, the Topp’s formula,
which does not take into account the soil texture and composition, appears to underestimate the
soil real permittivity, specially for those soils with porosities below 40%.
In practice, the choice of one or other model depends ultimately on the available information
about subsurface materials and the particular circumstances, but in general, it is fair to expect
that any of these approaches will provide acceptable results for typical soils in the GPR bandwidth
of interest for the present application.
4.5
Spatial Variability of Soils: Fluctuations in Electromagnetic
Parameters
Soils are characterized by a high degree of spatial variability due to the combined effect of physical,
chemical or biological processes that operate with different intensities and at different scales.
The parametrization of these models can be expressed as follows
s(r) = s̄ + ∆s(r),
(4.57)
where s is a model parameter (either elevation h, relative permittivity ǫr , or conductivity σ),
r denotes the location on the computational domain, and s̄ and ∆s (either ∆h, ∆ǫ, or ∆σ)
are the deterministic mean value of the background and the stochastic component of the model
parameter respectively (often mapped to a mathematical function such as the exponential or
gaussian distribution). The two statistical parameters that are important to characterize this
variability are the standard deviation of the considered parameter, also called root mean square
(RMS) and abbreviated as σ, and the correlation length l. For the data vector of an arbitrary
parameter s = si , with i=1,2, ..., N and N the number of elements in the sample, the RMS is
defined by this equation:
v
!
u
N
u 1
X
(si − s̄)2
(4.58)
σ=t
N −1
i=1
56
4: Physical and Geophysical Background
where
s̄ =
N
1 X
si
N
(4.59)
i=1
is the mean or average of the sample.
The simulated random data can be fitted into several models. In this thesis we will only study
normal distributed random distributions for all the variables considered, i.e., surface elevation,
electrical permittivity and conductivity.
4.5.1
Correlation Length and Statistical Considerations
The correlation length is a measure of the range over which fluctuations in one region of space
are correlated with those in another region. Values for a given property at distances beyond the
correlation length can be considered purely random, i.e., there is no further statistical relationship
between them. To statistically describe the variation of a given property with separation it is
necessary to determine either its variogram or its autocorrelation and from these functions the
correlation length can be easily determined. Both values behave contrarily and are directly
related.
The theoretical variogram 2γ(d) is a function describing the degree of spatial dependence of
a spatial random variable s(r). It is defined as the expected squared increment of the values
between locations separated a distance d [Wackernagel, 2003], i.e, it describes the difference of
data with increasing distance.
N (r)
1 X
(s(ri ) − s(ri + d))2
2γ(d) =
N (r)
(4.60)
N (r)
1 X s(ri )s(ri + d)
ρ(d) =
N (r)
σ2
(4.61)
i=1
where i is the sample index, N the number of pairs which are separated by the distance vector
d and γ(r) itself is called the semivariance or semivariogram (when represented against d). If
the semivariance only depends on the absolute lag distance d but not on the direction, the
semivariogram is considered to be isotropic and an only one omnidirectional semivariogram can
be used.
The correlation or covariance describes the degree of similarity between random variations, and
the autocorrelation for a random variable is the similarity between its values as a function of
their separation. The variation in the value of the autocorrelation coefficient as the distance
between the two points increases is referred to as the autocorrelation function. The normalized
autocorrelation function ρ, takes this form in the discrete form:
i=1
The above definitions require the random variable to be intrinsically stationary which implies
that the mean is constant. The property of ρ(d) is then independent of r and depends only on
d. When d=0, the numerator coincides with the sample variance σ 2 .
The correlation length is usually defined as the displacement d when ρ(d) is equal to 1/e. In the
following we will express the correlation length of the soil inhomogeneities and surface roughness
according to this definition.
57
4.5: Spatial Variability of Soils: Fluctuations in Electromagnetic Parameters
4.5.2
Rough Air-Ground Interface
The contribution of the roughness element depends on surface characteristics. It has been demonstrated that changes in surface roughness influence substantially the magnitude of the backscatter
energy.
The roughness of a surface can be represented by a number of parameters, but we need to characterize the roughness at the scale of the interactions that are occurring. It is well known that
scattering effects are a response to interactions with surface features that are of the order of the
wavelength of the signal (Rayleigh criterion). Likewise for the soil electrical parameters, we will
model surface roughness assuming normal distributed random height oscillation with a characteristic RMS and horizontal correlation length. The standard deviation of the height of the surface
indicates to what degree discrete values of surface elevation above a reference plane vary, i.e.,
describes the topography. Correlation length shows the horizontal scale at which these height
changes are produced, which is a direct indication of terrain roughness degree.
In Fig. 4.9 two of the soil/surface scenarios generated for our simulations are depicted and below
them, in Fig. 4.10, we display the autocorrelation functions of their dielectric spatial fluctuations.
The influence of the soil variability and surface roughness over the target scattering response and
the GPR antenna performance are analyzed in the next chapters.
Figure 4.9 – Two generated scenarios with rough surface and inhomogeneous soil.
Correlation length ∼ 5cm
Horizonal correlation
Vertical correlation
0.8
0.6
0.4
1/e
0.2
0
0
0.01
0.02
d(m)
0.03
0.04
0.05
Autocorrelation
Autocorrelation
Correlation length ∼ 2cm
Horizontal correlation
Vertical correlation
0.8
0.6
0.4
1/e
0.2
0
0
0.02
0.04
d(m)
0.06
0.08
Figure 4.10 – Autocorrelation function of the permittivity distribution for both inhomogeneous scenarios.
58
4: Physical and Geophysical Background
5
A 2D Parametric Study of the Scattering
by Small Objects
Great things are done by a series of small things brought together
Vincent van Gogh
This chapter contains several two-dimensional (2D) modeling results in frequency and time domain. The gathered information may be of great help to better visualize and understand the
different target responses both in free space, and when they are shallow buried in soil.
The electromagnetic scattering by a target is usually represented by its ‘echo area’ or Radar
Cross Section (RCS) (σ). The echo area or RCS is defined as “the area intercepting the amount
of power that, when scattered isotropically, produces at the receiver a density that is equal to the
density scattered by the actual target” [Balanis, 1989]. For a 2D object the scattering parameter
is referred to as the Scattering Width (SW) or alternatively as the the radar cross section per
unit length. In equation form the scattering width of a target take the form of
| Es |2
,
(5.1)
σ2D = lim
ρ→∞ | Ei |2
where ρ is the distance from the target to observation point and Ei , Es are the incident and
scattered electric fields respectively.
The RCS of a target is most easily viewed as the product of three factors: projected cross section,
reflectivity and directivity. Thus, apart from the mentioned electromagnetic contrast between
the target and the background material, target’s shape plays a decisive role to understand its
scattering behaviour.
The SW or RCS of targets are defined under homogeneous plane wave illumination, which represents the mathematical simplest form of EM excitation. A plane wave front can be assumed when
the target is placed in the far field of the source. The corresponding models are easy to implement
59
60
5: A 2D Parametric Study of the Scattering by Small Objects
and fast to compute, making plane wave illumination a convenient approach to the GPR modeling
problem. In particular, the study of the radar cross section of basic shapes may be an interesting
starting point to analyze the GPR scattering because such study gives insight into the scattering
from more complex objects and because many basic shapes are good approximations to some real
landmines. Moreover, the display of the EM field distribution over the domain surrounding the
target, in particular when the target is buried in soil, will provide a general understanding of the
whole scattering process.
In this chapter only representative data sufficient to illustrate the important parameter dependencies of EM propagation and radar cross section, are presented.
5.1
COMSOL Electromagnetic Module
The Electromagnetic Module is an optional add-on package for COMSOL Multiphysics adapted
to a broad range of electromagnetic problems [COMSOL, 2005] that we have used in most of the
simulations presented along this thesis.
The Electromagnetic Module contains a set of application modes that handles static, timedependent, time-harmonic, and eigenfrequency/eigenmode problems which fall into three main
categories: statics, quasi-statics and high-frequency analysis, which are available for harmonic
and transient analysis in 2D and 3D.
One major difference between the quasi-static and high-frequency modes is that the design of the
modes depends on the ‘electrical size’ of the structure. This dimensionless measure is the ratio
between the largest distance between two points in the structure divided by the wavelength of
the electromagnetic fields. The quasi-static modes are suitable for simulations of structures with
an electrical size in the range up to 1/10. The physical assumption of these situations is that the
currents and charges generating the fields vary so slowly in time that the electromagnetic fields
are practically the same at every instant as if they had been generated by stationary sources.
In the case of GPR simulations, the variations in time of the sources are more rapid and it becomes necessary to use the full Maxwell application modes for high-frequency electromagnetic
waves. They are appropriate for structures of electrical size 1/100 and larger. Thus, an overlapping range exists where you can use both the quasi-static and the full Maxwell application modes.
Independently of the structure size, the Electromagnetics Module accommodates any case of nonlinear, inhomogeneous, or anisotropic media, which is a very useful characteristic for modeling
wave propagation in realistic media. It also handles materials with properties that vary as a function of time as well as frequency-dispersive materials. The functions or data tables to describe
the media parameters can be interpolated and imported into COMSOL via Matlab or COMSOL
script. The boundary setting dialog box adapts to the current application in the module and you
can select application specific boundary conditions, which might require some parameters to be
specified.
Finally, and for further postprocessing calculations and visualization, the solution can be directly
exported to script.
61
5.2: PDE Formulation
5.2
PDE Formulation
For this first 2D approach to the GPR problem, we consider a situation where there is no variation
in the z direction, and the electromagnetic field propagates in the modeling x-y plane. To carry
out the simulations we use the In-plane waves application mode for the case of Transverse Electric
(TE) waves. A TE wave has only one electric component in the z direction, and the magnetic
field lies in the modeling plane. Thus, for this case the time-harmonic fields can be written:
E(x, y, t) = Ez (x, y, t) = Ez (x, y)ez ejωt
H(x, y, t) = Hx (x, y, t)ex + Hy (x, y, t)ey = [Hx (x, y)ex + Hy (x, y)ey ]ejωt .
(5.2)
From the assumption of time-harmonic fields, the time-dependent wave equation
∇ 2 E − µ r µ 0 ǫc
∂2E
=0
∂t2
becomes a Helmholtz type equation
∇2 E + µr µ0 ǫc ω 2 E = 0,
(5.3)
where we have introduced the complex permittivity ǫc = ǫ0 ǫr − i ωσ and E is the total vector
electric field. For TE waves, Eq.5.3 can be simplified to a scalar equation for Ez ,
σ
−1 2
k0 Ez ,
(5.4)
µr ∇ Ez − ǫr − j
ωǫ0
where k02 = ω 2 ǫ0 µ0 is the wave number of the free space.
Equation 5.4 is the PDE equation that will be solved along this chapter assuming plane wave
illumination and for frequencies within the bandwidth of interest.
5.3
The Boundary Conditions
The first question to address when dealing with an scattering problem is the proper selection of
the boundary conditions to truncate the numerical domain. This topic was briefly introduced in
section 3.5; for a detailed description of the mathematical definition and implementation of PML
and the ABC available in COMSOL see Appendix A.
In the first simulations we analyze the performance of the absorbing boundary conditions for scattering problems in free and a medium half-space. To investigate their ability to prevent unwanted
reflections we solve a simple scattering problem in free space applying either an ABC or adding
PML. In this model, we consider the incident electrical field E0 to be a linearly polarized plane
wave which travels in the direction parallel to y-axis: E0 = (0, 0, eik0 y ). The wave propagates from
top to bottom through air and is scattered by a dielectric target in different directions. Figure
5.1 show the scattered electric field with and without PMLs for 3 different frequencies. To define
the PMLs we introduce new subdomains around the air domain as described in [Berenger, 1994]
representing absorbing layers. In the corners the PMLs absorb the waves in both the x and y
directions, on the sides only the waves propagating in x direction are absorbed and on the top
62
5: A 2D Parametric Study of the Scattering by Small Objects
Figure 5.1 – Amplitude of the scattered field by a dielectric circular cylinder (r=6cm) with PML
(top) and ABC (bottom) for f=1,2,3 GHz.
and bottom the waves propagating in y direction are absorbed. Then, on the top boundary of the
layers we use a low-reflecting condition with Ez = 1V /m to generate a plane wave propagating
downwards. On the bottom border we apply a low-reflecting boundary condition with a source
field of zero. Finally, on the left and right boundaries we need to assume a perfectly magnetic
conductor (PMC) boundary condition, because the wave is propagating parallel to these boundaries and therefore the magnetic field is normal to them. The achieved results are quite similar
(see Fig. 5.2) but the PMLs absorb the scattered energy with a minimum of reflections while
without them some clear reflections arise for all the three cases. If there was no dielectric object
inside the domain, the low-reflecting and PMC boundary conditions would be enough because
the waves would only propagate in y direction and the low-reflecting boundary would absorb the
plane wave perfectly. Otherwise, with a scatterer present, the PMLs are needed to absorb the
scattered wave.
In Fig. 5.2, we quantify the error introduced by the application of low-reflecting boundary conditions and PMLs to truncate the numerical domain. The relative error measure for each frequency
value (calculated after applying the mean of the relative error per pixel over the whole 1x1m
spatial domain) is obtained by comparison of the simulated scattered field by a 5cm radius metallic sphere in free space (placed in the middle of the domain) with the corresponding analytical
solution. The already expected good performance of the PMLs is confirmed by these results. In
particular, we can see that the value of the error associated with the PMLs is clearly smaller than
that generated by the presence of low-reflecting boundaries. It remains below 0.1% for all the
frequencies evaluated and it grows slightly with the frequency. On the other hand the error due
to the low-reflecting boundary condition is always above 0.15% and follows an opposite trend: it
63
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
Error introduced by the boundary conditions
1
Low−Reflecting Boundaries
PMLs
Error (%)
0.8
0.6
0.4
0.2
0
0
1
2
3
Frequency (Hz)
4
5
6
9
x 10
Figure 5.2 – Relative error introduced by the boundary conditions.
decreases with increasing frequency.
The example presented before corresponds to a homogeneous unbounded medium but many real
life applications involve stratified media where the interfaces among the layers are unbounded.
As far as we know, there does not exist any theoretical result that guarantees the accuracy in
problems where the interface between two different media continues through the PML domain.
Even without theoretical support, in the literature there exist a wide range of works dealing with
stratified media which claim good numerical accuracy from the PML technique. The mentioned
numerical accuracy is demonstrated here by comparing the performance of ABC and PML in the
half-space simulation illustrated in Fig. 5.3, where we see that the PMLs are still working well.
In the present thesis, we apply PML to all the simulations in frequency domain.
5.4
Scattering by Circular and Rectangular Cylinders in Frequency Domain
The RCS of a target for the most general case of a bistatic configuration, is function of the
polarization of the incident wave, the polarization of the received wave, the angle of incidence, the
angle of observation, the geometry of the target (target shape and size), the electrical properties
of the target and the frequency of operation [Balanis, 2005].
In the next sections we study the scattered field behaviour for several objects (infinite in zdirection) under different conditions and assuming plane wave illumination (traveling downwards
in y-direction as described in the previous section). The targets will be initially considered in
free space, and afterwards buried at shallow depth in two representative soils.
5.4.1
Free Space
First, we analyze the scattered electric field in free space by targets of distinct geometry and
composition and for frequencies in the range of interest: 1, 2 and 3 GHz.
The computed scattered field by different objects is displayed in Fig. 5.4, 5.5 and 5.6. The field
64
5: A 2D Parametric Study of the Scattering by Small Objects
Figure 5.3 – Amplitude of the scattered field by a tilted metallic rectangular cylinder with PML
(top) and ABC (bottom) for f=1,2,3 GHz.
f=2Ghz
f=1Ghz
0
f=3Ghz
0
0.5
y(m)
y(m)
−0.4
−0.4
0
0
x(m)
−0.2
−0.4
0
−0.6
−0.6
−0.6
−0.5
−0.5
0
−0.2
−0.2
0
0.5
y(m)
0.5
−0.5
−0.5
0.5
f=1Ghz
−0.8
0
x(m)
−0.5
−0.5
0.5
f=2Ghz
0.5
−0.8
0
x(m)
0.5
f=3Ghz
0.5
0.5
0
−0.4
0
−0.6
−0.8
−0.5
−0.5
−1
0
x(m)
0.5
−0.5
y(m)
−0.6
0
0
−0.2
y(m)
y(m)
−0.4
0
−1
−0.8
−1.5
−1
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
−2
0
x(m)
0.5
Figure 5.4 – Amplitude of the scattered field by a metallic (top) and a dielectric (bottom) circular
cylinder (r=2.5cm) in free space.
65
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
f=1Ghz
f=2Ghz
−1
−1.6
−1.5
−2
0
x(m)
−2
0
−2.2
−2.5
−2.4
−3
−1
−1.5
y(m)
0
0.5
−1.4
−1.8
−0.5
−0.5
f=3Ghz
0.5
y(m)
y(m)
0.5
−2
0
−3
−3.5
−0.5
−0.5
0.5
f=1Ghz
0
x(m)
−4
−0.5
−0.5
0.5
0.2
0
0.5
x(m)
f=3Ghz
f=2Ghz
0.5
−2.5
0.5
0.5
0.2
−0.2
0
−0.4
−0.4
−0.6
−0.6
y(m)
−0.2
0
0
0
y(m)
y(m)
0
−0.5
0
−1
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
0
x(m)
0.5
Figure 5.5 – Amplitude of the scattered field by an empty (top) and a water filled plastic pipe
(r=5cm) (bottom) in free space.
amplitude values are given in dB (in order to enhance the visualization) and they are calculated
by subtraction of the incoming field from the total field.
According to the scattering theory already introduced in Section 4.2, when the particle size is
small in comparison with the wavelength (as it is the case for the 1GHz wave impinging on 2.5cm
radius circular cylinder in Fig. 5.4 or a thin rectangular cylinder in Fig. 5.6), we get close to
the the Rayleigh region and the scattering is almost symmetric about the plane normal to the
incident direction. When the size of the object becomes of the order of the wavelength, i.e, for
higher frequencies, it appears a forward radiation lobe, which is clearly visible at the mentioned
figures for the 2GHz and 3GHz illumination. For bigger and more complex objects at the same
frequencies, like the pipes in Fig. 5.5, the scattering behaviour gets more complicated giving rise
to secondary radiation lobes. It can be noticed that in the case of the metallic circular cylinder,
the radiation characteristic and the amount of energy scattered upwards is the same or at least
very similar for all the frequencies. This does not happen when the scatterer is a dielectric object.
In Fig. 5.7 we compare the amount of energy that is scattered back when the above referred
plane wave is impinging on objects of diverse shapes and dimensions. The presented results were
taken at a distance of 10cm above the top surface of the target and circular and rectangular objects of different sizes are investigated. Hence, this can be considered a kind of target cross section
calculation in the near field, which is the actual region of interest for our particular application.
As stated for the field distribution figures, the metallic circular cylinder presents an almost constant behaviour for frequencies above the Rayleigh limit. All the other objects present resonances
whose amplitude and frequency depend on the object dimensions and the dielectric material.
The analytical solution to the canonical case of the scattering by circular cylinders is given in
Appendix B together with an illustration for three of the cases represented in Fig. 5.7. By com-
66
5: A 2D Parametric Study of the Scattering by Small Objects
f=1Ghz
f=2Ghz
0.5
f=3Ghz
0.5
0.5
0
0
−0.5
0
−1
y(m)
−0.4
0
y(m)
y(m)
−0.2
−0.5
0
−1
−0.6
−1.5
−0.8
−0.5
−0.5
0
x(m)
−0.5
−0.5
0.5
0
x(m)
−2
0.5
−0.5
−0.5
f=2Ghz
f=1Ghz
−0.2
−0.2
0
−0.6
−0.4
y(m)
−0.4
y(m)
0
0.5
0.5
−0.4
−0.6
0
x(m)
f=3Ghz
0.5
0.5
y(m)
−1.5
−0.6
0
−0.8
−0.8
−1
−0.8
−1
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
−1.2
−1
0
x(m)
−0.5
−0.5
0.5
0
x(m)
0.5
Figure 5.6 – Amplitude of the scattered field by a plastic (top) and half-plastic (bottom) rectangular
cylinder (length w=10cm, height h=4cm) in free space.
0.7
0.6
Metallic circ. cylinder r=2.5cm
Plastic circ. cylinder r=2.5cm
Empty pipe r=5cm
Plastic circ. cylinder r=5cm
Plastic rect. cylinder lxh=10x4cm
Plastic rect. cylinder lxh=10x2cm
Norm ScattEz
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
f(Hz)
4
5
6
9
x 10
Figure 5.7 – Backscattered electric field by different objects in free space and plane wave excitation;
receiving point at a distance of 10cm.
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
67
Figure 5.8 – Scattering by a buried target illuminated by a downward propagating plane wave.
paring the figures from both analytical and simulated field calculations, we see that the resulting
values are in very good agreement.
5.4.2
Wet and Dry Soils
The following step is to investigate 2D dielectric targets when they are shallow buried in the
ground. The computational domain is now composed of air (y > 0), ground (y < 0) and the airground interface (y = 0) and we consider once more the incoming field to be a linearly polarized
plane wave traveling from top to bottom in y-direction (see Fig. 5.8).
In the presence of a half-space, the backscattered signal is the sum of the scattered field at the
air-ground interface and the scattering by the buried object. In order to eliminate this interface
contribution the model output is processed with a background subtraction. The PML have been
also adapted to the ground electrical parameters, but there are still a few reflections coming back
from the boundaries, in particular produced at the air-soil discontinuity. Nevertheless, their effect
over the scattered field is negligible and it will not be further considered here.
Two soil types which are representative of real wet and dry soils have been chosen for our simulations. For the wet soil which is a lossy medium, we assume ǫr = 10 and σ = 50mS/m. The dry
soil, on the other hand, has ǫr = 5 and σ = 1mS/m, i.e., the attenuation losses will be negligible
due to its low conductivity. The targets are buried 10cm deep (respect to their top boundary,
Fig. 5.8) and we consider again 1GHz, 2GHz and 3GHz operating frequencies. The scattered field
distribution for both soil types is presented in Fig. 5.9. Here we observe again that unlike for the
other targets, the backwards scattering pattern and amplitude by the buried metallic cylinder is
almost constant for all the frequencies. In the case of wet soil and particularly for the frequency
of 3GHz, there are some artifacts only present for this target that seem to be connected with a
high frequency resonance effect but they cannot be well explained. For the rest of the dielectric
objects (see Figures 5.10-5.14), it is clear that there are oscillations, and neither higher frequen-
68
5: A 2D Parametric Study of the Scattering by Small Objects
f=1Ghz
f=2Ghz
f=3Ghz
0.5
−0.5
−1
−1
−1
−1.5
0
−2
−2
−0.5
−0.5
0.5
f=1Ghz
−2.5
−3
0
x(m)
−0.5
−0.5
0.5
0.5
−1
−1
0
−2
y(m)
−1
y(m)
−2
0.5
f=3Ghz
0.5
0
0
x(m)
f=2Ghz
0.5
y(m)
−2
−3
−3
0
x(m)
−1.5
0
−2.5
−2.5
−0.5
−0.5
y(m)
−0.5
−1.5
0
0.5
−0.5
y(m)
y(m)
0.5
−2
0
−3
−3
−3
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
−4
0
x(m)
0.5
Figure 5.9 – Amplitude of the scattered field by a buried metallic circular cylinder (r=2.5cm) in
dry(top) and wet (bottom) soil.
cies nor bigger permittivity contrasts are always associated with more energy radiated back to
the surface. Regarding the downward radiation, secondary radiation lobes appear in all the cases,
which increase both with frequency and with permittivity, i.e., in wetter soils. The width and
intensity of these lobes respect to the main lobe depend on the geometry and composition of
the object and the background soil parameters. We can also observe from the amplitude of the
lobes in dry and wet soils, that as expected, due to the higher conductivity, the energy in wet
soil is dissipated much faster than in dry one, which decreases the penetration depth of the waves
significantly.
For buried dielectric targets, the scattering in wet soil is in general more intense than in dry soil
because of the higher dielectric contrast with the background soil, but as we see, this signal is
rapidly attenuated in its way to the target and back to the receiver. Therefore, for targets that
are buried a few centimeters deep in moist soils, there is a trade-off between the higher dielectric
permittivity of the wet soil, which increases the magnitude of the scattered signal by a plastic
target, and the higher conductivity, which produces a faster attenuation.
When we represent the scattered amplitude against frequency (see Fig. 5.16), the resonant behaviour of the buried targets becomes slightly more complicated than in free space. The behaviour
of the scattered field by the metallic cylinder is again almost constant, but when buried in soil
there are multiple little oscillations present which might be due to a numerical error or reflections
at the domain boundaries. However, when we look at the overall shape of the signatures for each
particular target, we observe that the number of main resonances and their positions are the same
for both types of soils. Moreover, all the objects present also the same number of resonances as
in free space except for the thinner rectangular cylinder, which now shows several resonances,
while in free space contains just one.
69
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
f=1Ghz
f=3Ghz
f=2Ghz
0.5
0.5
0.5
−1
−1
−1
−2
0
−2
y(m)
0
y(m)
y(m)
−1.5
0
−2
−2.5
−3
−3
−3
−0.5
−0.5
0
x(m)
−0.5
−0.5
0.5
f=1Ghz
0
x(m)
−0.5
−0.5
0.5
f=2Ghz
0.5
0
x(m)
0.5
f=3Ghz
0.5
0.5
−1
−1
−1
−2
0
−2.5
y(m)
−2
0
y(m)
y(m)
−1.5
0
−2
−3
−3
−0.5
−0.5
0
x(m)
−3
−4
−3.5
−0.5
−0.5
0.5
0
x(m)
−0.5
−0.5
0.5
0
x(m)
0.5
Figure 5.10 – Amplitude of the scattered field by a buried dielectric cylinder (r=2.5cm) in dry (top)
and wet (bottom) soil.
f=1Ghz
f=3Ghz
f=2Ghz
0.5
0.5
0.5
0
0
−0.5
−1
0
−2
−1
y(m)
−1.5
0
y(m)
y(m)
−1
0
−2
−2
−2.5
−0.5
−0.5
0
x(m)
0.5
−3
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
0
x(m)
0.5
−3
Figure 5.11 – Amplitude of the scattered field by a buried empty pipe (r=5cm) in dry (top) and wet
(bottom) soil.
70
5: A 2D Parametric Study of the Scattering by Small Objects
f=1Ghz
f=3Ghz
f=2Ghz
0.5
0
0
−1
−0.5
−0.5
−1.5
0
x(m)
0.5
0
−0.5
y(m)
y(m)
−0.5
0
0.5
−1
0
−1.5
−2
−2
−2.5
−2.5
−0.5
−0.5
0
x(m)
0.5
−1
y(m)
0.5
0
−2
−3
−0.5
−0.5
0
x(m)
0.5
Figure 5.12 – Amplitude of the scattered field by a buried plastic pipe full of water (r=5cm) in dry
(top) and wet (bottom) soil.
Figure 5.13 – Amplitude of the scattered field by a buried plastic rectangular cylinder (w=10cm,
h=4cm) in dry (top) and wet (bottom) soil.
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
71
Figure 5.14 – Amplitude of the scattered field by a buried half-plastic rectangular plate (w=10cm,
h=4cm) in dry (top) and wet (bottom) soil.
Inhomogeneous soils and rough surface
The following simulations depict the scattered signal from targets buried in dry and wet nondispersive inhomogeneous soils. For this analysis we will show the real part of the scattered field
instead of the absolute value since the effect of the inhomogeneities deforms totally the radiation
lobes that were visible in the amplitude representations for homogeneous soils and displaying the
real part, the propagation of the waves and their distortion can be better distinguished.
In the considered models, the relative permittivity takes values according to both configurations
described in section 4.5.2. Generally, the inhomogeneities closer to the ground surface have higher
conductivities due to the presence of organic material. Therefore, conductivities in the range 0.010.05S/m have been selected in both cases. The results are processed applying average background
removal.
As we can notice in Figures 5.17 to 5.20, the effect of the surface roughness is in both cases more
remarkable than the effect of the soil inhomogeneity. For the second configuration, where the
topographic variability is stronger than for the first one, we see that the field distribution seems
the same for both targets while for the first configuration the difference due to the object scattering
is more evident. This is because in the second case the contribution to the scattered field coming
from the target is negligible in comparison with the scattering produced at the rough surface.
On the contrary, when the surface is flat, the contribution of the targets in comparison with
that coming from the inhomogeneity is more important in wet soils, due to a higher permittivity
contrast of the objects with the background soil.
72
5: A 2D Parametric Study of the Scattering by Small Objects
0.7
0.6
Norm ScattEz
0.5
Metallic circ. cylinder r=2.5cm
Plastic circ. cylinder r=2.5cm
Empty pipe r=5cm
Plastic circ. cylinder r=5cm
Plastic rect. cylinder wxh=10x4cm
Air−plastic rect. cylinder wxh=10x4cm
0.4
0.3
0.2
0.1
0
0
1
2
3
f(Hz)
4
5
6
9
x 10
Figure 5.15 – Backscattered electric field by different objects buried in dry soil and plane wave
excitation; receiving point at a 10cm height above the surface.
0.5
0.45
0.4
Metallic circ. cylinder r=2.5cm
Plastic circ. cylinder r=2.5cm
Empty pipe r=5cm
Plastic circ. cylinder r=5cm
Plastic rect. cylinder wxh=10x4cm
Air−plastic rect. cylinder wxh=10x4cm
Norm ScattEz
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
f(Hz)
4
5
6
9
x 10
Figure 5.16 – Backscattered electric field by different objects buried in wet soil and plane wave
excitation; receiving point at a 10cm height above the surface.
73
5.4: Scattering by Circular and Rectangular Cylinders in Frequency Domain
f=2Ghz
f=1Ghz
f=3Ghz
0.5
0.5
0.5
0
0
0
0.5
y(m)
0
1
0.5
y(m)
y(m)
0.5
0
0
−0.5
−0.5
−0.5
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
−1
0
x(m)
0.5
−0.5
−0.5
0
x(m)
0.5
Figure 5.17 – Real part of the scattered field by a buried empty pipe (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous dry soil
with flat surface.
Figure 5.18 – Real part of the scattered field by a buried empty pipe (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous dry soil
with rough surface.
74
5: A 2D Parametric Study of the Scattering by Small Objects
f=1Ghz
f=3Ghz
f=2Ghz
0.5
0.4
0.5
0
0
−0.2
−0.4
0
x(m)
0.5
−0.5
f=1Ghz
−0.5
−0.8
0
x(m)
−0.5
0.5
f=2Ghz
0.5
0
0
0
x(m)
0.5
f=3Ghz
0.5
0.2
0.5
0.5
0.5
y(m)
y(m)
−0.5
−0.6
−0.5
0
0
y(m)
−0.5
0
0
−0.4
−0.2
−0.5
y(m)
0
y(m)
y(m)
0.2
0
0.5
0.5
0.2
0
0
−0.2
−0.5
−0.5
−0.4
0
x(m)
0.5
−0.5
−0.5
−0.5
0
x(m)
0.5
−0.5
−0.5
−0.5
0
x(m)
0.5
Figure 5.19 – Real part of the scattered field by a buried empty pipeline (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous wet soil
with flat surface.
Figure 5.20 – Real part of the scattered field by a buried empty pipeline (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous wet soil
with rough surface.
5.5: Signatures of Circular and Rectangular Cylinders in Time Domain
5.5
75
Signatures of Circular and Rectangular Cylinders in Time
Domain
Once we have calculated and studied the behaviour of the scattered wave for an input signal
with single frequency values, we can extend our modeling to the case of broadband illumination
according to the real external source we have. In order to obtain the broadband response of the
system, we decompose the broadband applied signal I(w) into a sum of narrowband contributions
Ii which are defined by the electrical parameters at the central frequency of each of them. Thereby,
the excitation can be approximated by the following sum,
I(ω) =
n
X
Ii (ω).
(5.5)
i=0
When the bandwidth of these components is narrow enough, this approximation is accurate.
To extend the problem to the whole bandwidth, we just need to apply the model for one frequency
to each of the narrowband components with their respective central frequency and input signal.
Then, we obtain a system of Helmholtz equations:
− ∇2 Ei − ki2 (x, y)Ei = fi ,
in
Ω
(5.6)
2 )E .
where ki2 (x, y) = wc2 µǫ(x, y) + iwc µσ(x, y) and the source fi = (ki2 − k0i
0i
This problem can be directly solved with COMSOL applying the parametric solver over the
desired frequency band. Thus, the target scattering for every spectral component can be easily
calculated.
In this thesis we assume the material electrical parameters σ and ǫ to be constant with frequency
(non-dispersive), which seems to be a good approximation for high frequencies.
For our numerical experiments, we consider a frequency band of 0.1-6GHz and 60 contributions,
so we obtained a spectral resolution of 0.1GHz. Then, we modify these contributions applying a
linear frequency-domain filter to each input plane wave signal to describe the frequency-dependent
excitation:
I(ω) = Bi (ω)Ii (ω),
(5.7)
where Bi is the transfer function.
The Gaussian bandpass filter represents properly the solution for an impulse input for most of
the systems; we apply thus a gaussian window centered at 2GHz with 2GHz bandwidth.
5.5.1
Synthetic Radargrams
When we have the frequency domain response for a normalized gaussian broad-band input signal,
we can compute the system response in time domain. This is done by applying the discrete Fourier
transform to the scattered field spectral distribution at the receiver positions.
In the following section we will display some radargrams obtained with this method.
76
5: A 2D Parametric Study of the Scattering by Small Objects
Pipe with water in free space
Metallic circ. cylinder in free space
Plastic circ. cylinder in free space
0
1
1
1
1
2
2
2
2
3
3
t(ns)
0
t(ns)
0
t(ns)
t(ns)
Empty pipe in free space
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
1
0
x(m)
2
0.2
3
6
0.4
4
5
−0.4
−0.2
0.02
0
x(m)
0.04
0.2
0.06
0.4
0.08
6
−0.4
−0.2
0.02
0
x(m)
0.04
0.2
0.06
0.4
0.08
6
−0.4
−0.2
0.01
0
x(m)
0.02
0.2
0.03
0.4
0.04
−3
x 10
Figure 5.21 – Backscattered amplitude by an empty plastic pipe, a water filled pipe (r=5cm), a
metallic circ. cylinder and a plastic circ. cylinder (from left to right).
Free Space
For the case of objects in free space, we calculate the scattered field every 1cm over an aperture
synthesized along the x-axis a few centimeters above the target. In this way, we obtain a simulated
radargram or B-scan. The scattering scenarios and the investigated objects are the same as in
the previous section. Fig. 5.21 depicts the corresponding radargrams for the scattered signal
in free space. As expected, the wave reflected by the different targets has an hyperbolic shape
[Daniels, 1996]. The shape of the hyperbola is in general a function of the distance to the object
and the propagation velocity in the medium above the object. Hyperbola fitting techniques can
help to get an estimation of the background soil permittivity if the object depth is known.
In these radargrams, the top and bottom reflections for the three non-metallic objects considered
are clearly resolved due to the broad bandwidth of the signal. As it can be noticed, the highest
reflectivity is related to the metallic circular cylinder while the lowest corresponds to the empty
pipe that is just produced by its thin plastic cover. The pipe filled with water has the highest
permittivity (ǫwater = 81), which makes the wavelength of the signal nine times smaller than in
free space. It is then required very small grid to get enough accuracy inside the pipe, and this
numerical error is probably producing the artifacts present in the associated radargram. These
artifacts are also present when we consider this pipe buried in soil in the next calculations.
Another interesting phenomenon that can be observed in the right image of Fig. 5.21 is that the
bottom reflection (sometimes referred in literature as the glory wave) is stronger than the top
reflection (or specular wave). This is an effect than can happen for certain dielectric objects in free
space as demonstrated in [Cloude et al., 1996], where the authors study the different scattering
mechanisms by dielectric cylinders.
77
5.5: Signatures of Circular and Rectangular Cylinders in Time Domain
Plastic circ. cylinder in dry soil
Metallic circ. cylinder in wet soil
Plastic circ. cylinder in wet soil
0
1
1
1
1
2
2
2
2
3
3
t(ns)
0
t(ns)
0
t(ns)
t(ns)
Metallic circ. cylinder in dry soil
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
0.01
0
x(m)
0.02
0.2
0.03
6
0.4
−0.4
0.04
−0.2
1
0
x(m)
0.2
3
4
2
6
0.4
−0.4
5
−0.2
0.005
0
x(m)
0.01
0.2
0.015
6
0.4
−0.4
0.02
−0.2
1
2
0
x(m)
3
0.2
4
0.4
5
−3
6
−3
x 10
x 10
Figure 5.22 – Backscattered amplitude by a metallic and a plastic circular cylinder (r=2.5cm) buried
in dry and wet soil.
Pipe with water in dry soil
Empty pipe in wet soil
Pipe with water in wet soil
0
1
1
1
1
2
2
2
2
3
3
t(ns)
0
t(ns)
0
t(ns)
t(ns)
Empty pipe in dry soil
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
0.005
0
x(m)
0.01
0.2
0.015
0.4
0.02
6
−0.4
−0.2
0.01
0
x(m)
0.02
0.2
0.03
0.4
0.04
6
−0.4
−0.2
5
0
x(m)
10
0.2
6
0.4
15
−0.4
−0.2
5
−3
x 10
0
x(m)
10
0.2
0.4
15
−3
x 10
Figure 5.23 – Backscattered amplitude by a plastic empty pipe and a pipe full with water buried in
dry and wet soil.
Wet and Dry Soils
Figures 5.22 to 5.24 correspond to the backscattered signals by several targets buried in wet and
dry soil respectively. The receiver positions are located 10cm above the interface and the excitation is again a normalized gaussian modulated incoming plane wave.
For most of the targets we are again able to distinguish top and bottom reflections, except for
two of the objects: the circular metallic cylinder, since the energy is totally reflected at its top
boundary; and the half air-plastic cylinder, whose vertical dimension lies in the limit of the signal
range resolution. As the waves travel faster through air than in plastic, the top and bottom
reflections are superimposed in the latter case, while for a cylinder of the same dimension but
just made of plastic, we can still distinguish both reflections. We observe that in all the cases
the bottom reflection intensity decreases respect to the top reflection intensity when the targets
are buried in wet soil and, as expected, the hyperbolas are slightly delayed in time and narrowed.
78
5: A 2D Parametric Study of the Scattering by Small Objects
Air−plastic rect. cylinder in dry soil
Plastic rect. cylinder in wet soil
Air−plastic rect. cylinder in wet soil
0
1
1
1
1
2
2
2
2
3
3
t(ns)
0
t(ns)
0
t(ns)
t(ns)
Plastic rect. cylinder in dry soil
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
0
x(m)
5
0.2
10
6
0.4
−0.4
−0.2
15
0.02
0
x(m)
0.2
0.04
0.4
6
−0.4
0.06
−0.2
0.005
0
x(m)
0.01
0.2
0.015
6
0.4
0.02
0.025
−0.4
−0.2
0.01
0
x(m)
0.02
0.2
0.03
0.4
0.04
−3
x 10
Figure 5.24 – Backscattered amplitude by a plastic rectangular cylinder and air-plastic rectangular
cylinder buried in dry and wet soil.
σ
=1mS/m, Disc r=2.5cm, Receiver h=10cm
−3 soil
−3
x 10
8
εsoil=5
εsoil=10
10
εsoil=15
8
6
4
0
2
4
d (cm)
6
8
10
Norm ScattEz
Norm ScattEz
12
x 10
ε
soil
=5, Disc r=2.5cm, Receiver h=10cm
σsoil=10mS/m
σsoil=50mS/m
6
σsoil=100mS/m
4
2
0
0
2
4
d(cm)
6
8
10
Figure 5.25 – Maximum scattered amplitude by a dielectric cylinder (r=2.5cm) in different soils.
There is also an evident difference between the scattering patterns of rectangular cylinders and
the rest of the targets which are circular. The scattering patterns are in the latter case clearly
flattened, and these are the type of echoes that we will mostly observe for typical buried mines.
Regarding the amplitude we see that a higher contrast between the object and the background
does not always mean an stronger backscattering, due to the higher attenuation connected with
a wetter soil. Which of both effects dominates will depend on the depth of the scatterer.
In the plots displayed in Fig. 5.25 and 5.26, we show the amplitude behaviour depending on the
target depth, the soil parameters and the object dimensions and composing material. As we can
see, the attenuation of the signal with depth increases rapidly when the conductivity increases,
and for depths below ∼4cm the attenuation gets dominant over the contrast. On the other hand,
an increase in the permittivity does not accelerate the attenuation with depth, which in this case
decays soft and constantly for the three permittivities considered.
The next illustrations (Fig. 5.27 to 5.29) correspond to the simulated radargrams for the scattering by targets buried in inhomogeneous dry and wet soil for both cases, with rough and flat
interface. The soil models are the ones previously presented in Fig. 4.9. For the dry soil scenario
we have an average permittivity of 5 and conductivity of 5mS/m with a standard deviation of
0.8 and a correlation length of 2cm, and the wet soil scenario presents an average permittivity of
10 and conductivity of 50mS/m with a standard deviation of 1.2 and a correlation length of 5cm.
Apart from the inhomogeneity we make these models more realistic adding surface roughness:
79
5.5: Signatures of Circular and Rectangular Cylinders in Time Domain
εsoil=7, σsoil=10mS/m, Disc d=−5cm, Receiver h=10cm
εsoil=7, σsoil=10mS/m
0.02
PEC
Plastic
Air
Water
0.06
Norm ScattEz
Norm ScattEz
0.08
0.04
0.02
0
2
4
6
r(cm)
8
0.015
0.01
0.005
10
r=2cm
r=4cm
r=6cm
r=8cm
5
10
15
h(cm)
20
Figure 5.26 – Maximum scattered amplitude for different cylinder radius and different receiver
heights.
Empty pipe in inhom. dry soil, flat surface
Plastic rectangular plate in inhom. dry soil, flat surface
0
0
Empty pipe in inhom. drysoil, rough surface
Rect. cylinder in inhom. dry soil, rough surface
0
1
1
2
2
2
2
3
3
t(ns)
1
t(ns)
1
t(ns)
t(ns)
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
0.005
0
x(m)
0.01
0.2
0.015
0.4
0.02
6
−0.4
−0.2
5
0
x(m)
10
0.2
6
0.4
15
−0.4
−0.2
0.02
0
x(m)
0.04
0.2
0.06
6
0.4
0.08
−0.4
−0.2
0.02
0
x(m)
0.04
0.2
0.06
0.4
0.08
−3
x 10
Figure 5.27 – Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic rect.
cylinder (w=10cm, h=4cm) in inhomogeneous dry soil with flat and rough surface.
related to the dry soil, we consider a randomly varying surface with a mean height variation of
±1cm with 2cm correlation length and for the wet soil the variation in height is ±2cm with 5cm
correlation length. We don’t present here the results for conductivity variations because we have
observed that the effects caused by permittivity fluctuations of the models are much stronger than
the effects caused even by considerable conductivity variations with exception of very conductive
soils (σ > 0.05S/m).
The analysis of the simulations reveal that if the soil permittivity is heterogeneous, the form and
the absolute traveltime of the hyperbolas change notably due to the velocity variations. Another
consequence of the heterogeneity are numberless reflections which interfere with the signals from
the targets. These effects worsen when the surface is not flat.
If the contrast of the objects to the soil is high enough, as is the case for moist soil, and we don’t
introduce roughness, the target signal is still visible. As the contrast gets smaller, the objects
become more difficult to detect or cannot be detected any more. In the simulations with both
soils (Figures 5.27 and 5.28), we can clearly see the diffraction hyperbolas, but particularly for the
dry soil case, they appear rather distorted. When we add surface roughness, the hyperbolas are
hardly to distinguish even after applying time gating (subtracting early arrival times to eliminate
partially the surface contribution to the radargram). In both cases, but in particular for the
wet soil environment, the surface roughness contribution is much more important than the soil
80
5: A 2D Parametric Study of the Scattering by Small Objects
1
1
1
1
2
2
2
2
3
3
Rectangular plate in inhom. wet soil, rough surface
0
t(ns)
Empty pipe in inhom. wet soil, rough surface
0
t(ns)
Rectangular plate in inhom. wet soil, flat surfce
0
t(ns)
Empty pipe in inhom. wet soil, flat surface
t(ns)
0
3
3
4
4
4
4
5
5
5
5
6
−0.4
−0.2
2
4
0
x(m)
6
0.2
8
10
6
0.4
−0.4
12
−0.2
0
x(m)
5
0.2
10
6
0.4
−0.4
15
−3
−0.2
0.05
0
x(m)
0.2
0.1
6
0.4
−0.4
0.15
−0.2
0.05
0
x(m)
0.2
0.1
0.4
0.15
−3
x 10
x 10
Figure 5.28 – Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic rect.
cylinder (w=10cm, h=4cm) in inhomogeneous wet soil with flat and rough surface.
Empty pipe in hom. dry soil, rough surface
Plastic rect. cylinder in hom. wet soil, rough surface
1
1
1.5
1.5
1.5
2
2
2
2
2.5
2.5
2.5
2.5
3.5
3
t(ns)
t(ns)
3.5
1.5
3
3
t(ns)
3
t(ns)
Empty pipe in hom. wet soil, rough surface
Plastic rect. cylinder in hom. dry soil, rough surface
1
1
3.5
3.5
4
4
4
4
4.5
4.5
4.5
4.5
5
5
5
5
5.5
5.5
5.5
5.5
6
−0.4
−0.2
0.005
0
x(m)
0.01
0.2
0.015
0.4
0.02
6
−0.4
−0.2
0.005
0
x(m)
0.01
0.2
0.4
0.015
0.02
6
−0.4
−0.2
0.01
0.02
0
x(m)
0.03
0.2
0.04
0.4
0.05
6
−0.4
−0.2
0.01
0.02
0
x(m)
0.03
0.2
0.04
0.4
0.05
Figure 5.29 – Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic rect.
cylinder (w=10cm, h=4cm) in homogeneous dry and wet soil with rough surface applying time gating.
5.5: Signatures of Circular and Rectangular Cylinders in Time Domain
81
heterogeneity contribution due to the larger fluctuation in height of this configuration.
Summarizing, the simulation results reveal that in general the contribution of the surface roughness is clearly stronger than the contribution of the heterogeneity even for slight height variations.
Only when the contrast is large enough and the target is not very shallowly buried (in this case
its diffraction hyperbola usually gets totally masked by the rough surface scattering), it may be
possible to distinguish the target.
82
5: A 2D Parametric Study of the Scattering by Small Objects
6
GPR Antenna Modeling in Frequency
Domain
Essentially, all models are wrong, but some are useful
George Box
GPR systems often operate in direct contact with the ground and very close to the target. When
the target is so close to the radar, it interacts with the reactive fields of the antenna (i.e., it is
in the near field region of the antenna), and only accurate models would reflect this mode of
operation and produce reliable results. The antenna is probably the most important component
of the modeling scheme as it has a significant influence on the nature of the propagating waves
and therefore, the final recorded signal; it also determines the detection capacity of the system.
Hence, accurate GPR simulations are fundamental to understand and characterize real antenna
performance.
The early GPR 2D simulations ignored the antenna concept, using excitations modeled as simple plane waves, line sources or infinitesimal sources (Hertzian dipoles) [Liu & Chen, 1991],
[Moghaddam et al., 1991], [Wang & Tripp, 1996]. Later, fully 3D simulations of typical GPR
antennas were published by some authors. In [Bourgeois & Smith, 1996] a GPR antenna system consisting of shielded and resistively loaded bow-tie antennas fed by 1D transmission lines
was modeled. Other models with unshielded finite length dipoles situated directly above surface were introduced in the following years by [Gürel & Oğuz, 2000], [Texeira & Chew, 2000],
[Holliger & Bergman, 1998] and although reasonably successful, they did not simulate the antenna directivity and ground coupling with enough accuracy for near-surface high resolution
applications. Recently, there have been some contributions which have modeled more realistic shielded bow-ties or dipole antennas obtaining satisfactory results in modeling the coupling, propagation and reflection response of the GPR wave [Uduwawala & Norgren, 2004],
[Lampe & Holliger, 2005], [Warren & Giannopoulos, 2009]. Nevertheless, most of these studies
83
84
6: GPR Antenna Modeling in Frequency Domain
were either focused only on the antenna modeling itself or they assumed some ideal conditions
to describe soil parameters, surface roughness or target’s internal structure. In many cases these
simplifications are a good compromise, but when high resolution and accuracy are required, some
of these assumptions may not be acceptable any more.
When dealing with GPR antenna modeling, although it is possible to conduct very realistic simulations using FEM and in particular COMSOL simulation tool, it is not required to model in
detail all the hardware components of the antenna system to achieve accurate results. This information is in any case not available for most commercial systems and such a complex model
would increase unnecessarily the computational demand.
In this chapter, we intend not only to find a satisfactory model to represent the antenna system
but to get a better knowledge about the influence of the most important parameters on the performance of an UWB antenna in near field, and to state some guidelines for the GPR antenna
modeling problem in near-surface applications. This analysis is accomplished in frequency domain, as it is desirable to select a design which shows a good behaviour in terms of radiation
pattern, power density and impedance along the frequency band of our commercial system.
6.1
Bow-Tie Antenna
As mentioned in Chapter 3, the GPR equipment employed in our investigation is a time domain
UWB system manufactured by ERA Technology with a nominal central frequency of 2Ghz and
2Ghz bandwidth. The antenna system consists of two bow-ties placed side by side (Fig. 6.1)
and enclosed in a shielding box designed to isolate the antenna from external interference and
to enhance the radiation downwards. The antenna casing is presumably filled with an absorbing
material in order to eliminate internal reflections. As already stated, the spatial dependence of
the radiation pattern from a shielded bow-tie antenna will be different from that of a bow-tie
in free space and it is necessary to understand how the effective field pattern, beamwidth and
impedance change for a real configuration. However, the exact shape and size of the bow-ties
inside the antenna unit is unknown and we will create an approximate model which fulfills our
system specifications and matches the measurements.
Figure 6.1 – Model of the GPR antennas (with a transmitter and a receiver bow-tie, EM absorbing
material and metallic shielding.)
85
6.2: Antenna Feed
In order to reach a satisfactory design and gain knowledge about GPR antenna radiation characteristics, in the following sections we analyze the whole modeling process step by step, investigating the effect of various antenna parameters and every element introduced in the model.
First, the antenna is modeled in free space from the simplest case of the bow-ties alone to the
most realistic one where the antennas are surrounded by a metallic reflector filled with absorbing material. Secondly, we analyze the antenna-medium energy coupling when the antenna is
operating in contact or at a certain height above the ground. In this way, we will gather enough
understanding to develop a satisfactory model to simulate the actual GPR antenna illumination.
However, this model still has to be optimized via time domain analysis and comparison with
measurements.
6.2
Antenna Feed
Ports are a unique type of boundary condition, which are defined on a 2D planar surface and allow
energy to flow into and out of a structure. The lumped port (port excitation) boundary condition
is recommended only for surfaces internal to the geometric model. It allows excitation in terms of
voltage potential and is considered the interface between voltages/currents and electric/magnetic
fields. The port is based upon transmission line theory and the feed point must be defined in a
way similar to a transmission line feed, thus the gap must be much less than the wavelength.
The relationship between the electric field and the input voltage is defined as
Z
Z
(6.1)
V = Edl = (E · ul )dl,
l
l
where l corresponds to the line length between the terminals in which the port is placed. The
input voltage at the port is set to 1V and the impedance should be set to fit the estimated antenna
impedance (as shown in the following section). A good matching reduces the reflections back to
the cable and maximizes the radiated energy. The ringing is thereby also reduced.
6.3
Antenna Radiation Pattern and Impedance
To describe the electromagnetic behaviour and performance of an antenna, it is necessary to
consider a series of parameters which will determine the antenna characteristic; in particular,
along this chapter we focus on the analysis of the radiation pattern, the power density and
the antenna impedance for different antenna models (for detailed parameter descriptions see
Definitions).
When we display the radiation patterns, we are representing the radiated power density (or the
electric or magnetic field) at a constant distance in 2D cuts of the principal E-plane and Hplane (defined in Chapter 4). The pattern observations are usually made on a sphere of constant
radius extending into the far field. Then, the radiated power density corresponds to the radial
component of the time-averaged Poynting vector which is proportional to |E|2 , since in far field
electric and magnetic fields are perpendicular. More specifically, the radiation power density can
86
6: GPR Antenna Modeling in Frequency Domain
be expressed in terms of the far-zone electric field by [Balanis, 2005]
Wrad =
1
|E|2 ,
2η
(6.2)
with η the intrinsic impedance of the medium. This magnitude is also directly related to the
directivity and gain of the antenna, which are sometimes preferred parameters to measure the
antenna performance. Since in practice the difference between maximum and minimum power
values is very large, it is quite common to express the power density or electric field in decibels
(dB) and refer to relative power/amplitude patterns according to:
Relative power density (dB) = 10log10
W
|E|
= 20log10
.
Wmax
|E|max
(6.3)
In the formula above, the power expression is normalized respect to its maximum angular value
for a given radius. In Fig. 4.5 a 2D-cut together with a three dimensional normalized plot of the
average power density of a bow-tie dipole was already illustrated. However, in the present chapter
we show all the simulated radiation patterns without applying any normalization in order to keep
the information about the amplitude. It must be mentioned that the distances at which we
evaluate the radiation patterns, are convenient for investigation purposes in landmine detection,
where we are interested in objects located within a region up to 20-25cm far from the antennas.
More precisely, we have analyzed the power patterns at a radial distance d of 25cm from the
antennas in free space and 20cm in dielectric medium. Such distances can be assumed in the far
field region of the antenna, although for the higher frequencies of the bandwidth they are very
close to the boundary with the Fresnel region or even inside (see Definitions). However, we have
observed from these simulations that the reactive component of the power density is insignificant
and overall behaviour of the EM fields corresponds to that characteristic of the far field, being
the power density associated with the fields predominantly real. Thus, the equality between the
power density and the electric field defined by Eq. 6.2 still holds and we use this relationship to
derive the power patterns in the following sections.
6.4
Antenna Contribution
6.4.1
The Antenna Flare Angle
Firstly, we analyze the impact of modifying the antenna flare (or aperture) angle between typical
values. Figures 6.2 and 6.3 show some results from the corresponding simulations. These illustrations demonstrate that the input impedance as well as the power patterns are not very sensitive
to small changes in flare angle. The impedance decreases slowly when increasing the angle but
the difference is not significant. From the plot of the radiated power density, we observe that for
all the frequencies within the bandwidth, the behaviour of the three considered antennas looks
almost the same and hence, the three radiation patterns are very similar. It is only appreciated
that the radiated power plateau is achieved at a slightly lower frequency when the flare angle
increases; this is an expected result since the antenna bandwidth increases with flare angle.
We have seen that realistic variations of the flare angle produce slight effects on the frequency
87
6.4: Antenna Contribution
Radiated power in forward direction
15
10
20*log10(normE)
5
0
−5
−10
−15
48°
59°
70°
−20
−25
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
9
x 10
Figure 6.2 – Radiated power in forward direction (d=25cm) for different flare angles.
48°
59°
70°
400
400
400
200
200
200
0
0
0
−200
−200
−200
−400
−400
−400
−600
0
2
f(Hz)
−600
0
4
9
x 10
2
f(Hz)
Resistance (Ohm)
−600
0
4
9
x 10
Reactance (Ohm)
Figure 6.3 – Impedances for 3 different flare angles.
2
f(Hz)
4
9
x 10
88
6: GPR Antenna Modeling in Frequency Domain
4cm
6cm
90
90
120
150
10
5
0
−5
−10
−15
120
60
150
30
180
0
330
210
330
210
90
10
−10
330
300
240
90
10
60
0
−10
−10
150
30
−20
0
330
210
0
330
210
300
300
240
180
0
330
210
1.5GHz
2GHz
300
240
270
1GHz
30
−20
180
270
10
120
60
150
30
180
240
0
0
−20
30
270
120
60
0
60
210
300
240
10
5
0
−5
−10
−15
180
270
120
150
150
30
0
270
90
90
120
60
180
300
240
8cm
10
5
0
−5
−10
−15
270
2.5GHz
3GHz
Figure 6.4 – Radiation pattern (d=25cm) for different box heights and frequencies=1, 1.5, 2, 2.5 and
3Ghz, E-plane (top), H-plane (bottom).
domain characteristic of the antennas that are irrelevant for our analysis. Therefore, from now
on this parameter is fixed at the intermediate value of 60◦ and will not be further investigated in
the next simulations.
6.4.2
The Antenna Shielding
In this section we add a rectangular shielding enclosure to our 2 bow-tie antenna model. The
dimensions of this metallic casing are 8cm long and 4cm wide and for the height we have tested
different values varying from 2cm to 10cm. We have also considered five frequencies within the
bandwidth of interest. As it can be appreciated in Fig. 6.4, the presence of the shielding has
a noticeable effect on the radiation characteristics in both E- and H-plane. The corresponding
radiation patterns are no more symmetric and the energy is efficiently directed downwards for
all the antenna heights and frequencies considered except for the 1GHz case. This is because
the associated wavelength is much bigger than the shielding height. Then, when increasing
the distance to the shielding, the amount of energy radiated backwards starts to decrease and
the forward directivity gets better. Besides, the radiation patterns are narrower than for the
unshielded case and therefore, in the standard acquisition mode, a shielded antenna illuminates a
significantly smaller volume of the subsurface (higher directivity and gain) than the corresponding
unshielded antenna, which also improves the horizontal resolution. When we look at the radiated
89
6.4: Antenna Contribution
Radiated power in forward direction
20
15
10
20*log10(normE)
5
0
−5
−10
−15
2cm
4cm
6cm
8cm
10cm
−20
−25
−30
0
0.5
1
1.5
2
2.5
Frequency (Hz)
3
3.5
4
9
x 10
Figure 6.5 – Radiated power in forward direction (d=25cm) for different box heights.
power density diagrams displayed in Fig. 6.5, we observe that there is not a big change for the
diverse heights. In general, shorter antenna casing heights tend to enhance the high frequency
gain while greater ones enhance the low-frequency contribution. More precisely, we noted that
for the shortest height (2cm), the bandwidth was slightly reduced but the rest of the simulated
curves from 4cm to 10cm were very similar. Hence, a box height of 4cm is chosen for the following
simulations.
6.4.3
The Absorbing Material
The presence of a shielding reflector produces signal reverberation (ringing) inside the box. A
common solution to overcome the problem of antenna ringing is to fill the shielding box with
absorbing material. Adding a dielectric absorbing material increases the antenna electrical size
and gives rise to remarkable effects on the radiation patterns that need to be investigated and
accounted for. In Fig. 6.6 and 6.7 we display the radiation patterns when we consider absorbing
materials of relative permittivities 6, 8, 10 and 12 and conductivities taking the values of 0.2,
0.3 and 0.4S/m for each of them. In all the cases, the E-plane radiation patters are broadened
substantially. By increasing the permittivity of the absorbing material, the ratio of lower to upper
half-space maximum amplitude decreases, i.e., the antenna becomes less directive.
To better show the effect of the absorber ǫr and σ over the whole bandwidth, Fig. 6.8 illustrates
the radiated power in forward direction for diverse absorber parameters. As expected from the
analysis of the radiation patterns, the radiated power (and consequently, the gain) in forward
direction decreases as the absorber permittivity rises. This trend holds true over the full frequency band. Moreover, the oscillations in the forward radiated power along the frequency band
are softened for increasing conductivity, which favours the impedance matching. Due to this
90
6: GPR Antenna Modeling in Frequency Domain
εr=6, σ=0.2S/m
90
120
150
3
1
−1
−3
−5
−7
−9
−11
−13
εr=6, σ=0.3S/m
90
60
120
30
150
180
0
210
240
90
3
1
−1
−3
−5
−7
−9
−11
−13
60
120
30
150
180
0
210
330
εr=6, σ=0.4S/m
60
30
180
0
210
330
240
300
3
1
−1
−3
−5
−7
−9
−11
−13
330
240
300
300
270
270
270
εr=8, σ=0.2S/m
εr=8, σ=0.3S/m
εr=8, σ=0.4S/m
90
90
120
150
3
1
−1
−3
−5
−7
−9
−11
−13
60
120
30
180
150
0
210
150
30
180
0
330
210
240
330
2.5GHz
3GHz
εr=10, σ=0.4S/m
90
3
1
−1
−3
−5
−7
−9
−11
−13
60
120
150
30
180
0
330
210
300
240
300
270
2GHz
90
30
0
240
300
120
60
180
εr=10, σ=0.3S/m
60
3
1
−1
−3
−5
−7
−9
−11
−13
210
330
1.5GHz
εr=10, σ=0.2S/m
150
150
270
1GHz
3
1
−1
−3
−5
−7
−9
−11
−13
30
0
240
300
90
120
180
270
120
60
210
330
240
90
3
1
−1
−3
−5
−7
−9
−11
−13
3
1
−1
−3
−5
−7
−9
−11
−13
60
30
180
0
330
210
300
240
300
270
270
270
εr=12, σ=0.2S/m
εr=12, σ=0.3S/m
εr=12, σ=0.4S/m
90
120
150
3
1
−1
−3
−5
−7
−9
−11
−13
90
60
120
30
180
150
0
210
330
240
120
60
150
30
180
0
210
330
240
300
90
3
1
−1
−3
−5
−7
−9
−11
−13
270
1.5GHz
2GHz
30
0
330
210
240
300
60
180
300
270
270
1GHz
3
1
−1
−3
−5
−7
−9
−11
−13
2.5GHz
3GHz
Figure 6.6 – Radiation pattern (d=25cm) in free space for variable ǫbox and σbox , E-plane.
91
6.4: Antenna Contribution
εr=6, σ=0.2S/m
90
120
150
4
2
0
−2
−4
−6
−8
εr=6, σ=0.3S/m
90
60
120
30
150
180
0
210
90
60
120
30
150
180
0
210
330
240
4
2
0
−2
−4
−6
−8
εr=6, σ=0.4S/m
60
30
180
0
210
330
240
300
4
2
0
−2
−4
−6
−8
330
240
300
300
270
270
270
εr=8, σ=0.2S/m
εr=8, σ=0.3S/m
εr=8, σ=0.4S/m
90
90
120
150
4
2
0
−2
−4
−6
−8
60
120
30
180
150
0
210
150
30
180
0
330
210
240
330
240
4
2
0
−2
−4
−6
−8
2.5GHz
3GHZ
εr=10, σ=0.4S/m
90
60
120
150
30
180
0
330
210
300
240
300
270
2GHz
90
30
0
300
120
60
180
εr=10, σ=0.3S/m
60
4
2
0
−2
−4
−6
−8
210
330
1.5GHz
εr=10, σ=0.2S/m
150
150
270
1GHz
4
2
0
−2
−4
−6
−8
30
0
240
300
90
120
180
270
120
90
60
210
330
240
4
2
0
−2
−4
−6
−8
4
2
0
−2
−4
−6
−8
60
30
180
0
330
210
300
240
300
270
270
270
εr=12, σ=0.2S/m
εr=12, σ=0.3S/m
εr=12, σ=0.4S/m
90
120
150
4
2
0
−2
−4
−6
−8
90
60
120
30
180
150
0
210
330
240
4
2
0
−2
−4
−6
−8
120
30
180
150
0
210
330
240
300
90
60
270
1.5GHz
2GHz
30
0
210
330
240
300
60
180
270
1GHz
4
2
0
−2
−4
−6
−8
300
270
2.5GHz
3GHz
Figure 6.7 – Radiation pattern (d=25cm) in free space for variable ǫbox and σbox , H-plane.
92
6: GPR Antenna Modeling in Frequency Domain
σ=0.3S/m
σ=0.4S/m
5
0
0
0
−5
−10
ε =6
−15
ε =8
r
r
ε =10
r
−20
−5
−10
ε =6
−15
ε =8
r
r
ε =10
r
−20
ε =12
1
2
3
f (Hz)
−5
−10
ε =6
−15
ε =8
r
r
ε =10
r
−20
ε =12
r
−25
0
20*log10(normE)
5
20*log10(normE)
20*log10(normE)
σ=0.2S/m
5
ε =12
r
r
4
5
−25
0
1
9
x 10
2
3
f (Hz)
4
5
−25
0
9
x 10
1
2
3
f (Hz)
4
5
9
x 10
Figure 6.8 – Radiated power in forward direction (d=25cm) for variable absorber ǫbox and σbox .
reason, a high conductivity material would be in principle preferred to model the absorber of
our commercial antenna. Figures 6.9 and 6.10 display the computed impedances for the different
absorber parameters considered. From these results, we see that after a first peak, the real part
of the impedance reaches a constant value which decreases with rising permittivity values. This
behaviour can be roughly explained applying transmission line (TL) theory to estimate the input
impedance at the open end of the shielding box.
A bulk absorbing material (no variation in the transverse plane, which means that the field
depends on transverse components only through a transient factor) of permittivity ǫr and conductivity σ can be modeled with a transmission line as displayed in Fig. 6.11. One end of the
transmission line is terminated in a load impedance ZL , and the input impedance at the other
end is for a lossy material [Pozar, 2005]:
Zin = Z0
ZL + Z0 tanh(kd)
,
Z0 + ZL tanh(kd)
(6.4)
where k is the complex propagation constant defined in Eq. 4.25, Z0 is the characteristic
impedance of the absorber defined in Eq. 2.4 and d its thickness. If we assume a PEC shielding
(short circuit, ZL = 0), Eq. 6.4 is transformed into
Zin = Z0 tanh(kd).
(6.5)
We have applied Eq. 6.5 (which is based on the model represented in 6.11) to estimate the input
impedance for different absorber permittivity ǫr and thickness. The results are depicted in Fig.
6.12 and 6.13.
As we can see in these figures, the impedance values obtained applying TL theory are very close to
the simulated ones showed in Fig. 6.9 and 6.10, which demonstrates that this can be an effective
method for a first evaluation of the impedance behaviour of different absorber configurations
(for example, the more complex case of a layered absorber) avoiding the performance of time
consuming 3D simulations for every single setup. Nevertheless, in the measurement there is a
peak at lower frequencies which is more pronounced for low conductivities and that cannot be
explained by our simple model. This peak is most probably due to the influence of the lateral
walls of the shielding box, which becomes more important for low frequencies and for the lower
conductivities, i.e., lower attenuation values. Then, this behaviour is not present in Fig. 6.12
93
6.4: Antenna Contribution
ε =6, σ =0.2S/m
r
ε =6, σ =0.3S/m
r
r
ε =6, σ =0.4S/m
r
r
200
200
200
150
150
150
100
100
100
50
50
50
0
0
0
−50
−50
−50
0
1
2
3
4
f(Hz)
5
0
1
2
3
f(Hz)
9
x 10
4
5
0
1
r
2
Resistance (Ohm)
3
4
f(hz)
9
x 10
Reactance (Ohm)
5
9
x 10
Figure 6.9 – Impedances for ǫbox = 6 and variable σbox .
ε =10, σ
r
ε =10, σ =0.3S/m
=0.2S/m
box
r
ε =10, σ =0.4S/m
r
r
200
200
200
150
150
150
100
100
100
50
50
50
0
0
0
−50
−50
0
1
2
3
f(Hz)
4
5
9
x 10
0
r
−50
1
2
3
f(Hz)
Resistance (Ohm)
4
5
9
x 10
Reactance (Ohm)
0
1
2
3
f(Hz)
Figure 6.10 – Impedances for absorber ǫr = 10 and variable σr .
Figure 6.11 – Transmission line model of the antenna box.
4
5
9
x 10
94
6: GPR Antenna Modeling in Frequency Domain
80
Imag(Z ) (Ohm)
Real(Z ) (Ohm)
200
100
in
in
150
50
0
0
2
4
f(GHz)
6
ε=4
8
ε=6
ε=8
60
40
20
0
0
2
4
f(GHz)
ε=10
ε=12
ε=14
6
8
Figure 6.12 – Input impedance for different permittivities, σ = 0.4S/m.
200
60
in
Imag(Z )
in
Real(Z )
150
100
50
0
0
2
4
f(GHz)
l=1cm
6
l=2cm
40
20
8
0
0
2
l=4cm
l=6cm
l=8cm
4
f(GHz)
6
8
l=10cm
Figure 6.13 – Input impedance for different absorber thicknesses, ǫ = 6, σ = 0.4S/m.
because in our model the transversal dimension has been considered infinite and this assumption
is not true in the real case.
6.4.4
The Receiver
To investigate the effects of the nearby receiving antenna on the transmitted radiation pattern,
we computed the responses of a model comprising a shielded transmitter bow-tie antenna with
60◦ flare angle and the corresponding receiver antenna juxtaposed in the perpendicular-broadside
mode. The receiver antennas were placed 4cm apart from the transmitter antennas and we considered the three different cases already analyzed for the transmitter antenna alone in the previous
sections: the bow-ties without any casing, and the bow-ties with empty and absorber filled metallic casing. The radiation patterns (Fig. 6.14) reveal that the receiver antennas have only minor
impact on the energy transmitted into the subsurface for all the three cases. Nevertheless, the
effect of the receiver is evident in the H-plane where the radiation pattern appears clearly biased
to the right side. This happens due to the asymmetry of the arrangement and the presence of
the shielded receiver on the left side of the transmitter.
95
6.4: Antenna Contribution
Tx+Rx
90
90
10
120
5
60
120
0
150
30
−5
150
−10
180
0
210
60
180
0
210
330
240
300
90
10
60
30
60
0
−10
150
30
−5
150
−20
−10
180
0
210
180
0
300
330
240
270
180
1.5GHz
300
2GHz
0
210
330
240
270
1GHz
30
−10
210
330
5
120
0
−5
30
−10
270
120
0
240
−5
150
300
90
10
150
30
270
5
60
0
330
240
5
120
0
300
90
90
60
180
270
120
10
5
0
−5
−10
−15
−20
210
330
240
Tx+Rx + Shielding +
Absorber (εr=6, σ=0.4S/m)
Tx+Rx + Shielding
300
270
2.5GHz
3GHz
Figure 6.14 – Radiation pattern in free space (d=25cm) of two bow-ties alone, with shielding box
and with absorbed filled shielding box, E-plane (top) and H-plane (bottom).
96
6.5
6: GPR Antenna Modeling in Frequency Domain
Soil Influence
When the antennas are located near the surface, the antenna current distribution and consequently the radiation pattern are significantly affected. It is known, that the medium generally
acts as a low pass filter modifying the spectrum of the transmitted signal as a function of its
electromagnetic properties. Additionally, if the medium is dissipative, an exponential damping
reduces rapidly the field intensity limiting drastically the wave penetrating depth. Hence, it is
very important that ground-coupled antennas maximize the power radiated into the material
medium, and in particular, that they concentrate the electromagnetic field in the forward direction (in our case defined at polar angles near θ = 270◦ ). The investigation of the radiation
behaviour for different soil types and antenna elevations becomes thereby of great interest to
understand the antenna performance in a realistic situation.
6.5.1
Soil Parameters
In this section we explore the radiation characteristics of the antennas coupled directly into a
dielectric nonconductive medium surface (antenna height h=0cm). The GPR scene is modeled
assuming no ground surface roughness and three different media with permittivity ǫr = 4, 7, 10
and conductivity σ = 0.001, 0.01, 0.05S/m respectively. These parameters are representative of
dry, slightly wet (medium) and wet soil. First, and for comparison purposes, we display the
E-plane and H-plane radiation patterns and the forward radiated power density corresponding to
the dipoles alone and with shielding box. Figure 6.16 shows that with increasing soil permittivity,
the resonance frequency is displaced to lower values. The important reduction of power amplitude
related to wet soil is due to its higher conductivity (higher attenuation).
In Figures 6.17 and 6.18, we consider two antenna models with two different absorbers, displaying
again the E-plane and H-plane radiation patterns for both cases. From these figures, we observe
that for wetter soil, the radiation coupled into the ground is rapidly attenuated, in particular
when the conductivity reaches 0.05S/m this effect becomes more prominent.
In general, the amount of energy transmitted to the soil will depend on the EM contrast between
the antenna and the soil: when the permittivity of the soil is equal or very close to the permittivity
of the absorber more energy will be coupled. This is an important issue to take into account when
selecting an absorber material to design a GPR antenna. For instance, the dry sand present on
the shallow subsurface has typically a permittivity equal to 4-5 and wetter sand can reach values
between 8-10. If we consider this fact and the rest of the results derived from the simulations, a
permittivity of 6-7 for the absorber material seems to achieve a good performance and it could
be a good compromise. However, it still needs to be validated through time domain simulations.
Respect to the radiation distribution, we see that when the antennas are in contact with soil,
there is a significant difference in comparison to the patterns of the same dipoles in free space.
In E-plane, several lobes appear, and their intensity strongly depends on the frequency and
soil/absorber contrast. As we can see for some cases there is a principal lobe in forward direction,
for others the radiation is split into two intense lobes, and some other times there may be three
97
6.5: Soil Influence
Bow−tie, dry soil
Bow−tie, medium soil
90
90
13
8
3
−2
−7
120
150
120
60
150
30
180
0
330
210
150
30
0
330
210
60
30
180
0
330
210
300
240
270
9
4
−1
−6
−11
120
180
300
240
Bow−tie, wet soil
90
15
10 60
5
0
−5
−10
300
240
270
270
Bow−tie + PEC casing, dry soil
Bow−tie + PEC casing, medium soil
Bow−tie + PEC casing, wet soil
90
90
16
120
12
120
60
6
150
30
−18
180
0
330
210
180
0
330
210
300
1.5GHz
2GHz
2.5GHz
Bow−tie, medium soil
90
120
60
−2
150
30
−12
−22
180
0
330
210
13
8
3
−2
−7
−12
−17
90
60
2
−8
150
30
30
−18
330
210
180
0
330
210
300
240
12
120
60
0
270
3GHz
Bow−tie, wet soil
180
300
240
300
270
90
150
330
210
240
Bow−tie, dry soil
8
30
0
270
1GHz
60
180
300
240
270
120
150
30
−8
−14
240
8
3
−2
−7
−12
−17
2
−4
150
90
120
60
300
240
270
270
Bow−tie + PEC casing, dry soil
Bow−tie + PEC casing, medium soil
Bow−tie + PEC casing, wet soil
90
90
18
120
8
90
120
60
8
150
30
−12
180
0
330
210
300
240
150
30
−12
180
0
330
210
300
240
270
180
0
330
210
1.5GHz
2GHz
300
240
270
1GHz
30
−12
−22
−22
−22
60
−2
−2
−2
150
8
120
60
270
2.5GHz
3GHz
Figure 6.15 – Radiation patterns (d=20cm) in medium half-space of the bow-ties alone and enclosed
in a PEC casing. Dry (left), medium (center) and wet soil (right), E-plane (half-top)
and H-plane (half-bottom).
98
6: GPR Antenna Modeling in Frequency Domain
Radiated power in forward direction
15
20*log10(normE)
10
5
0
−5
Bow−tie, dry soil
Bow−tie, medium soil
Bow−tie, wet soil
Bow−tie + PEC casing, dry soil
Bow−tie + PEC casing, medium soil
Bow−tie + PEC casing, wet soil
−10
−15
0
0.5
1
1.5
2
2.5
3
Frequency (Hz)
3.5
4
4.5
5
9
x 10
Figure 6.16 – Radiated power in forward direction (d=20cm) for of the bow-ties alone and enclosed
in a PEC casing above different soils.
lobes, whose relative intensity change with frequency and soil, being not always the central lobe
the main one. The H-plane patterns present differences as well but in general, they are more
homogeneous and directive.
For convenience, in the next section the antenna impedance curves corresponding to the previous
models will be also displayed and briefly commented too.
6.5.2
Antenna Height
The height of the antenna above the ground has little influence on the antenna input impedance.
This is demonstrated in Fig. 6.19, where the input resistance and input reactance are plotted as
functions of frequency for antenna heights of 1cm, 3cm and 6cm above 3 different soils together
with the impedance when the antenna lies directly on the ground. Although the overall shapes
of the curves are similar, the main peaks and troughs are shifted to lower frequencies when the
antennas are placed on the surface, and the reason for this effect is that the antennas become
electrically longer as they “sense” the higher permittivity and effective smaller wavelengths in the
underlying half-space. Moreover, the maximum values of the input impedance curves are reduced
due to the lower wave impedance of the soil in comparison to that of the free space. When
they are elevated 1cm, 3cm or 6cm the curves are very similar to each other and also almost
identical to their free-space analogs. The impact on the radiation pattern is on the other hand
very important. As we can see in Fig. 6.20 and 6.21, when we increase the antenna height the
amount of the energy that penetrates the soil decays drastically in all the cases, i.e., the coupling
with the soil worsens, and the backward lobes become bigger. Nevertheless, when the antenna is
slightly elevated (1-2cm for the frequency band used: 1-3GHz), the antenna directivity towards
99
6.5: Soil Influence
εr=6, σ=0.2S/m, dry soil
εr=6, σ=0.2S/m, medium soil
90
90
120
150
12 60
7
2
−3
−8
−13
120
30
150
180
0
210
90
120
30
150
180
0
210
330
240
10 60
5
0
−5
−10
−15
5
60
0
−5
−10
−15
−20
30
180
0
210
330
240
300
εr=6, σ=0.2S/m, wet soil
330
240
300
300
270
270
270
εr=6, σ=0.4S/m, dry soil
εr=6, σ=0.4S/m, medium soil
εr=6, σ=0.4S/m, wet soil
90
120
150
12 60
7
2
−3
−8
−13
90
120
30
180
150
0
210
90
120
30
180
270
1.5GHz
2GHz
2.5GHz
ε =10, σ=0.2S/m, medium soil
90
12 60
7
2
−3
−8
−13
r
120
150
30
180
0
330
210
240
330
10 60
5
0
−5
−10
−15
150
30
330
240
εr=10, σ=0.2S/m, wet soil
90
0
210
3GHz
120
180
300
300
270
90
150
0
240
300
εr=10, σ=0.2S/m, dry soil
120
30
180
270
1GHz
5
60
0
−5
−10
−15
−20
210
330
240
300
150
0
210
330
240
10 60
5
0
−5
−10
−15
5
60
0
−5
−10
−15
−20
30
180
0
330
210
300
240
300
270
270
270
εr=10, σ=0.4S/m, dry soil
εr=10, σ=0.4S/m, medium soil
εr=10, σ=0.4S/m, wet soil
90
90
120
150
12 60
7
2
−3
−8
−13
120
30
180
150
0
210
330
240
10 60
5
0
−5
−10
−15
30
180
150
0
210
330
240
300
90
120
270
0
210
330
240
300
1.5GHz
2GHz
30
180
270
1GHz
5
60
0
−5
−10
−15
−20
300
270
2.5GHz
3GHz
Figure 6.17 – Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m. Dry (left), medium (center) and wet soil (right), E-plane.
100
6: GPR Antenna Modeling in Frequency Domain
εr=6, σ=0.2S/m, dry soil
εr=6, σ=0.2S/m, medium soil
90
90
120
150
12 60
7
2
−3
−8
−13
120
30
150
180
0
210
90
120
30
150
180
0
210
330
240
10 60
5
0
−5
−10
−15
5
60
0
−5
−10
−15
−20
30
180
0
210
330
240
300
εr=6, σ=0.2S/m, wet soil
330
240
300
300
270
270
270
εr=6, σ=0.4S/m, dry soil
εr=6, σ=0.4S/m, medium soil
εr=6, σ=0.4S/m, wet soil
90
120
150
12 60
7
2
−3
−8
−13
90
120
30
180
150
0
210
90
120
30
180
270
1.5GHz
2GHz
2.5GHz
εr=10, σ=0.2S/m, medium soil
90
12 60
7
2
−3
−8
−13
120
150
30
180
0
330
210
240
330
10 60
5
0
−5
−10
−15
150
30
330
240
εr=10, σ=0.2S/m, wet soil
90
0
210
3GHz
120
180
300
300
270
90
150
0
240
300
εr=10, σ=0.2S/m, dry soil
120
30
180
270
1GHz
5
60
0
−5
−10
−15
−20
210
330
240
300
150
0
210
330
240
10 60
5
0
−5
−10
−15
5
60
0
−5
−10
−15
−20
30
180
0
330
210
300
240
300
270
270
270
εr=10, σ=0.4S/m, dry soil
εr=10, σ=0.4S/m, medium soil
εr=10, σ=0.4S/m, wet soil
90
90
120
150
12 60
7
2
−3
−8
−13
120
30
180
150
0
210
330
240
10 60
5
0
−5
−10
−15
30
180
150
0
210
330
240
300
90
120
270
0
210
330
240
300
1.5GHz
2GHz
30
180
270
1GHz
5
60
0
−5
−10
−15
−20
300
270
2.5GHz
3GHz
Figure 6.18 – Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m. Dry (left), medium (center) and wet soil (right), H-plane.
101
6.5: Soil Influence
Dry soil
Medium soil
Wet soil
150
150
150
100
100
100
50
50
50
0
0
0
−50
0
1
2
3
f(Hz)
4
5
−50
0
9
x 10
1
2
3
f(Hz)
4
5
9
x 10
−50
0
1
2
3
f(Hz)
Resistance (h=0cm)
Resistance (h=1cm)
Resistance (h=3cm)
Resistance (h=6cm)
Reactance (h=0cm)
Reactance (h=1cm)
Reactance (h=3cm)
Reactance (h=6cm)
4
5
9
x 10
Figure 6.19 – Input impedance for different antenna heights and soil types.
the soil increases since the referred coupling is enhanced. This behaviour can be recognized when
we compare the radiation patterns for elevated antennas (heights equal to 1, 3 and 6cm from
left to right) displayed in figures 6.20 and 6.21 with the patterns corresponding to the antennas
placed just above the soil (previously shown in figures 6.17 and 6.18).
When we look at the forward radiated power energy for different elevations (illustrated in Fig.
6.22) this effect becomes even more clear being the height of 1cm the most suitable one to get
a high energy coupling and best matching (flat along the whole frequency band of interest) for
all the investigated antenna setups. For elevations of 3cm or above less energy penetrates the
soil and the bandwidth tends to reduce. Similar results were reported by [Smith, 1984] in the
80’s, where after investigating the performance of a half-wave dipole located at different heights
above a dielectric medium half-space, the elevation for the best coupling was estimated to be at
λ/10cm, being λ the operating frequency of the dipole. These results confirm such conclusion,
since for an antenna with a central frequency of 2GHz, the corresponding wavelength is equal to
15cm; hence, according to [Smith, 1984], an elevation of 1.5cm would be optimum for our impulse
GPR system.
6.5.3
Interface Roughness
In the next and last simulations, the flat air-ground interface of the homogeneous half-space model
is replaced by random topographic fluctuations described by a zero-mean normal distribution
with a standard deviation of 0.5cm for the first scenario and 1cm for the second scenario and a
correlation length equal to 3cm. We have considered dry and wet soil for both configurations.
The resulting radiation patterns reveal distortions in both planes with respect to the reference
patterns for the flat half-space. However, the overall shape is still approximately the same. The
same effects due to the antenna elevation are visible here. Again the backward radiation lobe
grows for wetter soil and the energy transmitted to the soil is notably reduced.
These scenarios represent just a couple of examples of more complex and realistic cases. The soil
inhomogeneity could also be included in these models but, as we have seen in 2D case, the impact
over the field distribution is in general less remarkable; then, we will not show more examples
adding inhomogeneities in this thesis.
102
6: GPR Antenna Modeling in Frequency Domain
εr=6, σ=0.2S/m, h=1
εr=6, σ=0.2S/m, h=3
90
90
120
150
11
60
6
1
−4
−9
−14
120
30
180
150
0
210
εr=6, σ=0.2S/m, h=6
90
60
120
30
180
150
0
210
330
300
240
8
3
−2
−7
−12
60
30
180
0
210
330
330
300
240
4
2
0
−2
−4
−6
−8
−10
300
240
270
270
270
εr=6, σ=0.4S/m, h=1
εr=6, σ=0.4S/m, h=3
εr=6, σ=0.4S/m, h=6
90
90
120
150
11
60
6
1
−4
−9
−14
120
150
30
180
0
330
210
240
8
3
−2
−7
−12
90
60
120
180
0
330
210
300
240
270
1.5GHz
2.5GHz
εr=10, σ=0.2S/m, h=3
90
ε =10, σ=0.2S/m, h=6
r
90
4
150
30
120
60
150
30
−6
−11
0
330
240
3GHz
−1
180
210
300
270
90
150
330
210
240
2GHz
120
180
0
330
210
300
30
0
300
εr=10, σ=0.2S/m, h=1
10
60
5
0
−5
−10
−15
60
180
270
1GHz
120
150
30
4
2
0
−2
−4
−6
−8
−10
240
4
2
0
−2
−4
−6
−8
−10
−12
60
30
180
0
330
210
300
240
300
270
270
270
εr=10, σ=0.4S/m, h=1
εr=10, σ=0.4S/m, h=3
εr=10, σ=0.4S/m, h=6
90
120
150
10
60
5
0
−5
−10
−15
90
30
150
120
60
30
−6
150
−11
0
330
300
240
4
−1
180
210
90
120
180
0
330
210
300
240
270
1.5GHz
2GHz
60
30
180
0
330
210
300
240
270
1GHz
4
2
0
−2
−4
−6
−8
−10
−12
270
2.5GHz
3GHz
Figure 6.20 – Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox = 0.2, 0.4S/m
and antenna heights h=1cm (left), h=3cm (center) and h=6cm (right), E-plane.
103
6.5: Soil Influence
εr=6, σ=0.2S/m, h=1cm
εr=6, σ=0.2S/m, h=3cm
90
90
120
5
60
90
120
2
30
−15
150
5
120
0
30
−18
180
0
210
180
0
300
180
0
210
330
300
240
30
−10
210
330
60
−5
150
−28
−25
240
60
−8
−5
150
εr=6, σ=0.2S/m, h=6cm
330
300
240
270
270
270
εr=6, σ=0.4S/m, h=1cm
εr=6, σ=0.4S/m, h=3cm
ε =6, σ=0.4S/m, h=6cm
90
90
120
5
60
90
120
2
150
30
−15
180
0
330
210
0
330
210
240
1.5GHz
0
330
210
240
2.5GHz
εr=10, σ=0.2S/m, h=3cm
90
90
1
εr=10, σ=0.2S/m, h=6cm
3
120
60
150
30
−12
−7
150
30
−19
0
330
210
180
0
330
210
300
240
30
−12
−29
180
60
−2
−9
−22
240
3GHz
90
120
60
300
270
2GHz
−2
30
180
300
εr=10, σ=0.2S/m, h=1cm
150
−5
270
1GHz
60
−10
180
300
8
0
150
30
−18
270
120
5
120
−28
−25
240
60
−8
−5
150
r
180
0
330
210
300
240
300
270
270
270
εr=10, σ=0.4S/m, h=1cm
εr=10, σ=0.4S/m, h=3cm
εr=10, σ=0.4S/m, h=6cm
90
8
120
90
90
120
60
1
−2
150
30
−12
150
30
−19
−7
150
0
330
210
300
180
0
330
210
300
240
270
180
0
330
210
1.5GHz
2GHz
300
240
270
1GHz
30
−12
−29
180
60
−2
−9
−22
240
3
120
60
270
2.5GHz
3GHz
Figure 6.21 – Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox = 0.2, 0.4S/m
and antenna heights h=1cm (left), h=3cm (center) and h=6cm (right), H-plane.
104
6: GPR Antenna Modeling in Frequency Domain
PEC casing (no Abs.), dry soil
20
20
10
10
20*log10(normE)
20*log10(normE)
No casing, dry soil
0
−10
h=0cm
h=1cm
h=3cm
h=6cm
−20
0
1
2
3
−10
h=0cm
h=1cm
h=3cm
h=6cm
−20
4
f(Hz)
0
5
0
1
5
5
20*log10(normE)
20*log10(normE)
10
0
−5
−10
h=0cm
h=1cm
h=3cm
h=6cm
−15
−20
2
3
0
−5
−10
h=0cm
h=1cm
h=3cm
h=6cm
−15
−25
0
5
1
5
20*log10(normE)
20*log10(normE)
10
0
−5
−10
h=0cm
h=1cm
h=3cm
h=6cm
−15
−20
f (Hz)
4
5
9
x 10
εr=10, σ=0.4S/m, dry soil
5
3
3
f (Hz)
x 10
10
2
2
9
εr=10, σ=0.2S/m, dry soil
1
5
9
x 10
−20
4
f (Hz)
−25
0
4
εr=6, σ=0.4S/m, dry soil
10
1
3
f(Hz)
εr=6, σ=0.2S/m, dry soil
−25
0
2
9
x 10
0
−5
−10
h=0cm
h=1cm
h=3cm
h=6cm
−15
−20
4
5
9
x 10
−25
0
1
2
3
f (Hz)
4
5
9
x 10
Figure 6.22 – Radiated power in forward direction (d=20cm) for different antenna elevations without
PEC casing, with empty PEC casing and with PEC casing + absorber of ǫbox = 6, 10
and σbox = 0.2, 0.4S/m.
105
6.5: Soil Influence
ε =6, σ=0.4S/m, h=3cm, rough1, dry soil
εr=6, σ=0.4S/m, h=3cm, rough1, wet soil
r
90
120
150
4
2
0
−2
−4
−6
−8
−10
−12
90
60
120
150
30
60
30
−8
−13
180
0
180
330
210
240
2
−3
0
330
210
300
240
300
270
270
εr=6, σ=0.4S/m, h=3cm, rough2, dry soil
εr=6, σ=0.4S/m, h=3cm, rough2, dry soil
90
120
150
4
2
0
−2
−4
−6
−8
−10
−12
90
120
60
30
150
60
30
−8
−13
180
0
180
330
210
0
330
210
300
240
2
−3
300
240
270
270
1GHz
1.5GHz
2GHz
2.5GHz
3GHz
Figure 6.23 – Radiation pattern (d=20cm) in medium half-space with rough interface, E-plane.
εr=6, σ=0.4S/m, rough1, dry soil
90
120
150
4
−1 60
−6
−11
−16
−21
εr=6, σ=0.4S/m, rough1, dry soil
90
120
30
180
150
0
210
30
180
0
210
330
240
3
−2 60
−7
−12
−17
−22
−27
300
330
240
300
270
270
ε =6, σ=0.4S/m, rough2, dry soil
ε =6, σ=0.4S/m, rough2, wet soil
r
90
120
150
4
−1 60
−6
−11
−16
−21
r
90
120
150
30
180
0
180
330
210
240
3
−2 60
−7
−12
−17
−22
−27
0
330
210
300
240
270
30
300
270
1GHz
1.5GHz
2GHz
2.5Ghz
3GHz
Figure 6.24 – Radiation pattern (d=20cm) in medium half-space with rough interface, H-plane.
106
6: GPR Antenna Modeling in Frequency Domain
7
GPR Antenna and Target Responses in
Time Domain
There is nothing like looking, if you want to find something. You certainly usually find
something, if you look, but it is not always quite the something you were after
J.R.R Tolkien
7.1
Time domain Characteristics of GPR antennas
In Chapter 6 we have investigated the frequency domain characteristic of several antenna configurations in order to obtain a better understanding of the radiation behaviour of a GPR antenna
and the influence of different parameters on its performance. The gained knowledge may be
useful to build an appropriate model to represent the antenna. However, when UWB systems
directly radiate fast transient pulses rather than employing a continuous wave carrier, the effect of the antenna on the transmitted waveform becomes a critical issue and typical frequency
dependent parameters such as radiation pattern and gain get less meaning. Moreover, these parameters need to be described over the whole frequency band, thus becoming less convenient to
characterize UWB time domain systems. On the other hand, parameters like ringing, received
signal amplitude, impulse response duration, etc. get more important. A good time domain
performance for the specific application, which means producing good quality raw data before
processing, is a primary requirement of an UWB antenna.
In general every time-invariant linear system is completely described in the time domain by its
Impulse Response h(t), which mathematically relates the input signal x(t) to the output y(t) of
the system by a convolutional integral that can be expressed by the convolution operator ⊗:
y(t) = h(t) ⊗ x(t).
107
(7.1)
108
7: GPR Antenna and Target Responses in Time Domain
In frequency domain this relationship is given by the Fourier transform; the convolution becomes
then a multiplication and the impulse response the so called transfer function.
The impulse response of a given system will result of the combined contribution of the radar
electronics (pulse generator, receiver, digitizer, etc.), the antennas and the shallow soil. The need
to use separate transmit and receive antennas causes a convolution of both radiation patterns
and the effective waveform recorded is dependent of the characteristics of both dipoles, not only
the transmitter. The soil has to be included in the response because, as we have seen for groundcoupled antennas, the antenna response changes as a function of the properties of the soil in
the antennas’s reactive near-field region. The referred combined system response can be either
approximated analytically [Scheers, 2001] or estimated via numerical simulation, being the latter
the selected procedure for this investigation.
Since the purpose of this study is to obtain highly precise target signatures, we need to introduce
a realistic target illumination in the simulations as well as an accurate model of the entire GPR
scenario. This is accomplished in the first part of the chapter through the design and optimization
of a GPR antenna model which must exhibit analogous time domain response to the measured
one. Once an adequate model for the antenna is defined, the second part is devoted to the analysis
of the target response for different system and environmental parameters.
7.1.1
Definition of source pulses
The gaussian and the monocycle (which is given by the first derivative of the Gaussian function)
are naturally wide bandwidth signals, with the center frequency and the bandwidth completely
dependent of the pulse’s width. In practice, the center frequency of the pulse fc is approximately
the reciprocal of the pulse’s length and the bandwidth is approximately equal to the center
frequency. Thus, for a 0.5ns pulse width, the center frequency and the half power bandwidth
(shown with red dashed lines in Fig. 7.1) are approximately 2GHz. In time domain, these pulses
are mathematically described as follows:
V (t, f c, A) = Ae−2[πfc (t−τ )]
2
√
2
V (t, f c, A) = 2 eπfc A(t − τ )e−2[πfc (t−τ )]
(7.2)
(7.3)
where A determines the peak amplitude and τ = 1/fc .
Then, to simulate the real system the transmitter port is excited using 0.5ns gaussian and monocycle pulses. Figure 7.1 shows both pulses in time and frequency domain after normalization.
These pulses are the most common waveforms considered in literature to simulate the feed of
impulse GPR antennas. Here the power spectral density (PSD) describes how the power of the
signal is distributed with frequency.
7.1.2
Optimization of the GPR Model
In order to choose a particular model to accurately represent the real antenna, we carry out a
parametric study of diverse antenna features that play a role on its time domain response. In
109
7.1: Time domain Characteristics of GPR antennas
Gaussian pulse, width=0.5ns
Power spectral density of the Gaussian
0
1
0.9
−3
0.7
−4
PSD in dB |yf|
Amplitude yt
−2
0.8
0.6
0.5
0.4
0.3
−6
−8
−10
0.2
−12
0.1
0
0
0.5
1
t(ns)
1.5
−14
2
Monocycle, width=0.5ns
0.6
−2
−3
−4
0.4
−6
PSD in dB |yf|
Amplitude yt
0.8
0.2
0
−0.2
−16
−0.8
−18
1.5
5
−12
−0.6
1
t(ns)
4
−10
−14
0.5
2
3
f (Ghz)
−8
−0.4
0
1
Power spectral density of the Monocycle
0
1
−1
0
2
−20
0
1
2
3
f (Ghz)
4
5
Figure 7.1 – Applied source pulses in time and frequency domain.
110
7: GPR Antenna and Target Responses in Time Domain
particular, we analyze the impact of those parameters on the simulated crosstalk (direct signal
between antennas recorded at the receiver) which is an indication of the system impulse response
for every configuration. The simulations in this section are carried out in free space, since we
pretend to optimize the model by direct comparison with laboratory measurements. The resulting
crosstalks are all normalized to the maximum signal amplitude for each figure.
Received Signal Waveform Before any other consideration we display below the actual
crosstalk between transmitter and receiver recorded by our commercial GPR system without
applying any preprocessing (see Fig. 7.2).
Measurement of the antenna crosstalk in free space
1
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.2 – Measurement of the antenna crosstalk in laboratory.
In the next sections, we attempt to find a two bow-tie model which produces a crosstalk response
as close as possible to the measured one (in width and shape) and which fits the system’s physical
dimensions. At the same time this model should meet the antenna specifications (antenna type
and overall geometry, bandwidth, source pulse length) and demonstrate a good performance
according to the previous analysis in frequency domain. We accomplish this investigation in a
parametric way.
Antenna Flare Angle
First, we explore the influence of the antenna flare angle over the crosstalk. To do it, we initially
consider both bow-ties alone, transmitter and receiver, without any shielding box in order to
avoid adding other effects to the synthetic results. We perform the computation for 3 different
flare angles which were already considered in the frequency domain analysis of Chapter 6. The
bow-tie length is maintained constant and temporarily equal to 3.3cm from the antenna open
end to the center. We feed the transmitter with a gaussian and a monocycle pulse as described
in the section above. From the curves in Fig. 7.3 we can clearly infer that increasing the angle
has a positive effect on the impulse response. More precisely, the ringing amplitude decreases
respect to the crosstalk amplitude for increasing flare angles. The width of the crosstalk also
grows for bigger angles (since the lower frequencies are more efficiently radiated), getting closer
to the measured crosstalk (Fig. 7.2), in particular when applying a gaussian pulse. However, it
is still clearly narrower.
111
7.1: Time domain Characteristics of GPR antennas
Effect of the flare angle
1
49°
60°
70°
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
1
49°
60°
70°
0.5
Amplitude
6
−9
x 10
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.3 – Simulated crosstalks for different flare angles for a gaussian pulse (top) and a monocycle
(bottom) pulse.
Antenna length
To conduct this calculation we maintain a constant flare angle equal to 60◦ . We have selected this
value because we don’t want to exceed the actual antenna physical dimensions when increasing
the length; however, for the final model we may choose another angle. We observe that as
the antenna becomes longer, the crosstalk amplitude grows and broadens (Fig. 7.4). This effect
happens because a bigger antenna size is associated with lower frequencies, which is translated into
a wider pulse. The shape of the pulse is also slightly modified but the ringing is not significantly
affected with the size change.
Cable Impedance
We see that an increase of the impedance reduces the ringing. It also affects the crosstalk slightly,
being the influence a bit more significant when the excitation at the feed port is a gaussian pulse
(Fig. 7.5). This is probably because the impedance mismatch between the radiating element and
the cable affects more to the frequencies below the antenna central frequency (∼ 2GHz), and the
gaussian contains stronger lower spectral components than the monocycle (Fig. 7.1).
112
7: GPR Antenna and Target Responses in Time Domain
Effect of the antenna length
Amplitude
1
la=27mm
la=31mm
0
la=35mm
l =39mm
a
−1
−2
0
1
2
3
4
5
t(s)
Amplitude
1
la=27mm
la=31mm
0
la=35mm
la=39mm
−1
−2
0
6
−9
x 10
1
2
3
4
5
t(s)
6
−9
x 10
Figure 7.4 – Simulated crosstalks for different antenna lengths and constant flare angle for a gaussian
pulse (top) and a monocycle (bottom) pulse.
Effect of the cable impedance
1
Z=50Ω
Z=80Ω
Z=110Ω
Z=140Ω
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
1
Z=50Ω
Z=80Ω
Z=110Ω
Z=140Ω
0.5
Amplitude
6
−9
x 10
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.5 – Simulated crosstalks for different cable input impedances for a gaussian pulse (top) and
a monocycle (bottom) pulse.
113
7.1: Time domain Characteristics of GPR antennas
Effect of the source pulse central frequency
1
f=1.5GHz
f=2GHz
f=2.5GHz
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
1
f=1.5GHz
f=2GHz
f=2.5GHz
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.6 – Simulated crosstalks for a gaussian pulse (top) and a monocycle pulse (bottom) with
different central frequencies.
Source Pulse Frequency
The effect of decreasing the source pulse central frequency is as expected to broaden the pulse
width. The overall shape of the crosstalk does not change very much but the amplitude of the
peaks changes with varying frequency, becoming larger for lower frequencies. On the other hand,
late-time ringing remains at the same level respect to the peak amplitude for all the cases (Fig.
7.6).
Shielding Box
The addition of a a shielding box to the model has a positive effect since it isolates the antennas
from other external radiation sources, reduces the direct coupling, and directs the radiated energy
downwards. Hence, the presence of the shielding has a strong impact on the radiation pattern,
which was already reported in Chapter 6.
In this section we illustrate the effect on the crosstalk when the shielding box size (height h, length
l and width w) is varied. A sketch of the antenna model with the corresponding dimensions is
shown in Fig. 7.7.
114
7: GPR Antenna and Target Responses in Time Domain
Effect of the cavity height
1
h=2cm
h=4cm
h=8cm
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
h=2cm
h=4cm
h=8cm
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.8 – Simulated crosstalks for different cavity heights for a gaussian pulse (top) and a monocycle (bottom) pulse.
Figure 7.7 – Sketch of the antenna head with shielding box.
As expected we observe that in general the ringing with shielding box is clearly stronger than
without box due to the reflections at the cavity walls (Fig. 7.8, Fig. 7.9 and Fig. 7.10). In Fig.
7.8, we can also see that the height variation does not influence significantly the pulse shape.
Only in the case of 2cm box height we see a small difference, but for the other three values the
crosstalk does not reveal any noticeable change. Therefore, we will assume a 4cm high box for
the next simulations and the final model.
On the other hand, when we modify the cavity size in width (Fig. 7.9) and length (Fig. 7.10), we
observe some changes in the pulse shape and amplitude. The variations are more significant for
width than for length changes and follow an opposite trend: for increasing width the amplitude
grows while for increasing length, the amplitude decreases. This behaviour is probably due
to the omnidirectional pattern of the bow-tie, which predominantly radiates in the direction
115
7.1: Time domain Characteristics of GPR antennas
Effect of the cavity length
1
l=36mm
l=40mm
l=44mm
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
l=36mm
l=40mm
l=44mm
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.9 – Simulated crosstalks for different cavity lengths for a gaussian pulse (top) and a monocycle (bottom) pulse.
perpendicular to the polarization axis of the antenna. Moreover, it seems that the effect of the
dimension change for both directions is stronger when the feeding pulse is gaussian. As expected
we also observe from the figures that the ringing is clearly stronger than without shielding due
to the reflections at the cavity walls.
Absorber
In order to reduce the ringing introduced by the metallic walls, we consider the effect of filling the
shielding box with absorbing material. The high conductivity of this material will attenuate the
waves reflected by the walls inside the box and hence, reduce the associated ringing. However,
as we have seen in Chapter 6, the drawback of having such a high conductivity is a remarkable
reduction of the antenna gain. In general, the value of the absorber conductivity will affect to the
shape of the crosstalk, in particular changing significantly the ratio between the first and second
peaks. Another effect of the absorber is to increase the electrical size of the antenna due to its
permittivity different from that of free space; the consequence is a broadening of the crosstalk as
the permittivity grows (Fig. 7.11 and Fig. 7.12).
We know that when the absorber parameters change, the antenna impedance will vary but in a
real situation the impedance of the cable has a fix value. Therefore, for simplicity, we will assume
a typical cable impedance of 50Ω in the next simulations.
116
7: GPR Antenna and Target Responses in Time Domain
Effect of the cavity width
1
w=40mm
w=44mm
w=48mm
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
w=40mm
w=44mm
w=48mm
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.10 – Simulated crosstalks for different cavity widths for a gaussian pulse (top) and a monocycle (bottom) pulse.
The Optimized Model
After performing a comprehensive analysis to investigate the impact of various parameters on the
crosstalk signal we are in the right position to choose the best antenna model to obtain realistic
signatures of the different targets to be used for further processing.
Before presenting the final model, let’s summarize the results obtained from the parametric study:
1. for increasing flare angle the crosstalk is lengthened and the clutter reduced,
2. for increasing size the amplitude grows slightly, the pulse is also widened and the clutter
level stays almost constant,
3. an increase of the cable impedance reduces the ringing,
4. a decrease in the central frequency of the pulse broadens the crosstalk and increases the
amplitude,
5. adding a shielding introduces more ringing,
6. changing the height of the metallic enclosure does neither affect significantly the pulse shape
nor its amplitude,
7. modifying the length and particularly the width of the metallic enclosure has a slight effect
in both, the amplitude and shape of the pulse,
8. and finally filling the metallic cavity with absorbing material eliminates the late-time ringing
without introducing other artifacts and broadens the pulse the more with growing absorber
permittivity.
117
7.1: Time domain Characteristics of GPR antennas
Effect of the absorbing material (σ=0.2S/m)
1
ε =6
0.5
ε =8
Amplitude
r
r
0
ε =10
−0.5
εr=12
r
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
1
ε =6
Amplitude
0.5
r
εr=8
0
ε =10
r
ε =12
−0.5
r
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.11 – Simulated crosstalks for different absorbing materials (ǫr variable and σ = 0.2S/m)
for a gaussian pulse (top) and a monocycle (bottom) pulse.
Effect of the absorbing material (σ=0.4S/m)
1
ε =6
r
Amplitude
0.5
ε =8
r
ε =10
0
r
ε =12
r
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
1
ε =6
0.5
ε =8
Amplitude
r
r
ε =10
0
r
ε =12
r
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.12 – Simulated crosstalks for different absorbing materials (ǫr variable and σ = 0.4S/m)
for a gaussian pulse (top) and a monocycle (bottom) pulse.
118
7: GPR Antenna and Target Responses in Time Domain
Source Pulse
Gaussian
Monocycle
ǫabs
r
7.2
7.2
σ abs (S/m)
0.14
0.39
fc (GHz)
2
1.65
bow-tie lxw (mm)
72.9x42.83
86.6x41.1
Box lxw (mm)
80.5x94.2
94.7x95.9
Table 7.1 – The parameters of the optimized models.
The optimized parameters for each of the models are summarized in the Table 7.12. It must be
noted that these are not the only models possible, since other combinations of bow-tie dimensions/flare angle and absorber parameters may also produce a similar crosstalk. However, and
based of the observed performance in frequency domain, we have selected a configuration with
an absorber material of ’low’ permittivity to have a better energy coupling into the soil, instead
of for instance decreasing the antenna dimensions and increasing the absorber permittivity. The
simulated crosstalks after optimization of the antenna model are compared with the measured
crosstalk in Fig. 7.13.
Simulated and measured crosstalks
1
SIM Gaussian (97.9% corr.)
SIM Monocycle (98.8% corr.)
Measured
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.13 – Simulated crosstalks after optimization for Gaussian and Monocycle pulses compared
with the measured crosstalk.
7.1.3
Field Distributions for Different Antenna Configurations
In this section we display some time snapshots of the electric field norm for several GPR antenna
models (analyzed in the previous sections) above dry soil. More precisely, the soil beneath the
dipoles has a permittivity of ǫr = 5 and a conductivity of σ = 0.001S/m. The illustrations are
in logarithmic scale (dB) and the field is represented along both planes, parallel (E-plane) and
perpendicular (H-plane) to the bow-tie polarization direction.
Figures 7.14 and 7.15 show the temporal evolution in the case of transmitter and receiver side
by side without any shielding in two situations: placed on the surface and at 6cm elevation
respectively. The limits of the color scale are the same for all the snapshots to better recognize
the temporal evolution of the energy distribution. Due to this, the space where the energy is out
of the range of the scale is empty. The snapshots display spherical waves in the upper and lower
half-space, being the wavelength in soil shorter and the propagation velocity slower than in air due
to the higher permittivity. When comparing both situations we observe a clearly better energy
coupling into the soil when the antennas are placed directly on the interface, i.e., more energy
7.2: Target Scattering Analysis
119
penetrates the soil for surface laid antennas.
Figures 7.16 and 7.17 represent the snapshots
of the field when the antennas are shielded by a metallic enclosure without any absorber inside
and again placed above the interface and 6cm high respectively. In these cases the radiation is
efficiently focused in downward direction, being the fraction of energy radiated upwards notably
lower and the directivity into the soil better than for the unshielded case. In particular, the
energy coupling into the soil is larger when the antennas are located on the surface.
If we add an absorber (ǫr = 7.2, σ = 0.39S/m) less energy is radiated outside the enclosure due to
the wave attenuation produced by the filling material. This attenuation avoids the signal reflected
at the walls of the metallic shielding to be reradiated producing undesired ringing.
In Figures 7.18, 7.19 and 7.20 we compare different elevations (0cm, 2cm and 6cm) for this
antenna model. As we can see, at 2cm the energy coupling into the soil is slightly worse than at
0cm, which is demonstrated not only by the amount of energy that penetrates the soil, but by
the higher intensity of the reflected energy traveling in the upward direction. On the other hand,
the directivity improves when the antennas are elevated 2cm or 6cm, confirming the expected
results from the frequency domain simulations in Chapter 6.
Finally, and for comparison purposes, Fig. 7.21 exhibits the wave propagation for the same
antenna model but this time with an absorber material of ǫr =10 (and the same σ as before).
For this permittivity value, more energy is trapped within the casing and the radiation intensity
towards the soil is less strong than for an absorber with an smaller permittivity. This is an already
expected behaviour from the radiation patterns analysed in Chapter 6.
In all the cases with a shielding enclosure, we can also observe that while the field distribution
in E-plane is totally symmetric, in H-plane it is clearly affected by the presence of the receiver
antenna.
7.2
Target Scattering Analysis
The rest of this chapter is devoted to the investigation of the sensitivity of the target scattering
signatures to the different parameters that configure a GPR environment. For this analysis we
will consider the backscattered signal by a collection of objects: a plastic sphere, a metallic sphere
(both of r=2.5cm), an empty cylinder, a water filled cylinder (r=4cm, l=30cm), two mine-like
targets representative of the PMA2 and Type72 AP mines and two small plastic cylinders with
the same dimensions as the two mine-like respectively but with no internal structure. One of
the mine-like targets has an air-gap on the top and dielectric filling on the bottom and the other
one is just full of dielectric material and has an small piece of metal in the middle (Fig. 7.22).
These targets are located either in free space or buried a few centimeters in wet or dry soil
and the antenna system is placed at few centimeters above the ground. In our simulations we
vary the target dimensions, position and orientation as well as the antenna height. The cases
with inhomogeneous soil and rough interface are also considered. This setup covers a broad and
significant set of scenarios and we are able to analyze the influence of different antenna-target-soil
parameters over the corresponding signatures.
The signatures illustrated in the following sections are normalized with respect to the maximum
amplitude obtained for each configuration.
120
7: GPR Antenna and Target Responses in Time Domain
Figure 7.14 – Snapshots for Tx and Rx alone (without shielding and absorber) on the surface (dry
soil), E-plane (left) and H-plane (right).
7.2: Target Scattering Analysis
121
Figure 7.15 – Snapshots for Tx and Rx (without shielding and absorber), at height=6cm above dry
soil, E-plane (left) and H-plane (right).
122
7: GPR Antenna and Target Responses in Time Domain
Figure 7.16 – Snapshots for Tx and Rx with metallic shielding on the surface (dry soil), no absorber,
E-plane (left) and H-plane (right).
7.2: Target Scattering Analysis
123
Figure 7.17 – Snapshots for Tx and Rx with metallic shielding, no absorber, at height=6cm above
dry soil, E-plane(left) and H-plane (right).
124
7: GPR Antenna and Target Responses in Time Domain
Figure 7.18 – Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m on
surface, E-plane (left) and H-plane (right).
7.2: Target Scattering Analysis
125
Figure 7.19 – Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m, at
height=2cm above dry soil, E-plane (left) and H-plane (right).
126
7: GPR Antenna and Target Responses in Time Domain
Figure 7.20 – Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m, at
height=6cm above dry soil, E-plane (left) and H-plane (right).
7.2: Target Scattering Analysis
127
Figure 7.21 – Snapshots for Tx and Rx with shielding and absorber of ǫr = 10, σ = 0.39S/m, at
height=6cm above dry soil, E-plane (left) and H-plane (right).
128
7: GPR Antenna and Target Responses in Time Domain
Figure 7.22 – Models of a mine Type 1 (left) and Type 2 (right).
7.2.1
Source Pulse Influence
Before any further analysis, we firstly compare the effects on both mine-like targets’ signatures
when we consider the previously introduced gaussian and monocycle pulses to feed the transmitter
antenna. The picture in Fig. 7.23 depicts the corresponding signatures. The shape of the echoes
is rather similar but not equal, and the waveforms are slightly longer for the gaussian source. This
difference comes from the different spectral content of both excitation pulses (see Fig. 7.1). The
higher spectral components associated to the monocycle generate shorter responses and increase
the vertical resolution (as it can be more clearly recognized from the scattering signature of the
target Type 1).
7.2.2
Frequency Influence
As it can be seen in Fig. 7.24, the effect of changing the pulse central frequency (and consequently
the pulse width) is the change of the vertical resolution of the antenna. In particular, for the
higher frequency and the biggest target considered (Type 1), the top and bottom reflections start
to separate from each other and there is a clear difference between the signatures. On the other
hand, in the case of the target Type 2, its height is still too short and all three signatures look
very similar even for the highest frequency. Finally, for both targets, there is a slight reduction
of the echo length for increasing frequencies, which was already expected.
7.2.3
Target Influence
i. Target size
We analyze here the effects of changing the size of the two mine-like targets in horizontal and
vertical dimension, i.e., its diameter and height. In Fig. 7.25 and Fig. 7.26 we display the
signatures of the two targets when their horizontal and vertical dimension are modified 0.75, 1.5
129
7.2: Target Scattering Analysis
Effect of different source pulses
Type1 (gaussian)
Type1 (monocycle)
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
1
Type2 (gaussian)
Type2 (monocycle)
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.23 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for gaussian and monocycle source pulses.
Effect of the frequency
Type1 f=1.5GHz
Type1 f=2GHz
Type1 f=2.5GHz
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
6
−9
x 10
1
Amplitude
5
Type2 f=1.5GHz
Type2 f=2GHz
Type2 f=2.5GHz
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.24 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2(bottom)
when the transmitter is excited with monocycle pulses of different central frequencies.
130
7: GPR Antenna and Target Responses in Time Domain
Effect of horizontal size change
no xyscale
xyscale=0.75
xyscale=1.5
xyscale=2
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
no xyscale
xyscale=0.75
xyscale=1.5
xyscale=2
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.25 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
when their horizontal dimensions are modified.
and 2 times its original size. The figures show that the diameter of the target has a remarkable
effect on the signature magnitude, increasing considerably its value as the diameter increases.
In contrast, the shape of the signature seems to be independent of the diameter and remains
almost the same in both cases. On the contrary, targets’ height alters the shape of the echoes
displacing the position of the maxima and minima: the higher is the target, the longer will be
the corresponding signature. And when the height of the target considered is big enough, the
top and bottom reflections will be differentiated, as it is clearly recognized in the example where
Type 2 target’s height is taken the double of original one.
ii. Target Tilt and Orientation
In this section the sensitivity of the scattering response to the orientation and position of the
targets with respect to the GPR antennas is investigated. In the simulations presented until
now, the targets were always located just below and parallel to the antenna unit. The next
figures illustrate the influence of target horizontal displacements as well as target rotations on
the scattering signatures. Fig. 7.27 depicts the amplitude of the scattered signal by mine-like
targets of Type 1 and Type 2 when they are displaced 4 centimeters along both directions x and
y. Figure 7.28 shows the scattered amplitudes when the targets are tilted 20◦ , 40◦ and 60◦ respect
to the x-axis. We observe that little lateral displacements affect just very slightly to the echo
shape. When the target is just below the antenna unit (i.e., in the middle between transmitter
and receiver), the magnitude of the signature is a bit stronger, and when the target is displaced
4cm in both x and y directions the signatures are very similar because of the size of the antenna
131
7.2: Target Scattering Analysis
Effect of vertical size change
no zscale
zscale=0.75
zscale=1.5
zscale=2
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
1
no zscale
zscale=0.75
zscale=1.5
zscale=2
0.5
Amplitude
6
−9
x 10
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.26 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
when their vertical dimension is modified.
beamwidth and the equal length of the wave travel paths.
On the other hand and as it could be expected, Fig. 7.28 evidences that the target orientation
with respect to the incident fields has a stronger impact on its signature. This impact is bigger
for the smaller target, because the relative change when it is tilted, is bigger. Its apparent crosssection to the illuminating waves becomes smaller and then the amplitude decreases with tilt. For
the Type 1 this effect is not so evident because its vertical dimension is longer than the horizontal
one, and then the change of the echo is not very relevant.
iii. Target Shape and Contrast
This section shows the signatures for a collection of targets to study the effects of the different
geometries and the influence of their internal structure on the signatures. In addition to the
mine-like targets we consider here the targets that were previously described at the beginning of
this analysis (see Section 7.2) are: two cylinders with the same dimensions as the two mine-like
targets respectively, but made only of plastic (with the same electromagnetic parameters as the
plastic cover of the mines); two cylindrical pipes filled with water parallel to the antenna polarization direction (x) and perpendicular to it (y); and one metallic and one plastic sphere.
As we can see from Fig. 7.29, slight modifications in target internal structure don’t produce an
appreciable impact on the object signature. This is the case of target Type 1 and the corresponding homogeneous cylinder, where we recognize that the small metallic piece and the little
contrast between the plastic cover and the TNT, don’t make any big difference between both
signatures. On the other hand, in the case of the target Type 2 and the corresponding homoge-
132
7: GPR Antenna and Target Responses in Time Domain
Effect of lateral displacements
Type1
Type1(y=−4cm)
Type1(y=+4cm)
Type1(x=+4cm)
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
6
−9
x 10
1
Amplitude
5
Type2
Type2(y=−4cm)
Type2(y=+4cm)
Type2(x=+4cm)
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.27 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for horizontal displacements. The displacement x=-4cm is not displayed because due
to the configuration symmetry the signature is the same.
Effect of the tiltness
Type1 0°
Type1 20°
Type1 40°
Type1 60°
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
Type2 0°
Type2 20°
Type2 40°
Type2 60°
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.28 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for different tilt angles respect to the horizontal.
133
7.2: Target Scattering Analysis
Effect of the geometry and material
1
Type1
Plastic cylinder(1)
Water pipe(co)
Water pipe(cross)
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
0.6
Type2
Plastic cylinder(2)
Metallic sphere
Plastic sphere
Amplitude
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.29 – Amplitude of the scattered signal by different objects in comparison with the signature
by mine Type 1 (top) and mine Type 2 (bottom).
neous plastic cylinder, we observe that the void present of the mine-like target has a remarkable
effect on the signature, increasing the amplitude and changing slightly its shape. Regarding the
other objects, we recognize a big difference between the pipe echoes and all the others due to the
very high permittivity contrast of the water with the (background) free-space. This fact, apart
from lengthening the signature (due to the slower propagation velocity of the waves in water),
generates a very intense top reflection (because the most of the energy is reflected at the top
of the pipe); the second reflection is in comparison much weaker. With respect to the spheres,
we observe that their signatures are similar in length to the one scattered by the target Type 2,
since their vertical size is also rather similar. However, their shape (equal for both spheres, and
just stronger for the metallic one than for the plastic one) is slightly different to the shape of the
signatures produced by the cylindrical objects, being the first minimum much more intense than
the second one, while for the cylindrical objects this contrast between minima is not so strong.
7.2.4
Soil Contribution
This section focuses on the influence over the signatures of diverse model parameters when the
target is buried in soil.
134
7: GPR Antenna and Target Responses in Time Domain
Effect of the depth
0.4
Type1 d=3cm
Type1 d=8cm
Type1 d=13cm
Amplitude
0.2
0
−0.2
−0.4
0
1
2
3
t(s)
4
6
−9
x 10
1
Type2 d=3cm
Type2 d=8cm
Type2 d=13cm
0.5
Amplitude
5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.30 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil at 3, 8 and 13cm. The value of the amplitude is normalized for both
objects at every depth.
i. Target depth
Here the responses of both mine-like targets buried in dry and wet soil at different depths (3, 8
and 13cm) were simulated. As we can note from the Fig. 7.30 and Fig. 7.31, the width of the
signatures rises with the depth for both targets and soils due to the low-pass filter effect of the soil
(and lower frequencies are associated to longer widths). On the other hand, the changes in overall
shape when comparing the signatures of each target in dry and wet soil are not relevant and just
the arrival times of the echoes for the different depths change due to the different propagation
velocity in dry and wet soil. Another effect that can be noticed in both soils and for both types
of targets is that there is a polarity reversal for the case of 13cm target depth in comparison with
the other two smaller depths investigated. This effect needs further investigation.
ii. Dielectric Contrast
An increase in soil permittivity (usually related to a higher moisture content) results in general
in an increase of the response magnitude due to the bigger contrast. However, for highly lossy
soils, an increase in the permittivity can produce a decrease in the echo amplitude due to the
higher associated attenuation.
In Fig. 7.32 and Fig. 7.33 we can see a change in the polarity of the reflected signal in the case
of the water pipe with respect to the other targets, i.e, the polarity of the scattered wavelet is
the opposite, which is an effect that can happen when the emitted pulse encounters an interface
135
7.2: Target Scattering Analysis
Effect of the depth
Type1 d=3cm
Type1 d=8cm
Type1 d=13cm
Amplitude
0.4
0.2
0
−0.2
−0.4
−0.6
0
1
2
3
4
t(s)
5
6
8
−9
x 10
1
Type2 d=3cm
Type2 d=8cm
Type2 d=13cm
0.5
Amplitude
7
0
−0.5
−1
0
1
2
3
4
t(s)
5
6
7
8
−9
x 10
Figure 7.31 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in wet soil at 3, 8 and 13cm. The value of the amplitude is normalized for both
objects at every depth.
Objects buried (d=2.5−3cm) in dry soil
1
Type1
Plastic cylinder(1)
Water pipe(co)
Empty pipe(co)
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
t(s)
4
6
−9
x 10
Type2
Plastic cylinder(2)
Metallic sphere
Plastic sphere
0.5
Amplitude
5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.32 – Amplitude of the scattered signal by different shallow buried objects in dry soil.
136
7: GPR Antenna and Target Responses in Time Domain
Objects buried (d=7.5−8cm) in wet soil
Type1
Plastic cylinder(1)
Water pipe
Air Pipe
Amplitude
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
6
−9
x 10
1
Amplitude
5
Type2
Plastic cylinder(2)
Metallic sphere
Plastic sphere
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.33 – Amplitude of the scattered signal by different shallow buried objects in wet soil.
between two different materials.
The polarity flip of the reflected pulse occurs only when the impedance of the second medium
(or object) is higher (i.e., lower permittivity) than the first medium, being then an indication of
the transition from low velocity to higher velocity across a boundary. This phenomenon, which
can be easily deduced from the Fresnel equations when calculating the corresponding reflection
coefficient, is the reason why the polarity of the reflected signals by the water pipe and the
metallic sphere (whose permittivites are not lower than that of the background) are not reversed
and are opposite to the polarity of the rest of the considered targets.
The changes in the signal polarity can be used for detecting certain underground structures or
cavities.
iii. Soil Type and Water Content
Here we will analyze the effect of changing the soil permittivity and conductivity on the amplitude
and shape of the scattered signatures. As it was expected the reflected pulse amplitude increases
with permittivity due to the highest contrast between the target and the background soil (Fig.
7.34). At the same time it broadens slightly because the bandwidth of the incoming pulse is
reduced due to the low-pass filter effect of the soil. There is also a delay of the scattered signal
associated to the lower propagation velocity for increasing permittivity values. The overall shape
of the signal does not change very much, but the last tail associated with the ringing grows
notably when the permittivity increase.
When changing the conductivity (Fig. 7.35), we observe that the received signal strength slightly
137
7.2: Target Scattering Analysis
Effect of the permittivity, σ=1mS/m
0.6
εr=5
0.4
ε =7
Amplitude
r
ε =10
0.2
r
ε =15
r
0
−0.2
−0.4
0
1
2
3
t(s)
4
5
1
εr=5
ε =7
r
0.5
Amplitude
6
−9
x 10
ε =10
r
ε =15
0
r
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.34 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil with different permittivities and constant conductivity σ = 1mS/m.
Effect of the conductivity (εr=7)
Amplitude
0.4
σ=1mS/m
σ=5mS/m
σ=10mS/m
σ=50mS/m
0.2
0
−0.2
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
σ=1mS/m
σ=5mS/m
σ=10mS/m
σ=50mS/m
0.5
0
−0.5
−1
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.35 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil of different conductivities and constant permittivity ǫr = 7.
138
7: GPR Antenna and Target Responses in Time Domain
Effect of the antenna height above dry soil
Amplitude
0.4
h=2cm
h=4cm
h=6cm
h=8cm
0.2
0
−0.2
0
1
2
3
t(s)
4
5
6
−9
x 10
Amplitude
1
h=2cm
h=4cm
h=6cm
h=8cm
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.36 – Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for different antenna heights above dry soil.
decreases with the growing conductivity (more attenuation), which usually is related to a higher
moisture content. Nevertheless, this difference is really small due to the shallow depth of the
targets. For depths up to 10cm the influence will be still small for such conductivity values but
above 50mS/m, the attenuation starts to be more important. The good news are that even wet
soils usually don’t reach higher values. Regarding the shape of the scattered signals, we don’t see
any significant change either.
iv. Antenna Height
As expected, the increase of the antenna-soil distance yields a decrease in the magnitude of the
scattered signal, i.e., the energy coupling into the soil worsens with antenna elevation (Fig. 7.36).
The amplitude reduction is faster for the first 4cm.
v. Soil Heterogeneity
We have modeled two inhomogeneous soils, one representative of dry soil and the other one representative of wet soil. In the configuration with dry soil, we have assumed relative permittivities
with a mean value of 5 and standard deviation 0.8 (i.e., in the range ǫr ∼ 2 − 8), and for the
wet soil with a mean value of 10 and standard deviation 1.2 (in the range ǫr ∼ 7 − 13), both of
them according to a normal gaussian distribution as described in section 4.5. Generally, inhomogeneities closer to the ground surface have higher conductivities due to the presence of organic
139
7.2: Target Scattering Analysis
Effect of soil inhomogeneity
l=1cm
l=2cm
l=3cm
l=4cm
l=5cm
0.6
Amplitude
0.4
0.2
0
−0.2
−0.4
0
1
2
3
t(s)
4
5
1
Amplitude
6
−9
x 10
l=1cm
l=2cm
l=3cm
l=4cm
l=5cm
0.5
0
−0.5
0
1
2
3
t(s)
4
5
6
−9
x 10
Figure 7.37 – Amplitude of the scattered signal by mine Type 1 buried in inhomogeneous dry (top)
and wet soil (bottom) for different correlations lengths.
material, and conductivities in the range σ = 0.01 − 0.04S/m can be expected. However, since we
have seen that conductivity variations up to 50mS/m do not have a relevant impact on the signatures (at least for shallow buried objects), we consider here a constant conductivity of 1mS/m
for the dry soil and 50mS/m for the soil with a higher moisture content.
In Fig. 7.37 we illustrate the signatures of the Type 1 target when it is buried 3 centimeters in
the above mentioned soils. We have changed the correlation length from 1 to 5cm for both cases
to study the influence of this parameter on the corresponding signatures. In the pictures above,
we observe that for dry soil the influence of the inhomogeneity on the signature is much more
important than in wet soil. This happens because the target permittivity (∼3) lies within the
range of the background soil permittivity variability being the contrast rather small, while in the
case of wet soil the target permittivity stays always considerably below the background permittivity. In addition to this, we also see that the deviation from the target response in homogeneous
soil grows slowly as the correlation length increases from 1 to 5cm, and the reason is that the
soil inhomogeneity becomes the same order in size as the target, producing reflections of similar
intensity. If we consider higher correlation lengths this tendency will continue but above a certain
value, the effect will decrease again since then, the soil will become apparently homogeneous in
comparison with the target dimensions.
More specifically, the correlation between the signature of the Type 1 target in homogeneous
dry soil and in inhomogeneous dry soil varies from 97.1% for 1cm to 80.5 % for 5cm correlation
length. In wet soil, on the other hand, the change is very slight, going from 99.7% for 1cm to
95.7 % for 5cm correlation length.
140
7.2.5
7: GPR Antenna and Target Responses in Time Domain
Summary and Some Guidelines to Create a Representative Signature
Database
With the acquired knowledge about the different parameters influence, we have the required
information to create a representative database. This must be large enough to represent a majority
of scenarios but at the same time not contain redundancy or unnecessary information, which would
increase the processing time and slow down the recognition process.
Next, we summarize the obtained results and present some guidelines for building a convenient
database:
1. When considering different source pulses (typically a gaussian or a monocycle), we obtain
slightly different echoes in shape and width even for pulses of the same length. The difference in the signatures might be more important if the vertical dimension of the object
is similar to the pulse vertical resolution, since in this case top and bottom reflections can
be distinguished and the different spectral content of the source pulses may produce rather
distinct signatures. Then, it is important to choose the source pulse correctly.
2. The same happens when we apply sources with different central frequencies. If we consider
a deviation above 20%, the difference on the echo shape may be significant depending on
the target vertical dimension due to the effect stated before. In addition, the signature will
logically elongate for a smaller central frequency and compress for a higher central frequency.
Nevertheless, for small changes (up to 15-20% the central frequency) most probably there
will be no impact in shape and just a irrelevant increase or decrease of the signature width.
And we will assume that the actual pulse length of our commercial system corresponds to
the length given in the specification with an accuracy not below a 20%.
3. The object size is a important point when building the database, in particular the vertical
dimension. As we have seen a change in the vertical dimension of the objects alters completely the shape of the signature: the amplitude and position of the maxima and minima
are modified. The change in diameter affects mostly to the signature amplitude, but it can
also produce a slight change in shape. Thus, it is crucial to include signatures for objects of
different sizes. From the simulations we can affirm that changes above 15-20% in vertical
dimension should be at least represented in the database. For horizontal dimension 25%
would be sufficient.
4. Regarding the lateral displacements of the target respect to the antennas, we have seen
that the signature shape does not change significantly in shape for displacements of the
order of the size of the object, and just slightly in amplitude for the shorter travel path
(object in the middle between transmitter and receiver). Then, this point does not need to
be considered and including the simulated signature for only one position would be enough
to achieve a good correlation. If the object is rounded (for example an sphere) it would be
better to make an average of the simulated signatures for various positions.
5. The target tilt might introduce a significant difference of its echo, which will be more or
less remarkable depending on the object geometry. In general, for objects with a horizontal
7.2: Target Scattering Analysis
141
dimension larger than the vertical dimension, the impact of tilt will be more important.
Hence, according to the simulation, it would be recommended to introduce at least two
signatures per object, one for no tilt, and a second one for 45◦ tilt angle.
6. The target material need to be considered specially for those objects whose vertical dimension is enough large (or their permittivity is high enough) to make their top and bottom
reflection visible for the considered illumination. Then, the appearance of the complete
signature will be rather different depending on the composition. For very small objects
respect to the pulse length, a bigger or smaller contrast will just affect to the amplitude
and not the shape, and it is not necessary to consider different materials in this case. For
the case of internal structure, it does need to be taken into account when the dimension
of the internal structures are of the order of the vertical resolution, and when the contrast
between the materials in contact is at least of 25% or above.
7. Another relevant issue to take into account is the target depth. The signatures differ
significantly when we consider the same object at different depths. The signatures are
clearly elongated with depth due to the low-pass filter effect of the soil and it seems to
occur a polarity change when the target is deep enough. This effect happens in both soils
considered. Then, to account for this, we need to include the signatures of the targets
at different depths, at least every 3cm until the depth of interest for landmine detection
(maximum 20cm).
8. The soil permittivity affects mainly to the signature amplitude due to the change in the contrast. Apart from the amplitude change, the signature widens with increasing permittivity
and the late-time ringing grows, in particular for soil permittivities above 10. Then, it is a
good idea to include the signatures for different soils, at least one signature for soils with
permittivity below 10, typically 5 for dry sand, and 2 or 3 signatures more for permittivity
10 and bigger. Nevertheless, the permittivity of most common soils does not get over 20,
then it is not necessary to consider values above 20.
With respect to the conductivity, we have seen that this factor does not modify the shape
of the signature, at least not for typical values of the conductivity, and it is only responsible
of a decrease of the amplitude when it grows due to the increasing attenuation with conductivity. Then, it seems not necessary to pay too much attention to the conductivity when
creating the database; we have just to take into account that if the target is 5cm or deeper,
conductivities of ∼ 100mS/m can produce an important decrease of the target amplitude
making it very difficult to detect.
9. The antenna height produces a decrease in the signature amplitude since the coupling to the
soil worsens. The overall shape and the length of the signatures don’t change significantly
but the last tail of the signatures is modified when changing the height. Then this issue
should be also taken into account. However, since this parameter is known, one can directly
reduce the signatures to consider in the database when doing the automatic recognition if
they are separated into groups associated to each simulated antenna height.
10. The soil inhomogeneity affects to the signatures, and the magnitude of the impact depends
on the correlation length of the inhomogeneity and its standard deviation. If the variable
142
7: GPR Antenna and Target Responses in Time Domain
values of the permittivity are far from the object permittivity (for example permittivities
varying between 10-15 and object permittivity of 3), the object signature will not be strongly
affected by the inhomogeneity. If the object permittivity lies on the interval of variation,
the signature will be clearly distorted unless the correlation length is very small. As we have
observed, only for correlation lengths similar to the dimension of the object, the impact may
become significant. Correlation lengths of less than the half of the size of the considered
object don’t need to be taken into account (unless the standard deviation is very high,
which is not realistic).
However, considering that the soil inhomogeneity distribution is not exactly known, we
cannot model the exact response of the target in such scenarios. Hence, when we are
dealing with objects buried in inhomogeneous soil, it is more feasible to assume than the
soil is homogeneous, and reduce the threshold for the similarity constraint with the synthetic
signatures depending on the estimated soil variability degree.
8
Experimental Analysis and Validation
The true method of knowledge is experiment.
William Blake
In this last chapter we study the potential application of synthetic GPR target responses in
buried landmine detection/recognition and clutter suppression. The proposed methodology is a
combined approach that comprises an energy based detection algorithm and a cross-correlation
based identification algorithm. Basically, the latter consists in a shape comparison between measured and simulated reference signals and it can be implemented before conducting the detection
as an additional filtering step in the form of a similarity constraint. As described in previous
chapters, to obtain accurate one-dimensional temporal signatures for such a comparison, we need
to incorporate an optimized antenna model as well as realistic CAD representations of the targets
into our simulations.
To evaluate the performance of this approach, we carried out a measurement campaign in the test
field of the Leibniz Institute for Applied Geophysics (LIAG) Hannover where different test mines
were buried in inhomogeneous sandy soil. In the next sections the test targets, the experimental
setup and the test site are described. Then, GPR data preprocessing is briefly discussed and the
applied postprocessing algorithms are introduced. In particular, we describe in detail the proposed cross-correlation based identification technique together with the energy-based detection
algorithm, as well as the combined strategy. Next, we show the achieved similarity degree (correlation coefficient) comparing some simulated and measured signatures in free space. Finally, the
results obtained from the application of the individual algorithms and the combined methodology
to a collection of surveys acquired in Hannover using a little dataset of simulated waveforms, are
presented and analyzed.
143
144
8: Experimental Analysis and Validation
Figure 8.1 – PMN mine simulant (top), Type-72 mine simulant (middle) and ERA test mine (bottom) employed in the measurements and the corresponding CAD models.
8.1
Test objects
As explained in Chapter 1, there are a wide variety of non-metallic AP mines that come in
several sorts of sizes and shapes. Some of the mines have small cavities, metal springs or trigger
mechanisms of a few centimeters which, in some cases, will increase the amplitude of the reflected
signal. The contribution of small metal components inside the landmines is not significant
at frequencies around or below 1 GHz. For investigation purposes a collection of test objects
can be employed. The biggest difference is the size and shape of the mines, the metal trigger
mechanisms and clamping rings.
A Standard Test Target (STT) is a simulant or surrogate landmine used in the test of landmine
detection equipments. They are intended to interact with countermine systems in a way
representative of, or identical to, that of a real landmine or landmine category. A simulant
landmine (SIM) is an STT that has features or characteristics representative of a ‘category’ of
landmine types, but does not replicate any specific landmine type or model. A STT that lacks
some (one or more) features or characteristics of an actual landmine class is called a Surrogate
Landmine (SUM) [ITOP, 1999]. In this study we are going to present some results for two
landmine simulants (PMN, Type-72) and a Standard Test Target (ERA) (see Fig. 8.1).
Typical values for the electrical properties of the materials used in STT are provided in Table
8.1. They are given on the basis of the mine descriptions available. These electrical properties
are used along with the physical model to generate the numerical simulations of the GPR response.
MATERIAL
3110 RTV Silicone rubber
Bakelite
ǫr
2.20 @ 100 kHz
3.5-4.5 @3GHz
σ (S/m)
10−12
1.4 x
0.01-0.07 @3GHz
Source
Dow Corning
von Hippel (1966)
145
8.2: Test site description
Plastic (mine body)
Rubber
Metal
Wood
TNT
Beeswax
2-4 @3GHz
2-3 @3GHz
NA
1-2 @ 3GHz
2.9 @1GHz
2.4 @3GHz
0.001-0.1 @3GHz
0.001-0.1 @3GHz
1-5 x 107
0.003-0.03
0.0029 @1GHz
0.003 @3GHz
von Hippel (1966)
von Hippel (1966)
von Hippel (1966)
von Hippel (1966)
Table 8.1 – Electrical properties of materials used in mine construction.
8.2
Test site description
A picture of the test field is shown in Fig. 8.2 with arrows pointing to the lane where the targets
are buried. A detailed layout of the test lane and test targets (red points) is displayed in the left
side of the picture. Here the red squares correspond to the 1x1m area scanned in every survey.
The targets in the left line (the red points with odd numbers) are buried approx. 10cm depth
and the targets in the right line (the red points with even numbers) lie approx. 15cm depth.
The bold lines with the numbers starting in 200 and 300 respectively, indicate the plastic rails at
the borders of the area. These points will be our reference to measure the antenna offset of the
different surveys. The zero point of the coordinates is located on the right corner of the left rail
and it is marked by a red circle.
Time Domain Reflectometry (TDR) probes were used at each scanned area to determine the
dielectric constant and soil water content. The theory behind TDR probes is very similar to
ground penetrating radar. TDR probes measure the dielectric constant of the soil and use this
measurement to calculate the soil water content using Topps equation (see Chapter 4). It is
claimed that the volumetric water content of soils can be determined with this method to an
accuracy of 2% and a precision of 1% [Hillel, 1998]. The texture of the mineral soil, which
characterizes the distribution of the grain size at the test lane, was sandy (see Fig. 8.3) and highly
inhomogeneous due to the presence of organic material and changing moisture content; hence, the
electrical parameters and in particular the permittivity presented a substantial variability. The
dielectric constant was measured at three different days in August and September with a TDR
along 12m long line every 10cm. The average value oscillated between 4.6 in August to 10.1 in
September with ∼ 15% standard deviation and a correlation length of ∼20cm. The days of the
campaign the average permittivity was 7.3, a value that lies in the middle.
8.3
Methodology
The interpretation of GPR data can be significantly improved by the use of several data preprocessing algorithms as well as advanced postprocessing techniques adapted to the particular
application.
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8: Experimental Analysis and Validation
Figure 8.2 – Test area layout and targets’ position (left) and test field with buried test mines indicated by arrows (right).
Figure 8.3 – Texture triangle set after laboratory analysis, clay=1%, silt=6.7%, sand=92.3%
8.3: Methodology
8.3.1
147
Preprocessing
In the next sections we describe some basic preprocessing methods which are typically applied to
the GPR data, either offline or in real time during the acquisition process.
i. Stacking
Averaging, or stacking is used to reduce the random noise and consequently increase the SNR,
by averaging several samples (A-scans) together (10 or 20 for the ERA system).
ii. DC Component Removal
A common feature in commercial GPR systems is the presence of a continuous or low frequency
component, namely DC component, in the recorded A-scans so that the averaged level of the signal
amplitude is shifted to a value different from zero. The appearance of this component is usually
linked with both inductive effects and limitations on the system’s dynamic range [Annan, 2003].
DC levels often vary depending on the medium below the antenna and its height, so this component might change slightly from one trace to another along a continuous profile. The DC
component removal is necessary for a correct visualization and for the subsequent data processing because otherwise, the results may be significantly distorted.
iii. Time-Varying Gain
It has been shown that EM waves can be rapidly attenuated as they propagate through different
materials. The response from a target can therefore be much smaller in amplitude that the direct
wave. To clearly display both these responses a time varying gain function is often applied to
the GPR data. This method is usually only applied for visualization since it modifies the target
response.
iv. Frequency Filtering
Frequency filtering is a common signal processing technique and when correctly applied can
substantially enhance features present in GPR data. Typically, simple low and high-pass filters
are used and can be applied vertically to each A-scan or horizontally across a B-scan. A “devow”
high-pass filter is commonly applied to remove very low frequency components which can be
related to antenna tilt and inductive phenomena [Annan, 2003]. Similarly, a low-pass filter can
be run to reduce high-frequency noise. We will apply here a digital band-pass filter over the raw
data, typically for the bandwidth 100MHz-5GHz.
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8: Experimental Analysis and Validation
v. Clutter suppression algorithms: background removal
As mentioned before, the main problem for detection of AP mines with GPR is the presence of
clutter. The clutter, which is defined as any electromagnetic phenomenon not associated with
targets, cannot in general be treated as white additive noise. This significantly complicates the
issue of clutter suppression. Most of the algorithms for clutter removal are based on the background subtraction in various forms. The main idea of the algorithms of this type is the definition
of a background model and its removal from the measured signals.
By far the most popular techniques are average and moving average background subtraction.
However, these methods should be used carefully where features of interest are planar interfaces, since such responses can be removed by this sort of filtering [Daniels, 1996]. Other approaches that have shown good potential to take the target/background decision and reduce the
clutter are for example the wavelet transform [Carevic, 2000], independent component analysis
[Karlsen et al., 2002] or system identification [Brooks et al., 2000].
vi. Hilbert transform
Recorded amplitudes and arrival times of the reflected signals are the first information used to
interpret GPR data. However, the phase information is sometimes more sensitive to subsurface
changes than the amplitude and an equivalent complex-valued signal is desirable.
A Hilbert Transform can be applied to decompose the recorded real-valued signal into its magnitude (by envelope detection), local phase, or local frequency components (the derivative of phase),
which allows the phase to be reconstructed from its amplitude. The figure below depicts an example of the original and the resulting signal after applying the Hilbert transform to measured
1D data.
Original signal versus Hilbert−transformed signal
Amplitude
Original signal
Envelope
0
0
0.2
0.4
0.6
t (s)
0.8
1
1.2
−8
x 10
Figure 8.4 – Recorded 1D data before and after applying the Hilbert transform.
8.3.2
Postprocessing
To overcome the problem of landmine detection and classification in realistic scenarios there are
several postprocessing techniques.
For the detection alone a model of the background can be defined and all the reflections that
8.3: Methodology
149
clearly differ from the estimated background signal are declared as targets. Some algorithms
that have shown good potential to localize anomalies and reduce the clutter are for example the
already mentioned wavelet transform and related techniques [Carevic, 2000].
The approach followed in this thesis for the detection is, to establish a clutter level according to
the average amount of scattered energy at each depth, and the detection is called when a cluster
or a single pixel supersedes significantly this level. Another common method for target detection is based on performing a statistical binary hypothesis testing. However, it must be noted
that such techniques are not capable of discriminating between a landmine and other reflectors
present in soil (such as munition fragments, roots, stones, etc.), hence elevating the false alarm
rate and becoming necessary additional processing to identify anomalies. A possible strategy
relies on defining a target model given by a target feature vector and look for it in the data
[Cosgrove et al., 2004], [Gader et al., 2001a], [Gader et al., 2001b], [Kovalenko et al., 2007]. The
target feature vectors may be based on a single 1D waveform (A-scan) or on characteristic 2D
or 3D target traces spread along several B-scans or C-scans (typically hyperboloid-like). The
search of scattering features in the data can be implemented in different ways, including fuzzy
logic approaches, neural networks, Markov models or Support Vector Machines.
In order to accelerate the detection process and make it real time, our recognition approach is
based on the most simple feature vector, i.e., a single 1D waveform. Accordingly, the processing
methodology presented here consists of an energy based detection algorithm which takes into
account the amplitude information of the signatures and a target identification algorithm which
considers the shape of the scattered waveforms. More specifically, we will see that a similarity
measure (via cross-correlation) between measured and simulated temporal responses can be combined with the detection procedure to suppress image clutter and help in target recognition.
The complete procedure will be described in more detail in subsequent sections.
i. Energy based detection algorithm
The detection algorithm introduced in this section is based on scattered energy information.
It is performed over the absolute value of the preprocessed 3D data (Hilbert transformed or
not). Basically, it consists in normalizing the pixel energy per C-scan; then, we apply an energy
threshold (required SCR constraint in dB) per pixel respect to the average amplitude value per
C-scan and pixel (estimated clutter level at every depth) in order to find the most “brilliant”
pixels and reject the ones below this level; finally, we sum all these energy contributions over
an interval of the sampled time vector, thus obtaining the so called 2D detection map. This
algorithm can be formulated as follows:

h
i2
|I(t)|
|I(x, y; t)| if 10log
>SCR
EΩ (|I(t)|)
|I(x, y; t)| =
(8.1)
0,
otherwise
where |I| is the absolute value of the recorded signal per pixel, x, y are the spatial coordinates and
t the arrival times, SCR corresponds to the Signal-to-Clutter ratio in dB, E(·) is the expectation
operator and Ω ⊂ (x, y) is the slice area.
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8: Experimental Analysis and Validation
Next, the aforementioned detection map can be calculated by the following sum over depth:
D(x, y; T ) =
t2
X
t=t1
|I(x, y; t)|,
(8.2)
with T = t2 − t1 being the width of the time window considered for the detection.
It is possible to obtain 3D images of the detected targets if we perform the sum given by Eq. 8.2
for successive and very short time intervals along the whole desired investigation depth, so that
we obtain a collection of detection maps for successive depth intervals which can be displayed in
a 3D plot. This kind of representation would allow the viewer to localize the objects inside the
whole imaging domain, giving him an impression of the target’s real depth.
As an example, Fig. 8.5 displays the results after applying the detection algorithm to two of
the surveys acquired in the LIAG Hannover test field. The preprocessing applied is simply a
digital bandpass frequency filter and DC removal. No background removal by average subtraction
is performed due to the high degree of inhomogeneity of the upper subsurface. Instead, we
window the traces in time to eliminate partially the surface reflection and the antenna crosstalk.
Nevertheless, in case of the PMA-2 simulants in survey SVY-151, the target scattering was
enough strong to result in clear detections even when summing over the entire time axis without
any windowing.
The performance of the energy-based detection algorithm can be improved using an Inverse
Distance Weighted IDW averaging technique as shown in [Gonzalez-Huici & Giovanneschi, 2013].
In this way, the energy information within the neighbourhood of each individual pixel is considered
before performing the energy thresholding given by Eq. 8.1.
SVY−151, Detection (Threshold=10dB)
5
5
10
10
y(steps)
y(steps)
SVY−146, Detection (Threshold=10dB)
15
20
15
20
25
25
20
40
60
x(steps)
80
100
20
40
60
80
100
x(steps)
Figure 8.5 – Detection maps for the surveys 146 and 151. The targets in the middle are ERA test
mines (left) and PMA-2 simulants (right).
As a measure of the detectability of the different test mines in the LIAG test field, we have
also calculated the Receiver Operating Characteristic (ROC) curves related to the surveys with
buried mines of the same class separately. The definition of a ROC curve within the landmine
151
8.3: Methodology
ROC Curve after Detection
1
0.9
0.8
0.7
Pd
0.6
0.5
0.4
0.3
0.2
PMA2
PMN
ERA
Type−72
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
FAR
0.07
0.08
0.09
0.1
Figure 8.6 – ROC curves after applying the detection algorithm to the surveys with buried PMA-2,
PMN, ERA and Type-72 targets respectively.
detection context was already presented in Chapter 2. The probability of detection (Pd) and the
False alarm rate (FAR) are generated based on a pixel-wise calculation according to the following
expressions:
Number of detected mines
Number of mines
Number of false alarms
FAR =
Number of pixels outside halos
Pd =
(8.3)
(8.4)
The target impact area assumed for a true detection was 12cmx12cm in all cases.
In Fig. 8.6 we show the obtained results. The points of the curves correspond to different SCR
values for the detection algorithm. More detection results for other surveys are displayed in the
following sections.
ii. Phase-shift and Stolt migration
As already stated in Chapter 2, due to the beam-width of transmit and receive antennas and the
differences in round-trip travel time of the pulse caused by the movement of the antenna along the
measurement line, the reflections from scatterers will appear as hyperbolic curves in the recorded
data.
These hyperbolic structures can be migrated (focused) into the real position of the corresponding
scatterer via different migration techniques [Gazdag, 1978], [Schneider, 1978], [Stolt, 1978]. To
apply these algorithms successfully a correct estimation of the velocity structure of the propagation medium, i.e., the dielectric permittivity of the soil, needs to be done.
The phase-shift migration is a Fourier transform based technique which is also referred to as
152
8: Experimental Analysis and Validation
frequency-wavenumber migration (f-k migration). It makes use of the wave equation to backpropagate the received signal into the soil back to the scattering source, and obtain an image of
the subsurface reflectors. For the implementation a monostatic setting is assumed (which is a
valid approximation when the transmitter and receiver are close to each other); and we will only
consider the 2D case, but the extension to 3D is straightforward. In this procedure, a 2D Fourier
transform over the spatial components and time is calculated first via FFT as formulated below:
Z Z
D(kx , z0 = 0, ω) =
d(x, z0 = 0, t)e−jkx x e−jωt dxdt
(8.5)
where d is the data matrix, ω is the angular frequency, z0 is the antenna vertical position and kx
is the horizontal wave number. Here, it is assumed that d satisfies the wave equation.
Then, to determine the field at a range of depths, a phase shift is applied, which is dependent on
the propagation constant. This phase shift operation is an extrapolation along z-axis.
!
r
4ω 2
(8.6)
− kx2 ∆z ,
D(kx , z1 , ω) = D(kx , z0 = 0, ω) exp j
v2
being v the EM wave velocity which is given by Eq. 2.1. In this way, by recursively extrapolating the field in steps ∆z and using the result of each step as input for the next iteration the
frequency-wavenumber distribution of the field can be reconstructed [Gu et al., 2004].
Finally, the migrated data is obtained via the inverse Fast Fourier Transform iFFT of the
wavenumber data over kx and ω for the imaging condition t=0,
Z Z
1
d(x, z = z1 , t = 0) = 2
D(kx , z = z1 , ω)ejkx x dkx dω,
(8.7)
4π
where z1 is the vertical length of the migrated scene. In order the iFFT to be applicable to solve
the above integral, the data matrix D previously needs to be evenly mapped into the k-space via
interpolation.
This technique can handle velocity variations with depth by making the propagation velocity
a function of z and assuming it is constant for each step ∆z. When the velocity is assumed
constant along z, it is known as Stolt migration or Stolt mapping [Stolt, 1978]. This method does
not account for lateral velocity changes. Several interpretations of the Stolt migration exist in
seismic processing literature. A good overview is given in [Yilmaz, 2001].
For validation purposes, we first display the migration of the scattered energy by two of the
considered test mines in free space (see Fig. 8.7). It is interesting to observe that for the case of
the PMN on the right, we can even roughly recognize its shape, and the horizontal length is well
reconstructed. The vertical length is a bit longer than the actual height of the simulant, but this
is because the target, which is not a point scatterer, has a component material with a relative
permittivity higher than 1 and the algorithm assumes constant permittivity in range equal to 1
for free space.
In Figures 8.8 to 8.11, we show processed images after applying Stolt migration to different B-scans
measured in the field. In particular, in the middle of each image we can see four different buried
test mines, from the most difficult to detect, the Type-72 in survey SVY-142, to the easiest one,
the PMN in survey SVY-140. On the top of each figure, we represent the preprocessed raw data
only for comparison; both plots in the middle illustrate the corresponding migrated data (original
153
8.3: Methodology
Type 72 and PMN in air (migrated 8dB)
0
0.2
0.2
Distance (m)
Distance (m)
Type 72 and PMN in air (migrated)
0
0.4
0.6
0.6
0.8
0.8
1
0.4
−0.4
−0.2
0
x (m)
0.2
0.4
1
−0.4
−0.2
0
x (m)
0.2
0.4
Figure 8.7 – Migrated image with two targets in free space. Left: Type 72 and PMN simulants;
right: same as before with background noise removal (8dB). Survey SVY-188.
and filtered) and the plot on the bottom displays the energy contrast of the image for different
permittivity values.
For all these reconstructions in soil, the reason why we have selected
different values of the background permittivity is because we are applying an adaptive Stolt
migration algorithm that automatically estimates the permittivity which maximizes the energy
contrast of the scene. Then, the resulting permittivities are different for each survey because
the soil is highly inhomogeneous. A suitable quantitative measure of the above mentioned image
contrast is given by [Cumming & Wong, 2005]:
C=
E(| I |2 )
,
[E(| I |)]2
(8.8)
where I is the pixel magnitude and E(·) is the expectation operator. According to this definition
we obtain the contrast curves illustrated on the bottom of each figure.
The adaptive algorithm used here, basically looks for the permittivity value which after the
application of Stolt migration produces an image with a minimum number of pixels with energy
above a given threshold. Thereby, it is not necessary to have a priori information about the
permittivity of the soil. We explain this method in detail in [Gonzalez-Huici, 2011].
For the illustrated migration results the only preprocessing applied to the raw data has been again
DC correction and bandpass filtering. By a subsequent time gating we have removed partially
the antenna crosstalk and surface reflection, increasing the contrast to the object. To suppress
some background clutter, a 8dB energy filter has been additionally applied to each survey.
In all the B-scans the targets are visible and their depth and size are quite well reconstructed.
Moreover, the obtained permittivities are in agreement with the measured ones employing a
TDR. Nevertheless, since the targets are shallowly buried, it is not possible to remove completely
the surface and upper subsurface reflections, and these contributions are clearly visible in the
migrated image. As the wavelength decreases in soil with respect to free space, we are able to
distinguish top and bottom reflection for the biggest mine simulants, i.e., PMN and PMA-2. It
must be noted that the Stolt migration assumes waves traveling through a homogeneous medium;
hence, the obtained permittivity values correspond to the effective permittivity of the complete
travel path, accounting not only for the soil propagation but also for the air propagation segment
due to the antenna elevation.
It can be demonstrated that a preliminary migration of the input data for an energy-based
154
8: Experimental Analysis and Validation
−9
Time (s)
x 10
SVY−139, Raw Data
1
2
3
4
5
−0.5
0
X (m)
0.5
soil
Migrated Data (epsr =5.8)
Depth (m)
0.1
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Migrated Data (8dB)
Depth (m)
0.1
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Contrast
Contrast Function
4.9
4.8
3
4
5
εsoil
6
7
r
Figure 8.8 – From top to bottom: raw data with a PMA-2 simulant, migrated image, migrated image
with background noise removal (8dB), contrast function. Survey SVY-139.
155
8.3: Methodology
−9
Time (s)
x 10
SVY−140, Raw Data
1
2
3
4
5
−0.5
0
X (m)
0.5
soil
Depth (m)
Migrated Data (epsr =5.4)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Depth (m)
Migrated Data (8dB)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Contrast
Contrast Function
5.2
5.1
3
4
5
εsoil (m)
6
7
r
Figure 8.9 – From top to bottom: raw data with a PMN simulant, migrated image, migrated image
with background noise removal (8dB), contrast function. Survey SVY-140.
156
8: Experimental Analysis and Validation
−9
Time (s)
x 10
SVY−142, Raw Data
1
2
3
4
5
−0.5
0
X (m)
0.5
soil
Depth (m)
Migrated Data (epsr =4.6)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Depth (m)
Migrated Data (8dB)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Contrast
Contrast Function
4.9
4.8
3
4
5
εsoil
6
7
r
Figure 8.10 – From top to bottom: raw data with a Type-72 simulant, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-142.
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8.3: Methodology
−9
Time (s)
x 10
SVY−143, Raw Data
2
4
6
−0.5
0
X (m)
0.5
soil
Depth (m)
Migrated Data (epsr =4.2)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Depth (m)
Migrated Data (8dB)
0.2
0.3
0.4
−0.5
0
X (m)
0.5
Contrast
Contrast Function
4.9
4.8
3
4
5
6
εsoil
r
Figure 8.11 – From top to bottom: raw data with an ERA test target, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-143.
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8: Experimental Analysis and Validation
detection algorithm like the one presented before, may improve notably the detection rate of
antipersonnel landmines (see for example [Feng & Sato, 2004]) but this is not the focus of this
thesis.
iii. Cross-correlation based identification algorithm
A measure of the similarity between a given signal u(t) and a reference v(t) is well-known by
means of their cross-correlation function,
Z
Ruv (τ ) = u(t − τ )v(t)dt = u(−t) ∗ v.
(8.9)
This function shows the analogy between two non-identical waveforms as a function of the time
shift τ between them and may reveal similarities undetectable by other techniques. The crosscorrelation can be approximated by the sampling method:
Ruv (τ ) =
N
1 X
u(n∆t − τ )v(n∆t).
N
(8.10)
n=1
From the previous definition, we observe that the cross-correlation is essentially an averaged sum
of the term by term product of one waveform and the delayed version of the second waveform.
It depends linearly on the magnitude of the input signal u(t). But, for our purpose, we are
just interested in the shape information since the amplitude of the signal or its energy does
not represent a feature for the target model. Then, a normalization of the input and reference
waveforms needs to be performed before the comparison of the waveforms. However, to carry out
the input signal normalization is not a trivial issue because the target response usually does not
possess the highest amplitude in the trace, in particular if the direct coupling and the surface
reflection are not removed. Then, as we only wish to keep the shape information, the input signal
should be windowed in time and normalized before carrying out the similarity measure:

 u(T ) , for t ≤ T ≤ t
1
2
ū(t, T ) = max{|u(T )|}
(8.11)
0,
otherwise
where ū(t, T ) is the new input signal for the cross-correlation calculation and T is the time
window.
Applying this method, the discrete cross-correlation of a normalized portion of a measured Ascan and a high-quality normalized reference wavelet (which is obtained by accurate numerical
modeling) is determined. Then, the maximum absolute value of the cross-correlation vector
determines the correlation coefficient ρuv between both waveforms:
ρuv = max{|Ruv |}.
(8.12)
Measurements vs. Simulations To test the achieved similarity degree between simulated
echoes using accurate CAD models of the targets and actual measurements, we carry out some
tests in laboratory with the test mines displayed in Fig. 8.1. The cross-correlation is computed
159
8.3: Methodology
for a sequence 2N-1 time shifts T (m − N ) where m = 1, ..., 2N − 1 and N is the length of the
vectors u and v.
Figure 8.12 depicts the comparison of the normalized synthetic and recorded responses for the
considered test mines in free space and the corresponding cross-correlation measure between both
waveforms. The scattering signatures are obtained subtracting the simulated/measured signal
without target from the scattered simulated/measured signal with target for each of the cases.
The material electrical parameters to simulate the test mines are taken from [Hippel, 1995].
Normalized Amplitude
Type72 simulant signature, simulation vs. measurement
1
measurement
SIM
0.5
0
−0.5
−1
0
0.5
1
1.5
2
t(s)
2.5
3
3.5
4
−9
x 10
Max(Xcorr)=97.5%
Cross−Correlation
1
0.5
0
−0.5
−1
0
50
100
150
N
Normalized Amplitude
PMN simulant signature, simulation vs. measurement
1
measured
SIM
0.5
0
−0.5
−1
0
0.5
1
1.5
2
t(s)
2.5
3
3.5
4
−9
x 10
Max(Xcorr)=95.0%
Cross−Correlation
1
0.5
0
−0.5
−1
0
50
100
150
N
Normalized Amplitude
ERA testmine signature, simulation vs. measurement
1
measurement
SIM
0.5
0
−0.5
−1
0
0.5
1
1.5
2
t(s)
2.5
3
3.5
4
−9
x 10
Max(Xcorr)=96.6%
Cross−Correlation
1
0.5
0
−0.5
−1
0
50
100
150
N
Figure 8.12 – Cross-correlation between measured and simulated signatures for a Type-72 simulant
(top), a PMN simulant (middle) and an ERA test target (bottom).
The different plots reveal a high correlation coefficient between simulations and measurements for
160
8: Experimental Analysis and Validation
all the three objects analyzed. Even for the PMN, which is the one with most complex internal
structure, a correlation coefficient of 95% was achieved.
Class maps The class map is a classified image comprised of a mosaic of pixels, which are colorcoded according to the target class they belong to. To elaborate this sort of map we perform the
cross-correlation trace by trace with all the reference waveforms present in the database. The
values of the correlation coefficient obtained for each reference signature in the survey area define
a data level (slice) C k for that survey area. Besides, the pixels where these coefficients are below
a certain similarity threshold are set to zero, so that the data level for a reference target k (class)
is given by:

ρk (x, y) if ρk (x, y) >Threshold
(8.13)
C k (x, y) =
0,
otherwise
where k is the corresponding reference class for that data level. Each individual pixel of the
final map is then classified based on this similarity information. For those samples with a zero
in all the data levels, the corresponding cell is declared as empty, i.e., the assigned Class is
“No object”. For the rest of pixels, the class declaration Class corresponds to the most similar
reference target, which is the object from the dataset whose synthetic signature has the highest
correlation coefficient (maximum similarity) with the considered trace:
Class (x, y) = max {C k (x, y)}.
(8.14)
k
As an example, Figures 8.13 and 8.14 depict two of the computed class maps for one survey with
buried PMA-2 simulants (not present in our database) and another one with buried ERA test
targets. The computed detection maps are also illustrated for comparison purposes.
SVY−139, Detection (SNR=8dB)
SVY−139, Class Map, Threshold=86%
Type72_d5_wet.txt
Type72_d5_dry.txt
5
5
PlasticCyl_d5_dry.txt
PlasticCyl_d10_dry.txt
PMN_d5_wet.txt
10
y(steps)
y (steps)
10
PMN_d5_dry.txt
PMN_d10_wet.txt
15
15
PEC25_d10_wet.txt
ERA_d5_wet.txt
20
20
ERA_d10_wet.txt
ERA_d10_dry.txt
25
25
20
40
60
80
100
20
40
60
80
100
No Object
x (steps)
x(steps)
Figure 8.13 – Detection map (left) and class map of the survey SVY-139. The target in the middle
is a PMA-2 simulant, which is a rectangular plastic mine (height=3.5cm, lengthxwidth=6x14cm).
161
8.3: Methodology
SVY−147, Detection (SNR=8dB)
SVY−147, Correlation Map pixel per pixel (Threshold=85%)
Type72_d5_wet.txt
Type72_d5_dry.txt
5
5
PlasticCyl_d5_dry.txt
PlasticCyl_d10_dry.txt
PMN_d5_wet.txt
10
y (steps)
y(steps)
10
15
PMN_d5_dry.txt
PMN_d10_dry.txt
15
ERA_d5_wet.txt
ERA_d5_dry.txt
20
20
ERA_d10_wet.txt
ERA_d10_dry.txt
25
20
40
60
80
x(steps)
100
25
20
40
60
80
100
No Object
x (steps)
Figure 8.14 – Detection map (left) and class map of the survey SVY-147. The target in the middle
is an ERAtest mine.
In the middle of both images there are aggregates of coloured pixels, which correspond with the
position of the real targets. From the results, we can affirm that the class maps demonstrate
the presence of buried objects and may be able to identify them correctly. In the directory of
the database considered for the correlation calculation, there were 20 simulated waveforms for
different landmine simulants and clutter objects buried in dry and wet soil at two different depths.
For instance, none of the metallic objects in the dataset are declared in the maps, since their
waveforms are not enough similar to any of the traces in the surveys. In the case presented in
Fig. 8.13, there is no reference waveform in the synthetic dataset for the mine PMA-2 and the
mine is wrongly declared but detected. This happens because the similar dimensions of the mines
and all the clutter objects considered give rise to rather similar echoes. In particular, the bottom
mine is identified as a Type-72 which has almost the same height as the actual PMA-2. Note the
similarity of the detection for this survey and for the survey SVY-151 displayed in Fig. 8.5 (right).
Both correspond to the same scanning area, but the measurements were taken in different days.
In the second example, we have selected a survey with a particularly high noise level. As we can
see in Fig. 8.14, after the detection algorithm alone, it is difficult to distinguish the bottom mine.
However, in the correlation map an ERAtest mine is declared, which agrees with the reality.
iv. Detection-after-Recognition
Sometimes the energy-based detection algorithm alone does not work well since the backscattered
energy by the landmine-like target is of the same order or below the clutter level. This will happen
especially for very weak scatterers or for inhomogeneous soils where the clutter average value is
high. The incorporation of a previous processing or filtering step, may help to improve this
limitation, suppressing part of the clutter and hence, increasing the SCR. More specifically, we
propose a combined Detection-after-Recognition strategy, that is illustrated in the block diagram
below (see Fig. 8.15).
162
8: Experimental Analysis and Validation
Figure 8.15 – Block diagram of the proposed methodology.
In this schematic overview of the combined approach, we see the input data on the top (measured
d and simulated s), the main algorithms (for identification and detection) in blue and finally the
resulting outputs, the class and detection maps, are highlighted in yellow. We observe that the
input for the detection algorithm is either the raw or migrated data (only detection case), or the
filtered (after the similarity constraint) data (Detection-after-Recognition case). In particular,
the filter looks for the positions (x, y) in the input data matrix corresponding to those A-scans
which show a correlation coefficient above a certain threshold. The rest of the traces in the
data matrix are filled with zeros. Another possibility could be to take those coordinates (x, y)
(already determined by computing the cross-correlation between the traces in the raw data and
a given synthetic echo) and use them to filter the migrated data, i.e., to cancel all the migrated
A-scans except for those at such positions. Afterwards, the detection algorithm will search in
the filtered data for the pixels/clusters containing enough energy to be relevant according to Eq.
8.2. The similarity filter employed is defined as follows:
|I(x, y; T )| =

|I(x, y; T )|
0,
if ρk (x, y) >Threshold
otherwise
(8.15)
where |I(x, y; T )| represents again the absolute value of the time windowed recorded signal and
ρk (x, y) is the corresponding correlation coefficient with the reference waveform associated with
the object k. In this way, we eliminate much of the clutter contributions and the estimated clutter
level is accordingly lowered, resulting in a more efficient landmine detection performance.
163
8.3: Methodology
SVY−140, Corr. with PMN (Threshold=80%) + Detect. (SNR=16dB)
5
5
10
10
y(steps)
y(steps)
SVY−140, Detection (SNR=10dB)
15
20
15
20
25
25
20
40
60
x(steps)
80
100
20
40
60
80
100
x(steps)
Figure 8.16 – Detection map without (left) and with similarity filtering (PMN) (right) over a survey
area with buried PMN simulants.
Figures 8.16-8.22 display the improvement obtained by the combined strategy (right) in comparison with the application of the detection algorithm alone (left) for a collection of representative
surveys carried out in LIAG Hannover. The selected time window for each survey remains the
same as when we apply the detection alone. We can clearly appreciate that the clutter reduction
is significant in all cases, even those with a buried Type-72. Moreover, the SCR, is now notably
higher and the detection rate is not penalized. In difficult detection cases (weak scatterers), like
the surveys SVY-147 and SVY-157, where the bottom test mines were hardly visible applying the
detection algorithm alone, the performance of the combined algorithm is particularly satisfactory:
the mines are now visible, i.e., the detection rate is notably enhanced. Furthermore, the achieved
results are even more promising considering that the test field was highly inhomogeneous and the
buried test mines were shallow and non-metallic.
Finally, to evaluate the potential of this
method to discriminate between targets we present below the ROC curves obtained after applying
the combined strategy to the surveys with buried PMN, ERA and Type-72 mines respectively.
Figures 8.26, 8.24 and 8.25 represent the ROC curves for each group of surveys. The discrimination threshold is the correlation coefficient. We also display the ROC curves after applying only
detection for a better comparison.
In Fig. 8.26 we display the ROC curves for the biggest of the targets, the PMN simulant. In
this case, since the mine contrast is high, the detection algorithm gives a very low false alarm
rate (< 1%) for a detection rate of 100 %. For such a situation, applying a similarity filter does
not improve the curve in any case. This result makes sense since such a filter enhances certain
samples, removing some others that don’t fulfill the similarity criterion. But the cross-correlation
calculation is done respect to a simulated reference wavelet in ideal conditions (homogeneous soil)
for a CAD model of the target (which is not an exact copy) and for a given depth. As we have seen
in a previous section, we never get a correlation coefficient between simulation and measurement
above 99%. Then, the sensitivity of the similarity measure is below the true positive rate already
achieved with the detection algorithm alone. However, applying the correlation filter we are able
164
8: Experimental Analysis and Validation
SVY−141, Detection (SNR=10dB)
SVY−141, Corr. with Type−72 (Threshold=81%), Detect. (SNR=25dB)
10
10
y(steps)
5
y(steps)
5
15
15
20
20
20
40
60
80
100
20
40
x(steps)
60
80
100
x(steps)
Figure 8.17 – Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants.
SVY−157, Detection (SNR=10dB)
SVY−157, Corr. with Type−72 (Threshold=84%), Detect. (SNR=24dB)
10
10
y(steps)
5
y(steps)
5
15
15
20
20
20
40
60
x(steps)
80
100
20
40
60
80
100
x(steps)
Figure 8.18 – Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants.
165
8.3: Methodology
SVY−142, Corr. with Type−72 (Threshold=83%), Detect. (SNR=20dB)
5
5
10
10
y(steps)
y(steps)
Detection (SNR=8dB)
15
20
15
20
25
25
20
40
60
80
100
20
40
x(steps)
60
80
100
x(steps)
Figure 8.19 – Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants.
SVY−143, Corr. with ERA (Threshold=80%) + Detect. (SNR=20dB)
5
5
10
10
y(steps)
y(steps)
SVY−143, Detection (SNR=8dB)
15
20
15
20
25
25
20
40
60
x(steps)
80
100
20
40
60
x(steps)
80
100
Figure 8.20 – Detection map without (left) and with similarity filtering (ERA) (right) over a survey
area with buried ERA test targets.
166
8: Experimental Analysis and Validation
SVY−145, Corr. with ERA (Threshold=80%) + Detect. (SNR=19dB)
5
5
10
10
y(steps)
y(steps)
SVY−145, Detection (SNR=8dB)
15
20
15
20
25
25
20
40
60
80
100
20
40
x(steps)
60
80
100
x(steps)
Figure 8.21 – Detection map without (left) and with similarity filtering (ERA) (right) over a survey
area with buried ERA test targets.
SVY−147, Corr. with ERA (Threshold=89%) + Detect. (SNR=18dB)
5
5
10
10
y(steps)
y(steps)
SVY−147, Detection (SNR=9dB)
15
20
15
20
25
25
20
40
60
x(steps)
80
100
20
40
60
80
100
x(steps)
Figure 8.22 – Detection map without (left) and with similarity filtering (ERA) (right) over a survey
area with buried ERA test targets.
167
8.3: Methodology
ROC curves for SVY−140 and SVY−168
1
0.9
0.8
0.7
Pd
0.6
0.5
0.4
0.3
0.2
Detection
Corr. with PMN + Detection
Corr. with ERA + Detection
Corr. with Type−72 + Detection
0.1
0
0
0.02
0.04
0.06
FAR
0.08
0.1
0.12
Figure 8.23 – ROC curves after applying the detection algorithm to the surveys with buried PMN
simulant.
to classify the targets. We can clearly observe that the true positive rate is the highest for the
PMN mine while for Type-72 the ROC curve shows the worst values, since the latter is the most
dissimilar simulant.
A second example is illustrated in Fig. 8.24. Now, the true positive rate applying just the detection algorithm is still satisfactory but worse than in the case of the PMN mine, which is a logical
result since the mine is smaller. In this case, after the application of the correlation filter, the
true positive rate is improved when the comparison is made with an ERA test mine, which is
the actual mine. For the other two simulants the curve gets slightly worse. In this case the mine
discrimination is again successful.
The third example in Fig. 8.25 corresponds to the smallest and most difficult to detect simulant,
the Type-72. We can see that in this particular case, the detection performance is very unsatisfactory. Applying energy-based detection exclusively, all the mines cannot be detected even for a
low threshold (which gives rise to a high false alarm). On the other hand, when we perform the
correlation step, the ROC curves are substantially improved even for a correlation with a false
mine. This is because the mines are relatively similar (in size, shape and composition) objects
whose signatures don’t differ too much between each other. Then, when the energy detector
performance is so bad, a similarity filtering with any little plastic cylindrical object will most
probably improve the result. In particular, when we apply cross-correlation with the Type-72
reference waveform, the obtained ROC curve is the best one, achieving now 100% detection rate.
At this point, we must remark that the synthetic reference waveforms considered to carry out the
comparison were just a first approximation and most probably not enough to be representative.
They were computed for a few targets buried either 5cm or 10cm deep and antenna height of 5cm.
The soil assumed for the simulation was homogeneous (either dry: ǫr = 5, σ = 0.001S/m or wet:
168
8: Experimental Analysis and Validation
ROC curves for SVY−143, SVY−145, SVY−160 and SVY−161
1
0.9
0.8
0.7
Pd
0.6
0.5
0.4
0.3
0.2
Detection
Corr. with PMN + Detection
Corr. with ERA+ Detection
Corr. with Type−72 + Detection
0.1
0
0
0.01
0.02
0.03
FAR
0.04
0.05
0.06
Figure 8.24 – ROC curves after applying the detection algorithm to the surveys with buried ERA
test mine.
ROC curves for SVY−141, SVY−142, SVY−157 and SVY−158
1
0.9
0.8
0.7
Pd
0.6
0.5
0.4
0.3
0.2
Detection
Corr. with PMN + Detection
Corr. with ERA + Detection
Corr. with Type−72 + Detection
0.1
0
0
0.02
0.04
0.06
0.08
FAR
0.1
0.12
0.14
Figure 8.25 – ROC curves after applying the detection algorithm to the surveys with buried Type-72
simulant.
8.4: A GUI for automatic landmine detection and recognition
169
ǫr = 7, σ = 0.01S/m) while in the test field the soil permittivity values measured with the TDR
sensor changed between 4.5 and 11. Moreover, the soil roughness made impossible to maintain a
constant antenna height. Then, the obtained results could probably have been better if a more
accurate and large collection of waveforms had been considered. Nevertheless, the goal here was
not to build a complete database, but to demonstrate the potential of adding a similarity filter
based on accurate simulations to reduce the false alarm rate and classify the targets.
Summarizing, the procedure presented seems to work well as classifier even for an inhomogeneous
scenario like the one considered for the test. In particular, it achieves a clear false alarm reduction
respect to only energy detection when the ROC curve from the latter lies above 2% false alarm
for 100% detection rate, i.e., for those landmines that are most difficult to detect (small and
low-contrast). These are the situations where more complicated GPR postprocessing algorithms
become imperative and this method seems to bring at least part of the solution to the problem.
The presented strategy and examples were also introduced in [Gonzalez-Huici & Uschkerat, 2010]
and [Gonzalez-Huici, 2012]; in the aforementioned [Gonzalez-Huici & Giovanneschi, 2013], we
have proposed an extension of the methodology to enhance the imaging results incorporating
an IDW averaging technique.
8.4
A GUI for automatic landmine detection and recognition
A Graphical User Interface for target detection and recognition has been additionally developed.
The GUI allows reading, visualizing and processing the GPR data in an automatic and userfriendly way. The interface is connected to a representative but still small database with the
scattered waveforms by the previously presented landmine simulants and some canonical clutter
objects made of metal or plastic. The echoes correspond to the scatterers in free space and buried
in three different soils (dry, medium and wet soil) at three different depths (5cm, 10cm and 15cm).
All the standard preprocessing methods along with the energy based detection algorithm, crosscorrelation based identification technique and the combined Detection-after-Recognition strategy
has been implemented, being possible to visualize the resulting detection and class maps for each
survey. The similarity measure is performed with the reference objects included in the database
folder selected by the user. Some other reconstruction (adaptive migration and tomographic
inversion) algorithms are being incorporated. A screen shot of the GUI after the application of
the combined methodology to one of the surveys acquired in LIAG Hannover is displayed below.
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8: Experimental Analysis and Validation
Figure 8.26 – Screen shot of the GUI.
9
Conclusions
When you reach the top of a mountain, keep climbing
Zen Proverb
In this thesis we have described the process of full forward modeling a real GPR scenario and
have presented several simulations in the context of radar antenna characterization and target
scattering signature retrieval and interpretation.
The modeling problem was firstly addressed in 2D, investigating several simple and more complex
soil/target configurations and assuming ideal plane wave illumination. After this initial approach,
we faced the task of accurately modeling a real UWB GPR antenna and we performed a comprehensive parametric study (in frequency domain) of the antenna radiation characteristic and
performance for several antenna parameters/setups in free space and above different soil types.
This analysis was helpful to explore the parameters that control the antenna directivity, gain and
bandwidth. It may be also useful for setting an upper limit on the capability of a particular GPR
system given a particular configuration. Based on the acquired knowledge, we were able to built
an adequate model to simulate our commercial GPR system.
The above antenna model was optimized by comparing and fitting the simulated GPR responses
in time domain with some free space laboratory measurements. Furthermore, the impact on the
radar echoes of the antenna system characteristics, the target size and geometry, and the soil
conditions were fully investigated, including an analysis of the effects of soil inhomogeneity as
well as surface roughness. To finish this part of the investigation an overall summary along with
some guidelines to elaborate a representative database were highlighted. Summarizing, this study
supplied a deep physical understanding of the complex scattering phenomena involved and the
influence of fundamental parameters on the soil and target radar response.
In the last and most practical part of this thesis and considering the model presented before,
we simulated the scattering signatures of several test mines and clutter objects (in free space
and buried in typical soils) in order to investigate the potential of accurate modeling to identify
171
172
9: Conclusions
the GPR returns and discriminate between scatterers. To do this, we defined a combined detection and recognition strategy based on the incorporation of a similarity filter with synthetic
reference waveforms into an energy based detection scheme. This algorithm was applied to some
experimental data acquired in a prepared test field and we demonstrated that this method can
substantially reduce the clutter and increase the true detection rate for those mines most difficult
to detect. The technique showed also satisfactory results in target recognition.
9.1
Future work
There is still much work to do in order to bring the acquired knowledge to a practical demining
system that supersedes the performance of a single metal detector. The new system should be
reliable, considerably faster and present a better detection characteristic than a standard sensor in order to compensate for the higher costs of the new system itself and operator training
time/costs.
Possible future work would be to develop a complete and well-structured target database (with
the corresponding synthetic reference signatures or some other characteristic features) that should
be integrated in the GPR system together with an optimized detection/recognition algorithm to
automatically search for the signatures/features of mine-like targets and detect/identify them.
A k-nearest neighbour (K-NN) based approach could be also incorporated within the processing
chain to reduce different mine declarations in neighboring pixels to a single mine type per cluster.
Even if a complete and efficient classification of the detected objects would be the final goal to
achieve, it is at least desirable to be able to discriminate correctly between mines and other clutter objects, which would reduce significantly the false alarms and accelerate the demining labors.
For this task, self-learning techniques such as Support Vector Machines or Neural Networks could
be considered [Yang & Bose, 2005], [Massa et al., 2005], [Parekh et al., 2000], even though these
type of approaches usually require large training sets (and for this application are not easy to
get).
Furthermore, to make the algorithm more robust and assist the classification of “difficult” targets,
other recognition methods can be added to the main classification procedure, like algorithms for
detecting specific shapes or materials that do not try to reconstruct or identify every anomaly in
soil, but they just look for certain characteristic scattering patterns (1D or 2D) in the recorded
signals.
As we have mentioned, data migration (via for example Stolt migration) applied prior to the
detection algorithm may notably improve the detection rate of landmines. The Stolt migration
can be made adaptive as shown in [Gonzalez-Huici, 2011] in order to avoid introducing a priori
information (often unknown) about soil parameters into the algorithm. However, when considering migration for an homogeneous medium (as it is the case presented here) some artifacts and
image blurring may happen due to the vertical velocity variations associated with the different
material layers. Then, another task might be to extend the adaptive Stolt migration to a layered
medium, making it adaptive for every single layer, where each layer thickness is jointly with the
background permittivity automatically estimated and optimized to get a satisfactory focusing.
Other focusing techniques could be also further investigated in order to not only improve
9.1: Future work
173
the detection but also to approximately localize and reconstruct the object shape in 3D.
Among them, presently we are paying special attention to the backprojection technique
[Ribalta & Gonzalez-Huici, 2013a]. The 3D visualization of the focused information, may help
the operator to interpret the recorded data and in general, enhance the detection and recognition
of the scatterers.
Another possibility to identify or reconstruct the scatterers may be to build an efficient tomographic inversion algorithm, which would allow to obtain the electromagnetic contrast function of
the subsurface through the measurements taken above the surface. In fact, inversion is the only
technique capable of reconstructing not only the shape but also the electromagnetic parameters
(and then, the composition) of the scatterers. However, this is a typical illposed problem (there
are many possible solutions which satisfy the integral equation to be inverted) and the more data
we collect (frequencies, illumination angles and traveling paths), the more we reduce the solution
domain, i.e., the more accurate solution we will achieve.
There are a few approaches to this problem that employing certain approximations (Born Approximation) and making certain assumptions (antennas are Hertzian dipoles, the soil is nonmagnetic,
either the conductivity contrast dominates over permittivity contrast or vice versa) can reach satisfactory solutions in some cases but in general this is a rather complex mathematical problem
and is still open.
For obvious reasons this method cannot produce automatic results from single or small collection
of A-scans, and it is hardly applicable in real time for demining labors.
All the topics discussed and investigated along this thesis are presented within the landmine detection and identification scope. Nevertheless, the proposed algorithm for detection-after-recognition
of landmines may also be of interest in other application areas, like for instance in civil infrastructure sector (to recognize pipes, cables, and other utilities and their filling materials).
174
9: Conclusions
Appendices
175
Appendix
A
Boundary Conditions in COMSOL
In the next sections we describe the formulation of the boundary conditions applied in the models
presented in this thesis. The definition of the complete list of boundary conditions available in
the Electromagnetic Module of COMSOL Multiphysics is out of the scope of this appendix. Such
a description can be found in [COMSOL, 2005].
A.1
Absorbing Boundary Conditions
Truncating the computational domain without introducing large errors due to the reflections at
the boundaries in one of the great challenges in numerical modeling.
When solving radiation problems with open boundaries, special low-reflecting or absorbing boundary conditions have to be applied at the borders of the model geometry, which should lie in
the order of a few wavelengths away from any source. The Electromagnetic Module offers two
closely related types of absorbing boundary conditions, the scattering boundary condition and the
matched boundary condition. The former can handle plane, spherical and cylindrical waves and
is perfectly absorbing for an incident plane wave, whereas the latter is perfectly absorbing for
guided modes, provided that the correct value of the propagation constant is supplied. However,
in many scattering and antenna modeling problems, you cannot describe the incident radiation
as a plane wave with a well-known propagation direction. In such cases, the use of Perfectly
Matched Layers (PMLs) may be considered. PMLs are pure mathematical constructs that are
not physically realizable, and by a suitable choice of the PML parameters it is in principle possible
to minimize the energy reflected at the borders of the modeling domain.
A.1.1
Perfectly Matched Layers
In the seminar work of Berenger (1994) a new type of material ABC was introduced, referred as
perfectly matched layer (PML).
177
178
Appendix A: Boundary Conditions in COMSOL
The original PML technique is based on a split-field formulation of the Maxwell’s equations
while other PML approaches like the Generalized Perfectly Matched Layer (GPML) presented
by [Fang & Wu, 1996]GPML are based on more compact stretched-coordinate formulations
[Chew & Weedon, 1994] or combinations of both formulations. A more physical approach is
the uniaxial PML (UPML), which does not involve the splitting of fields. In this formulation
the absorbing material is a uniaxial anisotropic material involving permittivity and permeability
tensors.
The original PML formulation can be deduced from Maxwell’s equations by introducing a
complex-valued coordinate transformation under the additional requirement that the wave
impedance should remain unaffected for any frequency and any angle of incidence. For the
implementation, it is more practical to describe the PML as an anisotropic material with losses
and for simplicity, it is assumed that the wave is entering the PML from an isotropic medium.
To define a PML, we have to add an additional modeling domain outside the boundaries that we
would like to be absorbing. The PML can have arbitrary thickness and is specified to be made
of an artificial absorbing material. The material has anisotropic permittivity and permeability
that match the permittivity and permeability of the physical medium outside the PML in such a
way that there are no reflections. The subdomain representing the PML has anisotropic material
parameters:
ǫ = ǫ0 ǫr L
(A.1)
µ = µ0 µr L
where L is a rank 2 tensor. The values of the relative permittivity and permeability are those of
the physical domain. For a PML that is parallel to one of the Cartesian coordinates, L becomes
diagonal


Lxx 0
0


L =  0 Lyy 0 
0
0 Lzz
where
Lxx =
sy sz
,
sx
Lyy =
sz sx
,
sy
Lzz =
sx sy
sz
. The parameters sx , sy and sz are the complex-valued coordinate scaling parameters. By
assigning suitable values to these, you can obtain a PML that absorbs waves traveling in a
particular direction. The values below represent a PML that attenuates a wave traveling in the
x direction:
sx = a − bi, sy = 0, sz = 0,
where a and b are arbitrary positive real numbers. For an x PML, the attenuation of a propagating
wave over a distance ∆x is given by the x component of the wave vector kx , and the imaginary
part of sx :
(A.2)
|E| = |E0 |e−bkx ∆x .
The real part of sx affects how fast an evanescent wave decays in the PML. From the practical
viewpoint, it is necessary to resolve the e−1 attenuation length in the PML with at least a couple
of elements since a poorly resolved PML give rise to unwanted reflections. In the absence of
evanescent waves, an imaginary part of −i for sx in the PML results in approximately the same
179
Appendix A: Boundary Conditions in COMSOL
requirement on the mesh density as that for a propagating wave outside the PML (10 linear elements per wavelength). In many cases, due to the problem geometry, it is more convenient to use
cylindrical or spherical PML. This is the case for the 3D antenna radiation simulations presented
in this thesis, where we have applied spherical PMLs. Because the basic coordinate system of
COMSOL Multiphysics is Cartesian, we give the Cartesian tensor component for spherical PMLs.
Using spherical coordinates (r, θ, φ), it takes the form:
2
 2
r̃ sθ sφ
r̃ sθ sφ
2
2
2 φ + s (cos2 θ cos2 φ + sin2 φ)
2 θ − 1) cos φ sin φ
sin
θ
cos
sin
θ
+
s
(cos
r
r
2
 r22 sr
2r sr
 r̃ sθ sφ
r̃ sθ sφ
2
2
2
2
2
2
2 θ − 1) cos φ sin φ
L =  r2 sr sin θ sin φ + sr (cos θ sin φ + cos φ)
sin
θ
+
s
(cos
r
2
r sr 
2
r̃ sθ sφ
r̃ 2 sθ sφ
sin
θ
cos
θ
cos
φ
sin θ cos θ sin φ
−
s
−
s
r
r
2
2
r sr
r sr
2

r̃ sθ sφ
−
s
sin
θ
cos
θ
cos
φ
r
2
r2 sr

r̃ sθ sφ
,
−
s
sin
θ
cos
θ
sin
φ
r
2

r sr
r̃ 2 sθ sφ
r 2 sr
cos2 θ + sr sin2 θ
where
r̃ =
r
sr (r ′ )dr ′
0
.
A.1.2
Z
Scattering Boundary Condition
This boundary condition is used when we want a boundary to be transparent for a scattered
wave. The boundary condition is also transparent for an incoming plane wave. The wave types
that this boundary can handle are:
E = Esc e−jk(n·r) + E0 e−jk(k·r)
e−jk(n·r)
E = Esc √
+ E0 e−jk(k·r)
r
E = Esc
e−jk(n·r)
+ E0 e−jk(k·r)
rs
Scattered plane wave
Scattered cylindrical wave
Scattered spherical wave
The field E0 is the incident plane wave which travels in the direction k. Note that the boundary
condition is transparent for plane waves with any incidence angle. For the boundary to be
perfectly transparent it is important that the boundary represent an open boundary. If the wave
enters a guided structure where only certain modes are excited, this boundary condition will give
reflections. For such boundaries that do not represent a physical boundary, it is more adequate
to use the so called matched boundary condition. This boundary was not applied in any of the
models presented in this thesis and it will not be described here.
A.2
Interface Boundary Conditions
In order to minimize the problem size it is convenient to replace thin layers with boundary
conditions. For example we have replaced materials with high conductivity, like the antenna
180
Appendix A: Boundary Conditions in COMSOL
sheets, by the perfect electric conductor boundary condition.
A.2.1
Perfect Electric Conductor
The perfect electric boundary condition sets the tangential component of the electric field to zero:
n × E = 0.
This is a special case of the electric field boundary condition, which allows to specify the tangential
component of the electric field.
In the transient case this boundary condition is used to set the tangential component of the
magnetic vector potential to zero. Since the perfect electric conductor boundary condition
n × E = −n ×
∂A
=0
∂t
implies that
n × A = A(t = 0)
Normally the initial condition for A on a perfect electric conductor is zero. In the cases where
the initial boundary condition is different from zero the magnetic potential boundary condition
can be used.
A.2.2
Continuity
The continuity boundary condition is the natural boundary condition ensuring continuity of the
tangential components of the electric and magnetic fields:
n × (E1 − E2 ) = 0
n × (H1 − H2 ) = 0
Appendix
B
Plane Wave Scattering by Simple
Canonical Objects
In this Appendix we introduce the analytical solutions to some canonical scattering problems
which are of interest for validation and comparison with some of the simulated results presented
in Chapter 5. In particular, we will show the solution to the scattering by a metallic and a
dielectric circular cylinder and by a metallic and a dielectric sphere for an incoming plane wave.
The complete derivations can be found in [Balanis, 1989] and [Harrigton, 2001].
B.1
Scattering by Circular Cylinders
Many practical scatterers can be represented by cylindrical structures and due to its simplicity
and well known solutions, circular cylinders correspond to one of the most important types of
scattering geometries. For instance, they are widely employed to represent typical radar scatterers such as fuselage of airplanes and missiles. In particular, in cylindrical coordinates (z, ρ, φ) the
solutions are expressed in terms of the products of Bessel and Hankel functions and exponential
functions. In this section only infinite cylinders and two dimensional cases are considered; the
scattered field from finite length cylinders is calculated by transforming the corresponding solutions for infinite length applying approximate relationships.
Assuming an uniform plane wave (with only one electric component in the z direction as defined
in Section 5.2) normally incident upon a perfectly conducting cylinder of radius a, the scattered
electric field takes the form [Harrigton, 2001]:
Ezs = E0
∞
X
j −n an Hn(2) (kρ)ejnφ
(B.1)
−Jn (ka)
(B.2)
n=−∞
with the coefficient an given by
an =
(2)
Hn (ka)
181
,
182
Appendix B: Plane Wave Scattering by Simple Geometrical Objects
where k is the wavenumber of the free space, ρ is the distance to the measurement point, and n is
the order of the Bessel function of first kind Jn and the Hankel function of the second kind H (2) .
In the more general case of a dielectric circular cylinder it can be demonstrated that the scattered
field is equal to [Balanis, 1989]:
Ezs
= E0
∞
X
an Hn(2) (kρ)ejnφ
(B.3)
n=−∞
and now the coefficient an is of the form:
an = j
−n
Jn′ (ka)Jn (kc a) −
p
ǫr /µr Jn (ka)Jn′ (kc a)
(B.4)
p
(2)′
ǫr /µr Jn′ (kc a)H (2) (ka) − Jn (kc a)Hn (ka)
∂
.
with kc the wavenumber of the dielectric medium and ′ = ∂(ka)
An example of the resulting analytical solutions is displayed in the next illustration. It represents
the scattered fields by one metallic cylinder and two dielectric cylinders of different radius at a
distance of 10cm from the top of the object (and the z-polarised plane wave propagating from
top to bottom along the xy-plane).
Scattering by a dielectric circular cylinder at 10cm
0.7
Norm ScattEz
0.6
Metallic circ. cylinder r=2.5cm
Plastic circ. cylinder r=2.5cm
Plastic circ. cylinder r=5cm
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
f(Hz)
4
5
6
9
x 10
Figure B.1 – Analytical scattered field by different objects in free space and plane wave excitation;
receiving point at a distance of 10cm.
The results represented in this figure can be directly compared with the simulated analogs previously displayed in Fig. 5.7 in Section 5.4. As we can observe, the agreement is very good.
B.2
Scattering by a sphere
Probably the most classic scattering problem if that of a plane wave scattering by a metallic
sphere. Due to its symmetry, the sphere is often employed as a reference scatterer for calculating the scattering properties of other targets. The scattered fields are obtained through the
183
B.2: Scattering by a sphere
formulation of the radial components of the magnetic and vector potentials [Harrigton, 2001] and
in spherical coordinates, the solutions are expressed in term of products of the Riccati-Bessel
functions, associated Legendre polynomials, and exponential functions.
Assuming that the electric field of the impinging plane wave is polarized in the x direction and
traveling along the z-axis, the scattered electric field components in spherical coordinates are
given by:
Ers = −jE0 cos φ
∞
X
′′
bn [Ĥn(2) (kρ) + Ĥn(2) (kρ)]Pn1 (cos φ)
n=1
∞ X
Pn1 (cos θ)
sin θ
n=1
∞ X
P 1 (cos θ)
E0
′
′
sin φ
jbn Ĥn(2) (kρ) n
− cn Ĥn(2) (kρ) sin θPn1 (cos θ)
Eφs =
kρ
sin θ
Eθs =
E0
cos φ
kρ
′
′
jbn Ĥn(2) (kρ) sin θPn1 (cos θ) − cn Ĥn(2) (kρ)
(B.5)
n=1
(2)
where Ĥn refers to spherical Hankel functions of the second kind, and
′
′′
′
∂
for spherical Hankel functions
∂(kρ)
∂
for spherical Hankel functions
=
∂(kρ)2
1 ∂
∂
=−
for associated Legendre functions
=
∂(cos θ)
sin θ ∂θ
=
(B.6)
When we have a conducting sphere the coefficients acquire the following form:
an = j −n
(2n + 1)
n(n + 1)
Jˆ′ (ka)
bn = −an
cn = −an
n
(2)′
Ĥn (ka)
(B.7)
Jˆn (ka)
(2)
Ĥn (ka)
with Jˆn the spherical Bessel function.
For the general case of the dielectric sphere an does not change, and bn and cn become notably
more complicated:
√
√
− ǫd µ0 Jˆn′ (ka)Jˆn (kd a) + ǫ0 µd Jˆn (ka)Jˆn′ (kd a)
an
bn = √
√
(2)
(2)′
ǫd µ0 Ĥn (ka)Jˆn (kd a) − ǫ0 µd Ĥn (ka)Jˆn′ (kd a)
√
√
− ǫd µ0 Jˆn (ka)Jˆn′ (kd a) + ǫ0 µd Jˆn′ (ka )Jˆn (kd a)
cn = √
an
√
(2)′
(2)
ǫd µ0 Ĥn (ka)Jˆ′ (kd a) − ǫ0 µd Ĥn (ka)Jˆn (kd a)
(B.8)
n
where a is now the radius of the sphere.
Below (see Fig. B.2) we illustrate some examples of the analytical Radar Cross Section for
different size and material spheres at 50cm distance. As it was observed in the previous example
for a cylindrical target, the number of resonances increases with the size of the scatterer.
184
Appendix B: Plane Wave Scattering by Simple Geometrical Objects
Scattering by a sphere at 50cm
2
10
0
Normalized RCS
10
−2
10
−4
10
Metallic sphere r=2.5cm
Metallic sphere r=5cm
Plastic sphere r=2.5cm
Plastic sphere r=5cm
−6
10
0
1
2
3
4
5
f (Hz)
6
7
8
9
10
9
x 10
Figure B.2 – Radar Cross Section by different spheres in free space and plane wave excitation; receiving point at a distance of 50cm.
Definitions
• Field Regions
The space surrounding an antenna is usually divided into three regions: reactive near-field,
radiating near-field (Fresnel region) and far-field (Fraunhofer region). Although no abrupt
changes in the fields are appreciated as the boundaries are crossed, there are remarkable
differences among them. There boundaries between the regions are not unique but there
are various criteria commonly used to separate the regions [Balanis, 2005].
Reactive near-field region is defined as “the portion of the near-field region immediately
surrounding the antenna wherein the reactive field predominates”. For most antennas the
p
outer boundary of this region is considered to be at a distance R = 0.62D 3 /λ from the
antenna surface, where λ is the wavelength and D is the largest dimension of the antenna.
Radiating near-field (Fresnel) region is “the region between the reactive near-field and the
far-field wherein radiation fields predominate and wherein the angular distribution of the
fields is dependent upon the distance from the antenna. If the antenna has a maximum
dimension that is small compared to the wavelength, this field region may not exist”. The
p
inner boundary corresponds to the distance R = 0.62D 3 /λ and the outer is assumed to
be at R = 2D 2 /λ where D is the largest dimension of the antenna (to be valid D must be
large compared to the wavelength, D > λ). In this region the field pattern is, in general, a
function of the radial distance and the radial field component may be significant.
Far-field (Fraunhofer) region is “that region of the field of an antenna where the angular field
distribution is practically independent of the distance from the antenna. If the antenna has
a maximum overall dimension D, the far-field is usually assumed to be at distances greater
than 2D 2 /λ. In physical media, if the maximum dimension D is large compared to π/|γ|,
the far-field region can be assumed to start approximately at a distance |γ|D 2 /π from the
antenna, γ being the propagation constant in the medium”. In this region, the the E- and
H-field components are perpendicular to each other and transverse to the radial direction
of propagation, and the angular distribution is independent of the radial distance r, i.e.,
the r variations are separable from those of the angular directions (see Fig. 9.3).
• Radiation Pattern, Power Density and Poynting Vector
An antenna radiation pattern is defined as a graphical representation of the radiation prop185
186
Figure 9.3 – Coordinate system and relationship between E and H in the far field.
erties of the antenna as a function of space coordinates [Balanis, 2005]. In most cases, the
radiation pattern is determined in the far-field region. The most significant radiation property if the 2D or 3D spatial distribution of radiated energy as a function of the observer’s
position along a path or surface of constant radius. A trace of the received power at a
constant radius is called the power pattern and a graph of the angular distribution of the
electric (or magnetic) field is called an amplitude pattern.
The average power density radiated by an antenna is quantified by the time-averaged Poynting vector
1
Wav = [W(x, y, z; t)]av = Re[E × H∗ ] (W/m2 ),
2
where W is the instantaneous Poynting vector defined as W = E × H and for time harmonic
variations, the complex fields E and H are related to their instantaneous counterparts E
and H by
E(x, y, z; t) = Re[E(x, y, z)ejωt ],
H(x, y, z; t) = Re[H(x, y, z)ejωt ].
Since the Poynting vector is a power density, the average total power radiated by an antenna,
Prad , can be calculated by integrating the normal component of the Poynting vector over
the entire close surface
ZZ
ZZ
1
Prad =
Wav · n̂da =
Re(E × H∗ ) · ds.
2
In the above equation, while the real part of the expression inside parenthesis represents the
average real power density, the imaginary part represents the reactive (stored) power density
associated with the electromagnetic fields, i.e., the real part accounts for the radiative losses
and the imaginary part for the resistive losses.
• Directivity and Gain
The directivity of an antenna can be defined as the ratio of the radiation intensity in a
given direction from the antenna to the radiation intensity averaged over all directions.
The radiation intensity which is the power radiated by an antenna per unit solid angle
can be obtained by just multiplying the above defined radiated power density Wrad by the
Definitions
187
square of the distance. The average radiation intensity is equal to the total power radiated
by the antenna divided by 4π. If the direction is not specified, the direction of maximum
radiation intensity is implied. In mathematical form:
D=
U
4πU
,
=
U0
Prad
where U is the radiation intensity in a given direction and U0 if the radiation intensity of
an isotropic source.
The gain of an antenna is closely related to the directivity, but it takes into account the
antenna efficiency as well as its directional capabilities. Absolute gain of the antenna in
a given direction is defined as the ratio of the intensity in that direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated
isotropically. The radiation associated to the isotropically radiated is equal to the power
accepted by the antenna divided by 4π. However, in most cases we deal with relative gain,
which is defined as the ratio of the power gain to the power gain of a reference antenna in a
given direction. The power input must be then the same for both antennas. The reference
antenna can a dipole, horn, or any other whose gain is known or can be calculated. But in
most cases, the reference antenna is a lossless isotropic source. Thus
G(θ, φ) =
4πU (θ, φ)
,
iso
Pin
which is often expressed in terms of decibels and stated dBi to indicate that the antenna is
compared to an isotropic radiator.
Here again, when the direction is not indicated, the power gain is normally considered in
the direction of maximum radiation.
In the above formula, the total input power Pin is directly related to the total radiated power
Prad by the antenna radiation efficiency η (dimensionless): Prad = η ∗ Pin . For an ideal
lossless radiator the efficiency equals 1. Both directivity and gain are unitless measures.
• Impedance
The input impedance (Zin ) is the ratio between the voltage and currents at the antenna
port. It is a complex quantity that varies with frequency as
Zin (f ) = Rin (f ) + jXin (f ),
where f is the frequency, the real part Rin is the antenna’s total feed point resistance
including both radiation (Rr ) and loss (Rl ) terms, and the imaginary part Xin is the antenna’s feed point reactance. For efficient power transfer and antenna radiation the input
impedance needs to be matched to the internal impedance of the rest of the network. The
input impedance can be used to determine other related parameters such as the reflection
coefficient (Γ) and the voltage standing wave ratio (VSWR).
• Signal-to-Noise Ratio (SNR) is defined as the power ratio between a signal (meaningful
information) and the background noise (unwanted signal), where both the signal and the
noise power must be measured at the same or equivalent point of a system:
Asignal 2
Psignal
=
,
SN R =
Pnoise
Anoise
188
where P is the average power and A the RMS amplitude (for example, in the energy based
detection algorithm presented in Chapter 8, the RMS voltage). Because many signals have
a very wide dynamic range, SNRs are often expressed using the logarithmic decibel scale.
In decibels, the SNR is defined as:
SN RdB = 10log10
Asignal
Anoise
2
= 20log10
Asignal
Anoise
.
In our case the amplitude of the noise is calculated as the average RMS amplitude of the
voltage over a given scan area and the signal amplitude is evaluated pixel per pixel.
• Dynamic range is defined as the ratio between the peak transmitted signal power and
the minimum detectable peak power entering the receiver antenna. This number quantifies
the maximum amount of loss that the radar signal can have, and still be detectable in the
receiver, being a key measure for the performance of the radar system. It is dependent on
the radar integration time and it is normally given in dB [Hamran, 2010].
• Adaptivity is an active research area within FEM and it is particularly effective in fluid
flow, heat transfer, and structural analysis. Generally, there are two types of adaptation:
h-adaptation (mesh refinement), where the element size varies while the orders of the shape
functions are kept constant; p-adaptation, where the element size is constant while the orders of the shape functions are increased (linear, quadratic, cubic, etc.). Adaptive remeshing
(know as r-adaptation) employs a spring analogy to redistribute the nodes in an existing
mesh without adding new nodes; the accuracy of the solution is limited by the initial number of nodes and elements. In mesh refinement (h-adaptation), individual elements are
subdivided without altering their original position. The use of hp-adaptation includes both
h- and p-adaptation strategies and produces exponential convergence rates. Both mesh
refinement and adaptive meshing are present in COMSOL.
Acronyms
ABS
AP
AT
A/D
CO
dB
EM
FD
FDTD
FEM
FFT
GO
GPR
GTD
GUI
GPML
IDW
LIAG
LNA
MD
Absorbing Boundary Condition
Antipersonnel
Antitank
Analog to Digital
Common Offset
decibel
Electromagnetic
Finite Difference
Finite Difference Time Domain
Finite Element Method
Fast Fourier Transform
Geometrical Optics
Ground Penetrating Radar
Geometrical Theory of Diffraction
Graphical User Interface
Generalized Perfectly Matched Layers
Inverse Distance Weighted
Leibniz Institute for Applied Geophysics
Low Noise Amplifier
Metal Detector
189
MoM
MS
MWR
PEC
PSD
PML
RCS
RF
ROC
RS
RMS
SAR
SCR
SNR
STT
TDR
TNT
UWB
UXO
VNA
Method of Moments
Mie Scattering
Method of Weighted Residuals
Perfect Electric Conductor
Power Spectral Density
Perfectly Matched Layer
Radar Cross Section
Radio Frequency
Receiver Operation Characteristics
Rayleigh Scattering
Root Mean Square
Synthetic Aperture Radar
Signal to Clutter Ratio
Signal to Noise Ratio
Standard Test Target
Time Domain Reflectometry
Trinitrotoluene
Ultra Wideband
Undexploded Ordnance
Vector Network Analyser
190
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List of Figures
2.1
Distribution of the AP mines and UXO in the world. Source ICBL [ICBL, 2009]. .
8
2.2
ROC curves (Photo RAND). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Various AP blast landmines (Photo GICHD). . . . . . . . . . . . . . . . . . . . . . 11
2.4
Block diagram of a time domain UWB GPR. . . . . . . . . . . . . . . . . . . . . . 17
2.5
Common Offset acquisition mode.
2.6
Preprocessed A-scans in time (top) and frequency domain (bottom) for a metallic
and a plastic sphere of radius r=5cm. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7
B-scan and hyperbola formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8
Measured B-scans, received amplitude with a metallic sphere (left), with a metallic
sphere after average background removal (middle) and with a plastic sphere after
background removal (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9
Measured C-scans (raw data) at different time instants. Recorded amplitude with
a plastic and a metallic sphere situated at the same distance from antenna head. . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.10 View of GPR Transmitter/Receiver Head. . . . . . . . . . . . . . . . . . . . . . . . 22
2.11 Test field and SPRScan Radar in the LIAG (Hannover). . . . . . . . . . . . . . . . 23
3.1
Discretization scheme of the Yee cell. The six components of the EM field are
discretized in a staggered grid and referenced by the spatial coordinates x, y and z
directions, respectively. In addition to the spatial staggering the components of the
magnetic field are also offset in time from those of the electric field by a half-time
step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2
2D and 3D FEM Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
199
200
4.1
Incidence, reflection and refraction angles of an electromagnetic plane wave at the
interface between two dielectric media. . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2
Vertical resolution against pulse length for different media.
4.3
Horizontal resolution versus pulse length for different media. . . . . . . . . . . . . . 44
4.4
Modeled amplitude and phase of the scattered electric field by a dielectric circular
cylinder of 5 cm (top), 10cm (middle) and 25cm (bottom) radius applying ABC at
the borders, f=0.6Ghz (λ = 50cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5
Radiation pattern of the three dipoles in the far field (left) and 3D radiation pattern
of the bow-tie dipole (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6
Radiation patterns of the dipole antennas at different frequencies: 1.5, 2 and 2.5GHz. 50
4.7
Comparison between measured and CRIM modelled complex, frequency-dependent
permittivity of sandy soil with 20% water content and < 2% clay content
[Cassidy, 2009]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8
Real and imaginary parts of the relative effective permittivity for sandy soil against
frequency (left) and water content fraction (right). . . . . . . . . . . . . . . . . . . 55
4.9
Two generated scenarios with rough surface and inhomogeneous soil. . . . . . . . . 57
. . . . . . . . . . . . . 43
4.10 Autocorrelation function of the permittivity distribution for both inhomogeneous
scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1
Amplitude of the scattered field by a dielectric circular cylinder (r=6cm) with PML
(top) and ABC (bottom) for f=1,2,3 GHz. . . . . . . . . . . . . . . . . . . . . . . . 62
5.2
Relative error introduced by the boundary conditions. . . . . . . . . . . . . . . . . 63
5.3
Amplitude of the scattered field by a tilted metallic rectangular cylinder with PML
(top) and ABC (bottom) for f=1,2,3 GHz. . . . . . . . . . . . . . . . . . . . . . . . 64
5.4
Amplitude of the scattered field by a metallic (top) and a dielectric (bottom)
circular cylinder (r=2.5cm) in free space. . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5
Amplitude of the scattered field by an empty (top) and a water filled plastic pipe
(r=5cm) (bottom) in free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.6
Amplitude of the scattered field by a plastic (top) and half-plastic (bottom) rectangular cylinder (length w=10cm, height h=4cm) in free space. . . . . . . . . . . . 66
5.7
Backscattered electric field by different objects in free space and plane wave excitation; receiving point at a distance of 10cm. . . . . . . . . . . . . . . . . . . . . . 66
List of Figures
201
5.8
Scattering by a buried target illuminated by a downward propagating plane wave.
67
5.9
Amplitude of the scattered field by a buried metallic circular cylinder (r=2.5cm)
in dry(top) and wet (bottom) soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.10 Amplitude of the scattered field by a buried dielectric cylinder (r=2.5cm) in dry
(top) and wet (bottom) soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.11 Amplitude of the scattered field by a buried empty pipe (r=5cm) in dry (top) and
wet (bottom) soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.12 Amplitude of the scattered field by a buried plastic pipe full of water (r=5cm) in
dry (top) and wet (bottom) soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.13 Amplitude of the scattered field by a buried plastic rectangular cylinder (w=10cm,
h=4cm) in dry (top) and wet (bottom) soil. . . . . . . . . . . . . . . . . . . . . . . 70
5.14 Amplitude of the scattered field by a buried half-plastic rectangular plate
(w=10cm, h=4cm) in dry (top) and wet (bottom) soil. . . . . . . . . . . . . . . . . 71
5.15 Backscattered electric field by different objects buried in dry soil and plane wave
excitation; receiving point at a 10cm height above the surface. . . . . . . . . . . . . 72
5.16 Backscattered electric field by different objects buried in wet soil and plane wave
excitation; receiving point at a 10cm height above the surface. . . . . . . . . . . . . 72
5.17 Real part of the scattered field by a buried empty pipe (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous dry
soil with flat surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.18 Real part of the scattered field by a buried empty pipe (r=5cm) (top) and an airplastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous dry
soil with rough surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.19 Real part of the scattered field by a buried empty pipeline (r=5cm) (top) and an
air-plastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous wet
soil with flat surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.20 Real part of the scattered field by a buried empty pipeline (r=5cm) (top) and an
air-plastic rectangular cylinder (w=10cm, h=4cm) (bottom) in inhomogeneous wet
soil with rough surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.21 Backscattered amplitude by an empty plastic pipe, a water filled pipe (r=5cm), a
metallic circ. cylinder and a plastic circ. cylinder (from left to right). . . . . . . . . 76
5.22 Backscattered amplitude by a metallic and a plastic circular cylinder (r=2.5cm)
buried in dry and wet soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
202
5.23 Backscattered amplitude by a plastic empty pipe and a pipe full with water buried
in dry and wet soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.24 Backscattered amplitude by a plastic rectangular cylinder and air-plastic rectangular cylinder buried in dry and wet soil. . . . . . . . . . . . . . . . . . . . . . . . 78
5.25 Maximum scattered amplitude by a dielectric cylinder (r=2.5cm) in different soils.
78
5.26 Maximum scattered amplitude for different cylinder radius and different receiver
heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.27 Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic
rect. cylinder (w=10cm, h=4cm) in inhomogeneous dry soil with flat and rough
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.28 Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic
rect. cylinder (w=10cm, h=4cm) in inhomogeneous wet soil with flat and rough
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.29 Backscattered amplitude by a buried empty pipeline (r=5cm) and an air-plastic
rect. cylinder (w=10cm, h=4cm) in homogeneous dry and wet soil with rough
surface applying time gating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1
Model of the GPR antennas (with a transmitter and a receiver bow-tie, EM absorbing material and metallic shielding.) . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2
Radiated power in forward direction (d=25cm) for different flare angles. . . . . . . 87
6.3
Impedances for 3 different flare angles. . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4
Radiation pattern (d=25cm) for different box heights and frequencies=1, 1.5, 2,
2.5 and 3Ghz, E-plane (top), H-plane (bottom). . . . . . . . . . . . . . . . . . . . . 88
6.5
Radiated power in forward direction (d=25cm) for different box heights. . . . . . . 89
6.6
Radiation pattern (d=25cm) in free space for variable ǫbox and σbox , E-plane. . . . 90
6.7
Radiation pattern (d=25cm) in free space for variable ǫbox and σbox , H-plane. . . . 91
6.8
Radiated power in forward direction (d=25cm) for variable absorber ǫbox and σbox .
6.9
Impedances for ǫbox = 6 and variable σbox . . . . . . . . . . . . . . . . . . . . . . . . 93
92
6.10 Impedances for absorber ǫr = 10 and variable σr . . . . . . . . . . . . . . . . . . . . 93
6.11 Transmission line model of the antenna box. . . . . . . . . . . . . . . . . . . . . . . 93
6.12 Input impedance for different permittivities, σ = 0.4S/m. . . . . . . . . . . . . . . 94
List of Figures
203
6.13 Input impedance for different absorber thicknesses, ǫ = 6, σ = 0.4S/m. . . . . . . . 94
6.14 Radiation pattern in free space (d=25cm) of two bow-ties alone, with shielding
box and with absorbed filled shielding box, E-plane (top) and H-plane (bottom). . 95
6.15 Radiation patterns (d=20cm) in medium half-space of the bow-ties alone and enclosed in a PEC casing. Dry (left), medium (center) and wet soil (right), E-plane
(half-top) and H-plane (half-bottom). . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.16 Radiated power in forward direction (d=20cm) for of the bow-ties alone and enclosed in a PEC casing above different soils. . . . . . . . . . . . . . . . . . . . . . . 98
6.17 Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m. Dry (left), medium (center) and wet soil (right), E-plane. . . . . . . . 99
6.18 Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m. Dry (left), medium (center) and wet soil (right), H-plane. . . . . . . . 100
6.19 Input impedance for different antenna heights and soil types. . . . . . . . . . . . . 101
6.20 Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m and antenna heights h=1cm (left), h=3cm (center) and h=6cm (right),
E-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.21 Radiation pattern (d=20cm) in medium half-space for ǫbox = 6, 10 , σbox =
0.2, 0.4S/m and antenna heights h=1cm (left), h=3cm (center) and h=6cm (right),
H-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.22 Radiated power in forward direction (d=20cm) for different antenna elevations
without PEC casing, with empty PEC casing and with PEC casing + absorber of
ǫbox = 6, 10 and σbox = 0.2, 0.4S/m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.23 Radiation pattern (d=20cm) in medium half-space with rough interface, E-plane. . 105
6.24 Radiation pattern (d=20cm) in medium half-space with rough interface, H-plane. . 105
7.1
Applied source pulses in time and frequency domain. . . . . . . . . . . . . . . . . . 109
7.2
Measurement of the antenna crosstalk in laboratory. . . . . . . . . . . . . . . . . . 110
7.3
Simulated crosstalks for different flare angles for a gaussian pulse (top) and a
monocycle (bottom) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4
Simulated crosstalks for different antenna lengths and constant flare angle for a
gaussian pulse (top) and a monocycle (bottom) pulse. . . . . . . . . . . . . . . . . 112
204
7.5
Simulated crosstalks for different cable input impedances for a gaussian pulse (top)
and a monocycle (bottom) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.6
Simulated crosstalks for a gaussian pulse (top) and a monocycle pulse (bottom)
with different central frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.8
Simulated crosstalks for different cavity heights for a gaussian pulse (top) and a
monocycle (bottom) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.7
Sketch of the antenna head with shielding box. . . . . . . . . . . . . . . . . . . . . 114
7.9
Simulated crosstalks for different cavity lengths for a gaussian pulse (top) and a
monocycle (bottom) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.10 Simulated crosstalks for different cavity widths for a gaussian pulse (top) and a
monocycle (bottom) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.11 Simulated crosstalks for different absorbing materials (ǫr variable and σ = 0.2S/m)
for a gaussian pulse (top) and a monocycle (bottom) pulse. . . . . . . . . . . . . . 117
7.12 Simulated crosstalks for different absorbing materials (ǫr variable and σ = 0.4S/m)
for a gaussian pulse (top) and a monocycle (bottom) pulse. . . . . . . . . . . . . . 117
7.13 Simulated crosstalks after optimization for Gaussian and Monocycle pulses compared with the measured crosstalk. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.14 Snapshots for Tx and Rx alone (without shielding and absorber) on the surface
(dry soil), E-plane (left) and H-plane (right). . . . . . . . . . . . . . . . . . . . . . 120
7.15 Snapshots for Tx and Rx (without shielding and absorber), at height=6cm above
dry soil, E-plane (left) and H-plane (right). . . . . . . . . . . . . . . . . . . . . . . 121
7.16 Snapshots for Tx and Rx with metallic shielding on the surface (dry soil), no
absorber, E-plane (left) and H-plane (right). . . . . . . . . . . . . . . . . . . . . . . 122
7.17 Snapshots for Tx and Rx with metallic shielding, no absorber, at height=6cm
above dry soil, E-plane(left) and H-plane (right). . . . . . . . . . . . . . . . . . . . 123
7.18 Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m on
surface, E-plane (left) and H-plane (right). . . . . . . . . . . . . . . . . . . . . . . . 124
7.19 Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m,
at height=2cm above dry soil, E-plane (left) and H-plane (right). . . . . . . . . . . 125
7.20 Snapshots for Tx and Rx with shielding and absorber of ǫr = 7.2, σ = 0.39S/m,
at height=6cm above dry soil, E-plane (left) and H-plane (right). . . . . . . . . . . 126
List of Figures
205
7.21 Snapshots for Tx and Rx with shielding and absorber of ǫr = 10, σ = 0.39S/m, at
height=6cm above dry soil, E-plane (left) and H-plane (right). . . . . . . . . . . . 127
7.22 Models of a mine Type 1 (left) and Type 2 (right). . . . . . . . . . . . . . . . . . . 128
7.23 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for gaussian and monocycle source pulses. . . . . . . . . . . . . . . . . . . . . . . . 129
7.24 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2(bottom)
when the transmitter is excited with monocycle pulses of different central frequencies.129
7.25 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
when their horizontal dimensions are modified. . . . . . . . . . . . . . . . . . . . . 130
7.26 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
when their vertical dimension is modified. . . . . . . . . . . . . . . . . . . . . . . . 131
7.27 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for horizontal displacements. The displacement x=-4cm is not displayed because
due to the configuration symmetry the signature is the same. . . . . . . . . . . . . 132
7.28 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for different tilt angles respect to the horizontal. . . . . . . . . . . . . . . . . . . . 132
7.29 Amplitude of the scattered signal by different objects in comparison with the signature by mine Type 1 (top) and mine Type 2 (bottom). . . . . . . . . . . . . . . 133
7.30 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil at 3, 8 and 13cm. The value of the amplitude is normalized for
both objects at every depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.31 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in wet soil at 3, 8 and 13cm. The value of the amplitude is normalized for
both objects at every depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.32 Amplitude of the scattered signal by different shallow buried objects in dry soil. . . 135
7.33 Amplitude of the scattered signal by different shallow buried objects in wet soil.
. 136
7.34 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil with different permittivities and constant conductivity σ = 1mS/m.137
7.35 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
buried in dry soil of different conductivities and constant permittivity ǫr = 7. . . . 137
7.36 Amplitude of the scattered signal by mine Type 1 (top) and mine Type 2 (bottom)
for different antenna heights above dry soil. . . . . . . . . . . . . . . . . . . . . . . 138
206
7.37 Amplitude of the scattered signal by mine Type 1 buried in inhomogeneous dry
(top) and wet soil (bottom) for different correlations lengths. . . . . . . . . . . . . 139
8.1
PMN mine simulant (top), Type-72 mine simulant (middle) and ERA test mine
(bottom) employed in the measurements and the corresponding CAD models. . . . 144
8.2
Test area layout and targets’ position (left) and test field with buried test mines
indicated by arrows (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.3
Texture triangle set after laboratory analysis, clay=1%, silt=6.7%, sand=92.3% . . 146
8.4
Recorded 1D data before and after applying the Hilbert transform. . . . . . . . . . 148
8.5
Detection maps for the surveys 146 and 151. The targets in the middle are ERA
test mines (left) and PMA-2 simulants (right). . . . . . . . . . . . . . . . . . . . . 150
8.6
ROC curves after applying the detection algorithm to the surveys with buried
PMA-2, PMN, ERA and Type-72 targets respectively. . . . . . . . . . . . . . . . . 151
8.7
Migrated image with two targets in free space. Left: Type 72 and PMN simulants;
right: same as before with background noise removal (8dB). Survey SVY-188. . . . 153
8.8
From top to bottom: raw data with a PMA-2 simulant, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-139. . 154
8.9
From top to bottom: raw data with a PMN simulant, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-140. . 155
8.10 From top to bottom: raw data with a Type-72 simulant, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-142. . 156
8.11 From top to bottom: raw data with an ERA test target, migrated image, migrated
image with background noise removal (8dB), contrast function. Survey SVY-143. . 157
8.12 Cross-correlation between measured and simulated signatures for a Type-72 simulant (top), a PMN simulant (middle) and an ERA test target (bottom). . . . . . . 159
8.13 Detection map (left) and class map of the survey SVY-139. The target in the
middle is a PMA-2 simulant, which is a rectangular plastic mine (height=3.5cm,
lengthxwidth=6x14cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.14 Detection map (left) and class map of the survey SVY-147. The target in the
middle is an ERAtest mine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.15 Block diagram of the proposed methodology. . . . . . . . . . . . . . . . . . . . . . 162
List of Figures
207
8.16 Detection map without (left) and with similarity filtering (PMN) (right) over a
survey area with buried PMN simulants. . . . . . . . . . . . . . . . . . . . . . . . . 163
8.17 Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants. . . . . . . . . . . . . . . . . . . . . . . 164
8.18 Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants. . . . . . . . . . . . . . . . . . . . . . . 164
8.19 Detection map without (left) and with similarity filtering (Type-72) (right) over a
survey area with buried Type-72 simulants. . . . . . . . . . . . . . . . . . . . . . . 165
8.20 Detection map without (left) and with similarity filtering (ERA) (right) over a
survey area with buried ERA test targets. . . . . . . . . . . . . . . . . . . . . . . . 165
8.21 Detection map without (left) and with similarity filtering (ERA) (right) over a
survey area with buried ERA test targets. . . . . . . . . . . . . . . . . . . . . . . . 166
8.22 Detection map without (left) and with similarity filtering (ERA) (right) over a
survey area with buried ERA test targets. . . . . . . . . . . . . . . . . . . . . . . . 166
8.23 ROC curves after applying the detection algorithm to the surveys with buried
PMN simulant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.24 ROC curves after applying the detection algorithm to the surveys with buried ERA
test mine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.25 ROC curves after applying the detection algorithm to the surveys with buried
Type-72 simulant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.26 Screen shot of the GUI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.1 Analytical scattered field by different objects in free space and plane wave excitation; receiving point at a distance of 10cm. . . . . . . . . . . . . . . . . . . . . . . 182
B.2 Radar Cross Section by different spheres in free space and plane wave excitation;
receiving point at a distance of 50cm. . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3
Coordinate system and relationship between E and H in the far field. . . . . . . . . 186
208
List of Tables
2.1
Landmines around the world; - indicates insufficient data. Source ICBL. . . . . . .
2.2
Summary of Detection Technologies [MacDonald & R., 2003]. . . . . . . . . . . . . 13
4.1
Relative permittivity (ǫr ), conductivity (σ), velocity (v) and attenuation (α) from
[Davis & Annan, 1989]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.1
The parameters of the optimized models. . . . . . . . . . . . . . . . . . . . . . . . 118
8.1
Electrical properties of materials used in mine construction. . . . . . . . . . . . . . 145
209
9
210
Declaration
I hereby declare that the present work was done by myself and no other than the cited sources
were employed.
Ich erkläre hiermit, die vorliegende Arbeit selbständig und nur mit den angegebenen Hilfsmitteln
angefertig zu haben.
Bonn, November 2012.
Marı́a A. Gónzalez Huici
211

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