analytical modelling of ultra-wide band ground...

Analytical Modelling of Ultra-WideBand Ground Penetrating Radar forCharacterization of Subsurface Media

Subrata Maiti

Department of Electronics and Communication EngineeringNational Institute of Technology Rourkela

Rourkela, Odisha, India - 769 008

May 2017

Analytical Modelling of Ultra-WideBand Ground Penetrating Radar forCharacterization of Subsurface Media

Thesis submitted in partial fulfillment

of the Requirements for the degree of

Doctor of Philosophy

in

Electronics and Communication Engineering

by

Subrata Maiti

Roll no: 511EC407

Under the guidance of

Prof. Sarat Kumar Patra

&

Prof. Amitabha Bhattacharya

(Dept. of E&ECE, IIT Kharagpur)

Department of Electronics and Communication EngineeringNational Institute of Technology Rourkela

Rourkela, Odisha, India - 769 008

Dept. of Electronics & Communication Engineering

National Institute of Technology, Rourkela

Odisha-769 008, India.

May 29, 2017

Certificate

This is to certify that the work in the thesis entitled Analytical Modelling of Ultra-

Wide Band Ground Penetrating Radar for Characterization of Subsurface

Media by Subrata Maiti is a record of an original research work carried out under

our supervision and guidance in partial fulfillment of the requirements for the award of

the degree of Doctor of Philosophy in Electronics and Communication Engineering,

National Institute of Technology, Rourkela. Neither this thesis nor any part of it has been

submitted for any degree or academic award elsewhere.

Dr. Amitabha Bhattacharya Dr. Sarat Kumar Patra

(Co-Supervisor) (Supervisor)

Associate Professor, Dept. of E&ECE Professor, Dept. of ECE

IIT Kharagpur, West Bengal NIT Rourkela, Odisha

Dedicated to My Nation

Declaration of Originality

I, Subrata Maiti, Roll Number 511EC407 hereby declare that this dissertation entitled

Analytical Modelling of Ultra-Wide Band Ground Penetrating Radar for Characterization

of Subsurface Media presents my original work carried out as a doctoral student of NIT

Rourkela and, to the best of my knowledge, contains no material previously published or

written by another person, nor any material presented by me for the award of any degree

or diploma of NIT Rourkela or any other institution. Any contribution made to this

research by others, with whom I have worked at NIT Rourkela or elsewhere, is explicitly

acknowledged in the dissertation. Works of other authors cited in this dissertation have

been duly acknowledged under the section “References”. I have also submitted my original

research records to the scrutiny committee for evaluation of my dissertation.

I am fully aware that in case of any non-compliance detected in future, the Senate

of NIT Rourkela may withdraw the degree awarded to me on the basis of the present

dissertation.

May 29, 2017

NIT RourkelaSubrata Maiti

Acknowledgment

I am blessed with the support and encouragement from many people towards the

completion of this dissertation. I like to express my gratitude toward them.

Firstly, I like to express my warmest gratitude to my supervisor, Prof. Sarat Ku-

mar Patra, for his valuable advice, and support all along the course of this work. His

encouragement to collaborate with other people and institution helped me a lot.

I wish to express my warmest gratitude to my co-supervisor, Prof. Amitabha Bhat-

tacharya, for his valuable ideas and introduction to scientific work. His encouragement

and trust in me sustained my research work through difficulties.

I like to say a big ‘thank you’ to Dr. Sebastien Lambot, whose work has inspired me

to pursue my research in the field of GPR modelling. His e-mail replies were helpful to

refine my work.

I owe sincere gratitude to Prof. Kamalakanta Mahapatra, HOD of ECE department

and DSC chairman for his continuous support, and encouragement. I also like to express

special thanks to other members of DSC Prof. Santanu Kumar Behera, Prof. Prasanna

Kumar Sahu, Prof. Somnath Maity for reviewing my work and their valuable comments.

I wish to express my sincere gratitude to Prof. Sunil Kumar Sarangi, Ex-Director of

our institute for his trust on me, support, inspiration and valuable advices for my research

work at NITR.

Special thanks to Prof. Poonam Singh, Prof. Sukadev Meher, Prof. Umesh Chandra

Pati, Prof. Tarun Kumar Dan for their valuable advices.

I like to express my heartiest thanks to my friends and colleagues Debiprasad Priyabrata

Acharya, Shrishailayya M. Hiremath, Ajeet Kumar Sahu, Lakshmi Prosad Roy, Siddharth

Deshmukh, Sougata Kar, Ayas Kanta Swain, Samit Ari, Upendra Kumar Sahoo, Santos

Kumar Das, Manish Okade, Santanu Sarkar, Nihar Ranjan Mishra, Pramod Kumar Ti-

wari, for their encouragement, beautiful company to make my staying at NITR enjoyable

and memorable.

Thanks to my brother in law Sourav Das for reviewing a part of the mathematical

derivation. Thanks to my student Rashmi Ranjan Nayak for his help to conduct GPR

testing at NITR laboratory.

Thanks to my lab mate in IIT Kharagpur Ajith, Mallikarjun, Rakesh, Elango, Kuldeep,

Aditya daa, Biswanath daa, Pranab, Hemanta, Madhududan, Rajdeep, Shaibal for their

beautiful company and support during staying at IIT Kgp and GPR testing.

Thanks to my friends Achyut, Himadri, Pranoy, Suman, Kaushik, Tapas, Narayan,

Kalipada, Soumen, Chandan, Saikat, Naru Gopal, Debi Prosad, Atanu, Sanjeet, Anikh,

Kunal Kisore, and many Swyamsevak friends located in different places for their encour-

aging words, advices, and very good company.

I convey my deepest gratitude to my parents Sri Achintya Kumar Maiti and Srimati

Namita Maiti for their love, support, and guidance all the time. They are the sole motiva-

tor for me to join the PhD program. Many thanks to my sisters Sonali, Rupali, Barnali,

Lipica, in laws Amaresh, Mrinmoy, Shubham, Shovan, brothers Debabrata, Soumitra,

Asim, sister in laws Soma, Maumita, uncles Sri Amiya Maiti and Sri Apurba Maiti, un-

ties Smt. Niyati Maiti, Smt. Kabita Maiti, maternal uncles Sri Bibekandan Maji, Sri

Sandip Maji, unties Smt. Hena Maji, Smt. Sritikana Maji, parent in laws Sri Rangalal

Das and Smt. Kabita Das, brother in law Saikat, and many other unmentioned members

of my greater family for their support, encouragements, and well wishes. A special men-

tion to my son Eshitva, and cousin Twitty for their cute smile and innocence. Finally, I

like to wish special thanks to my dearest wife Soumi for her understanding, compassion,

and love to support this work.

Subrata Maiti

Abstract

Ground penetrating radar (GPR) is becoming an attractive sensor for quantitative

reconstruction of subsurface media. The accuracy and time-efficiency of GPR detection

largely depends on the inverse modelling scheme used. In most of the cases, both accu-

racy and processing speed can’t be achieved together because of the inherent limitation of

modelling GPR signal propagation in complex media. Full wave models (FWMs) are most

promising approaches to characterize multilayered media. However, they are computa-

tionally inefficient due to the requirement of significant integration time over singularity.

In this work, an FWM is developed to model a monostatic GPR in far-field con-

figuration. The subsurface media is assumed to be a multilayered one with each layer

being homogeneous having a specific electromagnetic property. The synthetic data based

on simulation and experimental study have demonstrated that the proposed FWM is

accurate enough to describe the subsurface media electrically.

Then three modified plane wave models (MPWMs) are derived based on the analytical

solution of the FWM to achieve similar accuracy and better computational speed than the

FWMs. MPWM-2 is the most accurate, and MPWM-3 is quite versatile to find response

due to multilayered media. Rigorous analysis has been carried out to show the similarity

between the proposed MPWMs and FWMs with high correlation across a broad frequency

spectrum and large ranges of media parameters.

The model inversion is achieved by novel layer stripping (LS) techniques followed by

a gradient-based method. The proposed models are validated by correctly detecting a

normal water layer and validating their accuracy with an existing FWM in the literature.

The testing of layered media in laboratory environment demonstrates that the proposed

MPWMs are as accurate as existing FWMs and computationally more efficient. The pro-

posed integrated approach of GPR detection having superior speed, and similar accuracy

as FWMs are valuable alternative for many real-time GPR applications.

Keywords: Ground penetrating radar ; Green’s function; GPR modelling ; in-

verse scattering ; SFCW radar ; ultra wide band radar ; layered media.

Contents

Contents xiii

List of Figures xvii

List of Tables xix

List of Abbreviations xxi

List of Symbols xxiii

1 Introduction 1

1.1 Motivation and Aim for the Research . . . . . . . . . . . . . . . . . . . . . 2

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Forward Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Inversion of Model and Optimization Methods . . . . . . . . . . . 6

1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 GPR Principle and Modelling 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Classification of GPR systems . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 GPR System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Unambiguous Range . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

xiii

2.3.3 Bandwidth and Operating Frequency Range . . . . . . . . . . . . . 20

2.3.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Antennas for GPR Application . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Modelling Earth’s Subsurface . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.1 Electromagnetic Properties of Materials . . . . . . . . . . . . . . . 24

2.5.1.1 Properties of water . . . . . . . . . . . . . . . . . . . . . 24

2.5.1.2 Volumetric mixing model . . . . . . . . . . . . . . . . . . 25

2.5.2 Characteristics of Wave in a Medium . . . . . . . . . . . . . . . . . 26

2.5.3 Reducing Modelling Complexity of Subsurface Media . . . . . . . . 27

2.6 Wave Propagation in Multilayered Media . . . . . . . . . . . . . . . . . . 28

2.6.1 Maxwell’s Equation in Frequency Domain . . . . . . . . . . . . . . 29

2.6.2 Plane Wave Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6.2.1 Reflection of Plane Wave from a Half space . . . . . . . . 31

2.6.2.2 Reflection from a Multilayered media . . . . . . . . . . . 33

2.6.3 A Point Source on Top of a Multilayered Media . . . . . . . . . . . 34

2.6.3.1 Fourier Transform for Representation in Spectral domain 35

2.6.3.2 An Approach for Solution in Spectral Domain . . . . . . 35

2.6.3.3 Finding Reflected Field due to Multilayered Media . . . . 38

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Development of Full Wave Model 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Modelling Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Derivation of FWM Green’s Function . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Formulation of Green’s Function . . . . . . . . . . . . . . . . . . . 45

3.3.2 Deriving Green’s Function in Spectral Domain . . . . . . . . . . . 46

3.3.3 Converting Spectral Domain Green’s Function to Spatial Domain . 48

3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Numerical Computation of FWM Green’s Function . . . . . . . . . . . . . 52

3.5 Validation of Proposed FWM . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Development of Modified Plane Wave Models 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Derivation of MPWMs by Simplification of FWM . . . . . . . . . . . . . . 61

4.2.1 Simplification of FWM-2 for a Two-layered and Three-layered Media 62

4.2.2 Obtaining a Generalized Formula for Green’s Function . . . . . . . 65

4.2.3 Representation of Green’s Function by a Compact Formula . . . . 68

4.3 Comparative analysis of the MPWMs . . . . . . . . . . . . . . . . . . . . 70

4.4 Relation of Models with Friis Transmission Equation . . . . . . . . . . . . 74

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Inversion Strategy for the Proposed Models 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1 Test setup for Calibration . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.2 Far-field Distance of Antenna . . . . . . . . . . . . . . . . . . . . . 80

5.2.3 Theory of Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.4 Results of Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Model Inversion Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Accuracy and Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.1 Response Surface Plots . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.2 Effect of Antenna Height Measurement Error . . . . . . . . . . . . 88

5.4.2.1 Effect of Calibration Error on Model Inversion . . . . . . 88

5.4.2.2 Effect of Height Measurement Error on Model Inversion . 94

5.4.3 System Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.4 Summary of Accuracy and Stability Analysis . . . . . . . . . . . . 97

5.5 Layer Stripping Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5.1 Layer Stripping by Approximate Method (LSAM) . . . . . . . . . 98

5.5.2 Layer Stripping by Inversion Method (LSIM) . . . . . . . . . . . . 100

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Results of Laboratory Experiments 103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Test Setup and Experimental Environment . . . . . . . . . . . . . . . . . 105

6.3 Single-layered Water Testing . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4 Single-layered Sand Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Two-layered Media Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1 Experiment at IIT KGP . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.2 Experiment at NITR . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Conclusion 121

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2 Limitation of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3 Scope for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

References 127

Disseminations of Work 138

Author’s Biography 141

List of Figures

2.1 General block diagram of a GPR system. . . . . . . . . . . . . . . . . . . 16

2.2 Reflection and transmission of plane wave at an interface. . . . . . . . . . 32

2.3 N -layered media with description of the coordinate system. . . . . . . . . 33

3.1 (a) Linear transfer function model (LTFM) representing the VNA-antenna-

multilayered medium system [1]. (b) The virtual source and receiver point

at antenna phase center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Plot of the spectral domain Green’s functions at 1 GHz frequency. . . . . 53

3.3 Comparison between FWM-1 and FWM-2 for ϵr = 81 and σf = 0 in σ− h

plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Green’s function plots for FWMs . . . . . . . . . . . . . . . . . . . . . . . 56

4.1 Example 2nd order reflections from interface z2 and z3 . . . . . . . . . . . 66

4.2 Changes of MPWM-2 Green’s function with order of reflection (No). . . . 67

4.3 Example three-layered media. . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Comparison between MPWMs and FWM-2 for ϵr = 81 and σf = 0 in σ−h

plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1 Block diagram of GPR calibration setup. . . . . . . . . . . . . . . . . . . . 80

5.2 Linear transfer functions (LTFs) and Green’s functions Gxx (ω) extracted

by calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xvii

5.3 Response surface plots (MPWM-1). . . . . . . . . . . . . . . . . . . . . . 89



5.6 Effect of uniform measurement error (UME) on Green’s function. . . . . . 92

5.7 Effect of uniform measurement error (UME) on model inversion. . . . . . 93

5.8 Effect of antenna height measurement error on model inversion (Sh = 0 mm). 95



6.1 Block Diagram of GPR Testing Setup. . . . . . . . . . . . . . . . . . . . . 105

6.2 Experimental setup at IIT KGP . . . . . . . . . . . . . . . . . . . . . . . 106

6.3 Experimental setup at NITR . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Compare measured and modeled Green’s functions for single-layered water. 109

6.5 Compare measured and modeled Green’s functions for single-layered sand. 113

6.6 Compare measured and modeled Green’s functions for two-layered media

(tested at IIT KGP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.7 Compare measured and modeled Green’s functions for two-layered media

(tested at NITR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

List of Tables

3.1 Comparison between FWM-1 and FWM-2 Applied to 1L Media . . . . . . 55

4.1 Comparative Analysis of Models Applied to 1L Media . . . . . . . . . . . 71

4.2 Computational Efficiency of Models Applied to 1L Media . . . . . . . . . 71

4.3 Comparative Analysis of Models Applied to 2L Media . . . . . . . . . . . 74

4.4 Computational Efficiency of Models Applied to 2L Media . . . . . . . . . 74

6.1 Water layer parameters estimated by different models . . . . . . . . . . . 110

6.2 Sand layer parameters estimated by different modelling methods . . . . . 112

6.3 Two-layered media parameters estimated by different modelling methods

(IIT KGP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4 Two-layered media parameters estimated by different modelling methods

(NITR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xix

List of Abbreviations

DR Dynamic Range

IP Intercept Point

MDS Minimum Detectable Signal

NF Noise Figure

SFDR Spur Free Dynamic Range

1-D One-dimensional

2-D Two-dimensional

3-D Three-dimensional

ADC Analog to Digital Conversion

AGC Automatic Gain Control

AGC Automatic Gain Control

CG Conjugate Gradient

EFIE Electric Field Integral Equation

EM Electromagnetic

FDTD Finite Difference Time Domain

FEM Finite Element Method

FFT Fast Fourier Transform

FWM Full Wave Model

GI Galvanized Iron

GPR Ground Penetrating Radar

IF Intermediate Frequency

xxi

IFFT Inverse Fast Fourier Transform

LNA Low Noise Amplifier

LS Layer Stripping

LTF Linear Transfer Function

LTI Linear Time-invariant

MFIE Magnetic

Field Integral Equation

MoM Method of Moments

MPWM Modified Plane Wave Model

NDT Non-destructive Testing

PEC Perfect Electric Conductor

PRF Pulse Repetition Frequency

PRI Pulse Repetition Interval

PWM Plane Wave Model

RADAR Radio Detection and Ranging

RCS Radar Cross Section

SFCW Stepped Frequency Continuous Wave

SNR Signal to Noise Ratio

TE Transverse Electric

TM Transverse Magnetic

UWB Ultra Wide Band

VNA Vector Network Analyzer

List of Symbols

a,A Scalar

a, A Vector

γ Propagation constant

α Attenuation constant

β Phase constant

c Velocity of EM wave in vacuam

v Velocity of EM wave in a media

λ Wave length of EM wave

ϵ Absolute permittivity

ϵr Relative permittivity

µ Absolute permeability

µr Relative permeability

σ Conductivity

f frequency

ω Angular frequency

i imaginary unit, i =√−1

J Electric current density

M Magnetic current density

E Electric field

H Magnetic field

xxiii

D Electric flux density

B Magnetic flux density

G Green’s function

r(x, y, z) Point in a Cartesian coordinate system

r(ρ, ϕ, z) Point in a Cylindrical coordinate system

n Normal to a surface

t time

τ pulse width

T Temperature in degree kelvin

CHAPTER1Introduction

Contents

1.1 Motivation and Aim for the Research . . . . . . . . . . . . . . 2

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Forward Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Inversion of Model and Optimization Methods . . . . . . . . . . 6

1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 11

1

1.1 Motivation and Aim for the Research

This chapter begins with a brief introduction on GPR modelling, its importance and

challenges in the subsurface detection and surveying. Subsequently, a literature review on

GPR modelling, its inversion schemes, and factors influencing its accuracy are presented.

This discussion leads to the identification of major problems in the field of GPR modelling.

Then objective and scope of the work are highlighted. The chapter ends with a brief details

of the thesis organization.

1.1 Motivation and Aim for the Research

Ground penetrating radar (GPR) is a noninvasive method for surveying shallow sub-

surface. It uses electromagnetic (EM) wave to reconstruct the properties of subsurface

media and identify the presence of objects, its location, and shape. The propagation of

EM wave in subsurface media can be completely described by Maxwell’s equations and its

constitutive relations proposed a long time back. However, finding a solution of Maxwell’s

equations applied to an inhomogeneous media bounded by a complex boundary is a dif-

ficult problem. The complexity further increases when the media/ object is placed very

close to the transmitter-receiver antennas. The high attenuation of an electromagnetic

wave in ground media, the changing spatial and temporal electromagnetic behavior of the

ground with environmental conditions, the electromagnetic similarities of physically dif-

ferent objects make GPR detection and estimation a challenging problem. The data sets

obtained by GPR are very complex, application specific and require a lot of expertise to

interpret correctly. The understanding of EM wave propagation in heterogeneous media

and representing it by a compact mathematical model are the key difficulties for accurate

interpretation of the GPR data set.

Various types of analytical and numerical models have been developed to simulate

signal propagation in the GPR environment. They can be classified in two categories:

approximate models and full wave models (FWMs). Approximate models are developed

for fast detection. However, they lack accuracy. The primary focus of FWMs is to im-

prove the accuracy of the GPR detection. FWMs can be implemented by numerical,

analytical, and hybrid techniques. Numerical techniques are versatile in nature, and they

make little assumptions on the type of structure. This results in high accuracy at the

2

1.2 Literature Survey

cost of enormous computational complexity. Analytical models on the other hand, are

problem specific. In these models, the complexity of 3-D Maxwell’s equations are reduced

by using various approximation for antenna, media, and objects under investigation. The

hybrid techniques exploit the advantages of both and optimize the accuracy and timing

requirements. A good GPR forward model is an extremely valuable tool for enhancing

knowledge in various associated research areas of GPR. Inversion of a model for prac-

tical data is a more difficult method which requires advanced mathematical techniques,

and engineering processes. An accurate and time efficient forward model gives a lot of

advantage in implementation of real-time inversion process.

The success of an inverse modelling scheme significantly depends on its accuracy,

computational speed, and versatility to apply for different antennas and subsurface con-

figurations. Therefore, the primary goal of the research in GPR modelling is to achieve

good accuracy and computational speed with commonly available resources so that many

real-time applications of this sensor become feasible.


GPR has become one of the most popular noninvasive techniques for quantitative esti-

mation of media properties and real-time, high-resolution imaging of sallow subsurface

objects. Unfortunately, the interpretation and use of GPR data are difficult owing to

the heterogeneities in the host media. It is practically difficult to make a versatile model

that can incorporate various types of antennas and subsurface profiles. Often there are

assumptions to simplify the problem to improve the computational efficiency of GPR de-

tection at the cost of accuracy. In this section, we present a review of various GPR models,

and their inversion schemes applied for quantitative estimation of media parameters.

1.2.1 Forward Modelling

The amplitude and phase of the GPR signal depend on the antenna orientation, and the

path traversed by the signal. For accurate interpretation of GPR data, it is necessary to

account for the antenna radiation pattern, and its vector characteristics [2]. This results

in increasing complexity in modelling antenna-subsurface media system.

3


The numerical methods have received significant interest from the researchers since

early 1990s. With the rapid enhancement of computational resources, numerical modeling

of GPR has been improved to account for higher dimensionality, complex environments,

increased size, and more sophistication [3]. The numerical methods are much versatile for

the types of structures they can handle since they make little assumptions on geometry,

and material composition. The finite difference time domain (FDTD) [4–8], finite element

method (FEM) [9, 10], method of moment (MoM) [11, 12] are popularly used for GPR

system modeling. The gprMax [4,5] is quite popular software for simulating several types

of GPR environment with popular antennas. This software is also useful to simulate com-

plex GPR scenario, test and validate new analytical models for their accuracy [13], useful

for testing new signal and image processing techniques [14,15], and so on. The drawback

of numerical techniques is that they often suffer from low computational efficiency [16].

Therefore, they are not suitable for the real-time applications.

Analytical modeling is achieved with some simplifying hypotheses on the nature of

the structure. This results in problem specific but efficient, accurate, and often robust

solution. Most approximate methods assume subsurface as homogeneous media, and

wave propagation to be a plane wave in nature. The common reflection method [16–18]

is simple, and most commonly used technique applied for approximate modelling of GPR

signal propagation. It computes the reflection, and transmission coefficients based on the

principle of plane wave propagation in media. For many years, the common midpoint

(CMP) [19–22] has been popularly used for GPR detection. However, it suffers from

processing delay as it requires several traces for single profile measurement. Several

researchers have applied the amplitude-variation-with-offset (AVO) dependent reflectivity

analysis [23] or dispersive amplitude and phase versus offset (DAPVO) [24] for thin bed

characterization based on common-midpoint (CMP) geometry acquisition. Since, only

part of the information i.e. mostly travel time is considered, these methods are not

suitable for accurate quantitative reconstruction of media properties [25].

The spectral inversion method [18, 26, 27] is derived based on the spectral analysis

of the reflection coefficient obtained for a layered media by applying common reflection

method. This has originated from a seismic method which uses spectral decomposition

4


technique, and apriori knowledge of the media to improve images of thin layers with

a thickness lower than a quarter wavelength. Puryear and Castagna used this method

successfully to find layer thicknesses [26]. Zhong-lai Huang and Jianzhong Zhang [18] have

further developed the method to apply for estimating electrical properties of a highway

pavement. This method has also been applied successfully for estimating layered media

parameters with low conductivity profile [18].

Among various analytical modelling techniques, electric field integral equation (EFIE)

and magnetic field integral equation (MFIE) based formulations are most accurate tech-

niques [28–32]. In these techniques, the scattered fields are related to electric or magnetic

dipoles at the source with the integral equation. Gentili and Umberto Spagnolini [31] have

modeled a horn antenna in terms of an array of frequency-independent source dipoles and

then calculated the returned electric field by solving Maxwell’s equation propagating

through multilayered media. However, the antenna and media coupling effects are not

accounted to relate the subsurface media response with the measured backscattered fields.

Until recently, the development of FWMs [1, 33, 34] for real-time applications was

limited. Solving Maxwell’s equation for wave propagation in a three dimensional (3-D)

inhomogeneous media bounded by complex boundaries is a difficult task. However, it is

possible to simplify an FWM for a typical GPR scenario by applying valid assumptions

and approximation. Lambot et al. [1] have proposed an FWM for detecting layered

media using monostatic off-ground SFCW GPR. In this scheme, all the effects of antenna,

and its interaction with ground subsurface are modeled by a linear transfer function

model, and response due to multilayered media is modeled by 3-D Green’s function.

This scheme also has been successfully applied for estimating and monitoring soil water

contents [35,36]. Various theoretical and experimental analysis [37,38] have demonstrated

that the proposed FWM is one of the most accurate schemes for characterization of layered

media in the far-field as well in the near-field conditions. Kalogeropoulos et al. [34] have

proposed an FWM for bistatic off-ground GPR. This method is based on finding the

shape and amplitude of an effective wavelet. The method has been successfully applied

to assess the chloride and moisture content in concrete. The computational complexity

of these FWMs is still high compared to other simplified models based on plane wave

5


approximation. Moreover, applying FWM for near-field application is much more difficult

as it involved effective modelling of antenna radiation characteristics. As an example

Lambot et al. [38] have extended the far-field model to near-field one by representing

antenna in terms of a set of point dipoles with increasing complexity of calibration and

computation [39].

Layer stripping (LS) has been used for many years for approximate and fast calculation

of electrical parameters of the layered media [20, 34, 40–42]. This method is suitable for

high-speed data processing [43, 44]. Various research works [45, 46] have demonstrated

that the LS is very much suitable for inspecting road pavements which composed of fixed

number layers with well-known material and geometric characteristics. In this method,

GPR processing is done in the time domain for reconstruction of each layer in step-by-step

starting from the top layer. The amplitudes and timing information of returned pulses

are used to reconstruct the electrical properties of the corresponding layer. In most of

the cases, there are common assumptions like plane wave propagation and neglecting

the presence of multiple reflections. Main drawback of LS is the recursive formulation

which causes accumulation of error. It also yields a significant error in media parameter

estimation [42] in the presence of lossy dispersive media.

One of the key driving factors for the development of accurate and efficient forward

models is the difficulty in processing the GPR data which is application specific, com-

putationally intensive, and under the exclusive domain of expert users. Therefore, an

accurate forward model is a very useful tool for correct interpretation of GPR data, and

also useful for many other prospective areas of GPR research.

1.2.2 Inversion of Model and Optimization Methods

Inversion is a process of reconstructing the model parameters based on the measurement

scattered field data, collected outside the probed volume and excited by known source [47].

An electromagnetic forward model calculates the scattered field based on the knowledge

of subsurface scenario. Whereas, the inversion process consists of reconstructing the

underground scenario based on the measurement response. In GPR context, successful

model inversion is an important objective to reconstruct the electromagnetic parameters

6


of media under investigation, or to detect, locate, and image an object buried in the

media. For imaging applications, the media property has to be known at first. Mostly

the inverse problems are ill-posed [48], having more difficulty to solve compared to the

forward modelling. The ill-posedness leads to the impossibility to retrieve all the details of

the subsurface scenario [48, 49]. Moreover, the inverse problem becomes nonlinear, when

the forward problem is nonlinear. Linearity is generally defined by the superposition

theorem

F (ax1 + bx2) = aF (x1) + bF (x2) (1.1)

where a, b are scalar quantities. In reality, the EM model inversion problem is both ill-

posed and nonlinear. Consequently, most of the model inversion process is affected by

the problems due to the nonuniqueness, and existence of local minima [1,50].

The inversion process starts with the GPR system calibration followed by measure-

ment of the Green’s function i.e. the response due to subsurface media. The media pa-

rameters are inverted by optimizing an objective function generally defined as to compare

the measured and modelled Green’s function. The accuracy and computational efficiency

of model inversion depend on the complexity of forward model, objective function def-

inition, presence of noise and interference, etc. In the presence of noise, the inversion

process become more challenging. Therefore, there must be a mechanism to filter out

the noise and clutter from the GPR measurement data. In this section, a brief review of

various model inversion schemes applied for quantitative estimation of media parameters

is presented.

The model-based inversion schemes have shown better accuracy compared to direct

methods like layer stripping [42], analytical approximate techniques [51, 52] for inverting

electromagnetic parameters of the media when measurement data is inaccurate and af-

fected by noise. The model-based method try to match the measurement data with the

synthetic data, and there by find the optimum model parameters. However, they are time-

inefficient due to the requirement of time-consuming optimization schemes. The optimiza-

tion schemes are classified into local and global techniques. Examples of local optimization

schemes include quasi-Newton techniques [53], Gauss-Newton techniques [54], gradient-

based methods [41,44,55], Neadler-Mead techniques [1]. The methods are relatively fast,

7


require previous knowledge of the media, and may lead to inaccuracy while being trapped

in a local minima. Thus, they are recommended when sufficient amount of prior informa-

tion is available on the media. Global techniques like genetic algorithm (GA) [56,57], par-

ticle swarm optimization (PSO) [58–60], stochastic hill-climbing algorithm (SHA) [18,27],

global multilevel coordinate search (GMCS) [1,38], direct search [25,61] are popular tech-

niques which have been used successfully for inverting electromagnetic parameters. While

the global search based methods are successful in reducing the false alarm rate [62, 63],

the process is slower compared to the local optimization techniques. Both deterministic

and stochastic optimization methods are computationally burdening while applying in

global optimization for nonlinear models [47]. The stochastic techniques like GA, PSO,

SHA have to be rigorously verified by the statistical yield analysis so as to get reliable

result.

The modern trend is to apply hybrid techniques with a combination of global search

and local technique to enhance the speed of optimization. For layered media application,

layer stripping (LS) can be applied for pre-estimation of parameters. Based on the quality

of GPR data and outcome of LS, it is possible to apply a local optimization technique or

global one by narrowing down the search boundary. This, in turn, reduces the inversion

time significantly. It is also possible to enhance the speed of global optimization by

implementing intelligent search by using the knowledge of material properties.

For invertibility of a model, it is necessary to check the uniqueness of solution in ideal

case (synthetic model) as well in practical case (in the noisy scenario). There must be

a unique solution for the media parameters under investigation for the optimum point

of the objective function. It is also very important to analyze the model stability on

large variations of model parameters, measurement and calibration errors, medium at-

tenuation, system noise, external interference, and variation of electronics with changing

atmospheric conditions and repeated measurements [64–67]. An elaborate analysis on the

effect of calibration error and instability of instruments has been studied for a monostatic

SFCW GPR system based VNA platform, and a commercial TD GPR system [64]. It

is reported that the inaccuracy of calibration has a severe impact on the accuracy of

extracted Green’s function, and its effect increases with increasing frequency. The effect

8

1.3 Problem Statement

of calibration error is more for the directive horn antenna compared to that of the Bow-

tie antenna. The system stability over repeated measurements can be affected by cable

bending, variations and drifts in instruments, loss of instrument calibration, etc. The

analysis and quantification of uncertainties originated due to error in measurement and

calibration process have been investigated by Patriarca et al. [65]. There are two kinds

of errors, i.e. gross error and random error, which can add during the process of cali-

bration and measurement. The source of gross errors like presence of external scattering

objects, misalignment of antenna, and inaccuracy in measurement setup can be controlled

by improving the measurement setup and environment. On the contrary, the sources of

random errors like instrument drift, imprecision due to analog-to-digital (A/D) conver-

sion, thermal noise can’t be controlled so easily. It’s observed that the uncertainties in

GPR estimation can be significantly reduced by averaging set of GPR calibration param-

eters, following specific measurement protocol, accurate instrument calibration, accurate

antenna height measurement, and so on [65].

1.3 Problem Statement

The literature survey has revealed important problems in the area of GPR modelling.

They are described below.

� The approximate models based on plane wave assumption help faster implementa-

tion of GPR system at the cost of accuracy. Thus, these models have little use for

quantitative estimation of subsurface media.

� The numerical implementations of FWM provide accurate solution for EM wave

propagation in a heterogeneous media for a large variety of GPR scenarios. However,

their applications are limited for real-time scenarios because of large computational

complexity.

� The analytical FWMs require simplifying assumptions in order to apply it for a

practical GPR application. So far, significant work has been done to implement

FWM for monostatic far-field GPR configuration, resulting in tremendous accuracy

to describe GPR data. However, they are still time-inefficient compared to the other

9

1.4 Objectives and Scope

simplified models based on plane wave approximation. Therefore, achieving accu-

racy and computational efficiency simultaneously in GPR modelling is a challenging

task.

� Inversion of EMmodel is further challenging because of nonlinearity and ill-posedness.

Pre-estimation of parameters helps to apply the optimization efficiently. However,

pre-estimation of media parameters is a challenging task to realize.

� Noise and errors during GPR measurement introduce instability in GPR system,

thereby cause uncertainty in the detection and estimation. Testing and verifying

accuracy and stability of a model are difficult tasks because of the existence of

deterministic and random error between actual and predicted scenario.

1.4 Objectives and Scope

The most fundamental aspects of an electromagnetic modelling is, how accurately it can

account for the three-dimensional (3-D) variations of media and its complex boundary

conditions. Most of the models suffer from inaccuracy because of the unrealistic as-

sumptions and simplifications. In this respect FWMs are very promising to achieve good

accuracy. However, they suffer from inefficiency due to the large time requirement for

computation. Therefore, achieving accuracy and computation speed simultaneously is a

difficult task. Accordingly, the objective of this research work has been to advance the

state of analytical model so that both, accuracy and time efficiency are achieved simulta-

neously for a class of problems. To achieve this goal, the following specific objectives are

set with clearly defined boundaries.

� To develop analytical forward models, which are accurate and computationally ef-

ficient for monostatic far-field GPR configuration.

� To develop an inversion method corresponding to the proposed forward model for

efficient quantitative reconstruction of layered media.

� To formulate a suitable technique for pre-estimation of media parameters.

10

1.5 Thesis Organization

� To validate the proposed models through laboratory testing using well-known media,

and also to compare their performances with existing model.

� To perform the accuracy and stability analysis of models in presence of various

sources of uncertainties.


The thesis compromises of seven chapters. Following this chapter on introduction, the

remaining thesis is organized as follows:

Chapter 2 discusses the principle of time domain and frequency domain GPRs. Im-

portant system parameters are described, and are related to the qualitative performance

specifications. This is followed by a discussion on characteristics of electromagnetic wave

interacting with heterogeneous subsurface media.

Chapter 3 proposes an FWM which can describe monostatic far field GPR data

accurately. Its performance in terms of accuracy and computational speed is compared

with an well-established FWM in the literature.

Chapter 4 proposes three time-efficient models named as MPWM-1, MPWM-2, and

MPWM-3 which are derived based on the analytical solution of FWM. A comprehensive

analysis is carried out to compare the accuracy and computational efficiency of MPWMs

with FWMs. Finally, promises of new models are highlighted.

Chapter 5 focuses on inversion approach for the model to estimate the layered media

parameters. The chapter begins with discussion on the GPR calibration process. Then,

objective function is defined for model inversion. A discussion is presented to illustrate the

stability of models with various types of uncertainties during calibration and measurement

process. Finally, two improved techniques of layer stripping are proposed for approximate

estimation of the layered media parameters.

Chapter 6 presents results of experiments with practical laboratory measurement

data. The models are validated by characterizing a well-known media, and comparing

the inversion outcomes with an existing model in literature. The performances of MPWMs

are compared with the FWMs. The outcome of experiments is summarized.

11


Chapter 7 summarizes the thesis listing the contributions from all the chapters,

discusses limitation of the work, and provides directions for future research works.

12

CHAPTER2GPR Principle and Modelling

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Classification of GPR systems . . . . . . . . . . . . . . . . . . . 15

2.3 GPR System Parameters . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Unambiguous Range . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Bandwidth and Operating Frequency Range . . . . . . . . . . . . 20

2.3.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Antennas for GPR Application . . . . . . . . . . . . . . . . . . 22

2.5 Modelling Earth’s Subsurface . . . . . . . . . . . . . . . . . . . 23

2.5.1 Electromagnetic Properties of Materials . . . . . . . . . . . . . . 24

2.5.2 Characteristics of Wave in a Medium . . . . . . . . . . . . . . . . 26

2.5.3 Reducing Modelling Complexity of Subsurface Media . . . . . . . 27

2.6 Wave Propagation in Multilayered Media . . . . . . . . . . . . 28

2.6.1 Maxwell’s Equation in Frequency Domain . . . . . . . . . . . . . 29

2.6.2 Plane Wave Solution . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6.3 A Point Source on Top of a Multilayered Media . . . . . . . . . . 34

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

13

2.1 Introduction

This chapter presents an overview of GPR principle, practical features of GPR sys-

tems, electromagnetic (EM) properties of materials, and interaction of waves with inho-

mogeneous media. Various terms and definitions that are used throughout the thesis are

also explained here. The system parameters are defined, and their limitations due to

component specifications are also discussed. Then EM properties of subsurface media is

discussed with a focus to understand properties of soil and water, which are very common

in GPR applications. Subsequently, a brief discussion is presented to explain EM wave

propagation in inhomogeneous media. The chapter ends with a brief summary.

2.1 Introduction

GPR which is a part of the family called nondestructive testing (NDT) methods, is also

known as surface penetrating radar (SPR). It is designed primarily to investigate the

earth’s shallow subsurface. GPR uses conventional radar principle to find location, shape,

property of the object buried underground. Though, the 1st use of GPR was reported

in 1910 in a patent by Leimbach and Lowy [16], it received wide attention during the

1970s. During this time, researchers used EM signal to measure the depth of ice [68],

investigate the property of minerals [69], probing of rock and coal [70, 71], and so on.

Annan [72] provided a detailed history of GPR developments from 1950 to 2000. Today

there are numbers of commercial GPR solutions available for inspection of road, bridge

and concrete structure; detection and mapping of buried utilities, from GPR developers

like MALA, GSSI, Sensor & Software Inc, US Radar Inc., UTSI Electronics Ltd., IDS

GeoRadar, Groundradar Inc., etc. With the advancement of radar hardware, more under-

standing of signal propagation in subsurface media, the prospective applications are also

growing. Various research works over past 50 years have demonstrated that the GPR can

be applied for numerous applications like archaeology, agriculture, civil engineering, geo-

physical characterization, mine engineering, landmine detection, planetary exploration,

and so on [16,47,73–76].

Understanding of EM wave propagation and its interaction with the media is very

important to interpret the GPR data set. Based on the media property and application

requirements, various design parameters like range, resolution, dynamic range, scanning

14

2.2 Classification of GPR systems

speed are defined. The various components of a GPR system like antenna, RF trans-

mitter, receiver, ADC have their limitation on performance specification related to the

operating bandwidth, IF bandwidth, sensitivity, spurious response, maximum power, etc.

The designing of a GPR system is a challenging task with the limitation of component

specifications and stringent application requirements.

A good amount of GPR research has been dedicated to the physical characterization

of subsurface media. This is accomplished by quantitative reconstruction of the EM

properties of media, and then mapping those parameters into the physical properties of

media. Its primary purpose is to characterize the media for various applications related

to civil, structural quality, mechanical strength, chemical composition, quality and purity

of materials, soil water contents, environment pollution, etc. This is also an essential part

for imaging, detection, and localization of buried target. Unless the channel (i.e. media)

is characterized with sufficient accuracy, the identification of target can’t take place.

A very good knowledge of EM theory is essential to understand wave propagation in

inhomogeneous subsurface media. Most of the GPRmodels suffer from inaccuracy because

of strongly simplifying assumptions on properties of media, EM wave, and its source, and

receiver. Therefore, a good foundation on EM theory applied to GPR problem is very

essential for accurate and efficient implementation of a GPR system.

This chapter introduces basic working principle of GPR. Various system parameters,

and their impact on radar performances are also discussed. In addition, a brief discussion

is presented on types of antennas and their performance parameters influencing GPR

operation. Following this, a discussion on interaction of EM wave with complex media

highlights the difficulty of modelling them, and explains approaches for simplifying their

models. Subsequently, a discussion is presented on solution of wave propagation through

a multilayered media. Finally, a summary is presented at the end.


A typical GPR system consists of transmitter, receiver and signal processing and control

unit as shown in Figure 2.1. The current trend in GPR is to use ultra-wide band (UWB)

for its operation. Based on the principle of operation, the UWB GPR systems can be

15


classified into following categories.

1. Time domain GPR or pulsed GPR

2. Stepped frequency continuous wave (SFCW) GPR or synthetic pulse GPR

Figure 2.1: General block diagram of a GPR system.

The pulsed GPR sends a pulse at a prescribed pulse repetition frequency (PRF) into

the ground and then detects the reflected pulses. Contrary to this, an SFCW GPR

decomposes the electromagnetic pulse into frequency domain and radiates the discrete

frequencies sequentially. Then it receives the train of single frequency scattered signals. If

we assume the soil, and buried target response is a response due to a linear time-invariant

(LTI) system, the pulsed and SFCW GPRs become theoretically equivalent [47]. Today

majority of the commercial GPR systems are of pulsed GPR type. The pulsed GPR

has the advantage of simplicity and low-cost implementation. However, it suffers from

the undesired ringing effect, limited power of transmitted pulse, and resolution limited

by pulse width. There are developments of frequency domain GPRs reported in various

literatures [77–80]. The SFCW GPR has the advantage of high signal to noise ratio

(SNR) because a single frequency is illuminated by a non-overlapping time period. It has

the flexibility to achieve high resolution by implementing UWB transmitter and receiver,

16

2.3 GPR System Parameters

and high dynamic range by restricting noise floor by setting intermediate frequency (IF)

bandwidth. Moreover, frequency domain allows faster derivation of Maxwell’s equation as

convolution is not required to derive the composite transfer function for the GPR system.

The SFCW radar supports the features like frequency selective processing, time-frequency

analysis, polarimetric processing, etc. In other words, research on SFCW GPR is moving

towards a software GPR [79]. Instead of all these advantages, the SFCW GPR suffers

from high scanning time and range side lobes in time domain processing [81].

The GPR system can be classified into the monostatic (single antenna) and the bistatic

(two antennas) based on the system configurations. Both the configurations have their

advantages and disadvantages. The depth-resolution degrades as the separation of bistatic

antennas increases and target becomes closer to the antenna system. Monostatic GPR

system is simple to implement. However, they are not suitable for CMP survey method

which is popularly used for GPR imaging. Again the antenna can be placed touching the

ground for higher penetration depth, alternatively it can be placed off the ground. The

monostatic off ground radar has the advantage of achieving high scanning speed [31] with

reduced depth of penetration.


2.3.1 Unambiguous Range

The unambiguous range of GPR is limited by available transmitter power and receiver

sensitivity. The GPR system design parameters are specified based on required range of

the target. Often it’s a limitation of the transmitter (power amplifier spec) and receiver

(sensitivity), which finally restrict the radar range. For a radar system realized by aperture

antennas, the received power (Pr) due to a target with an effective radar cross section

(RCS) σt located in far-field of the antenna can be calculated by the Friis equation:

Pr =PtGtGrλ

2σt(4π)3D4L

(2.1)

where, Gt and Gr are the gains of transmitter and receiver antennas, D is the distance

of the target, and L is the propagation and other system losses together. The loss L is

17


due to the several factors like antenna efficiency, antenna mismatch, attenuation loss in

the material, etc. The minimum power (Prmin) which can be detected by the receiver is

given by

Prmin = FsyskTB(SNR)min (2.2)

where Fsys is the radio frequency (RF) receiver system noise figure, (SNR)min is the

minimum SNR require at the demodulator, kT is the thermal noise floor expressed in

mW/Hz. Thus, the radar unambiguous range Dmax for a point source and far-field

configuration can be expressed as

Dmax =

[PtGtGrλ

2σt(4π)3PrminL

] 14

. (2.3)

For pulse radar application, the Dmax is also limited by the pulse repetition interval (PRI)

τPRI.

Dmax =vτPRI

2(2.4)

where v is the speed of EM wave in the media. For SFCW radar,

Dmax =v(N − 1)

2B=

v

2∆f(2.5)

where B is the total bandwidth, N is the number of frequency steps, ∆f is the frequency

step. When an SFCW radar is also gated, the radar range depends on the gated repetition

interval (GRI) similar to the PRI in pulsed radar.

2.3.2 Dynamic Range

The dynamic range (DR) is the useful signal level range the GPR receiver can process

with a particular information quality. The DR may be defined as the difference in power

level between the input 1-dB compression point and the noise floor of the receiver system.

DR = P1dB − PMDS (2.6)

18


where P1dB is the 1-dB compression point in dBm and PMDS is the minimum detectable

signal (MDS) of the receiver given by following relation:

PMDS = −174 + 10log(B) + NFSYS (2.7)

where, NFSYS is the noise figure of the receiver system. Thermal noise calculated at a

reference of 290◦K is -174 dBm/Hz. Sometimes, the spurious-free dynamic range (SFDR)

has got more significance than the normal linear dynamic range. SFDR is the range where

the receiver’s spurious response are with in tolerable limit. It is limited by MDS and the

maximum input power for which inter-modulation distortion becomes unacceptable.

SFDR =2

3[IP3O − PMDS −GSYS]

=2

3[IP3O + 174dB− 10log(B)−NFSYS −GSYS] (2.8)

where IP3O is the third order output intercept point of the receiver. The system dy-

namic range is also limited by the sampling dynamic range. The sampling dynamic range

depends on the number of bits in ADC as given below:

DRADC = 20log(2N ) (2.9)

where N is the number of bits in ADC. A 16 bit ADC will have theoretically 96 dB of

sampling dynamic range. The maximum and minimum range of the target is determined

by the radar system dynamic range. This can be enhanced by using an amplifier at the

transmitter and a good low noise amplifier (LNA) followed by an automatic gain control

(AGC) [82] in the receiver chain. There are many other factors which influence the actual

dynamic range of a GPR receiver.

The SFCW GPR can achieve excellent dynamic range by reducing MDS by lowering

IF bandwidth of the receiver. However, this benefit can be achieved in frequency domain

processing. While converting the GPR data in time domain, the actual dynamic range is

affected by the range side-lobes generated due to inverse fast Fourier transform (IFFT)

[78, 81]. By selecting appropriate synthetic pulse like Hann window, the dynamic range

19


of radar can be significantly improved at the cost of reduced resolution for a receiver

with fixed ADC. S E. Hamran et al. [78] has demonstrated that the synthetic pulse radar

(SFCW radar) has potential to achieve a 40 dB higher dynamic range compared with an

equivalent impulse radar. Further, by using a higher transmitter power and operating in

range/ time gated mode, the SFCW radar can achieve a theoretical dynamic range of 200

dB or higher.

2.3.3 Bandwidth and Operating Frequency Range

The choice of center frequency and bandwidth of a GPR system is an important issue, and

it depends primarily on the type of application. For each application, a set of constraints

can be addressed by choosing appropriate frequency band. The parameters influencing

the frequency range are: size of the object, required depth resolution, maximal penetra-

tion depth, and properties of the media. The GPR can achieve better resolution when

the bandwidth of system is high. It can support more depth of penetration when the

frequency of operation is very low, i.e. in the range of few MHz. The attenuation is a

function of electrical loss and scattering loss. Both the losses increase with increasing fre-

quency. The operating frequency f0 is chosen based on desired depth of penetration. The

bandwidth has a direct relation with the radar resolution. The need for good resolution

has led to the development of Ultra-wide band (UWB) GPR. The fractional bandwidth

in this case can be larger than 100%. The fractional bandwidth is defined by ratio of

the center frequency (fc) to the total band width (B). According to the U.S. Federal

Communications Commission (FCC), UWB refers to radio technology with a bandwidth

exceeding the lesser of 500 MHz or 20% of the arithmetic center frequency. The band

width (B) for pulse GPR is defined as inverse of the pulse duration (τp) i.e.

B =1

τp. (2.10)

For SFCW GPR the same is defined as

B = fh − fl (2.11)

20


where fh is the highest frequency of operation and fl is the lowest frequency of operation.

2.3.4 Resolution

GPR resolution defined in the direction of range is called as range resolution, and in

the perpendicular to range is called as cross-range resolution. Range resolution (∆D) of

radar is defined as the minimum resolvable separation distance between two nearby point

targets. This is defined by Rayleigh’s criterion [83] as following [84]:

∆D =v

2B=

vτp2

(2.12)

where v is the propagation velocity, and B, τp are respectively, the bandwidth and time

duration of the pulse. This is calculated based on -4 dB time width of main lobe for

the sinc pulse (rectangular in frequency domain). This theoretical resolution value is

affected by the frequency selective dispersion of transmitter, receiver, ground and target.

In another approach, the worst case range resolution (∆D) is defined by considering half

power (-3 dB in power)/ half width (-6 dB in voltage) [85] as

∆D =1.39v

2B. (2.13)

This is an empirically derived formula. The actual resolution can be better than this

value. According to another definition [73], the received pulses should be separated by

at least half the time duration of the pulse. Based on this logic, theoretical limit of the

range resolution is

∆D =vτp4

. (2.14)

According to the same logic, the cross-range resolution, also known as the lateral resolu-

tion (∆l) depends on the velocity (v), pulse width (τp), and distance (D) of the target

from antennas [73]. The theoretical limit of this is

∆l =

√vDτp2

. (2.15)

21

2.4 Antennas for GPR Application

The lateral resolution also depends on the antenna foot print which is again a function

of radiation pattern. In most cases, the worst case lateral resolution is determined by

the pulse width rather than the antenna radiation pattern. Recent development of signal

processing technique claimed that it is possible to achieve a depth resolution better than

(2.14) with time separation and band width product becoming close to 0.1 [86].

2.4 Antennas for GPR Application

The antennas are selected for GPR applications based on important criteria such as, large

fractional bandwidth, low time side-lobes, low cross-coupling levels, impact of the host

media over the radiation pattern of antenna, and so on [16]. Hence, the classes of antenna

which can meet this stringent requirements are limited. As GPR antenna operates very

close to the ground and sometimes in contact with it, they need to be measured not only

in free space but also in a realistic ground environment. To accurately predict the phase

and amplitude of GPR responses for near-surface, near-field targets, it is essential that a

forward model contains a realistic description of the antennas.

The performance of a GPR antenna on the surface of ground can be significantly

affected by various factors. Due to the so-called coupling or proximity effect, the electrical

impedance of an antenna on the surface of a dielectric medium can be different from that

when the antenna is in the air. The change in impedance has been reported as the main

cause for the drop in centre frequency (fc) of the antenna. The antenna characteristics like

linear phase, constant gains and same polarization over the GPR operating bandwidth

help to reduce signal processing complexity and improve target detection. The ringing

effects originated within antenna as well in the interface between antenna and media

need to be either reduced or corrected by signal processing techniques to ensure good

resolution. High directivity of antenna helps to transmit more EM energy in narrow solid

angle as well as minimise the interference with unwanted objects. The GPR antenna

should be light and compact for ease of operation in variety of applications.

The most commonly used antennas for GPR applications are Dipole, Horn, Vivaldi,

Bow-tie, Spiral, etc. The Horn antennas are used for the off-ground configuration. They

can handle high amount average or pulsed power causing sufficient penetration of GPR

22

2.5 Modelling Earth’s Subsurface

signal in the host media. TEM horn antenna is very useful for time-domain GPR appli-

cation as they are non-dispersive, directive and can have UWB characteristics [87–90].

Among Dipole antennas, Bow-tie is most popular because of its simplicity and relatively

UWB characteristics. It was proposed by Brown and Woodward in 1952 [91]. Recently

more complicated shape, R-C loading are used to design Bow-tie antenna [92–95] to

achieve higher bandwidth and reduced ringing effect. The Vivaldi antenna has few good

features like high directivity, linear polarization, easy feeding mechanism, etc. [96]. The

array antennas [11, 97] are very helpful for faster data collection by increasing the scan-

ning area. They have significant advantages for several civil engineering and geoscience

applications [98]. Moreover, many advanced signal processing features like simultaneous

multi-offset measurements, polarimetry processing can be implemented with array archi-

tecture. Hence, array antenna helps effective GPR imaging and fast characterization of

subsurface media.


The knowledge of EM properties of subsurface media is very important for correct inter-

pretation of signal propagation, thereby significant for both, design and reliable operation

of a GPR system. The penetration depth and resolution are the functions of GPR sys-

tem operating bandwidth, and complex propagation constant of soil. The attenuation

constant of soil increases with increasing water, soluble salt, and clay contents [99]. With

increasing water content, the soil becomes significantly dispersive [100]. Depending on the

operating frequency and soil composition, the penetrating depth may vary a lot [101,102].

The EM characterization of materials at radio frequencies has started long before GPR

development. Numerous scientists have made path-breaking discoveries and contributed

immensely on the EM characterization of materials and its interaction with RF signal.

GPR takes benefit of these research outcomes. The pioneer work of Debye [103] is still

relevant to the GPR research. In this section, the properties of subsurface media, and the

characteristics of EM wave propagating through media are discussed. This knowledge is

very helpful to model the subsurface in a realistic way, and prepare laboratory models for

GPR simulation, and testing.

23


2.5.1 Electromagnetic Properties of Materials

The EM properties of materials are electric permittivity (ϵ), electric conductivity (σ), and

magnetic permeability (µ). Most of the subsurface materials are nonmagnetic in nature,

and are classified based on the conductivity, and dielectric constant profile of material.

The electric conductivity, and dielectric constant are expressed as complex quantity as

σ = σ′ + jσ′′, and ϵ = ϵ′ − jϵ′′ respectively. Therefore, the total current density, J in a

homogeneous media can be expressed as the summation of conduction current density, Jc

and displacement current density, Jd as following:

J = Jc + Jd = jωϵeE (2.16)

where

jωϵe = σ + jωϵ =(σ′ + ωϵ′′

)+ j

(σ′′ + ωϵ′

), (2.17)

ϵe is the effective dielectric constant, and its value depends on frequency of EM wave.

2.5.1.1 Properties of water

Water is a homogeneous media, and its frequency-dependent electrical parameters are

well defined by various research works [103–106]. The complex dielectric constant (ϵe) of

water is a function of EM wave frequency (f), temperature (T ), and salinity (S). The

characteristics of ϵe is accurately described by Klein-Swift model [104] below 10 GHz

microwave frequency [105]. This model [104] is based on single relaxation Debye [103]

equation as given below:

σ (f) = σs +ϵs − ϵ∞

1 +(

ffr

)2 ( f

fr

)2πfϵ0 (2.18)

where fr is the temperature dependent relaxation frequency of water. For pure water

(salinity S = 0), fr is given by (2.19) [106], ϵs is the temperature- (T in degree Kelvin)

dependent static permittivity given by (2.20), and ϵ∞ is the permittivity at infinite fre-

24


quency with value as 4.9, σs is the static ionic conductivity of water.

fr (T ) = 2π/(1.1109× 10−10 − 3.824× 10−12T + 6.938× 10−14T 2 − 5.096× 10−16T 3).

(2.19)

ϵs (T ) = 88.045− 0.4147T + 6.295× 10−4T 2 + 1.075× 10−5T 3. (2.20)

2.5.1.2 Volumetric mixing model

It is well known that the soil materials are significantly dispersive in the operating fre-

quency band of GPR because of frequency dependency of ϵe. The frequency dependency

is usually described by the Debye relaxation equation [103] as given below:

ϵe (f) = ϵe,∞ +ϵe,0 − ϵe,∞

1 + j ffr

(2.21)

where f is the frequency and fr is the relaxation frequency of the material, ϵe,0 is the

static permittivity, and ϵe,∞ is the permittivity at infinite frequency.

There are many empirical formulas that can be considered as fundamental mixing

models. These models calculate the effective permittivity of a media from the knowledge

of its component parts. Among these models, the most popular one is Topp’s model [107].

This model fits a third order polynomial to the measured permittivity of a soil (sandy/

loamy soil’s) based on TDR experiments. The estimation of this model matches well

with the TDR measurements over an approximate frequency range of 10 MHz to 1 GHz

across a wide range (5-50%) of volumetric water contents (VWC). The soil’s electrical

properties change based on VWC in it. It is observed that water’s dielectric constant (ϵr)

is highly correlated with the VWC in it [107]. The relation between ϵr and volumetric

water content has been described by Ledieu et al. [108] as following:

θ = a√ϵr + b (2.22)

where a and b are the soil specific empirical parameters, their values depend on the type of

25


soils used. The electric conductivity can be derived from below mentioned equation [109]:

σ (θ) =(aθ2 + bθ

)σw + σt (2.23)

where σw is the soil solution electric conductivity (S/m), σt is the electric conductivity of

dry soil (S/m), and a and b are the soil specific empirical parameters.

Another popular mixing model is the complex refractive index model (CRIM) [110],

and it is represented by Equation (2.24). This model can calculate the complex permit-

tivity based on the knowledge of material’s permittivities and their fractional volume.

ϵe,mix =

(N∑i=1

fi√ϵi

)2

(2.24)

where ϵe,mix is the complex bulk effective permittivity of the mixture, fi is volume fraction

of the ith component, and ϵi is complex permittivity of the ith component.

2.5.2 Characteristics of Wave in a Medium

It is easy to find the attenuation, phase constant, velocity of a harmonic plane wave

propagating through a uniform, homogeneous, isotropic media. The complex propagation

constant (γ) is denoted by

γ = α+ jβ =√

iωµ(σ + iωϵ) (2.25)

where ω is the angular frequency of wave in radians per second (rad/s), α is the attenuation

constant expressed in nepers per meter (Np/m) or decibels per meter (dB/m), and β is

the propagation constant in radians per meter (rad/m).

α = ω

√√√√µϵ

2

[√1 +

( σ

ωϵ

)2− 1

](2.26) β = ω

√√√√µϵ

2

[√1 +

( σ

ωϵ

)2+ 1

]. (2.27)

26


The velocity (v) of a plane wave traveling in a homogeneous media is found as

v =ω

β=

1√µϵ2

[√1 +

(σωϵ

)2+ 1

] . (2.28)

The loss of plane wave due to propagation in a homogeneous media is directly related to

the attenuation constant α. It is helpful to find how far the GPR signal can penetrate in

a subsurface media and likely amplitude of a reflection from any interface between two

media or target with a particular radar cross section (RCS). After having knowledge of

complex propagation constant, it is possible to have initial estimation of target depth,

and resolution. It is also observed that, the α and β are functions of frequency. This

causes dispersion of pulse propagating through a media, and loss of information unless

this effect is properly accounted. The assumption of plane wave fails as the antenna

is placed very close to the media. Again the host media is generally inhomogeneous in

nature. As a result, the realistic assumptions of media, and its complex boundary are of

prime importance for accurate signal propagation modelling.

2.5.3 Reducing Modelling Complexity of Subsurface Media

Numerous research works done on the characterization of various types of soils are help-

ful for accurate description of EM properties of soil. This knowledge is also helpful

for the accurate numerical solution of EM wave propagation in a heterogeneous media.

However, this also increases the computation cost of GPR modelling. Especially model

inversions become very computational intensive work. Therefore, it is essential to reduce

the complexity of EM modelling of subsurface media for real-time GPR application on

both qualitative and quantitative detection.

Based on GPR applications and GPR system configurations, the subsurface media can

be assumed as a homogeneous media [19, 74], linearly varying [55], multilayered media

[1,31]. It can be modelled as a multilayered media if the particle size is smaller compared

to the antenna foot-print and range-resolution of the GPR system [31]. These assumptions

help to reduce the computational complexity of GPR modelling to a great extent.

The roughness of earth’s surface is one of the major sources of clutter in various

27

2.6 Wave Propagation in Multilayered Media

subsurface sensing applications [111]. The roughness can cause significant degradation

of signal quality, and needs to be taken into account in data processing. However, the

effect of roughness can be neglected if the surface protuberances satisfy the Rayleigh’s

criterion. For monostatic radar, a surface is called as smooth if surface protuberances

are lower than the one-eighth of wavelength [112] i.e. h < hc where hc = λ/8. It is also

possible to select an operating frequency band to minimize the effect of surface roughness

based on the application requirements. It is experimentally shown that the GPR can

be used with good accuracy for quantitative estimation of water contents if Rayleigh’s

criterion is satisfied [113].

The soil media has significant dispersive characteristics. As the media is moisturized

by water, its characteristics change according to the volumetric mixing models. For a

limited frequency band, real part of ϵe can be assumed as constant, and imaginary part i.e.

conductivity (σ) can be assumed as a linear function of the frequency for sand [100,114].

σ (f) = σc + σr (f − fc) (2.29)

where σc is the static electric conductivity (S/m) at center frequency fc, and σr is the

linear variation rate (S/m/GHz). These simplifications help to reduce the complexity of

GPR modelling without compromising much accuracy till a certain amount of VWC [100].


Layered media are the simplest kind among inhomogeneous media. They have been stud-

ied by various research works [28–30, 32, 115–117] because of huge applications related

to the study of subsurface media, microwave and millimeter wave integrated circuits

(MMIC’s), micro-electromechanical systems (MEMS), radio frequency (RF) printed cir-

cuit board (PCB), high speed digital circuits, etc. For the homogeneous media, closed

form solutions are available for the scalar and dyadic Green’s function [118–121]. For

planar layered media, the EM property variation of it has to be incorporated into the

Green’s function(s) itself. Generally, evaluation of the Green’s functions for layered me-

dia is more difficult compared to the homogeneous media. This is primarily due to the

28


unavailability of a closed form Green’s function in spatial domain. Mostly, the compu-

tation of Green’s functions is involved with lot of analytical processing as well numerical

computation. However, the solution of EM wave for planar layered media can be reduced

to a one-dimensional problem, and many mathematical elegant technique can be used for

finding a solution. Therefore, planar layered media assumption is a favorite choice for the

Geophysicists [18,26,33,41,122].

2.6.1 Maxwell’s Equation in Frequency Domain

The Maxwell’s equations in time harmonic differential form [118, 119, 123, 124] can be

written as following:

∇×H− η (ω)E = J (2.30) ∇×E+ ζ (ω)H = −M (2.31)

∇.D = ρe (2.32) ∇.B = ρm (2.33)

where ζ (ω) and η (ω) are the media’s EM parameters defined as ζ (ω) = iωµ and η (ω) =

σ + iωϵ = iωϵe. E and H are, respectively, the electric and magnetic field vectors at

frequency ω rad/s at a position vector r (x, y, z). J and M are, respectively, the electric

and magnetic specific source current densities at ω rad/s. D and B are, respectively, the

electric and magnetic flux densities at ω rad/s at location r. For isotropic media, D = ϵE

and B = µH. ρe and ρm are, respectively, the electric and magnetic specific volumetric

source charge densities at frequency ω rad/s. The time-dependence is implicit with an

exp(iωt) dependence in the formulation. Therefore, time harmonic form of electric field

can be expressed as

E(x, y, z, t) = Re[E(x, y, z, ω)eiωt

]. (2.34)

All other fields, source currents, and charges can be expressed by following (2.34).

To solve the EM fields at a point, suitable boundary conditions need to be applied.

They are mentioned below.

n× (H1 −H2) = Js (2.35) (E1 −E2)× n = Ms (2.36)

29


n · (B1 −B2) = ρsm (2.37) n · (D1 −D2) = ρse. (2.38)

where Js, Ms are, respectively, the electric and magnetic specific line current densities;

ρse, ρsm are, respectively, the electric and magnetic specific surface charge densities.

Some time Js, Ms are called as impressed currents for which boundary conditions are to

be satisfied at the sources [123]. These Maxwell’s equations along with its constitutive

relations, and EM boundary conditions can be solved for finding EM fields at any point

in a space.

2.6.2 Plane Wave Solution

Consider source-free, homogeneous, isotropic medium characterized by a scalar complex

electric permittivity ϵe and a scalar magnetic permeability µ. The Maxwell’s equations

discussed in previous subsection can be modified as following:

∇×H− η (ω)E = 0 (2.39) ∇×E+ ζ (ω)H = 0 (2.40)

∇.E = 0 (2.41) ∇.H = 0 (2.42)

The EM bundary conditions are

n× (H1 −H2) = 0 (2.43) (E1 −E2)× n = 0 (2.44)

n · (B1 −B2) = 0 (2.45) n · (D1 −D2) = 0. (2.46)

Solving (2.39)-(2.42), wave equations for E and H can be derived as following:

∇2E− γ2E = 0 (2.47)

∇2H− γ2H = 0 (2.48)

30


where γ is the propagation constant of media defined by

γ = α+ iβ =√

iωµ(σ + iωϵ) =√

ζη = ik. (2.49)

Here α and β are, respectively, the attenuation and phase constants of media, and k is

the propagation constant in another form. Equations (2.47), (2.48) are popularly known

as Helmholtz equations. A solution for electric field can be obtained in the following

form [119]:

E = E0e−ik.r (2.50)

where k is the propagation vector, and r is the position vector given by following relations:

k = kxx+ kyy + kz z (2.51)

r = xx+ yy + zz (2.52)

k.r = kxx+ kyy + kzz. (2.53)

Similarly, a solution of magnetic field can be found by solving the Heltmholtz equation

(2.48) and it is related to the electric field by the following formula:

H =ik×E

ζ=

k×E

Z(2.54)

where Z (=√

ζη ) is the impedance of media. Equation (2.50) is the electric field repre-

sentation of a general plane wave propagating in the direction of propagation vector k.

The plane wave is the characteristic of a wave for which amplitude and phase, both are

constant in a plane (k.r =constant) perpendicular to the propagation vector.

2.6.2.1 Reflection of Plane Wave from a Half space

The simplest example of a planar layered media is a half space with two regions having

constant EM properties. An EM wave in a homogeneous region can be decomposed into

transverse electric (TE) and transverse magnetic (TM) waves with an axis of reference.

Let us take axis of reference as z-axis ( Figure 2.2). Therefore, vertical electrical field (Ez)

31


(a) TE polarization (b) TM polarization

Figure 2.2: Reflection and transmission of plane wave at an interface.

corresponds to the TM wave (Figure 2.2b), and vertical magnetic field (Hz) corresponds

to the TE wave (Figure 2.2a). Note that, symbol ‘⊙

’ denotes out of the page and ‘⊗

’

into the page. Now applying EM boundary conditions and solving the set of equation,

the Fresnel reflection coefficient can be obtained [116,118] as

rTE1,2 =

ζ2k1z − ζ1k2zζ2k1z + ζ1k2z

=µ2k1z − µ1k2zµ2k1z + µ1k2z

(2.55)

rTM1,2 =

η2k1z − η1k2zη2k1z + η1k2z

=ϵe2k1z − ϵe1k2zϵe2k1z + ϵe1k2z

. (2.56)

where k1z and k2z are the vertical wave numbers of respectively, top and bottom regions.

Other terms have as usual meaning with subscript ‘1’and ‘2’ representing the top and

bottom layers respectively. It is important to note that, rTE1,2 denotes the reflection co-

efficient for the electric field, and rTM1,2 denotes the reflection coefficient for the magnetic

field. Similarly, the transmission coefficients (τ) can be found as

τTE1,2 =

2ζ2k1zζ2k1z + ζ1k2z

(2.57) τTM1,2 =

2η2k1zη2k1z + η1k2z

. (2.58)

For normal incidence of plane wave k1z = k1 = ω√µ1ϵ1e and k2z = k2 = ω

√µ2ϵ2e. By

simplifying (2.55) or (2.56), the reflection coefficient can be written as following.

r1,2 = rTE1,2 = −rTM

1,2 =Z2 − Z1

Z2 + Z1(2.59)

32


Figure 2.3: N -layered media with description of the coordinate system.

where Z2 (=√

ζ2η2) and Z1 (=

√ζ1η1) are the intrinsic impedances of respectively, top and

bottom layers. The transmission coefficient can be found as

τ1,2 =2Z2

Z2 + Z1. (2.60)

2.6.2.2 Reflection from a Multilayered media

A planar, inhomogeneous media with EM properties (µ, ϵ, and σ) varying in vertical

direction (z) can be modelled as multilayered media (Figure 2.3). In this case EM property

of each layer is constant. For an isotropic, source-free media, and EM property (µ, ϵ, σ)

varying in one direction, the solution of wave can be decomposed into transverse electric

(TE) and transverse magnetic (TM) waves. This decomposition helps to find the reflection

and transmission of plane wave at the various interfaces of multilayered media. Applying

the EM boundary conditions and solving the set of equations [116,118,119], the generalized

global reflection coefficients at any interface zn for a N -layered media can be found as

following:

RTEn =

rTEn,n+1 +RTE

n+1exp(−2ikn+1,zhn+1)

1 + rTEn,n+1R

TEn+1exp(−2ikn+1,zhn+1)

(2.61)

rTEn,n+1 =

ζn+1kn,z − ζnkn+1,z

ζn+1kn,z + ζnkn+1,z(2.62)

33


RTMn =

rTMn,n+1 +RTM

n+1exp(−2ikn+1,zhn+1)

1 + rTMn,n+1R

TMn+1exp(−2ikn+1,zhn+1)

(2.63)

rTMn,n+1 =

ηn+1kn,z − ηnkn+1,z

ηn+1kn,z + ηnkn+1,z(2.64)

where kn,z is the vertical wave number of nth layer, ζn = iωµn, and ηn = σn + iωϵn

(= iωϵe,n). Equations (2.61),(2.63) are in recursive loop. Assuming there is only down

going wave at the bottom most layer, the loop stops at the lowest interface (zN−1). In

this case RTMn = rTM

N−1,N and RTEn = rTE

N−1,N .

For perpendicular incidence of plane wave, the global reflection coefficient (Rn) is

modified by the following formula:

Rn =rn,n+1 +Rn+1exp(−2ikn+1hn+1)

1 + rn,n+1Rn+1exp(−2ikn+1hn+1)=

rn,n+1 +Rn+1exp(−2γn+1hn+1)

1 + rn,n+1Rn+1exp(−2γn+1hn+1)(2.65)

where

rn,n+1 =Zn+1 − Zn

Zn+1 + Zn. (2.66)

Observe that, no superscripts are used to represent the local and global reflection co-

efficient terms for normal incidence of plane wave. These notations will be followed

throughout the thesis. Equation (2.65) is an important formula used for approximate

GPR detection of multilayered media.

2.6.3 A Point Source on Top of a Multilayered Media

When a point source is on the top of a layered media, it generates a spherical wave around

it. Best way to solve this problem is to represent the wave in spectral domain in terms

of summation of plane waves. Then it is possible to represent spectral domain waves in

terms of TE-type (Ez = 0,Hz = 0) and TM-type (Ez = 0,Hz = 0) plane waves. It

would be easier to describe the transmission and reflection of TE and TM-type waves

through multilayered media. Then final solutions can be obtained in spatial domain by

applying suitable inverse Fourier transform. For horizontally placed layered media, each

layer is parallel to x − y plane, and the layer electrical property is independent of the

space variables x and y. Taking advantage of this property, the Fourier transform for all

the fields can be defined with respect to transverse coordinates x and y.

34


2.6.3.1 Fourier Transform for Representation in Spectral domain

The Fourier transformation of a scalar function f(x, y, z) in spectral domain parameters

kx, and ky can be defined as

f (kx, ky, z) =

∫ +∞

−∞

∫ +∞

−∞exp(ikxx+ ikyy)f (x, y, z) dxdy. (2.67)

The inverse Fourier transform to the spatial domain is defined by

f (x, y, z) =

(1

2π

)2 ∫ +∞

−∞

∫ +∞

−∞exp(−ikxx− ikyy)f (kx, ky, z)dkxdky. (2.68)

Note that, the z coordinate along the stratification of layered media is not affected by the

transformation.

2.6.3.2 An Approach for Solution in Spectral Domain

A point current source can be of two kinds: the electric current (J) and the magnetic

current (M). They can also be obtained by applying boundary conditions described by

Equations (2.35), (2.36) at source point. Let us assume that a point source is located at

the origin (O) of the Cartesian coordinate system (in Figure 2.3). It may consists of only

electric type source, or only magnetic type, or both types. The J and M can be expressed

as following:

J(x, y, z, ω) = J(ω)δ(x, y, z) (2.69) M(x, y, z, ω) = M(ω)δ(x, y, z). (2.70)

In spectral domain these sources can be expressed as

J(kx, ky, z, ω) = J(ω)δ(z) (2.71) M(kx, ky, z, ω) = M(ω)δ(z). (2.72)

Here we like to introduce few notations as following. A is the spectral representation of

any variable A. Transverse vectors are kT = {kx, ky} and xT = {x, y}T . The horizontal

vector partial derivative is defined as ∂T = {∂x, ∂y}, and ∂T is to be replaced by −ikT .

The short-hand notation ∂x, ∂y, ∂z are the partial derivatives.

35


In a homogeneous media, the Maxwell’s equations (2.30) and (2.31) can be written in

spectral domain as

∂z z× H− ikT × H− ηE = J (2.73)

∂zz× E− ikT × E+ ζH = −M. (2.74)

These equations can be separated as following:

zηEz = −zJz − ikT × H (2.75)

zζHz = −zM sz + ikT × E (2.76)

ηET = ∂zz× HT − ikT × zHz − JT (2.77)

ζHT = −∂z z× ET + ikT × zEz − MT (2.78)

By solving Equations (2.75)-(2.78) the horizontal components of the fields are eliminated

resulting in following equations for the vertical components:

(∂z∂z − Γ2

)Ez = ζJz + η−1∂z

(ikT .JT − ∂zJz

)− z.

(ikT × MT

)(2.79)

(∂z∂z − Γ2

)Hz = ηMz + ζ−1∂z

(ikT .MT − ∂zMz

)+ z.

(ikT × JT

)(2.80)

where Γ =√

kT .kT + γ2. These equations can be solved by following the solution of

modified Helmholtz equation as

Ez ={−ζJz − η−1∂z

(ikT .JT − ∂zJz

)}G (z) +

{z.(ikT × MT

)}G (z) (2.81)

Hz = −{z.(ikT × JT

)}G (z) +

{−ηMz − ζ−1∂z

(ikT .MT − ∂zMz

)}G (z) (2.82)

where G (z) = exp(−Γ|z|)2Γ is the solution of modified Helmholtz equation

(∂z∂z − Γ2

)G (z) = −δ(z). (2.83)

Let’s represent the vertical components of electric and magnetic fields i.e. Ez and Hz in

36


terms of tensor Green’s functions as following:

Ez = GEJz · J+ GEM

z · M (2.84)

Hz = GHJz · J+ GHM

z · M. (2.85)

By simplifying (2.84) and (2.84), the tensor Green’s functions can be expressed separately

in terms of contributions due to the transverse and vertical components of the electric

and magnetic currents (J, M) as

GEJz,T =

(ikT

ηΓsign(z)

)G (z) (2.86)

GEJz,z =

(k2ρηG (z)− δ(z)

η

)z (2.87)

GHJz,T = − (ikT × xT ) G (z) (2.88)

GEMz,T = (ikT × xT ) G (z) (2.89)

GHMz,T =

(ikT

ζΓsign(z)

)G (z) (2.90)

GHMz,z =

(k2ρζG (z)− δ(z)

ζ

)z. (2.91)

Here kρ =√

k2x + k2y and sign(z) is defined by relations

sign(z) =

−1 for z < 0

0 for z = 0

1 for z > 0

(2.92)

It can be observed that a vertical electric current causes generation of only TM waves,

and a vertical magnetic current causes only TE waves. Whereas, horizontal components

of electric and magnetic currents contribute both TM and TE waves. After some steps

of derivations by using (2.77) and (2.78), the transverse components of the electric field

37


can be expressed in terms of the vertical components of electric and magnetic fields as

ET = − ikT

k2ρ∂zEz +

ζ

k2ρikT × zHz +

ζ

k2ρJT +

ζ

k2ρz× MT (2.93)

HT = − η

k2ρikT × zEz −

ikT

k2ρ∂zHz −

η

k2ρz× JT +

η

k2ρMT . (2.94)

Therefore, once the vertical components of electric and magnetic fields are known, the

horizontal components can be obtained from Equations (2.93) and (2.94). In principle,

the horizontal components of EM fields are contributed by both TE and TM waves. Now

combining all the solutions, total field components for a propagating wave can be written

in a matrix form as given below:

Ex

Ey

Ez

Hx

Hy

Hz

=

−k2x+ηζη −kxky

ηikxη Γsign(z)

−kxkyη −k2y+ηζ

ηikyη Γsign(z)

ikxη Γsign(z)

ikyη Γsign(z)

k2ρη

0 Γsign(z) −iky

−Γsign(z) 0 +ikx

iky −ikx 0

G (z)

Jx

Jy

Jz

+

0 −Γsign(z) iky

Γsign(z) 0 −ikx

−iky ikx 0

−k2x+ηζζ −kxky

ζikxζ Γsign(z)

−kxkyζ −k2y+ηζ

ζikyζ Γsign(z)

ikxζ Γsign(z)

ikyζ Γsign(z)

k2ρζ

G (z)

Mx

My

Mz

(2.95)

Equation (2.95) gives complete solution of electric and magnetic fields in spectral domain

for a point source in a homogeneous medium.

2.6.3.3 Finding Reflected Field due to Multilayered Media

Based on Equation (2.95), it is possible to write the expression for vertical component

of electric and magnetic fields. As wave propagates through a multilayered media it en-

38


counters multiple reflections and refractions at different interfaces of the media. Based on

TE and TM decomposition of wave, it is possible to find generalized reflection coefficients

for them. A general solution for the vertical electric and magnetic fields in the region

(0 < z < z1) (in Figure 2.3) can be written as

Ez = f ez

[exp (−Γ1z) +RTM

1 exp (Γ1 (z − 2z1))]

(2.96)

Hz = fhz

[exp (−Γ1z) +RTE

1 exp (Γ1 (z − 2z1))]

(2.97)

where fez and fh

z are the arbitrary functions to be evaluated from (2.95). They are the

functions of J, M, and EM properties of media. RTM1 is the transverse magnetic global

reflection coefficient, and RTE1 is the transverse electric global reflection coefficient at

interface z1 accounting for all reflections from the multilayered interfaces. By follow-

ing Equations (2.61)-(2.64), the general expressions of RTMn and RTE

n can be written as

following:

RTEn =

rTEn,n+1 +RTE

n+1exp(−2Γn+1hn+1)

1 + rTEn,n+1R

TEn+1exp(−2Γn+1hn+1)

(2.98)

rTEn,n+1 =

ζn+1Γn − ζnΓn+1

ζn+1Γn + ζnΓn+1(2.99)

RTMn =

rTMn,n+1 +RTM

n+1exp(−2Γn+1hn+1)

1 + rTMn,n+1R

TMn+1exp(−2Γn+1hn+1)

(2.100)

rTMn,n+1 =

ηn+1Γn − ηnΓn+1

ηn+1Γn + ηnΓn+1(2.101)

where Γn (=√

k2ρ − k2n = ikn,z), kn is free space propagation constant of nth layer with

relation k2n = −ζnηn, ζn = iωµn, and ηn = σn + iωϵn (= iωϵe,n).

As usual the horizontal components of EM fields can be obtained from (2.93) and

(2.94). Based on the definition of Green’s function for a GPR configuration, it is possible

to derive a compact expression of it by finding the ratio between received signal and

transmitted signal. Important point to observe here is that the Green’s function i.e.

the response due to subsurface media is a function of EM properties of all the layers.

The spatial domain solutions can be obtained by applying inverse Fourier transform as

explained in Equation (2.68). This process is involved with integration over an infinite

39

2.7 Summary

path having singularities. Thus, a special attention is required to compute this integral

either analytically or numerically [118].

2.7 Summary

GPR is a nondestructive method for investigation of subsurface media and buried ob-

jects. Its prospectus to apply for various civilian and military applications are growing

over the days. In this section, the principle of GPR, and its system level important pa-

rameters have been discussed. The designing of sophisticated UWB GPR system has

got many challenges originated from complex media properties, limitations of component

specification, accuracy and efficiency of signal processing algorithm, etc. The discussion

on antenna has highlighted its importance and desired characteristics for developing an

UWB GPR system. The EM properties of subsurface media, and its interaction with

propagating waves have been discussed highlighting requirements of simplification to re-

duce the complexity of GPR modelling. A brief study has been presented on solution of

EM waves originated due to a point source and then reflected by a multilayered media.

The approach is generic which accounts for the complete EM properties of layered media

and can be extended for the various types of sources.

Despite development of several GPR models and sophisticated algorithm, achieving

accuracy and computation speed together is still a challenging problem. In subsequent

sections, we will focus on development of accurate and time-efficient GPR models, their

characterization and validation by synthetic and practical experiments.

40

CHAPTER3Development of Full Wave Model

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Modelling Assumptions . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Derivation of FWM Green’s Function . . . . . . . . . . . . . . 44

3.3.1 Formulation of Green’s Function . . . . . . . . . . . . . . . . . . 45

3.3.2 Deriving Green’s Function in Spectral Domain . . . . . . . . . . 46

3.3.3 Converting Spectral Domain Green’s Function to Spatial Domain 48

3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Numerical Computation of FWM Green’s Function . . . . . . 52

3.5 Validation of Proposed FWM . . . . . . . . . . . . . . . . . . . 53

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

41

3.1 Introduction

This chapter proposes a full wave model (FWM) for monostatic far-field GPR config-

uration applied to multilayered media. 3-D Maxwell’s equations are solved with suitable

boundary conditions to derive a compact formula for the FWM Green’s function. The

accuracy of model is compared with an existing FWM in literature.

3.1 Introduction

Modelling ground penetrating radar (GPR) while the antenna is very close to the ground

subsurface is a complex problem. The accuracy of modelling scheme is very important

for qualitative and quantitative estimation. Unrealistic assumptions for simplifying the

problem of GPR signal propagation through inhomogeneous subsurface media lead to the

inaccuracy in estimation of target information. In this respect, FWMs are very useful

models to describe the GPR data accurately. In the field of monostatic far-field GPR

configuration, a major breakthrough has been done by Lambot et al. [1]. In this scheme,

all the effects of antenna, and its interaction with ground subsurface are modeled by

a set of linear transfer functions (LTFs). The response due to multilayered media is

modeled by a 3-D Green’s function. The subsequent research works with this model

have demonstrated that the FWM is highly accurate to describe GPR data, and suitable

for detecting the medium electrical properties in several hydrophysical and engineering

applications [35, 63, 125, 126]. The model also capable of describing various types of

antennas like horn, bowtie, loop, vivaldi in far-field configuration [13,64,127]. Theoretical

and experimental analysis [13] has shown that the FWM is one of the most accurate

schemes for characterization of layered media in the far-field configurations.

In this work, an FWM is proposed for monostatic far-field GPR configuration. The

model is based on the assumption of aperture antenna as a point source, and calculation

of received scattered electric field due to unit transmit electric field at antenna phase

centre. A compact analytical expression of the Green’s function is derived to represent

the response due to multilayered media. Our derivation is based on earlier work by S.

Lambot [116]. Subsequently, the proposed FWM is validated by a comparative analysis

carried out between the proposed model and the FWM [1] by Lambot et al. over a wide

bandwidth and parameters vector space.

42

3.2 Modelling Assumptions

In this chapter, first, the assumptions for monostatic GPR modelling is discussed.

Then, Maxwell’s equations are solved with suitable boundary condition to derive the

FWM expression for multilayered media. Subsequently, a brief discussion is presented on

efficient computation of the FWM integral. Finally, the model is validated by a synthetic

study.

3.2 Modelling Assumptions

An SFCW radar can be emulated with help of a vector network analyzer (VNA), and

an antenna. VNA having high dynamic range and very narrow IF bandwidth is an ideal

setup to implement SFCW radar in a laboratory environment. The various assumptions

for the monostatic off-ground GPR applied to inhomogeneous media are discussed in

earlier works [1, 31]. The antenna can be assumed as a point source and receiver as long

as the target is placed in the far-field of antenna. Since, antenna introduces frequency

dependent gain, delay, and internal reflections due to mismatch, the antenna effect in the

GPR environment needs to be modelled accurately. These model parameters should be

evaluated by proper calibration process before actual GPR testing is carried out. In this

case, the VNA, antenna, and subsurface are modelled as suitable combinations of linear

systems [1] as shown in Figure 3.1a. The frequency-domain reflection coefficient S11 (ω)

(a) (b)

Figure 3.1: (a) Linear transfer function model (LTFM) representing the VNA-antenna-multilayered medium system [1]. (b) The virtual source and receiver point at antennaphase center.

43

3.3 Derivation of FWM Green’s Function

measured by VNA can be expressed as following:

S11 (ω) =Y (ω)

X(ω)= Hi (ω) +

Ht (ω)G↑xx(ω)Hr (ω)

1−Hf (ω)G↑xx(ω)

(3.1)

where, X(ω) and Y (ω) are the transmit and receive signals at the VNA reference plane;

Hi (ω) is the return loss of the antenna, Ht (ω) and Hr (ω) are, respectively, the transmit

and receive transfer functions of the antenna, and Hf (ω) represents the feedback loss

transfer function. G↑xx(ω) is the Green’s function representing the air-subsurface system.

The subsurface media is largely inhomogeneous and is difficult to model. However, it can

be modelled as a multilayered media if particle size is smaller compared to the antenna

foot-print and the range resolution of GPR system [31]. Moreover, the received scattered

wave in this GPR configuration has mainly traveled in the vertical (z) direction. The

horizontal variation of subsurface media property is expected to have negligible effect on

the Green’s function. Therefore, air-subsurface media can be modelled as multilayered

media with N homogeneous layers separated by N − 1 interfaces (shown in Figure 2.3).

Each layer is characterized by frequency-dependent complex dielectric constants (ϵe = ϵ+

σiω ) and its thickness (h). For nonmagnetic materials, which mostly occur in earth surface,

permeability is assumed to be constant with free space value (µ0). The electrical property

of materials mostly follow the Debye relation [103]. However, for limited frequency band,

real part of ϵe is assumed to be constant and conductivity (σ) is assumed to be a linear

function of the frequency as given below.

σ (f) = σc + σr (f − fc) (3.2)

where σc is the static conductivity (S/m) at center frequency fc and σr is the linear

variation rate of conductivity (S/m/GHz).


A most convenient technique to calculate the far-field of an aperture antenna is to define

electric field equivalent magnetic current density or magnetic field equivalent electric

44


current density or both types of currents at the antenna aperture. A point current source

generates spherical wave around it. For multilayered horizontal media distributed over

infinite length and width, it is easier to solve the fields in spectral domain by splitting

them into a set of transverse electric (TE) fields and another set of transverse magnetic

(TM) fields. It would be easier to calculate the reflections, refractions of plane waves due

to propagation through planner media by following the process discussed in Section 2.6.

After calculating the spectral domain Green’s function, it can be converted to spatial

domain by suitable transformation. The oscillating Bessel’s function originated during

conversion from spectral domain to spatial domain causes the numerical computation to

be a time-consuming task. By assuming antenna as a point source and receiver at the

same location, this problem can be largely simplified.

3.3.1 Formulation of Green’s Function

The source and receiver part of the antenna is located at the antenna phase center at

the origin O of the coordinate system. We assume that only x-directed electric field

Etxp exists at the antenna phase center located at the origin O of the coordinate system

(Figure 3.1b). To calculate the far-field due to antenna, let us find the fictitious magnetic

and electric current densities, respectively,Ms and Js by applying Huygen’s principle [128]

(pp. 575− 581)

Ms = −2n× xEtxp = −2Et

xpy (3.3)

and equivalent electric current density

Js = 0. (3.4)

The n is acting towards the direction of EM wave propagation (in z-direction). The radi-

ated far-field due to this equivalent magnetic source is to be derived by solving Maxwell’s

equations and suitable boundary conditions. The Green’s function G↑xx(ω) is defined

as the ratio between back scattered x-directed electric field, and transmitted x-directed

45


electric field at antenna phase center at frequency ω rad/s

G↑xx(0, ω) =

Erx,ρ=0,z=0

Etxp

. (3.5)

3.3.2 Deriving Green’s Function in Spectral Domain

Here multilayered media (in Figure 2.3) is placed at the far-field of antenna. We wish

to compute the reflected scattered fields (E,H) at the receiver antenna phase center

(r(x, y, z) = 0) due to specified current distributions (J,M) at transmitter antenna phase

center. From Section 3.3.1, it is clear that there is only magnetic current at the source.

Electric and magnetic fields are governed by Maxwell’s equations and its time harmonic

differential form are obtained from (2.30) and (2.31) as following:

∇×H− η (ω)E = 0 (3.6)

∇×E+ ζ (ω)H = −Ms (3.7)

where ζ (ω) and η (ω) are the media’s EM parameters defined as ζ (ω) = iωµ and η (ω) =

σ + iωϵ. E (= E(x, y, z, ω)) and H (= H(x, y, z, ω)) are, respectively, the electric and

magnetic field vectors at frequency ω rad/s. Ms (= Ms(x, y, z, ω)) is the magnetic specific

source currents at frequency ω rad/s, and it is located at the origin (O) of the Cartesian

coordinate system. This current can be expresses as

Ms(x, y, z, ω) = Ms(ω)δ(x, y, z). (3.8)

In a homogeneous media, the Maxwell’s equations (3.6) and (3.7) can be represented

in spectral domain as

∂zz× H− ikT × H− ηE = 0 (3.9)

∂zz× E− ikT × E+ ζH = −Ms. (3.10)

Now following the procedure mentioned in Section 2.6.3, the transverse components of

the electric and magnetic fields can be expressed in terms of the vertical components of

46


electric and magnetic fields as following:

ET = − ikT

k2ρ∂zEz +

ζ

k2ρikT × zHz +

ζ

k2ρz× Ms

T (3.11)

HT = − η

k2ρikT × zEz −

ikT

k2ρ∂zHz +

η

k2ρMs

T (3.12)

where kρ =(k2x + k2y

)1/2. The vertical components of electric and magnetic fields can be

derived as

Ez = z.(ikT × Ms

T

)G (z) (3.13)

Hz ={−ηM s

z − ζ−1∂z

(ikT .M

sT − ∂zM

sz

)}G (z) (3.14)

where G (z) = exp(−Γ|z|)2Γ is the solution of modified Helmholtz equation. Therefore, all

the electric and magnetic fields can be expressed in terms of contribution due to TE and

TM types of wave. After combining them, the total electric and magnetic fields for a

propagating wave can be expressed in matrix form as given below:

Ex

Ey

Ez

Hx

Hy

Hz

=

0 −Γsign(z) iky

Γsign(z) 0 −ikx

−iky ikx 0


ζikxζ Γsign(z)


ζikyζ Γsign(z)

ikxζ Γsign(z)

ikyζ Γsign(z)

k2ρζ

G (z)

M s

x

M sy

M sz

(3.15)

where G (z) = exp(−Γ|z|)2Γ . For propagating wave towards positive z-direction, the solution

can be written as

Ex

Ey

Ez

Hx

Hy

Hz

=

0 −Γ iky

Γ 0 −ikx

−iky ikx 0


ζikxζ Γ


ζikyζ Γ

ikxζ Γ

ikyζ Γ

k2ρζ

exp(−Γz)

2Γ

M s

x

M sy

M sz

. (3.16)

47


Since, there is only y-directed magnetic source term M sy as observed in (3.3), the

general solution for the vertical electric and magnetic fields in the region (0 < z < z1) are

expressed as

Ez =ikxM

sy

2Γ

[exp (−Γ1z) +RTM

1 exp (Γ1 (z − 2z1))]

(3.17)

Hz =ikyM

sy

2ζ

[exp (−Γ1z) +RTE

1 exp (Γ1 (z − 2z1))]

(3.18)

where RTM1 is the transverse magnetic global reflection coefficient, and RTE

1 is the trans-

verse electric global reflection coefficient at interface z1 accounting for all reflections from

the multilayer interfaces. They are defined by Equations (2.98)-(2.101) in Section 2.6.3.

For the monostatic SFCW radar with single TEM horn antenna, the emitter and receiver

both are assumed to be located at the antenna phase center at z = 0. By substituting

(3.17) and (3.18) in (3.11), the x-directed electric field at phase center is computed as

Ex,z=0 =1

2k2ρ

[−2ζ − k2ρ

]My +

1

2k2ρ

[k2xR

TM1 − k2yR

TE1

]exp (−2Γ1z1) My. (3.19)

Considering only the backscattered field(Er

x,z=0

), and using the relation Ms = −2Et

xpy

from (3.1), (3.19) is simplified to

Erx,z=0 =

1

2k2ρ

[k2xR

TM1 − k2yR

TE1

]exp (−2Γ1z1)

(−2Et

xp

). (3.20)

Accordingly, the response due to the multilayered medium is defined as

G↑xx (kρ, ω) =

Erx,z=0

Etxp

=1

k2ρ

[k2yR

TE1 − k2xR

TM1

]exp (−2Γ1z1) . (3.21)

3.3.3 Converting Spectral Domain Green’s Function to Spatial Domain

The spectral domain Green’s function obtained in Equation (3.21) can be converted to

the spatial domain easily in the cylindrical coordinate. Since, the layered media has no

variation along x and y directions, the transition to cylindrical coordinate reduces one

dimension of integration. Let us introduce cylindrical coordinate systems for both spatial

domain and spectral domain as following:

48


x = ρ cos(ϕ) (3.22) y = ρ sin(ϕ) (3.23)

ρ =√

x2 + y2 (3.24) ϕ = tan−1(yx

)(3.25)

kx = kρ cos(kϕ) (3.26) ky = kρ sin(kϕ) (3.27)

kρ =√

kx2 + ky

2 (3.28) kϕ = tan−1

(kykx

)(3.29)

.Using relations (3.26)-(3.29), the spectral domain Green’s function (3.21) can be written

as

G↑xx (kρ, ω) =

[RTE

1 sin2(kϕ)−RTM1 cos2(kϕ)

]exp (−2Γ1z1) . (3.30)

Based on (2.68), the spatial domain Green’s function is obtained from the spectral domain

expression (3.30) as

G↑xx (ρ, ϕ, z = 0, ω) =

(1

2π

)2 ∫ +∞

−∞

∫ +∞

−∞G↑

xx (kx, ky, z = 0, ω)exp(−ikx.x− iky.y)dkxdky.

(3.31)

Converting to cylindrical domain, and using the identities dkxdky = kρdkϕdkρ and

cos(kϕ)cos(ϕ) + sin(kϕ)sin(ϕ) = cos (kϕ − ϕ),

G↑xx (ρ, ϕ, z = 0, ω) =

(1

2π

)2 ∫ ∞

0

∫ 2π

0

[RTE

1 sin2(kϕ)−RTM1 cos2(kϕ)

]exp (−2Γ1z1)

exp(−ikρρ cos(kϕ − ϕ))kρdkϕdkρ. (3.32)

Let us split the integration (3.32) into two equations as following:

G↑xx (ρ, ϕ, z = 0, ω) = I1 + I2 (3.33)

49


where

I1 =

(1

2π

)2 ∫ ∞

0

∫ 2π

0RTE

1 sin2(kϕ) exp (−2Γ1z1) exp(−ikρρ cos(kϕ − ϕ))kρdkϕdkρ

(3.34)

I2 =

(1

2π

)2 ∫ ∞

0

∫ 2π

0−RTM

1 cos2(kϕ) exp (−2Γ1z1) exp(−ikρρ cos(kϕ − ϕ))kρdkϕdkρ.

(3.35)

Now changing the integration variable kϕ to θ = kϕ − ϕ, (3.34) is modified to

I1 =

(1

4π2

)∫ ∞

0RTE

1 exp (−2Γ1z1) kρdkρ

∫ 2π−ϕ

−ϕsin2(θ + ϕ)exp(−ikρρ cos(θ))dθ.

(3.36)

Use the identity for the Bessel function of 1st kind

Jn(kρρ) =(−i)−n

2π

∫ 2π

0exp(−ikρρ cos(θ))cos(nθ)dθ. (3.37)

The inner integration of (3.36) can be written as

1

2π

∫ 2π−ϕ

−ϕsin2(θ + ϕ)exp(−ikρρ cos(θ))dθ =

1

2π

∫ 2π

0sin2(θ + ϕ)exp(−ikρρ cos(θ))dθ

=1

2[J0(kρρ) + J2(kρρ)cos(2ϕ)] . (3.38)

Now using (3.38), (3.36) can be expressed as

I1 =

(1

4π

)∫ ∞

0[J0(kρρ) + J2(kρρ)cos(2ϕ)]R

TE1 exp (−2Γ1z1) kρdkρ. (3.39)

Similarly, (3.35) is simplified as

I2 =

(−1

4π

)∫ ∞

0[J0(kρρ)− J2(kρρ)cos(2ϕ)]R

TM1 exp (−2Γ1z1) kρdkρ. (3.40)

50


Combining I1 and I2, the spatial domain Green’s function in (3.33) is expressed as

G↑xx (ρ, ϕ, z = 0, ω)

=1

4π

∫ ∞

0

[J0 (kρρ)

{RTE

1 −RTM1

}+ J2 (kρρ) cos 2ϕ

{RTE

1 +RTM1

}]exp (−2Γ1z1) kρdkρ.

(3.41)

Let us write the spatial domain Green’s function in cylindrical coordinate in terms of

simplified spectral domain Green’s function as following:

G↑xx (ρ, ϕ, z = 0, ω) =

1

4π

∫ ∞

0G↑

xx (kx, ky, z = 0, ω)kρdkρ =1

4π

∫ ∞

0G↑

xx (kρ, ω)kρdkρ

(3.42)

where

G↑xx (kρ, ω) =

[J0 (kρρ)

{RTE

1 −RTM1

}+ J2 (kρρ) cos 2ϕ

{RTE

1 +RTM1

}]e−2Γ1h1 . (3.43)

Here ϕ = arctan( yx

), h1 = z1 − z0 is the thickness of 1st layer media. For the specific

monostatic configuration (ρ = 0, z = 0), the spectral domain Green’s function is further

simplified to a compact expression as

G↑xx (kρ, ω) =

[RTE

1 −RTM1

]e−2Γ1h1 . (3.44)

It can be observed from (3.42) that the spatial domain Green’s function G↑xx (ω) requires

to compute 1-D semi-infinite integration to find the response due to layered media at a

frequency ω rad/s.

3.3.4 Discussion

The FWM Green’s function proposed by (3.42) and (3.44) relates the received electric

field with the transmitted electric field with complex ratio at any frequency. Therefore,

the proposed FWM Green’s function is a unit-less frequency-dependent quantity, which

can be easily related to the VNA measured S-parameter. By simplifying (3.42) and

(3.44) it can be proved that, the FWM represents Friss transmission equation for signal

propagation. The supporting experimental analysis is presented in Section 4.4.

51

3.4 Numerical Computation of FWM Green’s Function

Lambot et al. [1] have derived a similar Green’s function by assuming TEM horn

antenna as an infinitesimal electric dipole. It is defined as a ratio of x-directed electric

field at receiver and x-directed electric current density at source. The expression of

spectral domain Green’s function is

G↑xx (kρ, ω) =

[RTM

n

Γn

ηn−RTE

n

ζnΓn

]e−2Γnhn . (3.45)

It can be proved that, these FWMs are highly correlated, and differ by a scale factor. We

denote the FWM proposed by [1] as FWM-1, and the proposed one in this work as FWM-

2. These models are versatile in finding response due to multilayered media accurately.

However, they have got inefficiency due to slow computational speed.

3.4 Numerical Computation of FWM Green’s Function

A special care is required for efficient numerical computation of semi-infinite integration

to transform the spectral domain Green’s function to spatial domain. There is singularity

on the integration path. A synthetic model of a two-layered subsurface media is consid-

ered to analyze the singularity of the FWM Green’s function. The top layer is the half

space air media, and the 2nd layer is a perfect electric conductor (PEC). Spectral domain

Green’s functions of FWM-1 [1] and FWM-2 at a fixed frequency (1 GHz) are plotted

in Figures 3.2a and 3.2b respectively. Contrary to FWM-1, the spectral domain Green’s

function of FWM-2 does not have singularity due to pole while integrating in real path

of kρ. Whereas it has got strong oscillation behavior for kρ < k1 similar to the FWM-1

Green’s function. In order to do fast integration, the integration path should avoid the

integrand singularities and, the function oscillation should be minimized. It is observed

that by applying constant phase path of integration [129], the integration becomes fast

reducing the computational time significantly. After applying constant phase path it can

be observed in Figures 3.2c and 3.2d that, the oscillation are reduced for both the Green’s

functions.

52

3.5 Validation of Proposed FWM

0 50 100 150−400

−200

0

200

400

kρ

real

(Gxx

)

0 50 100 150−400

−200

0

200

400

kρ

Im(G

xx)

(a) FWM-1 in real kρ.

0 50 100 150−4

−2

0

2

4

kρ

real

(Gxx

)

0 50 100 150−4

−2

0

2

4

kρ

Im(G

xx)

(b) FWM-2 in real kρ.

0 50 100 150

−100

0

100

kρ

real

(Gxx

)

0 50 100 150

−100

0

100

kρ

Im(G

xx)

(c) FWM-1 in constant phase path.

0 50 100 150

−2

0

2

kρ

real

(Gxx

)

0 50 100 150

−2

0

2

kρ

Im(G

xx)

(d) FWM-2 in constant phase path.

Figure 3.2: Plot of the spectral domain Green’s functions at 1 GHz frequency.


To validate the proposed FWM, a comparative analysis between the FWMs, i.e., FWM-

1 [1] and FWM-2 is carried out for a single-layered media over a large parameters vector

space and a wide frequency band. The single-layered media is bounded by half space

air media at top and PEC at the bottom. It is observed that the Green’s functions of

FWM-1 and FWM-2 differ by a constant amplitude factor, and a phase shift of 1800.

After compensating the phase shift, it is found that both the models are highly correlated

in time domain. After compensating both, phase shift and amplitude ratio, the frequency

domain Green’s functions have a very small %RMS difference (%RMS diff).

%RMS diff between two frequency domain Green’s functions, and %cross correlation

coefficient (%CCC) between two time domain Green’s functions are computed by following

53


formulas.

%RMS diff = 100×

√√√√√√∑Nf

j=1

∣∣∣G↑1xx(ωj)−G↑2

xx(ωj)∣∣∣2∑Nf

j=1

∣∣∣G↑2xx(ωj)

∣∣∣2 (3.46)

%CCC = 100×

∑Ntj=1

(g↑1xx(tj)− g↑1xx(tj)

)(g↑2xx(tj)− g↑2xx(tj)

)√∑Nt

j=1


)2∑Ntj=1


)2 . (3.47)

g↑1xx(tj) and g↑2xx(tj) are the averages of time domain Green’s functions 1 and 2 respectively.

Nf and Nt are the number of points, respectively, for the frequency and time domain

Green’s functions. Here, Green’s functions are computed at 101 points with frequency

spacing of 40 MHz over the wide frequency band of 0.5 to 4.5 GHz. Then IFFT is

applied with 4096 point to compute the time domain Green’s functions. Layered media

parameters are varied over the parameter vector space of 2 ≤ ϵr ≤ 81, 10 ≤ σ ≤ 104mS/m,

and 1 ≤ h ≤ 103cm. To cover such a wide range of space, the parameters values are varied

exponentially to compute total 4851 points (11 along ϵr, 21 along σ and 21 along h). The

layered media is bounded by half space air media on the top with thickness h1 = 35 cm,

and PEC at the bottom.

The FWMs Green’s functions are compared, and the results are presented in Table 3.1.

It is observed that, %RMS diff between the FWMs in the mentioned parameter vector

space is lower than 0.7662, and the worst case %CCC between them is greater than

99.99899. Therefore, FWM-1 and FWM-2 have similar behavior in frequency and time

domain. The time required to compute single Green’s functions over 101 frequency points

and averaged over 1000 times running in an 1.93 GHz corei3 2 GB RAM laptop are

presented in the last row of Table 3.1. The results show that the FWMs perform similarly

in terms of computational efficiency.

Figure 3.3 shows the surface plots of %RMS diff and %CCC in σ − h plane while

ϵr value is fixed at 2. The worst case %RMS diff (Figure 3.3a) is observed for ϵr = 2,

σ = 63.0957 and h = 10 cm. The point is marked by a star. The worst case %CCC

(Figure 3.3b) is located at slightly different location, and is also marked by a star. Fig-

ures 3.4a and 3.4b present the plots of spatial domain Green’s functions for both the

54

3.6 Summary

Table 3.1: Comparison between FWM-1 and FWM-2 Applied to 1L Media

ϵr Worst case %RMS diff in σ − h plane Worst case %CCC in σ − h plane

2 0.7662 0.999989916 0.5363 0.999998381 0.3541 0.9999990

Processing time in milliseconds for computing Gxx(ω) at 101 frequency pointsFWM-1 FWM-2

2112.7 2103.0

log10

h (cm)

log 10

σ (m

S/m

)

εr=2, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%RMS Diff.

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

(a) %RMS difference.

log10

h (cm)

log 10

σ (m

S/m

)

εr=2, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%CCC99.999

99.9991

99.9992

99.9993

99.9994

99.9995

99.9996

99.9997

99.9998

99.9999

(b) %Cross correlation coefficient.

Figure 3.3: Comparison between FWM-1 and FWM-2 for ϵr = 81 and σf = 0 in σ − hplane.

models for the parameter vector at which worst case %RMS diff is observed. It can be

observed that the amplitude plots differ by a scale factor whereas, the phase plots match

perfectly. Figures 3.4c and 3.4d present the plots of both the Green’s functions respec-

tively, in frequency and time domain after compensating the amplitude factor and phase

shift. It can be observed that, both the Green’s functions curves have merged very well

and it’s difficult to distinguish them in plots.

3.6 Summary

The FWMs are accurate to describe the electromagnetic wave propagation in inhomoge-

neous media. The proposed FWM (FWM-2) has been derived by solving 3-D Maxwell’s

equations, and applying EM boundary conditions. FWM-2 Green’s function relates the

received electric field with the transmitted electric field. The response due to the sub-

surface media can be easily related to the VNA measured S11(ω) by suitable arithmetic

55

3.6 Summary

0.5 1 1.5 2 2.5 3 3.5 4 4.50

500

1000

1500

Frequency (GHz)

|Gxx

|

FWM−1

0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−2

0

2

4

Frequency (GHz)

∠G

xx(r

ad)

(a) Gxx(ω) for FWM-1

0.5 1 1.5 2 2.5 3 3.5 4 4.50

2

4

6

8

Frequency (GHz)

|Gxx

|

FWM−2

0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−2

0

2

4

Frequency (GHz)

∠G

xx(r

ad)

(b) Gxx(ω) for FWM-2

0.5 1 1.5 2 2.5 3 3.5 4 4.50

2

4

6

8

Frequency (GHz)

|Gxx

|

FWM−1 FWM−2

0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−2

0

2

4

Frequency (GHz)

∠G

xx(r

ad)

(c) Gxx(ω) for FWM-1 and FWM-2

0 1 2 3 4 5 6−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time (ns)

g xx(t

)

FWM−1 FWM−2

(d) gxx(t) for FWM-1 and FWM-2

Figure 3.4: Green’s function plots for FWMs

operation with linear transfer functions as mentioned in (3.2). This is important for the

physical interpretation of far-field GPR data by using a full wave model.

The spectral domain Green’s function is represented by a compact formula that can

calculate the response due to multilayered media in an iterative loop. The spatial domain

Green’s function is obtained by a semi-infinite integration. The computational efficiency

of the FWM has been enhanced by reducing 3-D integration to just 1-D by layered-media

and point source assumptions. The efficiency has been further improved by numerical

computation of integration over a constant phase path. These features make proposed

FWM (FWM-2) useful for accurate and efficient characterization of layered media.

Based on the comparative analysis presented in Section 3.5, it is observed that FWM-

2 is highly correlated to FWM-1 [1] over a wide frequency band and large parameter

vector space. It is expected that FWM-2 would be as accurate and versatile as FWM-1

56

3.6 Summary

in finding response due to multilayered media with various kind of antennas in far-field

configuration. Next chapter focuses on techniques to reduce the computational complexity

of FWM-2 to propose time efficient models.

57

3.6 Summary

58

CHAPTER4Development of Modified Plane

Wave Models

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Derivation of MPWMs by Simplification of FWM . . . . . . 61

4.2.1 Simplification of FWM-2 for a Two-layered and Three-layered

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 Obtaining a Generalized Formula for Green’s Function . . . . . . 65

4.2.3 Representation of Green’s Function by a Compact Formula . . . 68

4.3 Comparative analysis of the MPWMs . . . . . . . . . . . . . . 70

4.4 Relation of Models with Friis Transmission Equation . . . . . 74

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

59

4.1 Introduction

This chapter presents the modified plane wave models (MPWMs) which are compu-

tationally efficient. FWM-2 proposed in the previous chapter is simplified by analytical

technique to yield MPWM-1, MPWM-2, and MPWM-3. A comparative analysis with

FWM-2 shows that the MPWMs are very time-efficient, and MPWM-2 and MPWM-3

are as accurate as FWM-2.

4.1 Introduction

In majority of EM modelling, the accuracy and computational speed are two contradic-

tory goals to be achieved. In Chapter 3, it has been demonstrated that a compact formula

for FWM can be obtained for layered media, and monostatic far-field configuration. The

computational efficiency of the FWM (FWM-2) has been enhanced by reducing integra-

tion dimension from 3-D to 1-D, and by following constant phase path of the integration

variable. Still, the computational complexity of FWM-2 is significantly high compared

to the other simplified models based on plane wave assumptions. The inversion of FWM

is a computationally intensive work because of nonlinearity, and slow speed of forward

model computation. As a result, GPR implementation for many real-time applications

like road, bridge inspection would be time-consuming process.

On the contrary, many researchers have opted for the simplified modelling approaches

based on plane wave approximations [17,18,20,21,27]. These models are capable of reveal-

ing limited information on the media under investigation. The major drawback of these

models is inaccuracy due to unrealistic assumptions. The plane wave assumptions are not

valid when the source and media are located in finite distance. Therefore, modifications

are essential to improve these schemes. There are popular assumptions to improve the

accuracy of plane wave models (PWMs) by introducing a spreading factor according to

the distances of the targets [42].

In this work, the integral expression of FWM-2 is simplified to yield numerically

efficient modified plane wave models i.e. MPWM-1, MPWM-2, and MPWM-3. Among

these, MPWM-2 is most accurate, and MPWM-3 is simplest to apply for multilayered

media. Their expressions are similar to the PWM, with a modification of multiplication

factors to account for losses due to spreading, reflections, and refractions at multilayered

60

4.2 Derivation of MPWMs by Simplification of FWM

interfaces. A comparative analysis demonstrates that MPWM-2 and MPWM-3 are highly

correlated with FWM-2.

The chapter begins with a discussion on plane wave model for layered media. FWM-2

integral is solved for an example two-layered and three-layered media to yield compu-

tationally efficient formulas of MPWM-1 and MPWM-2. Subsequently, a generalized

formula is obtained to represent them for multilayered media. Then, by applying suit-

able boundary conditions, a compact iterative formula for multilayered media is obtained

to yield MPWM-3. A comprehensive analysis is carried out to extract the important

characteristics of MPWMs, and compare their performance with FWM-2. At the end an

analysis is presented to correlate the GPR model with the Friis transmission equation.

The chapter concludes with a brief summary.


The FWM-2 has been discussed in Chapter 3. It is possible to simplify FWM-2 by

suitable analytical method to increase its computation efficiency. The process involves

modification of the common reflection method with an appropriate complex spreading

factor obtained after simplifying FWM-2 integral. The common reflection method is

based on plane wave propagation at normal direction of a planar multilayered media. This

method is found in various literatures [16, 18, 27], and is mostly applied for approximate

estimation of the layered media’s electrical parameters while conductivity is negligible.

The first order reflection(r1n,n+1

)from nth interface (zn) is obtained by solving (2.65) as

(r1n,n+1

)= rn,n+1

n−1∏j=1

(τj,j+1τj+1,j)n∏

j=1

exp(−2γjhj)

= rn,n+1

n−1∏j=1

(1− (rj,j+1)

2) n∏

j=1

exp(−2γjhj). (4.1)

where rn,n+1 is the local reflection coefficient, and τn,n+1 is the transmission coefficient of

plane wave at nth layer interface zn as given below:

rn,n+1 =Zn+1 − Zn

Zn+1 + Zn(4.2)

61


τn,n+1 = 1 + rn,n+1 =2Zn+1

Zn+1 + Zn(4.3)

where Zn =√

ζnηn

is the impedance of nth layer media. r1n,n+1 is the local reflection

coefficient (rn,n+1) translated to the origin O i.e. it accounts for the losses and phase shift

due to two way travel of EM wave from layer 1 to n. Higher order reflections from any

interface can be found by simplifying (2.65). The Green’s function due to a multilayered

media can be calculated by summing all the reflections from all interfaces.

The plane wave assumption is valid when source is located at infinite distance from

the target to be illuminated. For finite distance, (4.1) needs to be modified to make it

accurate. Therefore, it is proposed to simplify the expression of FWM-2 presented in

Chapter 3, and use the result obtained to modify the common reflection method.

4.2.1 Simplification of FWM-2 for a Two-layered and Three-layered

Media

FWM-2 Green’s function can be written from Equations (3.42)-(3.44) as

G↑xx(ω) =

1

4π

+∞∫0

[RTE

1 −RTM1

]e−2Γ1h1kρdkρ. (4.4)

Here, Γ1 =√

k2ρ − k12 =

√k2ρ + γ12, and RTE

1 , RTM1 are the global reflection coefficients

for TE and TM waves represented by Equations (2.98)-(2.101).

Let us consider the case of two-layered media. A two-layered media can be created with

half space air media (σ1 = 0 and ϵr1 = 1) followed by a media having either conductivity

(σ2) infinity or thickness (h2) infinity. Here the expression of global reflection coefficients

for TE and TM can be obtained as RTE1 = rTE

1,2 and RTM1 = rTM

1,2 from Equations (2.98)-

(2.101). Therefore, (4.4) modifies to

G↑xx(ω) =

1

4π

+∞∫0

[rTE1,2 − rTM

1,2

]e−2h1

√kρ

2+γ12kρdkρ. (4.5)

It can be observed in (4.5) that, e−2h1

√kρ

2+γ12 is a highly oscillation function, and[rTE1,2 − rTM

1,2

]changes slowly with respect to the integration variable kρ. Now apply-

62


ing method of stationary phase [118] (pp. 79− 82), the integral in (4.5) can be simplified

to

G↑xx(ω) =

r1,22π

+∞∫0

e−2h1

√kρ

2+γ12kρdkρ. (4.6)

It can be noted that, ddkρ

(√kρ

2 + γ12)

= 0 at kρ = 0. Again at kρ = 0, rTE1,2 = r1,2 =

−rTM1,2 , where r1,2 is plane wave reflection coefficient at z1 interface for normal incidence

as defined in (4.2). Applying integration by parts repeatedly, (4.6) can be simplified to

G↑xx(ω) =

r1,22π

[e−2γ1h1

2h1γ1

+e−2γ1h1

4 (h1)2

]. (4.7)

Now, the case of a three-layered media is considered. In this case RTE1 and RTM

1 can

be expressed as following:

RTE1 =

rTE1,2 +RTE

2 exp(−2Γ2h2)

1 + rTE1,2 R

TE2 exp(−2Γ2h2)

=rTE1,2 + rTE

2,3 exp(−2Γ2h2)

1 + rTE1,2 r

TE2,3 exp(−2Γ2h2)

= rTE1,2 + rTE

2,3

(1− rTE

1,22)e−2Γ2h2 + . . . (4.8)

and

RTM1 =

rTM1,2 +RTM

2 exp(−2Γ2h2)

1 + rTM1,2 RTM

2 exp(−2Γ2h2)

=rTM1,2 + rTM

2,3 exp(−2Γ2h2)

1 + rTM1,2 rTM

2,3 exp(−2Γ2h2)

= rTM1,2 + rTM

2,3

(1− rTM

1,22)e−2Γ2h2 + . . . (4.9)

Using expressions of RTE1 and RTM

1 in (4.4), Green’s function for a three-layered media

can be written as

G↑xx(ω) =

1

4π

+∞∫0

[(rTE1,2 − rTM

1,2

)+{rTE2,3

(1− rTE

1,22)− rTM

2,3

(1− rTM

1,22)}

e−2Γ2h2 + . . .]

× e−2Γ1h1kρdkρ. (4.10)

63


Rearranging the terms we get

G↑xx(ω) =

1

4π

+∞∫0

[rTE1,2 − rTM

1,2

]e−2Γ1h1kρdkρ

+1

4π

+∞∫0

[rTE2,3

(1− rTE

1,22)− rTM

2,3

(1− rTM

1,22)]

e−2(Γ1h1+Γ2h2)kρdkρ

+Higher order terms. (4.11)

Again by applying method of stationary phase followed by integration by parts, (4.11)

can be simplified as

G↑xx(ω) =

r1,22π

{1

2h1γ1

+1

4 (h1)2

}e−2γ1h1

+r2,3

(1− r21,2

)2π

1

2(h1γ1

+ h2γ2

) +

(h1

γ31+ h2

γ32

)4(h1γ1

+ h2γ2

)3 e−2(γ1h1+γ2h2)

+Higher order terms. (4.12)

It may be noted that, terms with higher than h2 variation are neglected from the analytical

expression of the integral with highly oscillating term in (4.11) as they contribute little

to the Green’s function i.e.

+∞∫0

e−2(Γ1h1+Γ2h2)kρdkρ = e−2(γ1h1+γ2h2)

1

2(h1γ1

+ h2γ2

) +

(h1

γ31+ h2

γ32

)4(h1γ1

+ h2γ2

)3 + . . .

. (4.13)

In (4.12), the 1st term signifies contribution due to 1st order reflection (O11) from interface

z1, which can be represented as

O11 =

r1,22π

{1

2h1γ1

+1

4 (h1)2

}e−2γ1h1 =

r11,22π

{1

2h1γ1

+1

4 (h1)2

}(4.14)

where r11,2 expression is given by (4.1). The superscript of O11 denotes the order of reflec-

tion coefficient, and subscript denotes interface number. Similarly, the 2nd term signifies

64


contribution due to 1st order reflection (O12) from interface z2 i.e.

O12 =

r12,32π

1

2(h1γ1

+ h2γ2

) +

(h1

γ31+ h2

γ32

)4(h1γ1

+ h2γ2

)3 . (4.15)

4.2.2 Obtaining a Generalized Formula for Green’s Function

Based on simplified expression of FWM-2 presented in (4.12) for a three-layered media,

and using (4.1) to find plane wave reflection due to multilayered media, the 1st order

reflection from the interface zn can be generalized as

O1n =

(r1n,n+1

2πi

) 1

2∑n

j=1 hj/γj+

(∑nj=1 hj/γ

3j

)4(∑n

j=1 hj/γj

)3. (4.16)

Here division of i (√−1) is introduced to have phase matching with (4.1). In fact division

of i is important to maintain proper phase relation between the received and transmitted

waves, and thereby to measure accurate delays of reflected signals from different layers.

Let us rewrite (4.16) to understand its feature and obtain a more generalized version to

represent multilayered media

O1n =

rn,n+1

n−1∏j=1

(1− (rj,j+1)

2) 1

2πi

1

2∑n

j=1 hj/γj+

(∑nj=1 hj/γ

3j

)4(∑n

j=1 hj/γj

)3

×

n∏j=1

exp(−2γjhj)

= L1Rn

L1Snexp

n∑j=1

−2γjhj

(4.17)

where L1Rn

= rn,n+1∏n−1

j=1

(1− (rj,j+1)

2)and L1

Sn= 1

2πi

(1

2∑n

j=1 hj/γj+

(∑n

j=1 hj/γ3j )

4(∑n

j=1 hj/γj)3

).

L1Rn

signifies the losses due to reflections and refractions at different interfaces, and L1Sn

signifies the spreading loss for traveling the path∑n

j=1 2hj . A generalized formula to

find reflection coefficients due to mth order reflections from interface zn can be written as

following:

Omn =

Pmn∑

k=1

Lm,kRn

Lm,kSn

exp

n∑j=1

−2γjajhj

(4.18)

65


where aj are +ve integer constants having values more than one, and are related by

following inequality:

n+m− 1 ≤n∑

j=1

aj ≤ (n− 1)m+ 1. (4.19)

Pmn is the total number of possible ways the inequality (4.19) is satisfied for mth order

reflection from interface zn. For kth possible way, Lm,kRn

is the losses due to reflections

and refraction in multiple interfaces, and Lm,kSn

is the spreading loss. The Lm,kSn

can be

calculated by the following expression:

Lm,kSn

=1

2πi

1

2∑n

j=1 ajhj/γj+

(∑nj=1 ajhj/γ

3j

)4(∑n

j=1 ajhj/γj

)3. (4.20)

Lm,kRn

depends on the path followed by the ray. For a limited number of layers, it can be

easily visualized. For example, Figure 4.1 shows how 2nd order reflections can occur from

interfaces z2 and z3. It shows that there is one way (P 22 = 1) the 2nd order reflection can

originate from the interface z2. On the contrary, there are three ways (P 23 = 3) the 2nd

order reflections can originate from the interface z3. Based on same principle, Pm2 = 1

Figure 4.1: Example 2nd order reflections from interface z2 and z3

for the mth order reflection from interface z2, and Lm,1R2

can be found as

Lm,1R2

=[r2,3

(1− r21,2

)(r2,1r2,3)

(m−1)]. (4.21)

66


0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

15

20

Frequency (GHz)

|Gxx

|

0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−2

0

2

4

Frequency (GHz)

∠G

xx(r

ad)

FWM−2MPWM−2, N

o=5

MPWM−2, No=20

(a) Frequency domain plot.

1 2 3 4 5 6−5

−4

−3

−2

−1

0

1

2

3

4

5

Time (ns)

g xx(t

)

FWM−2MPWM−2, N

o=5

MPWM−2, No=20

(b) Time domain plot.

Figure 4.2: Changes of MPWM-2 Green’s function with order of reflection (No).

The overall Green’s function contributed by reflections with maximum order No from all

the layer interfaces (z1 to zN−1) can be expressed as

G↑MPWMxx (ω) =

No∑k=1

N−1∑j=1

Okj . (4.22)

Let us denote the above model (4.22) as modified plane wave model (MPWM) to differ-

entiate it from the plane wave propagation expression described in (4.1). From (4.20),

it is clear that the spreading loss has two components. The 1st term decays with 1h and

2nd term with 1h2 . The Green’s function obtained by considering only 1

h variation term is

denoted as MPWM-1 where as MPWM-2 for considering both 1h and 1

h2 terms. The No

value should be decided best on accuracy requirement of the GPR system.

Figure 4.2 explains how the frequency and time response of MPWM-2 for an example

water layer becomes closer to FWM-2 as No value is increased from 5 to 20. It is clear

that the MPWMs don’t require time-consuming integration, and express the reflections

from multilayer interfaces and their higher order terms seperately. However, as we con-

sider for higher order reflections from multilayer interfaces, computation of reflection and

transmission loss Lm,kRn

becomes difficult with number of possible paths Pmn increasing ex-

ponentially. The problem becomes further worse for modelling multilayered media having

thin layers, with higher order reflections contributing the Green’s function significantly.

Neglecting higher order terms results error in computing the Green’s function. It is pos-

67


sible to restrict the number of higher order terms based on the GPR system dynamic

range. For example, if the GPR system has a dynamic range of 70 dB, the reflection

terms which are 70 dB lower than the maximum allowable input signal can be neglected

for computing the Green’s function. Again, the number of terms can be restricted by

applying a time window, in case, the GPR data is processed in the time domain. For the

synthetic example presented in Figure 4.2, reflected terms upto 2nd order are sufficient to

represent the water layer, if a time window is defined over 0 to 2.6 ns. These techniques

can significantly reduce the computational complexity of MPWM-1 and MPWM-2.

4.2.3 Representation of Green’s Function by a Compact Formula

In the previous sub-section we have found a generalized formula for the MPWMs Green’s

function which has a complexity that increases with the number of layers. Here we like to

simplify the formula by applying EM boundary conditions at the interfaces. Let us solve

it for an example three-layered (3L) media as shown in Figure 4.3. It is assumed that the

Figure 4.3: Example three-layered media.

radar wave is propagating in +z direction with spreading, reflection, transmission losses in

the spatial domain described by (4.17). As per assumptions mentioned in Section 3.3.1,

an electric field has to exist in the x-direction only. The wave in the 1st layer can be

written as a linear superposition of transmitted and reflected wave [118] (pp. 48− 52) as

following.

E1x = A1

[L+S (z) e−γ1z +RTEM

1 L−S (z) e−2γ1z1+γ1z

](4.23)

where, RTEM1 is the reflection coefficient of the incident wave at the interface z1; L

+S (z)

and L−S (z) denote the spreading losses of down-going and up-going waves, respectively,

and they are functions of the total travel path. A1 is an unknown constant. Similarly the

68


wave in the 2nd layer is represented as

E2x = A2

[L+S (z) e−γ2z + r2,3L

−S (z) e−2γ2z2+γ2z

](4.24)

where r2,3 is the reflection coefficient for a down-going wave at the interface z2 and A2 is

an unknown constant. The wave in the layer 3 is written as

E3x = A3L+S (z) e−γ3z (4.25)

where A3 is an unknown constant. Since, the layer 3 extends to infinity, there is only

down-going wave present in the region. The unknown parameters A1, A2, and A3 are

related to each others based on the boundary conditions at the interfaces. Based on these

relations, RTEM1 can be evaluated.

It can be observed that the down-going wave at layer 2 is a summation of the trans-

mitted down-going wave from layer 1, and the up-going wave from layer 2 reflected at

interface z1. Therefore, the boundary condition at the interface z1 can be written as

A2L1S1e−γ2z1 = A1L

1S1e−γ1z1τ1,2 +A2r2,3r2,1L

1S2e−2γ2z2+γ2z1 . (4.26)

It may be noted that, at z = z1, L+S (z) is taken as L1

S1and L−

S (z) is taken as L1S2.

L1S1

and L1S2

spreading losses are calculated based on (4.20) for a travel path of 2h1 and

2h1 + 2h2 respectively. The logic is that, the Green’s function is contributed by the

reflection coefficients for which reference point is z = 0 for all the received signals. Now

observe that, the up-going wave at layer 1 is contributed by the reflected down-going wave

in layer 1 at interface z1, and the transmitted up-going wave from layer 2. Therefore, the

boundary condition at interface z1 can be expressed as

A1RTEM1 L1

S1e−γ1z1 = A1L

1S1e−γ1z1r1,2 +A2r2,3τ2,1L

1S2e−2γ2z2+γ2z1 . (4.27)

Now finding A2 from (4.26) and replacing its expression in (4.27), and using the identities

69

4.3 Comparative analysis of the MPWMs

rn,n+1 = −rn+1,n and τn,n+1 = 1 + rn,n+1, RTEM1 can be expressed as

RTEM1 =

L1S1r1,2 + r2,3L

1S2e−2γ2(z2−z1)

L1S1

+ r1,2r2,3L1S2e−2γ2(z2−z1)

. (4.28)

Here, r2,3 is the local reflection coefficient at the interface z2. RTEM1 is the generalized

reflection coefficient of radar wave due to three-layered media. For more than three-layered

geometry, (4.28) can be generalized by just replacing r2,3 with RTEM2 as following.

RTEM1 =

L1S1r1,2 +RTEM

2 L1S2e−2γ2h2

L1S1

+ r1,2RTEM2 L1

S2e−2γ2h2

(4.29)

where h2 = z2−z1. ForN -layered media separated by (N−1) interfaces, any RTEMn can be

evaluated by the iterative formula (4.29) with recursive loop stopping at RTEMN−1 = rN−1,N .

Now, the overall Green’s function for modified plane wave model (MPWM) is written as

Gxx(0, ω) = RTEM1 L1

S1e−2γ1h1 = RTEM

1

1

2πi

{1

2h1γ1

+1

4h12

}e−2γ1h1 . (4.30)

Equation (4.30) can be used for calculating the response due to layered media having any

number of layers with ease of computation. Let us denote this model as MPWM-3. This

is a novel scheme to represent response due to multilayered media by a compact iterative

formula in the spatial domain with no requirement of integration.


Based on the comparative analysis presented in Section 3.5, it has been demonstrated

that FWM-1 [1] and FWM-2 are highly correlated. In this section, a similar exercise is

carried out for a synthetic single-layered (1L) media, and a two-layered (2L) media to

compare the accuracy and time-efficiency of the MPWMs.

The Green’s functions are as usual computed at 101 points over the frequency band

of 0.5 to 4.5 GHz. Then 4096 points IFFT is applied to compute the time domain

Green’s functions. The 1L media parameters are varied over the parameter vector space

of 2 ≤ ϵr ≤ 81, 10 ≤ σ ≤ 104mS/m, and 1 ≤ h ≤ 103cm. To cover such a wide range of

parameter vector space, the parameters are varied exponentially to compute total 4851

70


Table 4.1: Comparative Analysis of Models Applied to 1L Media

Worst case %RMS diff with FWM-2 in σ − h plane

ϵrMPWM-1 MPWM-2 MPWM-3

No = 5 No = 20 No = 5 No = 20

2 4.4075 4.4096 0.1376 0.1351 0.690316 6.0538 2.6394 5.5798 0.0674 1.246081 21.2627 2.5535 21.1270 0.6580 1.3156

Worst case %CCC with FWM-2 in σ − h plane

ϵrMPWM-1 MPWM-2 MPWM-3

No = 5 No = 20 No = 5 No = 20

2 99.91482 99.91476 99.99995 99.99995 99.9977116 99.80958 99.96207 99.84174 99.99998 99.9925881 98.03462 99.96406 98.06755 99.99815 99.99265

Table 4.2: Computational Efficiency of Models Applied to 1L Media

Processing time in milliseconds for computing Gxx(ω) at 101 frequency points

MPWM-1 MPWM-2MPWM-3 FWM-1 FWM-2

No = 5 No = 20 No = 5 No = 20

6.2 7.0 6.3 8.2 13.1 2112.7 2103.0

points (11 along ϵr, 21 along σ, and 21 along h). The media is bounded by half space

air media on the top with thickness h1 = 35 cm, and PEC at the bottom. The MPWMs

Green’s functions are compared with FWM-2 and results are presented in Table 4.1.

The order of reflections (No) is considered as 5 and 20 for computing MPWM-1 and

MPWM-2. Only the worst case %RMS diff and %CCC values obtained for the ϵr values

2, 16 and 81 over σ − h plane are presented. The %RMS diff and %CCC are defined by

Equations (3.46) and (3.47) in the previous chapter. The comparative analysis for the ϵr

in between 16 and 81 are not presented as the worst case values are better in this range of

the parameter. Table 4.1 shows that, MPWM-2 and MPWM-3 are highly correlated with

FWM-2, and MPWM-2 is more accurate compared to MPWM-3 as we consider for No

as 20. The worst cases for %RMS diff and %CCC are observed for ϵr = 81. In Table 4.2,

the computational efficiency of all the models are highlighted. The average timing for

computing single Green’s functions is calculated by running the program 1000 times in

an 1.93 GHz core i3 laptop. This result demonstrates that the MPWMs are extremely

time-efficient compared to the FWMs.

Figure 4.4 shows the surface plots of %RMS diff, and %CCC between the MPWMs

and FWM-2 in σ−h plane while ϵr value is fixed at 81. The top figures (Figure 4.4a, 4.4b)

71


are meant for MPWM-1, the middle figures (Figure 4.4c, 4.4d) for MPWM-2, and the

bottom figures (Figure 4.4e, 4.4f) for MPWM-3. For MPWM-1 and MPWM-2, the worst

case %RMS diff and %CCC are observed at high ϵr, lowest σ and h. For MPWM-3, the

worst cases occur for lowest σ and h = 63.1 cm. It signifies that as conductivity (σ)

decreases, the difference between MPWMs and FWM-2 increases. MPWM-1 is the least

accurate among three MPWMs. The worst case %RMS diff and %CCC regions are little

different for MPWM-2 and MPWM-3. This happens because of approximation used for

deriving the iterative formula for MPWM-3 as mentioned in Section 4.2.3. In general,

MPWM-2 and MPWM-3 are highly correlated with FWM-2 over the wide parameter

vector space.

Following this single-layer characterization, same exercise is carried out for a 2L media

bounded by half space air media, and a PEC at the bottom. In this case, the No value is

considered as 4 with total 22 terms require to be computed for MPWM-1 and MPWM-2

Green’s function. At first, the top layer media parameters are varied in a same process

as discussed for single-layered media while keeping the bottom layer parameters constant

as ϵr3 = 6.2, σc3 = 20 mS/m, σr3 = 22mS/m/GHz, and h3 = 10 cm. Following this

the bottom layer media parameters are varied while the top layer parameters are kept

constant at ϵr2 = 2.33, σc2 = 20 mS/m, σr2 = 22 mS/m/GHz, and h2 = 20.5 cm. The

results of comparative analysis are presented in Table 4.3. Now it can be observed that,

MPWM-3 is more accurate compared to MPWM-2. The difference between MPWM-2 and

FWM-2 is further increased with variation of bottom layer parameters. This is due to the

fact that, more number of higher order reflections become significant from the z3 being

interface between second layer and PEC. The processing speeds of MPWMs are again

observed (Table 4.4) to be significantly higher compared to the FWMs. As we consider

the higher order reflections for MPWM-1 and MPWM-2, the complexity further increases

with number of terms and computational time increasing exponentially. For example with

No = 5, MPWM-2 Green’s function need to compute 38 terms, and its computational

time observed is more compared to MPWM-3. These analysis concludes that MPWM-3

is more suitable to represent a multilayered media, even though its accuracy is slightly

lower than MPWM-2 in ideal case.

72


log10

h (cm)

log 10

σ (m

S/m

)

εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%RMS Diff.

4

6

8

10

12

14

16

18

20

(a) %RMS diff for MPWM-1

log10

h (cm)

log 10

σ (m

S/m

)

εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%CCC

98.2

98.4

98.6

98.8

99

99.2

99.4

99.6

99.8

(b) %CCC for MPWM-1

log10

h (cm)

log 10

σ (m

S/m

)

εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%RMS Diff.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(c) %RMS diff for MPWM-2

log10

h (cm)

log 10

σ (m

S/m

)εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%CCC

99.9982

99.9984

99.9986

99.9988

99.999

99.9992

99.9994

99.9996

99.9998

(d) %CCC for MPWM-2

log10

h (cm)

log 10

σ (m

S/m

)

εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%RMS Diff.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(e) %RMS diff for MPWM-3

log10

h (cm)

log 10

σ (m

S/m

)

εr=81, σ

r=0

0 0.5 1 1.5 2 2.5 31

1.5

2

2.5

3

3.5

4

%CCC

99.993

99.994

99.995

99.996

99.997

99.998

99.999

(f) %CCC for MPWM-3

Figure 4.4: Comparison between MPWMs and FWM-2 for ϵr = 81 and σf = 0 in σ − hplane.

73

4.4 Relation of Models with Friis Transmission Equation

Table 4.3: Comparative Analysis of Models Applied to 2L Media

Worst case %RMS diff with FWM-2 in σ − h plane

ϵrMPWM-1 (No = 4) MPWM-2 (No = 4) MPWM-3

Top layer Bottom layer Top layer Bottom layer Top layer Bottom layer

2 0.7113 0.4193 0.6731 0.4246 0.2703 0.678216 0.2342 6.5997 0.1845 6.6215 0.1471 0.962581 1.9927 21.2247 1.9252 21.1802 0.2493 1.0082

Worst case %CCC with FWM-2 in σ − h plane

ϵrMPWM-1 (No = 4) MPWM-2 (No = 4) MPWM-3

Top layer Bottom layer Top layer Bottom layer Top layer Bottom layer

2 99.99965 99.99922 99.99776 99.99916 99.99965 99.9977216 99.99974 99.75942 99.99984 99.7559 99.99989 99.9953581 99.98003 97.97628 99.98148 97.97705 99.99969 99.99513

Table 4.4: Computational Efficiency of Models Applied to 2L Media

Processing time in milliseconds for computing Gxx(ω) at 101 frequency pointsMPWM-1 (No = 4) MPWM-2 (No = 4) MPWM-3 FWM-1 FWM-2

15.3 18.2 22.6 4592.7 4554.6

4.4 Relation of Models with Friis Transmission Equation

The MPWMs are obtained by simplifying FWM-2. MPWM-1 is the most approximated

model among the MPWMs. Let us consider the case of reflection by an infinite size

perfect electric conductor (PEC) plate located at distance h1 from the antenna. In this

case, r1,2 = −1. MPWM-1 Green’s function is written as

G↑xx(ω) =

−1

2πi

e−2h1γ1

(2h1/γ1). (4.31)

For free space with air media, γ1 = iβ1 = iω√µ0ϵ0 =

2πiλ1

. The (4.31) can be simplified to

G↑xx(ω) =

−e−iβ1(2h1)

2h1λ1. (4.32)

The LTFs relates the Green’s function with the VNA measured S11 (ω) by Equation (3.1).

By neglecting Hi (ω) and Hf (ω) in (3.1), and using the Green’s function expression from

(4.32), the S11 (ω) is written as

S11 (ω) = Ht (ω)G↑xx(ω)Hr (ω) = −Ht (ω)Hr(ω)

e−iβ1(2h1)

2h1λ1. (4.33)

74

4.5 Summary

Let us use the Friis transmission equation to find relation between the reflected electric

field (Erx(ω)) and transmitted electric field (Et

x(ω)) for the target as infinite size PEC.

For plane wave incidence on a PEC, reflection coefficient is -1. Because of infinite size

reflector, we can assume that a mirror image of source is present at an equal distance

from the reflector at opposite side of the source. The ratio of received and transmitted

power can be written as following:

Pr(ω)

Pt(ω)=

λ12GtGr

(4π(2h1))2 (4.34)

where Gt and Gr are, respectively, the transmit and receive antennas power-gains. The

ratio of reflected and transmitted electric field (Erx(ω) and Et

x(ω)) can be written as

Erx(ω)

Etx(ω)

= −λ1Ht′(ω)Hr

′(ω) e−iβ1(2h1)

4π(2h1)= −λ1

2

4πHt

′(ω)Hr

′(ω)

(e−iβ1(2h1)

2h1λ1

)(4.35)

where Ht′(ω) and Hr

′(ω) are the complex gains of transmitter and receiver antennas,

respectively. For monostatic configuration Ht′(ω) = Hr

′(ω), and they are related with

the antenna power gain (Gt(ω)) as

∣∣∣Ht′(ω)∣∣∣2 = ∣∣∣Hr

′(ω)∣∣∣2 = Gt(ω). (4.36)

By correlating (4.33) and (4.35), it can be observed that

Ht (ω)Hr (ω) = H (ω) =λ1

2

4πHt

′(ω)Hr

′(ω) . (4.37)

Now, λ12

4π

∣∣∣Ht′(ω)Hr

′(ω)∣∣∣ (=λ1

2

4π Gt(ω)) is equivalent to the antenna aperture area at fre-

quency ω. Therefore, H (ω) (=Ht (ω)Hr (ω)) represents the antenna aperture area, and

4πλ1

2 |H (ω)| is the antenna gain.

4.5 Summary

In this chapter, FWM-2 has been analytically solved to yield three modelling schemes

i.e. MPWM-1, MPWM-2, and MPWM-3. Rigorous comparative analysis has shown

75

4.5 Summary

that all these models are computationally very efficient compared to the FWMs. The

MPWMs are found to be at least 160 times time-efficient compared to the FWMs for

the example media configurations. Among the MPWMs, MPWM-1 is most approximate

one. Its accuracy is limited in the parameter vector space. MPMW-2 is the closest

with the FWMs, as higher order reflection coefficients are considered. As the number of

layer increases, the modelling complexity of MPWM-1 and MPWM-2 increases, and the

computational efficiency degrades. This issue has been finally solved by the introduction

of MPWM-3. MPWM-3 is represented by a compact formula that can calculate the

reflections due to multilayered media by iterative loop. Therefore, it is most suitable

among the MPWMs to represent the multilayered media. These models with superior

performance have a significant advantage over FWMs for GPR applications related to the

multilayered media.

The proposed models i.e. FWM-2 and MPWMs have been correlated with the Friss

transmission equation. This helps to interpret the monostatic GPR data in a realistic way

with the help of VNA measured reflection coefficients and standard antenna parameters.

76

CHAPTER5Inversion Strategy for the

Proposed Models

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1 Test setup for Calibration . . . . . . . . . . . . . . . . . . . . . . 79

5.2.2 Far-field Distance of Antenna . . . . . . . . . . . . . . . . . . . . 80

5.2.3 Theory of Calibration . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.4 Results of Calibration . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Model Inversion Approach . . . . . . . . . . . . . . . . . . . . . 85

5.4 Accuracy and Stability Analysis . . . . . . . . . . . . . . . . . 85

5.4.1 Response Surface Plots . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.2 Effect of Antenna Height Measurement Error . . . . . . . . . . . 88

5.4.3 System Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.4 Summary of Accuracy and Stability Analysis . . . . . . . . . . . 97

5.5 Layer Stripping Approaches . . . . . . . . . . . . . . . . . . . . 98

5.5.1 Layer Stripping by Approximate Method (LSAM) . . . . . . . . 98

5.5.2 Layer Stripping by Inversion Method (LSIM) . . . . . . . . . . . 100

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

77

5.1 Introduction

This chapter presents the inversion approach for the proposed models. The measure-

ment Green’s function is extracted by the calibration process. In order to invert the

models, an objective function is defined. Then the inversion is realized by applying gra-

dient method. The optimization process by gradient method requires initial information

on layered media parameters, which is obtained by novel layer stripping methods. The

chapter also includes a limited analysis on synthetic data to understand the accuracy and

stability of the models.

5.1 Introduction

Inversion of GPR model is an important process for geophysical characterization of sub-

surface media. The success of an inverse modelling scheme primarily depends on its speed

and accuracy to invert the model parameters. Its success also depends on the ability to

work with required accuracy in the presence of noise, interference, and measurement error.

The system calibration is an integral part of a model based inverse scattering scheme. It

helps to reduce the effect of measurement error, and filter out uncorrelated noise from

the data [100]. Often the global optimization schemes [1,18,25,58] are applied to invert a

model having nonlinearity and multiple minima. The gradient-based methods [41,44,55]

work efficiently while sufficient initial information is available. The initial information

on model parameters is very important to decide the starting point of an optimization

process. It is also beneficial for successful, reliable and efficient implementation of both

types of optimization schemes. However, extracting the initial information on unknown

media is a difficult problem.

Layer stripping (LS) [20, 34, 41, 42] is a popular approach to find initial information

on layered-media parameters. In this method, the GPR processing is done in time do-

main for reconstruction of each layer in step-by-step starting from the 1st layer. In most

cases, there are common assumptions like plane wave propagation, and no presence of

multiple reflections. In a recent work, Caorsi et al. [42] have tried to improve this method

by applying the dispersion correction, energy-based detection method, super resolution

technique, etc. However, the technique is based on the assumption of lossless media. The

main drawback of the existing techniques is the accumulation of error due to recursive

78

5.2 Calibration Method

formulations. The LS technique proposed in recent works [34] is suitable for the charac-

terization of single-layered media when other layers above and below are known. In this

work, we have improved the LS technique by using the fast and accurate models MPWMs,

and by using a time domain inversion (TDI) approach. The proposed method is capable

of finding approximate values for the electrical parameters of a multilayered media.

The sensitivity and stability analysis are performed on synthetic data to investigate

the well-posedness of an inverse modelling scheme. These analysis heavily depend on

the knowledge of expected measurement and modelling errors. Quantifying these errors,

though not impossible, is a difficult task due to the existence of deterministic and random

errors. For SFCW monostatic GPR, the uncertainty in inverted model parameters is

largely originated due to the calibration error, height measurement error, instrument

instability with respect to time and environment conditions, noise, interference, and so

on [64,65]. It is reported that the calibration error contributes to error in Green’s function

extraction and limit the usable bandwidth of the SFCW GPR system. The antenna height

measurement error causes tremendous uncertainties on the reconstruction of layered media

parameters.

In this chapter, first, the calibration method for SFCW monostatic GPR is discussed.

It partially helps to reduce the error in the extracted Green’s function based on GPR

measurement. Then, model inversion approach is explained in Section 5.3. Subsequently,

a brief analysis is carried out on the accuracy and stability of the proposed models based

on synthetic data experiments in Section 5.4. Two novel LS methods are proposed in

Section 5.5 to find the initial values for the layered media parameters. The chapter

concludes with a brief summary.


5.2.1 Test setup for Calibration

The model of laboratory test setup used for GPR calibration is presented in Figure 5.1.

There are two setups created at two different places i.e. IIT Kharagpur (IIT KGP) and

NIT Rourkela (NITR). A detailed discussion on both the setups is presented in Section 6.2.

79


Here we present the configuration details of the NITR setup. It is assembled with a VNA

(E5071C, Agilent Technologies), Dual ridged horn antenna (BBHA 9120A, Schwarzbeck

Mess-Elektronik), and a wooden tank (240 cm × 180 cm × 40 cm) having a metal plate

(180 cm × 120 cm) at the bottom. The maximum output power of VNA is fixed at 10

dBm. It is operated in the frequency range of 800 MHz to 4000 MHz with a step of 4

MHz / 5 MHz (number of frequency bins are 801/641) to conduct the GPR experiment.

An averaging factor of 10 is set to improve the signal to noise ratio of the measurement.

A 3-meter length 50 ohm RF cable is used to connect VNA output port with the antenna

connector. The VNA along with the RF cable is calibrated by standard OSM kit to

bring the reference measurement plane at cable and antenna interface. The antenna

BBHA 9120A is a linearly polarized dual-ridged horn antenna with aperture dimensions

of (24.5 cm × 14.2 cm), and length of 22 cm. It has got isotropic gain of 6 to 14 dBi,

and nominal operating frequency range of 1 to 5 GHz. However, it can be useful in the

frequency range of 0.8 GHz to 10 GHz.

Figure 5.1: Block diagram of GPR calibration setup.

5.2.2 Far-field Distance of Antenna

The far-field distance is calculated based on 2D2

λ from the antenna phase centre. D is

the maximum dimension of the antenna aperture at the phase centre. The location of

phase centre is a frequency dependent variable. For ultra-wide band (UWB) GPR, the

phase centre is equivalent to the virtual source point of the pulse [130, 131]. This is

80


calculated by extrapolating the inverse of PtP (peak to peak) versus twice of the antenna

height (h) by time domain processing of Green’s function due to an infinite size PEC.

The phase centre location is the point where 1/PtP becomes zero. Based on the analysis

presented in earlier work by Lambot et al. [1] for a similar antenna, the phase centre is

located at approximately 7.2 cm from the antenna aperture centre inside the antenna for

an operating bandwidth of 1-2 GHz. The far-field distance is calculated as 34 cm from the

antenna phase centre at highest (2 GHz) operating frequency. Based on this method [1]

we have tried to calculate the phase centre location of the antenna resulting in different

values ranging from 2 cm to 10 cm with repetition of experiments. Small size of the metal

plate, presence of external scatterers, and uncontrolled environment cause errors in our

experiments. In a recent research work [13], it is demonstrated that the point source and

far-field assumptions are valid for a minimum distance of 1.2 times maximum dimension of

the antenna aperture for all types of antennas. This hypothesis is validated by numerical

and laboratory experiments with a vivaldi antenna. For the dual-ridged horn antenna

with aperture dimension (24.5 cm×14.2 cm) the minimum distance is around 34 cm from

the aperture. We have selected the antenna heights close to this value based on best GPR

estimation accuracy achieved with our assembled testing set up. The selection of far-field

distance is partially validated by the outcomes of calibration experiments presented in

Section 5.2.4.

5.2.3 Theory of Calibration

The GPR system calibration are carried out by taking reflection coefficient (S11(ω)) mea-

surements while placing antenna at different heights above a large size metal plate. Ideally

the antenna height (h1) should be function of frequency as the location of antenna phase

center changes with frequency. However, this point can be fixed at a convenient location,

for example at the center of antenna aperture. A study by Khan Zaib Jadoon et al. [36]

have shown that the linear transfer function model used to represent the antenna behav-

ior inherently accounts for the gain and delay due to frequency dependent antenna phase

center location through the calibration process. Denoting the different antenna heights

with index variable k varying from 1 to n, Equation (3.1) relating linear transfer functions

81


(LTFs) with Green’s function can be written as

S11,k (ω) = Hi (ω) +Ht (ω)G

↑xx,k(ω)Hr (ω)

1−Hf (ω)G↑xx,k(ω)

. (5.1)

This can be modified and rewritten as following [65]:

S11,k = Hi + S11,kG↑xx,kHf +G↑

xx,k (H −HiHf ) (5.2)

where H = HtHr. For different antenna heights (k = 1 to n), the Green’s functions G↑xx,k

for different models are well defined by respective GPR signal model. The S11,k can be

measured by VNA at each frequency bins. To solve three unknown transfer functions,

at least three set of equations are required. However, it is recommended to solve an

overdetermined set of equations with n value at least 6 [65] to reduce the uncertainty in

LTFs estimation. Equation 5.2 can be expressed in matrix form as given below:

b = Ax (5.3)

where

b = (S11,1, ..., S11,k, ..., S11,n)T (5.4)

A =

1 S11,1G↑xx,1 G↑

xx,1

......

...

1 S11,kG↑xx,k G↑

xx,k

......

...

1 S11,nG↑xx,n G↑

xx,n

(5.5)

x = (Hi,Hf ,H −HiHf )T . (5.6)

Therefore, the unknown parameters (LTFs) vector is computed in the least-square sense

as

x =(AHA

)−1AHb (5.7)

82


where the symbolH denotes the Hermitian. This process helps to reduce the uncertainties

in LTFs extraction. During calibration, the antenna heights from metal plate are to be

measured accurately. Due to manually adjustable antenna stand, our height measurement

has an accuracy of approximately ±1.5 mm. This in turn contributes the error in Green’s

function extraction.

5.2.4 Results of Calibration

The GPR system calibration was carried out by taking S11(ω) measurements with the

antenna at different heights above the metal plate. Here FWM-2 was used to calculate

metal plate Green’s functions G↑xx,k. The unknown LTFs were calculated by solving (5.7).

Figure 5.2 presents the plots of LTFs with frequency. The calibrated antenna return loss

(Hi (ω)) as well as three free space measurements values are presented in Figure 5.2a. It

can be observed that the calibrated and free space measurements for Hi (ω) closely follow

in lower frequencies, and differ towards higher frequencies. Figure 5.2b presents the plot

of feedback loss transfer function (Hf (ω)). It is observed that Hf (ω)) is much lower

than Hi (ω). Figure 5.2c presents plots for4πλ2H (ω) amplitude and phase. The amplitude

plot is compared with the antenna manufacturer supplied gain measurement data. It has

been proved analytically (in Section 4.4) that, the MPWMs and FWM-2 are related to

the Friis transmission equation. The amplitude of H (ω) (= Ht (ω)Hr (ω)) is actually the

frequency dependent antenna aperture area. Again a close matching is observed between

the 4πλ2 |H| and gain measurement data till approximately 2 GHz. Figure 5.2d presents the

amplitude plots of extracted Gxx (ω) for metal plates placed at different distances. Since,

the effect of millimetric inaccuracy of height measurements is more towards increasing

frequency, the fluctuation of Gxx (ω) is observed to be more towards higher frequencies.

This calibration error limits the usable bandwidth [64] for the GPR system. Therefore,

we have selected the GPR processing bandwidth of 1.2 GHz from 0.9 to 2.1 GHz for

optimum GPR performance. In few experiments we have also opted for frequency range

of 0.8 to 2.0 GHz based on optimum detection performance on laboratory testing data. It

may be noted here that, the calibration results presented in this section are the outcome

of experiment conducted at NITR setup. The calibration parameters obtained by testing

83


at IIT KGP are having similar characteristics with degraded quality due to the limitation

of setup as explained in Section 6.2.

1 1.5 2 2.5 3 3.5 4−30

−20

−10

0

20lo

g 10|H

i|

1 1.5 2 2.5 3 3.5 4−4

−2

0

2

4

∠H

i

Freq (GHz)

Calibrated Meas. 1 Meas. 2 Meas. 3

(a) The return loss Hi.

1 1.5 2 2.5 3 3.5 4−45

−40

−35

−30

−25

20lo

g 10|H

f|

1 1.5 2 2.5 3 3.5 4−4

−2

0

2

4

∠H

f (ra

d)

Freq (GHz)

(b) The feedback loss Hf

1 1.5 2 2.5 3 3.5 40

5

10

15

10log(4π|H

|

λ2)

1 1.5 2 2.5 3 3.5 4−4

−2

0

2

4

∠H

(ra

d)

Freq (GHz)

Calibrated dataManufacturer data

(c) The antenna gain 4πλ2H

1 1.5 2 2.5 3 3.5 42

4

6

8

10

12

14

16

18

20

22

Freq (GHz)

|Gxx

|

30 cm32 cm34 cm36 cm38 cm40 cm

(d) The Green’s functions Gxx (ω)

Figure 5.2: Linear transfer functions (LTFs) and Green’s functions Gxx (ω) extracted bycalibration.

It may be noted here, the close matching between the calibrated and measured (sup-

plied by manufacturer) far-field gains in the lower frequency band partially validate the

selection of far-field distance. In fact, the LTFs extracted by the calibration process are

expected to have better matching with the standard antenna parameters after improving

measurement setup, using bigger size of the PEC, etc.

84

5.4 Accuracy and Stability Analysis

5.3 Model Inversion Approach

To invert the models, an objective function is defined in least square sense as following.

ϕ (b) =∣∣∣G↑∗

xx (ω)−G↑xx(ω,b)

∣∣∣T ∣∣∣G↑∗xx (ω)−G↑

xx(ω,b)∣∣∣ (5.8)

where G↑∗xx (ω) and G↑

xx(ω,b) are the vectors containing measured and simulated Green’s

functions respectively. The parameter vector b needs to be estimated by minimizing the

objective function ϕ (b). Like FWMs, the proposed models i.e. MPWMs are nonlinear

having multiple minimas for the objective function in the parameter vector space. There-

fore, a special mathematical treatment is required for efficient inversions of the models.

There are two approaches to invert these models based on the GPR testing application

and environment. In the first approach, the gradient-based method can be used to opti-

mize the objective function. The gradient-based method works efficiently while sufficient

information on the initial model parameters is available. The preliminary information

can be obtained by using LS technique. The second approach is to apply a robust global

optimization scheme to invert the models. The global optimization schemes are slow but

reliable for inverting large number of parameters in the presence of noise and measure-

ment errors. The preliminary information is still useful to decide the initial starting point,

and narrow down the search boundary in the parameter vector space. In this research

work, our main focus is to improve the computational efficiency of GPR detection while

retaining the accuracy as good as FWMs. We have developed improved LS techniques

for pre-estimation of parameters. Then Matlab-based gradient method is used to invert

the models efficiently.


The accuracy of model inversion depends on the ability to extract the good quality Green’s

function from GPR measurement on media under test. There are various sources of

errors and noise in the process of acquisition of antenna characterization parameters and

measurement data. At different stages of the GPR calibration and measurement process,

85


this errors and noise can enter the chain, and propagate towards the end of the process.

They finally yield uncertainties in the model parameters estimation. It is very important

to understand the relation between inversion accuracy with the various kind of noise

and error for two reasons. The first one is to improve the GPR detection accuracy by

improving system design, and the measurement process. The second one is to define the

GPR estimation accuracy based on the model performance and existence of noise level.

An elaborate analysis of the SFCW GPR system stability over the repeated measure-

ments, and calibration error are presented by M. R. Ardekani and S. Lambot [64]. It is

reported that the millimetric inaccuracy of the calibration setup has a severe impact on

the accuracy of extracted Green’s function, and its effect increases with increasing fre-

quency. To improve the stability of GPR system, a time gap after the VNA switched on

is recommended before starting the measurement. The uncertainties of GPR detection

can be significantly reduced by accurate VNA calibration, taking an average of linear

transfer functions, estimating accurate antenna height, and so on [65]. We have realized

the GPR system in the laboratory using VNA. The same calibration approach [65] is used

to extract the measured Green’s function. It is expected that the effect of measurement

error, noise, instruments instability on FWM-2 to be similar as FWM-1 [1] as the fre-

quency averaged %RMS difference is very small, and the correlation coefficient is close to

unity between the models over a wide parameter vector space. Here a limited analyses are

carried out with synthetic data to analyze the performance of MPWMs in the presence

of few important sources of measurement errors, and uncertainties.

5.4.1 Response Surface Plots

The response surface plots are the 2-D contour plot of the objective function with respect

to two model parameters while other remain constants. It partially reveals the uniqueness

of inversion, presence of local minima, parameters sensitivities, and their correlations,

etc. The analysis on FWM-1 [116] has revealed that the model is highly nonlinear with

objective functions having local minima. There are inverse relations between ϵr and h,

and no correlation between σ and other two parameters. In this work, a similar analysis

is carried out for the MPWMs for a single-layered media.

86


The logarithm of objective function (ϕt (b)) is plotted for the synthetic error-free data,

and actual measurement data with respect to three different combinations of parameter

vector space i.e. ϵr − h, ϵr − σ, and h − σ. The synthetic model selected is equivalent

to a laboratory model of a single-layered sand for which measurement data and GPR

estimation results are available. The parameter vector space is selected as 2 < ϵr < 22,

0 < h < 100 cm, and 14 < σc < 114 mS/m (static conductivity). The optimum point for

the single-layered model exists inside the parameter vector space. Figures 5.3, 5.4, and 5.5

present the response surface plots for the three different models, respectively, MPWM-1,

MPWM-2, and MPWM-3. The left side plots (a,c,e) are for the synthetic data, and right

side plots (b,d,f) are for the measured data. These plots reveal that the objective function,

ϕ (b) has got unique global minima (shown by white ∗ in figures) for the corresponding

true values of the parameters combination. The nonlinearity and oscillation lead to the

presence of multiple minimas of ϕ (b) over the multidimensional parameter vector space.

This analysis reveals that there is requirement of a robust optimization technique to invert

these models.

It is also observed that the sensitivity of parameters are different, and their relations

with each other are unique. The plots for ϵr − σ and h − σ show that these parameters

are uncorrelated, which is helpful for their inversion. However, with the region of minima

extended parallel to the σ-axis proves that the objective function is less sensitive to it.

The plot for ϵr − h shows that there is an inverse relation between these parameters.

This is a disadvantage to have a unique solution while GPR data is affected by noise and

measurement errors. Again, the parameter sensitivity and their correlations also depend

on the range of parameter. For example, at high value of ϵr, it is highly sensitive with

small changes in h. Whereas, for high value of σ, h and ϵr are not at all sensitive in

the region other than global minima. In fact, high attenuation is a problem for GPR

detection as it causes loss of signal, and also loss of information.

The response surface plots for the real data have got a similar shape like the plots

for the synthetic data with exactly same location of minimas. The regions of global

minima are expanded. These properties yield more uncertainty for obtaining correct

parameter values by inversion while GPR data is corrupted by measurement error and

87


external interferences. The same position of global minima supports the invertibility

of the models. Even though there are measurement error, various sources of noise, the

solution is unique and same as the synthetic model. This fact also indicates that there is

a filtering effect in the calibration process to retrieve the Green’s functions by minimizing

the error. All these analyses prove that the proposed models are invertible, and there is

requirement of a robust optimization scheme to invert these models.

5.4.2 Effect of Antenna Height Measurement Error

The error in antenna height measurement has a severe impact on the monostatic GPR

calibration and estimation. The accuracy of GPR detection of material parameters de-

pends on the accurate estimation of linear transfer functions (LTFs) i.e. Hi(ω), Ht(ω),

Hr(ω), and Hf (ω) representing the antenna effect. To calculate these transfer functions,

complex reflection coefficient S11(ω) are measured by the VNA while keeping antenna at

different heights above a big size metal plate. During the calibration process, the antenna

heights are measured manually. A measuring instrument also can take the height mea-

surements automatically. In both the cases, there are limited precision on measurement

accuracy. Therefore, error in extraction of the LTFs is unavoidable. However, this error

can be minimized to a certain extent by improving the measurement setup.

During GPR detection, the antenna height can be measured either manually or by

time domain (TD) processing of GPR signal. In both the cases, the error is unavoidable.

The accuracy of height measurement by TD signal processing depends on the size of IFFT,

dispersion effect, and accuracy of LTFs. In this section, the effect of height measurement

on a single-layered media is presented by observing the accuracy of GPR inversion on

synthetic data.

5.4.2.1 Effect of Calibration Error on Model Inversion

We took a synthetic model of single-layered (1L) media with ϵr = [1, 5.8, 6], h =

[32.33, 10,∞] cm, σc =[0, 17, 1× 1010

]mS/m, and σr = [0, 22, 0] mS/m/GHz. The GPR

processing bandwidth was selected as 0.9 to 2.1 GHz. Total 61 points were computed for

calculating frequency domain Green’s function using the models MPWMs, and FWM-2.

88


h (cm)

ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(a)

h (cm)

ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(b)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(c)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(d)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(e)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(f)

Figure 5.3: Response surface plots (MPWM-1).

89


h (cm)

ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(a)

h (cm)ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(b)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(c)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(d)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(e)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(f)


90


h (cm)

ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(a)

h (cm)

ε r

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(b)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(c)

σ (mS/m)

ε r

20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

(d)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)0

0.1

0.2

0.3

0.4

0.5

(e)

h (cm)

σ (m

S/m

)

0 5 10 15 20

20

30

40

50

60

70

80

90

100

110

log10

(Φ)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

(f)


91


1 1.2 1.4 1.6 1.8 2

x 109

0

5

10

|Gxx

|

S

h=0 mm

Sh=0.2 mm

Sh=0.4 mm

Sh=0.6 mm

Sh=0.8 mm

Sh=1.0 mm

Sh=1.2 mm

Sh=1.4 mm

Sh=1.6 mm

Sh=1.8 mm

Sh=2.0 mm

1 1.2 1.4 1.6 1.8 2

x 109

−5

0

5

∠G

xx (

rad)

1 1.2 1.4 1.6 1.8 2

x 109

0

0.5

1

1.5

Frequency (GHz)

RM

S e

rror

(G

xx)

Figure 5.6: Effect of uniform measurement error (UME) on Green’s function.

The antenna transfer functions i.e. LTFs were calculated based on one set of S11(ω)

measurement data with antenna at different heights above the large size metal plate in

the laboratory. The S11(ω) for the 1L media was calculated by using (5.1), and one set

of LTFs extracted by the calibration process. Assuming the initial height measurements

to be a correct set, the uniform measurement error (UME) with standard deviation (Sh)

varying from 0 to 2 mm with a step size of 0.2 mm was added with the measurement

antenna heights. For each set of height vector, new transfer functions (LTFs) were calcu-

lated leading to a new measured Green’s function G↑xx(ω) for the target by using (5.1).

Figure 5.6 describes the plot of Green’s functions affected by the height measurement

error with Sh varying from 0 to 2 mm for the model FWM-2. It shows that the height

measurement error has more impact towards higher frequencies. The similar outcomes are

observed for the MPWMs Green’s functions (plots not presented). Then inversion soft-

ware was run each time (for different Sh) to find the optimized parameter values for all

four models i.e. MPWM-1, MPWM-2, MPWM-3, and FWM-2. MPWM-1 and MPWM-2

were computed with maximum order of reflection coefficient N0 as 10.

Total four parameters i.e. sand layer thickness (h2), its relative dielectric constant

(ϵr2), static conductivity (σc2), and conductivity variation coefficients (σr2) were opti-

mized by the inversion process. The antenna height (h1) was fixed at actual height of the

92


0 0.5 1 1.5 29.5

10

10.5

Sh (mm)

h2 (

cm)

MPWM−1 MPWM−2 MPWM−3 FWM−2

0 0.5 1 1.5 25

6

7

Sh (mm)

εr2

0 0.5 1 1.5 216

18

20

Sh (mm)

σc2

(m

S/m

)

0 0.5 1 1.5 220

25

30

Sh (mm)

σ r2(f

) (m

S/m

.GH

z)0 0.5 1 1.5 2

0

5

10

Sh (mm)

%R

MS

err

or

Input synthetic data

0 0.5 1 1.5 20

5

10

Sh (mm)

%R

MS

err

or

Inverted model

Figure 5.7: Effect of uniform measurement error (UME) on model inversion.

antenna i.e. at 32.33 cm. The gradient technique was used to invert the model with start-

ing point as the actual model parameters. Figure 5.7 describes the optimized values for

the parameters with change of Sh. The plots indicate that the measurement error leads

to significant error in GPR estimation of material parameters values. The layer thick-

ness (h2) decreases, and dielectric constant (ϵr2) increases. The increase in frequency

dependent conductivity is more compared to the static conductivity. The MPWMs, and

FWM-2 almost follow the similar trend. The bottom most plots of Figure 5.7 present the

%RMS error for different Sh values. The plot of %RMS error for the input synthetic data

versus Sh shows that, as the UME with more Sh is introduced, the extracted Green’s

function deviates from the error free synthetic Green’s function resulting increase in the

%RMS error. The plot of inverted models demonstrate the %RMS error between the

inverted model and respective input synthetic data for different Sh. The plot shows that

the %RMS error decreases as it is compared with the input noisy data, and corresponding

inverted model. The shifting of optimum point in the process of inversion also causes re-

duction of error between the input and inverted models. All the models perform similarly

to invert the model parameters resulting %RMS error almost same.

93


5.4.2.2 Effect of Height Measurement Error on Model Inversion

The same 1L media used for UME analysis was selected for analyzing the effect of antenna

height measurement error. The synthetic data affected by the UME of standard deviations

(Sh) 0 mm, 1 mm, and 2 mm were selected for this analysis. Now the antenna height

(h1) error (∆h) varying from -1 to +1 mm with a step size of 0.2 mm was introduce for

inverting the models. The results of inversions are presented in Figures 5.8, 5.9, and 5.10,

respectively, for Sh = 0 mm, Sh = 1 mm, and Sh = 2 mm. It is clear from the plots that

as antenna height h1 deviate from the actual one, the inverted parameters also change

from the original values. The plots also demonstrate that the calibration error along

with antenna height measurement error during GPR detection can contribute significant

amount of uncertainty in quantitative estimation of media parameters.

The bottom most plots for all the figures present the %RMS error for the input

synthetic data, and for the inverted models. Here input data remained same i.e. affected

by UME of fixed Sh value for respective models. In Figure 5.8, the %RMS error plot

versus ∆h shows that the antenna height estimation error causes significant changes in

the Green’s functions and shifting of optimum point in the parameter vector space. Based

on observation on the plots for Sh = 1 mm (Figures 5.9) and Sh = 2 mm (Figures 5.10),

it can be explained that the calibration error, and the height measurement error during

GPR detection, both are not additive. The %RMS error for the inverted models changes

little for different Sh values. It signifies that as more amount of UME is added with

the synthetic GPR data, the inversion process try to optimize towards different points in

the parameters vector space to reduce the error between measured and modelled Green’s

functions.

In conclusion, it can be commented that the both types of height measurement er-

rors i.e., UME during calibration, and antenna height measurement error during GPR

detection contribute error in the parameters estimation. The %RMS error between the

measured and modelled Green’s function can’t be a sole performance parameters to judge

the accuracy of the inverted parameters.

94


−1 −0.5 0 0.5 19.5

10

10.5

∆h (mm)

h2 (

cm)


−1 −0.5 0 0.5 15

6

7

∆h (mm)

εr2

−1 −0.5 0 0.5 116

18

20

∆h (mm)

σc2

(m

S/m

)

−1 −0.5 0 0.5 120

25

30

∆h (mm)

σ r2(f

) (m

S/m

.GH

z)

−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or


−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or

Inverted model

Figure 5.8: Effect of antenna height measurement error on model inversion (Sh = 0 mm).

−1 −0.5 0 0.5 19.5

10

10.5

∆h (mm)

h2 (

cm)


−1 −0.5 0 0.5 15

6

7

∆h (mm)

εr2

−1 −0.5 0 0.5 116

18

20

∆h (mm)

σc2

(m

S/m

)

−1 −0.5 0 0.5 120

25

30

∆h (mm)

σ r2(f

) (m

S/m

.GH

z)

−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or


−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or

Inverted model


95


−1 −0.5 0 0.5 19.5

10

10.5

∆h (mm)

h2 (

cm)


−1 −0.5 0 0.5 15

6

7

∆h (mm)

εr2

−1 −0.5 0 0.5 116

18

20

∆h (mm)

σc2

(m

S/m

)

−1 −0.5 0 0.5 120

25

30

∆h (mm)

σ r2(f

) (m

S/m

.GH

z)

−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or


−1 −0.5 0 0.5 10

5

10

∆h (mm)

%R

MS

err

or

Inverted model


5.4.3 System Dynamic Range

The system dynamic range of GPR depends on the difference between receiver gain com-

pression point, and minimum detectable signal (MDS) (Section 2.3.2). Again the MDS

depends on the thermal noise floor of the system. The thermal noise affects the GPR sys-

tem performance especially when the received signal is close to the noise floor of receiver.

The received scattered signal degrades due to the spherical divergence, media attenua-

tion, multiple reflections and refractions at multilayer interfaces, etc. Our GPR test setup

consists of a VNA (E5071C of Agilent), and a dual ridged horn antenna having gain of

6-14 dBi over the operating frequency 0.8-5.0 GHz. The GPR operating bandwidth was

selected as 800 MHz to 2100 MHz. The intermediate frequency bandwidth (IFB) of the

VNA was fixed at 3 kHz. At room temperature, the VNA has a noise power spectrum

density of -123 dBm/Hz i.e. noise floor of -88.3 dBm. The VNA can transmit maximum

upto 10 dBm CW signal. In absence of an external signal amplifier, the maximum signal

expected from a subsurface media at a typical far-field distance of 35 cm is limited to -3

dBm at 1 GHz frequency. At higher frequency, this reflected signal will further degrade

due to the increase in spreading loss. The typical dynamic range achievable by this VNA

96


is around 85 dB with the maximum output power and IFB of 3 kHz. The actual system

dynamic range are generally still worse than this ideal figure due to the various sources

of errors, and interferences.

The available maximum dynamic range can be exploited in frequency domain pro-

cessing. Many cases processing of data is done in time domain to extract the preliminary

layer information. The frequency domain data is converted to time domain by applying

IFFT, and a synthetic pulse is generated at the transmitter to observe the returned pulses.

Depending on the type of synthetic pulse, the IFFT causes range side lobes to generate.

This reduces dynamic range significantly [81]. By a proper choice of pulse shaping, the

range side lobes can be reduced at the cost of resolution. In this case the user has to

make a choice between the dynamic range, and range resolution.

5.4.4 Summary of Accuracy and Stability Analysis

In this work, we have presented a brief analysis on the sources of errors, and uncertainties

during measurement and calibration process with the help of synthetic data. Efforts have

been made to quantify their effect. However, this analysis is not sufficient to predict the

errors on the measurement scenarios. The off-ground monostatic GPR signal modelling

has few important assumptions. The antenna is assumed to be a point source, signal

to be propagating mainly in normal directions, media under investigation to be planner

with each layer being homogeneous in electromagnetic properties. However, there is al-

ways differences with the ideal setup requirements for accurate representation of a model,

and the setup implemented in the laboratory. Moreover, many sources of error are not

under the control due to the limitation of experimental setup, instrumentation support,

and uncontrolled environment. The origin of uncertainties are of two kinds, i.e. the er-

rors which can be controlled by improving measurement process, and the random error

which can’t be controlled. A major source of errors is the antenna height measurement

errors during GPR calibrations and detection. This effect has been discussed in length

in Sections 5.4.2.1 and 5.4.2.2. There are also other sources of errors due to the finite

size of metal plate used for calibration, antenna axis misalignment from the vertical axis

during up and down movement, presence of external scattering objects, etc. In many

97

5.5 Layer Stripping Approaches

cases there are limitation of GPR setup to control these error. However, it is possible

to reduce them significantly by developing automated precision test setup, and placing

it in a control environment. The random errors originate due to the noise and insta-

bility in VNA i.e in the transceiver electronics used for the measurement. It’s observed

that the instability of VNA [64] causes loss of calibration which degrades the accuracy

of LTFs extracted by the GPR calibration process. A simple bend in cable also affect

the VNA calibration parameters thereby increase the noise floor. In absence of an ideal

experimental setup, environment, and ground truth measurement data, the calculation

of modelling error and quantification of uncertainties, though not impossible, are dif-

ficult tasks. It is recommended to follow certain measurement protocols e.g., accurate

antenna height measurement, accurate calibration of VNA, absence of undesired external

scatterers, averaging the LTFs representing antenna effect, etc.


The layer stripping (LS) is used to extract the electrical and geometrical properties of

layered media in step-by-step for each layer starting from the top layer. The process starts

with time-domain processing of the measured Green’s function g↑∗xx(t). First, the timing

information (ti) and amplitude values for all the 1st order significant reflections (Ai) are

evaluated. Following this the amplitude and timing information are used to find layered

media parameters approximately. This section presents two LS approaches suitable for

reconstructing layered media.

5.5.1 Layer Stripping by Approximate Method (LSAM)

The proposed LS named as layer stripping by approximate method (LSAM) is based on

the approach proposed by Kalogeropoulos et al. [34], and the most simplified GPR model

MPWM-1. It has the limitation of finding layered media parameters while conductivity

(σ) of layers is high. However, it can find all the electrical parameters (ϵr, h and σ) of a

single-layered media. For any N -layered media, 1st layer is the air media. Its thickness

h1 i.e. height of the antenna can be easily evaluated by the relation h1 = (c×t1)2 . Here c

is the velocity of EM wave at free space, t1 is the time delay of the 1st reflection. Generic

98


step for extracting layered media is explained below.

Step 1: Synthetically generate a Green’s function for a layered media with the PEC

placed at z1 interface by using MPWM-1. Determine the peak reflection, Apec1 due to the

PEC by analyzing the time domain Green’s function g↑xx (t). Since r1,2 = −1 for PEC

at z1, comparing the synthetic Apec1 with measured 1st order reflection A1 from z1, the

following expression can be written based on (4.16):

− A1

Apec1

= r1,2 =Z2 − Z1

Z2 + Z1. (5.9)

Neglecting the conductivity parameters of the 2nd layer, (5.9) can be simplified as

r1,2 =

√ϵr1 −

√ϵr2√

ϵr1 +√ϵr2

. (5.10)

Now the ϵr,2 can be evaluated from (5.10). The thickness h2 is estimated by following

relation:

h2 =c× (t2 − t1)

2β2/β1(5.11)

where β1 and β2 are the propagation constants of free space for 1st layer (air) and 2nd

layer respectively; t1 and t2 are the time of arrival for 1st order reflection from z1 and z2

respectively. Neglecting the σ2, (5.11) is simplified as

h2 =c× (t2 − t1)

2√ϵr2

. (5.12)

For negligible conductivity, all the layers can be reconstructed by repeating the step-1.

In presence of significant conductivity, this process will cause accumulation of error in

estimating layered media parameters.

The conductivity of a layer can be estimated if the layer is bounded by a PEC or a

known media at the bottom i.e. all the electrical parameters including conductivities of

bottom layer are known. Assuming that the parameters for 1st to nth layer are known,

and (n + 2)th layer is either a PEC, or a known media. To extract the conductivity of

(n+ 1)th layer media, the following step need to be followed.

Step 2: Find the 1st order reflection Apecn+1 due to zn+1 by processing the time domain

99


Green’s function. Now the ratio between Apecn+1 and An can be written based on MPWM-1

as following:

Apecn+1

An=

rn+1,n+2

(1− r2n,n+1

)rn,n+1

∑nj=1 hj/γj∑n+1j=1 hj/γj

× exp(−2αn+1hn+1). (5.13)

rn+1,n+2 can be calculated approximately as the (n+ 2)th layer parameters are known.

αn+1 can be approximated at center frequency as

αn+1 =σc,n+1

2√ϵr,n+1

Z1 (5.14)

where σc,n+1 is the effective conductivity of (n+ 1)th layer at center frequency (fc) and Z1

is the free space impedance of air media. σc,n+1 is evaluated by solving (5.13) and (5.14).

Then update the thickness hn+1 by (5.11) using the newly obtained value of σc,n+1.

LSAM is a fast approach for reconstructing layered media parameters while conduc-

tivity is negligible. It is also capable of finding conductivity at center frequency for a

single-layered media bounded by known layers at top and bottom.

5.5.2 Layer Stripping by Inversion Method (LSIM)

The proposed method is based on common LS technique, and the time domain inversion

(TDI) focusing on a limited number of reflections to reconstruct the property of one layer

at a time. The objective function in time domain(ϕt (b)

)is defined in least square sense

as following:

ϕt (b) =∣∣∣g↑∗xx (t)− g↑xx(t,b)

∣∣∣T ∣∣∣g↑∗xx (t)− g↑xx(t,b)∣∣∣ (5.15)

where g↑∗xx (t) and g↑xx(t,b) are, respectively, the measured data and simulated models

defined over a time window. The method is named as layer stripping by inversion method

(LSIM). LSIM is capable of estimating approximate values for the parameters ϵr, h, σc,

and σr. The fast processing capability of MPWMs is exploited to invert the parameters

of each layer at a time. The following steps are to be performed for realizing LSIM.

Step 1: The 1st layer being air media, ϵr1 = 1, and σc1 = σr1 = 0. Now generate

a Green’s function for the media having 1st layer property as ϵr1 , σc1 , σr1 , and h1, and

100


a PEC at layer 2. Find the reflection Apec1 from the interface z1 by the time-domain

processing. Then compare A1 with Apec1 , and use the following approximate formula to

find the ϵr2 :

− A1

Apec1

= r1,2 =

√ϵr1 −

√ϵr2√

ϵr1 +√ϵr2

. (5.16)

Step 2: Apply the TDI with a time window focusing on A1 to invert the parameters

vector b consists of five parameters i.e. σc1 , σr1 , ϵr2 , σc2 , and σr2 . Initial values for σc2 , and

σr2 can be chosen either arbitrarily or equated to the values of σc1 , and σr1 respectively.

Step 3: Update the ϵr2 based on the inversion result obtained in Step 2. Find the

thickness h2 by the following relation:

h2 =c× (t2 − t1)

2β2/β0. (5.17)

Now repeat the step 1 to 3 for next layer, and repeat them till all the layers are evaluated.

The ability to find the accurate values for current layer parameters σcn , σrn , and the

next layer parameters ϵrn+1 , σcn+1 , and σrn+1 depends on the sensitivity of time domain

objective function (ϕt (b)) on these parameters. We have done a response surface analysis

to find the sensitivity of ϕt (b) for a 3L media with respect to the variation of parameters

ϵr2 , σc2 , h2, ϵr3 , and σc3 . Here the ϕt (b) is bounded by a time window focusing on 1st

order reflection from the interface z2. It has revealed that the ϕt (b) is less sensitive

to σ in the lower range of parameter values. Better quality of signal is required to

resolve conductivities accurately. The ϕt (b) being more sensitive to ϵr and h, they can

be retrieved accurately by the TDI method. For good quality signal, the TDI can be

implemented by gradient method for faster implementation of the LSIM. However, it is

recommended to use a global optimization technique to improve the accuracy of GPR

detection as the initial estimation in Step 1 may not be in global basin. To improve the

accuracy of inversion in Step 3, the ϕt (b) can be defined over a time-window focusing on

two consecutive reflections i.e. one from the current interface and other from the previous

interface.

101

5.6 Summary

5.6 Summary

The accuracy of GPR estimation is directly related with the accuracy of measurement,

calibration, forward model computation, and inversion process. Quantifying the modeling

error is a tedious job because of difference between the ideal and actual measurement

model, and existence of uncontrolled random error during the process of measurement.

In this work, the measurement Green’s functions have been extracted based on the linear

transfer functions measured by the calibration process. Solving an over determined set of

equations based on GPR measurements on PEC helps to reduce the measurement error

to some extent.

The primary focus of the accuracy and stability analysis is to verify the invertibil-

ity, and to quantify the impact of few major sources of measurement errors. This study

has revealed that there are unique solutions to the models inversions for both, synthetic

and measurement data. However, with expansion of the global minima region for prac-

tical measurement data, finding optimum solution is a challenge. This issue needs to be

addressed by proper optimization method. The GPR estimation of subsurface param-

eters is majorly affected by the calibration and height measurement errors. Therefore,

steps should be taken to improve the height measurement accuracy of the monostatic

GPR setup in off-ground configuration. Though, this analysis is not sufficient for ac-

curate quantification of the modelling error, model stability, system dynamic range, an

approximate knowledge has been obtained on them. Based on the discussion presented

in Section 5.4, few important steps related to the measurement process can be taken to

reduce the error and thereby improve GPR system stability.

The inversion approach with the help of LS followed by gradient method can be very

effective to estimate electrical parameters of an unknown layered media. Out of the two

methods proposed on LS, LSIM is suitable for complete characterization of multilayered

media, and LSAM is time-efficient method, suitable for characterization of lossless media.

LSAM is also capable of complete characterization of a single-layer bounded by known

layers at top and bottom. The next chapter focuses on validation of the inverse modelling

approach by laboratory testing on layered media.

102

CHAPTER6Results of Laboratory

Experiments

Contents

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Test Setup and Experimental Environment . . . . . . . . . . . 105

6.3 Single-layered Water Testing . . . . . . . . . . . . . . . . . . . 108

6.4 Single-layered Sand Testing . . . . . . . . . . . . . . . . . . . . 111

6.5 Two-layered Media Testing . . . . . . . . . . . . . . . . . . . . 112

6.5.1 Experiment at IIT KGP . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.2 Experiment at NITR . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

103

6.1 Introduction

This chapter presents laboratory testing results of GPR detection on layered media.

The proposed models are validated by testing an well-known media, and comparing the

inversion accuracy with an existing FWM [1] in literature. The performances of the

MPWMs are compared with the FWMs in terms of accuracy and computational efficiency.

6.1 Introduction

In Chapter 3, an FWM (FWM-2) has been proposed, and validated by comparing its

accuracy with an existing FWM (FWM-1 [1]). Subsequently, the MPWMs have been

proposed in Chapter 4 based on analytical solutions of FWM-2. The MPWMs have been

validated by observing %RMS diff and %CCC with FWM-2 by synthetic experiments.

In this chapter, the proposed models are validated by the laboratory experiments.

The chapter begins with a discussion on the testing setup, environment, and process

used for the GPR experiments. The GPR detection process starts with the calibration,

followed by measurement on media under test, and finally, software simulation to invert

the models to retrieve the media parameters. Based on the analysis presented in Section

4.3, it is understood that the MPWMs are very fast to compute layered media Green’s

function. MPWM-2 and MPWM-3 are highly correlated with FWM-2. Laboratory exper-

iments are conducted to validate the proposed models, and compare their performances

with the FWMs. Normal water being well-known media is used to validate the proposed

models. The proposed models are also validated by comparing their accuracy with an

existing model FWM-1 [1]. The testings on a single-layered (1L) sand, and a two-layered

(2L) media demonstrate the advantages of MPWMs over FWMs. The performance of

MPWMs is compared with the FWMs in terms of accuracy and computational efficiency.

The accuracy of model inversions is quantified by the %RMS error and %CCC between

the measured and modelled Green’s functions. They are defined by relationships (3.46)

and (3.47) respectively.

The model inversions are carried out by applying LS technique followed by a Matlab-

based gradient method. LSAM is used for extraction of initial information on single-

layered media. Whereas, LSIM is found suitable for approximate characterization of

two-layered media. These LS techniques help to realize GPR inversion with the help of

104

6.2 Test Setup and Experimental Environment

time-efficient gradient method. Finally, the chapter concludes with a summary of overall

observation and limitation of our study.


The model of laboratory SFCW GPR setup used for layered media testing is presented

in Figure 6.1. It is assembled with a VNA (E5071C, Agilent Technologies), a dual ridged

horn antenna (BBHA 9120A, Schwarzbeck Mess-Elektronik), and a wooden tank filled

with material under test. The antenna is mounted on a manual scanner with adjustable

height. A metal plate is kept at the bottom of the tank to control the boundary of the

layered media. The frequency range of 800 MHz to 4000 MHz is swept with a step of 4

or 5 MHz to conduct the GPR experiment. The testing was conducted in two places i.e.

one at the rooftop of E&ECE department, IIT Kharagpur (IIT KGP) and other in the

GPR laboratory, NIT Rourkela (NITR). The whole setup is kept in a place having no

control on the experimental environment. In both the places, same models of VNA and

antenna were used for testing. However, there are important differences in the test setups

and environments as explained below.

Figure 6.1: Block Diagram of GPR Testing Setup.

At IIT KGP, the scanner is made up of wooden bar, and two wooden stands (Fig-

ure 6.2). The wooden tank has a size of (138.5 cm × 98.5 cm × 30 cm). The metal

105


Figure 6.2: Experimental setup at IIT KGP

Figure 6.3: Experimental setup at NITR

106


plate used for calibration, is made up of GI sheet. Its dimension is (122 cm × 81 cm).

These dimensions are smaller compared to the recommended size based on the analysis

presented by Patriarca et al. [65]. The antenna is fixed with a wooden bar. The setup

has a limitation that, the antenna’s horizontal and vertical alignment get’s disturbed as

it is moved up and down. The whole set up was kept in the rooftop under the open sky.

During calibration, the measurement of antenna heights were taken manually by a ruler.

This process contributes maximum height measurement error of ±1.5 mm. Beside this,

there were fluctuation of temperature because of the outdoor environment, and variation

of sunlight during GPR measurements.

The other setup (Figure 6.3) has been recently developed inside the GPR lab at NITR.

Here the antenna scanner is made up of a wooden bar supported by two iron pillars. With

a better vertical scanner, the antenna can be fixed at any height with resolution of less

than 1 mm. As both sides of the wooden bar don’t move equally with good precision

during vertical movements, the antenna height measurements need to be carried out

again by the ruler. Beside this, there are also issue related to the antenna misalignment.

The height measurement accuracy has not improved with the new setup. The wooden

tank used at NITR has dimensions of (240 cm × 180 cm × 40 cm). The metal plate

used for calibration is made up of semi-hard copper material. It has got dimension of

(180 cm×125 cm). These dimensions, though bigger than the IIT KGP setup, still shorter

compared to the recommended size [64, 65]. The copper plate has multiple bents which

couldn’t be corrected during experiments. The ceiling is covered by absorber materials

to simulate the open half-space. However, side walls aren’t covered by the absorber. The

setup at NITR has got some improvement over the IIT KGP setup in terms of higher

dimension of metal plate, and wooden box, lower variation of environment conditions due

to indoor setup etc. However, the bending in copper plate surface, antenna alignment

issue, and prevailing height measurement issue have contributed significant error during

the GPR calibration and testing.

107

6.3 Single-layered Water Testing


The primary aim of this testing is to validate the proposed models by reconstructing

water layer thickness and antenna height as water’s electrical parameters are well-known.

Single-layered water testing was conducted in both the places i.e. IIT KGP and NITR.

The NITR testing data having better quality is presented here for the analysis. The GPR

measurement was conducted inside the laboratory to collect the S11 data while keeping

the antenna above a normal water layer. The water layer thickness was measured as 3.2

cm. A metal plate was kept at the bottom of the layer to control the boundary con-

dition. During experiment, the temperature of water layer was measured as 27 degree

centigrade. As discussed in Section 2.5.1.1, the electrical parameters of normal water can

be obtained from Klein-Swift model [104]. The temperature-dependent relative dielectric

constant ϵs2 was calculated based on (2.20). This constituted an input (i/p) parameter

for model inversions. The frequency- and temperature-dependent conductivity (σ2(f))

were obtained by (2.18). With nonavailability of standard conductivity meter, we consid-

ered static conductivity (σs2) as optimization parameter for GPR inversion. Total three

parameters i.e. antenna height (h1), water layer thickness (h2), and its static conductiv-

ity (σs2) were optimized by using the models MPWM-1, MPWM-2, MPWM-3, FWM-2,

and FWM-1 [1]. MPWM-1 and MPWM-2 Green’s functions were computed by consid-

ering maximum up to 20th order reflection. Total 51 frequency points were considered

in 1200 MHz bandwidth (800 MHz to 2000 MHz) for model inversion to achieve good

detection accuracy. LSAM was utilized to obtain the preliminary information on water-

layer parameters. Subsequently, the final inversion was carried out by a Matlab-based

gradient method. Here we present the GPR inversion results for the measurement data

with antenna height set to 35 cm.

Table 6.1 presents the results of inversions. It is observed that all the models are able

to invert the antenna height, and water layer thickness with remarkable accuracy with

errors much lower than the manual height measurement tolerance. LSAM estimated an-

tenna height also has inaccuracy due to the finite sampling frequency used for converting

the Green’s function from frequency to time domain, calibration error, and dispersion in

108


0.8 1 1.2 1.4 1.6 1.8 20

5

10

|Gxx

|

0.8 1 1.2 1.4 1.6 1.8 2−5

0

5

∠G

xx (

rad)

0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Frequency (GHz)

RM

S E

rror

(G

xx)

Meas. MPWM−1 MPWM−2 MPWM−3 FWM−1 FWM−2

(a) Green’s functions in frequency domain.

0 5 10 15−5

−4

−3

−2

−1

0

1

2

3

4

5

Time (ns)

g xx(t

)

Meas.MPWM−1MPWM−2MPWM−3FWM−1FWM−2

(b) Green’s functions in time domain.

Figure 6.4: Compare measured and modeled Green’s functions for single-layered water.

109


Table 6.1: Water layer parameters estimated by different models

Modelused

h1(cm)

h2(cm)

ϵs2σs2

(mS/m)Run time(second)

%RMSerror

%CCC

LSAM 35.105 2.734 79.59 427.708 0.962 - -MPWM-1 35.040 3.158 i/p 383.623 0.346 9.633 99.880MPWM-2 35.039 3.160 i/p 384.294 0.257 9.635 99.714MPWM-3 35.039 3.159 i/p 384.564 0.580 9.618 99.714FWM-1 35.039 3.159 i/p 386.722 121.153 9.578 99.713FWM-2 35.039 3.159 i/p 384.540 95.728 9.615 99.713

the reflected signal. However, it is interesting to observe the water layer parameters ob-

tained by inversions of different models. All the models have yielded antenna height with

a tolerance lower than ±0.01 mm. The estimated water layer thickness has a tolerance

of ±0.01 mm. Though, the static conductivity estimations by the MPWMs, FWMs, and

even LSAM [34] matches closely, the value seems to be high. One possible reason can be

attributed to source of water, the bore-well water which typically has high iron contents

in the locality may be a reason. Again, with almost 9.7% frequency averaged %RMS er-

ror, the estimated conductivity may not be very accurate one. By observing the inverted

water layer parameters, %RMS error, and %CCC, it can be seen that the proposed MP-

WMs, and FWM-2 are as accurate as FWM-1 [1] which is an well established model in

the literature. It may also be noted here, %RMS error for LSAM is not computed since,

it is an intermediate step for the complete inversion process. Figure 6.4 presents plot

of the measured and modeled Green’s functions for the water layer in frequency domain

and time domain. The frequency domain plot (Figure 6.4a) shows remarkable agreement

of phase characteristics, and partial agreement of amplitude characteristics between the

measured and modeled Green’s functions. This difference can be largely attributed to the

calibration inaccuracy of manually operated GPR setup as well as small size of the PEC

and wooden box. The time domain plot (Figure 6.4b) shows that the synthetic data are

closely matching with the measurement data for all the models even for the higher order

reflections.

110

6.4 Single-layered Sand Testing

6.4 Single-layered Sand Testing

A single-layered media was created at the roof-top facility of IIT KGP by placing wet sand

in the wooden box. The metal plate placed at the bottom formed the PEC boundary. The

sand layer was prepared homogeneously with uniform thickness of approximate 10 cm.

Dry sand was mixed with normal water of around 9% volume to increase its dielectric con-

stant. After the GPR experiment, simulation was conducted for all the modeling schemes

to estimate the electrical parameters of sand under test. In this case, 61 frequency points

were selected over the operating bandwidth of 1200 MHz (from 900 MHz to 2100 MHz)

for GPR processing. Total five parameters were estimated by the GPR inversion. These

include height of the antenna from sand surface (h1), sand layer thickness (h2), relative

dielectric constant (ϵr2), conductivity (σc2) at center frequency (fc), and conductivity

variation coefficients (σr2). For MPWM-1, and MPWM-2, maximum up to 10th order

reflection was considered to calculate the Green’s function. A quick analysis of the media

under investigation has shown that 10th order reflection causes reflected signals till 100

dB lower than the strongest signal considered for computing the Green’s function.

Table 6.2 presents the results of GPR inversion, frequency averaged %RMS error, and

%CCC between the measured and modelled Green’s functions. It can be observed that the

MPWMs are as accurate as the FWMs to estimate electrical and geometrical parameters

of single-layered sand. The %RMS error and %CCC for all the models are comparable

with small difference in fraction number. LSAM has yielded approximate values for the

media parameters. The timing efficiency gained by MPWMs are enormous. Figure 6.5

presents plot of measured and modeled Green’s functions in frequency and time domain.

It can be observed (in Figure 6.5a) that the phase response is reproduced remarkably well

by the models inversions. However, there is significant amplitude error resulting in RMS

error between the measured and modeled Green’s functions. The time domain plot (in

Figure 6.5b) shows remarkable agreement between the measured and modeled Green’s

functions even for the higher order reflection coefficients. During wet sand testing, the

GPR measurement need to be conducted as soon as the layer is moisturized, and ready

for the testing. Once the surface gets dried, the reflection from it reduces causing error in

111

6.5 Two-layered Media Testing

Table 6.2: Sand layer parameters estimated by different modelling methods

Modelused

h1(cm)

h2(cm)

ϵr2σc2

(mS/m)σr2

(mS/m/GHz)Run time(second)

%RMSerror

%CCC

LSAM 32.649 9.052 6.774 14.790 - 0.983 - -MPWM-1 32.328 10.069 5.798 16.996 22.411 0.400 11.001 99.406MPWM-2 32.324 10.046 5.837 17.046 22.576 0.402 11.029 99.602MPWM-3 32.325 10.022 5.860 17.019 22.843 0.932 11.118 99.595FWM-1 32.324 10.061 5.815 17.070 22.352 309.431 11.057 99.602FWM-2 32.324 10.024 5.859 17.295 22.377 185.616 11.056 99.603

estimation of the media parameters. The roof-top arrangements, and manual sand mixing

technique have resulted significant amount of delay between the sample preparation, and

GPR measurement.


6.5.1 Experiment at IIT KGP

Following the single layer measurement as discussed in Section 6.4, two-layer testing was

conducted. The two-layered media was created at the roof-top laboratory facility by

placing dry wood powder above the wet sand layer used in single-layer testing. Thickness

of the wood powder layer was approximately 20.5 cm. As usual the metal plate was

kept at the bottom. The total number of parameters estimated here are seven, which

include 1st layer thickness (h2), relative dielectric constants (ϵr2 and ϵr3), conductivities

(σc2 and σc3) at center frequency (fc) and conductivity variation coefficients (σr2, and

σr3). The antenna height (h1) from surface of 1st layer and 2nd layer thickness (h3)

were considered as known parameters, and were fixed at manual measurement values of

33 cm and 10 cm respectively. As the quality of the measurement data was poor, the

number of parameters were restricted to reduce the computational complexity as well

as the uncertainties of the inverted media parameters up to certain extent. For both

MPWM-1 and MPWM-2, maximum up to 4th order reflection from the interface z2 and

z3 were considered to calculate the Green’s functions. With 4th order reflections under

consideration, reflected signals till 70 dB lower than the strongest signal were considered.

The GPR estimation results are presented in Table 6.3. From the result it is observed

that LSIM has failed to estimate the media parameters with sufficient accuracy. The

112


1 1.2 1.4 1.6 1.8 20

5

10

|Gxx

|

1 1.2 1.4 1.6 1.8 2−5

0

5

∠G

xx (

rad)

1 1.2 1.4 1.6 1.8 20

1

2

Frequency (GHz)

RM

S E

rror

(G

xx)



0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

3

4

Time (ns)

g xx(t

)



Figure 6.5: Compare measured and modeled Green’s functions for single-layered sand.

113


Gradient method has failed to optimize the parameters with starting point as LSIM

output. Then we applied LSAM to invert the 1st layer parameters. LSAM has able

to estimate the σc2 along with ϵr2 and h2 for the top sand layer as the bottom layer

electrical properties have been known by estimation of single-layered sand presented in the

previous section (Section 6.4). As usual it is observed that the MPWMs are highly time-

efficient compared to the FWMs. A very good similarity is observed among the estimated

layer parameters by all five models. The relative dielectric constants are evaluated with

a tolerance of ±0.031, 1st layer thickness with ±0.22 mm. However, with percentage

RMS errors higher than 21 for all the models, it is quite possible that models inversions

yield error in parameter estimation. By comparing Table 6.2 and Table 6.3 for wet sand

parameters, it can be observed that the change in inverted ϵr values are limited to 8%,

where as much higher changes are observed for the conductivities (σ) parameter. Due

to manual setup, the uniform thickness and homogeneity of two-layered media couldn’t

be achieved accurately. The imperfect model of two-layered media along with calibration

inaccuracy have contributed to high RMS error. It may also be noted that, the surface of

the wood powder layer couldn’t be made uniform. The surface of the wet sand layer was

getting dry during single-layered sand testing. This can be a strong reason for observing

the difference between the estimated dielectric constants for the single-layer and two-

layer testings. The plots of measured and modeled Green’s functions in frequency and

time domain are presented in Figure 6.6. Similar errors for the frequency and time

response of Green’s functions are observed across all the models. A close observation on

the frequency domain plot (Figure 6.6a) reveals that there are certain frequency points

where mismatch between the measured and modeled Green’s functions are visible. In

few cases abrupt transitions are visible on the plot of measured Green’s function. The

external interferences in form of cell tower signal, reflections from external scatterers are

also sources of distortion in the measured data.

6.5.2 Experiment at NITR

A two-layered media was created by placing two sand layers with thickness of top and

bottom, respectively, 25 cm and 13 cm approximately. The original sand sample was mois-

114


1 1.2 1.4 1.6 1.8 20

2

4

|Gxx

|

1 1.2 1.4 1.6 1.8 2−5

0

5

∠G

xx (

rad)

1 1.2 1.4 1.6 1.8 20

0.5

1

Frequency (GHz)

RM

S E

rror

(G

xx)



0 2 4 6 8 10 12−1.5

−1

−0.5

0

0.5

1

1.5

Time (ns)

g xx(t

)



Figure 6.6: Compare measured and modeled Green’s functions for two-layered media(tested at IIT KGP).

115


Table 6.3: Two-layered media parameters estimated by different modelling methods (IITKGP)

Modelused

h2(cm)

ϵr2σc2(mS/m)

σr2(mS/m/GHz)

ϵr3σc3(mS/m)

σr3(mS/m/GHz)

Run time(second)

%RMSerror

%CCC

LSIM 23.243 2.095 40.000 10.296 4.296 27.921 69.803 80.125 - -LSAM 21.714 2.391 25.734 - 5.800 17.000 - 2.149 - -MPWM-1 20.375 2.334 26.212 23.172 6.303 27.148 22.199 1.663 21.732 95.202MPWM-2 20.415 2.329 26.055 23.462 6.314 27.285 20.906 1.984 21.892 95.451MPWM-3 20.411 2.329 26.082 23.376 6.316 27.064 21.698 2.457 21.769 95.471FWM-1 20.367 2.339 26.159 23.528 6.317 27.473 20.990 765.349 21.901 95.406FWM-2 20.419 2.328 26.068 23.422 6.314 27.159 21.275 524.429 21.815 95.466

turized with unknown volumetric water content (VWC). The bottom layer was subjected

to approximate VWC of 15% to increase its dielectric constant. The top layer was pre-

pared with the original sample. Both the layers are placed in an areas of (180 cm×180 cm)

inside the wooden box. A metal plate (180 cm × 125 cm) was placed at the bottom of

the wet sand layer to form PEC boundary. Total seven parameters were estimated by

the model inversion process. These include 1st layer thickness (h2), relative dielectric

constants (ϵr2 and ϵr3), static conductivities (σc2 and σc3), and conductivity variation

coefficients (σr2, σr3). The 2nd layer thickness (h3) is taken as known parameter, and

is fixed at 13 cm. During the inversion simulation, the height of antenna (h1) was fixed

at LSIM measurement value 34.729 cm. Maximum up to 4th order reflection from the

interface z2 and z3 were considered to calculate the MPWM-2 Green’s function. With

4th order reflections under consideration, the reflected signals till 46.2 dB lower than the

strongest signal were considered for computing the Green’s function.

The GPR estimation results are presented in Table 6.4. Now it can be observed that,

LSIM is able to estimate media parameters with sufficient accuracy to apply the gradi-

ent method. We could observe remarkable similarities of the inverted media parameters,

%RMS error, and %CCC among the models. The relative dielectric constants have toler-

ance of ±0.019, and 1st layer thickness has ±0.74 mm. These figures will be further tight

if we consider only MPWM-3 and FWMs. Again it is observed that the MPWMs are

significantly time-efficient compared to the FWMs. MPWM-3 yields better %RMS error

compared to MPWM-1, and MPWM-2. With the RMS error close to 16%, it is quite

possible that the model inversion yields error in reconstructing layered media parame-

ters. Along with the limitation of measurement setup, the imperfect laboratory model

116

6.6 Discussion

Table 6.4: Two-layered media parameters estimated by different modelling methods(NITR)

Modelused

h2(cm)

ϵr2σc2

(mS/m)σr2(mS/m/GHz)

ϵr3σc3

(mS/m)σr3(mS/m/GHz)

Run time(second)

%RMSerror

%CCC

LSIM 25.770 2.732 6.103 1.803 8.920 32.445 0.000 82.229 - -MPWM-1 24.917 2.939 4.086 6.078 9.165 12.324 8.983 1.591 16.537 97.404MPWM-2 24.864 2.954 4.078 6.027 9.173 12.382 9.399 1.952 16.014 98.085MPWM-3 24.769 2.977 3.989 6.323 9.176 12.141 9.061 2.541 15.354 98.066FWM-1 24.792 2.973 4.112 6.192 9.171 12.242 8.791 707.430 15.576 98.116FWM-2 24.849 2.959 4.070 6.243 9.170 12.138 8.715 597.734 15.650 98.113

of two-layered media also contributed to error in the inversion. The uniform thickness

and homogeneity of the two-layered media couldn’t be obtained by manual setup. Fig-

ure 6.7 shows the plots of measured and modeled Green’s functions in frequency and time

domain. Compared to the IIT KGP testing (Figure 6.6), better matching between the

measured and modelled Green’s functions are observed in this experiment.

6.6 Discussion

Due to nonavailability of a standard equipment to verify the testing data, the accuracy

of models has been demonstrated by comparing their performance with a proven model

FWM-1 [1], and by reconstructing a water layer. By testing a water layer, single-layered

sand, and two-layered media using two different setups, it has been demonstrated that,

the MPWMs are as accurate as the FWMs, and are very time-efficient. Though, MPWM-

3 has lower time-efficiency compared to MPWM-1, and MPWM-2, it outperforms them

by its ability to accommodate with increasing number of layers, and increasing order of

reflections under consideration.

Although, the time-efficiency achieved by the MPWMs are enormous, all these models

are extracted based on the LTFs calibration which is a time-consuming task. Future

research work should focus to simplify the calibration process of monostatic GPR.

The inversion method proposed here is based on finding initial parameters values by

LSIM. The proposed LSIM can estimate complete electrical parameters of the layered

media as long as the measurement error is within certain limit. It is also very sensitive to

the antenna height measurement error. Though, a hard bound has not been evaluated,

it’s observed that, LSIM fails to yield sufficient accuracy as the %RMS error between

117

6.6 Discussion

0.8 1 1.2 1.4 1.6 1.8 20

2

4

6|G

xx|

0.8 1 1.2 1.4 1.6 1.8 2−5

0

5

∠G

xx (

rad)

0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

Frequency (GHz)

RM

S E

rror

(G

xx)



0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (ns)

g xx(t

)



Figure 6.7: Compare measured and modeled Green’s functions for two-layered media(tested at NITR).

118

6.7 Summary

measured and modeled Green’s function becomes closer to 20%. LSIM takes around 82

seconds to reconstruct the two-layered media parameters using MPWMs, as direct search

followed by gradient method is applied to implement the time domain inversion in two

stages. LSIM can be realized with the FWMs. The FWMs would take computational

time more than 300 to 500 times compared to the MPWMs. Therefore, the MPWMs help

faster implementation of LSIM. In many applications, calibration and initial parameters

evaluation are required once in a while. In such cases, the time-efficiency gained by the

MPWMs will be substantial resulting in the GPR detection time for a two-layered media

to be in the order of 2-3 seconds.

6.7 Summary

This chapter has presented laboratory experimental results to validate the proposed mod-

els. The GPR estimation of water layer has demonstrated that the proposed models have

ability to reproduce the antenna height and water layer thickness by using well known

EM model of normal water. The inverted parameters are also very close to the results of

FWMs. The experimental outcomes of single-layered sand and two-layered media have

demonstrated that the proposed MPWMs are very time-efficient, and as accurate as the

FWMs. With detection time of the order of few seconds for a two-layered media, the

MPWMs have potentials to be used for many real-time applications.

The GPR inversion has been successfully implemented by efficient gradient method.

Though, the gradient-based inversion scheme is efficient, its success in finding the global

optimum point depends on the initial guess that needs to be in the global basin. It

has been also observed that LSAM is capable of finding the parameters approximately

for single-layered media, and LSIM can find the same for a two-layered media tested at

NITR laboratory. As measurement error increases beyond a certain limit, LSIM fails to

work. In such case, the gradient method fails to invert the model with required accuracy.

Therefore, a robust global optimization scheme is essential for inverting GPR data affected

by high measurement error and noise.

The GPR measurement results at two different experimental setup have demonstrated

that, it is possible to reduce the measurement error by increasing the dimensions of the

119

6.7 Summary

box containing material under test, size of the PEC, and by improving the measurement

process.

120

CHAPTER7Conclusion

Contents

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2 Limitation of the Work . . . . . . . . . . . . . . . . . . . . . . . 124

7.3 Scope for Future Research . . . . . . . . . . . . . . . . . . . . . 125

121

7.1 Conclusions

The chapter summarizes overall contributions of the thesis. Some important conclu-

sions are outlined based on the work reported in the thesis. Following this, limitation and

future scope of the research are discussed.

7.1 Conclusions

This dissertation has presented a full wave model (FWM-2) for GPR application and

proposed three computationally efficient modified plane wave models (MPWMs) which

are essentially the analytical solutions of FWM-2. The inversion of models has been

successfully carried out by novel layer stripping methods and gradient-based method. A

stability study carried out with synthetic data has partially revealed the accuracy of all

these models in presence of measurement error and noise. Following is the sequential

conclusions based on the individual contributions presented at the end of each chapter.

GPR is a noninvasive method for characterizing and imaging earth’s subsurface. De-

spite several break through in the field of GPR modelling, the accuracy and speed of

computation are the key driving factors for modern research in developing novel mod-

elling scheme. The analytical FWM has the potential to offer good accuracy, and better

computational speed compared to the existing numerical FWM.

Discussion on system parameters and interaction of EM wave with inhomogeneous

media in Chapter 2 has highlighted the importance of accurate and efficient model for

real-time GPR detection. A brief review of EM theory applied to planar layered media has

given basic approach to develop a GPR model incorporating EM properties of subsurface.

With a motivation to develop accurate and efficient model, an FWM (FWM-2) has

been proposed in Chapter 3 to represent response due to multilayered media in monostatic

and far-field configuration. Here the Green’s function is expressed in terms of the ratio

between received and transmitted electric fields. A closed form integral formula is derived

to represent the forward model accounting the EM properties of layered media. Subse-

quently, its accuracy has been verified by comparing with an existing full wave model

based on synthetic experiment. However, the proposed FWM-2 is still time-inefficient

compared to the approximate models based on plane wave assumptions. Therefore, a

new approach was required to address this issue.

122

7.1 Conclusions

In view of above, novel modelling schemes have been proposed in Chapter 4. Using

an asymptotic method, the FWM-2 Green’s function has been simplified to the MPWMs

which don’t require time-consuming integration. MPWM-2 is the most accurate among

MPWMs, and MPWM-3 is the simplest model to apply for multilayered media. The

problem of MPWM-1 and MPWM-2 becoming inaccurate and inefficient with increasing

number of layers has been solved with introduction of MPWM-3. In conclusion, a compu-

tationally efficient, completely closed form formula is derived to represent layered-media

Green’s function in spatial domain to propose MPWM-3.

Chapter 5 has analyzed the model inversion process. Any inverse modelling scheme

requires a calibration method for the extraction of high quality measurement data. Here

the GPR calibration has been carried out by repeated measurements on a metal plate

with variation in the antenna height. By using an over-determined set of equations,

the uncorrelated errors are minimized, and better quality of the measurement Green’s

functions have been extracted. Based on the outcome of calibrations, most of the linear

transfer functions have been related to the standard antenna parameters defined for the

free space measurements. These analyses are important for the physical interpretation of

monostatic far-field radar data with the FWM or any other existing models. Subsequently,

the model inversion approach has been proposed by defining the objective function in

least square sense. Though, it is difficult to estimate the accuracy of models due to

the imperfect measurement environment, it is possible to analyze the effect of major

sources of error with the help of synthetic data. In this work, the effect of antenna

height measurement error has been analyzed with the observation that it causes significant

uncertainties in the GPR estimation. The response surface analysis has partially revealed

the uniqueness of solution, the property of objective function, relations among the model

parameters, existence of local minimas, etc. The gradient method is efficient to invert

a model. However, this method requires a good initial parameter vector which can be

evaluated by novel layer stripping techniques proposed in this work. Therefore, accuracy

of a model inversion depends on multiple factors such as accuracy of the model, precession

of the measuring equipments, variation of setup and environment, process and protocols

followed during measurements, etc.

123

7.2 Limitation of the Work

The laboratory experiments (Chapter 6) have been planned with an aim to validate

the proposed models in terms of accuracy and computational speed. The testing of water

layer has been used to partially validate the proposed models. The laboratory testing of

single-layered and two-layered media have demonstrated that the MPWMs are capable of

reconstructing multilayered media with accuracy as good as the FWMs while maintaining

significant advantage of computational speed. It has been also observed that the proposed

LS techniques are capable of finding complete electrical parameters approximately for a

layered media. This initial information has helped to apply gradient method for inverting

layered media parameters efficiently.

An accurate and time-efficient model is an extremely useful tool for various GPR ap-

plications. In this work, significant improvements have been introduced in GPR modelling

to describe monostatic far-field data. With remarkably high computational efficiency and

detection accuracy, the proposed integrated approach of model inversion has the potential

to become a valued alternative for many real-time GPR applications including reconstruc-

tion of layered media.

7.2 Limitation of the Work

The work presented in the thesis has got certain limitations in terms of completeness

in the laboratory experiments, analysis of model performances, etc. Some of these are

discussed below.

The inversion method proposed here is based on finding initial parameters by LSAM

and LSIM. Though gradient-based inversion schemes are efficient, their success of finding

the optimum point depends on the initial guess that needs to be in the global basin. The

proposed LSIM can find complete electrical parameters of multilayered media as long as

measurement error is within a certain limit. Therefore, there is a limitation of applying

this approach for inversion in the presence of significant error and noise.

With nonavailability of standard equipment to verify the testing data, the accuracy of

models has been demonstrated by comparing their performance with an well-established

FWM [1]. The accuracy is also verified by testing a water layer. It would be more

interesting to compare the accuracy of inverted soil parameters with a standard measuring

124

7.3 Scope for Future Research

instrument.

The experimental setup has got few limitation. The dimensions of sandboxes are small

compared to the recommended dimension. In addition to this, there is an issue with

antenna mounting which causes misalignment of the antenna. Moreover, the sand mixing

was done manually without having proper facility of mixing, drying, or moisturization.

All these errors have contributed significant amount of calibration and measurement error.


The following points discuss some possible future research directions for the work reported

in this thesis.

The Green’s function for the FWM is simplified due to the monostatic radar configu-

ration with source and receiver antenna to be the same point. In this case, the oscillating

Bessel’s function disappears. However, many GPR configuration are of bi-static type.

Therefore, it is essential to develop the similar models for bi-static GPR configuration.

In many cases, the GPR antennas are placed very close to the ground to increase

the penetration depth. Modelling GPR for this type near-field configuration has many

challenges. Most importantly, the antenna can no longer be assumed as a point source

and receiver. It’s radiation pattern has to be accounted for accurate modelling of GPR

system. The future work can explore the possibility of extending the proposed models for

near-field GPR configuration.

Although, the time-efficiency achieved by the MPWMs are significant, all these models

are extracted based on the LTFs calibration which is a time-consuming task. The future

work should focus on reducing the calibration complexity. Again the computational com-

plexities of forward models and their inversions are measured by time taken by a laptop

(1.93 GHz core i3 2 GB RAM) while applied for various types of planner medium. Better

method may be adopted to quantify this parameter for all the models.

The proposed LSIM works well as long as the measurement error is within a certain

limit. For higher error, global optimization schemes are recommended. The future work

should focus on developing a robust global optimization scheme for inverting models while

GPR data is corrupted by measurement error and noise.

125


126

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Disseminations of Work

Journals

Accepted

1. S. Maiti, S. K. Patra, and A. Bhattacharya, “GPR modeling for rapid character-

ization of layered media,” Progress In Electromagnetics Research B, vol. 63, pp.

217-232, 2015.

2. S. Maiti, S. K. Patra, and A. Bhattacharya, “A Modified Plane Wave Model for Fast

and Accurate Characterization of Layered Media,” IEEE Trans. Microw. Theory

Techn. (Accepted in Jan 2017)

Conferences

1. S. Maiti, S. K. Patra, and A. Bhattacharya, “Improving GPR signal modelling for

efficient characterization of multi-layered media,”16th International Conference on

GPR, Hong Kong, June 2016 (Presented).

2. S. Maiti, S. K. Patra, and A. Bhattacharya, “Modeling GPR signal for fast and

accurate characterization of layered media,”Antennas & Propagation Conference

(LAPC), 2015 Loughborough. IEEE, 2015.

3. S. Maiti, S. K. Patra, and A. Bhattacharya, “Modelling GPR for characterization

of subsurface EM properties,”IEEE MTT-S International Microwave and RF Con-

ference., New Delhi, IEEE, 2013.

139

Author’s Biography

Subrata Maiti completed his Bachelor of Engineering (B.E) from Electronics and Com-

munication Engineering Department of National Institute of Technology (NIT, previously

REC), Durgapur, India in 2000, and Master of Technology (M. Tech) with specialization

Microwave Engineering from the Department of Electrical Engineering, Indian Institute

of Technology (IIT) Kanpur in 2002. He served C-DOT Bangalore as a Research Engi-

neer during 2002 to 2005 and Kyocera Wireless India Limited, Bangalore as a Hardware

Engineer and Senior Hardware Engineer during 2005 to 2007. He joined NIT Rourkela

in the Department of Electronics and Communication Engineering as an Assistant Pro-

fessor and Institute research scholar in year 2011. Till date, he is a PhD scholar and

serving NITR Rourkela as an Assistant Professor. His research interests include radio

frequency (RF) engineering, electromagnetic modelling, inverse scattering, ground pene-

trating radar (GPR), GPR signal processing. He can be contacted at: [email protected],

& [email protected].

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