TIME REVERSAL MICROWAVE METHODS FOR SOLVING INVERSE
PROBLEMS IN NON-DESTRUCTIVE EVALUATION
By
Solimar Reyes Rodriguez

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Electrical and Computer Engineering
2011

ABSTRACT
TIME REVERSAL MICROWAVE METHODS FOR SOLVING INVERSE
PROBLEMS IN NON-DESTRUCTIVE EVALUATION
By
Solimar Reyes Rodriguez
Non-Destructive Evaluation (NDE) methods are use to inspect a material and
components without damaging their usefulness. NDE is used in diﬀerent industries
to inspect the reliability of critical components, product quality, or detect material failure. The techniques used in NDE range from simple visual methods to microwaves,ultrasound, x-rays, and thermography. The key problem in NDE is the
inverse problem which involves reconstructing defect proﬁles using the information in
the output signal of the system. Inverse problem solutions in NDE can be classiﬁed
as model-based and system-based approach. In model-based approach, an accurate
forward model is used in an iterative framework to estimate the defect shape that
minimizes the error between the measured and simulated signals. However, this approach results in repeated executions of a three dimensional forward model in each
iteration, making it computationally demanding.
This thesis presents a direct approach to inversion using principles of time reversal.
Time reversal focusing is based on the fact that when a wave solution is reversed in
time and back propagated it comes to focus at the source. Research on time reversal
techniques of ultrasound ﬁelds has demonstrated the reliability of the technique for
detecting small defects in complex geometries. This thesis uses a computational model
to study the feasibility of applying principles of time reversal to microwave NDE data
for solving the inverse problem of defect detection in dielectric materials.
Microwave NDE methods are well suited for inspection of dielectric materials because electromagnetic waves can propagate through and interact with such materials.
The interaction is inﬂuenced by the electrical and magnetic properties of the material

and hence the response of this interaction contains information of discontinuities of
permittivity in the material.
A two-dimensional ﬁnite diﬀerence time domain (FDTD) model, for simulating the
propagation of forward and time reversed wave ﬁelds is developed. A dielectric slab
used in the simulations is illuminated by a gaussian modulated pulse. The measured
microwave NDE measurements are recorded, time reversed and propagated backwards
once through the FDTD model to highlight the scatterer/defect. Maxima in the
energy image indicates the location of the defect. Simulation results demonstrate the
ability of the technique to accurately detect defects in dielectric and lossy dielectric
materials. Experiments were performed in free space to validate the FDTD model.
The signals recorded in the experiment were time reversed and input to the FDTD
model. Errors in source position, using the model, were attributed to experimental
measurements.

DEDICATION
This thesis is dedicated to my husband Carlos Eduardo Nino Baron. He oﬀered me
unconditional love and support through my Ph.D. studies. I also dedicate this thesis
to my parents Samuel Reyes Saldana and Aida Rodriguez Rodriguez for all their
support, love and for teaching me the importance of education.

iv

ACKNOWLEDGMENT
Diﬀerent people helped me complete my Ph.D. degree. First of all, I have to thank
my advisor, Dr. Lalita Udpa for all her invaluable guidance through my research,
course work, and writing. I would also like to thank my graduate studies committee:
Dr. Robert McGough, Dr. Satish Udpa, Dr. Wright, and Dr. Rothwell for their
time, guidance and suggestions. Special thanks to Dr. Rothwell, who let me used
his laboratory and equipment to perform the experiments that were necessary to
complete this thesis and for his guidance after the experiments were completed.
I would also like to thank Benjamin Crowgey for all his time and for his help
with the experiments. He taught me how to use the equipment and helped me with
the experimental set up. I like to thank Raoul Ouedraogo for his help analyzing the
experimental data.
Next, I would like to thank Dr. Barbara O’Kelly, Dr. Percy Pierre, the Sloan
foundation, and the Graduate School for all their support, guidance, and funding
through my graduate studies. I am a Harriet G. Jenkins Fellow and this fellowship
provided me with funding for my graduate studies. I would like to thank this program
for all the ﬁnancial support and all the wonderful experiences that I was exposed to
while being part of this program. I was also a Co-op student at NASA Glenn Research
Center. The center provided me ﬁnancial support over the summers and invaluable
work experiences. I will always be grateful for the opportunities that the center gave
me.
Finally, I would like to thank my parents, my friends, colleges at the MSU NDE
laboratory and my beloved husband, Carlos Eduardo Nino Baron for all their support.
They were with me every step of this incredible journey and I will always be grateful.

v

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

1 Introduction . . . . . . . . . . . . . . . . .
1.1 Introduction to Non-Destructive Evaluation . .
1.2 Microwave NDT/NDE principles . . . . . . . .
1.3 Forward and Inverse Problem . . . . . . . . . .
1.4 Model-Based Approach . . . . . . . . . . . . . .
1.5 Time Reversal approach for inverse problems . .
1.6 Contributions . . . . . . . . . . . . . . . . . . .

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2.1 Introduction . . . . . . . . . . . . . . .
2.2 Maxwell Equations . . . . . . . . . . .
2.3 FDTD Spatial and Time Discretization
2.4 Absorbing Boundary Conditions . . . .
2.5 Source . . . . . . . . . . . . . . . . . .

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3 Inverse Problem Solution Using Time Reversal Techniques . . .
3.1 Introduction to Inverse Problems in NDE/NDT . . . . . . . . . . . .
3.1.1 Calibration Techniques . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Indirect Inversion . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Previous Research: Time Reversal Technique used in Ultrasound and
Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Previous Research: Time Reversal Used to Solve the Inverse Problem
3.4 Proposed Approach to Solve the Inverse Problem in NDE/NDT: Time
Reversal Technique using an FDTD Model . . . . . . . . . . . . . . .
3.4.1 Mathematical Background . . . . . . . . . . . . . . . . . . . .
3.4.2 Time Reversal Focusing . . . . . . . . . . . . . . . . . . . . .

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4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Validation of Time Reversal Analysis . . . . . . . . . . . . . . . . . .
4.2 Application of Time Reversal for Defect Imaging . . . . . . . . . . . .
4.3 Test 1: Two scatterers in Dielectric Medium . . . . . . . . . . . . . .
4.4 Test 2: Single perfect electric conductor (PEC) Scatterer in Dielectric
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Test 3: Two Scatterers in Dielectric Medium- Resolution of the Model
4.6 Test 4: Two Scatterers in Dielectric Medium - Reconstruction with noise
4.7 Test 5: Two Scatterers in Dielectric Medium- Two Sources with Noise.

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4.8 Test 6: Minimum number of sensors. . . . . . . . . . . . . .
4.9 Test 7: Time Reversal Applied to Lossy Dielectric Medium. .
4.9.1 Two well separated defects. . . . . . . . . . . . . . .
4.9.2 Closely placed defects. . . . . . . . . . . . . . . . . .
4.10 Test 8: Eﬀect of errors in receiver measurements. . . . . . .
4.10.1 Error in time measurements. . . . . . . . . . . . . . .
4.10.2 Error is Transducer Position. . . . . . . . . . . . . .
4.11 Test 9: Results with diﬀerent sensor locations . . . . . . . .
4.12 Test 10: Distances from the source to the sensor array. . . .

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6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5 Experiment .
5.1 Experiment
5.2 Experiment
5.3 Experiment

Bibliography

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Set Up . . .
Calibration .
Results . . .

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vii

90

LIST OF FIGURES

1.1 General NDT/NDE system

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2

1.2 Inverse problem description . . . . . . . . . . . . . . . . . . . . . . .

7

1.3 Forward problem description. (a) Forward problem classiﬁed as the
direct problem. (b) Forward problem classiﬁed as the signal detection
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4 Iterative process to solve the inverse problem . . . . . . . . . . . . . .

9

2.1 Spatial Grid for the FDTD algorithm.

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16

2.2 FDTD Time Discretization Scheme. . . . . . . . . . . . . . . . . . . .

17

3.1 Model based iterative method to solve the inverse problem. . . . . . .

24

4.1 Time reversal cavity that shows the receiver array location and the
propagation of the Gaussian pulse. ”For interpretation of the references to color in this and all other ﬁgures, the reader is referred to the
electronic version of this dissertation.” . . . . . . . . . . . . . . . . .

32

4.2 Progress of electric ﬁeld as a function of time when the electric ﬁeld is
propagated backwards in the model. . . . . . . . . . . . . . . . . . . .

33

4.3 A block diagram indicating the procedure that was followed. . . . . .

34

4.4 Simulations results for two scatterers in a dielectric medium. (a) Forward propagating wave from source. (b) Focusing of time reversed
wave on scatterers. (c) Energy images indicating defects location. . .

35

4.5 Simulations results for single PEC scatterer in a dielectric medium. (a)
Forward propagating wave from source. (b) Focusing of time reversed
wave on scatterer. (c) Energy image indicating defect location. . . . .

37

4.6 Simulations results for 2 scatterers in a dielectric medium 2.27λ apart.
(a) Focusing of time reversed wave on scatterers. (b) Energy image
indicating location of defects. . . . . . . . . . . . . . . . . . . . . . .

38

viii

4.7 Simulations results for 2 scatterers in a dielectric medium 6.28λ apart
with noise. (a) Focusing of time reversed wave on scatterers. (b)
Energy image indicating location of defects. . . . . . . . . . . . . . .

39

4.8 Simulations results for 2 scatterers in a dielectric medium 6.28λ apart
with noise. (a) Focusing of time reversed wave on scatterers. (b)
Energy image indicating location of defects. . . . . . . . . . . . . . .

40

4.9 Simulations results for two scatterers in a dielectric medium. (a) Forward propagating wave from the two sources. (b) Focusing of time
reversed wave on scatterers. (c) Energy images indicating defects location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.10 Simulations results for 1 scatterer in a dielectric medium with permittivity of 4. The number of sensors used is 50. (a) Time reversed waves
focus on the scatterer. (b) Energy image indicating maximum energy
at defect location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.11 Simulations results for 2 scatterer in a dielectric medium 2.27λ apart.
The number of sensors used is 50. (a) Time reversed waves focus on
the scatterers. (b) Energy image indicating maximum energy at defects
location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.12 Simulations results for 2 scatterers in a dielectric medium when noise
(SNR=10) is added to the recorded signals. The number of sensors
used is 50. (a) Time reversed waves focuses on the 2 scatterers. (b)
Energy image indicating maximum energy at defect location. . . . . .

45

4.13 Simulations results for 2 scatterers in a dielectric medium when noise
(SNR=2) is added to the recorded signals. The number of sensors used
is 50. (a) Time reversed waves focuses on the 2 scatterers. (b) Energy
image indicating maximum energy at defect location. . . . . . . . . .

46

4.14 Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.05. (a) Time reversed waves focuses on the 2 scatterers. (b)
Energy image indicating maximum energy at defect location. . . . . .

47

4.15 Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.1. (a) Time reversed waves focuses on the 2 scatterers. (b) Energy
image indicating maximum energy at defect location. . . . . . . . . .

48

ix

4.16 Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.2. (a) Attenuated time reversed waves. (b) Energy image indicating maximum energy at defect location. (c) Energy image indicating
maximum energy location. . . . . . . . . . . . . . . . . . . . . . . . .

49

4.17 Simulations results for 2 scatterers 2.27 λ apart in a lossy dielectric
medium when σ=0.05. (a) Attenuated time reversed waves. (b) Energy
image indicating maximum energy at defect locations. . . . . . . . . .

50

4.18 Attenuation eﬀects from the conductivity (σ=0.08) of the material for
2 scatterers 2.27 λ apart. (a) Comparison between signal from sensor
337 for a lossy dielectric and a dielectric. (b) Signals from sensor 337
for a lossy dielectric. (c) Signals from sensor 337 for a pure dielectric.

51

4.19 Simulations results for 2 scatterers 2.27 λ apart in a lossy dielectric
medium when σ=0.08. (a) Attenuated time reversed waves. (b) Energy
image indicating maximum energy at defect locations. . . . . . . . . .

52

4.20 Energy image for 2 scatterers 2.27 λ apart in a lossy dielectric medium
when σ=0.2 and ε=2.9. . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.21 Individual A-scan signals at the receiver when the time delay is modiﬁed. 54
4.22 Energy image when signals at all positions were delayed by random
quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.23 Individual A-scan signals at the receiver when the time delay is modiﬁed. 55
4.24 Energy image when 46 signals were modiﬁed. . . . . . . . . . . . . . .

55

4.25 Individual A-scan signals at the receiver when the time delay and amplitude are modiﬁed. . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.26 Energy image when 46 signals were modiﬁed. . . . . . . . . . . . . . .

56

4.27 Energy image when time delay is simulated using sensor position error. 57
4.28 Energy image when 11 sensors from a distance of -1.5 to 1.5 m were
used in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.29 Snapshot from movie showing the place where the waves focus. . . . .

58

4.30 Individual A-scan signals at the receivers. . . . . . . . . . . . . . . . .

59

x

4.31 Energy image when 6 sensors from a distance of -1.5 to 1.5 m were
used in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.32 Snapshot from movie showing the place where the waves focus. . . . .

60

4.33 Individual A-scans signals at the receiver. . . . . . . . . . . . . . . . .

60

4.34 Individual A-scans signals at the receivers. . . . . . . . . . . . . . . .

61

4.35 Snapshot from movie showing the place where the waves focus. . . . .

61

4.36 Energy image when 5 sensors from a distance of -0.6 to 0.6 m were
used in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.37 Individual A-scan signals at the receivers. . . . . . . . . . . . . . . . .

62

4.38 Snapshot from movie showing the place where the waves focus. . . . .

63

4.39 Energy image when 6 sensors from a distance of -0.5 to 0.5 m were
used in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.40 Individual A-scan signals at the receivers. . . . . . . . . . . . . . . . .

63

4.41 Snapshot from movie showing the place where the waves focus. . . . .

64

4.42 Energy image when 7 sensors from a distance of -0.45 to 0.45 m were
used in the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.43 Energy image when the sensors are 1 m away from the source location. 65
4.44 Energy image when the sensors are 0.7 m away from the source location. 65
4.45 Energy image when the sensors are 0.5 m away from the source location. 65
4.46 Energy image when the sensors are 0.2 m away from the source location. 66
5.1 Experiment Set Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2 Transmitter antenna placed on the arch range. . . . . . . . . . . . . .

69

5.3 Receiver antenna on cart. . . . . . . . . . . . . . . . . . . . . . . . .

70

xi

5.4 Experiment performed at a distance of 1.96 m between transmitter and
receiver antennas. (a) Picture of experiment when antennas are 1.96
m away. (b) Drawing depicting the distance between the antennas. .

71

5.5 Experiment performed at a distance of 2.90 m between transmitter and
receiver antennas. (a) Picture of experiment when antennas are 2.90
m away. (b) Drawing depicting the distance between the antennas. .

72

5.6 Experiment set up when antennas are 1.96m away.

. . . . . . . . . .

73

5.7 Experiment set up when antennas are 2.90m away.

. . . . . . . . . .

74

5.8 Plywood panel used to move and avoid shift in the receiver antenna. .

75

5.9 Conducting plate placed between transmitter and receiver antenna. .

76

5.10 Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna is 1.96 m away from receiver antenna. . . . . . . . . . . . . . .

76

5.11 Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna is 1.96 m away from receiver antenna. . . . . . . . . . . . . . .

77

5.12 Energy image for experiment performed at a distance of 1.96 m where
the energy maximum occurs at (0.053, -0.2740). . . . . . . . . . . . .

77

5.13 Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna is 2.90 m away from receiver antenna. . . . . . . . . . . . . . .

78

5.14 Energy image for experiment performed at a distance of 2.90 m where
the energy maximum occurs at (0.0920, -0.3850). . . . . . . . . . . .

78

5.15 Experimental signals at a distance of 2.90 m for positions 1 (60 cm),
21 (0 cm), and 41 (-60 cm). . . . . . . . . . . . . . . . . . . . . . . .

79

5.16 Triangle formed by the measurements taken at the experiment set up.

79

5.17 Triangle formed by the measurements taken at the experiment set up.

80

5.18 Triangle constructed to calculate coordinates of true sensors positions
at the experiment set up. . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.19 Energy image from simulation of experiment set up. . . . . . . . . . .

81

xii

5.20 Experimental signals at a distance of 1.96 m for positions 1 (60 cm),
21 (0 cm), and 41 (-60 cm). . . . . . . . . . . . . . . . . . . . . . . .

82

5.21 Triangle formed by the measurements taken at the experiment set up.

83

5.22 Triangle formed by the measurements taken at the experiment set up.

83

5.23 Triangle constructed to calculate coordinates of true sensors positions
at the experiment set up. . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.24 Energy image from simulation of experiment set up. . . . . . . . . . .

84

xiii

Chapter 1
Introduction
1.1

Introduction to Non-Destructive Evaluation

Detection of defects in critical components and materials without compromising
the integrity of the sample is an important area of research in Non-Destructive
Testing/Non-Destructive Evaluation (NDT/NDE). NDT/NDE is deﬁned by the American Society of Nondestructive Testing as a test that inspects an object, material or
system without damaging the usefulness of the material. Figure 1.1 describes a general NDT/NDE system where the specimen or sample is illuminated with a speciﬁc
form of energy, the type of energy depends on the nature of the material that is being
analyzed, the signals that are produced from the interaction of this energy and material is picked up by the receiving sensor. These signals contain information about
the material properties and defects within the material. The signals are processed for
material characterization and defect detection purposes.
There are diﬀerent forms of energy used to illuminate the sample that is inspected
and an appropriate method is selected based on the nature of the application. Ultrasound, Eddy currents, Shearography, Magnetic Particle Testing, X-Rays, and Microwaves are some examples of the diﬀerent techniques and energy sources used in
1

Figure 1.1: General NDT/NDE system

NDT/NDE for the inspection of materials. Ultrasound techniques can determine the
thickness of a material and the location of a discontinuity or defect by measuring
the time of ﬂight at the reﬂections from the defects seen as distinct peaks in the signal that is received. The disadvantages of this technique is that it needs a couplant
to couple the energy eﬃciently into the material and secondly, it is diﬃcult for the
acoustic signals to penetrate and interact inside porous materials [1]. Eddy current
techniques are useful in detecting surface cracks, and near surface cracks, measuring
material thickness, coating thickness, and conductivity of samples for material identiﬁcation and heat damage detection. Eddy current methods are not well suited to
inspect dielectric materials because eddy currents can only be induced in conductive
materials [1]. Magnetic Particle Testing uses magnetic ﬁelds and iron ﬁlings to detect
defects in ferromagnetic materials. As with eddy current techniques the problem with
Magnetic Particle Testing is that it cannot be used to inspect dielectric materials.
Radiography is a technique that uses gamma or X-radiation to detect defects. Defects are identiﬁed by changes in the image caused by discontinuities/changes in mass
density or attenuation constant of materials being inspected. However X-rays present
a health hazard, and is not suitable for surface defects. Shearography is an imaging
2

technique that uses interferometry to obtain real time measurements of changes to a
target. It is used to evaluate composites and to detect disbonds, delaminations, and
porosity.
Microwave techniques well suited for dielectric materials which allow electromagnetic waves to propagate through and interact with non-conducting structures which
is useful for inspection and characterization of composite and dielectric materials [2].
The increasing importance of microwave NDE is largely due to increasing utilization
of composite materials in several industries ranging from aviation and automotive
to civil infrastructure. Microwaves are also used as an alternate method to inspect
new upcoming dielectric and composite materials that are being used in the industry
to replace metal. These materials are lighter, stiﬀer, stronger and electrically insulating and new NDE methods need to be developed for reliably characterizing these
materials [3].

1.2

Microwave NDT/NDE principles

Microwave spectrum ranges from 300 MHz to 30 GHz with wavelengths ranges from
1,000 to 10 mm [1]. Microwaves interact better with dielectric and composite materials
because they can penetrate deep into these materials. The material-energy interaction
is aﬀected by the loss factor associated with dielectric materials and the frequency
at which the experiment is conducted [1, 4, 3]. The key advantages of microwave
techniques are; it is a non contact method, does not require a couplant as in ultrasound
NDE, it is monostatic, and it does not require high power [1].
Microwaves can oﬀer sub-wavelength resolution even though the wavelength is in
the centimeter range. Near-ﬁeld Microwave measurements have shown resolution in
the micrometer range for a wavelength of 3cm in free space when monitoring thickness
variation in a sample and a resolution of λ/10 separation between two defects. This
3

is because lateral resolution in the near ﬁeld of the diﬀerent probes that are used is
determined by the physical characteristics of the probes. Spatial resolution in the
other hand will depend on the type of probe that is selected [3]. The availability of
a variety of probes and sensor arrays allows inspection of large areas in an optimal
manner.
There are diﬀerent ways to conduct microwave experiments. Measurements could
be done in-contact or non-contact modes. Both of these measurements can be done
in transmission or reﬂection mode. Further, microwaves are also polarized and this
characteristic is useful to increase measurement sensitivity to defects of a speciﬁc
orientation [3].
Microwaves as stated before have many applications in the area of dielectric material characterization. Some of the applications in this area are dielectric mixture
constituent determination, porosity evaluation in polymers, moisture measurements,
and cure monitoring of resin binders, rubber products and cement-based materials
[1]. Microwaves are also useful to determine changes in dielectric sheets thickness
and also to detect small defects, thin disbonds, delaminations and the location of
these in composites [1]. Defects on metals surfaces, fatigue crack detection and sizing
under paint in metals, corrosion under paint, and corrosion precursor pitting impact
damage on carbon based composite are some examples of the inspection capabilities
of microwaves [1]. Microwaves have also been used for biological applications [1].
Cement-based materials are commonly inspected using microwaves NDT/NDE.
These materials are used extensively in the construction industry and for this reason
their inspection is crucial for security and quality control reasons. In [5] the reﬂection
and transmission properties of cement-based materials were analyzed. To determine
properties of this material at the ﬁrst stages of the hydration process and that in order
to perform long time monitoring of hardened cement-based materials transmission
measurements were used. They also found that because of the porosity of the cement
4

materials, higher transmission coeﬃcient measurements correspond to initial high
water content in the material. The authors found that the amplitude of reﬂection and
transmission coeﬃcients was related to the permittivity of the material. It was shown
in [5] that the reﬂection coeﬃcient depends on the real part of the permittivity and
that the transmission coeﬃcient depends on the imaginary part of the permittivity.
Microwave measurements of reﬂection and transmission coeﬃcients can be done in
the near-ﬁeld and far-ﬁeld. Arunachalam et al. [6] presents a feasibility study about
far-ﬁeld microwave frequency domain reﬂection measurements to produce images of
cement-based materials. They produced good images of planar mortar, concrete, and
steal bar inside concrete samples using simple post-processing techniques. Boise et
al. [7] presents the use of microwave near ﬁeld reﬂection measurements to evaluate
the material content of concrete. Where an open-ended rectangular waveguide probes
measure the reﬂection coeﬃcient of diﬀerent concrete samples with diﬀerent material
constituents to determine the compressive strength of concrete. The paper concludes
that the reﬂection measurements obtained from the samples follow diﬀerent statistical
distributions. At 10 GHz the specimens that were measured followed a Gaussian
distribution and at this frequency there was also a correlation with concrete coarse
aggregate to cement ratio. At 3 GHz the results followed a uniform distribution and
the mean of the reﬂection coeﬃcient at this frequency had a correlation with water
to cement ratio [7].
Carbon ﬁber reinforced polymer (CFRP) is a composite material that is used to
reinforce structures such as bridges. This materials need to be inspected to determine if they become disbonded from the structure to which they were attached. In
order to inspect these materials a dual-polarized near-ﬁeld microwave reﬂectometer
was used in [8]. This technique takes advantage of the anisotropy presented in unidirectional CFRP laminates. Microwaves reﬂect of the CFRP when the polarization
of the electric ﬁeld is normal to the ﬁber dimensions and they penetrate the material
5

when the polarization is parallel to the ﬁber dimensions [8]. The authors use these
properties to generate images of the CFRP using perpendicular polarization to detect
disbonds in the material and parallel polarization to remove inﬂuences due to surface
roughness, and standoﬀ distance change from perpendicular polarization [8]. This
technique provided images in which disbonds locations and sizes could be seen.
Layered composite materials are also evaluated with NDT/NDE techniques with
good results. Zoughi et al. [9, 10] developed techniques to detect the presence of corrosion under thin layers of paint and primer, which are considered dielectrics. It was
found in this research that the phase of the reﬂection coeﬃcient at high frequencies
is useful to detect corrosion. In this research open-ended rectangular waveguides are
used to achieve detection of corrosion under layers of paint and primer.
Delamination cracks in thermal barrier coatings (TBCs) are also a problem in the
aerospace industry. TBCs cover the components of gas turbines to avoid overheating
of these parts [11]. The authors found that at around 55 to 65 GHz, the amplitude
and phase of the transmission coeﬃcient contain information about location and
characterization of the defects.

1.3

Forward and Inverse Problem

In an inverse problem, the transfer function of a system is estimated. The transfer
function describes the system characteristics. By analyzing the output and input of
the system, the transfer function can be found; this is the central principle of the
inverse problem[12].
In NDT/NDE inverse problem, an energy source is used to illuminate a sample
and the response signal is analyzed to detect and characterize defects in the sample
[13]. Figure 1.2 illustrate an inverse problem system. The input and output are
known but the system parameters are unknown. Usually the input is an excitation
6

signal from a transmitting transducer and the output is the response of the system to
the excitation signal. Using these two known signals a system can be identiﬁed [14].
In NDT/NDE the structure of the object under test is known but internal parameters
such as defects, their location, and shape characteristics need to be estimated.

Figure 1.2: Inverse problem description

The inverse problem is in general ill-posed. An ill-posed problem is one where
the solutions are non unique and non continous [12]. The fact that the solutions
are unstable and non unique is a problem because the information contained in the
output signal corresponds to more than one defect proﬁle [15].
Methods to overcome these issues with inverse problems solutions are a major area
of interest in NDT/NDE. One of the techniques used to solve the inverse problem is
the iterative approach. The iterative approach involves the solution of the forward
problem using forward models, which are discussed in section 1.4.
Unlike inverse problems, forward problems are well-posed. In contrast to an illposed problem, well-posed problems are stable, which means that the solution depends
continuously on the given input data. Well-posed problems generate accurate and
unique solutions. There are two types of forward problems: direct problem and
signal detection problem, illustrated in ﬁgure 1.3. Figure 1.3 (a) describes a direct
problem. In a direct problem the output of the system is found because the input
and the transfer function of the system are known. Figure 1.3 (b) describes a signal
detection problem. In a signal detection problem the input is found when the output
and the system are known [14].
7

(a)

(b)
Figure 1.3: Forward problem description. (a) Forward problem classiﬁed as the direct
problem. (b) Forward problem classiﬁed as the signal detection problem.

1.4

Model-Based Approach

Iterative approaches, that involve the solution of the forward problem, are used to
solve the inverse problem. Figure 1.4 describes the overall model-based process. The
algorithm starts with an estimate of the system and solves the forward problem to
obtain the output signal. The error between the measured signals and the simulated
output signals is minimized iteratively by updating the defect proﬁle. If the error
calculated is above a given threshold, the defect parameters are updated and the
process is repeated until the error is below the threshold. When the error is below
the threshold, the defect parameter is the desired solution [16]. The iterative approach
requires an accurate forward model. Forward models are numerical models based on
integral and diﬀerential equations.
There are diﬀerent numerical methods used to solve the inverse problem. Finitediﬀerence method (FDM), boundary element method (BEM), and the well known
ﬁnite-element method (FEM) are examples of numerical methods used to create computational models. However, use of a computational numerical model in an iterative
framework results in exorbitant computational resources. This thesis presents a noniterative model-based method for solving the inverse problem.
8

Figure 1.4: Iterative process to solve the inverse problem

1.5

Time Reversal approach for inverse problems

Time reversal focusing is based on the fact that when a wave solution is reversed in
time and back propagated it comes to focus at the source. In acoustics the propagation
equation for a pressure ﬁeld in a lossless medium is given by equation 1.1
(− )
→
−
→
∇p
1 ∂ 2p
∇•
−
=0
ρ
ρc2 ∂t2

(1.1)

where p(r, t) is the pressure ﬁeld ρ(r) is the density and c(r) is the sound velocity.
Because of invariance under Time Reversal both p(r, t) and p(r, −t) are solutions of
the propagation equation. This property can be exploited to use wave measurements
for imaging sources or scatterers /defects in a test specimen.
Time Reversal experiments in ultrasound are usually performed using a time reversal mirror that is made of an array of transmit-receive transducers. In [17] experiments were performed to detect kidney stones in aberrating media. These experiments
proved to be successful in detecting the stones and focusing on a small portion of them
through aberrating media. Time Reversal had also been used to detect ﬂaws in thin
plates by increasing the signal to noise ratio [18]. It is very diﬃcult to detect ﬂaws
using Lamb waves but time reversal, because of the invariance property, reduces the
9

dispersion of Lamb waves allowing the ﬂaws on thin plates to be detected.
Time reversal of microwave ﬁelds has also been studied. Kosmas et al. have
developed a ﬁnite diﬀerence time domain (FDTD) time reversal as a matched ﬁlter
approach for microwave breast cancer detection [19, 20]. Experiments done in microwave time reversal are presented in [20]. This thesis presents a feasibility study,
using computational model, of applying principles of time reversal to microwave NDE
data for solving the inverse problem of defect detection in dielectric materials.

1.6

Contributions

The major contribution of this work is development of an FDTD forward model and
time reversal techniques to solve the inverse problem of defect detection in Microwave
NDE. The main advantage combining these two techniques, time reversal and forward
models, is that an iterative solution to the inverse problem is not needed. Usually
iterative methods, where the minimization of an error is required, can encounter
convergence problems, are time consuming and computationally expensive.
The dissertation consists of the following chapters. Chapter 2 describes the ﬁnite
diﬀerence time domain formulation and boundary conditions applied. Time reversal
theory is discussed in Chapter 3. Validation of time reversal analysis is discussed in
Chapter 4. The application of time reversal techniques for defect imaging and the
results obtained with diﬀerent defects and in the presence of noise are also presented
in Chapter 4. Concluding remarks are given in Chapter 5. Future work is presented
in Chapter 6.

10

Chapter 2
Forward Model
2.1

Introduction

In this thesis a numerical model for solving the forward problem of microwave propagation through a material is developed. A brief description of the numerical model
based on ﬁnite diﬀerence time domian (FDTD) method is presented in this chapter.
In FDTD, the time-dependent Maxwell’s equations are discretized using centered
two point ﬁnite diﬀerences to approximate the spatial and time derivatives. The
formulation uses absorbing boundary conditions. Simulation results for several NDE
geometries are presented in Chapter 4.

2.2

Maxwell Equations

The underlying equation governing NDE inspection can be derived from Maxwell’s
equations. The Maxwell-Ampere’s and Maxwell-Faraday’s laws for time varying electromagnetic ﬁelds are:
⃗
∂D
⃗
⃗
=∇×H −J
∂t

(2.1)

⃗
∂B
⃗
= −∇ × E
∂t

(2.2)

11

The divergence equations are:
⃗
∇·D =ρ

(2.3)

⃗
∇·B =0

(2.4)

⃗
⃗
In linear, isotropic, time-invariant, nondispersive media, the ﬂuxes, D and B, and the
⃗
⃗
ﬁelds strengths, E and H, are related by material parameters.

⃗
⃗
D = εE

(2.5)

⃗
⃗
B = µH

(2.6)

Substituting equation (2.5) in equation (2.1)

ε

⃗
∂E
⃗
⃗
=∇×H −J
∂t

(2.7)

After substituting equation (2.6) in equation (2.2)

µ

⃗
∂H
⃗
= −∇ × E
∂t

where the variables are:
⃗
E: Electric Field Intensity (V /m)
⃗
H: Magnetic Field Intensity (A/m)
⃗
D: Electric Flux Density (C/m2 )
⃗
B: Magnetic Flux Density (T )
⃗
J: Electric Current Density (A/m3 )
ρ: Electric Charge Density (C/m3 )
12

(2.8)

ε: Permittivity

µ: Permeability

In free space and a charge free environment, Maxwell’s equations are:

⃗
∇·E =0

(2.9)

⃗
∇·B =0

(2.10)

⃗
∂B
⃗
= −∇ × E
∂t

(2.11)

⃗
∂E
⃗
=∇×B
∂t

(2.12)

µε

By applying diﬀerent vector identities to these equations, the wave equation for
source-free uniform lossless media is obtained. The ﬁrst step is to ﬁnd the curl of
equation (2.2).
(

(

)

⃗
∇× ∇×E =∇×

⃗
∂B
−
∂t

)
(2.13)

The curl of the curl is given by:
(

)
(
)
⃗ = ∇ ∇ · E − ∇2 E
⃗
⃗
∇× ∇×E

(2.14)

Producing the equation (2.13)
(

)
(
)
⃗ − ∇2 E = − ∂ ∇ × B
⃗
⃗
∇ ∇·E
∂t

(2.15)

Substituting equations (2.9) and (2.12) into equation (2.15) gives the electromag13

netic wave equation in terms of the electric ﬁeld
∂2 ⃗
E
∂t2

(2.16)

1 ∂2 ⃗
E
c2 ∂t2

(2.17)

⃗
∇2 E = µε

⃗
∇2 E =

⃗
where c is the wave propagation speed given by c = √1 and E is the electric ﬁeld.
µε
Following the same steps the electromagnetic wave equation in terms of the magnetic ﬁeld can be found. The ﬁrst step is to ﬁnd the curl of equation (2.12).
(

(

)

⃗
∇× ∇×B =∇×

⃗
1 ∂E
c2 ∂t

)
(2.18)

After ﬁnding the curl of equation (2.12) and using the deﬁnition of the curl of the
curl given in (2.14)
(

)
(
)
⃗ − ∇2 B = 1 ∂ ∇ × E
⃗
⃗
∇ ∇·B
c2 ∂t

(2.19)

Substituting equation (2.4) and equation (2.2) into equation (2.19) gives
1 ∂
⃗
−∇2 B =
c2 ∂t

(

⃗
−∂ B
∂t

)
(2.20)

Rearranging the terms in equation (2.20) produces the electromagnetic wave equation in terms of the magnetic ﬁeld
(

1 ∂2
∇2 −
c2 ∂t2

)

→
−
B =0

(2.21)

⃗
where c is the speed of light, and B is the magnetic ﬂux density.
The FDTD algorithm is based on the solution of Ampere’s and Faraday’s laws
14

using centered two point ﬁnite diﬀerences to approximate the spatial and time derivatives in these equations. The derivatives are replaced by
f (z0 + ∆z/2) − f (z0 − ∆z/2)
df
=
dz z=z
∆z
0

(2.22)

After the derivatives in the Ampere’s and Faraday’s laws are approximated by
centered two point ﬁnite diﬀerences, the equations to solve the magnetic and electric
ﬁelds are expressed in terms of the past magnetic and electric ﬁelds values. The
magnetic and electric ﬁelds are then solved one time step into the future and the
process is repeated until all the ﬁelds are calculated over the desired duration.

2.3

FDTD Spatial and Time Discretization

Transverse magnetic (TM) polarization was used for these simulations. This means
that there is no magnetic ﬁeld in the z direction. For this case Maxwell’s curl equations
reduce to
Ex = Ey = 0,Hz = 0

(2.23)

∂Hy ∂Hx
∂Ez
=
−
∂t
∂x
∂y

(2.24)

∂Hx
∂Ez ∂Hy
∂Ez
=−
,µ
=−
∂t
∂y
∂t
∂x

(2.25)

ε
µ

The spatial FDTD discretization is achieved by approximating the spatial derivatives
by two-point centered diﬀerences. For example the spatial FDTD approximation to
equation (2.25) is given by
µ ∂Hx = − ∂Ez
∂t
∂y
1 ]−H [i,j+ 1 ]
Hx [i,j+ 2
x
2 = Ez [i,j+1]−Ez [i,j]
−µ
∆t
∆y
15

(2.26)

After the two-point centered diﬀerence is applied to the spatial derivatives it could
be noticed that the electrical ﬁeld nodes are located at integer spatial steps and that
the magnetic ﬁeld nodes are located at half integer spatial steps. The location of the
electric and magnetic ﬁelds in the two diﬀerent grids is shown in Figure 2.1.

Figure 2.1: Spatial Grid for the FDTD algorithm.

Centered ﬁnite-diﬀerence approximations are also used to discretize the ﬁelds in
time domain. After the approximations are made the electric ﬁeld exists at integer
time steps and the magnetic ﬁelds are oﬀset half time step from the electric ﬁelds.
The ﬁelds are advanced in time using the same ﬁelds but at a diﬀerent instant in
time, the past instant in time in the grid, and also using the opposite ﬁeld type at
an intermediate instant in time. The solution of the ﬁelds depends only on past ﬁeld
values. Figure 2.2 shows the time discretization scheme.
After replacing the spatial and time derivatives in Faraday’s and Ampere’s equations using centered ﬁnite-diﬀerences, the equations for the magnetic and electric
16

Figure 2.2: FDTD Time Discretization Scheme.

ﬁelds used in the FDTD algorithm for TM polarization are:

Hx

Hy

1
n+ 2

1
n+ 2

[

] 1−
1
i, j +
=
2
1+

σi ∆t
]
1[
1
2µ
n− 2
H
i, j +
σi ∆t x
2
2µ
∆t
1
(E n [i, j + 1] − Ez n [i, j])
−
σi ∆t µ∆y z
1 + 2µ

[
] 1−
1
i + ,j =
2
1+

σi ∆t
[
]
1
2µ
n− 1
2 i + ,j
H
σi ∆t x
2
2µ
1
∆t
+
(E n [i + 1, j] − Ez n [i, j])
σi ∆t µ∆x z
1 + 2µ
17

(2.27)

(2.28)

σ∆t
n+1 [i, j] = 1 − 2ε E n [i, j]
Ez
σ∆ z
1 + 2ε t

}
{
[
]
]
1[
∆t
n+ 1
n+ 2
2 i + 1 , j − Hy
i − 1, j
Hy
 ε∆
2
2
x

1

{
}
+
[
]
[
]
σ∆t 
∆t
n+ 1
n+ 1

1
1 + 2ε
− ε∆
Hx 2 i, j + 1 − Hx 2 i, j − 2
2
y








(2.29)
Stability conditions have to be met in order to obtain accurate solutions for the
equations that are approximated using ﬁnite diﬀerences. It is necessary that the ﬁeld
solutions do not change signiﬁcantly from one point in the grid to the other. To
achieve this the grid steps must be the same in each direction, ∆x = ∆y [21, 22]. For
√
the two dimensional FDTD code the stability condition is given by c∆t ≤ ∆x .
2

2.4

Absorbing Boundary Conditions

FDTD is used to solve time-domain electromagnetic-ﬁeld equations in an unbounded
space. In order to limit the domain in which the ﬁeld is computed, a mesh of limited
size and boundary conditions are used. Absorbing boundary conditions are used on
the outer surface of the mesh to accurately model the unbounded surroundings in the
computational domain. There are diﬀerent kinds of absorbing boundary conditions
that are used in FDTD simulations to avoid ﬁelds reﬂections. The one selected for
these simulations falls into the One-way wave equations, speciﬁcally the ones developed by Mur [23]. One-way wave equations consist of a partial diﬀerential equation
that supports wave propagation in certain directions only. For example, if the computational domain is deﬁned in the area of x > 0 then FDTD equations already
deﬁned are used but at the boundaries of the computational domain, such as x=0,
18

one-way wave equations are used. In this case absorbing boundary conditions are
needed for the H-ﬁeld components that are tangential to the boundary of interest.
For the outgoing wave the ﬁrst order boundary condition is given by
(

1
∂x − c0 −1 (1 − (c0 sy )2 − (c0 sz )2 ) 2 ∂t

)
Wx=0 = 0

(2.30)

where sy , sz are the inverse velocity components, c is the speed of light and W is
the Hx or Hy for TM to z. This equation solves for W on the outer surface, which
is consistent with an outgoing wave. An approximation of this equation is made in
order to account for the fact that the angle of incidence of the wave approaching x=0
is unknown. After the approximation is made the ﬁnite diﬀerence approximation
equations used to implement absorbing boundary conditions in the FDTD code are:

Hy

n+ 1
2

[i, 0] = Hy

n− 1
2

c ∆ − ∆x
[i, 1] + 0 t
c0 ∆t + ∆x

c ∆ − ∆x
n+ 1
n− 1
2 [1, j] + 0 t
Hx 2 [0, j] = Hx
c0 ∆t + ∆x

2.5

(
Hy

n+ 1
2

[i, 1] − Hy

n− 1
2

)
[i, 0]

(2.31)

(

)
1
1
n+
n− 2
Hx 2 [1, j] − Hx
[0, j] (2.32)

Source

There are diﬀerent kinds of sources. Additive source is the one used in these simulations. The additive source is introduced in the FDTD algorithm through Ampere’s
law, which is one of Maxwell’s equations that contain the current density term. The
⃗
current density J has two diﬀerent meanings, the conduction current due to the ﬂow
of charge in a material due to an electric ﬁeld or the current of a source. For this
⃗
case J represents the current generated by a source and in this case is deﬁned to be
a gaussian or quasi-gaussian pulse. The source could be applied to one or more than
19

one node.
⃗
∂E
1
1⃗
⃗
= ∇×H − J
∂t
ε
ε

(2.33)

The equation that is used in the FDTD algorithm after the centered ﬁnite-diﬀerence
approximations are applied is

σ∆t
n+1 [i, j] = 1 − 2ε E n [i, j]
Ez
σ∆ z
1 + 2ε t

{
}
]
]
1[
1[
∆
n+ 2
n+ 2
1
i + 1 , j − Hy
i − 2, j
 ε∆t Hy
2
x

1

{
}
+
]
]
1[
1[
σ∆t 
1 + 2ε  − ∆t Hx n+ 2 i, j + 1 − Hx n+ 2 i, j − 1
2
2
ε∆y
−








∆t n+ 1
2 [i, j]
Jz
ε
(2.34)

This chapter presented the underlying governing equations for microwave NDE
and formulation of the FDTD method. The diﬀerential equations that describe Ampere’s and Faraday’s laws are approximated using two-point centered diﬀerences. And
absorbing boundary conditions are imposed. The source conﬁgurations used in the
Forward Model were also presented. The next chapter presents how the Forward
Model is used to solve the inverse problem. The Forward Model is validated in Chapter 4.

20

Chapter 3
Inverse Problem Solution Using
Time Reversal Techniques
3.1

Introduction to Inverse Problems in NDE/NDT

Defect locations or proﬁles in NDE/NDT, requires the solution of inverse problems
using information in the measured signal, that is provided by experimental or model
based data [24]. Inverse problems are mathematically ill posed because the uniqueness
and stability of the solution can not be guaranteed [25]. There are diﬀerent proposed
techniques to solve the inverse problem. Some of the techniques used to solve inverse
problems are calibration techniques, direct, and indirect approaches [24].

3.1.1

Calibration Techniques

Calibration techniques involve the creation of calibration curves that are used to
obtain defect information. Features in experimental data are acquired for diﬀerent
geometrical parameters of known defects. The geometric parameters include defect
characteristics such as width, depth, length, angle of inclination to the surface, and
the distance of the defect to the surface. In order to obtain the calibration curves
21

the defect characteristics are changed one at changes in signal features vs. defect
characteristic being changed are plotted. This method can only be used if the defects
shapes are known beforehand. This means that if the defect shapes are diﬀerent from
the ones used to create the calibration curves the method is not useful to solve the
inverse problem [26].

3.1.2

Direct Methods

Electromagnetic NDE problems assuming linear and isotropic conditions can be modeled by equation (3.1):
(∇ + k 2 )h(⃗ ) = f (⃗ )
x
x

(3.1)

where f (⃗ ) is the source in terms of current densities or charges, and h represents the
x
electric or magnetic ﬁeld that is used to ﬁnd the defects in the material. Equation (3.1)
represents the wave equation when the wavenumber is k 2 > 0. Assuming linearity,
equation (3.1) can be expressed in integral form by equation (3.2)
∫
h(⃗ ) =
x

G(⃗ , ⃗ ′ )f (⃗ ′ )dx,
x x
x

(3.2)

v
where G(⃗ , ⃗ ′ ) is the Green’s function. A discrete-space version of equation (3.2) is
x x
given by:
{h} = {G} {f } .

(3.3)

In direct methods, for solving the inverse problem, the matrix presented in equation
(3.3) is solved for f , where f represents the source or defects. The disadvantage of
this method is that produces an unreliable solution due to ill conditioning of matrix
G [26]. A small error or change in G causes a large error or change in f due to the
high condition number of matrix G. The direct approach makes the solution to the
inverse problem for defect detection unreliable.
22

3.1.3

Indirect Inversion

There are two types of indirect methods used to solve the inverse problem:
phenomenological/model-based and algorithm/signal classiﬁcation based.
Phenomenological/model based approaches are based on the underlying physics of the
NDE method. A forward model simulating the physics is used in an iterative processes
where the forward problem is solved in each iteration. The process starts assuming
an initial defect proﬁle and a signal, corresponding to this defect characteristic, is
generated using a forward model. After this signal is generated using the model, it
is compared to an experimental signal. If the error between this two signals does
not meet an established threshold, the process is repeated assuming a diﬀerent defect
proﬁle, which is adjusted using the error information [24]. This process is illustrated in
ﬁgure 3.1. It is important to choose the appropriate forward model in order to obtain
reliable results. A major drawback of this approach is that solving a numerical model
in each iteration makes it computationally demanding and the method can take very
long times.
In the non-phenomenological approach, the inverse problem is formulated as a
defect classiﬁcation problem. The measured signal is classiﬁed onto a set of diﬀerent
known types of sources or defect signals. Algorithmic/signal classiﬁcation-based approach require a training database consisting of a collection of defect signals from a
variety of defects. This data bank is used to train the algorithm to detect and identify
diﬀerent classes of defects in materials using features in the experimental data [26].
The method presented in this work, to solve the inverse problem, is model-based.
Model-based methods depend on the interaction of the ﬁeld, which is illuminating
the sample, with the diﬀerent ﬂaws in the material. Model-based methods that are
used in an iterative manner are time consuming and computationally expensive. The
advantage of the model-based method, presented in this work, to solve the inverse
23

Figure 3.1: Model based iterative method to solve the inverse problem.
problem is that is not an iterative method.

3.2

Previous Research: Time Reversal Technique
used in Ultrasound and Microwaves

Extensive work has been done in time reversal, speciﬁcally in the ultrasound ﬁeld.
Basic principles and the validation of the technique with experimental results are
presented in [27, 17]. The signals created after time reversal are focused signals that
have characteristics of the defect and the medium in which the wave propagates.
Time reversal, using ultrasound, is used to ﬁnd defects in materials because it is able
to reduce speckle noise due to the fact that the time reversed signal that comes from a
defect has speciﬁc properties of the defect [28]. After the signals are processed, using
normalized correlation coeﬃcient and deviation histogram techniques, the defect can
be detected and distinguished from the speckle noise zone.
A variety of materials have been inspected using time reversal techniques. An
important application of this technique is inspection of titanium. Titanium is a noisy
medium, therefore strong ultrasonic speckle noise is induced by the polycrystalline
24

microstructure. It also contains hard-α defects. These defects are hard to detect
because of the low reﬂectivity caused by acoustic impedance mismatch, furthermore
these defects have irregular and unknowns shapes. Time reversal techniques overcome
these diﬃculties because it is a process that is matched to the defect shape and to
the propagation medium. It also compensates for distortions due to the propagation
through inhomogeneous media.
Chakroun et al. [29] experiments were conducted with titanium using iterative
time reversal which showed that defect and speckle noise signals had diﬀerent characteristics. They developed a new noise reduction technique using deviation histogram
which allows distinguishing between noise and defect signal. In [30], time reversal
technique is optimized to avoid false identiﬁcation of defects, to reduce noise amplitude in the signal, and it is used to detect defects in titanium billets. In order
to optimize the technique two new refocusing processes were developed, iterative
matched ﬁlter and iterative coherent summation. The matched ﬁlter technique allows the detection of defects of the size 0.6mm at a 140mm depth in a titanium billet
of 250mm.
Defects in duralumin alloy have also been detected using time reversal techniques.
In [31] a complex geometry of this material with a 2 mm ﬂat bottom hole defect was
studied. The defect in the material was easily detected using time reversal techniques.
Study of time reversal of ultrasound ﬁelds, has demonstrated the capability of the
technique for detecting small defects in complex geometries and for distinguishing
between defects and speckle noise.
Time reversal is also used to detect defects in thin plates using lamb waves. Lamb
waves are very useful to inspect plates because of their guided characteristic. Their
disadvantage is that they are very dispersive and this characteristic limits their use
for non destructive evaluation testing. Time reversal allows the use of lamb waves
for defect detection because it compensates for the disadvantage of lamb waves. The
25

eﬀect of time reversal in the detection of ﬂaws using lamb waves is validated with
experimental data in [18].
In [32, 33] elastic waves are propagated and they interact with a circular cylinder
tube. The ﬁelds are recorded, time reversed, and propagated in a FDTD model in
order to ﬁnd defects in this circular cylinder tube. This work presents the simulation
results with experimental data. The results show that the signals that are propagated
in the FDTD model focus on the location where the actual circular cylinder tube had
the defect. FDTD and time reversal had also been used in microwave breast cancer
detection. In [19] a feasibility study is performed using FDTD as the model used
to detect breast cancer. The paper presents simulation results that show the time
reversed signals focusing on a tumor inside a breast model.

3.3

Previous Research: Time Reversal Used to
Solve the Inverse Problem

Time reversal has been used to solve the inverse problem. The work presented in [34]
proposes a time domain topological gradient as an inverse method for ultrasound.
The topological gradient is a function of the forward and adjoint problems. The
adjoint problem mathematical solution is based on the physical time reversal operation. Applying time reversal techniques to the solve the adjoint problem prevent the
method to fall into a local minimum, which will eventually lead to a non-physical
solution. This is due to refocusing properties that times reversal provides.
In [35] the information of the scattered ﬁeld was used to characterize a spatially
extended scatterer. In the presence of a spatially extended scatterer there are two
types of waves that are generated after the waves interact with the defect. The ﬁrst
reﬂected wave corresponds to the target geometry and the rest of the reﬂected waves
correspond to circumferential and volume waves that propagate around or inside
26

the scatterer. The inverse problem for this case, consists in the characterization of
circumferential waves. Circumferential waves are related to the normal modes of
vibration of the cylinder and these waves also represent its acoustical mark.

3.4

Proposed Approach to Solve the Inverse Problem in NDE/NDT: Time Reversal Technique
using an FDTD Model

The propagation of electromagnetic waves through a material or free space is given
⃗
by equation (3.4), a second order partial derivative equation, where E is the electric
ﬁeld and c is the speed of light. Because of invariance under Time Reversal, both
E(r, t) and E(r, −t) are solutions to the wave equation presented in 3.4. Invariance is
a characteristic of the wave equation that allows the system to be unchanged under
certain transformations, in this case a time transformation. Time reversal focusing is
based on the fact that when a wave solution is reversed in time and back propagated
it comes to focus at the source. Time reversal principle is only valid for a lossless
propagation medium. When the propagation medium contains frequency-dependant
attenuation, invariance and time reversal properties are invalid due to the fact that
the wave equation contains an odd order time derivative operator [27].
(

2
2− 1 ∂
∇
c2 ∂t2

3.4.1

)

→
−
E =0

(3.4)

Mathematical Background

In time reversal acoustic experiments the receivers, which are transducers, are known
as time reversal mirrors (TRMs). In microwave experiments, the TRM is a single or
an array of antennas. The antenna or the array of antennas is placed in the far ﬁeld
27

of the source [36]. The ﬁrst step in a time reversal experiment is to illuminate the
sample using a time domain pulse emitted from the transmitter antenna at position
⃗s . The ﬁelds incident on the target, which is located at position ⃗, are [37, 38]:
r
r

E inc (⃗, t) = p(t) ∗ G(⃗, ⃗s , t) =
r
r r

∫
p(τ )G(⃗, ⃗s , t − τ )dτ
r r

(3.5)

where p(t) is the time domain pulse and G(⃗, ⃗s , t) represents the Green’s function of
r r
the background medium, which is the solution to equation (3.6).

⃗
⃗ ⃗
∇ × ∇ × G − k 2 G = Iδ

(3.6)

The incident ﬁeld in the frequency domain is given by equation (3.7),
E inc (⃗, ω) = P (ω)G(⃗, ⃗s , ω)
r
r r

(3.7)

where P (ω) is the spectrum of the pulse.
The second step consists on recording the scattered ﬁeld on the receiving antenna
or array. The scatter ﬁeld measured at receiver i is given by equation (3.8),

E scatter (⃗i , ω) = kb 2 (ω)
r

∫
E(⃗, ω)G(⃗i , ⃗, ω)O(⃗, ω)d⃗
r
r r
r
r

(3.8)

where ⃗i is the location of the ith receiver, E(⃗, ω) is the total ﬁeld resulting from
r
r
the incident ﬁeld. The total ﬁeld, inside the scatterer, is a function of O(⃗, ω), which
r
makes equation (3.8) non-linear [39]. The scattered ﬁeld can also be represented using
the Born approximation in order to linearize equation (3.8). The Born approximation
states that the eﬀect of the incident ﬁeld on the targets is small and a ﬁrst approximation of the total ﬁeld, E(⃗, ω), in equation (3.8) can be set equal to the incident
r
ﬁeld [28, 40]. The resulting equation is given by:
28

E scatter (⃗i , ω) = kb 2 (ω)
r

∫

E inc (⃗, ω)G(⃗i , ⃗, ω)O(⃗, ω)d⃗
r
r r
r
r

(3.9)

O(⃗, ω) is the object function to be imaged and is given by equation (3.10),
r

O(⃗, ω) =
r

k 2 (⃗, ω)
r
−1
kb 2 (⃗, ω)
r

(3.10)

where the wavenumbers of the total medium, including background and targets, and
background medium alone are k 2 ,k 2 b , respectively.
The third step in the time reversal process is to time reverse the scattered ﬁelds
that were recorded by the receiver antennas. In order to achieve time reversal the
medium has to be reciprocal and non-absorbing. In the frequency domain time reversal is performed by phase conjugating the received ﬁeld at each receiver position.
After the ﬁelds are time reversed, they are propagated from the receiver location.

3.4.2

Time Reversal Focusing

Time reversal focusing is similar to matched-ﬁlter techniques used in signal processing.
A matched ﬁlter is an ideal linear ﬁlter which maximizes the signal to noise ration
(SNR) of the ﬁltered signal. For example, in an acoustic experiment, a pulse δ(t),
at position ⃗s , is propagated. The TRM, in this case a transducer, that is located
r
at ⃗j , will record an impulse response h(t). Due to reciprocity, this means that
r
G(⃗r , ⃗s , t) = G(⃗s , ⃗r , t), h(t) is also the impulse response that will be recorded at
r r
r r
the source location if the pulse δ(t) was sent from the transducer location.
The time reversed signal at ⃗s is given by equation (3.11), and it is a common
r
equation use to describe matched ﬁlter [41].

h(t) ∗ h(−t)
29

(3.11)

The result of this convolution will reach its maximum at time t = 0 [27]. The
maximum is equal to the energy of the signal recorded by the transducer and is given
by equation (3.12).

∫

h2 (t)dt

(3.12)

There are two qualities of the time reversal process that can be described using
match ﬁlter theory. The ﬁrst, is the coherence interference that the waves present
when more than one transducer is used. Equation (3.13) represents the time reversed
signal at the source when an array is used,

ϕ(⃗s , t) =
r

r=N
∑

hr (t) ∗ hr (−t).

(3.13)

r=1
Each signal in the sum add constructively and reach a maximum at t = 0. Uncorrelated contributions that occur at a diﬀerent time will interfere destructively.
The second, is that time reversal techniques are not susceptible to noise. If the
signal that is transmitted from the receiver location has noise in it hr (−t) + nr (t)
the time reversed signal is then given by equation (3.14).
r=N
∑

hr (t) ∗ hr (−t) +

r=1

r=N
∑

hr (−t) ∗ nr (t)

(3.14)

r=1

A sharp peak occurs when the time reversed signal hr (−t) matches the medium
impulse response. Noise, which is not match to the medium, produces low-level
background noise that is diﬀerent from the sharp peak that is produced by the time
reversed signal [41].

30

Chapter 4
Results

4.1

Validation of Time Reversal Analysis

In order to validate the code and the implementation of a time reversal process, a
time reversal cavity was analyzed. The progress of the electric ﬁeld as a function of
time was considered. The excitation source is a 2 GHz center frequency Gaussian
modulated pulse. The parameters used in the simulations are listed in table 4.1. The
Gaussian pulse is propagated in free space and the electric ﬁeld was recorded on a
linear array of receivers. Figure 4.1 shows the simulation geometry. The electric ﬁeld
that was recorded was time reversed and propagated backwards from the receivers’
location using the same free space model.
Figure 4.2 shows the progression of the ﬁeld when is propagated backwards in
the model. The back propagated wave starts at time n=1 (20ps) (Fig. 4.2a) and
converges at the source at n=108 (2.16 ns) (Fig. 4.2 c). Note that the forward
propagating wave at n=75 (1.5 ns)(Fig. 4.2 b), which will focus at the source 66ns
later, is a time reversed replica of the wave at n=141 (2.82 ns) (Fig. 4.2 d), which
occurs 66ns after focusing at the source.
31

Table 4.1: Parameters used in the simulations.
Time step Cell Size
(∆t)
(∆)
20ps
1.2cm

Pulse Width
of Source
16 ∆t

Grid Size
249∆ X 249∆

Length
of array
50∆

Figure 4.1: Time reversal cavity that shows the receiver array location and the propagation of the Gaussian pulse. ”For interpretation of the references to color in this and
all other ﬁgures, the reader is referred to the electronic version of this dissertation.”

4.2

Application of Time Reversal for Defect Imaging

The principles of Time Reversal can be applied to experimentally measured data
for defect imaging. A schematic diagram that describes the procedure for imaging is
presented in Figure 4.3. Several cases of sample and defect materials were considered.
The simulation geometry considered was a dielectric slab with a relative permittivity (εr ) of 4 and the excitation source is a modulated Gaussian pulse of 2 GHz center
32

(a)

(b)

(c)

(d)

Figure 4.2: Progress of electric ﬁeld as a function of time when the electric ﬁeld
is propagated backwards in the model. (a) Backpropragated Fields at time 1. (b)
Backpropagated Fields at time 75. (c) Backpropagated Fields at time 108. (d)
Backpropagated Fields at time 141.

frequency. The parameters used in the simulations are listed in table 4.2. Using a
single excitation source the scattered electric ﬁeld was recorded on a linear array of
receivers; time reversed and applied to the simulation model. The model propagates
the time reversed signals backwards to highlight the defect. Energy of the propagated
ﬁeld was calculated by integrating the square of the magnitude of the ﬁeld and the
image of the ﬁeld was displayed. The maxima of the energy image was calculated
and its coordinates were compared to the coordinates of the defect.
33

Table 4.2: Parameters used in the simulations.
Time step Cell Size
(∆t)
(∆)
0.01ns
0.006m

Pulse Width
of Source
16 ∆t

Grid Size
673∆ X 673∆

Length
of array
500∆

Figure 4.3: A block diagram indicating the procedure that was followed.

4.3

Test 1: Two scatterers in Dielectric Medium

Two air voids in the dielectric slab were simulated. The scatterers had a radius of
0.05m and were placed 6.28λ apart. Figure 4.4a shows the forward propagating wave
through the dielectric. Figure 4.4b shows how the time reversed waves focus back
on the defects. The energy image in Figure 4.4c shows that the maximum energy
for defect 1 occurs at the point (-0.6720m, -0.9300m) while the defect location is at
34

(-0.7020m,-0.9780m). The maximum energy for defect 2 occurs at the point (1.062m,0.702m) while the defect location is at (1.0920m,-0.7320m). For defect 1, the distance
between the defect location and the energy maximum was 0.7557λ. The distance
between defect 2 location and the maximum energy is 0.5664λ. It can be observed
through a comparison between the coordinates of the maximum of the energy and
the true location of defects that the microwave time reversal technique accurately
identiﬁes the location of both defects.

(a)

(b)

(c)
Figure 4.4: Simulations results for two scatterers in a dielectric medium. (a) Forward
propagating wave from source. (b) Focusing of time reversed wave on scatterers. (c)
Energy images indicating defects location.

35

4.4

Test 2: Single perfect electric conductor (PEC)
Scatterer in Dielectric Medium

A single PEC scatterer in a dielectric slab was simulated. The scatterer had a radius
of 0.05m. Figure 4.5a shows how the forward propagating ﬁeld through the dielectric.
Figure 4.5b shows the time reversed wave focusing on the defect. The energy image in
ﬁgure 4.5c shows that the maximum energy for the defect occurs at the points (1.05m,0.6900m) while the defect location is at (1.0920m,-0.7320m). The error between the
defect location and the maximum energy location was 0.7930λ. The result from
the maximum energy coordinate leads us to the conclusion that the microwave time
reversal technique accurately identiﬁes the defect in the dielectric material.

4.5

Test 3: Two Scatterers in Dielectric MediumResolution of the Model

In order to study the resolution of time reversal imaging, two scatterers 2.27λ apart
were considered as shown in ﬁgure 4.6. Figure 4.6a shows how the time reversed wave
focus on the two defects. Figure 4.6b shows that the maximum energy coordinate and
the true location of defect coordinate are very similar. The true location of defect 1
was (-0.8640m,-0.972m) and the maximum energy coordinate was(-0.8460m,-0.930m).
Defect 2 was located at (-0.6960m,-0.9720m) while the energy coordinate for defect
2 was (-0.6720m,-0.9240m). For defect 1 the distance between the defect location
and the maximum energy location was found to be 0.6λ. For defect 2, the distance
between the defect location and maximum energy location was 0.7λ. The technique
is able to identify defects that are close together, which indicates the good resolution
of the model.
36

(a)

(b)

(c)
Figure 4.5: Simulations results for single PEC scatterer in a dielectric medium. (a)
Forward propagating wave from source. (b) Focusing of time reversed wave on scatterer. (c) Energy image indicating defect location.

4.6

Test 4: Two Scatterers in Dielectric Medium
- Reconstruction with noise

In order to evaluate the robustness of the method, the performance was evaluated
with noisy ”measurements”. The simulation geometry consists of two air voids in the
dielectric slab placed 6.28λ apart. Random noise with signal to noise ratio (SNR) of
10 was added to the signal measured by the linear array. Fig.4.7(a) shows the time
reversed wave focusing back on the defects. The energy image in Fig.4.7(b) shows
that the maximum energy for defect 1 occurs at the point (-0.6720m,-0.9300m) while
37

(a)

(b)
Figure 4.6: Simulations results for 2 scatterers in a dielectric medium 2.27λ apart. (a)
Focusing of time reversed wave on scatterers. (b) Energy image indicating location
of defects.

the defect location is at (-0.7020m, -0.9780m). The error between the location of the
defect and the maximum energy location was calculated to be 0.75λ. The maximum
energy for defect 2 occurs at the point (1.1062m,-0.6720m) while the defect location
is at (1.0980m, -0.7320m). For the second defect the error between the location
of the defect and the maximum energy location was 0.807λ. Figures 4.8 (a) and
(b) show the corresponding results when the SNR =2. Defect 1 was located at (0.7020m, -0.9780m) while the maximum energy occurs at (-0.6720m,-0.9300m). The
error between these locations was found to be 0.7547λ. Defect 2 was located at
(1.0920m, -0.7320m) while the maximum energy occurred at (1.1062m,-0.6720m).
The error between the location of this defect and the location where the maximum
38

energy occurs was calculated to be 0.82λ. These results clearly demonstrate the
robustness of the proposed method.

(a)

(b)
Figure 4.7: Simulations results for 2 scatterers in a dielectric medium 6.28λ apart with
noise. (a) Focusing of time reversed wave on scatterers. (b) Energy image indicating
location of defects.

4.7

Test 5: Two Scatterers in Dielectric MediumTwo Sources with Noise.

The purpose of this simulation was to improve the SNR of the image with additional
sources. It can be seen in Figure 4.9c that the SNR of the energy image is larger than
that of the image in Figure 4.7b. The maximum energy for defect 1 occurs at the
point (-0.6720m, -0.9420m) while the defect location is at (-0.7020m,-0.9780m). The
39

(a)

(b)
Figure 4.8: Simulations results for 2 scatterers in a dielectric medium 6.28λ apart with
noise. (a) Focusing of time reversed wave on scatterers. (b) Energy image indicating
location of defects.
maximum energy for defect 2 occurs at the point (1.08m,-0.666m) while the defect
location is at (1.0920m,-0.7320m).

4.8

Test 6: Minimum number of sensors.

The simulation geometry was a dielectric slab with a relative permittivity (εr ) of
4. The dielectric slab has one air void with a 0.05 m radius. In this simulation the
eﬀect of number of receivers on the time reversal process was studied to determine
the minimum number of receivers required in an experimental set up to ensure source
focusing. The number of receivers was reduced from 500 to 50. The signals from
40

(a)

(b)

(c)
Figure 4.9: Simulations results for two scatterers in a dielectric medium. (a) Forward propagating wave from the two sources. (b) Focusing of time reversed wave on
scatterers. (c) Energy images indicating defects location.

forward propagation are recorded, time reversed and back propagated. The wave
focuses on the defect as shown in ﬁgure 4.10a. The energy image in ﬁgure 4.10b
shows that the coordinates of the energy maximum and true coordinates of the defect
are similar. The defect coordinates are (-.7020,-.9780) and the maximum energy
coordinates are (-.6720,-9240). The error between the two locations is 0.8247λ. The
error between the true and estimated locations of the defect when 500 receivers were
used, is 0.7557λ. The error is higher when less sensors are used but the defect can
41

still be detected when 50 sensors are used. The capability of the technique with 50

(a)

(b)
Figure 4.10: Simulations results for 1 scatterer in a dielectric medium with permittivity of 4. The number of sensors used is 50. (a) Time reversed waves focus on the
scatterer. (b) Energy image indicating maximum energy at defect location.

receiving sensors was also tested for the test sample with two scatterers 2.27 apart
as shown in ﬁgure 4.11. Figure 4.11a shows the time reversed wave focusing on the
two defects. Figure 4.12a shows the energy image with the coordinates of the energy
maximum. The true coordinate of defect 1 is (-.7020,-.9780) and the coordinate of
defect 2 is (-.8700,-.9780). The error between true coordinate and energy coordinate
of defect 1 is 2.027λ and for defect 2, the error is 2.76λ. The error between the true
and estimated locations of defect 1 when 500 receivers were used, is 0.6λ and for
defect 2 is 0.7λ. The error is higher when less sensors are used but the resolution
42

capability of the model was not aﬀected when 50 sensors are used.
When a random noise of SNR=10 is added to the signal, the results obtained with
50 sensors were able to detect and locate the defects as shown in ﬁgure 4.12. Figure
4.12a shows the time reversed waves focusing on the two defects and Figure 4.12b
shows the energy image. The true coordinate of defect 1 is (-0.7020,-0.9780) and the
energy maximum coordinate is (-0.6720,-0.924). For defect 2, the true coordinate
is (1.0920,-0.7320) and the maximum energy coordinate is (1.062,-0.678). The error
between the true defect location and maximum energy coordinate for both defects,
1 and 2, is 0.82λ. Figure 4.13a and 4.13b show the corresponding results when SNR
= 2. The maximum energy coordinate for defect 1 is (-0.6720,-0.924) and the true
coordinate of the defect is (-0.7020,-0.9780). The error between these two coordinates
is 0.82λ. The maximum energy coordinate of defect 2 is (1.062,-0.678). The error
between these coordinates and the location of defect 2 is 0.82λ. The results show that
the model is robust to noise even when fewer sensors are used to record the signals.
The simulations performed with only 50 sensors produced error values higher than
those produced when 500 sensors were used. This increase in error values does not
aﬀect the capability of the method to detect the defects in the sample.

4.9

Test 7: Time Reversal Applied to Lossy Dielectric Medium.

4.9.1

Two well separated defects.

In this test a sample with ﬁnite conductivity and two scatterers was simulated. The
scatterers in the simulation have a radius of 0.05m. The simulations were performed
for diﬀerent conductivity values (σ). After the waves were time reversed they were
propagated, using the forward model, through the medium with the same ε but
43

(a)

(b)
Figure 4.11: Simulations results for 2 scatterer in a dielectric medium 2.27λ apart.
The number of sensors used is 50. (a) Time reversed waves focus on the scatterers.
(b) Energy image indicating maximum energy at defects location.

the σ=0. Figure 4.14 shows the simulation image for a lossy dielectric slab with
conductivity σ = 0.05 and relative permittivity ε = 4. This image shows how the time
reversed wave focuses on the defects. Figure 4.14a shows the energy image obtained
from the simulation. The true coordinate of defect 1 is (-0.7020,-0.9780) and for
defect 2 is (1.0920,-0.7320). The error between true coordinate and maximum energy
coordinate of defect 1 is 0.65λ and between true coordinate and maximum energy
coordinate of defect 2 is 0.716λ.
The conductivity values were increased to determine the maximum conductivity
value of the material under which the time reversal process can still yield the location
of the defect in a lossy test sample. When the conductivity value is chosen as σ = 0.1
44

(a)

(b)
Figure 4.12: Simulations results for 2 scatterers in a dielectric medium when noise
(SNR=10) is added to the recorded signals. The number of sensors used is 50. (a)
Time reversed waves focuses on the 2 scatterers. (b) Energy image indicating maximum energy at defect location.

the attenuation of the wave decreases the intensity values of the wave as shown
in ﬁgure 4.15a. The intensity of the wave is low and the time reversed wave does
not focuses in the defects. Figure 4.15b shows the energy image obtained from the
simulation. The true coordinate of defect 1 is (-0.7020,-0.9780) and for defect 2 is
(1.0920,-1.20). The error between true coordinate and the coordinate of maximum
energy of defect 1 is 0.4 λ. The error between true coordinate and maximum energy
coordinate of defect 2 is 0.24λ.
The simulation was repeated with sample conductivity σ = 0.2. The waveform
45

(a)

(b)
Figure 4.13: Simulations results for 2 scatterers in a dielectric medium when noise
(SNR=2) is added to the recorded signals. The number of sensors used is 50. (a) Time
reversed waves focuses on the 2 scatterers. (b) Energy image indicating maximum
energy at defect location.

does not focuses on the defects in ﬁgure 4.16a. Figure 4.16b shows the energy image
obtained from the simulation. The energy image shows that there is only one maximum, which occurs at (1.098,-1.224). The true defect coordinate is (1.0920,-1.2020)
and the error between the true coordinate and the coordinate of maximum energy is
0.3040λ. When this defect is moved to position (1.0920,-0.7320) neither of the defects
can be detected as shown in ﬁgure 4.16c. It can be concluded that at a conductivity
σ = 0.2 the attenuation is too severe for the technique to detect the defects.
46

(a)

(b)
Figure 4.14: Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.05. (a) Time reversed waves focuses on the 2 scatterers. (b) Energy image
indicating maximum energy at defect location.

4.9.2

Closely placed defects.

Two scatterers 2.27λ apart were considered to further test the accuracy of time reversal imaging, when lossy dielectrics were used as samples. A conductivity σ=0.05
and permittivity ε=4 were used in the simulation. The time reversed waves focus
on the defects as shown in ﬁgure 4.17a. Figure 4.17b shows the energy image and
it is seen that the maximum energy coordinates and the true coordinates of defects
are very similar. Defect 1 is located at (-0.7020m,-0.9780m) while the coordinate of
maximum energy is located at (-0.8520m,-0.9360m). The error between these two
coordinates is 2.08λ. Defect 2 is located at (-0.8700m,-0.9780m) while the maximum
47

(a)

(b)
Figure 4.15: Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.1. (a) Time reversed waves focuses on the 2 scatterers. (b) Energy image
indicating maximum energy at defect location.
energy coordinate is located at (-0.6780m,-0.9540m). The error between these two
coordinates is 2.68λ.
The simulation was performed next with a higher conductivity σ=0.08. In lossy
dielectrics, the waves are attenuated as ﬁgure 4.18a shows. Figures 4.18b and 4.18c
show that the wave is aﬀected by attenuation and not by dispersion. Due to attenuation the wave does not focus on the defect location, as shown in ﬁgure 4.19a. The
true coordinate of defect 1 and 2 are (-0.7020m,-0.9780m) and (-0.8700m,-0.9780m),
respectively. The error between true and estimated locations of defect 1 is 2.08λ.
The error between true and estimated locations of defect 2 is 2.68λ. The simulation
48

(a)

(b)

(c)
Figure 4.16: Simulations results for 2 scatterers in a lossy dielectric medium when
σ=0.2. (a) Attenuated time reversed waves. (b) Energy image indicating maximum
energy at defect location. (c) Energy image indicating maximum energy location.

49

(a)

(b)
Figure 4.17: Simulations results for 2 scatterers 2.27 λ apart in a lossy dielectric
medium when σ=0.05. (a) Attenuated time reversed waves. (b) Energy image indicating maximum energy at defect locations.

was then performed with a conductivity σ=0.2 and permittivity ε=2.9. Figure 4.20
shows that the energy maximum does not occur at the true defect location. When
the material has a conductivity σ=0.2 and a permittivity ε=2.9 the defects can not
be detected.
50

(a)

(b)

(c)

Figure 4.18: Attenuation eﬀects from the conductivity (σ=0.08) of the material for
2 scatterers 2.27 λ apart. (a) Comparison between signal from sensor 337 for a lossy
dielectric and a dielectric. (b) Signals from sensor 337 for a lossy dielectric. (c)
Signals from sensor 337 for a pure dielectric.

51

(a)

(b)
Figure 4.19: Simulations results for 2 scatterers 2.27 λ apart in a lossy dielectric
medium when σ=0.08. (a) Attenuated time reversed waves. (b) Energy image indicating maximum energy at defect locations.

Figure 4.20: Energy image for 2 scatterers 2.27 λ apart in a lossy dielectric medium
when σ=0.2 and ε=2.9.

52

4.10

Test 8: Eﬀect of errors in receiver measurements.

4.10.1

Error in time measurements.

Two diﬀerent test were conducted to determine eﬀect of error in time measurements
of the receiver array signals. The errors were introduced in the form of time delays
in the received signals which were then time reversed and back propagated using the
FDTD model. The simulation was performed in free space and the source was located
at position (0,0). The ﬁrst test consisted of changing the time delay in randomly selected receiver signals using pseudorandom values obtained from the standard uniform
distribution. The maximum time delay introduced to the signals was 175 ∆t and the
minimum value was 90 ∆t. The source ﬁeld ﬁeld was recorded at 51 sensors positions,
was time reversed after time delays were selectively introduced in some signals.
Signals at all 51 sensors positions, placed along the x axis, were time delayed by
random quantities. The resulting A-scan signals of each receiver in the array, after
introducing random time delays are shown in ﬁgure 4.21. The energy image in ﬁgure
4.22 shows the position where the maximum of the back propagated energy occurs.
The coordinates where the maximum occurs are (0.0060,-0.0840). The error between
the coordinate of maximum energy and the true source coordinate is 0.56λ. This
error is higher than the error of 0.08λ without any error in time delay.
The simulation was repeated but only 46 signals were modiﬁed. The 5 signals,
that were not modiﬁed, were at positions -0.6, -0.54, -0.48, -0.42, -0.36 m. The A-scan
signals of each receiver in the array, after introducing random time delays are shown
in ﬁgure 4.23. The energy image in ﬁgure 4.24 shows that the maximum energy
position occurs at (0.0060,0.0720). The error between the coordinates of maximum
energy and the true coordinates is 0.62λ.
53

Figure 4.21: Individual A-scan signals at the receiver when the time delay is modiﬁed.

Figure 4.22: Energy image when signals at all positions were delayed by random
quantities.

The number of sensors, which signals were delayed, in the next test was kept at
46. The same 46 signals used in the last test were used for this test. In this test
the amplitude of the 46 signals was modiﬁed by multiplying the signals by a factor
based on pseudorandom values obtained from the standard uniform distribution. The
factors varied from 0.90 to 0.99. The A-scan signals, after the time delay and amplitude were modiﬁed, are shown in ﬁgure 4.25. The coordinates where the energy
maximum occurs is (0.0060,0.0780), as shown in ﬁgure 4.26. The error increased to
54

Figure 4.23: Individual A-scan signals at the receiver when the time delay is modiﬁed.

Figure 4.24: Energy image when 46 signals were modiﬁed.

0.68λ when the amplitude of the signals was modiﬁed. The energy images presented
in this section show that changes in time delay in the signals greatly aﬀect focusing
of the back propagating wave and estimation of source location. In all the cases the
waves focus near the source location but not on the source location.
55

Figure 4.25: Individual A-scan signals at the receiver when the time delay and amplitude are modiﬁed.

Figure 4.26: Energy image when 46 signals were modiﬁed.

4.10.2

Error is Transducer Position.

The second type of test consisted in simulating an error in transducer position. When
the sensors in the x coordinate are not ideally placed at locations speciﬁed in the
simulation model, the use of the model in back propagation can result in errors. This
study was conducted with 40 sensors as is the case with the experiment performed.
The simulation was performed in free space. The sensors were located 2 m away
from the source position. The true coordinates of the source are (0.002,-0.022). The
56

estimated source position can be identiﬁed from the energy image, as shown in ﬁgure
4.27.

Figure 4.27: Energy image when time delay is simulated using sensor position error.
In conclusion, the results presented in these sections show that:
• Changes in amplitude and time delay increased detection error.
• Time delay diﬀerences created by position errors in the x coordinate did not
aﬀect the focusing capability of the model.
• The time delay due to position errors caused small changes in time delay.

4.11

Test 9: Results with diﬀerent sensor locations

Simulations, using diﬀerent number of sensors and location in space, were performed
to document the performance under diﬀerent spatial distributions of sensors. The
ﬁrst test was performed using 11 sensors, equally spaced, in the x coordinate from a
distance of -1.5 to 1.5 m. The energy image and a snapshot of the movie, presented
in ﬁgures 4.28, 4.29 respectively, show how the wave focuses on the source position.
The individual A-scan signals acquired in the sensor positions are shown in ﬁgure
4.30.
57

Figure 4.28: Energy image when 11 sensors from a distance of -1.5 to 1.5 m were used
in the simulation.

Figure 4.29: Snapshot from movie showing the place where the waves focus.
The number of sensors were decreased from 11 to 6. The sensors were placed, in
the x position, from -1.5 to 1.5 m. Reducing the number of sensors did not aﬀect the
focusing capabilities of the model, as shown in ﬁgures 4.31, 4.32. The A-scan signals
at the 6 sensors are shown in ﬁgure 4.33. The spatial extent of the receiver array was
reduced to determine if this change aﬀects the focusing capabilities of the model.
Next, the spatial extent of the receiver array was reduced to the area between -0.6
to 0.6 m. The number of sensors used was reduced to 5. Individual A-scans signals
at the receivers are shown in ﬁgure 4.34. The snapshot taken from the movie ﬁle, in
ﬁgure 4.35, shows the focusing point is wider than the one in ﬁgure 4.32. The energy
image in ﬁgure 4.36 shows the location of the defect but the energy is spread over the
58

Figure 4.30: Individual A-scan signals at the receivers.

Figure 4.31: Energy image when 6 sensors from a distance of -1.5 to 1.5 m were used
in the simulation.

location of the source.
The A-scan signals of the receiver signals when the spatial extent is reduced to
the area from -0.5 to 0.5 m, are shown in ﬁgure 4.37. The waves do not focus in the
source position as shown in ﬁgure 4.38. As a result of this, the energy is spread in
the depth direction although the location in x-direction is fairly accurate as seen in
ﬁgure 4.39. The same observations are made when the spatial distance of the receiver
array is reduced to the area from -0.45 to 0.45 m. The number of sensors used for
59

Figure 4.32: Snapshot from movie showing the place where the waves focus.

Figure 4.33: Individual A-scans signals at the receiver.

this test were 7. The A-scan are shown in ﬁgure 4.40. The waves do not focus at the
source location, as seen in ﬁgure 4.41 and in the energy image in ﬁgure 4.42.
The number of sensors used in the simulations did not aﬀect the focusing capability of the model. However, the distance covered by the sensors aﬀected the focusing
capabilities of the model. The source position can not be detected when the receiver
array covers a small area on the receiver plane that does not capture the wave characteristics of the data as seen when the distance covered by the sensors is reduced to
-0.6 to 0.6 m, to -0.5 to 0.5 m, and to -0.45 to 0.45 m.
60

Figure 4.34: Individual A-scans signals at the receivers.

Figure 4.35: Snapshot from movie showing the place where the waves focus.

4.12

Test 10: Distances from the source to the
sensor array.

Simulations, using diﬀerent distances from transmitter position to receiver position,
were performed to understand if this distance aﬀected the focusing capabilities of the
model. The distance between the transmitter and the central receiver, used for the
ﬁrst simulation, was 1 m. The number of sensors used for this simulation were 6.
The distance covered by the sensors was from -1.5 to 1.5 m, in the x axis, for all the
simulations. The energy image, in ﬁgure 4.43, shows that the wave focuses on the
61

Figure 4.36: Energy image when 5 sensors from a distance of -0.6 to 0.6 m were used
in the simulation.

Figure 4.37: Individual A-scan signals at the receivers.
source original location.
The distance between transmitter and the central receiver was decreased to 0.7
m. The number of sensors, used in all test in this section, is 11. The energy image,
in ﬁgure 4.44, shows the wave focusing on the source position. The simulation is
repeated decreasing the distance between transmitter and receiver to 0.5 m and 0.2
m. The energy image, for the 0.5 m case, in ﬁgure 4.45, shows that the maximum of
the energy occurs at the source location. The same results, for the 0.2 m case, can
62

Figure 4.38: Snapshot from movie showing the place where the waves focus.

Figure 4.39: Energy image when 6 sensors from a distance of -0.5 to 0.5 m were used
in the simulation.

Figure 4.40: Individual A-scan signals at the receivers.

63

Figure 4.41: Snapshot from movie showing the place where the waves focus.

Figure 4.42: Energy image when 7 sensors from a distance of -0.45 to 0.45 m were
used in the simulation.
be observed in ﬁgure 4.46. The distance between the transmitter and receivers does
not aﬀect focusing of the wave in the source location.

64

Figure 4.43: Energy image when the sensors are 1 m away from the source location.

Figure 4.44: Energy image when the sensors are 0.7 m away from the source location.

Figure 4.45: Energy image when the sensors are 0.5 m away from the source location.
65

Figure 4.46: Energy image when the sensors are 0.2 m away from the source location.

66

Chapter 5
Experiment
This chapter describes the experimental validation results of time reversal techniques
applied to experimental microwave NDE data. First, an experiment was set up to validate the simulation results obtained using the FDTD model. Transmission measurements from a microwave source in air were performed in time domain to validate the
simulation model. Two TEM horn antennas, a pulse generator, and an oscilloscope
were used in the experiment set up. One of the antennas was used as a transmitter
and the receiving antenna was moved to emulate a linear array of receivers that is
used in the simulations. The measured data was recorded, time reversed, and input
into the FDTD model to validate it. An energy image with the experimental data
was generated and compared to the energy image produced using simulated data.

5.1

Experiment Set Up

The experiment set up used is shown in ﬁgure 5.1. The experiment was performed in
free space constituting a homogeneous medium in the solution domain. Two H-1498
TEM horn antennas, manufactured by American Electronic Laboratories were used.
Each antenna had a bandwidth of 2 to 18 GHz so that the average wavelength, λ=0.1.
67

The transmitter antenna was placed on the arch range and the receiver antenna was
placed in a moving cart shown in ﬁgures 5.2 and 5.3, respectively. Two diﬀerent
experiments were performed. In the ﬁrst experiment the antennas were placed 1.96
m or 19.6λ away as shown in ﬁgure 5.4. In the second experiment the antennas were
placed 2.90 m or 29λ away as shown in ﬁgure 5.5.

Figure 5.1: Experiment Set Up.
The time domain ﬁeld detected by the receiving antenna is measured using a
HP54750A Hewlett Packard digital sampling oscilloscope (DSO). The oscilloscope
had a HP54753A vertical plug in module with a time-domain-reﬂectometer/timedomain-transmission (TDR/TDT) step generator built into channel 3. Channel 3, in
the oscilloscope, has a 18 GHz bandwidth and channel 2 has a 20 GHz bandwidth.
The (TDR/TDT) step generator produces a 200 mV step with a system rise time of
45 ps. These parameters are also presented in table 5.1. The HP 54753 triggers a
Picosecond Pulse Labs (PSPL) pulse generator, model 4015 and the driver output is
connected to a pulse head, model PSPL 4015RPH, that produces a -9V, 15 ps falltime
68

Figure 5.2: Transmitter antenna placed on the arch range.
Table 5.1: Oscilloscope Parameters.
Channel 3 Bandwidth
18 GHz

Channel 4 Bandwidth Step Amplitude
20 GHz

200 mV

Rise Time
of Step Generator
45ps

step pulse. The pulse head is connected to an impulse forming network, model PSPL
5208 DC, which receives the step and generates an approximately Gaussian pulse.
This pulse is fed into the transmitting antenna. The voltage signal from the receiver
antenna is applied by channel 4 on the oscilloscope. The signal was 4096 points long
and the sampling time is 1.22 ps. The signal was averaged over 256 cycles to improve
the SNR and the scale, on the oscilloscope was set to 3mV/div. The oscilloscope was
turn on an hour before starting the experiment in order to avoid equipment thermal
drift.
The receiver antenna was scanned using 3 cm step size to measure the electric
ﬁeld at 41 diﬀerent positions over 120 cm distance, as shown in ﬁgures 5.6 and 5.7.
The pulse was sent through the transmitter antenna every time the receiver antenna
was moved to a diﬀerent position. This procedure was followed because there was not
an antenna array available. In order to maintain the receiver antenna from shifting
to the sides a plywood panel was used. Two grooves were made on a plywood panel
69

Figure 5.3: Receiver antenna on cart.
to guide the cart carrying the receiver antenna as shown in ﬁgure 5.8.

5.2

Experiment Calibration

A calibration procedure was followed to minimize the eﬀects of diﬀerent reﬂections
from the room on the measurements. A conducting plate was placed between the
transmitter antenna and the receiver antenna in order to measure the background
clutter, as shown in ﬁgure 5.9. The background clutter includes reﬂections from the
ﬂoor, ceiling, walls, and the metal arch range. Absorbers were placed on the ﬂoor,
back walls, and in front of the cart, which holds the receiver antenna, to reduce the
diﬀerent reﬂections. After the background signal is obtained it is saved in the memory
of the oscilloscope and subtracted from each signal measured along the linear scan
positions of the receiver antenna.
70

(a)

(b)
Figure 5.4: Experiment performed at a distance of 1.96 m between transmitter and
receiver antennas. (a) Picture of experiment when antennas are 1.96 m away. (b)
Drawing depicting the distance between the antennas.

5.3

Experiment Results

In the ﬁrst experiment the distance between the transmitter antenna and the receiver
antenna was 1.96 m. This distance was measured from the transmitter antenna to
the receiver antenna placed in the center position. The ﬁeld was measured in the x
axis from -60 cm to 60 cm, as shown in ﬁgure 5.10. The signals, shown in ﬁgure 5.11,
were decimated 4 times in order to input them into the FDTD model so that the
value of ∆t changed from 1.22 ps to 4.88 ps. The stability criteria was followed in
√
the simulations (c∆t ≤ ∆x ) and the ∆x was chosen to be 0.003 m. The decimated
2
voltage signals were time reversed and used as input to the FDTD model using the
experimental distance of 1.96 m from the source location.
The energy of the electric ﬁeld in the back propagated electric ﬁeld is calculated
using the FDTD model. The position where the energy maximum occurs is estimated.
The energy image in ﬁgure 5.12, shows the position where back propagated ﬁelds focus
71

(a)

(b)
Figure 5.5: Experiment performed at a distance of 2.90 m between transmitter and
receiver antennas. (a) Picture of experiment when antennas are 2.90 m away. (b)
Drawing depicting the distance between the antennas.

on the source. Figure 5.12 shows the error between true (0,0) and estimated source
location coordinates. The experiment was repeated at 2.90 m in order to reproduce
the same error at a diﬀerent distance and understand the source of the error. The
electric ﬁeld recorded for 41 diﬀerent receiver positions is shown in ﬁgure 5.13. The
energy image in ﬁgure 5.14, shows a similar error in the estimation of the source
locations.
In order to understand the error in source location the experimental and simulated
signals were analyzed. It was observed that the simulated signals at the ﬁrst and last
sensors positions were symmetric about the center position of receiver but this was
not true of experimental signals. Experimental signals from positions -60 cm and 60
72

Figure 5.6: Experiment set up when antennas are 1.96m away.

cm in the x axis, for example, were compared and they were not identical, as shown
in ﬁgure 5.15. It is observed in ﬁgure 5.15 that signal at position 60 cm has a shorter
time delay than the signal at -60 cm. Theoretically this means that the distance
between the receiver antenna and transmitter antenna at position 60 cm is shorter
than the distance between the transmitter antenna and receiver antenna at position
-60 cm. The diﬀerence in the values of the time of ﬂight of signals at 60 cm and
-60 cm were subtracted and the velocity of the wave in free space (3 × 108 m/s) was
used to ﬁnd that the diﬀerence between the two sensors positions was 0.04 m. The
distances between the transmitter and receiver antenna were measured for positions
60 cm and -60 cm in the experiment set up. The distance between transmitter and
receiver antenna at position 60 cm is 2.9464 m and at position -60 cm the distance
was measured as to be 2.9972 m, producing a diﬀerence of 0.0508 m which is close
to the estimated error in distance between transmitter and receiver. The source of
error is in estimation of source location is therefore attributed to the error in receiver
location between simulation and experiment.
73

Figure 5.7: Experiment set up when antennas are 2.90m away.
The true distance was used next in the simulation in order to replicate the error
shown in the experiments. A geometrical analysis was conducted to determine the
location of the source. The distances that were measured in the experimental set up
were used in this analysis. A triangulation method was used, as shown in ﬁgure 5.16.

The value of H was expected to represent the shortest distance between transmitter
and receiver and intersect cathetus C in the middle or at 0 m. In the experimental
set up, the distance between the transmitter and receiver was measured to be 2.9015
m. The diﬀerence in the measurements between catheti A and B caused the value of
H to be diﬀerent from the distance measured in the experiment set up. In addition,
cathetus H does not intercept cathetus C in the middle as shown in ﬁgure 5.16.
Triangle formed by catheti B, H, and X1 shown in ﬁgure 5.17 is further analyzed.
The Y cathetus is drawn from the source position to the point where catheti C from
the triangle in ﬁgure 5.16 is divided in half. Catheti Y is found using Pythagoras
theorem and it divides catheti X1 in X3=0.6 m and X4=0.099 m. The waves focused
74

Figure 5.8: Plywood panel used to move and avoid shift in the receiver antenna.
at position (0.0920, -0.3850) when the experimental data was applied as input into
the FDTD model. Distance X4=0.099 corresponds to the x coordinate where focusing
occurred when the experimental data was used as input in the FDTD model. The
diﬀerence in the distances of catheti A and B in ﬁgure 5.16 explains the error in the
x coordinate.
The triangle in ﬁgure 5.18 is used to ﬁnd the real position of the sensors in the
experimental set up and to calculate the ﬁelds at this positions in the simulation.
The triangle formed by catheti a1, b1, and d1 is a right triangle and trigonometric
theorems are used to ﬁnd b1. Coordinate b is found to be 2.929 by adding catheti b1
and Y. Cathetus d1 is found using trigonometric theorems and it also corresponds to
coordinate a. The distance between coordinates (0,0) and coordinate (-0.5996,2.929)
is found to be 2.99 m which is the same distance that was measured in the experimental set up. The same procedure was followed to ﬁnd coordinates (0.5996,2.888). The
distance between coordinates (0,0) and (0.5996,2.888) is found to be 2.94 m which
75

Figure 5.9: Conducting plate placed between transmitter and receiver antenna.

Figure 5.10: Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna
is 1.96 m away from receiver antenna.

is the same distance that was measured in the experimental set up. The distance
between coordinates (-0.5996,2.929) and (0.5996,2.888) was found to be 1.2 m which
is the same distance that was covered by the receiver antenna.
Simulations replicating the experiment conditions were conducted to conﬁrm that
the asymmetry in experimental measurements at positions 60 cm and -60 cm were
responsible for the error in estimation of source location. A gaussian modulated pulse
with 2.8 GHz center frequency, was propagated in free space in the simulation. The 41
receivers were located at 17 diﬀerent positions in the y axis, at a distance of 2.883 m
to 2.931 m away from the source. The signals saved in the 17 diﬀerent positions, were
interpolated to ﬁnd the signals on the line between coordinates (-0.5996,2.929) and
(0.5996,2.888) in ﬁgure 5.18. These signals were then time reversed and propagated.
76

Figure 5.11: Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna
is 1.96 m away from receiver antenna.

Figure 5.12: Energy image for experiment performed at a distance of 1.96 m where
the energy maximum occurs at (0.053, -0.2740).

The energy image, in ﬁgure 5.19, shows the coordinates where the waves focuses.
The wave, using data from the simulation, focuses at coordinates (0.080,-0.55), while
when the experimental data is used, the wave focuses at coordinates (0.0920,-0.3850).
The focusing error, caused by a diﬀerence in the distance between the transmitter
and receiver at position 60 cm and -60 cm, was replicated.
77

Figure 5.13: Electric ﬁeld at sensors 1 to 41 for experiment where transmitter antenna
is 2.90 m away from receiver antenna.

Figure 5.14: Energy image for experiment performed at a distance of 2.90 m where
the energy maximum occurs at (0.0920, -0.3850).

The same procedure was followed in order to explain the source location error
obtained when the experiment was performed at a distance of 1.96 m. The experimental signals at position 60 cm and -60 cm, shown in ﬁgure 5.20, are not identical.
Signal 1 has a shorter time delay than signal 41. The time of ﬂight of the peak in
signal 1 is 2.034 ns and in signal 41, is 2.148 ns. The diﬀerence between the two is
78

Figure 5.15: Experimental signals at a distance of 2.90 m for positions 1 (60 cm), 21
(0 cm), and 41 (-60 cm).

Figure 5.16: Triangle formed by the measurements taken at the experiment set up.

0.114 ns, which multiplied by the velocity of the wave in free space, yields a distance
of 0.034 m. Triangulation techniques were used once again to explain the focusing
error, as shown in ﬁgures 5.21, 5.22, and 5.23, to ﬁnd the coordinates of the line
79

Figure 5.17: Triangle formed by the measurements taken at the experiment set up.

Figure 5.18: Triangle constructed to calculate coordinates of true sensors positions
at the experiment set up.

where the 41 sensors lie. Cathetus Y=1.9776 m, shown in ﬁgure 5.22, corresponds
to the shortest distance between transmitter and receiver. This distance is diﬀerent
from the measured distance of 1.96 m. Distance X4=0.06 m, shown in ﬁgure 5.22, is
the distance between catheti Y and H and corresponds to the error found, in the x
coordinate, when using the experimental data in the FDTD model. The error in the
x coordinate was 0.0530 m which is similar to X4. The coordinates, where the linear
array of sensors was placed, were found to be (-0.5997,-1.9949) and (0.5997,-1.96) as
shown in ﬁgure 5.23. The distance between these coordinates was found to be 1.2 m,
which corresponds to the distance measured in the experiment set up.
Simulations were performed to validate the FDTD model and replicate the error
in the experiment. The 41 receivers were located at 15 diﬀerent positions in the y
80

Figure 5.19: Energy image from simulation of experiment set up.

axis, at a distance of 1.96 m to 1.99 m away from the source. The signals, saved in
the 15 diﬀerent positions, were interpolated to ﬁnd the signals on the line between
coordinates (-0.5997,-1.9949) and (0.5997,-1.96) in ﬁgure 5.23. These signals were
then time reversed and propagated. The energy image, in ﬁgure 5.24, shows the
coordinates where the waves focus. The wave, using data from the simulation, focuses
at coordinates (0.0500,-0.2050), while when the experimental data is used, the wave
focuses at coordinates (0.0530,-0.2740). The focusing error, caused by a diﬀerence in
the distance between the transmitter and receiver at position 60 cm and -60 cm, was
replicated.
The experimental results presented in this chapter validates the concept of time
reversal presented earlier using simulation signals. As expected the estimation of
source location was in error. But the error was analyzed in detail and traced back to
errors in experimental measurements. It should also be noted that the experimental
geometry is three dimensional, whereas the FDTD model was only two dimensional.
81

Figure 5.20: Experimental signals at a distance of 1.96 m for positions 1 (60 cm), 21
(0 cm), and 41 (-60 cm).
This is also expected to contribute in error in source location. However the results
presented in this chapter clearly show that the experimental microwave NDE data can
be used in conjunction with simulation model to reconstruct the position of source or
defect in samples.

82

Figure 5.21: Triangle formed by the measurements taken at the experiment set up.

Figure 5.22: Triangle formed by the measurements taken at the experiment set up.

Figure 5.23: Triangle constructed to calculate coordinates of true sensors positions
at the experiment set up.

83

Figure 5.24: Energy image from simulation of experiment set up.

84

Chapter 6
Conclusions
In this dissertation we have proposed a model based inversion procedure, using principles of time reversal, for microwave NDE data. The formulation and simulation
results, demonstrating the feasibility of the approach, are described. A computational FDTD model, for microwave NDE has been developed. The model has been
used in conjunction with time reversal principles to solve the inverse problem. The
most important contribution of this work is that it oﬀers a model-based inverse problem solution in a non-iterative manner.
Simulations, testing the feasibility of the technique to detect single and multiple
defects in dielectric and lossy dielectric materials, were performed. Defects were
detected in lossy dielectric materials with conductivity values up to σ=0.1. The
resolution of the technique was also tested. The model was used to distinguish two
defects that were next to each other. Time reversal techniques were able to detect
defects that were 2.27λ apart. The technique is robust in presence of measurement
noise. Random noise with SNR=10 and 2 was added to the signals measured by the
linear array. Energy images provided accurate information about the defect location
for all the diﬀerent simulations. The error between the coordinates of maximum
energy and true locations of defects coordinates was small in all the simulations that
85

were performed.
Simulations, using diﬀerent sensor locations and diﬀerent distances between the
source and sensor array, were performed in order to document the performance under
diﬀerent spatial distributions of sensors. Source detection was aﬀected by the sensor
locations because wave characteristics of the data were not captured when the receiver array covered a small area on the receiver plane. The distance between source
and receiver array did not aﬀect source detection. Diﬀerent distance conﬁgurations
between source and central receiver were used in the simulations. The source position was detected for all diﬀerent distance conﬁgurations. The time delay of diﬀerent
signals was modiﬁed to document a source of error of the experiment. Source detection errors were higher when the time delay of the signals was modiﬁed. Time delay
caused by errors in sensor positions did not aﬀect source detection.
The FDTD model, used to solve the inverse problem of defect detection, was
validated using experimental data. Errors were found in the estimation of the source
location. The errors were attributed to the experimental measurements. Microwave
NDE data and the FDTD model can be used to ﬁnd defects and source locations.

86

Chapter 7
Future Work
This dissertation presents a non-iterative solution to the inverse problem of defect
detection using the FDTD model and time reversal microwave techniques. Simulation results demonstrated the ability of the technique to accurately detect defects
in dielectric and lossy dielectric materials. Simulations to characterize air voids in
dielectric materials, using the FDTD model and time reversal techniques, will be
performed. Several studies remain to be conducted.
• Energy images must be further analyzed using image processing techniques to
determine size and shape of defects.
• Detection and characterization of disbonds or delaminations in composite materials using the FDTD model and time reversal techniques must be explored.
The delaminations or disbonds can be simulated as extended defects. Energy
images can be analyzed to determine size and shape of the defects.
• Simulation results showed that source or defect location is aﬀected by the spatial
distribution of the sensor array. Simulations using a narrower pulse can be
performed to investigate if less sensor area can yield accurate determination of
source location.
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• Experiments, in free space, were performed to validate the FDTD model. Experiments, using samples with defects, must be performed to further validate
the approach. Advanced algorithms that use experimental data in conjunction
with the FDTD model to reconstruct the position of the defects in the sample,
must be developed.

88

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