GROUND PENETRATING RADAR BRIDGE DECK INVESTIGATIONS USINGCOMPUTATIONAL MODELING

A Dissertation Presented

by

Kimberly Marie Belli

to

The Department of Mechaical and Industrial Engineering

in partial fulfillment of the requirementsfor the degree of

Doctor of Philosophy

in the field of

Interdisciplinary Engineering

Northeastern UniversityBoston, Massachusetts

April 2008

AbstractInfrastructure in the United States is failing. According to a 2005 study by the American

Society of Civil Engineers, over a quarter of the bridges are structurally deficient or

functionally obsolete. Condition assessment without the assistance of subsurface sensing

techniques leads to poor detection and quantification of damage because much of the

damage and precursors to damage is hidden beneath the surface. Ground Penetrating

Radar (GPR) a popular choice for bridge deck assessment, depends on a subjective

process, which is the trained eye of a technician. The ability to simulate a GPR

investigation provides insight into the response from bridge deck elements, as well as

the interaction among the elements and changes due to the presence of an anomaly and

supports defect detection.

A subsurface modeling tool is developed with physical modeling components avail-

able for general applications but extended to meet specific requirements for geometric

modeling of civil infrastructure. The simulation component implements the 2-dimensional

Finite Difference Time Domain (FDTD) method for electromagnetic modeling. Comparisons

between 2D and 3D simulations show that, for bridge deck analysis, 2D modeling is

adequate for condition assessment.

A model-based assessment augments the conventional approach to analysis by using

iterative computational models to reconstruct the bridge deck in a healthy condition. To

identify areas of suspect condition, the response from the computed healthy deck can be

compared to the response collected in the field.

The effect of the presence of rebars on the scattering from an anomaly can be

significant, and is not easily removed from GPR data. In the computational model,

the strong scattering rebars are replaced with an excitation source that results in wave

propagation equivalent to the scattering from the rebar. This technique makes the GPR

bridge deck problem better suited to the traditional inversion algorithms that are often

complicated by strong scatterers.

Through experimentation, the GPR antenna can be characterized to determine a

virtual sensor for the 2D FDTD model. The resulting sensor allows for a significantly

smaller geometry, which saves time and computational resources while reducing differences

in propagation associated with using a 2-dimensional instead of 3-dimensional model.

AcknowledgmentsSeveral complementary funding sources supported the research presented in this disser-

tation. These sources are an NSF CAREER Development Award (CMS - 9702656), The

Gordon - CenSSIS NSF Engineering Research Center (EEC - 9986821), The Depart-

ment of Education GAANN Fellowship program (P200040251), and NSF research grant

(CMMI - 0600578). These sources of funding are greatly appreciated. The diversity

in their source as well as the opportunity to be funded as a GAANN Fellow provided

flexibility to explore various research areas. The generous donation of a GPR unit, data

acquisition system, and data sets to Gordon - CenSSIS from Geophysical Survey Systems,

Inc. supported field survey collection and verification. I want to express my gratitude to

the College of Engineering and, in particular, the Department of Civil and Environmental

Engineering for providing the cost share to the GAANN Fellowship.

Thank you to David Abramo for your interest in and comments on my dissertation, to

Laura Carey for being the first user of the modeling tools presented herein and to Bryan

Lavigne for your assistance in collecting experimental data. Thank you to He (Sophia)

Zhan for many thoughtful discussions and for your simulation assistance. Thank you

to Ronald Mourant and Jose Angel Martinez-Lorenzo for serving on my dissertation

committee. Your thoughtful comments and insights are most appreciated.

Thank you to Carey Rappaport for serving as my ECE adviser. I would not have

been able to complete this work without your guidance, expertise, and sense of humor.

It’s been a real pleasure to work with you.

Thank you to Sara Wadia-Fascetti for being my primary adviser. I couldn’t have

done it without you. I am grateful for all the opportunities you’ve presented me with,

the respect you’ve shown me, and the encouragement you’ve given me. I’m grateful for

the chance to work with you and for all we’ve been able to accomplish. Is this where I’m

suppose to say something about the naming of my first born? Thank you.

Thank you to my family and friends. Your love and support (and patience) is ap-

preciated. I love you all so much. I’d list everyone by name, but it would end up being

longer than the dissertation. I would like to thank Steve Richardson for going above and

beyond the call of duty and reflowing the cold solder joint on my iBook’s motherboard.

Thank you for helping to keep me sane.

Thank you to Mike Andrews. You’re wonderful and amazing and I couldn’t have

done this without you. Thank you for everything (even your red pen).

Kim BelliApril 17, 2008

Contents

1 Introduction 51.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Identified Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Background Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Development of the ISMME Tool . . . . . . . . . . . . . . . . . . 121.5.3 Condition Assessment Using the ISMME . . . . . . . . . . . . . . 131.5.4 Implementation of a Realistic Virtual Sensor . . . . . . . . . . . . 141.5.5 Summary & Contributions . . . . . . . . . . . . . . . . . . . . . . 14

2 GPR for Civil Infrastructure Applications 152.1 GPR Antenna Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Data Collected in the Field . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Overview of Electromagnetic Properties . . . . . . . . . . . . . . . . . . . 22

2.3.1 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.4 Wave Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.5 Wavelength & Feature Resolution . . . . . . . . . . . . . . . . . . 252.3.6 Intrinsic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.7 Environmental Influence . . . . . . . . . . . . . . . . . . . . . . . 26

3 Processing of GPR Data 293.1 Determination of Permittivity . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Determination of Layer Thickness & Feature Depth . . . . . . . . . . . . 333.3 Deterioration Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Two-Dimensional Electromagnetic Simulation with Finite DifferenceTime Domain Method 394.1 The 2D TMz Mode of Maxwell’s Equations . . . . . . . . . . . . . . . . . 404.2 Finite Difference Approximation of the First Derivative . . . . . . . . . . 42

1

4.3 Spatial and Temporal Interleaving of E- and H-Fields . . . . . . . . . . . 444.4 FDTD Formulation of the 2D TMz Mode . . . . . . . . . . . . . . . . . . 464.5 Stability of 2D FDTD Formulation . . . . . . . . . . . . . . . . . . . . . 494.6 General 2D FDTD Computer Implementation . . . . . . . . . . . . . . . 52

5 Perfectly Matched Layers for the 2D FDTD 575.1 2D PML Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Computer Implementation of PML for 2D FDTD . . . . . . . . . . . . . 62

6 Integrated Sensor & Media Modeling Environment 676.1 ISMME Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Programming in Object Oriented MATLAB . . . . . . . . . . . . . . . . 69

7 ISMME Physical Modeling Tool 737.1 3D Physical Scene Creation . . . . . . . . . . . . . . . . . . . . . . . . . 747.2 General Physical Modeling Components . . . . . . . . . . . . . . . . . . 75

7.2.1 The Shape Class . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.2.2 The Cuboid Class . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2.3 The Cylinder Class . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2.4 The Ellipsoid Class . . . . . . . . . . . . . . . . . . . . . . . . . . 827.2.5 The Random Shape Method . . . . . . . . . . . . . . . . . . . . . 847.2.6 General Physical Modeling Code Listing . . . . . . . . . . . . . . 85

7.3 Application-Specific Shape Components . . . . . . . . . . . . . . . . . . . 877.3.1 The Rebar Class . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.3.2 The Aggregate Class . . . . . . . . . . . . . . . . . . . . . . . . . 927.3.3 The Rough Surface Class . . . . . . . . . . . . . . . . . . . . . . . 947.3.4 Application-Specific Physical Modeling Code Listing . . . . . . . 96

7.4 Material Property Definition . . . . . . . . . . . . . . . . . . . . . . . . . 987.4.1 The Material Object Class . . . . . . . . . . . . . . . . . . . . . . 987.4.2 The Material Library Class . . . . . . . . . . . . . . . . . . . . . 100

7.5 Discretization of the 3D Volume: The Slice Class . . . . . . . . . . . . . 101

8 ISMME Sensor Modeling & Simulation Tool 1098.1 Input Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.2 The Sensor Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3 The Simulation Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148.4 Running the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.5 The Simulation Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.6 Sensor Modeling & Simulation Code Listing . . . . . . . . . . . . . . . . 121

9 Comparison of the Accuracy of 2D versus 3D FDTD 1259.1 Computational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.2 Numerical Simulation Experiments . . . . . . . . . . . . . . . . . . . . . 128

9.2.1 Excitation Filtering Process . . . . . . . . . . . . . . . . . . . . . 1289.2.2 Physical Model Examples . . . . . . . . . . . . . . . . . . . . . . 129

2

9.2.3 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.3 Numerical Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 1349.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

10 Model Based Assessment 14910.1 Example Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15110.2 Material Parameters Computation . . . . . . . . . . . . . . . . . . . . . . 15310.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

11 The Effect of Rebar on Scattering from an Anomaly 16311.1 Modeling Rebar as a Point Source . . . . . . . . . . . . . . . . . . . . . . 16511.2 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 167

11.2.1 Simulating Total Field GPR Data . . . . . . . . . . . . . . . . . . 16711.2.2 Building the Background Response . . . . . . . . . . . . . . . . . 16811.2.3 Locating Rebar to Replace with Point Source . . . . . . . . . . . 16911.2.4 Modeling the Healthy Bridge Deck . . . . . . . . . . . . . . . . . 17111.2.5 Determining the Effect of Rebar on Scattering from an Anomaly . 171

11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.3.1 Test Case A: Healthy Bridge Deck . . . . . . . . . . . . . . . . . . 17311.3.2 Test Case B: Air Void Above Rebar . . . . . . . . . . . . . . . . . 17411.3.3 Test Case C: Air Void Below Rebar . . . . . . . . . . . . . . . . . 17611.3.4 Test Case D: Single Shifted Rebar . . . . . . . . . . . . . . . . . . 17811.3.5 Test Case E: Higher Dielectric Around Rebar . . . . . . . . . . . 17911.3.6 Test Case F: Air Void at Asphalt/Concrete Interface . . . . . . . 18011.3.7 Experimental Summary . . . . . . . . . . . . . . . . . . . . . . . 182

11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

12 Determination of Simulated Antenna Excitation 18712.1 Antenna Characterization Experiment . . . . . . . . . . . . . . . . . . . 190

12.1.1 Experiment Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . 19012.1.2 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . 192

12.2 Determination of Relevant AntennaCharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19612.2.1 Phase Center Height & Separation . . . . . . . . . . . . . . . . . 19612.2.2 Isolation of Transmit Antenna Amplitude Factors & Time Delays 203

12.3 Assembling the Desired Direct Signal . . . . . . . . . . . . . . . . . . . . 20712.4 Implementation into the 2D FDTD . . . . . . . . . . . . . . . . . . . . . 21112.5 Translating from Virtual Sensor to Receiver . . . . . . . . . . . . . . . . 21412.6 Example Bridge Deck Simulation . . . . . . . . . . . . . . . . . . . . . . 21712.7 Summary & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

13 Summary and Contributions 21913.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21913.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

3

13.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Bibliography 223

4

Chapter 1

Introduction

The condition of U.S. roadways and bridges has declined due to undetected precursors to

deterioration and damage along with postponed maintenance. The American Society of

Civil Engineers determined that, in 2003, approximately 27.1% of U.S. bridges were

structurally deficient or functionally obsolete, and that it would cost approximately

$7.3 billion a year to maintain the current poor condition of the bridge inventory and

$9.4 billion a year for 20 years to eliminate deficiencies [1,2].

A challenge associated with improving roadways and bridges is that defects are

often hidden from view until the damage becomes too severe for cost- effective repair.

Deterioration is often a subsurface mechanism that increases in severity from within the

material before it reaches the surface. Visual inspections identify problems once they have

progressed to a point of severity. As a result, most civil infrastructure defects cannot be

identified, quantified, or diagnosed reliably without subsurface imaging. Quantitative

subsurface imaging can identify and diagnose hidden defects such as delaminations,

corrosion of reinforcing steel bars (rebars), air and water voids, and changes in material

strength before they have progressed to more significant forms of damage.

Knowledge about the condition of the subsurface aids in the management of bridge

planning and maintenance. Subsurface imaging of civil infrastructure requires sensing

5

through inhomogeneous materials that are difficult to access and are subjected to time-

variant environmental conditions. The ability to simulate a subsurface investigation via

a forward model provides insight into the response from bridge deck elements, as well as

interaction among the elements and changes due to the presence and relative position of

an anomaly.

1.1 Background

While damage to bridges can be caused by many factors including structural fatigue, the

focus here is on the corrosion of reinforcing steel and delamination in reinforced concrete

bride decks. Infiltration of water and chlorides through surface cracks can, over time,

corrode the steel reinforcement within the deck. As the rebar rusts, it expands and cracks

the surrounding concrete. The rebar becomes debonded from the concrete resulting in

decreased structural integrity. Delaminations are separations of the layers in the deck

and can result from conditions such as freeze/thaw cycles, corrosion of rebar or improper

finishing during the construction process. Delaminations typically occur between the

wearing surface, such as an asphalt overlay, and the concrete, and, if corroded rebar has

debonded, just above or below the grid of rebar. The majority of the deterioration is

happening under the deck surface.

By the time deterioration signs are observed at the deck surface the damage may be

too severe and require a more costly rehabilitation. In some cases, the damage my result

in structural failure before the deterioration signs are visible or noted at the surface. To

prevent accidents and to plan for proper maintenance, it is desirable to have an idea

of the condition of the bridge deck under the surface. This can be accomplished using

non-destructive testing.

The three primary non-destructive sensing modalities used in bridge deck condition

assessment include infrared thermography [3], mechanical wave testing [3–5], and ground

6

penetrating radar (GPR) [6]. In 1993, the Strategic High Research Program investigated

the potential for GPR [7] and the Federal Highway Administration has invested significant

funds in the development a high-resolution antenna array [3]. Data obtained from

each modality yield different diagnostic information. Infrared thermography provides

information near the surface and lacks data through the depth of the bridge deck. Both

mechanical and electromagnetic waves can provide information in the form of the B-scan

through the depth of the bridge deck with sensitivity to contrast in stiffness and electrical

properties, respectively. A B-scan is a useful representation of data because it shows

these changes as a function of distance across the bridge deck. GPR is advantageous as

it facilitates rapid data collections at near highway speeds.

Quantitative imaging with GPR is required to identify and diagnose hidden defects

such as delamination, voids, and changes in material properties signifying possible

change in material strength. In practical systems data interpretation for bridge deck

deterioration evaluation is based on detection of reflection and attenuation extremes

in the GPR signal produced by variations in moisture/chloride caused by freeze-thaw

processes and corrosion [7–12]. GPR has become a popular and effective technology for

subsurface assessment of reinforced concrete bridge decks. The technology is based on the

transmission of an electromagnetic wave into the medium of interest, here a reinforced

concrete bridge deck. The electromagnetic wave refracts and reflects as it encounters

different materials. The signal that returns to the antenna at the surface is rich with

information about the subsurface. GPR wavelengths are typically in the 500MHz to

2.5GHz frequency range for bridge deck investigations.

Early civil engineering researchers seeking to use GPR for bridge deck and pavement

assessment include Maser [13], Udaya et al. [14], and Al-Qadi et al. [15]. Research related

to reinforced concrete structures includes evaluating deck electromagnetic properties

of concrete systems [16–18]. For the purpose of condition assessment and system

management many researchers have been interested in quantifying concrete overlay

7

above reinforcing steel, pavement thickness, and concrete permittivity to assess concrete

quality [19]. Barnes and Trottier [20] found GPR to be effective compared to chain drag

and half-cell potential methods as it identified well bridges in need of repair when they

were 10-50% deteriorated. Several case studies have been performed to demonstrate the

potential of GPR to evaluate reinforced concrete systems [21, 22]. These studies use

ground-coupled antennas, which while making the steel reinforcements very visible for

interpretation do not collect data as rapidly as air coupled antenna systems. Improved

training for GPR service technicians and enhanced software is necessary as is better

understanding of the electrical properties of road materials and soil related to moisture,

strength, and deformation [23].

Conventional methods of processing and analyzing GPR data for civil infrastructure

are qualitative, using relative reflection amplitudes from subsurface boundaries or

reinforcing steel bars (rebars) as an indicator of health. Typically, numerous signals

collected over a distance along the bridge deck are combined to form a scan of the

bridge subsurface. While anomalies or possible defects can be evident in the scanned

image, quantification and interpretation of critical defects remains ambiguous. The

ability to simulate the GPR investigation of a bridge deck via a forward computational

electromagnetic model provides insight into the response from and complex interaction

between bridge deck elements, and into changes in the response due to the presence and

relative location of an anomaly. In a forward problem, the source and model are known

and the field due to the source is estimated. The inverse problem takes the field due to a

source and generates an approximation of the model and/or source. Unlike the solution

to the forward problem, the solution to the inverse problem can be non-unique.

Numerical techniques for electromagnetic wave propagation solve the field equations

throughout a particular model geometry. There are several accepted and well documented

techniques that can be used to simulate wave propagation from a GPR system. These

techniques include the finite element method, the finite difference method in the time and

8

frequency domain, the method of moments, and the transmission line matrix method [24].

The GPR antenna typically used in highway and bridge evaluation is a time domain

radar, and therefore it is beneficial to simulate wave propagation using Maxwell’s time

dependent curl equations. The GPR simulations presented have been computed via

a 2-dimensional Finite Difference Time Domain (FDTD) algorithm. The electric field

measurements output from the 2D FDTD are analogous to the output from the GPR

antenna, and the method is well suited to handle inhomogeneous materials.

1.2 Identified Needs

Conventionally, bridge deck layer thickness is determined by the arrival of the reflection

from the layer interfaces, and relative condition is determined from the reflection

amplitude of the reflection. While this process can be somewhat automated, it requires

a technician to review the data to ensure that the reflection selection algorithm is

functioning as expected. Additionally, detection of areas of suspect condition are done

via visual inspection of the GPR data. The time consuming and monotonous process

of reviewing GPR data often results in subjective selection of areas of suspect condition

and the ambiguous quantification and interpretation of these areas. Improved techniques

are required for assessing the condition of bridge decks using GPR.

Analyzing GPR data with the assistance of electromagnetic modeling has a lot of

potential. A forward model produces an estimation of the electric field for a given source,

physical geometry and set of material properties. This electric field is analogous to data

recorded by the GPR. There are many software packages for electromagnetic simulation,

but in order to make forward modeling viable for use in evaluating bridge deck condition a

simple to use tool is required. The simulation tool should be able to easily and accurately

represent and modify geometries and the electromagnetic properties of materials found

in civil infrastructure.

9

Inversion techniques attempt to use the data recorded by the GPR to approximate

the physical geometry and material properties of the bridge deck. Many physics-based

inversion techniques iteratively implement a forward model to arrive at a solution.

Therefore, forward modeling plays a major role in these approaches to reconstruct

the physical geometry and material properties of the bridge deck. The reconstructed

structure and/or determined material properties are integral to the condition assessment

of the bridge. More work needs to be done to apply existing inversion methods to the

evaluation of civil infrastructure. Traditional inversion techniques can be made more

difficult by the inclusion of strong scatterers, such as rebar. Techniques are required to

make the bridge deck problem better suited for inversion.

1.3 Objectives

This thesis will address the need for a civil infrastructure specific simulation tool

allowing easy model and parameter variation by developing an Integrated Subsurface

Media Modeling Environment (ISMME). ISMME implementation of civil infrastructure

components such as rebar, aggregate and rough surfaces is straightforward, and the

simulation and material parameters are easily varied within the modeling environment.

The ISMME tool can be extended beyond the applications presented within this

dissertation.

It will be shown that two-dimensional simulations of electromagnetic wave propaga-

tion can adequately capture scattering observed in three-dimensions. This will justify

the decision to use two-dimensional simulations in condition assessment at significant

computational cost and time savings.

Model based inversion will be used to advance traditional condition assessment

techniques. Once a model for the bridge deck in a healthy condition is determined,

the the GPR response can be removed from GPR data collected over a section of bridge

10

deck that may include anomalies. The difference in the two data sets can help to localize

areas of suspect condition.

A technique for replacing individual reinforcing bars with point sources will be

presented. This technique can be used to verify assumed location of the reinforcing steel

and will illustrate the important role of reinforcing steel on the field scattered from a

subsurface anomaly. Additionally, the resulting bridge deck contains multiple excitation

sources at the reinforcing steel locations instead of multiple strong scatterers. This makes

the GPR problem better suited for application of traditional inversion techniques, such

as the Born Approximation.

The development of a virtual sensor model will be presented. The virtual sensor

will result in a probing wave that propagates through freespace similar to an actual

GPR antenna. The virtual sensor model requires a significantly smaller computational

area, and is therefore less susceptible to differences between two- and three-dimensional

propagation.

1.4 Scope

The scope of this dissertation is to develop approaches for assessing the condition of

reinforced concrete bridge decks using computational modeling. The bridge deck model

is limited to a single array of reinforcing steel, and the materials are assumed to be

non-dispersive. As will be discussed in Chapter 2, this is a reasonable assumption.

With exception to comparisons made to three-dimensional simulated data, all simulations

are performed using a 2D Finite Difference Time Domain approximation of Maxwell’s

equations within a computational region terminated by a Perfectly Matched Layer to

absorb outgoing waves and approximate an open geometry.

11

1.5 Organization of Dissertation

1.5.1 Background Chapters

A detailed discussion of GPR, as applied to civil infrastructure applications, is presented

in Chapter 2 followed by a primer on GPR processing techniques in Chapter 3.

Prior to the presentation of the Integrated Subsurface Media Modeling Environment

for the simulation of GPR investigations of reinforced concrete bridge decks, an in-depth

discussion of the 2D FDTD implementation and boundary condition consideration is

presented in Chapters 4 & 5.

1.5.2 Development of the ISMME Tool

The Integrated Subsurface Media Modeling Environment (ISMME) concept was devel-

oped to address the forwarding modeling needs for the simulation of a GPR investigation

of a bridge deck. The ISMME, introduced in Chapter 6, is written in object-oriented

MATLAB and includes both physical modeling tools and simulation tools. While

developed for 2D FDTD GPR simulation of bridge decks, ISMME’s object-oriented

design is easily extended to the creation of physical models for other applications,

to the utilization of different computational modeling algorithms including different

dimensionality, and to the simulation of different sensing modalities. The physical

modeling tools presented in Chapter 7 and coded for the purposes of this research allow

for inclusion of simple shapes such as cuboids and ellipsoids, application specific shapes

such as grids of rebar, randomly sized and distributed objects such as aggregate, and

pseudo-random rough surfaces. The simulation tools described in Chapter 8 and coded

for the purposes of this research perform 2D FDTD modeling of electromagnetic wave

propagation. ISMME allows for easy adjustment of physical models, material properties

and simulation parameters.

12

1.5.3 Condition Assessment Using the ISMME

The ISMME tool can be used to enhance understanding of the GPR response of bridges

under different conditions. Chapter 9 explores the trade-offs between using a two-

dimensional computational model or using a three-dimensional computational model for

simulation. The full 3D model is accurate but computationally time-consuming and

complicated to implement. The 2D model generates results significantly faster with less

of a drain on computational resources, but the accuracy of the model, with regard to

the actual physical condition, is decreased due to invariance in the 3rd dimension. The

degree of accuracy of an air-coupled GPR 2D FDTD model of a bridge deck is evaluated

by comparison to a full 3D model.

The ISMME tool can be used to improve the analysis of GPR responses to determine

the condition of a bridge deck. Model based assessment presented in Chapter 10 is

an inversion technique that uses iteratively adjusted forward models to determine the

electromagnetic properties of materials and depth of layers such as asphalt and concrete

and reinforcing steel. Using the determined parameters, a model of a healthy deck can

be obtained. The response due to the healthy deck can be removed from field data to

highlight anomalies.

One of the difficulties in using conventional inversion methods on GPR data is the

strong scattering from rebars. In Chapter 11, the scattering from the rebar at different

locations and for different asphalt and concrete layer thicknesses is isolated. This response

can help locate the position of the rebar and remove the effect of not only the scattering

from the rebar, but also the rebar scattered field that is incident on layer boundaries.

Additionally, the presence of rebar has a significant influence on the signal scattered

from an anomaly. Attempts to subtract only the primary rebar reflection may appear to

highlight remaining subsurface features, but the complicated interactions including the

scattering from the rebar incident upon layer interfaces, scattering from other rebar, and

13

scattering from additional elements in the subsurface will remain. Attempts to remove the

primary rebar reflection from the entire GPR response may result in erroneous analysis

of the remaining data.

1.5.4 Implementation of a Realistic Virtual Sensor

In order to accurately simulate GPR investigation of a bridge deck, it is necessary to

have an appropriate sensor model to excite the system. Using data collected in the

field, a point source excitation can be derived that results in simulated layer reflections

with a high correlation to real-world data. However, when looking at non-layered objects

(such as rebar and voids), accuracy can be greatly improved by considering the antenna’s

directivity. In Chapter 12 results are presented from an experiment using an air-coupled

antenna to determine the radiation pattern of the antenna. The resulting radiation

pattern is implemented in the 2D FDTD model as a virtual sensor. In addition to

representing propagation from the actual antenna, the virtual sensor allows for a smaller

computational model than one using antenna phase centers and the wave propagation

is less subject to differences caused by 2D propagation over 3D. Comparisons are made

between data simulated using a point source and data simulated using the experimentally

determined line source.

1.5.5 Summary & Contributions

Chapter 13 summarizes the results of the work presented in this dissertation and restates

the contributions.

14

Chapter 2

GPR for Civil Infrastructure

Applications

This section establishes the fundamentals of GPR used in civil infrastructure applications.

Information about the typical GPR antenna is presented prior to a discussion of how the

antenna are used to collect data in the field. Finally, an overview of electromagnetic

properties of materials important to this work is discussed.

2.1 GPR Antenna Overview

From a high-level perspective, a GPR system primarily consists of a data collection unit,

a transmitting antenna and a receiving antenna. If the transmit and receive functions are

performed by the same antenna, the system is called mono-static, otherwise the system

is called bi-static. Frequently, GPR systems have both antennas combined in a single

housing. This is considered to be a bi-static system. The separation of the two antennas

is often fixed.

GPR systems are either ground-coupled or air-coupled. Ground-coupled antennas

are placed directly on the bridge deck, and then dragged over it. Air-coupled antennas

15

are often mounted on a vehicle that drives over the bridge. Since the signal from the

ground-coupled antenna does not travel through air, the majority of the energy from

the antenna is transmitted into the bridge deck. This results in more visible subsurface

features than the air-coupled system. However, a vehicle with an air-coupled antenna

can collect data fast enough to forgo lane closures.

An impulse radar is typically used for bridge or roadway applications. Impulse radars

transmit a very short pulse of energy as a probing wave, directed downward toward the

ground. As the transmitting antenna is not unidirectional, some of the pulse is recorded

by the receiving antenna. This record of the pulse is referred to as the direct coupled

response. In the case of the air-coupled antenna, there is a distinct waveform representing

the direct coupling, thus it can be removed from the recorded data. The propagating

wave does not interact with any material with electromagnetic properties different than

free space before arriving at the receiving antenna. However, with a ground-coupled

antenna, the propagating wave almost immediately reaches the surface of the bridge

and, therefore, the direct coupling signal is combined with the reflection from the surface

of the deck and is not separable.

For bridge and roadway investigation, the pulse transmitted by the antenna is wide-

band, with a center frequency approximately equal to its bandwidth. While influenced by

a variety of parameters including antenna characteristics and electromagnetic properties

of the materials being probed, it is reasonable to expect signal penetration depth of

0.50-0.75 meters [25].

Air-coupled GPR systems are usually polarized parallel to direction the survey vehicle

is moving [26]. In this configuration, the radar is more sensitive to metal targets (such as

rebar) that are perpendicular to the path of the vehicle. Rotating the antenna 90 degrees

increases its sensitivity to metal targets running parallel to the path of the survey vehicle.

In cases where the reflection from metal targets is obscuring non-metallic features, it may

16

be desirable to reduce the antenna sensitivity to these objects by considering polarization

effects in the orientation of the antenna [26].

2.2 Data Collected in the Field

GPR antennas are typically moved over the roadway or bridge deck in the direction of

traffic at a given transverse separation resulting in a series of longitudinal scans. For

example, traces are taken every three feet across the width bridge deck. As the GPR

moves along a bridge, measured changes in expected radar response are indicative of

the subsurface material electromagnetic properties. The cause of these changes may be

differences in the materials of the bridge deck such as asphalt, concrete, or rebar; or the

changes may indicate defects in the subsurface, such as contaminated concrete, voids,

etc. Changes to the physical structure of the bridge, such as deck or overlay thickness or

the bridge support system, will also be reflected in the radar responses.

A pulse is transmitted at regular intervals, governed by the speed of the vehicle. For

bridge and road way investigations, this may be approximately one scan every one-half-to-

two inches. The voltage of the response is discretized and recorded for a pre-determined

time duration. Typically for bridge deck investigations, this duration is 10-15ns, during

which 512 or 1024 samples of the voltage are recorded. This information can be plotted

as a function of voltage versus sample number and is referred to as a trace.

Consider the survey of a bridge deck in Lewiston, Maine, conducted by GSSI using

a 2.0GHz air-coupled antenna and a 1.6GHz ground-coupled antenna [27]. A typical

unprocessed trace of air-coupled data is presented in Figure 2.1. The first event recorded

in the air-coupled system is the probing wave propagating from the transmitter directly to

the receiver (direct coupling). This can be seen in Figure 2.1 beginning at approximately

sample number 25.

17

Other significant events corresponding to significant changes in electromagnetic

properties occur at points 1, 2, 3 and 4 depicted in Figure 2.1. Since the antenna is air-

coupled, it is reasonable to assume that Point 1 corresponds to the surface of the asphalt

(and bridge deck). Given that GPR is relatively high frequency and bandwidth (about

2GHz), it is likely that the asphalt/concrete interface will be visible and correspond to

Point 2. It is also fair to assume that Point 3 corresponds to the layer of reinforcing steel

(rebar) and Point 4 to the bottom of the bridge deck. These features will become more

evident in the following sections.

50 100 150 200 250 300 350 400 450 500

2

2.5

3

3.5

4

4.5

x 104

Am

plitu

de

Sample Number

Direct Coupling

1

23 4

Figure 2.1: Typical 2.0GHz Air-Coupled GPR Trace (Raw Data)

GPR receiving antennas record amplitude at a particular sampling rate, for a

particular time duration. With this information, it is possible to determine the time

between sample points, thus converting the horizontal axis in Figure 2.1 (sample number)

to time. For the air-coupled data collected in Lewiston, Maine, 512 samples were recorded

in 12 nanoseconds, yielding a time between samples of approximately 23.4 picoseconds.

Likewise, the ground-coupled data recorded 512 samples in 10 nanoseconds, yielding a

time between samples of 19.5 picoseconds. Note that the time scale is often referred to

as two-way travel time. It includes the time that the transmitting wave has to travel

to the target (surface for example) where it is reflected and then returns back to the

18

receiving antenna. This is an important consideration when computing the distance

between features and will be discussed in Section 3.2.

Additionally, the air-coupled trace is often presented with the direct coupling signal

removed. This is accomplished either with time gating or by subtracting the measured

direct coupling. For time gating, the plot is truncated wit the trace beginning after

the direct coupling. Alternatively, by first recording only the direct coupling signal,

typically done by pointing the antenna toward the sky in an unobstructed area, it may

be subtracted from the recorded trace data. The latter is preferable as it can be difficult

to obtain a clean direct signal in the field.

The the typical trace (or time history), a plot of voltage versus time, is presented

in Figure 2.2(a). Direct coupling has been removed using time gating, resulting in

the horizontal axis starting at approximately 2.5ns instead of 0ns. Time histories are

sometimes presented with the surface reflection occurring at 0ns. However, as will be

discussed in Chapter 3, since the reflection from a particular object cannot be reduced

to a single instance in time, this work assumes that the first sample occurs at 0ns.

3 4 5 6 7 8 9 10 11 12

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

x 104

Am

plitu

de

Time (nanoseconds)

Top of Asphalt

Asphalt / Concrete Interface

Rebar

Bottom of Deck

(a) 2.0GHz Air-Coupled Data

1 2 3 4 5 6 7 8 9 10

2

2.5

3

3.5

4

4.5

x 104

Am

plitu

de

Time (nanoseconds)

Direct Coupling & Top of Asphalt

Rebar

Bottom of Deck

Asphalt / Concrete Interface

(b) 1.5GHz Ground-Coupled Data

Figure 2.2: Typical GPR Trace (Time History) Plots

A typical ground-coupled radar trace is presented in Figure 2.2(b). Note that

the direct coupling cannot be removed from the trace because it is intermixed with

19

the reflection from the top of the asphalt. Additionally, the typical ground-coupled

antenna has a lower frequency and bandwidth that often results in reflections from the

asphalt/concrete interface that are not separable from the air/asphalt interface.

Data are collected along the bridge deck at a pre-defined scan rate. A survey wheel

mounted on the vehicle measures the distance traveled, controlling the transmission of

the probing wave. For example, the air-coupled data in Lewiston, Maine were collected

every 1.0 inch, and the ground-coupled data were collected every 0.5 inches. In this case,

every foot of bridge deck survey results in 12 air-coupled traces and 24 ground-coupled

traces.

For bridge deck surveys, most GPR data analysis is based on interpretation of a series

of GPR signals presented in what is referred to as a B-scan. B-scans allow a series of

traces taken at regular intervals to be viewed simultaneously on a plot with the trace time

along the vertical axis and the distance at which the trace was taken on the horizontal

axis. A matrix is then formed where each row represents a sample point (or time) and

column represents a trace. The value of the element of the matrix is the amplitude of

the voltage for the associated trace and sample point. Assigning an intensity to the

amplitude of the voltage allows the matrix to be displayed as an image.

Typical B-scans for the air-coupled and ground-coupled data are shown in Figures 2.3(a)

and 2.3(b). Note that the ground-coupled B-scan has a different color axis and time

scale than the air-coupled B-scan. Additionally, the B-scans shown here include the

rotated trace on the left of the image that corresponds to the trace at the center down

track position (at 24 inches). In Figure 2.3(a) the top of the asphalt can be seen

at approximately 4.00ns, the top of the concrete at approximately 5.00ns, the rebar

layer at approximately 6.25ns, and the bottom of the deck at approximately 9.25ns. In

Figure 2.3(b) the direct coupling and top of asphalt occurs at approximately 2.00ns, the

top of the concrete at approximately 3.25ns, the rebar layer at approximately 4.50ns, and

the bottom of the deck (barely visible) at approximately 7.75ns. Longitudinal changes

20

along a layer (represented by a horizontal band) are indicative of potential subsurface

problems.

3

4

5

6

7

8

9

10

Tim

e (n

s)

Down Track (inches)0 10 20 30 40

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

x 104

(a) 2.0GHz Air-Coupled Data

1

2

3

4

5

6

7

8

9

10

Tim

e (n

s)

Down Track (inches)0 10 20 30 40

1.5

2

2.5

3

3.5

4

4.5

x 104

(b) 1.5GHz Ground-Coupled Data

Figure 2.3: Typical GPR B-Scans

Ground-coupled GPR data are typically collected as a technician walks the GPR unit

across the bridge deck. The slower speed allows higher sampling of the bridge deck data.

The data in Figure 2.3(b) was collected at 1/2” intervals across the deck as opposed to

the 1” intervals used in the air-coupled data of Figure 2.3(a). Additionally, the ground-

coupled antenna is in contact with the bridge deck which allows more of energy to be

transmitted into the deck. The frequency of scans and increased energy arriving at the

rebar layer result in improved imaging of the rebar layer. However, the ground-coupled

antenna must have good contact with the bridge deck, and therefore works best over

smooth surfaces. Air-coupled antennas can perform surveys at much higher speeds. This

allows a survey to be conducted with out police details and lane closures. Additionally,

since the direct signal is separable from the surface reflection, information about the

electromagnetic properties along the surface can be extracted.

21

2.3 Overview of Electromagnetic Properties

An electromagnetic wave is affected by the electromagnetic properties of the materials

through which it propagates. In order to analyze the GPR data, and determine areas

of suspect condition, it is important that these properties are understood and that the

extent they are influenced by environmental conditions is appreciated.

The fundamental electromagnetic properties of materials: permeability, conductivity

and permittivity will be introduced. Wave velocity, wavelength and intrinsic impedance

are important to GPR data analysis (Chapter 3) and their computation, from the

electromagnetic properties, is presented. Finally, the influence of environmental

conditions on the electromagnetic properties of bridge deck components will be discussed.

2.3.1 Permeability

Magnetic permeability (µ), henceforth simply permeability, is a measure of the magnetic

polarization of a material. It is measured in henries per meter (H/m). The relative

permeability of a material is the ratio of its permeability to that of free space

(µ0 = 4π × 10−7 H/m). Therefore, relative permeability (µr = µ/µ0) is a unitless

quantity.

Ferromagnetic materials (µr ≫ 1) are considered to be magnetically lossy and may

have a frequency dependent permeability. In these cases, the permeability also has an

imaginary component [28]. Asphalt and concrete are non-ferromagnetic (µr ≈ 1) and,

therefore, it is assumed that the permeability of these materials is the same as for free

space (µr = 1). Rebar, however, is made from steel, and therefore a ferromagnetic

material.

22

2.3.2 Electrical Conductivity

Electrical conductivity (σ), henceforth simply conductivity, is a measure of a material’s

ability to carry an electric current. It can have a significant effect on the attenuation of

a radar signal. Conductivity is measured in siemens per meter (S/m). The conductivity

of free space is zero (σ0 = 0 S/m).

For civil infrastructure materials, the conductivity is assumed to be isotropic, having

the same value in each direction. Asphalt and concrete are considered to have low

conductivities on the order of 0.001 to 0.01 S/m for dry concrete and 0.01 to 0.1 S/m

for wet concrete [29]. There is no electric field present in materials characterized as good

conductors (such as rebar).

2.3.3 Permittivity

The permittivity (ǫ) of a material is a measure of the material’s ability to allow the forma-

tion of an electric field within it. Permittivity is measured in farads per meter (F/m), and

the permittivity of free space (ǫ0) is assumed to be the same as in a vacuum. Permittivity

of a vacuum is inversely proportional to the speed of light squared times the permeability

(ǫ0 = 1/c2µ0 ≈ 8.854×10−12 F/m). The relative permittivity (ǫr) of a material is the ratio

of its permittivity to that of free space (ǫr = ǫ/ǫ0), and is unitless. Relative permittivity

is also known as dielectric constant.

Although a material’s permittivity can be frequency dependent, experiments have

shown that for civil infrastructure materials at typical GPR frequencies, there is little

variation in permittivity [15, 16].

The Society of Exploration Geophysicists of Japan [29] found typical relative

permittivities of dry concrete between 4 and 10 and of wet concrete between 10 and

20. GSSI’s Handbook For RADAR Inspection of Concrete [26] lists typical relative

permittivities of construction materials to be between 4 and 12.

23

2.3.4 Wave Velocity

The angular frequency (ω) of an electromagnetic wave is measured in radians/second

(rads/s) and is proportional to its frequency (f , measured in Hertz): ω = 2πf .

The wave number, or propagation constant, (β) is measured in radians/meter and is

given by [30]:

β = ω

√

√

√

√

µǫ

2

[

√

1 +[ σ

ωǫ

]2

+ 1

]

(2.1)

For a material with no conductivity, Equation (2.1) can be reduced to:

β = ω√

µǫ (2.2)

The velocity at which a wave travels through a material (v) is measured in meters per

second (m/s) and is proportional to the angular frequency and inversely proportional to

the wave number [30]:

v =ω

β(2.3)

From Equations (2.3) and (2.2), the wave velocity of free space (c) is:

c =ω

β0

=ω

ω√

µ0ǫ0

=1√µ0ǫ0

≈ 3 × 108m/s (2.4)

For materials with no conductivity (σ = 0) and permeability equal to that of free space

(µr = 1) Equation (2.3) can be stated as:

v =ω

β=

1√µrµ0ǫrǫ0

=c√ǫr

(2.5)

24

2.3.5 Wavelength & Feature Resolution

A propagating wave repeats itself at a particular distance called the wavelength (λ).

Wavelength is measured in meters and is inversely proportional to the wave number:

λ =2π

β= 2π

v

ω=

v

f(2.6)

GSSI states that GPR antennas, when moving along the bridge deck features aligned

perpendicular to the direction of travel can typically be distinguished if they are spaced

at least one half of a wavelength apart (λ/2). When features are stacked on top of each

other, they can typically be distinguished if they are at least a quarter of a wavelength

apart (λ/4) [26]. Features that are closer to each other than these distances may appear

as a single feature in the GPR data.

Consider a 1GHz wave propagating through free space, then through concrete with

an assumed relative permittivity of 9 and no conductivity. The velocity of the wave

in free space is 3 × 108 m/s and the wavelength is c/f = 3 × 108/1 × 109 = 0.30m

(30.0cm). In concrete, the wave velocity is, from Equation (2.5), c/√

9 = 1 × 108m/s

and the wavelength is v/f = 1 × 108/1 × 109 = 0.10m (10.0cm). If the wave were to

increase in frequency to 2GHz, the wave velocities which are material dependent would

stay the same, but the wavelengths would be half as long: 15.0cm in free space and

5.0cm in concrete. At 2GHz, it is expected that targets in the horizontal direction would

be distinguishable if they were separated by more than 7.5cm in free space or 2.5cm in

concrete. It is expected that targets in the vertical direction would be distinguishable if

they were separated by more than 3.75cm in free space or 1.25cm in concrete.

25

2.3.6 Intrinsic Impedance

The intrinsic impedance (η) of a material is the ratio of the amplitude of the electric field

to that of the magnetic field, measured in ohms (Ω) and given by [28]:

η =

√

jωµ

σ + jωǫ(2.7)

It is important in determining the reflection coefficients in Section 3.1. The intrinsic

impedance of free space (η0) simplifies to:

η0 =

√

jωµ0

0 + jωǫ0

=

√

µ0

ǫ0

≈ 376.6 Ω (2.8)

In non-conductive materials (σ = 0) with a permeability equal to free space (µr = 1),

Equation (2.7) reduces to:

η =

√

1µ0

ǫrǫ0=

η0√ǫr

(2.9)

2.3.7 Environmental Influence

A bridge deck is exposed to environmental elements. Since concrete is porous, water

(from rain, for example) can infiltrate it, carrying with it chlorides from de-icing salt

used in cold climates. Water and chloride infiltration can significantly affect wave

propagation through the bridge deck. Additionally, advanced deterioration may introduce

delaminations or other air voids into the deck, weakening its structural integrity.

The relative permittivity of concrete is very low (approximately 4 to 12) compared

to water (which is about 81). Infiltration of water into the deck will therefore raise

the relative permittivity of the affected region. Additionally, the concrete will become

more dispersive, although typically without adding significant frequency dependence to

the permittivity. Recall that permittivity refers only to the real part of the complex

26

permittivity, and that the imaginary part, where losses are introduced, is contained in

the conductivity. The permittivity of concrete isn’t drastically different from free space

(4 to 12 compared to 1), therefore, the addition of air voids would lower the permittivity

of the deck, but most likely not drastically.

The static field conductivity comprises the portion of the overall conductivity that is

frequency inependent. The static field conductivity of distilled water is approximately

1 × 10−4 S/m compared to that of sea water which is approximately 4 S/m [28]. This

large difference is due to salt content of sea water. The conductivity of concrete (typically

0 to 0.01 S/m for dry concrete) will increase with the infiltration of salt, but will not be

drastically changed by the water itself. Due to the relatively small difference between the

conductivity of free space (0 S/m) and the conductivity of of dry concrete, the addition

of air voids into concrete is likely to have little effect on the conductivity of the deck.

However, in cases where the concrete has been infiltrated by water (and therefore has a

higher conductivity), the addition of air voids may lower the conductivity of the deck.

Generally speaking, it can be seen from Equations (2.1), (2.3) and (2.6), that as the

conductivity or permittivity increase, the wave number increases and the wave velocity

and wave length decrease. The cumulative effect of all environmental can be complex as

one effect may be increasing the wave velocity, while another effect is decreasing it.

27

28

Chapter 3

Processing of GPR Data

Traditional GPR data processing techniques rely on the amplitude and arrival time of

recorded peaks associated with subsurface features. Since the probing wave is not a

unit impulse, the reflection from a subsurface feature cannot be pinpointed in a time

history. As is often the case with GPR data, typically three peaks correlate to the

arrival of a reflection from a feature. For example, in Figure 2.2(a) the top of the

asphalt reflection is considered to be the valley, peak and valley between approximately

3.75ns and 4.25ns making it difficult to choose a time and amplitude to be used in depth

and permittivity computations. Different criteria can be used to determine the time at

which a reflection from a feature is considered to be received, but the center peak is

usually chosen because it has the highest amplitude and is easiest to identify. While this

algorithm is straightforward to implement for automated analysis, it is very important

that the selection criteria remain consistent. This ensures that the established arrival

time of one reflection relative to another reflection is as accurate as possible. For example,

to measure asphalt thickness the position of the top of the underlying concrete layer is

measured relative to the top of asphalt so the algorithm that determines the arrival time

of each response must be consistent.

29

It is also important to remember that the recorded reflection from an interface does

not occur at a single instance in time. During the time the received reflection was

recorded, the probing wave has actually traveled past the interface. If the media is

not homogeneous, the amplitude of the reflection from the interface will be influenced.

Therefore, the computed permittivity at the surface of a deck, is not solely the

permittivity at the surface but an effective permittivity over the distance the wave has

traveled through the media up to the point considered to be the time of reflection.

In addition to the well-established data processing techniques discussed later in this

section, researchers have recently developed notable processing techniques. Maser [13]

presents an automated interpretation for sensing in situ conditions of bridge decks

by matching measured signals to a knowledge base. Udaya et al. [14] developed an

empirical inversion strategy that accounts for reflection from concrete-embedded steel

reinforcements. A parametric study was performed to assess the impact of cracks on

the measured waveform. Udaya et al. [31] developed one-dimensional mixture models

to predict velocity and attenuation of electromagnetic waves in concrete as a function of

wave frequency and environmental effects such as temperature, moisture content, chloride

content, and concrete mix constituents. Such mixture models handle heterogeneous

systems by mixing dielectric properties of different materials into a single mixed system.

Houston et al. [32], interested in the use of GPR for bridge deck evaluation, measured

permittivity as a function of curing time with a 2GHz GPR. They detected air voids in

concrete test slabs as thin as 1/4”, but found some discrepancy in thickness and location

of the voids when compared to the slab diagrams. Feng et al. [33] used a forward

model and a regularized inversion for layered media to obtain reflection coefficients

of jackets applied to columns for seismic retrofitting. This application was a layered

media problem without strong scatterers and used a ground-coupled antenna capable of

spanning multiple frequencies. Loulizi et al. [34] developed a method to measure layer

thickness in flexible pavements by using a forward model to compute many frequency

30

responses. Layer thickness is obtained by comparing the incident frequency to the forward

model spectra. Steel rebars (not considered in Loulizi’s study) are strong scatterers that

would complicate the signal, making the spectra comparison extremely difficult. Shaw et

al. [35] developed a neural network approach to locate reinforcing steel in concrete systems

that uses a multi-layer perceptron network to identify hyperbolic contours evident in

GPR B-scans. Since this technique requires evidence of these curves in the image, it is

most promising for ground-coupled systems. Abenius and Strand [36] used FDTD and

gradient-based minimization to reconstruct the layered image of a known structure for

electromagnetic waves between 4 - 18 GHz. Halabe et al. presented a forward modeling

and time domain inversion approach [37,38] using concrete mix models [39] which focuses

on extracting the concrete porosity, saturation and salt content and the concrete cover

above the top layer of rebar. Here a one-dimensional model for layered media is assumed,

and, once the response to the layered material is determined, the rebar response (which

was determined experimentally) is superimposed.

Data analysis techniques discussed in upcoming chapters extend the traditional data

processing techniques, called the Surface Reflection Parameters Method [25, 40]. This

data analysis techniques is discussed and exemplified in this chapter. Data were collected

by GSSI along a bridge deck in Lewiston, ME as described in 2.2 using a 2GHz air

coupled horn antenna [27]. The methods for computing relative permittivity and layer

thickness and feature depth will be discussed followed by a description of the typical

representations of the results for analysis. Finally, an example of the Surface Reflection

Parameters Method will be presented.

3.1 Determination of Permittivity

According to Fresnel’s Equation, the reflection coefficient (ρ) expresses the relationship

between the reflected and incident energy of a plane wave. The reflection coefficient at the

31

interface of free space and a different media can be expressed as a function of the intrinsic

impedances of air (η0) and the second media (η1). Assuming a normally incident plane

wave on a planar interface the reflection coefficient is calculated using Equation (3.1).

ρ01 =η1 − η0

η1 + η0(3.1)

Assuming that the two media are lossless and the magnetic permeabilities are equal to

that of free space (as is the case with most non-ferromagnetic materials), Equation (3.1)

can be written in terms of the relative permittivities (dielectric constants) of the upper

(ǫr0 = 1) and lower (ǫr1) material given by Equation (3.2).

ρ01 =1 −√

ǫr1

1 +√

ǫr1(3.2)

Using Equation (3.2), the relative permittivity of second media can be solved for in terms

of the reflection coefficient.

ǫr1 =

(

1 − ρ01

1 + ρ01

)2

(3.3)

When computing the relative permittivity of a third layer, the wave interaction with

the first and second media and the spreading of the wave as it leaves the phase center

need to be considered. It is assumed that, at the point of normal incidence, the wave is

locally considered to be planar. The composite reflected signal from the three layers is the

product of the transmission coefficient of the first interface, the reflection coefficient of

the second interface, and the transmission coefficient back out through the first interface.

The difference between the transmission and reflection coefficients is unity, so T = 1+ ρ.

According the Snell’s Law, the waves in the second layer appear to be propagating from

a virtual point above the interface. If the transmitter/receiver is located h meters above

the interface, rays in the asphalt appear to be emanating from hv = h√

ǫ1. The wave

is reduced in amplitude due to the spreading of the wave through the asphalt layer

32

(of thickness d in meters) on the way to and from the asphalt/concrete interface. The

composite reflection coefficient is given by:

ρ12 comp = T01ρ12T101

√

1 + d/hv

=

(

1 +η1 − η0

η1 + η0

)(

η2 − η1

η2 + η1

)(

1 +η0 − η1

η0 + η1

)

1√

1 + d/hv

=4η0η1(η2 − η1)

(η1 + η0)2(η1 + η2)√

1 + d/hv

(3.4)

which can be solved for η2:

η2 =

(

4η0η1 + ρ12(η1 + η0)2√

1 + d/hv

4η0η1 − ρ12(η1 + η0)2√

1 + d/hv

)

η1 (3.5)

Equation (3.5) is used to compute η2 and then the relative permittivity of the second

layer is found using Equation (2.9). Permittivities from additional layers can be similarly

computed, but the composite reflected signal becomes increasingly complicated.

3.2 Determination of Layer Thickness & Feature

Depth

The velocity of propagation of an electromagnetic wave through a low-loss material (v)

can be computed as the ratio of the speed of light (c) to the square root of the relative

permittivity (ǫr) in the material:

v =c√ǫr

(3.6)

The data recorded from a GPR system are voltages of the received signal versus time.

If the velocity of propagation through a material is known, the depth to an object (d)

can be calculated from the recorded two-way travel time (t). Notice that since the wave

travels from the transmitter to the object and back to the receiver, the two-way travel

33

time is halved to obtain the actual time to the feature. Layer thickness can be computed

from Equation (3.7).

d = vt

2(3.7)

Additionally, if a feature such as rebar or an anomaly is identified in the trace, and the

wave velocities are known, the depth to the feature can be computed using Equation 3.7.

3.3 Deterioration Maps

After analyzing each time history recorded in a bridge deck investigation, the results

can be mapped to a plan view representation of the bridge deck. The primary

determination of the condition of the bridge deck using GPR data is done by investigating

relative changes across the deck. Deterioration maps can be computed in a variety of

ways. Popular maps include maps of the relative permittivity of the surface and the

relation between the surface reflection amplitude and a metal plate reflection’s amplitude

expressed in decibels (dB). Areas along the the bridge deck with comparatively high or

low relative permittivities or those with a higher magnitude are areas of concern. In a

healthy state, the electromagnetic properties of the bridge deck should be very similar

across the bridge deck. Localized differences in the properties may indicate problem

areas.

In addition to information obtained at the surface, if deeper interface reflections can

be identified, deterioration maps can be generated at those interfaces. It is also common

to generate deterioration maps from the rebar reflections or from the bottom of the bridge

deck. Again, areas of concern are identified relative to the overall results from the deck.

34

3.4 Examples

Part of a data collection process includes obtaining a calibration scan. One such scan

is the “bumper jump” record. This is recorded in the field by triggering the GPR while

it is positioned over a metal sheet. Additionally, while the GPR is recording, a person

jumps on the bumper of the vehicle. This calibration scan provides a perfect reflection at

a variety of antenna heights that may be encountered in the field. A typical calibration

scan is shown in Figure 3.1(a). Note that the horizontal axis is trace number not distance

because the GPR was stationary over the metal plate. The metal plate reflections occur

around 3.75ns. The variation in time reflects the variation of the GPR height due to the

person jumping on the bumper. A single trace from that scan is shown in Figure 3.1(b).

Trace Number

Tim

e (n

s)

200 400 600 800 1000 1200 1400

0

2

4

6

8

10

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

x 104

Direct Coupling

Metal Plate Reflections

(a) Typical Calibration Scan

0 2 4 6 8 10

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

x 104

Time (ns)

Am

plitu

de

(b) Typical Metal Plate Reflection

Figure 3.1: Typical Metal Plate Scan and Typical Trace

An example trace collected by GSSI along a bridge deck in Maine (see Section 2.2) is

shown in Figure 3.2 along with the calibration trace from Figure 3.1(a). The calibration

trace was taken with GPR height that best matches the height when the bridge deck

trace was recorded. This can be determined by the closest match between the time from

the direct signal to the surface and the times to the metal plate in the calibration traces.

35

The signals in Figure 3.2 have been shifted vertically to the origin and normalized to the

direct coupling response.

0 2 4 6 8 10

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Time (ns)

Am

plitu

de

Metal Plate ReflectionBridge Deck Trace

Figure 3.2: Typical Metal Plate and Bridge Deck Records

The reflection coefficient of the air/asphalt boundary is the amplitude of the reflection

from the interface divided by the amplitude of the perfect reflection [25]:

ρasph = − air/asphalt interface amp

air/metal plate interface amp= −1.6785

3.6436= −0.4607 (3.8)

Once the reflection coefficient has been computed using Equation (3.8), the relative

permittivity of the asphalt can be computed using Equation (3.3):

ǫr asph =

(

1 − ρasph

1 + ρasph

)2

=

(

1 + 0.4607

1 − 0.4607

)2

= 7.34 (3.9)

where the minus sign is applied because the reflection coefficient of the metal plate is -1.

From Figure 3.2, the asphalt/concrete interface occurs at approximately 4.9ns, or

about 1.1ns after the surface reflection. Using Equations (3.6) and (3.7), the asphalt

thickness can be computed:

dasph =ct

2√

ǫr asph=

3e8 × 1.1e − 9

2√

7.34= 0.061m (3.10)

36

The composite reflection coefficient (ρconc) from the asphalt/concrete interface can be

obtained using:

ρconc = −composite air/asphalt/concrete interface amp

air/metal plate interface amp= −0.2445

3.6436= −0.067 (3.11)

Assuming the transmitter is located at h = 50cm above the deck surface, Equation (3.5)

will solve for the intrinsic impedance of concrete (ηconc = η2) resulting in ηconc = 116.7Ω.

The relative permittivity can be computed using Equation 2.9:

ηconc =η0√

ǫr conc

=⇒ ǫr conc =

(

η0

ηconc

)2

=

(

376.6

116.7

)2

= 10.4 (3.12)

The rebar reflection in Figure 3.2 occurs at approximately 6.2ns, or 1.2ns after the

asphalt/concrete interface. The depth of the rebar in the concrete can be computed using

Equations (3.6) and (3.7):

drebar =ct

2√

ǫr conc=

3e8 × 1.2e − 9

2√

10.41= 0.056m (3.13)

Added to the asphalt thickness of 0.061m, the rebar is located approximately 0.117m

under the surface.

The amplitude ratio between the surface of the deck and the metal plate reflection

(LdB asph), is given by:

LdB asph = 20 log10

(

air/asph amp

air/metal plate amp

)

= 20 log10

(

1.6785

3.6436

)

= −6.7dB (3.14)

The ratio between the asphalt/concrete and metal plate reflection is given by:

LdB conc = 20 log10

(

composite asph/conc amp

air/metal plate amp

)

= 20 log10

(

0.2445

3.6436

)

= −23.5dB

(3.15)

37

The ratio between the rebar reflection and metal plate reflection is given by:

LdB rebar = 20 log10

(

rebar reflection amp

air/metal plate amp

)

= 20 log10

(

0.1594

3.6436

)

= −27.2dB (3.16)

An example deterioration map was presented by Parrillo and Roberts [27] for data

collected along the bridge in Lewiston, ME discussed in 2.2. It maps the reflection

amplitude of the rebar layer (LdB rebar) across the bridge deck. It is reproduced in

Figure 3.3.

More

Deteriorated

Less

Deteriorated

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

4

8

12

16

20

24

-44 -39 -38 -37 -36 -35 -34 -33 -32 -31 -30 -29 -28 -27 -26 -25 -24 -23 -22 -21 -20

Figure 3.3: Deterioration Map Based on Rebar Reflection Amplitudes (From [27])

38

Chapter 4

Two-Dimensional Electromagnetic

Simulation with Finite Difference

Time Domain Method

To simulate ground penetrating radar through a bridge deck in two dimensions the

Transverse Magnetic mode was chosen with wave propagation in the x− y plane (TMz).

This mode has no magnetic field in the direction perpendicular to the cross sectional

geometry. The 2D TMz mode must be derived from Maxwell’s equations. The solution

to the 2D TMz mode is obtained explicitly by approximating the partial differentials

using a finite difference formulation with attention paid to accuracy and stability.

The formulation is presented in the following sections. First, the 2D TMz mode is

derived from Maxwell’s Equations. Then, the finite difference approximation is presented.

Spatial and temporal discretization of the model is discussed prior to the application of

the finite difference approximation to the 2D TMz mode. Finally, stability conditions

are addressed and a computer implementation is presented.

39

4.1 The 2D TMz Mode of Maxwell’s Equations

Time-dependent Maxwell’s equations in differential form [28,41] are given by:

Faraday’s Law:∂B

∂t= −∇× E −M (4.1)

Ampere’s Law:∂D

∂t= ∇×H − J (4.2)

Gauss’ Law for E-field: ∇ · D = 0 (4.3)

Gauss’ Law for H-field: ∇ · B = 0 (4.4)

where B is magnetic flux density, t is time, E is the electric field, M is the equivalent

magnetic current density, D is the electric flux density, H is the magnetic field, and

J is the electric current density. Gauss’ Laws are implicit in the FDTD formulation of

Faraday’s and Ampere’s Laws so focus is directed to Equations (4.1) and (4.2). Assuming

materials are linear, isotropic and non-dispersive [41]:

D = ǫE

B = µH (4.5)

where ǫ and µ are the electrical permittivity and magnetic permeability respectively. The

electric and magnetic current densities are defined in terms of the electric and magnetic

fields, the electric conductivity (σ), the equivalent magnetic conductivity (σ∗, not to be

confused with a complex conjugate), and independent sources of E- and H-field energy

(Jsrc and Msrc, respectively) [41]:

J = Jsrc + σE

M = Msrc + σ∗H (4.6)

40

Substituting Equations (4.5) and (4.6) into (4.1) and (4.2):

∇×E = −µ∂H

∂t−Msrc − σ∗H

∇× H = ǫ∂E

∂t+ Jsrc + σE (4.7)

In rectangular coordinates, for a given vector A,

A = Axx + Ay y + Az z

∇× A = x

(

∂Az

∂y− ∂Ay

∂z

)

+ y

(

∂Ax

∂z− ∂Az

∂x

)

+ z

(

∂Ay

∂x− ∂Ax

∂y

)

Therefore, Equation (4.7) can be written as a set of six scalar equations:

∂Ez

∂y− ∂Ey

∂z= −µ

∂Hx

∂t− Msrc x − σ∗Hx

∂Ex

∂z− ∂Ez

∂x= −µ

∂Hy

∂t− Msrc y − σ∗Hy

∂Ey

∂x− ∂Ex

∂y= −µ

∂Hz

∂t− Msrc z − σ∗Hz (4.8)

∂Hz

∂y− ∂Hy

∂z= ǫ

∂Ex

∂t+ Jsrc x + σEx

∂Hx

∂z− ∂Hz

∂x= ǫ

∂Ey

∂t+ Jsrc y + σEy

∂Hy

∂x− ∂Hx

∂y= ǫ

∂Ez

∂t+ Jsrc z + σEz

For the 2D case, the partial with respect to z is zero. To investigate the TMz mode,

the x and y components of the E-field and the z component of the H-field are zero

41

(∂/∂z = Ex = Ey = Hz = 0) and (4.8) simplifies to:

∂Ez

∂y= −µ

∂Hx

∂t− Msrc x − σ∗Hx

∂Ez

∂x= µ

∂Hy

∂t+ Msrc y + σ∗Hy (4.9)

∂Hy

∂x− ∂Hx

∂y= ǫ

∂Ez

∂t+ Jsrc z + σEz

This is the 2D TMz formulation of Maxwell’s Equations to be evaluated using finite

differences.

4.2 Finite Difference Approximation of the First

Derivative

The finite difference (FD) method provides an approach to approximate the solution to

partial differential equations. For a discretized space, the finite difference equations relate

the value of a derivative at a given point to the values of surrounding points through a

linear combination of Taylor series expansions.

For a given function, f(x), finite difference equations can approximate the value of

the partial derivative at location xi (∂f(xi)/∂x) using the values of the function at the

surrounding points xi +∆x and xi −∆x. The Taylor series expansion of the surrounding

points is

f(xi + ∆x) = f(xi) + ∆xf ′(xi) +1

2!(∆x)2f ′′(xi) +

1

3!(∆x)3f ′′′(xi) + ... (4.10)

f(xi − ∆x) = f(xi) − ∆xf ′(xi) +1

2!(∆x)2f ′′(xi) −

1

3!(∆x)3f ′′′(xi) + ... (4.11)

42

The forward-difference formulation is obtained by looking forward (at xi + ∆x). Solving

Equation (4.10) for the first derivative:

f ′(xi) =f(xi + ∆x) − f(xi)

∆x+ O(∆x) (4.12)

The above formulation has approximation errors that are indicated by O(), on the order

of ∆x. For the forward-difference approximation of the first derivative

O(∆x) = − 1

2!(∆x)f ′′(xi) −

1

3!(∆x)2f ′′′(xi) + ... (4.13)

The backward-difference formulation is obtained by looking backward (at xi − ∆x).

Solving Equation (4.11) for the first derivative:

f ′(xi) =f(xi) − f(xi − ∆x)

∆x+ O(∆x) (4.14)

The central-difference formulation is obtained by looking forward and backward.

Subtracting Equation (4.11) from (4.10):

f(xi + ∆x) − f(xi − ∆x) = 2∆xf ′(xi) +2

3!(∆x)3f ′′′(xi) + ... (4.15)

and solving for the first derivative

f ′(xi) =f(xi + ∆x) − f(xi − ∆x)

2∆x+ O(∆x2) (4.16)

For small values of ∆x, it is seen from Equations (4.12), (4.14 and (4.16) that

the truncation error for a first derivative approximation using the central-difference

formulation is smaller than the forward- and backward-difference formulations. Due

43

to the reduced error, the central-difference formulation will be used to approximate the

partial derivatives in Maxwell’s equations.

4.3 Spatial and Temporal Interleaving of E- and H-

Fields

Yee’s finite-difference implementation of the coupled Maxwell’s curl equations [42] solve

the electric and magnetic fields in time and in space. A critical feature of Yee’s

formulation is the interleaving of the E- and H-fields in three dimensions such that every

E component is surrounded by four H components and, likewise, every H component is

surrounded by four E components [41]. The spatial interleaving, commonly referred to

as Yee’s Cube, can be seen in Figure 4.1. Cubes are stacked together to to fill the entire

computational area. Note that for a given magnetic field point there are four surrounding

electric field points (represented by a face of the cube in Figure 4.1). When the cubes are

stacked together, it is clear that each electric field point has four surrounding magnetic

field points.

In addition to interleaving the equations in space, Yee also interleaved them in time.

Consider the time steps to be a group of integers divided in half (for example 1/2, 1,

3/2, 2, 5/2, 3,...). The E-fields are computed at the integer time steps (1, 2, 3,...) and

the H-fields are computed at the non-integer time steps (1/2, 3/2, 5/2, ...).

The location of a field component in the x, y and z direction is a function of the

spatial steps (∆x, ∆y, ∆z) and an integer multiplier (i, j, k). Similarly the location in

time is a function of the time step (∆t) and an integer multiplier n. Any field component

can be located in space and time as Ez(i∆x, j∆y, k∆z, n∆t) and abbreviated as Enz (i,j,k).

44

(i,j,k)

Ex

Ex Ex

Ex

Ez

Ey

Ey

Ey

Ey

Ez

Ez

Ez

Hy

Hz

Hz

Hx

Hx

Hy

x

y

z

Figure 4.1: Representation of Yee’s Cube (After [42])

The interleaved spatial coordinate system for the 2D TMz mode can be seen in Figure

4.2. At integer time steps, the Ez field is computed (ex. n, n + 1, n + 2...) and at non-

integer time steps Hx and Hy are computed (ex. n + 1/2, n + 3/2...).

x

y

Hx (i,j-1/2) Ez (i,j)

Hy (i+1/2,j)

Hy (i-1/2,j)

Hx (i,j+1/2)

Figure 4.2: Representation of Coordinate System for 2D TMz Mode

45

4.4 FDTD Formulation of the 2D TMz Mode

As described in Section 4.3, the electric field is computed at integer time steps of n

and the magnetic fields are computed at non-integer time steps. The partial spatial

derivatives utilized in Equation (4.9) are estimated using the central difference (from

Equation (4.16) resulting in:

∂Ez

∂y=

Enz (i,j+1) − En

z (i,j)

∆y

∂Ez

∂x=

Enz (i+1,j) − En

z (i,j)

∆x

∂Hx

∂y=

Hn+1/2x (i,j+1/2) − H

n+1/2x (i,j−1/2)

∆y(4.17)

∂Hy

∂x=

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j)

∆x

Note that since the E- and H-fields are interleaved at ∆x/2 (or ∆y/2) intervals, the

spatial step in Equation (4.17) is half of that in the general Equation (4.16), and the 2

disappears from the denominator in the right hand side of (4.17). The partial temporal

derivatives in Equation (4.9) are also estimated using the central difference:

∂Ez

∂t=

En+1z (i,j) − En

z (i,j)

∆t

∂Hx

∂t=

Hn+1/2x (i,j+1/2) − H

n−1/2x (i,j+1/2)

∆t(4.18)

∂Hy

∂t=

Hn+1/2y (i+1/2,j) − H

n−1/2y (i+1/2,j)

∆t

46

When taking the finite difference formulation of a time-variant variable without a

derivative, use the time average of the points considered in the approximation of the

derivative (Equation 4.18):

Ez =En+1

z (i,j) + Enz (i,j)

2

Hx =H

n+1/2x (i,j+1/2) + H

n−1/2x (i,j+1/2)

2(4.19)

Hy =H

n+1/2y (i+1/2,j) + H

n−1/2y (i+1/2,j)

2

Equations (4.17), (4.18) and (4.19) are substituted into (4.9) with the condition that

the material parameters and independent sources change as a function of location.

Enz (i,j+1) − En

z (i,j)

∆y= −µ(i,j+1/2)

Hn+1/2x (i,j+1/2) − H

n−1/2x (i,j+1/2)

∆t

− Msrc x (i,j+1/2) − σ∗

(i,j+1/2)

Hn+1/2x (i,j+1/2) + H

n−1/2x (i,j+1/2)

2

Enz (i+1,j) − En

z (i,j)

∆x= µ(i+1/2,j)

Hn+1/2y (i+1/2,j) − H

n−1/2y (i+1/2,j)

∆t(4.20)

+ Msrc y (i+1/2,j) + σ∗

(i+1/2,j)

Hn+1/2x (i+1/2,j) + H

n−1/2x (i+1/2,j)

2

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j)

∆x−

Hn+1/2x (i,j+1/2) − H

n+1/2x (i,j−1/2)

∆y= ǫ(i,j)

En+1z (i,j) − En

z (i,j)

∆t

+ Jsrc z (i,j) + σ(i,j)

En+1z (i,j) + En

z (i,j)

2

Under the assumption that there are no independent sources of field energy (Jsrc z =

Msrc x,y = 0), the materials have a uniform magnetic permeability equal to that of free

space (µ = µ0, as is the case with most non-ferromagnetic materials) and the spatial

step is uniform (∆x = ∆y = ∆), Equation (4.20) is simplified and arranged to solve for

the temporally incrementing variables. Objects comprised of metal have conductivities

going toward infinity. Therefore, in the computation of the fields, the electric field at

47

the location of metal objects is set to zero consistent with electromagnetic theory rather

than computing the fields with the FDTD approximation.

Hn+1/2x (i,j+1/2) =

(

1 − σ∗

(i,j+1/2)∆t/2µ0

1 + σ∗

(i,j+1/2)∆t/2µ0

)

Hn−1/2x (i,j+1/2) −

∆t

µ0 ∆

(

1

1 + σ∗

(i,j+1/2)∆t/2µ0

)

...

(

Enz (i,j+1) − En

z (i,j)

)

Hn+1/2y (i+1/2,j) =

(

1 − σ∗

(i+1/2,j)∆t/2µ0

1 + σ∗

(i+1/2,j)∆t/2µ0

)

Hn−1/2y (i+1/2,j) +

∆t

µ0 ∆

(

1

1 + σ∗

(i+1/2,j)∆t/2µ0

)

...

(4.21)

(

Enz (i+1,j) − En

z (i,j)

)

En+1z (i,j) =

(

1 − σ(i,j) ∆t/2ǫ(i,j)

1 + σ(i,j) ∆t/2ǫ(i,j)

)

Enz (i,j) +

∆t

ǫ(i,j) ∆

(

1

1 + σ(i,j) ∆t/2ǫ(i,j)

)

...

(

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j) − H

n+1/2x (i,j+1/2) + H

n+1/2x (i,j−1/2)

)

From Taylor series expansion, 1 + x ≈ ex, and (4.21) simplifies to:

Hn+1/2x (i,j+1/2) = e−σ∗

(i,j+1/2)∆t/µ0H

n−1/2x (i,j+1/2) −

∆t

µ0 ∆e−σ∗

(i,j+1/2)∆t/(2µ0) [En

z (i,j+1) − Enz (i,j)

]

Hn+1/2y (i+1/2,j) = e−σ∗

(i+1/2,j)∆t/µ0H

n−1/2y (i+1/2,j) +

∆t

µ0 ∆e−σ∗

(i+1/2,j)∆t/(2µ0) [En

z (i+1,j) − Enz (i,j)

]

(4.22)

En+1z (i,j) = e−σ(i,j)∆t/ǫ(i,j)En

z (i,j) +∆t

ǫ(i,j) ∆e−σ(i,j)∆t/2ǫ(i,j)...

[

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j) − H

n+1/2x (i,j+1/2) + H

n+1/2x (i,j−1/2)

]

Equation (4.22) represents the 2D FDTD formulation of the TMz mode of Maxwell’s

time dependent curl equations.

48

4.5 Stability of 2D FDTD Formulation

Three types of errors need to be considered: modeling errors, roundoff errors and

truncation (or discretization) errors [28]. The first is caused by assumptions in

formulating the numerical approach, such as assuming that a physical system can

be modeled as linear partial differential equation. Roundoff errors occur due to the

computer’s precision. Truncation errors, for the finite difference approach presented here,

are caused primarily by the finite termed approximation for the Taylor series expansion

used in the central-difference formulation (Equations (4.10) and (4.11)).

The explicit solution of a differential equation is subject to numerical instability which

causes computed values to increase without bound when provided a bounded input. In

order to achieve stability in the system a small error at any point must result in a

smaller cumulative error. For the finite difference formulation, the time step is bound to

the spatial step via Courant’s Condition.

Assume that the TMz mode (Equation (4.9)) can be solved using a plane wave of

form [41]:

Enz (I,J) = Ez0e

j(ωn∆t−kxI∆x−kyJ∆y)

Hnx (I,J) = Hx0e

j(ωn∆t−kxI∆x−kyJ∆y) (4.23)

Hny (I,J) = Hy0e

j(ωn∆t−kxI∆x−kyJ∆y)

where j =√−1, k is the wave vector and ω is the angular frequency of the wave.

49

To consider the case where ∆x 6= ∆y, Equation (4.21) is restated with the assumption

that ∆x = ∆y removed:

Hn+1/2x (i,j+1/2) =

(

1 − σ∗

(i,j+1/2)∆t/2µ0

1 + σ∗

(i,j+1/2)∆t/2µ0

)

Hn−1/2x (i,j+1/2) −

∆t

µ0 ∆y

(

1

1 + σ∗

(i,j+1/2)∆t/2µ0

)

...

(

Enz (i,j+1) − En

z (i,j)

)

Hn+1/2y (i+1/2,j) =

(

1 − σ∗

(i+1/2,j)∆t/2µ0

1 + σ∗

(i+1/2,j)∆t/2µ0

)

Hn−1/2y (i+1/2,j) +

∆t

µ0 ∆x

(

1

1 + σ∗

(i+1/2,j)∆t/2µ0

)

...

(4.24)

(

Enz (i+1,j) − En

z (i,j)

)

En+1z (i,j) =

(

1 − σ(i,j) ∆t/2ǫ(i,j)

1 + σ(i,j) ∆t/2ǫ(i,j)

)

Enz (i,j) +

∆t

ǫ(i,j)

(

1

1 + σ(i,j) ∆t/2ǫ(i,j)

)

...

(

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j)

∆x−

Hn+1/2x (i,j+1/2) − H

n+1/2x (i,j−1/2)

∆y

)

Substitute Equation (4.23) into (4.24), first solving for Hx0:

Hx0ej(ωn∆t−kxI∆x−kyJ∆y)ejω∆t/2e−jky∆y/2 = Hx0e

j(ωn∆t−kxI∆x−kyJ∆y)e−jω∆t/2e−jky∆y/2

− ∆t

µ0∆yEz0e

j(ωn∆t−kxI∆x−kyJ∆y)(

e−jky∆y − 1)

(4.25)

Hx0(ejω∆t/2 − e−jω∆t/2) =

−∆tEz0

µ0∆y

(

e−jky∆y − 1

e−jky∆y/2

)

Hx0 2j sin (ω∆t/2) =∆tEz0

µ0∆y2j sin (ky∆y/2)

Hx0 =∆tEz0

µ0∆y

sin (ky∆y/2)

sin (ω∆t/2)(4.26)

Very similarly Hy0 is solved for:

Hy0 = −∆tEz0

µ0∆x

sin (kx∆x/2)

sin (ω∆t/2)(4.27)

50

Assuming a lossless homogeneous material (σ = 0), Ez0 is solved for:

Ez0ej(ωn∆t−kxI∆x−kyJ∆y)ejω∆t = Ez0e

j(ωn∆t−kxI∆x−kyJ∆y)

+∆t

ǫej(ωn∆t−kxI∆x−kyJ∆y)ejω∆t/2

[

Hy0

∆x

(

e−jkx∆x/2 − ejkx∆x/2)

− Hx0

∆y

(

e−jky∆y/2 − ejky∆y/2)

]

Ez0ejω∆t − 1

ejω∆t/2= +

∆t

ǫ

[

Hy0

∆x

(

e−jkx∆x/2 − ejkx∆x/2)

− Hx0

∆y

(

e−jky∆y/2 − ejky∆y/2)

]

Ez0 (−2j) sin

(

ω∆t

2

)

=∆t

ǫ(−2j)

[

Hy0

∆xsin

(

kx∆x

2

)

− Hx0

∆ysin

(

ky∆y

2

)]

Ez0 = − ∆t

ǫ sin (ω∆t/2)

[

Hy0

∆xsin

(

kx∆x

2

)

− Hx0

∆ysin

(

ky∆y

2

)]

(4.28)

Equations (4.26) and (4.27) are substituted into (4.28) to relate ∆t to ∆x and ∆y:

Ez0 = − ∆t2Ez0

ǫµ0 sin (ω∆t/2)

[− sin (kx∆x/2)

∆x sin (ω∆t/2)

sin (kx∆x/2)

∆x− sin (ky∆y/2)

∆y sin (ω∆t/2)

sin (ky∆y/2)

∆y

]

ǫµ0 sin 2(ω∆t/2)

∆t2=

sin 2(kx∆x/2)

∆x2+

sin 2(ky∆y/2)

∆y2(4.29)

This can be further simplified assuming the material is free space (ǫ = ǫ0) where the

velocity of the wave is c = 1/√

µ0ǫ0. This simplification results in the general form of

the numerical dispersion relation for Yee’s algorithm in two dimensions [41]:

[

sin (ω∆t/2)

c∆t

]2

=

[

sin (kx∆x/2)

∆x

]2

+

[

sin (ky∆y/2)

∆y

]2

(4.30)

To determine the range that the finite difference algorithm is stable, Equation (4.30) is

solved for ω.

sin (ω∆t/2) = c∆t

√

[

sin (kx∆x/2)

∆x

]2

+

[

sin (ky∆y/2)

∆y

]2

ω =2

∆tsin −1(ξ) where ξ = c∆t

√

[

sin (kx∆x/2)

∆x

]2

+

[

sin (ky∆y/2)

∆y

]2

(4.31)

51

It can be seen that ξ is bounded as follows:

0 ≤ ξ ≤ c∆t

√

1

(∆x)2+

1

(∆y)2(4.32)

Equation (4.31) is stable when ω is real. This occurs when ξ ≤ 1 and is commonly

referred to as the Courant Condition:

ξ = c∆t

√

1

(∆x)2+

1

(∆y)2≤ 1

∆t ≤ 1

c√

1(∆x)2

+ 1(∆y)2

In the special case where ∆x = ∆y = ∆, the Courant Condition becomes:

∆t ≤ ∆

c√

2(4.33)

The finite difference formulation of Equation (4.22) is numerically stable if the Courant

Condition is satisfied.

4.6 General 2D FDTD Computer Implementation

This section contains a general overview for implementing the 2D FDTD (Section 4.4)

in MATLAB. The implementation differs from the derivations presented earlier in two

important ways: 1) Elements of a matrix are indexed (row, column) and first element of

the matrix, located at the upper left corner, is indexed as (1,1), and 2) Index arrays must

be transformed into whole integer coordinates using the following mapping as shown in

Table 4.1.

52

Table 4.1: Field Index Mapping

Actual (Yee’s) Coordinate Implementation Coordinate

Hn+1/2x (i,j+1/2) HX(i,j,n+1)

Hn+1/2y (i+1/2,j) HY(i,j,n+1)

En+1z (i,j) EZ(i,j,n+1)

The material distribution matrix is considered to be the discretized physical model.

Figure (4.3) shows an example of a 3x3 material distribution matrix where element (2,2)

is of a different material. The EZ matrix is chosen to align with the material distribution

matrix.

The field computed in a particular element is considered to be at the center of the

element. A typical element (with actual field indices) can be seen in the ’Example

Element from Material Distribution Matrix’ in Figure (4.3). Field components that

index out of the actual material distribution area are considered to be in an imposed

Perfect Electrical Conductor (PEC).

Additionally, the HX components are computed at an infinitesimal distance left of

where the field is in Yee’s cube. Similarly, the HY components are computed at an

infinitesimal distance above where the field is in Yee’s cube. Since the EZ components

line up with the material distribution matrix, the HX and HY components fall precisely on

the boundary between materials. By slightly shifting the location of the field calculations,

the fields are computed inside an individual material rather than on material boundaries.

With a slight right or upward shift, HX and HY components have the same indices into

the material distribution matrix as EZ.

For any given time step, Figure (4.3) shows the mapping between the actual field

indices in the Material Distribution Matrix and the implementation coordinates in the

EZ, HY, and HX matrices.

53

PEC

PE

C

HY Matrix

EZ Matrix

HX Matrix

Hx (2,3/2) Ez (2,2)

Hy (5/2,2)

Hy (3/2,2)

Hx (2,5/2)

Example Element from

Material Distribution Matrix(Discretized physical model)

EZ(2,2)

HY(2,2)

HX(2,2)

Material Distribution Matrix

Figure 4.3: Field Component Mapping

Using this mapping, Equation 4.22 can be written in terms of coordinates for computer

implementation:

HX(i,j,n+1) = e−σ∗(i,j)

∆t/µ0HX(i,j,n) − ∆t

µ0∆e−σ∗

(i,j)∆t/(2µ0) [EZ(i,j+1,n) − EZ(i,j,n)]

HY(i,j,n+1) = e−σ∗(i,j)

∆t/µ0HY(i,j,n) +∆t

µ0∆e−σ∗

(i,j)∆t/(2µ0) [EZ(i+1,j,n) − EZ(i,j,n)]

EZ(i,j,n+1) = e−σ(i,j)∆t/ǫ(i,j)EZ(i,j,n) +∆t

ǫ(i,j)∆e−σ(i,j)∆t/(2ǫ(i,j))... (4.34)

[HY(i,j,n+1) − HY(i-1,j,n+1) − HX(i,j,n+1) + HX(i,j-1,n+1)]

54

Furthermore, assuming σ∗, σ and ǫ are matrices, the following constant matrices are

defined:

HX C1 = e−σ∗∆t/µ0 HX C2 =∆t

µ0∆e−σ∗∆t/(2µ0)

HY C1 = e−σ∗∆t/µ0 HY C2 =∆t

µ0∆e−σ∗∆t/(2µ0) (4.35)

EZ C1 = e−σ∗∆t/ǫ EZ C2 =∆t

ǫ∆e−σ∆t/(2ǫ)

Note that HX C1 = HY C1 and HX C2 = HY C2 here because materials of interest are

isotropic. As will be discussed in the next section, artificial materials comprising the

PML are anisotropic.

For a physical model of size (IE,JE), the field propagation can be written in terms

of MATLAB matrix operations looping over N time steps. The constant matrices

(Equation (4.35)) can be computed outside the time loop. Neglecting the excitation

of the system and result storage, the fields are computed as follows:

for n = 1:N

HX = HX C1.*HX - HX C2.*([EZ(:,2:end) zeros(IE,1)] - EZ);

HY = HY C1.*HY + HY C2.*diff([EZ; zeros(1,JE)]);

% Possible location of system excitation (4.36)

EZ = EZ C1.*EZ + EZ C2.*(diff([zeros(1,JE); HY]) - HX...

+ [zeros(IE,1) HX(:,1:end-1));

% Possible location to save results

end

55

56

Chapter 5

Perfectly Matched Layers for the 2D

FDTD

Every additional row or column in the FDTD model matrix costs additional compute

power. For economy, the simulated area is focused on an area of interest in the model.

The area of interest typically is an open boundary problem, where it is desirable for

the waves to propagate out of the area of interest with no reflection. The simulated

area is given absorbing boundaries to prevent waves from reflecting off the edge of the

computational area, effectively mimicking a window into an infinite simulation area.

An absorbing boundary is formed by the careful design of an artificial absorbing

material layer. Unlike some Absorbing Boundary Conditions (ABCs) which only absorb

waves at normal incidence, Berenger’s Perfectly Matched Layer (PML) formulation

absorbs waves traveling in free space at any frequency and at any incident angle [43].

An example of the effect of a PML can be seen in Figure 5.1. In this example, a

cylindrical wave travels in free space outward from a point source located at the ’x’

in Figure 5.1. Without an absorbing boundary condition, the wave reflects off of the

boundaries of the computational model as seen in Figure 5.1(a). With the addition

of a PML layer in Figure 5.1(b), the wave appears to continue past the edge of the

57

computational model with no reflection, as if the computational model was a window

into a much larger (effectively infinite) computational area.

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

−2

−1

0

1

x 10−4

(a) No Absorbing Boundary

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

−2

−1

0

1

x 10−4

(b) Addition of PML

Figure 5.1: Propagation of a Cylindrical Wave in Free Space

Implementation of an ABC, such as the PML, requires designing an artificial

impedance-matched layer that will not cause reflection at the boundary. Instead it

attenuates the outgoing waves. This design is investigated below.

5.1 2D PML Formulation

Implementation of a PML requires construction of an artificial material layer around the

computational area. Let the subscript p indicate material properties in the PML and

subscript 0 indicate those in free space. The magnetic and electrical conductivity will

be used to attenuate the outgoing waves and therefore ǫp and µp are considered to be

complex. The associated impedances in these materials are given by [41]:

η0 =

√

µ0

ǫ0

ηp =

√

µp(1 + σ∗/jωµp)

ǫp(1 + σ/jωǫp)(5.1)

58

The reflection coefficient of a normal incident wave impinging on the free space/PML

boundary is given by [41]:

Γ =η0 − ηp

η0 + ηp(5.2)

For no reflection to occur at the boundary from an incident wave (Γ = 0), η0 must

equal ηp. Setting the impedances in Equation (5.1) equal to each other and choosing the

absorbing layer such that ǫp = ǫ0 and µp = µ0, the following relationship is obtained:

σ

ǫ0

=σ∗

µ0

(5.3)

Provided the relationship in Equation (5.3) is met, there is no reflection from an wave

impinging on the PML at normal incidence.

Berenger showed that, for the 2D TMz case, the Ez field could be split into x and

y components such that Ez = Ezx + Ezy. For no independent sources of field energy

(Jsrc z = Msrc x,y = 0), Equation (4.9) becomes:

∂(Ezx + Ezy)

∂y= −µ

∂Hx

∂t− σ∗

yHx

∂(Ezx + Ezy)

∂x= µ

∂Hy

∂t+ σ∗

xHy (5.4)

∂Hy

∂x= ǫ

∂Ezx

∂t+ σxEzx

−∂Hx

∂y= ǫ

∂Ezy

∂t+ σyEzy

59

Following a process similar to the one presented in Section 4.4, the 2D FDTD

implementation of split field formulation in Equation (5.4) is

Hn+1/2x (i,j+1/2) = e−σ∗

y (i,j+1/2)∆t/µ0H

n−1/2x (i,j+1/2) −

∆t

µ0 ∆e−σ∗

y (i,j+1/2)∆t/(2µ0) [En

z (i,j+1) − Enz (i,j)

]

Hn+1/2y (i+1/2,j) = e−σ∗

x (i+1/2,j)∆t/µ0H

n−1/2y (i+1/2,j) +

∆t

µ0 ∆e−σ∗

x (i+1/2,j)∆t/(2µ0) [En

z (i+1,j) − Enz (i,j)

]

(5.5)

En+1zy (i,j) = e−σy (i,j)∆t/ǫ(i,j)En

zy (i,j) −∆t

ǫ(i,j) ∆e−σy (i,j)∆t/2ǫ(i,j)

[

Hn+1/2x (i,j+1/2) − H

n+1/2x (i,j−1/2)

]

En+1zx (i,j) = e−σx (i,j)∆t/ǫ(i,j)En

zx (i,j) +∆t

ǫ(i,j) ∆e−σx (i,j)∆t/2ǫ(i,j)

[

Hn+1/2y (i+1/2,j) − H

n+1/2y (i−1/2,j)

]

Note that for the computation of Hx and Hy, the split field components of Ez have been

added back together (Ez = Ezx + Ezy).

Using the split field approach, for a wave of particular frequency (ω) and angle of the

magnetic field (φ), the impedance for the PML media is computed as:

ηp =

√

µo

ǫ0

√

wx cos2 φ + wy sin2 φ (5.6)

where wx =1 − j(σ∗

x/µ0ω)

1 − j(σx/ǫ0ω)wy =

1 − j(σ∗

y/µ0ω)

1 − j(σy/ǫ0ω)

This derivation is shown in detail in [43]. If the PML media satisfies the relationship

in (5.3), wx and wy become unity for any frequency and ηp becomes η0 for any angle.

Therefore, by splitting the Ez field and requiring that (σx, σ∗

x) and (σy, σ∗

y) satisfy (5.3),

there is no PML boundary reflection at any angle and frequency.

The PML increases the size of the computational area by the addition of a border of

the PML media around the original computational area. This border of depth d units is

often, but not always, 8 units wide. Berenger initially presented his work with both a 4

and 8 layer thick PML.

60

A layout of the PML grid can be seen in Figure (5.2). The top and bottom of the grid

have σy = σ∗

y = 0. The media in this region absorbs waves with Ezx and Hy components

propagating along the x direction. The left and right areas have σx = σ∗

x = 0, so

absorbing waves with Ezy and Hx components propagating along the y direction. The

corners of the PML grid are subjected to loss in both directions.

wave

source in

vacuum

x

y

PML(0,0,!y,!*y)

PML(!x,!*x,0,0) PML(!x,!*x,!y,!*y)

PML(!x,!*x,!y,!*y)

PML(!x,!*x,!y,!*y)

PML(!x,!*x,!y,!*y)

PML(!x,!*x,0,0)

PML(0,0,!y,!*y)

PEC

Figure 5.2: Representation of 2D TMz FDTD Grid with PML (After [43])

In the PML it is assumed that ǫp = ǫ0 and µp = µ0 and given the constraint of the

relationship in (5.3), it is still necessary to choose a conductivity (σ) for the PML region.

As is typical with ABCs, a σmax is chosen and the value of the conductivity in the PML

gradually increases as a function of distance from the boundary to be σmax at the extent

of the PML [41].

Gedney defined an equation for σmax that minimizes reflection error and discretization

error [44]. In terms of the spatial resolution of the FDTD (∆) and the order of spatial

61

variation of the absorbing properties (m, typically chosen to be between 3 and 4) σmax

was determined to be:

σmax ≈ (m + 1)

150π∆√

ǫr(5.7)

There are several ways to grade the conductivity from 0 to σmax including linearly,

polynomially and geometrically [41]. Polynomial grading is a popular choice and is

presented here. In terms of the depth of the PML layer (d) and the distance from PML

boundary (x), the conductivity at a particular location is computed as [41]:

σx(x) =(x

d

)m

σx, max (5.8)

Note that when x = 0, σx = 0 and at x = d, σx = σx, max. Computation of σy is similar.

The propagation of fields through the PML adjacent to free space can be computed

using (5.5), with ǫ0 and µ0. Equation (5.8) provides σx and σy, and Equation (5.3) relates

the electrical conductivities to the magnetic conductivities(σ∗

x and σ∗

y).

Winton and Rappaport [45] presented an approach for specifying PML conductiviities

that considers the wavelength of the excitation signal. This approach has been shown

to work very well at matching waves incident upon the PML region at a wide range of

angles while minimizing reflections from the boundaries. This alternate assignment of

PML conducivities will be implemented in future work.

5.2 Computer Implementation of PML for 2D FDTD

This section contains a general overview for implementing the PML for the 2D FDTD

presented in (Section 5.1). It is presented for implementation in MATLAB in which

elements of a matrix are indexed (row, column) and first element of the matrix, located

at the upper left corner, is indexed as (1,1).

62

The PML implementation lends itself well to the matrix based operations in

Equation (4.36). Since the EZ field is split and the σ and σ∗ now have x and y components,

there is an increased memory requirement for the additional two σ matrices and the split

field EZx and EZy matrices over Equations (4.34, 4.35). The constant matrices in Equation

(4.35) become

HX C1 = e−σ∗y∆t/µ0 HX C2 =∆t

µ0∆e−σ∗y∆t/(2µ0)

HY C1 = e−σ∗x∆t/µ0 HY C2 =∆t

µ0∆e−σ∗x∆t/(2µ0) (5.9)

EZx C1 = e−σ∗x∆t/ǫ EZx C2 =∆t

ǫ∆e−σx∆t/(2ǫ)

EZy C1 = e−σ∗y∆t/ǫ EZy C2 =∆t

ǫ∆e−σy∆t/(2ǫ)

and the loop in Equation (4.36) becomes

for n = 1:N

HX = HX C1.*HX - HX C2.*([EZ(:,2:end) zeros(IE,1)] - EZ);

HY = HY C1.*HY + HY C2.*diff([EZ; zeros(1,JE)]);

% Possible location of system excitation (5.10)

EZx = EZx C1.*EZx + EZx C2.*(diff([zeros(1,JE); HY]

EZy = EZy C1.*EZy - EZy C2.*(HX - [zeros(IE,1) HX(:,1:end-1));

EZ = EZx + EZy

% Possible location to save results

end

The σx, σ∗

x, σy, and σ∗

y matrices of the original computational area must be

modified (or extended) to include the PML as indicated in Figure 5.2 and computed

63

in Equations (5.7, 5.8). However, difficulties in implementing this modification can occur

due to the interleaving of the E and H fields presented in Section 4.3.

Assume, for illustrative purposes, an implementation of an 3-layer PML (d = 3). A

3 cell thick boundary needs to be established for the PML on all sides of the computational

area. The conductivity grading extends from 0 at the PML boundary to σmax at the edge

of the computational area. Focusing on the left and right PML regions, the interleaving

of the fields will result in the fields being computed differently for σy and σ∗

y from one

side to the other.

It is required that the PML is, in this case, 3 cells wide. Figure 5.3 shows the

interleaving of EZy and Hx with the PML region overlaid. This is dependent on the

coordinate system presented in Section 4.6. On the right hand side, σ∗

y values (used in

the EZy computation) are computed (from Equation (5.3)) using σy determined by the

values of x = 0, 1, 2, 3, 4 in Equation (5.8). Recall that x is the distance from the PML

boundary. It can be seen that the PML on the left side of the EZy matrix fills four

elements. The values of σy (used in Hx computation) are computed from Equation (5.3)

using the values of x = 1/2, 3/2, 5/2. The PML on the left side of the HX matrix fills three

elements. Due to the interleaving of the fields, in order to get a 3 cell thick PML, the

electric and magnetic conductivities are swapped for the right hand side. The σy (and

therefore HX) computations use four whole number x values and four matrix elements,

and the σ∗

y (and therefore EZy) computations use the three half number x values and

three matrix elements. The top and bottom layers of the PML are similar and can be

seen in detail in Figure 5.4.

64

EZy(i,1)

HX(i,1)

EZy(i,1+d)

HX(i,d)

EZy(i,JE)

HX(i,JE)

3 05/2 3/2 1/212 0 33/2 5/2211/2

Distance from PML

boundary

EZy(i,JE-d+1)

HX(i,JE-d)

Figure 5.3: Left & Right PML Structure

Distance

from

PML

boundary

3

0

5/2

3/2

1/2

1

2

EZx(1,j)

EZx(1+d,j)

HY(1,1)

HY(1,d)

EZx(IE,j)

0

3

3/2

5/2

2

1

1/2

HY(IE,j)

EZx(IE-d+1,j)

HY(IE-d,j)

Figure 5.4: Top & Bottom PML Structure

65

For the mapping presented here, and for a computational region of size (IE,JE) the

regions of the conductivity matrices that are modified for the PML are:

σ∗

y for EZy(:,1:d+1:-1) computed using x = fliplr([0:d])

σ∗

y for EZy(:,JE-d+1:JE) computed using x = [0.5:d-0.5]

σy for HX(:,d) computed using x = fliplr([0.5:d-0.5])

σy for HX(:,JE-d:JE) computed using x = [0:d]

σ∗

x for EZx(1:d+1,:) computed using x = flipud([0:d]’)

σ∗

x for EZx(IE-d+1:IE) computed using x = [0.5:d-0.5]’

σx for HY(1:d,:) computed using x = flipud([0.5:d-0.5]’)

σx for HY(IE-d:IE) computed using x = [0:d]’

When implementing the PML it is very important to know the mapping of the fields in

Yee’s coordinates to the implementation coordinates (Figure 4.3) and carefully consider

the bounds of the PML and which elements of the σx,y and σ∗

x,y need to be modified.

66

Chapter 6

Integrated Sensor & Media

Modeling Environment

An Integrated Sensor and Media Modeling Environment (ISMME) has been developed

as an analysis tool capable of simulating subsurface sensing systems and environmental

parameters relevant to the sensing modality. The ISMME can represent complex

subsurface features and the randomness of these features, sensor models, along with

the integration of the sensors with the subsurface environment. Many applications

can benefit from ISMME modeling, simulation, and interpretation capabilities, and it

supports improved understanding of system behavior and the ability of a particular

modality to detect defects.

6.1 ISMME Overview

Several electromagnetic wave propagation code exist, but ISMME fills a need for a tool

that can easily physically represent the complex civil infrastructure environment. Its

focus on being able to easily vary the physical representation, material properties and

simulation parameters, and its extensibility to other applications and testing modalities

67

make ISMME a welcome addition. Civil infrastructure assessment, specifically the GPR

investigation of reinforced concrete bridge decks, is the initial application for the ISMME.

However in keeping with the vision of “Similar Solutions to Diverse Problems” held by

the Gordon Center for Subsurface Sensing and Imaging Systems (Gordon CenSSIS, a

National Science Foundation Engineering Research Center) its development is generalized

to support collaboration and to enable portability to other application domains. The

ISMME consists of two primary modules: physical modeling tools and simulation tools.

The interaction between the two components can be seen in Figure 6.1.

Create scene

(3D volume)

Add components in

physical model

Define material

properties

Discretize

3D volume

Create sensor

object

Create simulation

object

Run

simulation

PHYSICAL MODELING TOOLS SIMULATION TOOLS

Figure 6.1: ISMME Tool Overview

The physical modeling tools shown to the left in Figure 6.1 are used to create a

representation of a physical system which can be used with the ISMME simulation tools

or exported as a matrix to another program. The output of these tools, discussed in

detail in Chapter 7 is a volumetric representation which can be discretized into 1-,

2- or 3-dimensions. ISMME provides basic building blocks for creating the physical

representation. These building blocks include cuboid, ellipsoid and cylindrical shapes,

as well as a container, called a scene, to hold all the elements. These components are

discussed in 7.2.

The initial application for the ISMME requires a physical representation of a

reinforced concrete bridge deck. In addition to the basic physical modeling components,

ISMME includes components built from the basic building blocks but specific to the

reinforced bridge deck structure. These application-specific components, discussed in

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7.3, include rebar arrays, pseudo-random rough surfaces and interfaces, and aggregate

distributions that contain individual aggregate pieces having random size and location

within given parameters. The ISMME can be extended to include physical modeling

components specific to other applications.

The simulation tools are discussed in Chapter 8. These tools run a numerical

simulation of the discretized model created using the ISMME physical modeling tools

or one generated from another physical modeling tool. The initial application for the

ISMME requires a 2D FDTD simulation of electromagnetic wave propagation, presented

in detail in Chapters 4 and 5. Consistent with the implementation of the FDTD from

Chapter 4, the ISMME does not currently support dispersive materials. The ISMME

can be extended to include simulation tools for other sensing modalities such as acoustic

methods, other numerical techniques such as finite elements, and to use the FDTD in 1-,

3- or N-dimensions. With the exception of the 3D simulations in Chapter 9, the ISMME

tool was used to simulate the GPR investigations presented in Chapter 9 through 12.

6.2 Programming in Object Oriented MATLAB

The ISMME was developed using object-oriented MATLAB. Its large library of built-in

functions and resource management with automatic garbage collection, make MATLAB

a convenient choice over lower-level languages such as C or Fortran for technical and

scientific programming. Using object-oriented programming allows the code to be

easily extended. This section is not intended to be an in-depth discussion on object-

oriented design, but rather serves to provide the reader with the general background and

terminology of object-oriented design as implemented in MATLAB that is necessary to

understand the ISMME code. For a detailed discussion of object-oriented programming

in MATLAB the reader is referred to [46].

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MATLAB contains several built-in data types, called classes. These include classes

such as double precision numbers (’double’), characters (’char’) and structures (’struct’).

In addition, users can define their own classes that are inherently a subclass of a structure.

For example, a user may wish to define a cube class that contains attributes such as the

length of the cube sides, the material the cube is made of, and its location in space. Once

instantiated, all this information is stored in one object variable that can be operated

upon. Simply, classes can be used to keep all relevant information about an object in

one place.

An instance of a particular class is called an object. An object is created by calling

the class constructor. Once an object has been instantiated, it is in the user’s workspace

and can be operated upon. In object-oriented programming, operations or functions that

work with objects of a particular class are commonly referred to as methods.

The user can define methods for a particular class that overload or override existing

MATLAB functions. For example, if the user wanted addition of two cubes to be a single

cube that had side lengths equal to the sum of the lengths of the previous cubes, the cube

class’s add function could be overloaded to perform this task. When a method is called,

MATLAB first looks for the called method in the class of the object being operated on.

Therefore, if the add method is overloaded for the cube class its behavior doesn’t change

for other classes such as double.

The user can create a hierarchy of classes such that the lower class called child, inherits

all of the data fields (attributes) and methods from the higher class called the parent.

Since all user-defined classes are a subclass (child) of the MATLAB structure class, they

inherit attributes and methods from the structure class. The relationship between classes

defined through inheritance can be summarized as “is a”. If there exists an automobile

and a bicycle class that both inherit from a vehicle class, the automobile “is a” vehicle

and the bicycle “is a” vehicle.

70

User defined classes can also be created using aggregation. With aggregation, an

object contains other objects. This relationship can be summarized as “has a” or

“contains a”. For example, if there exists a place setting class, it would include a spoon

object; the place setting “contains a” spoon.

When calling a user-defined method in MATLAB, the first argument is the object

type that MATLAB uses to select the method to call. MATLAB does not pass variables

by reference. An object must be updated through an assignment statement and the

method must return the updated object.

A discussion of ISMME’s classes, including attributes and methods is presented in

subsequent sections. All of the classes presented, and the methods contained within the

classes, were developed by the author. The class diagrams presented will indicate the

inheritance and/or aggregation of the class. For simplicity, the class attributes but not its

methods, will be identified. Attributes identified in red are those modified or defined by

the class constructor. Some of the ISMME’s class types are prefixed with “IME ”. IME

is an abbreviated form of ISMME and used to avoid conflict with existing MATLAB

functions. For example, MATLAB has a cylinder function that is used to create a

cylinder object in the ISMME. To avoid conflict or accidental overloading, ISMME’s

cylinder class is “IME cylinder”.

There are code snippets throughout Chapters 7 and 8. When the code appears on a

shaded background or with a line number, it is part of an ongoing example within the

chapter. It is restated in its entirety at the end of the chapter. Other code snippets

represent general instantiation of objects or implementation of methods.

The following sections provide an overview and examples for using the ISMME. They

are not intended to be a comprehensive users guide. Significant effort was put into the

development of ISMME to permit straightforward implementation by the user.

71

72

Chapter 7

ISMME Physical Modeling Tool

The ISMME tool, introduced in Chapter 6, contains physical modeling tools for a

structure represent the system materials and defects with the primary modeling objective

to generate a physical representation of the system for use in discretization for simulation.

For this thesis the structure consists of a bridge deck and the system materials

include concrete, rebar and asphalt. The defects include delaminations, air voids, and

contaminated concrete.

The physical modeling tools are used to create a 3D representation of a system which

can be discretized into a one-, two- or three-dimensional mesh. These meshes can be

used in numerical techniques, such as finite differences, that approximate solutions to

partial differential equations. A 2D discretization of the 3D system results in a matrix

called a slice. While in this thesis, simulations are two-dimensional and performed on

2D slices, a 3D simulation is implementable on a series of multiple 2D slices and a 1D

simulation is implementable on a vector from the 2D slice. Parameters required for

sensing simulations (such as the electromagnetic properties of the materials) are defined

at the time the volumetric model is discretized.

The creation of the physical model, illustrated in Figure 7.1, requires three steps:

I. creation of the computational volume called the scene,

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II. creation of component objects and their insertion into the scene, and

III. discretization of the scene resulting in a slice (or slices) of the physical model

volume.

The component objects can be generalized, simple shapes, or more complex, application

specific shapes.

Create scene

(3D volume)

Add components in

physical model

Define material

properties

Discretize

3D volume

Create sensor

object

Create simulation

object

Run

simulation

PHYSICAL MODELING TOOLS SIMULATION TOOLS

Figure 7.1: ISMME Tool Overview

7.1 3D Physical Scene Creation

The Scene object developed as part of the ISMME tools is the container for the 3D

physical model. All objects included in the physical model are inserted into this container.

The scene constructor instantiates a scene object and takes as input the size of the

container, which will eventually become the computational area. By default, the X, Y

and Z extents of the scene object are entered in millimeters. A scene object is instantiated

as follows:

mScene = scene ( [X Y Z ] ) ;

The render method graphically displays the 3D scene and it’s contents. A 500mm

computational area can be created and rendered with the following code seen in

Figure 7.2:

74

Listing 7.1: General Physical Modeling Component Example, Part I

1 % Create empty scene 500mm x 500mm x 500mm

2 mScene1 = scene ( [ 500 500 5 0 0 ] ) ;

3 render (mScene1 ) ;

0100

200300

400500

0100

200300

400500

0

100

200

300

400

500

Z (

mm

)

X (mm)Y (mm)

Figure 7.2: Rendered Scene Object

7.2 General Physical Modeling Components

To define the physical model, shape objects are inserted into the scene to represent one

component of the scene, such as reinforcing steel or a layer of asphalt (cylinder or cube).

Given a shape called mShape, it is added to the scene using the scene’s add method or

with the overridden ’+’ operator. It can be removed from the scene using the remove

method or overridden ’-’ operator.

mScene = add(mScene,mShape); OR mScene = mScene + mShape;

mScene = remove(mScene,mShape); OR mScene = mScene − mShape; Once a

shape is inserted into a scene, it can be modified using the scene object. Changes to

the original shape object are not reflected in the scene.

75

7.2.1 The Shape Class

The Shape Class is an abstract base class: a class that is not itself instantiated but

is instead inherited by other classes. In MATLAB, there are no abstract classes, so

it is up to the implementor to only use objects that inherit from shape. The shape

constructor contains information on the shape description (ID), the material from which

the shape is constructed, grip location and description, and render information. The

class is shown in Figure 7.3. The shape class provides the base structure that all other

modeling components (general and application specific) inherit. The user only needs to

be concerned with the shape class when defining a new class of modeling component.

Each shape object is composed of a single material in the physical model. For example, a

two layer media would be two shapes, not a single shape made of two different materials.

Object ID

Material

Grip location

Grip description

Render information

ime_shape

Figure 7.3: Class Diagram for ime shape

When any object that inherits from the shape class is instantiated, the material from

which it is constructed (concrete for example) is identified. The shape class has a modify

method that includes a ’changeMat’ option that allows the user to change the object’s

material after instantiation or insertion into the scene. This modify method option can

be called on any object that has inherited from the shape class. It will be illustrated in

Section 7.2.2.

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The object ID of the shape is a user-specified string that must be unique within the

context of the scene that incorporates the shape.

7.2.2 The Cuboid Class

The Cuboid Class defines a rectangular parallelepiped object to be inserted into a

scene. The cuboid constructor inherits attributes from the shape class and, in addition,

requires data appropriate to a cuboid shape. It takes as input an object descriptor (ID),

the location of the corner (grip) in millimeters, the material from which the cube is

constructed, and the length of each of the sides. The attributes required to instantiate

a cuboid object are highlighted in the cuboid class diagram (Figure 7.4). The remaining

attributes are not passed in to the constructor but are assigned automatically. The

material properties will be discussed later, but at the scene level, only a string indicating

the material type is required. In general, a cuboid object is instantiated as follows:

mCube = ime cuboid ( id , g r ip (X,Y, Z) , mater ia l , s i d e length (X,Y,Z ) ) ;

Object ID

Material

Grip location

Grip description

Render information

ime_shape

Side length (X, Y, Z)

ime_cuboid

Figure 7.4: Class Diagram for ime cuboid

Consider a cuboid labeled as ’myBlock’ and constructed of concrete. The lower, front,

right corner (the cuboid grip) is located at X = Y = Z = 100mm. The cuboid has sides

100mm in the X-direction, 200mm in the Y-direction, and 50mm in the Z-direction. The

cuboid can be added to the scene using the following code. The resulting scene is shown

in Figure 7.5 (a).

77

Listing 7.2: General Physical Modeling Component Example, Part II

5 % Create cuboid & add to scene

6 mCube = ime cuboid ( ’myBlock’ , [ 100 100 100 ] , ’voidAir’ , [ 100 200 5 0 ] ) ;

7 mScene1 = mScene1 + mCube ;

8 render (mScene1 ) ;

The physical modeling tools include methods to move and resize the cuboid. The

cuboid can be modified either before or after insertion into the scene. The following

commands move or translate the cuboid. The operations have no visible effect until

render() is called.

newCube = modify ( cube ob ject , ’move’ , t r a n s l a t i o n (X,Y,Z ) ) ;

newCube = modify ( cube ob ject , ’moveTo’ , new gr ip l o c a t i o n (X,Y, Z ) ) ;

newCube = modify ( cube ob ject , ’resize’ , s ize in / decr eas e (X,Y,Z ) ) ;

newCube = modify ( cube ob ject , ’setsize’ , new s ize (X,Y,Z ) ) ;

If the cuboid has already been inserted into the scene, it can be modified by passing the

modify method the scene object and the cuboid ID rather than the cuboid object. The

method then outputs the updated scene object instead of the updated cuboid object.

For example,

newScene = modify ( scene obj , cube id , ’move’ , t r a n s l a t i o n (X,Y, Z ) ) ;

newScene = modify ( scene obj , cube id , ’moveTo’ , new gr ip (X,Y,Z ) ) ;

newScene = modify ( scene obj , cube id , ’resize’ , i n c r e a s e (X,Y,Z ) ) ;

newScene = modify ( scene obj , cube id , ’setsize’ , new s ize (X,Y, Z ) ) ;

Consider the previously created cuboid shown in Figure 7.5(a). The cuboid has

already been inserted into the scene but several changes are required. To move the

cuboid to a new location with a grip at X = Y = Z = 250mm, it needs to be resized so

all sides are 100mm. Additionally, the material should change from concrete to rebar.

Recall that the method to change the material of a component was discussed earlier in

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Section 7.2.1. This changes are done using the following code and shown in Figure 7.5(b).

Note that a copy of the original scene (called ’mSceneTmp’) is obtained, preserving the

original (’mScene1’).

Listing 7.3: General Physical Modeling Component Example, Part III

10 % Move , r e s i z e & change mate r i a l o f b l o c k ( copy mScene1 o b j e c t )

11 mSceneTmp = modify (mScene1 , ’myBlock’ ,’moveTo’ , [ 250 250 2 5 0 ] ) ;

12 mSceneTmp = modify (mSceneTmp , ’myBlock’ ,’setsize’ , [ 100 100 1 0 0 ] ) ;

13 mSceneTmp = modify (mSceneTmp , ’myBlock’ ,’changeMat’ , ’rebar’ ) ;

14 render (mSceneTmp ) ;

(a) Initial Cuboid Inserted into Scene (b) Modified Cuboid within Scene

Figure 7.5: Insertion and Modification to Cuboid Object

7.2.3 The Cylinder Class

The Cylinder Class defines a cylindrical object that can be inserted into a scene. The

cylinder constructor inherits attributes from the shape class and, in addition, fills in the

data appropriate to a cylindrical shape. It takes as input an object descriptor (ID),

the location of the cylinder centroid (grip), the material from which the cylinder is

79

constructed, the direction of the height of a cylinder, the height of the cylinder and the

radius as seen in Figure 7.6. It is instantiated as follows:

mCyl = ime cy l i nd e r ( id , g r ip (X,Y, Z) , mater ia l , . . .

he ight dir (X,Y, Z) , height , r ad iu s ) ;

Object ID

Material

Grip location

Grip description

Render information

ime_shape

Height direction

Height

Radius

ime_cylinder

Figure 7.6: Class Diagram for ime cylinder

A cylinder labeled ’myCylinder’ has the following parameters: a centroid at X =

400mm and Y = Z = 300mm, material of type rebar, a height of 200mm in the Z-

direction, and a radius of 50mm. The cylinder can be seen in Figure 7.7 (a). It is

constructed and added to the scene using the following code:

Listing 7.4: General Physical Modeling Component Example, Part IV

16 % Create c y l i n d e r & add to scene

17 mCyl = ime cy l i nd e r (’myCylinder’ , [ 400 300 300 ] , ’rebar’ ,’z’ , 2 00 , 50 ) ;

18 mScene1 = mScene1 + mCyl ;

19 render (mScene1 ) ;

As with the cuboid object, the cylinder object can be modified while it is in or out

of the scene. The ’move’ and ’moveTo’ options to the modify method are called the

same way as for a cuboid object. The ’resize’ and ’setSize’ options to the cylinder’s

modify method are similar to the cuboid object. Instead of taking a vector of side length

or change to side length, they take the new radius, and optionally, the new height. A

second form of the options incorporates an incremental change to the radius and height.

80

Calls to the cylinder’s modify method with the ’resize’ option outside the scene are

constructed as follows:

newCyl = modify ( cylinder ob ject , ’resize’ , in / decr eas e rad iu s ) ;

- OR -

newCyl = modify ( cylinder obj , ’resize’ , in / decr eas e [ rad he ight ] ) ;

Calls to the cylinder’s modify method with the ’setSize’ option outside the scene are

constructed as follows:

newCyl = modify ( cylinder ob ject , ’setSize’ , new rad iu s ) ;

- OR -

newCyl = modify ( cylinder ob ject , ’setSize’ , new [ rad iu s he ight ] ) ;

If a cylinder object has already been inserted into the scene, the modify method call

is invoked similarly to the cuboid’s modify method when it is in a scene. This modify

method takes the scene object and cylinder ID. It outputs a modified scene object instead

of a cylinder object. As an example, the modify method to change the radius of a cylinder

within a scene is called with the following structure:

newScene = modify ( scene obj , cylinder id , ’setSize’ , new rad iu s ) ;

Consider the previously created cylinder shown in Figure 7.7 (a). The cylinder has

been inserted into the scene but needs to be moved to a new location where the centroid

(cylinder grip) is 100mm lower than the current centroid (the X and Y location of the

grip stays the same). Additionally, the 50mm radius needs to be increased by 75mm and

the 200mm height needs to be decreased by 150mm. This is done using the following

code. The resulting scene is shown in Figure 7.7 (b). Note that a copy of the original

scene is obtained so that the original is preserved.

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Listing 7.5: General Physical Modeling Component Example, Part V

21 % Move & r e s i z e c y l i n d e r ( us ing copy o f scene o b j e c t )

22 mSceneTmp = modify (mScene , ’myCylinder’ ,’move’ , [ 0 0 −100]) ;

23 mSceneTmp = modify (mSceneTmp , ’myCylinder’ ,’resize’ , [ 7 5 −150]) ;

24 render (mSceneTmp ) ;

(a) Initial Cylinder Inserted into Scene (b) Modified Cylinder within Scene

Figure 7.7: Insertion and Modification to Cylinder Object

7.2.4 The Ellipsoid Class

The Ellipsoid Class defines an ellipsoidal object. The ellipsoid constructor inherits

parameters. In addition, the constructor fills in the data appropriate to a ellipsoidal

shape. It takes as input an object descriptor (ID), the centroid of the ellipsoid (grip),

the material the ellipsoid is constructed of and the radius in the X, Y, and Z directions

as seen in Figure 7.8. It is instantiated as follows:

mEll ip se = im e e l l i p s o i d ( id , g r ip (X,Y, Z) , mater ia l , r ad iu s (X,Y,Z ) ;

An ellipsoid, labeled as ’myEll’, has a centroid at X = 100mm and Y = Z = 300mm.

It is constructed of asphalt, and has a radius of 100mm, 50mm and 75mm in the X, Y,

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Radius (X, Y, Z)

ime_ellipsoidObject ID

Material

Grip location

Grip description

Render information

ime_shape

Figure 7.8: Class diagram for ime ellipsoid

and Z directions respectively. The ellipsoid can be seen in Figure 7.9 and is constructed

and added to the scene using the following code:

Listing 7.6: General Physical Modeling Component Example, Part VI

26 % Create e l l i p s o i d & add to scene

27 mEl l ip se = im e e l l i p s o i d ( ’myEll’ , [ 100 300 300 ] , . . .

28 ’asphalt’ , [ 100 50 7 5 ] ) ;

29 mScene1 = mScene1 + mEl l ip se ;

30 render (mScene1 ) ;

As with the cuboid and cylinder objects, the ellipsoid object can be modified while

it is in or out of the scene. The ’move’ and ’moveTo’ options to the modify method

are used the same way as for a cuboid object. The ’resize’ and ’setSize’ options to the

ellipsoid’s modify method are similar to the cuboid object. Instead of taking a vector of

side length or an incremental change to side length, it takes a vector of the new radius

or change to the new radius. As an example, the structure to move an ellipsoid to a new

location before it is inserted the scene is:

newEl l ip se = modify ( e l l i p s o i d ob ject , ’moveTo’ , new gr ip (X,Y,Z ) ) ;

The structure to increase or decrease the radii of an ellipsoid after it has been inserted

into a scene is:

newScene = modify ( scene obj , e l l id , ’resize’ , r a d i i change (X,Y,Z ) ) ;

83

Consider the previously created ellipsoid shown in Figure 7.9 (a). The ellipsoid has

been inserted into the scene and can be moved to a new centroid (grip) location of

(X,Y,Z) = (400,100,300) mm. The radii of the ellipsoid can be changed to 50mm, 25mm

and 150mm in the X, Y, and Z direction respectively. This can be done with the following

code. The resulting scene is shown in Figure 7.9 (b). Note that a copy of the original

scene is obtained so that the original is preserved.

Listing 7.7: General Physical Modeling Component Example, Part VII

32 % Move & r e s i z e e l l i p s o i d ( us ing copy o f scene o b j e c t )

33 mSceneTmp = modify (mScene1 , ’myEll’ ,’moveTo’ , [ 400 100 3 0 0 ] ) ;

34 mSceneTmp = modify (mSceneTmp , ’myEll’ ,’setSize’ , [ 5 0 25 1 5 0 ] ) ;

35 render (mSceneTmp ) ;

(a) Initial Ellipsoid Inserted into Scene (b) Modified Cylinder with Scene

Figure 7.9: Insertion and Modification to Ellipsoid Object

7.2.5 The Random Shape Method

The physical modeling tools also include a function to create a random general physical

modeling component contained within another component. The method takes as input

the type of object desired (’cuboid’, ’ellipsoid’, or ’cylinder’), the material the object is

84

constructed of, the container the random object will be placed in, the maximum and

minimum size of the object (size range of random object), and an object identifier.

newShape = ime random ( type o f shape , mater ia l , conta iner , . . .

max size , min size , id ) ;

example : % Create a random cuboid to be i n s e r t e d i n t o e x i s t i n g

% mCube o b j e c t wi th s i d e s between 25 & 50mm

randCube = ime random ( ’cuboid’ , ’asphalt’ , mCube, . . .

50 ,25 , ’randCuboid’ ) ;

This method can be useful for inserting an object, such as a void in a concrete deck,

where the user wishes to avoid specifying a particular size or location.

7.2.6 General Physical Modeling Code Listing

Sections 7.1 through 7.2.4 have detailed an example physical modeling scene created

using the general physical modeling components. The code used to create and display

the scenes in this example (shown previously as snippets in shaded boxes) is listed below

in its entirety.

Listing 7.8: General Physical Modeling Component Example

1 % Create empty scene 500mm x 500mm x 500mm

2 mScene1 = scene ( [ 500 500 5 0 0 ] ) ;

3 render (mScene1 ) ;

5 % Create cuboid & add to scene

6 mCube = ime cuboid ( ’myBlock’ , [ 100 100 100 ] , ’voidAir’ , [ 100 200 5 0 ] ) ;

7 mScene1 = mScene1 + mCube ;

8 render (mScene1 ) ;

10 % Move , r e s i z e & change mate r i a l o f b l o c k ( copy mScene1 o b j e c t )

85

11 mSceneTmp = modify (mScene1 , ’myBlock’ , ’moveTo’ , [ 250 250 2 5 0 ] ) ;

12 mSceneTmp = modify (mSceneTmp , ’myBlock’ , ’setsize’ , [ 100 100 1 0 0 ] ) ;

13 mSceneTmp = modify (mSceneTmp , ’myBlock’ , ’changeMat’ ,’rebar’ ) ;

14 render (mSceneTmp ) ;

16 % Create c y l i n d e r & add to scene

17 mCyl = ime cy l i nd e r (’myCylinder’ , [ 400 300 300 ] , ’rebar’ , ’z’ , 2 00 , 50 ) ;

18 mScene1 = mScene1 + mCyl ;

19 render (mScene1 ) ;

21 % Move & r e s i z e c y l i n d e r ( us ing copy o f scene o b j e c t )

22 mSceneTmp = modify (mScene1 , ’myCylinder’ ,’move’ , [ 0 0 −100]) ;

23 mSceneTmp = modify (mSceneTmp , ’myCylinder’ ,’resize’ , [ 7 5 −150]) ;

24 render (mSceneTmp ) ;

26 % Create e l l i p s o i d & add to scene

27 mEl l ip se = im e e l l i p s o i d (’myEll’ , [ 100 300 300 ] , . . .

28 ’asphalt’ , [ 100 50 7 5 ] ) ;

29 mScene1 = mScene1 + mEl l ip se ;

30 render (mScene1 ) ;

32 % Move & r e s i z e e l l i p s o i d ( us ing copy o f scene o b j e c t )

33 mSceneTmp = modify (mScene1 , ’myEll’ , ’moveTo’ , [ 400 100 3 0 0 ] ) ;

34 mSceneTmp = modify (mSceneTmp , ’myEll’ , ’setSize’ , [ 5 0 25 1 5 0 ] ) ;

35 render (mSceneTmp ) ;

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7.3 Application-Specific Shape Components

The physical modeling tool can build off of the general physical modeling components

(cuboids, cylinders and ellipsoids) to create application-specific modeling components.

Here, the focus is on components developed for civil infrastructure: these components

include rebar layers, collection of aggregate, or air voids, and rough surfaces and

interfaces.

These components will be described in the following sections by building an example

bridge deck. The example begins with a modeling area 2000mm long, 1000mm wide

and 1000mm deep. A 200mm thick concrete deck and 60mm this asphalt overlay are

added using cuboid objects as described in Section 7.2.2. This initial model, shown in

Figure 7.10, is created using the following code.

Figure 7.10: Concrete Deck & Asphalt Overlay Cuboids

87

Listing 7.9: Application Specific Physical Modeling Component Example, Part I

1 mScene2 = scene ( [ 2000 1000 1000 ] ) ;

2 deck = ime cuboid (’deck’ , [ 0 0 200 ] , ’concrete’ , [ 2000 1000 2 0 0 ] ) ;

3 over l ay = ime cuboid ( ’overlay’ , [ 0 0 400 ] , ’asphalt’ , [ 2000 1000 6 0 ] ) ;

4 mScene2 = mScene2 + deck + over l ay ;

5 render (mScene2 ) ;

7.3.1 The Rebar Class

The Rebar Class is an application specific class developed for civil infrastructure. It

builds off of either the cylinder class or the cuboid class depending on whether the user

wants a circular or square cross section of rebar. The rebar class generates a single object

that contains a series of cylinders or cuboids as defined by the rebar configuration. The

rebar constructor takes as input an object descriptor (ID), the previously defined object

that contains the rebar, the orientation and diameter of the rebars, the depth of cover

over the bars, and the rebar separation as seen in Figure 7.11. The rebar object is

instantiated as follows:

mRebar = rebar ( id , conta iner , d i r e c t i o n (X or Y) , diameter , . . .

cover , separat ion , cross s e c t i o n shape ( c i r c l e , square ) ) ;

Consider a concrete deck (the previously created object with a variable name deck)

that contains two perpendicular rebar arrays (forming a rebar mesh). The first array,

labeled ’barY’, is oriented in the Y direction, has a diameter of 16mm, a depth of cover

of 50mm, a bar separation of 120mm on center, and has a square cross section. The rebar

object can be created using the following code:

Listing 7.10: Application Specific Physical Modeling Component Example, Part II

7 % Bui ld rebar in y−d i r e c t i o n

8 mRebarY = rebar ( ’barY’ , deck , ’y’ , 16 ,50 ,120 , ’square’ ) ;

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Object ID

Material

Grip location

Grip description

Render information

ime_shapeSide length (X, Y, Z)

ime_cuboid

Container

Bar cover

Bar separation

Bar diameter

Bar shape

rebar

Height direction

Height

Radius

ime_cylinderOR

Figure 7.11: Class Diagram for rebar

The second array, labeled ’barX’, is oriented in the X direction, has a diameter of 13mm,

a depth of cover of 65mm, a bar separation of 200mm on center, and has a circular cross

section. The rebar object can be created using the following code:

Listing 7.11: Application Specific Physical Modeling Component Example, Part III

9 % Bui ld rebar in x−d i r e c t i o n

10 mRebarX = rebar ( ’barX’ , deck , ’x’ , 13 ,65 ,200 , ’circle’ ) ;

Both rebar objects can be added to the scene object using the following code. The

rendered scene is shown in Figure 7.12. Note that the cylindrical rebars running in the

X-direction appear to taper at the ends. As will be clear when the discretization of the

scene is discussed, there is actually no taper. This is due to requiring the cylinder to be

closed (or capped). Increasing the render (not the scene) resolution along the height of

the cylinder would alleviate this issue but result in potentially substantially longer render

times.

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Listing 7.12: Application Specific Physical Modeling Component Example, Part IV

11 % Add rebar to scene

12 mScene2 = mScene2 + mRebarY + mRebarX ;

13 render (mScene2 ) ;

Figure 7.12: Addition of Rebar Mesh

By default, the rebar class computes the number of rebars based on the bar separation

and the size of the container. The rebar array is then centered in the specified container.

However, an alternate instantiation of the rebar class allows the user to specify an

argument that identifies the lateral location of the first rebar. The rebar class then

computes the number of rebars and their locations from that starting point. For example,

to create a rebar object similar to mRebarY above, but that has the first rebar located

at X = 400mm, the following code could be used: the difference from Figure 7.12 can be

seen in Figure 7.13. Note that the previous rebar in the Y-direction have been removed

from the scene and the new array added.

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Listing 7.13: Application Specific Physical Modeling Component Example, Part V

15 % Change s t a r t i n g po in t o f y−d i r rebar ( us ing copy o f mScene2 )

16 newRebarY = rebar (’rebarY’ , deck , ’y’ , 16 ,50 ,120 , ’square’ , 4 00 ) ;

17 mSceneTmp = mScene2 − rebarY + newRebarY ;

18 render (mSceneTmp ) ;

Figure 7.13: Rebar Mesh with Specified Initial Rebar Location

The rebar class modify method allows for a few changes to rebar objects. Like the

previous objects, rebar objects can be modified before or after insertion into the scene.

They are exemplified here assuming modification of the object after insertion into the

scene. The entire rebar array can be shifted laterally relative to its current position by

passing in the shift (in mm by default).

mSceneTmp = modify ( scene ob ject , r ebar id , ’shift’ , s h i f t amount ) ;

example : % Sh i f t rebar running in the Y−d i r by 10mm in the X−d i r

mSceneTmp = modify (mScene , ’barY’ ,’shift’ , 1 00 ) ;

The amount of cover over the rebar can be increased or decreased.

mSceneTmp = modify ( scene ob ject , r ebar ob j ec t id , . . .

’changeCover’ , cover i n c r e a s e or dec r eas e ) ;

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example : % Lower the depth o f the Y−d i r rebar ( t h e r e f o r e i n c r e a s i n g

% the cover ) by 50mm

mSceneTmp = modify (mScene , ’barY’ ,’changeCover’ , 5 0 ) ;

The diameter of the rebars in the array can be changed by passing in a new rebar diameter.

Incremental increase to the rebar diameter is not supported. Note that the individual

bars are resized from their grip so the amount of cover may need to be adjusted.

mSceneTmp = modify ( scene ob ject , r ebar id , ’changeDiam’ , new diam ) ;

example : % Increase the diameter o f the Y−d i r rebar to 20mm

mSceneTmp = modify (mScene , ’barY’ ,’changeDiam’ , 2 0 ) ;

7.3.2 The Aggregate Class

The Aggregate Class is an application specific class developed for civil infrastructure.

It builds off of the ellipsoid class by generating a single object that contains a series of

ellipsoids of random size, randomly scattered throughout a container to fill a particular

volume. While this is often used to scatter aggregate through the computational area, it

can also be used to insert air voids or a collection of ellipsoids of any material. It should

be noted, that when there is a high concentration of small-sized aggregate, the time it

takes to insert all the pieces and to render the scene grows with the number of pieces

within the aggregate.

The aggregate class attributes are an object descriptor (ID), the previously defined

object that contains the aggregate, the minimum and maximum size (diameter) of the

aggregate (in mm by default), the volume percentage of the container the aggregate

occupies, and the material from which the aggregate is composed as shown in the class

diagram of Figure 7.14. The aggregate object is instantiated as follows:

agg = aggregate ( id , conta iner ,min size ,max size , percent , mate r i a l ) ;

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Container

Minimum radius

Maximum radius

Percent container volume

aggregate

Radius (X, Y, Z)

ime_ellipsoidObject ID

Material

Grip location

Grip description

Render information

ime_shape

Figure 7.14: Class Diagram for Aggregate

Consider an aggregate object labeled ’myAgg’ contained in the deck (a previously

defined concrete cuboid object). The size of each piece of aggregate is between 15mm

and 30mm and the aggregate comprises 5% of the deck. Additionally, the aggregate is

constructed of the material type aggregate (a defined material similar to concrete). The

same method could be used to insert air voids in the concrete. For air voids, use the

material type “air”. The aggregate object is created and inserted into the scene using

the following code. The resultant scene is shown in Figure 7.15.

Listing 7.14: Application Specific Physical Modeling Component Example, Part VI

20 % Create agg rega te o b j e c t

21 agg = aggregate ( ’myAgg’ , deck , 15 , 30 , 5 , ’aggregate’ ) ;

22 mScene2 = mScene2 + agg ;

23 render (mScene2 ) ;

The aggregate class constructor computes the volume of the container and the desired

volume of aggregate fill. In the example shown in Figure 7.15 the concrete deck volume

is 0.4m3 and the aggregate occupies 5% or 0.02m3 of the deck. The algorithm picks a

random location vector (within the container extents) and random radius vector (within

the user specified range) for each piece of aggregate, places the aggregate into the scene

and subtracts the volume of that aggregate chunk from the amount to fill. This process

can result in lengthy computational times and high memory requirements for the 3D

93

Figure 7.15: Addition of Aggregate

scene, not to mention lengthy render times, until the scene is discretized. The aggregate

object in this example (Figure 7.15) resulted in over 3,300 discrete aggregate pieces.

However, once the scene is discretized, as will be discussed in Section 7.5, the required

render time and storage space are equivalent to a scene that does not contain aggregate.

7.3.3 The Rough Surface Class

The Rough Surface class is an application specific class developed for civil infrastructure.

It transforms the upper surface of a cuboid (layer) from a smooth surface to a rough

surface. ISMME implements a Perlin Noise algorithm [47] to create a surface with a

pseudo-random appearance. The Perlin Noise algorithm generates noise values for each

position in space that change smoothly as the space is traversed. For other applications,

these noise values often represent color to create images such as clouds or textures (such

as wood grain or marbling) where the appearance of randomness is important. For more

information on Perlin Noise the reader is referred to several informative tutorials [48,49].

For the purpose of the rough surface class, Perlin Noise generates the elevation value

(Z coordinate) for a particular X & Y coordinate pair. This gives the appearance of a

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smoothly changing rough surface. The rough surface takes as input the previously defined

object to receive the rough surface, an object description (ID), the material that fills in

the dips in the rough surface, and the maximum peak and valley of the rough surface as

shown in Figure 7.16. The user more familiar with Perlin noise can pass in variables to

control the Perlin noise output such as interpolation smoothness and number of summed

octaves. The rough surface object is instantiated as follows:

rS = roughSurf ( conta iner , id , dip f i l l mater ia l , max peak ) ;

Container

Dip fill material

Maximum peak

roughSurf Object ID

Material

Grip location

Grip description

Render information

ime_shape

Figure 7.16: Class Diagram for roughSurf

Having a user-defined attribute that specifies the material to fill the surface dips allows

the rough surface to function as a rough interface. Consider the model in Figure 7.15.

For clarity, remove the aggregate from the model and add a rough surface, labeled as

’roughSurf’, onto the asphalt overlay (previously defined overlay object) with a maximum

peak of 12mm. Additionally, transform the interface between the asphalt and concrete

layers to a rough interface, labeled as ’roughInter’ with a maximum peak of 12mm. These

changes are made using the following code. The resultant scene is shown in Figure 7.17.

The rough surface and interface will be more obvious when the volume is discretized.

Note that “mScene3” is the scene object with the rough surface (and no aggregate) so

that the aggregate model, “mScene2”, is preserved for future examples.

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Listing 7.15: Application Specific Physical Modeling Component Example, Part VII

25 % Create rough sur face and rough a spha l t / concre te i n t e r f a c e

26 rS = roughSurf ( over lay , ’roughSurf’ ,’freespace’ , 1 2 ) ;

27 r I = roughSurf ( deck , ’roughInter’ ,’asphalt’ , 1 2 ) ;

28 mScene3 = mScene2 − agg + rS + r I ;

29 render (mScene3 ) ;

Figure 7.17: Addition of Rough Surface & Interface

7.3.4 Application-Specific Physical Modeling Code Listing

Sections 7.3.1 through 7.3.3 have detailed an example physical modeling scene constructed

using general and application specific physical modeling components. The code used to

create and display the scenes in this example (shown previously as snippets in shaded

boxes) is listed below in its entirety.

Listing 7.16: Application Specific Physical Modeling Component Example

1 mScene2 = scene ( [ 2000 1000 1000 ] ) ;

2 deck = ime cuboid (’deck’ , [ 0 0 200 ] , ’concrete’ , [ 2000 1000 2 0 0 ] ) ;

3 over l ay = ime cuboid (’overlay’ , [ 0 0 400 ] , ’asphalt’ , [ 2000 1000 6 0 ] ) ;

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4 mScene2 = mScene2 + deck + over l ay ;

5 render (mScene2 ) ;

7 % Bui ld rebar in y−d i r e c t i o n

8 mRebarY = rebar (’barY’ , deck , ’y’ , 16 ,50 ,120 , ’square’ ) ;

9 % Bui ld rebar in x−d i r e c t i o n

10 mRebarX = rebar (’barX’ , deck , ’x’ , 13 ,65 ,200 , ’circle’ ) ;

11 % Add rebar to scene

12 mScene2 = mScene2 + mRebarY + mRebarX ;

13 render (mScene2 ) ;

15 % Change s t a r t i n g po in t o f y−d i r rebar ( us ing copy o f mScene2 )

16 newRebarY = rebar (’rebarY’ , deck , ’y’ , 16 ,50 ,120 , ’square’ , 4 00 ) ;

17 mSceneTmp = mScene2 − rebarY + newRebarY ;

18 render (mSceneTmp ) ;

20 % Create agg rega te o b j e c t

21 agg = aggregate (’myAgg’ , deck , 15 , 30 , 5 , ’aggregate’ ) ;

22 mScene2 = mScene2 + agg ;

23 render (mScene2 ) ;

25 % Create rough sur face and rough a spha l t / concre te i n t e r f a c e

26 rS = roughSurf ( over lay , ’roughSurf’ , ’freespace’ , 1 2 ) ;

27 r I = roughSurf ( deck , ’roughInter’ , ’asphalt’ , 1 2 ) ;

28 mScene3 = mScene2 − agg + rS + r I ;

29 render (mScene3 ) ;

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7.4 Material Property Definition

Until this point, all of the materials discussed have only been the names of the materials:

“freespace”, “concrete”, “rebar”, etc. Creation of a 3D scene is not dependent on any

material characteristics; ISMME simply needs to know what material is being added to

the scene. Before the model can be discretized, information about the materials must be

defined. To this end, ISMME contains two types of objects: the material object itself and

a library of material types which will be used in simulation. Additionally, the ISMME

tool contains a default material library but the user will, most likely, want to modify the

properties of existing materials.

7.4.1 The Material Object Class

The Material Object Class contains information about a particular material used in the

physical model. It includes the name of the material (for example ’freespace’, ’concrete’,

’rebar’, etc.), a material ID number, and a tag attribute. For the purposes presented

here, it also contains electromagnetic properties of permittivity (eps, F/m), conductivity

(sig, mhos/m), and permeability (mu, H/m), a material ID number, and a tag attribute.

By default, the tag attribute is set to be the same as the material name. The material ID

plays an important role when the 3D physical model is to be discretized: it determines

which material ’cuts through’ which. Materials with a higher material ID cut through

materials with a lower ID. For example, by default, freespace has a material ID of 1 and

concrete has a material ID of 2. Therefore, when the model is discretized, the concrete

replaces the freespace. Similarly, rebar has a default material ID of 10, so the rebar will

cut through the concrete. This material ID gives the user the ability to control dominance

of various materials. In general, the Material Object class is instantiated as follows:

mMat = matObj (name , id , eps , s ig , mu) ;

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By default, the Material Object constructor returns a material object that has the same

properties as free space. The following two instantiations are equivalent:

mMat = matObj ;

mMat = matObj ( ’freespace’ , 1 , 8 . 85 e−12 ,0 ,4 e−7∗pi ) ;

Consider the creation of a concrete material object. The material ID should be 2 so the

concrete will cut through the freespace (this can be changed later if necessary). It should

have a relative permittivity of 9 with conductivity and permeability equal to free space.

Since sig and mu are the same as free space, the values do not need to be passed into the

material object constructor.

concMat = matObj (’concrete’ , 2 , 9∗8 .85 e−12);

The material object class has a modify method to allow the material ID, name

and electromagnetic properties to be changed. Assume that we want to create an

object similar to the previously created concrete object. Because the physical model

contains different types of concrete, this new material will be called “concrete2” instead

of “concrete”. Additionally, the new concrete should have a higher priority than the

previous concrete, assume a material ID of 7. A new object is created so that the

previous one is maintained. The name and material ID can be modified as follows:

concMat2 = modify ( concMat , ’changeName’ ,’concrete2’ ) ;

concMat2 = modify ( concMat2 , ’changeID’ , 7 ) ;

If it is decided that this new concrete is contaminated, the electromagnetic properties

should change. Assume that the relative permittivity of the new concrete is 13 and the

conductivity is 1e-3 mhos/m. Only the material properties being changed need to be

passed into the modify method. They can be passed in in any order but the property

name (’eps’, ’sig’, or ’mu’) should be passed in as a string and the value of that property

should immediately follow. These can be modified as follows:

concMat2 = modify ( concMat2 , ’modProp’ ,’sig’ ,1 e−3,’eps’ , 13∗8.85 e−12);

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7.4.2 The Material Library Class

The Material Library class is a list of all the materials used in the simulation. The

default library contains seven materials of interest to civil infrastructure application,

listed in Table 7.1. Note how there are two nearly identical materials, one is named

’freespace’ and the other ’voidAir’, that vary only by name and material ID. It was

discussed previously that this material ID, or priority, dictates which material will cut

through which when discretized. Free space has the lowest material ID (or priority) since

it primarily functions as the background to a physical model. However, in order to have

an air void included in the concrete, for example, it must have a higher material ID than

the concrete. Therefore the voidAir material object is included in the library. By default,

all materials take precedence over free space, but the air void cuts through all materials

except for rebar.

Table 7.1: Default Material Library Contents

ID Material NameRelative Conductivity Relative

Permittivity (mhos/m) Permeability1 freespace 1 0 12 concrete 9 0 13 aggregate 7 0 14 contamConc 12 1e-2 16 asphalt 5 0 19 voidAir 1 0 110 rebar 1e30 10e7 1

The default library can be obtained by passing no variables to the constructor. It can

be instantiated using either of the following:

mLib = matLib ;

mLib = matLib ( ) ;

Furthermore, the material library can be extended by the user.

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Once materials have been added to the library, their electromagnetic properties can

be modified using the ’modProp’ option to the modify method. The behavior is similar

to modifying the properties of a material object but the library object also gets passed

to the method. The “concrete” material object has a conductivity of 0 mhos/m. Assume

that the conductivity needs to change to 5e-4 mhos/m. This can be done using the

following:

mLib = modify (mLib , ’modProp’ , ’concrete’ , ’sig’ ,5 e−4);

7.5 Discretization of the 3D Volume: The Slice Class

In order to simulate wave propagation using the 2D FDTD method, the 3D model must

be discretized into a 2D slice. The 2D slice is a cross-section of the 3D model, represented

by a matrix of material ID numbers that index into the material library. The Slice Class

is responsible for generating a slice. The ISMME tool, as will be discussed later, also

provides functionality to create a 1D or 3D discretization.

The slice constructor takes as input the 3D scene object, the coordinate at which to

create the slice, the coordinate plane of the slice (x, y, or z), the resolution of the slice (in

mm by default), and an optional user defined material library object. The resolution of

the slice will be the resolution used in the FDTD simulation. If no resolution is specified,

1mm is chosen by default. If the material library object is not included in the constructor

call, the default library is assumed. In general, a slice object is instantiated as follows:

mSlice = im e s l i c e ( scene ob ject , s l i ce at , keep constant (x or y ) , . . .

r e s o l u t i on , mate r i a l l i b r a r y ob j ec t ) ;

Once a slice object has been obtained, it can be rendered in the same way as the 3D

scene object is: render(mSlice);.

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Additionally, the ISMME tool provides a means to view the location of the slice in

the context of the 3D volume. This can be done using the previewSlice method by

passing in the 3D scene, the coordinate at which to create the slice, the direction in which

the slice is constant (this is similar to the creation of the slice object but the resolution

and material library object are not required). The method will render the 3D scene plus

a translucent plane representing the 2D slice. The previewSlice method is called as

follows:

p r ev i ewS l i c e ( scene ob ject , s l i ce at , keep constant ( ’x’ or ’y’ ) ) ;

Recall the example presented in Section 7.2 (Listing 7.8). Consider discretizing the

model by taking a slice at Y=275mm with a resolution of 1mm. Assume the default

material library contains the correct materials and properties of the materials. The

preview of the slice (Figure 7.18(a)) and the slice object (Figure 7.18(b)) can be obtained

with the following:

p r ev i ewS l i c e (mScene1 , 275 , ’y’ ) ;

mSl ice1 = ime s l i c eOb j (mScene1 , 275 , ’y’ ) ;

r ender ( mSl ice1 )

If instead it is necessary to use the previously constructed, user-defined material library

object (“mLib”), the slice object can be instantiated as follows retaining the 1mm

resolution:

mSlice1 = ime s l i c eOb j (mScene1 , 275 , ’y’ , 1 ,mLib ) ;

The examples presented in Section 7.3 (Listing 7.16) require a discretization of 1mm

(default) and the default material library. For this example, preview the slice and slice

the model containing aggregate (Figure 7.15) at Y=400mm and the model with the

aggregate removed and rough surface added (Figure 7.17) at X=1000mm. Results can

be produced using the following code. The resultant scene is shown in Figure 7.19.

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(a) Physical Model (see Fig. 7.9(a))

X (mm)

Z (

mm

)

Slice at: y=275 mm, X−Res: 1 mm, Z−Res: 1 mm, Simulation Size: 501 x 501

100 200 300 400 500

100

200

300

400

500

(b) Slice at Y=275mm

Figure 7.18: Discretization of Example 1 Physical Models

p r ev i ewS l i c e (mScene2 , 400 , ’y’ ) ;

mSl ice2 = ime s l i c eOb j (mScene2 , 400 , ’y’ ) ; r ender ( mSl ice2 )

p r ev i ewS l i c e (mScene3 , 1000 , ’x’ ) ;

mSl ice3 = ime s l i c eOb j (mScene3 , 1000 , ’x’ ) ; r ender ( mSl ice3 )

Selection of resolution can be important in the simulation. If the resolution is low, the

shape of the inclusions can be changed in the discretization, and small enough inclusions

can be neglected. As an example, consider the aggregate physical model (mScene2,

Figure 7.15) sliced at X=1000mm using 1mm, 5mm and 10mm resolution. Results

generated with the following code can be seen in Figure 7.20.

p r ev i ewS l i c e (mScene2 , 1000 , ’x’ ) ;

mSl ice2a = ime s l i c eOb j (mScene2 , 400 , ’y’ , 1 ) ; r ender ( mSl ice2a )

mSlice2b = ime s l i c eOb j (mScene2 , 400 , ’y’ , 5 ) ; r ender ( mSlice2b )

mSlice2c = ime s l i c eOb j (mScene2 , 400 , ’y’ , 1 0 ) ; r ender ( mSl ice2c )

At 1mm resolution (Figure 7.20(b)), the cross section of the rebar is circular and the

aggregate elliptical. Recall that the rebars shown here (running in the X-direction) have a

diameter of 13mm. At 5mm resolution (Figure 7.20(c)), the aggregate and rebar become

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(a) Physical Model (see Fig. 7.15)

X (mm)

Z (

mm

)

Slice at: y=400 mm, X−Res: 1 mm, Z−Res: 1 mm, Simulation Size: 1001 x 2001

200 400 600 800 1000 1200 1400 1600 1800 2000

200

400

600

800

1000

(b) Slice at Y=400mm

(c) Physical Model (see Fig. 7.17)

Y (mm)

Z (

mm

)

Slice at: x=1000 mm, Y−Res: 1 mm, Z−Res: 1 mm, Simulation Size: 1001 x 1001

200 400 600 800 1000

200

400

600

800

1000

(d) Slice at X=1000mm

Figure 7.19: Discretization of Example 2 Physical Models

more irregularly shaped. At 10mm resolution (Figure 7.20(d)), the rebars are represented

as squares and the aggregate are very pixelated, with a majority of aggregate having

rectangular shape. Additionally, several of the smaller aggregate pieces have disappeared

due to the coarse discretization. While the level of detail required in a physical model is

very important, it is not the only requirement when selecting a spatial resolution. The

FDTD simulation parameters must meet stability and accuracy requirements detailed in

later sections.

The simulations presented will assume that the spatial resolution is the same in each

of the 2D directions (∆x = ∆z or ∆y = ∆z). However, the ISMME physical modeling

tools allow slices where this assumption is not valid. To obtain a 2D slice with varying

104

(a) Physical Model (see Fig. 7.15)

Y (mm)

Z (

mm

)

Slice at: x=1000 mm, Y−Res: 1 mm, Z−Res: 1 mm, Simulation Size: 1001 x 1001

200 400 600 800 1000

200

400

600

800

1000

(b) 1mm Resolution

Y (mm)

Z (

mm

)

Slice at: x=1000 mm, Y−Res: 5 mm, Z−Res: 5 mm, Simulation Size: 201 x 201

200 400 600 800 1000

200

400

600

800

1000

(c) 5mm Resolution

Y (mm)

Z (

mm

)

Slice at: x=1000 mm, Y−Res: 10 mm, Z−Res: 10 mm, Simulation Size: 101 x 101

200 400 600 800 1000

200

400

600

800

1000

(d) 10mm Resolution

Figure 7.20: Using Different Resolutions for 2D Slice at X=1000mm

resolution, a vector is passed to the slice constructor. Consider the physical model scene

and slice in Figure 7.18. Instead of a uniform spatial resolution of 1mm, the following

code can be used to generate a slice with a vertical resolution (∆z) of 10mm and a slice

with a vertical resolution of 20mm. In both slices, the horizontal resolution (∆x) is 1mm.

The results can be seen in Figure 7.21. Note that in the case of ∆x = ∆z = 1mm the

computational matrix is 501x501, when ∆z is changed to 10mm it is 51x501, and when

∆z is changed to 20mm it is 26x501.

105

mSlice1a = ime s l i c eOb j (mScene1 ,275 , ’y’ , [ 1 1 0 ] ) ; r ender ( mSl ice1a )

mSlice1b = ime s l i c eOb j (mScene1 ,275 , ’y’ , [ 1 2 0 ] ) ; r ender ( mSlice1b )

X (mm)

Z (

mm

)

Slice at: y=275 mm, X−Res: 1 mm, Z−Res: 10 mm, Simulation Size: 51 x 501

100 200 300 400 500

100

200

300

400

500

(a) ∆x = 1mm, ∆z = 10mm

X (mm)

Z (

mm

)

Slice at: y=275 mm, X−Res: 1 mm, Z−Res: 20 mm, Simulation Size: 26 x 501

100 200 300 400 500

100

200

300

400

500

(b) ∆x = 1mm, ∆z = 20mm

Figure 7.21: Discretization Using Non-square Elements

The modify method to the slice object allows the 2D matrix representing the physical

slice to be modified in two ways. Individual points in the matrix can be replaced or the

entire slice can be modified. An individual point can be replaced using the ’editSlicePt’

option to the modify method, with the following format:

mSlice = modify ( s l i ce obj , ’editSlicePt’ , [ row column ] , mate r i a l id ) ;

Note that when editing the matrix for an existing slice, the material ID must be used

since the model has already mapped the material name from the 3D physical model to the

material ID for the 2D discretization. Additionally, since the discretization is complete,

and the shape hierarchy resolved, the material numbers for freespace and voidAir can

be used interchangeably. As an example, to change point (row, column) = (200, 100) in

Figure 7.18(b) to freespace (material id=1), the following code could be used:

mSlice1 = modify ( mSlice1 , ’editSlicePt’ , [ 200 100 ] , 1 ) ;

At times, it may be useful to be able to replace an entire slice with another matrix.

This enables a matrix generated outside of the ISMME tool, perhaps as part of a different

106

simulation code, to be input into ISMME. The matrix must consist of material ID

numbers appropriate for ISMME so mapping between the two sets of material ID matrices

may be necessary prior to replacing the slice. For example, assume that the user has been

provided a matrix consisting of free space and concrete (called ’mMtx’) having a spatial

resolution of 5mm (so the physical separation between entries in the matrix is 5mm).

The matrix is such that everywhere there is free space, the matrix contains a 1 (which is

consistent with the ISMME default material ID), and everywhere there is concrete the

matrix contains a 4 (by default ISMME assumes concrete has a material ID of 2). A

quick mapping can be done to change the 4 to a 2:

[ rows , c o l s ] = find (mMtx0==4); % f ind a l l i n s t anc e s o f 4

mMtx( rows , c o l s ) = 2 ; % rep l a c e 4 wi th 2

The user can then obtain a blank slice object, insert the 2D matrix and set the resolution

(in mm by default) using the ’replaceSlice’ option to the modify method:

mSlice = im e s l i c e ;

mSl ice = modify ( mSlice , ’replaceSlice’ , mMtx, 5 ) ;

The slice object represents the discretized 3D physical model, including information

about its material properties. This is the final result of the ISMME physical modeling

tools. The 2D slice can then be used in the ISMME simulation tools. It also bears

mentioning that although the focus is on 2D simulation, the ISMME tool can also

construct 1D and 3D discretization of the physical model. These will be discussed briefly

before continuing on to the simulation tools.

Obtaining a 1D physical model from the 3D physical model scene is similar to the

previously described 2D slice. However, the constructor is “ime slice1d” instead of

“ime slice” and, instead of passing the X or Y direction that remains constant, the X and

Y coordinates of the slice get passed into the constructor. Similarly, the previewSlice

method can represent the 1D discretization. Consider the model in Figure 7.18 which

107

was sliced at Y=275mm. It is desirable to obtain a 1D discretization at X=150mm

and Y=275mm using the default resolution (1mm). The following code will display the

preview and rendered 1D discretization which can be seen in Figure 7.22:

x = 150; y = 275; % lo c a t i on o f the s l i c e in mm

p r ev i ewS l i c e (mScene1 , [ x y ] ) ;

mSlice1d = ime s l i c e 1d (mScene1 , [ x y ] ) ;

r ender ( mSlice1d ) ;

(a) Physical Model (see Fig. 7.9(a))

Z (

mm

)

Slice at: x=150 mm, y=300 mm, Res: [1 1 1] mm Simulation Size: 501 x 1

100

200

300

400

500

(b) Slice at X=150mm, Y=275mm

Figure 7.22: 1D Discretization

Obtaining a 3D discretization of the physical model is done with the “ime slice3d”

constructor by passing in the scene object and the resolution (in mm by default). For

example, a 3D discretization of “mScene1” with a 10mm resolution is obtained as follows:

r e s = 10 ;

mSlice3d = ime s l i c e 3d (mScene1 , r e s ) ;

108

Chapter 8

ISMME Sensor Modeling &

Simulation Tool

As introduced in Chapter 6 and summarized in Figure 8.1, the ISMME tool contains

a physical modeling component (discussed in Chapter 7) and a simulation component.

The simulation component of the ISMME tool provides structure to the data that are

required to execute a simulation. This structure includes information about the geometry

of model, the electromagnetic properties of the materials, and sensors contained within

the computational area.

Create scene

(3D volume)

Add components in

physical model

Define material

properties

Discretize

3D volume

Create sensor

object

Create simulation

object

Run

simulation

PHYSICAL MODELING TOOLS SIMULATION TOOLS

Figure 8.1: ISMME Tool Overview

The sensor modeling and simulation tools provide capability to construct a sensor

object that contains information about the type and location of the sensor and the

excitation signal, if applicable. ISMME combines these sensor objects with the previously

109

created slice object under a larger simulation object. The simulation object contains all

the information needed to run the simulation and collect the proper output.

For the example presented in this chapter, a simple model is created using the

following MATLAB code. The associated 2D slice at 2mm resolution is shown in

Figure 8.2. The reader should be reminded that the classes presented herein, and the

methods belonging to the classes, were developed by the author.

Listing 8.1: Simulation Tools Example, Part I

1 % Create s l i c e o f p h y s i c a l model

2 mScene = scene ( [ 1000 1000 1000 ] ) ;

3 deck = ime cuboid (’deck’ , [ 0 0 100 ] , ’concrete’ , [ 1000 1000 2 0 0 ] ) ;

4 over l ay = ime cuboid ( ’overlay’ , [ 0 0 300 ] , ’asphalt’ , [ 1000 1000 6 0 ] ) ;

5 mRebarY = rebar ( ’barY’ , deck , ’y’ , 16 ,50 ,120 , ’circle’ ) ;

6 mRebarX = rebar ( ’barX’ , deck , ’x’ , 13 ,65 ,200 , ’circle’ ) ;

7 mScene = mScene + deck + over l ay + mRebarY + mRebarX ;

8 mSlice = s l i c eOb j (mScene ,250 , ’y’ , 2 ) ;

X Grid Cooridnate

Z G

rid C

oorid

nate

100 200 300 400 500

100

200

300

400

500

Figure 8.2: Example Physical Model Slice

110

8.1 Input Signals

Each sensor that functions as a transmitter requires an excitation signal. In the ISMME

simulation tools, the user can specify this excitation function for each transmitter.

Alternatively, two built-in functions are provided to generate input signals. A Gaussian

pulse can be specified with the getGauss function. It requires the -3dB bandwidth

in hertz, the time the peak of the pulse should occur in seconds, the number of time

steps or the length of the desired signal, and the time step in seconds. The time step

should be the same as that which will be used in the simulation. The additional ’doPlot’

boolean variable triggers the generation of time domain and frequency domain plots of

the resulting signal. The function is called as follows:

mySig = getGauss ( bandwidth , peakTime , timeSteps , dt , doPlot ) ;

For example, to create a 2GHz bandwidth Gaussian pulse with a peak time of 1ns, a

time step of 2ps, and a duration of 3000 time steps use the following code:

mySig = getGauss (2 e9 , 1e−9, 3000 , 2e−12);

Similarly, a modulated Gaussian pulse can be obtained using the getModGauss

function. The only difference is that the center frequency of the modulated signal must

be passed in to the function. The function is called as follows:

mySig = getModGauss ( cen te r frequency , bandwidth , peakTime , . . .

t imeSteps , dt , doPlot ) ;

A 2GHz center frequency and 2GHz bandwidth Gaussian pulse with a peak time of 1ns,

a time step of 2ps, and a duration of 4000 time steps can be computed with the following

code. It results in the signal shown in Figure 8.3.

Listing 8.2: Simulation Tools Example, Part II

10 % Obtain de s i r e d e x c i t a t i o n s i g n a l

11 mySig = getModGauss (2 e9 , 2e9 , 1e−9, 4000 , 2e−12, 1 ) ;

111

0 1 2 3 4 5 6 7 8

x 109

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Frequency (Hz)

dB

Bandwidth

(a) Frequency Domain

0 1 2 3 4 5 6 7 8

−0.2

0

0.2

0.4

0.6

0.8

1

Time (ns)

Am

plitu

de

Pulse: CF = 2 GHz, BW = 2 GHz, PEAK TIME = 1 ns

(b) Time Domain

Figure 8.3: Example Excitation Signal

8.2 The Sensor Class

ISMME supports three types of sensors: hard transmitters, soft transmitters, and

receivers. Transmitters excite the model by specifying the electric field at the sensor

location. The electric fields are recorded at all sensor locations, regardless of whether

they are a transmitter or receiver. When a sensor is a hard transmitter, the total field

is specified as the value of the excitation signal for all time at the excitation point. Any

waves propagating toward the hard transmitter will get reflected. When the sensor is a

soft transmitter, the computed electric field is added to the value of the excitation at the

sensor point. When the input sensor is a receiver, there is no excitation at the sensor

location, but the electric field is recorded.

The Sensor Class defines sensor objects to be used in simulation. The attributes

required to instantiate a sensor object are the unique sensor descriptor (ID), the grid

coordinates of the sensor in the physical model slice, and the type of sensor. If the sensor

is a transmitter, the excitation signal is also required. In general, a sensor is instantiated

as follows:

s = sen sor ( ID , l o c a t i o n [ Z , X or Y] , . . .

type ( ’soft’ , ’hard’ or ’receive’ ) , s i g n a l ( i f t r an smi t t e r ) ) ;

112

Assume a transmitter is excited with the previously created modulated Gaussian pulse

and that it has the identifier ’transmit’. It is a soft transmitter located at X coordinate

200 and Z coordinate 125. A receiver is also be instantiated with an identifier ’recvr’ and

located at X coordinate 300 and Z coordinate 125. The sensor excitation is plotted in

the time domain using the render method. This can be done with the following code.

The results are shown in Figure 8.4:

Listing 8.3: Simulation Tools Example, Part III

12 s1 = sen sor ( ’transmit’ , [ 125 200 ] , ’soft’ ,mySig ) ;

13 s2 = sen sor ( ’recvr’ , [ 125 300 ] , ’receive’ ) ;

14 render ( s1 ) ;

500 1000 1500 2000 2500 3000 3500 4000

−0.2

0

0.2

0.4

0.6

0.8

1Sensor: transmit, Type: soft, Location: [125 200]

Time Step

Am

plitu

de

Figure 8.4: Rendered Sensor Object

The modify method allows attributes in sensor objects to be changed. A sensor

location can be moved using the ’moveTo’ option and new sensor coordinates. The type

of sensor can be changed using the ’changeType’ option along with either ’hard’, ’soft’

or ’receive’ to indicate the updated type. The excitation signal can be changed using

the ’replaceSig’ option and the new excitation. For example, the following code can be

used to create a new sensor, s3, from the previously instantiated sensor, s2. The sensor

is changed to a new height at Z grid coordinate 100. Its type is changed from a receiver

to a hard source with the same excitation as sensor s1.

113

s3 = modify ( s2 , ’moveTo’ , [ 100 2 0 0 ] ) ;

s3 = modify ( s3 , ’changeType’ ,’hard’ ) ;

s3 = modify ( s3 , ’replaceSig’ ,mySig ) ;

8.3 The Simulation Class

The Simulation class instantiates an object that combines slice objects (Section 7.5) and

the sensor objects (Section 8.2). This object contains all the information required to

run the simulation. The attributes required to instantiate a simulation object are an

output file name, the physical slice object, the time step for the simulation in seconds,

the number of time steps, and the sensor objects. In general, it is instantiated as follows:

mSim = s imu lat i on ( output f i l e name , s l i ce ob ject , time step , . . .

number o f time steps , s en sor ob j e c t s ) ;

The simulation object is displayed using the render method. The display looks very

similar to the results from the slice object render method because the physical model has

not changed. However, rendering a simulation object shows the location of the sensors.

In the rendered image, sensors that are receivers are indicated by ’o’, hard transmitters

are indicated by ’x’, and soft transmitters are indicated by ’+’.

Following the same example in this chapter, a simulation object is instantiated using

the previously defined slice object (mSlice) and the transmitter and receiver sensor objects

(s1 and s2). Assume a file name ’myOutput’, a time step of 2ps and a simulation run

of 4000 time steps. Multiple sensors can be input into a simulation object using a list

in curly braces, a MATLAB cell array. The simulation object is instantiated with the

following code. It is shown rendered in Figure 8.5:

114

Listing 8.4: Simulation Tools Example, Part IV

15 % Create the s imu la t i on o b j e c t

16 mSim = s imu lat i on (’myOutput’ , mSlice , 2e−12, 4000 , s1 , s2 ) ;

17 render (mSim ) ;

X Grid Cooridnate

Z G

rid C

oorid

nate

100 200 300 400 500

50

100

150

200

250

300

350

400

450

500

Figure 8.5: Example Simulation Object

The modify method allows several attributes of the simulation object to be changed.

The output name can be changed using the ’outputName’ option and the new file name.

The number of time steps can be changed using the ’timeSteps’ option with the new

number of time steps. The sensor list can be replaced using the ’replaceSensors’ option

and the new sensor list. Examples of these options are shown below:

mSim = modify (mSim, ’outputName’ , ’newOutput’ ) ;

mSim = modify (mSim, ’timeSteps’ , 5 00 ) ;

mSim = modify (mSim, ’replaceSensors’ , newSensor1 ) ;

The attributes of an individual sensor can also be changed once the simulation object

has been instantiated. A single sensor can be moved using the ’moveSensor’ option by

passing in the ID of the sensor to move and its new location. The type of a single sensor

can be changed using the ’changeSensorType’ option and the ID of the sensor and the

new sensor type (’hard’, ’soft’ or ’receive’). The excitation signal of an individual sensor

115

can be changed using the ’changeSensorSig’ option with the ID of the sensor and the new

signal. Examples of these options are shown below:

mSim = modify (mSim, ’moveSensor’ , ’transmit’ , [ 125 2 5 0 ] ) ;

mSim = modify (mSim, ’changeSensorType’ , ’transmit’ , ’hard’ ) ;

mSim = modify (mSim, ’changeSensorSig’ ,’transmit’ , newSignal ) ;

By default, the electric fields are recorded at all the sensor locations at each time

step. This output can be supplemented to include the fields recorded at every location

in the physical model at particular time steps, stored in a binary file. This is done

by modifying the simulation object with the ’saveAll’ option with a 3 element vector

of flags that enable the recording of the Ez, Hx and Hy fields. A 1 indicates that the

corresponding field should be recorded, and a 0 indicates that it should not be recorded.

For example, passing in [1 0 1] causes the simulation to record the Ez and Hy fields. By

default, fields specified in the ’saveAll’ attribute are recorded every 10 time steps. This

can be changed by passing the frequency (time steps) to record the field into the modify

method using the ’saveFreq’ option. For example, to modify the simulation object to

record the electric field every 50 time steps the following code is used:

Listing 8.5: Simulation Tools Example, Part V

19 % Save Ez f i e l d at a l l l o c a t i on s every 50 time s t e p s

20 mSim = modify (mSim, ’saveAll’ , [ 1 0 0 ] ) ;

21 mSim = modify (mSim, ’saveFreq’ , 5 0 ) ;

The modify method can also change the electromagnetic properties of the materials

included in the simulation. This can be particularly useful when running several

simulations on the same geometry while varying the electromagnetic properties. For

example, recall from Section 7.4.2 that the default conductivity (σ) of concrete is 0.

116

Once the simulation object has been instantiated, the conductivity can be be changed

to 5 × 10−4 using the following code:

mSim = modify (mSim, ’modProp’ , ’concrete’ , ’sig’ ,5 e−4);

8.4 Running the Simulation

With an instantiated simulation object, the simulation can be run simply with the run

method.

Listing 8.6: Simulation Tools Example, Part VI

23 run (mSim ) ;

By default, the ISMME simulation tools use the 2D FDTD algorithm (presented

in Chapter 4) and a computational area terminated by an eight layer PML (discussed

in Chapter 5). Recall that a limitation of the ISMME is that the implementation

presented in Chapter 4 assumes materials are non-dispersive. While materials can have

an associated conductivity, it is assumed not to be frequency dependent. When the run

method is called, the following general steps are performed:

I. Simulation parameters are checked to make sure the Courant condition (Section

4.5) is satisfied.

II. Material property matrices are created.

III. Excitation sources are set up. If the simulation duration is longer than the provided

excitation, the excitation is padded with zeros. If the simulation duration is shorter

than the provided excitation, only the required number of samples are taken from

the excitation.

IV. The PML parameters are determined from the material matrices.

V. The electric and magnetic fields are initialized.

117

VI. The FDTD equations in Equation (5.10) are looped through for each time step.

The Ez fields at all sensor locations are recorded. If applicable, the Ez, Hx, and

Hy fields at all locations in the model are written to a binary file.

8.5 The Simulation Output

After the simulation has finished running, relevant data is stored in the file name specified

in the simulation instantiation. MATLAB adds the ’.mat’ extension. In addition to

several variables used in the simulation, some variables of interest are stored in this file.

The original simulation object is stored as ’mSim’. This object can be reloaded from the

file, modified, and used to run a new simulation. The ’src’ variable contains the excitation

signal for each of the sensors specified in the simulation. The ’srcLoc’ variable contains

the location of each of the sensor in the simulation. The ’srcOut’ variable contains electric

field at every sensor location recorded at every time step. The variables ’src’, ’srcLoc’

and ’srcOut’ contain a series of row vectors with the order of rows consistent with their

order in the simulation object instantiation.

These variables can be easily loaded into MATLAB using the load function, passing

in the variable name. Assume that the previous example, the simulation with the file

name ’myOutput’ was run, and that a second simulation was run with similar parameters

but only containing free space. The second simulation results were stored in the file

’myOutputDC’, they provide the direct coupling signal to remove from the results from

the bridge deck case. Results can be plotted using the following code. They are shown

graphically in Figure 8.6(a).

Listing 8.7: Simulation Tools Example, Part VII

25 % load d i r e c t coup le output

26 dc = srcOut ;

27

118

28 % load response wi th deck and time s t e p

29 load myOutput srcOut dt

30 deck = srcOut ;

31

32 % crea te time vec tor in nanoseconds

33 t = [ 1 : s ize ( deck , 2 ) ] ∗ dt/1e−9;

34

35 % p l o t wi thout d i r e c t coup le s i g n a l

36 subplot ( 2 , 1 , 1 ) ; plot ( t , deck (1 , : ) − dc ( 1 , : ) , ’b’ ) ;

37 xlabel (’Time (nanoseconds)’ ) ; ylabel (’Amplitude’ ) ;

38 t i t l e (’Trace Recorded at Soft Transmitter’ ) ;

39

40 subplot ( 2 , 1 , 2 ) ; plot ( t , deck (2 , : ) − dc ( 2 , : ) , ’r’ ) ;

41 xlabel (’Time (nanoseconds)’ ) ; ylabel (’Amplitude’ ) ;

42 t i t l e (’Trace Recorded at Receiver’ ) ;

An alternate plot of the output data can be seen in Figure 8.6(b). The reflection

from the air/asphalt boundary occurs around 3.5ns, and the reflection from the

asphalt/concrete boundary occurs around 4.5ns. As expected, the responses recorded

at the soft transmitter and receiver are very similar for the layers. The reflection from

the layered media has a shorter path to travel from the surface to the transmitter than

from the surface to the receiver. This accounts for the earlier reflections in the signal

recorded at the transmitter. Neglecting the time shift, the signals begin to deviate with

the reflections from the rebar at about 5.5ns. This is due to differences in the structure

that the wave encounters for the different paths: one path from the transmitter to the

rebar layer and back to the transmitter, and one path from the transmitter to the rebar

layer to the receiver (see Figure 8.5).

119

1 2 3 4 5 6 7 8

−3

−2

−1

0

1

2x 10

−3

Time (nanoseconds)

Am

plitu

de

Trace Recorded at Soft Transmitter

1 2 3 4 5 6 7 8

−3

−2

−1

0

1

2x 10

−3

Time (nanoseconds)

Am

plitu

de

Trace Recorded at Receiver

(a) Responses at Two Different Sensor Locations

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

Time (nanoseconds)

Am

plitu

de

Trace Recorded at Receiver

Soft TransmitterReceiver

(b) Comparison of Sensor Responses

Figure 8.6: Example Output from the Simulation

Recall that the simulation object was modified to record the electric field everywhere

in the computational area every 50 time steps. These records were stored in a binary

file called ’myOutput Ez.bin’. If the magnetic fields, Hx & Hy, had been recorded,

’myOutput Hx.bin’ and ’myOutput Hy.bin’ would also exist. Information in these files

can be obtained using the extractMtx function:

mMtx = extractMtx ( fname , n ) ;

This function takes as input the binary file name (fname) and the number of the matrix

to be extracted (n). The extractMtx function outputs the value of the field everywhere

in the computational area. If the data was recorded every 50 time steps, and n is specified

as 2, the data represents the field everywhere at time step 100.

From the example simulation, snapshots of the value of the electric field everywhere

in the computational model for four times can be seen in Figure 8.7. The matrices for

time steps 1000 (n=20), 1500 (n=30), 2000 (n=40) and 2500 (n=50) correspond to 2, 3,

4 and 5 ns. They can be extracted using the following code:

120

44 mMtxA = extractMtx ( ’myOutput_Ez.bin’ , 2 0 ) ;

45 mMtxB = extractMtx ( ’myOutput_Ez.bin’ , 3 0 ) ;

46 mMtxC = extractMtx ( ’myOutput_Ez.bin’ , 4 0 ) ;

47 mMtxD = extractMtx ( ’myOutput_Ez.bin’ , 5 0 ) ;

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

X Grid Coordinate

Z G

rid C

oord

inat

e

−6

−4

−2

0

2

4

6

x 10−3

(a) T = 2.0ns

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

X Grid Coordinate

Z G

rid C

oord

inat

e

−6

−4

−2

0

2

4

6

x 10−3

(b) T = 3.0ns

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

X Grid Coordinate

Z G

rid C

oord

inat

e

−6

−4

−2

0

2

4

6

x 10−3

(c) T = 4.0ns

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

X Grid Coordinate

Z G

rid C

oord

inat

e

−6

−4

−2

0

2

4

6

x 10−3

(d) T = 5.0ns

Figure 8.7: Value of the Electric Field at Various Times

8.6 Sensor Modeling & Simulation Code Listing

This section has demonstrated how to create sensor and simulation objects using ISMME,

how to run simulations, and how to extract output. The code used to create and display

121

the scenes in this example (shown previously as snippets in shaded boxes) is listed

below in its entirety to show straightforward modeling using the ISMME tool. Similar

implementations are used to perform the 2D simulations in the following chapters.

Listing 8.8: Simulation Tools Example

1 % Create s l i c e o f p h y s i c a l model

2 mScene = scene ( [ 1000 1000 1000 ] ) ;

3 deck = ime cuboid (’deck’ , [ 0 0 100 ] , ’concrete’ , [ 1000 1000 2 0 0 ] ) ;

4 over l ay = ime cuboid (’overlay’ , [ 0 0 300 ] , ’asphalt’ , [ 1000 1000 6 0 ] ) ;

5 mRebarY = rebar (’barY’ , deck , ’y’ , 16 ,50 ,120 , ’circle’ ) ;

6 mRebarX = rebar (’barX’ , deck , ’x’ , 13 ,65 ,200 , ’circle’ ) ;

7 mScene = mScene + deck + over l ay + mRebarY + mRebarX ;

8 mSlice = s l i c eOb j (mScene ,250 , ’y’ , 2 ) ;

10 % Obtain de s i r e d e x c i t a t i o n s i g n a l

11 mySig = getModGauss (2 e9 , 2e9 , 1e−9, 4000 , 2e−12, 1 ) ;

12 s1 = sen sor (’transmit’ , [ 125 200 ] , ’soft’ ,mySig ) ;

13 s2 = sen sor (’recvr’ , [ 125 300 ] , ’receive’ ) ;

14 render ( s1 ) ;

15 % Create the s imu la t i on o b j e c t

16 mSim = s imu lat i on (’myOutput’ , mSlice , 2e−12, 4000 , s1 , s2 ) ;

17 render (mSim ) ;

19 % Save Ez f i e l d at a l l l o c a t i on s every 50 time s t e p s

20 mSim = modify (mSim, ’saveAll’ , [ 1 0 0 ] ) ;

21 mSim = modify (mSim, ’saveFreq’ , 5 0 ) ;

23 run (mSim ) ;

122

25 % load d i r e c t coup le output

26 dc = srcOut ;

28 % load response wi th deck and time s t e p

29 load myOutput srcOut dt

30 deck = srcOut ;

32 % crea te time vec tor in nanoseconds

33 t = [ 1 : s ize ( deck , 2 ) ] ∗ dt /1e−9;

35 % p l o t wi thout d i r e c t coup le s i g n a l

36 subplot ( 2 , 1 , 1 ) ; plot ( t , deck (1 , : ) − dc ( 1 , : ) , ’b’ ) ;

37 xlabel (’Time (nanoseconds)’ ) ; ylabel ( ’Amplitude’ ) ;

38 t i t l e ( ’Trace Recorded at Soft Transmitter’ ) ;

40 subplot ( 2 , 1 , 2 ) ; plot ( t , deck (2 , : ) − dc ( 2 , : ) , ’r’ ) ;

41 xlabel (’Time (nanoseconds)’ ) ; ylabel ( ’Amplitude’ ) ;

42 t i t l e ( ’Trace Recorded at Receiver’ ) ;

44 mMtxA = extractMtx ( ’myOutput_Ez.bin’ , 2 0 ) ;

45 mMtxB = extractMtx ( ’myOutput_Ez.bin’ , 3 0 ) ;

46 mMtxC = extractMtx ( ’myOutput_Ez.bin’ , 4 0 ) ;

47 mMtxD = extractMtx ( ’myOutput_Ez.bin’ , 5 0 ) ;

123

124

Chapter 9

Comparison of the Accuracy of 2D

versus 3D FDTD

Two dimensional models are quick to compute, but lack some important information.

In the bridge deck problem, for example, a three dimensional cylinder may be sliced

such that it will be represented as a rectangle in two dimensions, with its width varying

depending on the angle of the plane slicing it. Additionally, objects of finite extent

perpendicular to the slicing plane will be represented as having infinite extent in the

perpendicular direction. For example, rebars running parallel to a 2D slice would

be represented as a perfect electrical conducting (PEC) sheet with finite depth but

infinite width and length. Since electromagnetic waves would not pass through the rebar

layer, the rebars parallel to the 2D observation plane cannot be included in the 2D

model, and as a result, the scattering from parallel rebars is not simulated in the 2D.

Furthermore, a point source in 2D is actually a line source in 3D. The difference in

spreading between point and line sources will be addressed via excitation signal filtering

in the 2D simulations. It is expected that scenarios in which the 2D and 3D models

represent the same in-situ condition that the 2D results will best match the 3D results.

125

For the purposes of this chapter, this includes a layered media model (air, asphalt and

concrete) and rebar running perpendicular to the observation plane.

Using 2D simulations instead of 3D simulations saves significant computational time,

especially when performing multiple simulations to construct a B-scan or performing

model inversion techniques requiring iterative forward modeling. Using 2D simulations

to aid in the analysis of 3D field-collected data requires an examination of the accuracy

of 2D modeling by comparing FDTD simulations of reinforced bridge deck deterioration.

To this end, simulations for a healthy and a delaminated bridge deck are examined.

The healthy deck simulations are based on asphalt and concrete layered media both

with and without transverse and longitudinal rebar. Building off of the healthy bridge

deck model, three different delamination cases are presented representing different in-situ

deteriorated conditions. Results from the 2D and 3D models are presented and compared.

The accuracy trade-offs between the two models will be discussed and conclusions drawn

for which applications the 2D model may be adequate. The 2D simulations are run using

the ISMME tool (Chapter 6-8). The 3D simulations are run using code written by Adnan

Sahin, Carey Rappaport, John Talbot, Panos Kosmas and others that has been in use

at Northeastern University since 1997.

9.1 Computational Models

The 2D and 3D simulations presented implement Yee’s equations (Chapter 4). The

3D simulations can generate electric and magnetic fields with all polarizations. In

2D simulation, two modes are typically used - the transverse electric (TE) mode and

transverse magnetic (TM) mode. In the TE mode, there is no electric field perpendicular

to the computation plane; while in the TM mode, there is no magnetic field perpendicular

to the plane. The X-direction is considered to be along the length of the deck (the

direction in which vehicles would travel), the Y-direction is across the width of the deck

126

(across the traffic lanes) and the Z-direction represents the depth of the deck and air.

The X-Z plane composes the 2D physical model (distance along the deck versus depth).

Therefore, to simulate Y-polarized ground penetrating radar through a bridge deck in

two dimensions, the TM mode was chosen (TMy). GPR systems used for bridge deck

analysis are usually aligned polarized in the cross-track direction so that the strong

rebar scattering occurs periodically, as the GPR passes above a transverse rebar. With

longitudinal polarization, the GPR might be positioned directly above a longitudinal

rebar, which would obscure much of the sensing signal.

Both the 2D and 3D simulation code satisfy the accuracy requirement of having

at least 10 grid points per shortest wavelength. In the models presented containing

asphalt, concrete and air, the shortest wavelength occurs in concrete (represented with

a dielectric constant of 9). The highest frequency is considered to be 1.5GHz, which is

the center frequency (1.0GHz) plus 50% bandwidth, resulting in smallest wavelength of

6.7cm (Equations 2.5 and 2.6). Simulations presented herein use a spatial step of 0.6cm.

Stability of the FDTD formulation is met by ensuring that the spatial and temporal

resolution satisfy the Courant condition [50], which specifies the maximum time step

(∆t) in terms of spatial resolution (∆), the wave velocity (c) and the dimensionality

(D) of the computational model by: ∆t ≤ ∆/(c√

D). For the cases presented here, the

maximum wave velocity is equal to the speed of light through free space (≈ 3× 108m/s),

and square or cube grid elements are assumed (∆ = ∆x = ∆y = ∆z). The 3D Courant

condition is met for ∆ = 0.6cm if ∆t ≤ 11.5ps. The simulations presented here use

∆t = 5ps.

The simulations presented are for an open-region area and, therefore the compu-

tational grid must be truncated using an absorbing boundary condition to properly

represent an infinite scattering area. The 3D FDTD employs the first order and second

order Mur absorbing boundary condition (MUR’s ABC) [51] that includes the single

pole conductivity model [52, 53] for frequency dispersion medium based on the one-

127

way wave equation. The MUR’s ABC is straightforward to implement and does not

require storage of field components or extra ABC layers [54]. The 2D FDTD utilizes

Berenger’s Perfectly Matched Layer (PML) formulation as an absorbing boundary layer

(Chapter 5). The PML is an 8-layer region surrounding the computational area in which

artificial magnetic and electric losses are introduced to attenuate the signal and eliminate

reflection at the PML interface. In the 2D model used here, the PML layer has loss defined

by a polynomial grading [41] from 0 at the PML interface to a maximum loss (σmax) at

the extent of the computational model as given by Chapter 5, Equations (5.7) and (5.8).

9.2 Numerical Simulation Experiments

9.2.1 Excitation Filtering Process

Wave propagation in 3D is different from the propagation in 2D model. Consider the same

excitation function applied at a single grid point. In three dimensions, the wavefront is

spherical and the propagation is proportional to the inverse of the radial distance (≈ 1/r).

In two dimensions, the wavefront is circular and the propagation is proportional to the

inverse of the square root of the radial distance (≈ 1/√

r). The circular wave in 2D

is a cylindrical wave in 3D due to the inability to model variance in the Y-direction of

a 2D model. For the same excitation signal, this difference in propagation results in

different received pulse shapes. To overcome this difference and allow proper comparison

of the 2D and 3D responses, the 2D excitation signal was reconfigured to produce the

same response at a given receiver point over an infinite PEC plate as the 3D response

over the same plate. This process creates a filtered excitation signal that when used in

2D simulations results in a reflection from a PEC plate at the surface that matches the

reflection from the 3D simulation using the unfiltered excitation. However, it does not

account for the variation in propagation between 3D and 2D as the wave progresses into

128

the deck. It is expected that this variation will cause small discrepancies in reflections

that increase proportionally to depth in the deck.

The magnitude of the response collected in air depends on the media on both sides

of the layer interface in the bridge deck. Since the comparison between 2D and 3D will

be performed with GPR over different media (for example, concrete or asphalt), the

magnitude of the response will be different. Therefore, only the shape of the response

will be configured. The magnitude differences are not considered in the filtering process.

Simulations over a PEC (where the incident wave undergoes total reflection) were

performed in 2D and 3D using the same excitation (a pulse with center frequency and

bandwidth of either 1.0GHz or 0.5GHz). A filter was determined by taking the Fourier

transform of the responses and using the ratio of the 3D frequency response to the

2D frequency response. The product of the filter and the frequency domain excitation

is brought back into the time domain resulting in the new filtered excitation for the

2D simulations. This process is summarized in Figure 9.2. The excitation pulses for

the 1.0GHz and 0.5GHz center frequencies are shown in Figure 9.1a. The results from

a simulated 1.0GHz air-coupled antenna investigation of an air, asphalt and concrete

layered media (discussed in detail in the following sections) using the unfiltered excitation

in 2D and 3D and the filtered excitation in 2D are shown in Figure 9.1b. Note the

discrepancies in reflection amplitude caused by the variation in propagation between 3D

and 2D at the bottom surface of the deck (approximately 7.75ns).

9.2.2 Physical Model Examples

The 3D physical models are discretized from a 3D volume resulting in a 3D matrix

representation of the computational area. The model considered here has an overall size

of 157.8cm in each X-, Y- and Z-directions. For the 2D simulations, fields are computed

across the X-Z plane located at Y=79.2cm relative to the start of the computation model.

129

1 2 3 4 5 6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (ns)

Am

plitu

de

0.5GHz

2D Sig3D Sig

1 2 3 4 5 6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (ns)

Am

plitu

de

1.0GHz

2D Sig3D Sig

(a) Simulation Excitation Signals

3 4 5 6 7 8 9

−4

−3

−2

−1

0

1

2

x 10−4

Time (ns)

Am

plitu

de

Unfiltered 2D, NormalizedFiltered 2D Response3D Response

(b) Example Results from 2D Filtering

Figure 9.1: Simulation Excitation & 2D Filtering Results

2D FDTD system

h2(t)

3D FDTD system

h3(t)

y2(t)

y3(t)

x(t)

2D FDTD system

h2(t)

3D FDTD system

h3(t)

x(t)x’(t)

y3(t)f(t)

y3(t)x(t)

X( f )H2 ( f ) =Y2 ( f )

X( f )H3( f ) =Y3( f )

X( f )F( f )H2 ( f ) =Y 3( f )

F( f ) =Y3( f )/Y2 ( f )

x'(t) = F 1F( f )X( f )

Figure 9.2: Summary of Excitation Filtering Process where x(t) is the excitation, f(t) isthe filter, x′(t) is the configured excitation to the 2D FDTD system, Fourier transformFx(t) = X(f), and inverse Fourier transform x(t) = F−1X(f)

This 2D slice occupies the plane that includes the transmitter and receiver in the 3D

model, bisecting the delamination, when present.

130

Several physical models that represent in-situ conditions found in bridge decks are

examined. The base structure examined is an 18.0cm concrete bridge deck with a 2.4cm

asphalt overlay. The physical modeling parameters are summarized in Table 9.1.

Table 9.1: Physical Modeling Parameters

Asphalt Layer Thickness: 2.4cmAsphalt Properties: dielectric constant=5, conductivity=0,

relative permeability=1Concrete Layer Thickness: 18.0cmConcerte Properties: dielectric constant=9, conductivity=0,

relative permeability=1Transverse Rebar: diameter=1.8cm (approximately a #5 rebar)

spacing=12.6cm on center, concrete cover=4.8cmLongitudinal Rebar: diameter=1.2cm (approximately a #4 rebar)

spacing=20.4cm on center, concrete cover=6.6cmAir Void: Cylindrical air void with a 0.6cm thickness and

30.0cm diameter

The healthy and delaminated deck models investigated are summarized in Table 9.2.

The healthy models first compare only asphalt and concrete layers. Next, the models

include transverse rebars, and finally longitudinal rebars to form a rebar mesh are

incorporated. The healthy physical models are shown in Figure 9.3. The location of

the 2D slice is indicated in the representation of the 3D model. Both the 2D and 3D

simulation cases are modeling infinitely long and infinitely wide asphalt and concrete

layers, and infinitely long (in the Y-direction) transverse rebars. The 3D case also models

infinitely long (in the X-direction) longitudinal rebars. Note that due to discretization

steps of 0.6cm, the cross section of the transverse (oriented in the Y-direction) rebars

(1.8cm or 3 grid point diameter) is represented as a plus (+) and the cross section of

the longitudinal (oriented in the X-direction) rebars (1.2cm or 2 grid point diameter) is

represented as a square.

The unhealthy deck models place a consistently sized delamination at one of three

locations (Figure 9.4): between the asphalt and concrete layers, above the transverse

131

Table 9.2: Simulation Cases

CaseAsphalt & Cross Sectional Longitudinal

Air VoidConcrete Rebar Rebar *

A XB X XC X X XD X X X Between asphalt & concreteE X X X Above rebar arrayF X X X Below rebar array

* Longitudinal rebar is only modeled in the 3D simulations

rebar, or below the longitudinal rebar. In the 3D simulations, the delamination is modeled

as a disk (thickness 0.6cm, diameter 30.0cm). Since the 2D model is constructed along

the center of the delamination, the equivalent 2D representation is an infinitely long (in

the Y-direction) rectangular delamination. Note that Figures 9.4d, 9.4e and 9.4f are

presented in detail only. The entire 2D model is similar to Figure 9.3f differing only in

the region of interest.

9.2.3 Simulation Setup

In all simulations in this section, the transmitter and receiver pair are positioned with a

fixed separation of 3.0cm in the X-direction and the same Y or Z coordinates. In both the

2D and 3D cases, the transmitter and receiver occupy one grid point. As discussed earlier,

this results in a spherical wave in 3D and a cylindrical wave in 2D which is accounted for

by filtering the 3D excitation. The simulations approximate air-coupled antennas. They

are constructed with the transmitter/receiver pair 30.6cm above the surface of the deck

using an excitation of center frequency and bandwidth of 1.0GHz and of 0.5GHz. The

simulations are performed in a 263x263x263 computational region (157.8cm/0.6cm per

grid point) for a duration of 12.5ns (2500 time steps).

132

(a) Case A: No Rebar, 3D Model(b) Case B: Transverse Rebar,3D Model

(c) Case C: Rebar Mesh, 3DModel

X (cm),Down Track

Z (

cm),

Dep

th

0 25 50 75 100 125 1500

25

50

75

100

125

150Air

Asphalt

Concrete

Air

(d) Case A: No Rebar, 2D Model& Detail

X (cm),Down Track

Z (

cm),

Dep

th

0 25 50 75 100 125 1500

25

50

75

100

125

150

Rebar (typ.)

(e) Case B: Transverse Rebar, 2DModel & Detail

X (cm),Down Track

Z (

cm),

Dep

th

0 25 50 75 100 125 1500

25

50

75

100

125

150

(f) Case C: Rebar Mesh, 2DModel & Detail

Figure 9.3: Healthy Physical Models

The results of the simulations are presented as B-scans. B-scans are constructed by

moving the bi-static transmitter/receiver pair along the bridge deck (in the X-direction)

and taking a time history every 1.2cm (2 grid points). The B-scans start with the

transmitter located at the X grid point 89, and consist of 42 time histories spaced at

1.2cm, resulting in a final transmitter location of X grid point 171. As indicated in

Figure 9.3 this corresponds to the spatial range 53.4 to 102.6cm. The transmitter and

receiver are at the height Y=79.2cm (grid point 132). The B-scan image is formed by

stacking the time histories and by using intensity to represent amplitude.

Air-coupled GPR systems first record the transmitted signal as it propagates directly

to the receiver. This is referred to as direct coupling. This signal is easily measured,

and subtracted from the resulting B-scans. Additionally, the B-scans presented include

a time history plot on the left that corresponds to the column located at the center of

the scanned area (X=77.4cm).

133

(a) Case D: Delam. BetweenLayers, 3D Model

(b) Case E: Delam. AboveRebar, 3D Model

(c) Case F: Delam. Below Rebar,3D Model

X (cm),Down Track

Z (

cm),

Dep

th

60 70 80 90 1000

10

20

30

40

Delamination

(d) Case D: Delam. BetweenLayers, 2D Detail

X (cm),Down Track

Z (

cm),

Dep

th

60 70 80 90 1000

10

20

30

40

Delamination

(e) Case E: Delam Above Rebar,2D Detail

X (cm),Down Track

Z (

cm),

Dep

th

60 70 80 90 1000

10

20

30

40

Delamination

(f) Case F: Delam Under Rebar,2D Detail

Figure 9.4: Delaminated Physical Models

9.3 Numerical Simulation Results

In the cases presented, waves travel through asphalt (assigned dielectric constant of 5)

at approximately 13cm/ns. Given an asphalt thickness of 2.4cm, two-way travel time

through the asphalt layer is expected to be approximately 0.36ns. The wave travels

through concrete (assigned dielectric constant of 9) at about 10cm/ns. For a concrete

thickness of 18.0cm, the expected two-way travel time to the bottom of the deck is

approximately 3.6ns from the asphalt/concrete interface. The wave should interact with

the top surface of the transverse rebar with a two-way travel time of approximately 0.96ns

after the asphalt/concrete interface.

Case A: Asphalt & Concrete Layers Only. Using 1.0GHz modulation excitation,

the reflection from the asphalt and concrete layers are clearly visible in the B-scans

from the 2D and 3D simulations (Figure 9.5a and 9.5b). The horizontal banding

134

at approximately 3.75ns is due to the air/asphalt interface. The reflection from the

asphalt/concrete interface is not visually separable from that of the air/asphalt interface,

and the band at approximately 7.75ns is the reflection from the bottom of the concrete

deck (the concrete/air interface). While there is no visible difference in the B-scans,

differences become evident in the magnitude of deviation between 2D and 3D B-scans

(Figure 9.5c). Note that the deviation is presented on an enlarged amplitude scale relative

to the B-scans. The B-scan amplitudes range from −3.8× 10−3 to 3.5× 10−3 (relative to

the excitation amplitude), while the deviation is displayed on a 0 to 15×10−5 scale. Also

presented in Figure 9.5(c) is the column-wise Root Mean Square (RMS) deviation as a

function of position across the bridge deck, which shows the accumulated deviation over

time. In 2D, the air/asphalt and asphalt/concrete reflections match the 3D reflections

well. Noticeable variation occurs at the time of the reflection at the bottom of the

bridge deck (around 8ns). This expected variation is independent of down-track position

and is due to the wave propagating differently in 3D (proportional to 1/r) than in 2D

(proportional to 1/√

r).

Using a center frequency and bandwidth of 0.5GHz instead of 1.0GHz, the excitation

pulse is much wider and the reflections from the air/asphalt and asphalt/concrete

interfaces are entangled. The resulting B-scan shows one horizontal band instead of two

bands representing each interface reflection. This is seen in Figures 9.5d and 9.5e where

the air/asphalt and asphalt/concrete interfaces (around 4.0-5.5ns) are not separable.

The bottom of the deck can be seen again at approximately 8.0ns. As with the 1.0GHz

excitation, Figures 9.5d, 9.5e are visually similar, and the deviation (Figure 9.5f) shows a

good match toward the surface of the deck with small discrepancies due to the spreading

of the propagating wave further into the deck. The deviation amplitude scale range is

0 to 3 × 10−5.

135

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Figure 9.5: Air-coupled B-Scans for Case A: Asphalt & Concrete Only

Case B: Addition of Transverse Rebar. With a 1.0GHz excitation, the addition

of the transverse rebar array results in the B-scans shown in Figure 9.6a, 9.6b. Each

rebar will generate a characteristic hyperbola in the B-scan. The hyperbolas cause

horizontal banding at approximately 5.5ns and subsequent ringing due to constructive

and destructive interference. The deviation in B-scans for Case B 1.0GHz is shown in

Figure 9.6c. The most substantial deviation between the 2D and 3D cases with transverse

rebar occurs at the location of the individual reinforcing bars (four rebars are evident in

the RMS deviation as a function of down track position). The reflection from the rebars

is also subject to the variation in amplitude due to the difference in wave propagation

between the 2D and 3D. Since this effect is a function of radial distance, there is less of

a difference at the earlier time of the rebar reflection than at the time corresponding to

the bottom of the deck.

136

For 0.5GHz excitation, the transverse rebar array causes banding at approximately

6.25ns (Figure 9.6d, 9.6e). The rebar layer reflection is more horizontally uniform than

for the 1.0GHz B-scan, and there is almost no subsequent ringing from the rebar array.

This is due to the fact that the wavelength in concrete is 20.0cm at 0.5GHz and the rebar

separation distance is 12.6cm. The scattering from one rebar can not be distinguished

from another. The deviation between the 2D and 3D B-scans is shown in Figure 9.6f.

Due to the lower center frequency and bandwidth of the excitation, the peaks of the

norm of the RMS Deviation as a function of down track position correlate less to the

actual position of the individual rebars than with a 1.0GHz excitation. In some down

track locations, the constructive and destructive nature of the scattering leads to lower

RMS deviation than when there is no rebar present.

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Figure 9.6: Air-coupled B-Scans for Case B: Addition of Transverse Rebar

137

Case C: Addition of Longitudinal Rebar. Since the longitudinal rebar is not

added to the 2D simulations, the 3D rebar mesh results are compared to the 2D transverse

rebar results (Case B). Based on visual comparison, the 3D B-scan using the 1.0GHz

excitation with the rebar mesh (Figure 9.7a) varies from the 2D B-scan with transverse

rebar only (Figure 9.6a) as expected at just after 6ns. Note that the longitudinal rebars

are right beneath the transverse rebars. The deviation (Figure 9.7c) shows significant

variation at the time corresponding to the longitudinal rebar (approximately 6ns) which

is missing from the 2D model.

Using the 0.5GHz excitation, there is a slight difference in the 3D B-scans (Figure 9.7d)

compared to the case with transverse rebar only (Figure 9.6d), which corresponds to the

second layer of rebar at approximately 7ns. The deviation (Figure 9.7f) is very similar to

the Case B deviation (Figure 9.6f) but with an additional horizontal band corresponding

the the longitudinal rebar (approximately 7ns) and a higher overall RMS deviation as a

function of down track position.

Case D: Delamination Between Asphalt & Concrete Layers. Using a 1.0GHz

excitation, a delamination at the asphalt/concrete interface results in the B-scans shown

in Figures 9.8a and 9.8 b. The delamination between the asphalt and concrete causes a

hot spot at approximately 4.75ns that is most noticeable around 75-85cm down track.

There is a slightly higher peak and lower valley amplitude in the 3D than in the 2D. This

difference is due to the variation of the wave propagation in 3D versus 2D and to the

modeling of the disk shaped void as an infinitely long strip in 2D (due to invariance in the

Y-direction). This can be seen in the deviation of the B-scans for Case D (Figure 9.8c).

The RMS Deviation also shows the differences caused by the addition of longitudinal

rebar to the 3D simulation. Using the 0.5GHz excitation, the delamination is not visible

in the B-scans (Figure 9.8d, 9.8e), but the differences are evident in the deviation plot

(Figure 9.8f) at approximately 5ns and 65-90cm.

138

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Figure 9.7: Air-coupled B-Scans for Case C: Addition of Longitudinal Rebar

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Figure 9.8: Air-coupled B-Scans for Case D: Asphalt/Concrete Delamination

139

To further investigate the difference in 3D versus 2D modeling as they relate to the

condition assessment of civil infrastructure using GPR, the healthy B-scan background

signals (Case C) are removed from those with defects. The resulting B-scan represents

scattering from the defect, including primary scattering from the delamination due to the

probing wave and subsequent effects due to the delamination. The differences between

the B-scans of the model with an asphalt/concrete layer delamination and the healthy

deck (Case D - Case C) are shown in Figures 9.9a and 9.9b for 1.0GHz and Figures 9.9d

and 9.9e for 0.5GHz. Furthermore, the difference between these defect scattering B-scans

are presented in Figures 9.9c and 9.9f. While both 2D and 3D show the presence of an

anomaly, differences in the 2D versus 3D are evident in 9.9c and 9.9f. These differences

are due to the nature of the variation of the propagation of the wave and the differences

in model geometry in 3D versus 2D. The largest variation occurs in the vicinity of the

delamination.

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Figure 9.9: Removal of Healthy B-Scan from Delamination Between Layers (Case D-C)

140

As a final comparison for the presence of a delamination between asphalt and concrete

layers, the 2D healthy B-scan (Case B, Figure 9.6b, 9.6e) is removed from the 3D defect B-

scan (Case D, Figure 9.8a, 9.8d). Recall that since the longitudinal rebars aren’t included

in the 2D model, Cases B and C are equivalent in 2D. Assuming the 3D simulation data

match well with data that would be collected in the field, this comparison provides

insight into the validity of using a 2D model to construct a healthy deck representation

which can be removed from actual data to highlight anomalies. The results are shown in

Figure 9.10. In both the 1.0GHz and 0.5GHz cases, the delamination is identifiable but

differences in model geometry and wave propagation act as sources of clutter.

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Figure 9.10: Delamination Between Layers (3D, Case D) - Healthy Deck (2D, Case B)

Case E: Delamination Above the Transverse Rebar Array. Using a 1.0GHz

excitation, a delamination above the transverse rebar array results in the B-scans shown

in Figure 9.11a, 9.11b. The delamination causes a slight bowing of the banding at the

rebar layer (approximately 5.5ns). This is most noticeable around 75-85cm. The bowing

effect is caused by faster wave propagation in the area of the air void (the wave travels

faster in free space than in asphalt or concrete). The deviation of the B-scans for Case

E 1.0GHz is shown in Figure 9.11c. Previously, there was a horizontal band representing

the absence of longitudinal rebar in the 2D model. In the presence of a delamination at

the rebar layer, there is interference in this horizontal band (at about 6ns) in the area

of the void (75-85cm). Using a 0.5GHz excitation, the resulting B-scans (Figure 9.11d,

141

9.11e) are very similar to Case C, but there is slight bowing present at just after 6ns.

This bowing is also evident in the deviation (Figure 9.11f).

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Figure 9.11: Air-coupled B-Scans for Case E: Delamination Above Rebar Array

Removing the healthy deck background signals from Case E results in a defect

scattering B-scan similar to Figure 9.9, but with the significant differences occurring later

in time (at about 5.5ns) and with a more distorted shape. These differences occur due

to differences in model geometry and increased distance to the delamination over Case

D, and therefore more effects due to variation in propagation. Similar to Case D, the

removal of the healthy 2D data (Figure 9.6b, 9.6e) from the 3D data with a delamination

above the rebar mesh (Figure 9.11a, 9.11d) results in a noticeable anomaly presented in

Figure 9.13.

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Figure 9.12: Removal of Healthy B-Scan from Delamination Above Rebar (Case E-C)

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Figure 9.13: Delamination Above Rebar (3D, Case E) - Healthy Deck (2D, Case B)

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Case F: Delamination Below the Longitudinal Rebar Array. Using a 1.0GHz

excitation, a delamination below the longitudinal rebar results in the B-scans shown in

Figure 9.14a, 9.14b. The area of delamination is more difficult to identify but there

are slight differences around 6ns when compared to the healthy deck B-scans (Case C).

The effect of the air void under the rebar mesh is most noticeable in the reflection from

the bottom of the deck where the signal at approximately 8ns and 75-80cm is more

attenuated than the surrounding response. The deviation of the B-scans for Case F

1.0GHz is shown in Figure 9.14c. Using a 0.5GHz excitation, the delamination below

the rebar mesh results in the B-scans of Figure 9.14d, 9.14e. Of all the delamination

cases, the delamination below the longitudinal rebar is the most difficult to identify. In

the deviation plot (Figure 9.14f), there is a slight difference when compared to Case C

(Figure 9.7f) after approximately 7ns.

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Figure 9.14: Air-coupled B-Scans for Case F: Delamination Below Rebar Mesh

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Removing the healthy background B-scans (Case C) from the B-scans with a

delamination under the reinforcing mesh highlights the anomaly at around 6ns and 75-

85cm (Figure 9.15a-9.15b) using the 1.0GHz excitation. The difference between these

differences, shown in Figure 9.15c, are most pronounced at the time of the reflection from

the bottom of the deck (approximately 8ns). Results are similar, though less defined,

when the 0.5GHz excitation is used, with the greatest difference occurring at the time

of the reflection from the delamination (approximately 6.5ns). Removing the 2D healthy

model B-scan signal from the 3D model B-scan results in Figure 9.16. The delamination

is much more obscured than in Case E (Figure 9.13), but is somewhat noticeable in the

slight bowing of the horizontal band that corresponds with the longitudinal rebar not

modeled in 2D (at approximately 6ns).

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Figure 9.15: Removal of Healthy B-Scan from Delamination Below Rebar (Case F-C)

Simulating B-scans for six different physical models resulted in 504 3D simulations

(42 time histories per B-scan x 2 excitation configurations x 6 physical models). Since

longitudinal rebars are not modeled in the 2D cases, 420 simulations were performed.

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Figure 9.16: Delamination Below Rebar (3D, Case F) - Healthy Deck (2D, Case B)

Both simulations ran for 2500 time steps and used single precision computation. The 3D

FDTD code is written in FORTRAN and executed on a Unix cluster with 4GB of RAM.

The 2D FDTD code is written in MATLAB and executed on a 1.6 GHz PowerPC G5

(Mac OS X 10.3.9) with 3GB of RAM. The 3D simulations took, on average, 14.5 hours

per simulation (or 7 days and 3.6 computational hours per B-scan). The 2D simulations

took an average of 68.6 seconds per simulation (or 48.0 minutes per B-scan).

9.4 Conclusions

Two dimensional modeling of GPR adequately predicts how waves scatter from bridge

deck in three dimensions. The physical models presented in Case A (asphalt and concrete

only) and Case B (asphalt/concrete and transverse rebar) are invariant in the Y-direction.

It is shown that these cases match well, but the difference in propagation between the

3D and 2D point sources must be considered and is more pronounced deeper into the

bridge deck. As the physical models include more variance in the Y-direction, the RMS

Deviation between the B-scans increases. Case F, with a delamination under the rebar

mesh, has the highest deviation between 2D and 3D models, as well as the highest RMS

deviation as a function of down track position. These strongest differences are due to

the delamination being located under the strongest rebar scatterers, deep into the bridge

146

deck. Differences in wave spreading between the 2D and 3D are most pronounced for the

furthest scatterers.

Removal of the respective healthy bridge deck models (Case C) from the delaminated

models (Figures 9.9, 9.12, and 9.15) highlights scattering from the delamination and its

subsequent interactions with the deck structure. This subtraction process removes some

of the variation due to difference in wave propagation in 3D versus 2D. However, the

scatter from the delamination and subsequent interactions with the deck include effects

of the spreading variation. In all delaminated cases, the effect of scattering from the

delamination is visible and, while there is variation in amplitude between the 3D and 2D

models, the shapes of the resulting waveforms (including the time of the peaks) are well

matched.

An important concern in the selection of 2D versus 3D simulations is the validity of

the 2D model and geometry to capture the 3D nature of field collected data. Assuming

that the 3D FDTD simulations are analogous to data collected from an air-coupled GPR,

the 2D healthy deck B-scan (Case B, consisting of only asphalt, concrete and transverse

rebar) was removed from the delaminated 3D B-scans and shown in Figures 9.10, 9.13,

and 9.16. While there are differences between 2D and 3D simulations, the isolation

of anomalies in 3D data using 2D healthy deck data is very evident in the cases with

delamination between the asphalt and concrete layers (Case D) and above the rebar mesh

(Case E), and somewhat evident in the case with the delamination below the rebar mesh

(Case F). Traditional signal processing techniques have the potential to remove clutter

from these images, but even with no data processing, the application of 2D forward

modeling to detect anomalies in 3D data is very promising when one considers the time

saving.

For the condition assessment of civil infrastructure, it is important to be able to locate

anomalies in the subsurface. The marriage of forward modeling and traditional analysis

techniques has the potential to greatly aid in the solution of anomaly detection [55].

147

Performing forward modeling in 2D instead of 3D can provide fast results with less

computational load than 3D modeling. The reduced simulation time in 2D versus

3D makes real-time or close to real-time assessment possible, either by finding a best

fit match for the B-scan from a library of previously run simulations or by running

simulations in the field using a fast processor or a Field-Programmable Gate Array

(FPGA) implementation [56] of the FDTD code.

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Chapter 10

Model Based Assessment

Conventional Ground Penetrating Radar (GPR) analysis often relies on subjective

examination of the travel time and amplitudes of reflected impulses (Chapter 3). In

this section, a method is presented that augments the conventional data-driven approach

by using ISMME (Chapters 6 through 8) to predict the scattering from strong rebar

scatterers and pavement layer interfaces, through a more accurate reconstruction of the

geometric structure of a healthy bridge deck.

A model-based assessment technique is implemented, wherein iterative computational

models determine electromagnetic properties of the subsurface materials and the

subsurface layer thicknesses. These parameters are used to create a healthy model of

the bridge deck. The response from the healthy deck can be removed from the response

from a deck containing a defect as a way of highlighting the suspect area.

The process begins with a healthy bridge deck model constructed from a forward

modeling tool. An integrated modeling medium is developed and presented here to

simulate radar response through heterogeneous media. The created ’virtual’ healthy

bridge deck model is then excited with a radar wave. The response can be used to

determine areas of suspect condition. Finally, this forward model is integrated with an

inverse solution for the purpose of quantifiable subsurface characterization.

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Model based assessment, as applied here, results in a computational model of a

healthy bridge deck. The healthy bridge deck model is then subjected to a simulated

GPR investigation, and the response is compared to the ground truth response to better

identify areas of suspect condition.

Two procedures are used in this model based assessment to compute two parameters.

One procedure is used to determine relative permittivity of the layer or feature and

the second procedure is used to determine layer thickness and feature depth. The

assessment algorithm uses these properties to build and modify physical models to be

used in simulation. The objective is to obtain a model of a healthy bridge deck to use in

comparison to a model with possible defects.

The Nelder-Mead method is an unconstrained nonlinear optimization that searches for

a minimum of an objective function [57]. The amplitude of the ground truth signal at the

boundary location is extracted from the data. In order to determine the closest match

of simulation parameters to actual values, the difference in amplitude of the ground

truth and simulated layer boundary must be minimized. An objective function takes

the permittivity of the material as an input and outputs the difference between the

simulated amplitude and the ground truth amplitude at the boundary. The Nelder-

Mead method uses the initial guess of the permittivity as estimated using the approach in

Section 3.1, searching for the value of permittivity that minimizes the objective function.

This permittivity value, for which the simulation values most closely matches the actual

values, is henceforth referred to as the determined permittivity of the material.

Once the permittivity of a material, and therefore the propagation velocity through

that material, has been determined, the layer thickness can be calculated as discussed

in Section 3.2. This thickness and the material permittivity can be used to update the

physical model for simulation. After the simulation has been implemented, the time

at which the wave hits the boundary can be compared to the time from the ground

truth response. If the simulation peak occurs before the actual peak, the simulated layer

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thickness may need to be increased. If the simulation peak occurs after the actual peak,

the simulated layer thickness may need to be decreased.

10.1 Example Application

In the examples presented herein, the simulated bridge deck is a 188.0 mm thick reinforced

bridge deck with rebar buried 73.7mm under the surface of the concrete, spaced 203.2mm

on center with a diameter of 12.7mm. There is a 104.1mm asphalt overlay. A 12.7mm

thick and 142.24mm wide air void has been introduced into the concrete below the

reinforcing steel. The representative physical model is presented in Figure 10.1 with the

colocated transmitter and receiver indicated by the ’X’. The assumed electromagnetic

properties used to compute the example deck response are provided in Table 10.1 in

terms of the permeability of free space (µ0 = 1.2566 × 10−6 H/m) and the permittivity

of free space (ǫ0 = 8.85 × 10−12 F/m). The conductivity of the materials is assumed to

be zero.

0 200 400 600 800 1000 1200

0

100

200

300

400

500

600

700

800

900

X

Bridge Deck Model

Distance (mm)

Dep

th (

mm

)

Figure 10.1: Simulated Reinforced Concrete Bridge Deck Model with Void (ExampleBridge Deck)

The duration of the simulations presented is 18ns. The temporal resolution is 5.95ps

and the spatial resolution is 2.54mm. The transmitter and receiver are co-located

307.3mm above the bridge deck as indicated by the ’X’ in Figure 10.1. The simulation

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Table 10.1: Assumed Electromagnetic Properties

Material Permeability, µ Permittivity, ǫAir µ0 ǫ0

Asphalt µ0 5 ǫ0

Concrete µ0 10 ǫ0

Reinforcing Steel 3800 µ0 ∞

excitation is a spherical wave and is modeled as a soft source. The excitation can be seen

in Figure 10.2. This excitation signal produces a record of the transmitted signal as it

passes through air similar to measured GPR signals collected in the field.

0 2 4 6 8 10 12 14 16 18−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Time (nanoseconds)

Am

plitd

ue

Excitation Signal (Time Domain)

Figure 10.2: Excitation Signal of the Simulation in the Time Domain

Any air-coupled GPR system records the transmitted signal as it first passes through

the air. This direct coupling signal can be obtained in the field by pointing the antenna

toward the sky and recording the response. The direct coupling can be simulated with

a physical model comprised entirely of air. This simulated response can be seen in

Figure 10.3. The direct coupling is removed from the bridge deck response by subtracting

it from the bridge deck simulation. The result of the bridge deck simulation with direct

coupling removed is shown in Figure 10.4. The first arrival and highest amplitude peaks

discussed here are identified. However, it should be noted that peaks in responses of layer

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interfaces may not always be separable without removal of the signal from previous layer

interfaces.

Figure 10.3: Direct Coupling Response from Air Only Simulation

Figure 10.4: Example Bridge Deck Response (Bridge Deck with Air Void)

10.2 Material Parameters Computation

The electromagnetic wave used by a GPR system may not be a normally incident plane

wave on a planar interface between two media, as required to calculate the relative

permittivity (Section 3.1). In the simulated cases presented here, the electromagnetic

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wave is circular not planar. However, the approach in Chapter 3 for plane waves can

provide a reasonable approximation to the relative permittivity even with a circular

excitation. Any error in the computed relative permittivity of a material due to this

difference will result in erroneous propagation velocity and depth calculation errors.

In order to apply Equation (3.3), it is necessary to have the the reflection coefficient

and the relative permittivity of the upper material. Since the GPR system is air coupled,

the upper material is air. The lower material is asphalt as shown in Figure 10.1. The

reflection coefficient is computed for the simulated case just as it is obtained in the field:

a metal plate is placed on the deck surface to obtain a ’perfect’ reflection. The reflection

coefficient is the ratio of the response amplitude of the example bridge deck response at

the air-asphalt boundary to that of the amplitude of the metal plate simulation at the

air-plate boundary.

From the simulated bridge deck and metal plate responses, the reflection coefficient

can be computed from Equation (3.8) as follows:

ρ =bridge deck surface amplitude

metal plate surface amplitude=

−2.9079 × 10−4

7.6533 × 10−4= −0.38 (10.1)

Given that the upper material is free space (ǫr1 = 1), Equation (3.3) can be used to

calculate the relative permittivity of the asphalt.

ǫr2 =

(

1 − ρ

1 + ρ

)2

ǫr1 =

(

1 − (−0.38)

1 + (−0.38)

)2

∗ 1 = 4.95 (10.2)

The relative permittivity of asphalt used in the simulation was 5.0. In this case,

Equation (10.2) provides a good approximation of the asphalt permittivity. However, for

a given simulated (actual) permittivity, the closeness of fit of the computed permittivity

will vary with the center frequency and bandwidth of the excitation signal. In order to

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compute the thickness of the asphalt layer, the travel time between the air-asphalt and

asphalt-concrete boundary is required.

Given a sensor height of 307mm, the peak of the wave hits the air-asphalt boundary

approximately 2ns after the peak of the direct coupling, which, from Figure 10.3, is

approximately 2.7ns. Referencing Figure 10.4, the time that the peak excitation hits the

air-asphalt boundary is 4.84ns and the time at which the wave hits the asphalt-concrete

boundary is 6.40ns, resulting in a two-way travel time of 1.55ns. Given the computed

permittivity of asphalt in Equation (10.2), the asphalt layer thickness can be computed

from Equations (3.6) and (3.7).

d =c t

2√

ǫr2

= 104.6mm (10.3)

The actual thickness of the asphalt layer used in the simulation was 104.1mm.

Asphalt Permittivity Determination

The methods outlined in Chapter 3 and exemplified in Section 10.2 are applied to

the example bridge deck response; the estimated relative permittivity of the asphalt

is computed to be 4.95. A physical model for simulation is created consisting of asphalt

only. The physical model to determine relative permittivity of asphalt can be seen in

Figure 10.5(a), along with the response.

Using methods described in 3, the difference between amplitudes of the example

bridge deck response and the asphalt model in Figure 10.5(a) at the time of air-asphalt

boundary is minimized. This method uses the computed value of the relative permittivity

as the initial value to the minimizer. After 13 iterations, termination and convergence

tolerances (1×10−4) were satisfied resulting in a relative permittivity of asphalt equal to

5.01. The actual relative permittivity of asphalt used to model the example bridge deck

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0 200 400 600 800 1000 1200

0

100

200

300

400

500

600

700

800

900

X

Bridge Deck Model − Asphalt Only

Distance (mm)

Dep

th (

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)

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4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

4x 10

−4

Time (nanoseconds)

Am

plitd

ue o

f Ele

ctric

Fie

ld

Optimized Asphalt Permittivity

Simulated Ground TruthOptimized Asphalt Response

(b) Simulation Response

Figure 10.5: Response of Asphalt Only Model with Determined Permittivity

was 5. A comparison of the asphalt-only model and the example bridge deck model is

presented in Figure 10.5(b).

Asphalt Thickness & Concrete Permittivity Determination

Recomputing the example in Equation (10.3) using the determined relative permittivity

of 5.01, the thickness of the asphalt layer is found to be 104.1mm. Recall that the actual

layer thickness used to model the example bridge deck is 104.1mm and the calculated

asphalt thickness is 104.6 mm with the estimated asphalt permittivity of 4.95 from

Equation (10.2).

Equation (3.11) leads to an estimate of the reflection coefficient at the concrete

surface:

ρ =top concrete amplitude

metal plate surface amplitude=

−1.0852 × 10−4

7.6533 × 10−4= −0.1418 (10.4)

Given that the upper material is asphalt (determined ǫr2 = 5.01), Equation (3.5) can be

used to calculate the intrinsic impedance of concrete from which the relative permittivity

is computed. Using the determined ǫr2, the relative permittivity of the model’s concrete

is calculated to be 10.30. If the relative permittivity of the concrete was computed with

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the estimated permittivity (ǫr2 = 4.95) the resulting concrete relative permittivity would

be 10.16.

The physical model in Figure 10.5(a) can be improved using the computed asphalt

layer thickness and relative permittivity of concrete (ǫr3 = 10.30). The new model is

shown in Figure 10.6(a).

0 200 400 600 800 1000 1200

0

100

200

300

400

500

600

700

800

900

X

Bridge Deck Model − Asphalt & Concrete

Distance (mm)

Dep

th (

mm

)

(a) Simulation Model

4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

4x 10

−4

Time (nanoseconds)

Am

plitd

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f Ele

ctric

Fie

ld

Optimized Asphalt Thickness & Concrete Permittivity

Simulated Ground TruthOptimized Concrete Response

(b) Simulation Response

Figure 10.6: Response of Asphalt & Concrete Model with Determined Permittivities &Asphalt Layer Thickness

After the simulation is run with the initial approximation of the asphalt thickness and

relative permittivity of concrete, the time at which the wave hits the asphalt-concrete

boundary is compared to the corresponding time in the example bridge deck response.

This ensures that the thickness of the asphalt layer is correct.

To determine the relative permittivity of the concrete, an algorithm must minimize the

difference between amplitudes of the example bridge deck response and the asphalt and

concrete model in Figure 10.6(a) at the time of asphalt-concrete boundary. The relative

permittivity of concrete is used as the initial value to the minimizer. After 13 iterations,

termination and convergence tolerances (1 × 10−4) were satisfied, resulting in a relative

permittivity of concrete equal to 10.02. The actual relative permittivity of concrete used

to model the example bridge deck was 10. A comparison of the asphalt-concrete model

and the example bridge deck response is presented in Figure 10.6(b).

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Concrete Deck Thickness Determination

Equation (3.10) is used to calculate the depth of the concrete deck. The time at the

asphalt-concrete boundary is 6.40ns and the concrete-air boundary is 10.33ns, resulting

in a two-way travel time of 3.93ns. Using the determined relative permittivity of concrete

and considering the spatial resolution of the model, the depth of the concrete deck is

computed to be 185.4mm. The actual depth of the concrete deck used to model the

example bridge deck is 188.0mm. When the thickness of the concrete deck is computed

using the estimated concrete relative permittivity, the thickness is 185.1mm.

The physical model in Figure 10.7(a) incorporates the computed concrete deck

thickness. After the simulation is run with the initial approximation of the concrete deck

thickness and determined relative permittivity of concrete, the time at which the wave

hits the concrete-air boundary is compared to the corresponding time in the example

bridge deck response to ensure that the thickness of the concrete deck is correct. A

comparison of the concrete deck (with no rebar) model and the example bridge deck is

presented in Figure 10.7(b).

0 200 400 600 800 1000 1200

0

100

200

300

400

500

600

700

800

900

X

Bridge Deck Model − No Rebar

Distance (mm)

Dep

th (

mm

)

(a) Simulation Model

4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

4x 10

−4

Time (nanoseconds)

Am

plitd

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f Ele

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Optimized Deck with No Rebar

Simulated Ground TruthOptimized Concrete Response

(b) Simulation Response

Figure 10.7: Response of Bridge Deck (Determined Permittivities & Layer Thicknesses)

158

Determination of Rebar Location

The two-way travel time reflected from the rebar is 7.96ns, resulting in a depth of cover

of 73.7mm. The two-way travel time of the second reflection from the rebar is 9.01ns.

Using trigonometry, the horizontal travel time between the reinforcing steel is calculated

as 4.23ns. Considering the resolution of the model, this results in a horizontal separation

of the rebars of 200.7mm. The actual depth of rebar that was used to model the example

bridge deck is 73.7mm and the actual horizontal separation of the rebars is 203.2mm on

center. The diameter of the rebar is assumed to be 12.7mm. If the estimated relative

permittivity of the concrete is used in the computations of rebar location, the concrete

cover thickness is 73.7 mm and separation is 200.7mm.

Information about the depth and separation of the rebar is used to update the physical

model as shown in Figure 10.8(a). The response of the updated model is obtained and

the times at which the wave hits the rebars are compared to verify the rebar depth is

accurate.

0 200 400 600 800 1000 1200

0

100

200

300

400

500

600

700

800

900

X

Bridge Deck Model − Optimized Deck

Distance (mm)

Dep

th (

mm

)

(a) Simulation Model

4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

4x 10

−4

Time (nanoseconds)

Am

plitd

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f Ele

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Fie

ld

Optimized Reinforced Concrete Deck

Simulated Ground TruthOptimized Deck Response

(b) Simulation Response

Figure 10.8: Response of Healthy Bridge Deck Model

The model in Figure 10.8(a) is considered to be the determined healthy deck

model. This healthy deck response is compared to the example bridge deck response

in Figure 10.8(b). Note that Figure 10.8(b) is presented with a shorter time scale

159

(11.5ns instead of 18ns). This is done to give attention to the signal within the bridge

deck. The signal leaves the bridge deck at approximately 10ns. A summary of the

actual parameters, the computed estimates, and the determined minimization results is

presented in Table 10.2.

Table 10.2: Parameter Determination Summary

Parameter Actual Computed DeterminedAsphalt Relative Permittivity 5 4.95 5.01Asphalt Thickness (mm) 104.1 104.6 104.1Concrete Relative Permittivity 10 10.16 10.02Concrete Deck Thickness (mm) 188.0 185.1 185.4Rebar Cover (mm) 73.7 73.7 73.7Rebar Separation (mm) 203.2 200.7 200.7

The response due to the air void located very close to the rebar (Figure 10.1) is not

distinguishable from the response due to the rebar (Figure 10.4). Using the model based

assessment approach presented here to determine the expected response from a healthy

deck, and then comparing the response to the example bridge deck (Figure 10.8(b)),

discrepancies can be seen. These discrepancies include errors in feature thickness and

depth, errors in permittivity, and feature differences between the determined healthy

deck and the example bridge deck. There is a significant discrepancy between the signals

at approximately 8ns. This discrepancy occurs at the approximate location of the rebar

(see Figures 10.7(b) and 10.8(b)) but is not distinguishable from it. This discrepancy

corresponds to the presence of the air void.

In addition to investigating the responses at the center of the model, a simulated

response was obtained 14.7mm and 29.4mm to the right of the center of the physical

model, and therefore the center of the void. This response could be considered the next

signal recorded from a GPR system moving along a bridge deck. Using the determined

bridge deck model (Figure 10.8(a)), the simulated healthy deck response was obtained

at the same locations. The determined healthy deck responses were subtracted from the

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responses of the deck with the void yielding the results presented in Figure 10.9. The

common feature at approximately 6ns is an error in the determined permittivity of the

concrete. This feature is also noticeable in Figure 10.8(b). The discrepancy corresponding

to the presence of the small air void is, as expected, of lesser amplitude and occurring

later in time as the distance between the sensor and the void grows.

4 5 6 7 8 9 10 11 12−8

−6

−4

−2

0

2

4

6

8x 10

−5

Time (nanoseconds)

Am

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f Ele

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Fie

ld

Deck with Void − Healthy Deck Response

Sensor Centered over VoidSensor 14.7mm Off CenterSensor 29.4mm Off Center

Figure 10.9: Difference in Responses of the Healthy & Defective Bridge Deck Models atVarious Sensor Locations

For example, several simulations were run using the same relative permittivity of

asphalt and concrete presented Section 10.1 but with varying air void location. In

addition to the presented case with the air void located in the concrete below the rebar,

simulations were performed with the air void in the concrete above the rebar, between

the asphalt and concrete layers, and mid-way through the asphalt layer. Using the

approximate method, the relative permittivity of asphalt (actual value 5) varied from

4.95 to 5.30, the asphalt thickness (actual depth 104.1mm) varied from 101.6 mm to

109.2mm, and the relative permittivity of concrete (actual value 10) varied from 8.60 to

15.24. The results are summarized in Table 10.3.

161

Table 10.3: Relative Permittivity & Thickness Calculations for Various Void Locations

Asphalt Relative Asphalt Concrete RelativeVoid Location Permittivity Thickness (mm) Permittivity

(Actual=5) (Actual=104.1) (Actual=10)Concrete, Computed 4.95 104.6 10.16Below Rebar Determined 5.01 104.1 10.02Concrete, Computed 4.95 105.4 10.28Above Rebar Determined 5.01 104.1 10.03Between Asphalt Computed 4.95 109.4 15.24& Concrete Layer Determined 5.01 109.2 7.92Middle of Computed 5.30 102.3 8.60Asphalt Determined 4.98 106.7 10.96

10.3 Results

In the case presented, the Surface Reflection Parameter method (presented in Chapter 3)

seems to work well. However, this approximate method requires that the time and

amplitude of the layers are distinguishable. The accuracy of this method may change

with variation of the excitation signal, the relative permittivities of the materials and the

location of defects in the bridge deck.

The model based assessment approach is encouraging. Even in cases where the

approximate method may not provide as accurate results as presented here, it is still

adept at resolving the relative permittivities.

The methods presented herein allowed construction of a healthy bridge deck model

from one with a defect. While the air void is not distinguishable from the rebar above

it in the raw traces, by comparing the example bridge deck response to the healthy deck

response, an area of suspect condition can be identified in the vicinity of the rebar. These

areas of suspicion may help to locate defects.

162

Chapter 11

The Effect of Rebar on Scattering

from an Anomaly

The GPR probing wave scatters from subsurface features, including defects. The

amplitude and time of reflection are used to identify and locate the features. Metal

objects such as rebars reflect the entire wave impinging upon it. These strong scatterers

of the radiated field can obscure the weaker scattered signals from defects.

Much work has been done to identify the depth of the rebar and, to somewhat less

of an extent, the separation of rebars. If the reflection from the rebar in the measured

data can be removed or isolated it can focus previously obscured reflections from the

bottom of pavements in roadways and also from defects located within the reinforced

concrete decks of bridges. Li et al. [58] presented an approach for isolating and removing

the primary scattered field from the rebar when using a ground-coupled antenna in a

single media. However, the field scattered from a rebar is more complicated when an

air-coupled antenna is used over a multi-layered media which may include defects.

Conventional impulse GPR analysis requires examination of variations in the travel

time and amplitudes of impulses returned to the receiver as described in Chapter 3.

The method outlined in this chapter augments this data-driven approach by using

163

electromagnetic modeling to predict the scattering from strong rebar scatterers and

pavement layer interfaces. The scattering signals are then removed from the signal under

test, so that the remaining weak scattering accurately identify suspected hidden pavement

defects. The electromagnetic modeling is accomplished using the ISMME tool. Using

iterative perturbations of the model, a healthy bridge deck model can be constructed from

a forward modeling tool. The response from the healthy deck model can be removed from

the received GPR data (raw data) to determine areas of suspect condition.

The refined FDTD model enables the consideration of a inhomogeneous subsurface

including material layers, steel reinforcement, and any subsurface anomalies. Electro-

magnetic waves traveling through a medium will scatter when passing through materials

with contrasting dielectric constants. Steel is a perfect scatterer.

It’s conductivity goes toward infinity resulting in an intrinsic impedance near 0

(Equation (2.7)) and the reflection coefficient (Equation (3.1)) becomes -1. The entire

electromagnetic wave is reflected from it, and no part of the wave to passes through it.

With regularly spaced steel reinforcements, one can imagine a subsurface cluttered with

scatter. Unknown air and water filled anomalies add to the clutter. The objective in this

section is to quantify the expected subsurface scatter in a healthy reinforced concrete

system and to quantify the difference in clutter due to different deterioration scenarios.

An approach to model rebar as a point source in a layered background media will be

presented in Section 11.1. This approach is being investigated using simulated results

in a layered media. This approach is particularly useful with data collected using an

air-coupled antenna. When rebar spacing isn’t large enough or the scan density isn’t

high enough, rebar appear as a horizontal layer in the GPR data rather than discrete

hyperbolas.

In Section 11.2, the secondary effect of a rebar array on a field scattered from an

anomaly at various locations will be investigated. Differences between scattered fields

from the anomaly both considering and neglecting the presence of rebar will be presented,

164

demonstrating that the effect of rebar must be considered when attempting to identify

and quantify anomalies by the waves scattered from them.

Section 11.3 presents six examples of the processes outlined in Sections 11.1 and 11.2

with a discussion of the results following in Section 11.4.

11.1 Modeling Rebar as a Point Source

The air-coupled (vehicle mounted) horn antenna is typically mounted about 431.8mm

above the road surface. This mounting results in less focused energy at the bridge surface

and yields reflections from a closely-spaced rebar array which, when viewed as a B-scan,

can appear as a homogeneous stripe. In contrast, distinct hyperbolas from each rebar are

often visible in data collected by a ground coupled antenna that remains in contact with

the ground [27]. For this reason, conventional analysis often only identifies the depth of

the rebar array and not the individual rebars.

The conventional tools for imaging subsurface anomalies: migration, time reversal,

and Born approximation inversion, are based on single incident wave interaction with

regions of dielectric constant variations in an otherwise uniform background. In most

cases, the variations are imaged best when they have low dielectric contrast with the

surrounding medium. Strong scatterers such as metal rebar introduce complications for

inversion methods not only in the form of the strong obscuring clutter they generate

relative to the weak low-contrast anomaly, but also in the secondary waves they scatter,

incident on the anomaly. The clutter signals from the rebar can be computationally

modeled and subtracted from the observed signals, however, the effects of their scattered

waves on the anomaly must be compensated for in order to apply inversion methods. The

inversion strategy described above is beyond the scope of this thesis, but the forward time

domain model formalism establishes a basis for further investigation.

165

To compensate for the rebar scattered field that is incident on the anomaly, this field

must be isolated from the original GPR probing field. By modeling the scattering as

a hard point source excitation with appropriate timing, the rebar scattered field effects

on the anomaly can be analyzed separately, and removed from the total field scattered

by the anomaly. Subtracting this rebar-excited field from the total field leaves the field

due to an isolated anomaly - illuminated by the above-ground GPR source. The field

scattered from the anomaly will interact with the all of its surroundings, including rebar.

These interactions can be modeled for a presumably known geometry, as opposed to

the multiple interactions between the unknown anomaly position and the rebar. Second

order multiple scattering (from the anomaly to the rebar, and then back to the anomaly,

to be scattered again) can be assumed to be small for low-contrast scatterers.

The scattering that occurs when the transmitted wave encounters rebar can be

modeled using a hard point source in place of the rebar. This model is effective

because it isolates the wave reflected from the steel reinforcement from the rest of the

electromagnetic wave that propagated through the surface of the material. In this model,

the excitation that feeds the point source has the opposite sign of the computed field (as

a function of time) at the rebar location when no rebar is present (the field due only to

the wave traveling through free space, asphalt and concrete layers). If the only sources

are at the rebar locations, when the GPR transmitter is not transmitting, the response

recorded at the receiver is only the response of the scattering due to the rebar. Adjacent

rebars lead to multiple scattering, so all rebars must be identified and replaced with an

array of point sources. Replacing the rebar with a point source affords enhanced viewing

of scattering due to rebar separate from the background scattering. A void located in

the bridge deck will not only scatter the incident field, but will also scatter the field

previously scattered by the rebar.

166

11.2 Computational Procedure

Isolating the anomaly-scattered field allows better identification of possible defects. This

scattered field includes multiple wave interactions of the anomaly with both the rebar

array and the pavement layers. To accomplish this, a B-scan of the healthy deck model

is constructed and then subtracted from the measured GPR data.

The B-scan is constructed with a time history record (16ns) taken every 50.8mm

along a 609.6mm section of bridge deck. A B-scan is formed by stacking the time history

records, where the vertical axis is the two-way travel time, the horizontal axis is distance

along the bridge deck, and the color intensity of each pixel indicates the amplitude of the

electric field at that position (time vs. GPR position).

11.2.1 Simulating Total Field GPR Data

The simulations cover a duration of 16ns with a temporal resolution of 29.5ps. A two-

dimensional cross section of a typical three-dimensional physical model is discretized with

a uniform spatial resolution of 12.7mm. For all simulations, the excitation is a cylindrical

wave in 3D. In 2D the excitation is modeled as a hard point source in which the total

field is specified for all time at the excitation point. The chosen excitation signal is that

which produces the same received signal as measured GPR signals collected in the field

when reflected from a metal plate on the road surface. The receiver is located 25.4mm

to the right of the transmitter.

For the purpose of this work, synthetic data is used for simulation inputs, in place

of measured data. For each case presented, a model geometry is chosen and a B-scan

computed with FDTD simulation. The result is referred to as the total signal, and is

analogous to GPR data that would be collected in the field.

Representative physical models of healthy bridge decks include an asphalt overlay,

a concrete deck and a rebar array. The relative permittivity of asphalt is chosen to be

167

between 4 and 6, and the relative permittivity of concrete is chosen to be between 7 and

11. Conductivity is assumed to be negligible. An example of a discretized 2D physical

model and the simulated response of a GPR investigation can be seen in Figure 11.1.

The horizontal bands in the simulated response B-scan represent scattering from various

layers.

0

3

6

9

12

15

Tim

e (n

s)

Distance Along Deck (mm)0 100 200 300 400 500 600

Sig

nal I

nten

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(ar

bitr

ary

rela

tive

units

)

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02Air / Asphalt Boundary

Rebar Layer

Asphalt / Concrete Boundary

Concrete / Air Boundary

Figure 11.1: Healthy Deck Geometry with 88.9mm Asphalt, 228.6mm Concrete Layer,and 88.9mm Deep Rebar Array Spaced at 177.8mm, and a B-scan of Simulated TotalGPR Data

11.2.2 Building the Background Response

The background response is defined as the scattering from the uniform asphalt and

concrete layers from the probing wave. It contains no rebar response. The parameters

required to model the background are the asphalt and concrete layer thickness and

relative permittivities. These parameters can be determined from measured GPR data

alone using the Surface Reflection Parameters Method [40, 59] described in Chapter 3,

or by augmenting the method results by iteratively modifying the parameters in FDTD

models to ensure the background best fits the measured signal. Here, the parameters

are automatically calculated from a single trace computed as the mean of the simulated

total field B-scan. For more details on the process, the reader is referred to [55].

168

The values of the thickness and relative permittivity for asphalt and concrete are used

to construct the background geometry, from which the B-scan is generated. The B-scan

for the computed background for the model geometry shown in Figure 11.1 is presented

in Figure 11.2.

0

3

6

9

12

15

Tim

e (n

s)

Distance Along Deck (mm)0 100 200 300 400 500 600

Sig

nal I

nten

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(ar

bitr

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rela

tive

units

)

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Figure 11.2: B-scan of Computed Background without Rebar

11.2.3 Locating Rebar to Replace with Point Source

The depth of the rebar array layer drebar is determined using time analysis of the signal:

drebar =

(

T

2− hair

c− dasph

vasph

)

vconc (11.1)

where T is the two-way travel time, hair is the height of the GPR above ground, dasph and

vasph are the thickness and velocity of the asphalt layer and vconc is the wave propagation

velocity in concrete. Note that the velocities are given by the speed of light in free space

scaled by the square root of the relative permittivity of the medium (c/√

ǫr). After the

rebar array depth has been determined, a FDTD simulation is used to produce data at

various rebar array separations and positions. The actual rebar spacing is considered to

be the rebar spacing parameter from the simulation with a response that best fits the

actual response [55]. The rebar depth and spacing can thus be determined automatically.

It is assumed that, for a given section of bridge deck, these values remain constant. Areas

169

where the layer depth or spacing varies are considered anomalies. As is often the case with

air-coupled GPR data, the individual rebars cannot be identified in the B-scan image,

which means that conventional methods do not determine the rebar spacing. It is from

the rebar spacing computation that using forward modeling can advance conventional

processing methods.

In order to obtain the electric fields scattered by the rebar, the rebar is modeled

as a hard point source. The rebar source excitations are chosen to mimic a perfect

reflection: the negative of the electric field computed as a function of time in the

background (Figure 11.2) simulation at the specific locations. Additionally, a perfect

electrical conductor is placed at the location of the transmitter so that the response

recorded at the receiver accurately represents any field that may be scattered from the

transmitter.

The depth and separation of the rebar is used to locate the position of the individual

rebars to be replaced with point sources. A B-scan is then computed for scattering from

rebars with layers present in the background. This response due to a point source array

superimposed to the background B-scan is the representation of the healthy bridge deck

B-scan. Removal from the synthetic total field signal highlights potential defects or any

errors in assumptions to attention. The computed response of just the secondary field

due to scattering from rebar as modeled with the point source array of Figure 11.1 can

be seen in Figure 11.3.

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0

3

6

9

12

15

Tim

e (n

s)

Distance Along Deck (mm)0 100 200 300 400 500 600

Sig

nal I

nten

sity

(ar

bitr

ary

rela

tive

units

)

−5

0

5

10x 10

−3

Figure 11.3: B-scan of Point Source Array without External Source

11.2.4 Modeling the Healthy Bridge Deck

The healthy bridge deck B-scan is the summation of the background B-scan and the

point source array B-scan. Any discrepancies between the healthy bridge deck scan

and the simulated total GPR data are discussed. These discrepancies include errors in

assumptions about size and spacing of bridge components and material parameters as

well as response due to scattering from voids such as delaminations, debonded rebar and

areas of varying relative permittivity.

Since the transmitted wave encounters the rebar from the surface, it is only

approximately represented as a point source. However, given the discretization limits

of the FDTD, the results of the healthy deck representation (using the point source

array) are close to the healthy deck simulated total field response.

11.2.5 Determining the Effect of Rebar on Scattering from an

Anomaly

The secondary scattering from the rebar on the anomaly must be removed to determine

the effect of rebar on a field scattered from an anomaly. Here, this is accomplished in

simulation only because knowledge of the geometry of the anomalies are known only in

171

synthesized data. However, it may be implemented iteratively in inversion methods that

converge on the correct position and size of the anomaly.

Similar to the procedure presented in Section 11.1, the rebars are replaced by hard

point sources, the primary GPR source is replaced by a perfect conductor, and the

anomaly is included in the simulation. The resulting B-scan is the scattering from the

rebar on the bridge deck layers, the anomaly and other rebars in the array. Removing

both the background field with no anomaly as well as the hard point source array field

generated with the anomaly from the total field results in scattering from the anomaly

due to just the primary GPR illumination.

If instead, a simulation is implemented on a bridge deck model with an anomaly

but no rebar, the computed background field can be removed to present a B-scan of the

scattering from only the anomaly. This configuration is much easier to analyze and invert

since it consists of only uniform parallel layers, with no strong scatterers. When response

with the rebar scattered secondary field removed is sufficiently close to this latter case,

standard layered media processing can be confidently applied.

In the test cases below, the fractional error is presented as a means to assess how well

the B-scan computed neglecting the rebar array compares with the B-scan computed

considering the rebar array. The two methods are in agreement when the fractional

error is zero. Computing the fractional error between these two B-scans reveals the error

associated with computing the scattering from anomaly but neglecting the rebar for the

incident field. It also demonstrates how the rebar array can change the scattering of an

anomaly.

11.3 Examples

In addition to the healthy deck scenario presented in Section 11.2, five defect cases

are investigated using the same procedure. These cases include bridge deck models

172

with air voids, delamination, debonding, shifted rebar, and areas of higher permittivity

(simulating contaminated concrete). Each case represents a common in situ condition

that is difficult to discriminate using currently available interpretation algorithms. They

can be seen next to the healthy deck case as shown schematically in Figure 11.4. Note

that the scattering from the anomaly (Section 11.2.5) does not apply to the healthy deck

or the shifted rebar conditions. Thus, it is performed only on geometries with an included

air void or area of contamination (Cases B, C, E, and F).

Figure 11.4: Model Geometries for Test Cases A: Healthy Deck, B: Air Void Above Rebar,C: Air Void Below Rebar, D: Single Rebar Shifted Left, E: Area of Higher Permittivity,F: Debonding

11.3.1 Test Case A: Healthy Bridge Deck

Test Case A is the healthy reinforced concrete deck exemplified in Section 11.2. The

asphalt overlay is 88.9mm thick and has a relative permittivity of 5.7. The concrete is

228.6mm thick and has a relative permittivity of 9. The rebar is at a depth of 88.9mm and

has a separation of 177.8mm. The difference between the simulated total signal presented

173

in Figure 11.1 and the healthy deck data B-scans built using the process in Section 11.2

is shown in Figure 11.5. This case confirms that the secondary source replacement of

the rebar model is suitable, as the difference between the B-scans is less than 2 × 10−14,

which can be considered negligible.

0

3

6

9

12

15

Tim

e (n

s)

Distance Along Deck (mm)0 100 200 300 400 500 600

Sig

nal I

nten

sity

(ar

bitr

ary

rela

tive

units

)

−2

−1

0

1

2

x 10−14

Figure 11.5: B-Scan Difference Between Total Signal and Healthy Deck Data (Case A)

11.3.2 Test Case B: Air Void Above Rebar

A 12.7mm thick and 38.1mm wide air void is placed directly over the rebar, located

at a transverse distance of 431.8mm. The simulation parameters of this case are a

101.6mm asphalt overlay with a relative permittivity of 4.9, a 241.3mm concrete deck

with a relative permittivity of 9, and a rebar depth at 88.9mm separated by 127mm. The

simulated total GPR B-scan for this bridge deck configuration is shown in Figure 11.6.

The difference from the healthy deck data can be seen in Figure 11.7. This difference

represents the anomaly scattered field with additive clutter removed. However it does

still include multiple wave interactions with both the rebar array and the pavement

layers. Note that the peak of the wide hyperbola occurs at the location of the air void

(at approximately 431.8mm across the deck). The multiple interactions of the anomaly

with other subsurface structures can be observed in the various bands below and parallel

to the hyperbola.

174

0

3

6

9

12

15

Tim

e (n

s)

Distance Along Deck (mm)0 100 200 300 400 500 600

Sig

nal I

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(ar

bitr

ary

rela

tive

units

)

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Figure 11.6: B-Scan of Simulated Total Signal (Case B)

0

3

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Figure 11.7: B-Scan Difference Between Total Signal with Void Above Rebar and HealthyDeck Data (Case B)

The scattering from the air void located above the rebar when rebar is present, but

when the secondary rebar scattering illumination is removed (left image of Figure 11.8)

is similar to the scattering from the void when there is no rebar present (right image

of Figure 11.8) since the probing wave has not passed through the rebar array before

encountering the anomaly. The fractional error image (Figure 11.9) indicates that the

best agreement between the simple no rebar and processed cases occurs at the transverse

position of the anomaly. At several time instances after the initial anomaly interaction,

the B-scan records the signal ringing as waves pass through the rebar array, interacting

with layer interfaces and other rebars. This ringing is much less pronounced in the

no-rebar case.

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Figure 11.8: B-Scan of Scattering From an Air Void Above Rebar with Rebar (withSecondary Rebar Scattering Illumination Removed), & with No Rebar Present (Case B)

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Figure 11.9: Fractional Error between the Scattering From an Air Void Above Rebarwith Rebar (with Secondary Rebar Scattering Illumination Removed) and with No RebarPresent (Case B)

11.3.3 Test Case C: Air Void Below Rebar

A 12.7mm thick and 38.1mm wide air void is placed directly under the rebar located

at 431.8mm. The simulation parameters of layer thickness, rebar separation and

permittivity generated for this case are the same Case B. One cannot observe any

differences in the simulated total signal between this case and Case B (Figure 11.6). The

B-scans from the computed background with no rebar and point source array only are the

same as in Case B because the structure of the deck and the material parameters are the

same. The resulting difference from the healthy deck data can be seen in Figure 11.10.

Note that the peak of the hyperbola again occurs at the location of the air void (at

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approximately 431.8mm) but is slightly deeper and of much lower amplitude than the

response of scattering due to an air void of the same size located above the rebar.

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Figure 11.10: B-Scan Difference Between Total Signal with Void Below Rebar andHealthy Deck Data (Case C)

When the air void is located below the rebar, the scattering from the anomaly interacts

strongly with the rebar, reducing the observed signal above the round surface (left image,

Figure 11.11). When there is no rebar present (right image, Figure 11.11) the B-scan

anomaly response is much greater as indicated by the darker image. This holds true for

the initial scattering from the air void and the subsequent ringing and reflections. The

fractional error (Figure 11.12) is relatively large compared to the error with an air void

above the rebar (Figure 11.9).

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Figure 11.11: B-Scan of Scattering From Air Void Below Rebar with Rebar (withSecondary Rebar Scattering Illumination Removed), & with No Rebar Present (Case C)

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Figure 11.12: Fractional Error between the Scattering From an Air Void Below Rebarwith Rebar (with Secondary Rebar Scattering Illumination Removed) and with No RebarPresent (Case C)

11.3.4 Test Case D: Single Shifted Rebar

Using the same layers, rebar configuration and dielectric parameters as Case B, the rebar

located at 431.8mm is shifted 25.4mm to the left as seen in Figure 11.4. The simulated

total signal is so close to that of Case B (Figure 11.6) that the results are not repeated

here. The computed background with no rebar and point source array only B-scans are

the same as in Case B. Since the rebar spacing was assumed to be uniform, the location

of the point source used to generate the healthy deck results will not correspond to the

location of the actual rebar. The difference from the healthy deck data is shown in

Figure 11.13. The resulting discrepancy between the rebar and point source location

produces substantial differences between the simulated total signal and healthy model

data that look very dissimilar from differences seen in the cases with the air voids making

it possible to distinguish between defect cases.

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Figure 11.13: B-Scan Difference Between Total Signal with Single Shifted Rebar andHealthy Deck Data (Case D)

11.3.5 Test Case E: Higher Dielectric Around Rebar

Test Case E uses the same simulation parameters as in Case B but with the single rebar

located at 431.8mm surrounded by a 63.5mm circular area where the relative permittivity

of the concrete is 2 higher than that of the surrounding concrete, to simulate chemical

contamination near the rebar. Again, one cannot observe any differences in the simulated

total signal between this case and Case B (Figure 11.6). The B-scan for the computed

background with no rebar and point source array only are the same as in Case B. The

resulting difference from the computed healthy deck data can be seen in Figure 11.14.

This difference is similar to Case B (Figure 11.7), but because the permittivity of the

simulated contamination area is closer to the surrounding concrete than free space is to

concrete, the amplitude of the field along the hyperbola is lower in Case E than Case

B. There is an extra (positive) band in the B-scan hyperbola, most likely due to the

presence of a rebar in the middle of the simulated contamination area.

Removing the secondary rebar scattering illumination from Case E (left image,

Figure 11.15), does not have much affect relative to the simple difference B-scan

(Figure 11.14). Since the rebar is in the middle of the anomaly it would not be expected

to contribute significantly to the incident field on the surrounding anomaly. However, the

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Figure 11.14: B-Scan Difference Between Total Signal with an Area of Higher Permittivityand Healthy Deck Data (Case E)

internal rebar will greatly affect the anomaly scattering. This observation is confirmed

by comparison to the no rebar case, which is quite different (right image, Figure 11.15).

The magnitude of the fractional error (Figure 11.16) falls between the the error of the

case with the air void above the rebar (Figure 11.9) and the case with the air void below

the rebar (Figure 11.12).

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Figure 11.15: B-Scan of Scattering from Area of Higher Permittivity with Rebar (withSecondary Rebar Scattering Illumination Removed), and with No Rebar Present (Case E)

11.3.6 Test Case F: Air Void at Asphalt/Concrete Interface

Test Case F uses the same simulation parameters as Case B but includes a 12.7mm thick

and 38.1mm wide debonded area (air void) located just below the asphalt centered at

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0

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Figure 11.16: Fractional Error between the Scattering from an Area of HigherPermittivity with Rebar (with Secondary Rebar Scattering Illumination Removed) andwith No Rebar Present (Case E)

431.8mm as shown in Figure 11.4. The computed background is the same as in Case B

but, due to the particular geometry of the bridge deck and the air void, the automatic

computation of the rebar layer placed the depth at a grid point (12.7mm) higher than it

is in the model geometry (Figure 11.1). The difference between the simulated total data

and the computed background and point source array is shown on the left in Figure 11.17.

The checkered look of the difference gives warning that the originally assumed rebar array

depth is incorrect. When adjusted to be a grid point (12.7mm) lower, the results are

a closer match to the simulated total signal, and the hyperbolic shape of the scattering

from the debonded area is evident. The difference between the simulated total signal

and the healthy deck results (with correct rebar depth) is shown in Figure 11.18. If the

process outlined in Section 11.2 had resulted in a rebar depth 12.7mm lower than actual,

the difference between simulated total signal and healthy deck model (with incorrect

assumed rebar depth) would still produce a horizontal layer and would look like the

image on the right of Figure 11.17.

Due to the relatively large distance of the debonding anomaly from the rebar array,

the initial scattering from the debonded area with rebar present (left image, Figure 11.19)

is similar to the case with no rebar present (right image, Figure 11.19). The subsequent

ringing and reflection in the case with no rebar is a close match to the case with rebar.

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Distance Across Deck (mm)

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Figure 11.17: B-Scan Difference Between Total Signal with Debonding and Healthy DeckData with Rebar Layer Located Too High and Too Low (Case F)

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Figure 11.18: B-Scan Difference Between Total Signal with Debonding and Healthy DeckData with Correct Rebar Layer Location (Case F)

However, the case where rebar is present has an additional band of scattering at around

10 ns and the transverse location of the debonding. This is very visible in the fractional

error image (Figure 11.20). The fractional error in this case is lower than in the previous

cases (Figs. 11.9, 11.12, and 11.15).

11.3.7 Experimental Summary

The six examples presented represent common bridge deck conditions found in situ.

Subtle differences can be seen in the responses from various anomaly conditions. The

application of the methods detailed in Section 11.2 and summarized in Figure 11.21

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Figure 11.19: B-Scan of Scattering Off of a Debonded Area with Rebar (with SecondaryRebar Scattering Illumination Removed), and with No Rebar Present (Case F)

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Figure 11.20: Fractional Error between the Scattering from a Debonded Area with Rebar(with Secondary Rebar Scattering Illumination Removed) and with No Rebar Present(Case F)

reveals the interaction between rebar an anomaly. This interaction is important when

attempting to remove the rebar clutter from a background, with the objective of

identifying and quantifying anomalies.

11.4 Conclusion

The scattering from an array (or multiple arrays) of rebars due to the probing GPR wave

produces strong clutter when compared to scattering from anomaly targets. Multiple

interactions of the array itself and the bridge deck layers strengthen this clutter and

further obscure the target anomaly.

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Background

Signal (ex.

Fig. 2)

Scatter From

Rebar Array

(ex. Fig. 3)

Healthy

Deck Signal

Scatter From

Rebar With

Anomaly

Scatter From Anomaly due

to primary illumination &

considering the presence of

rebar (ex. Fig 8a)

Scatter From Anomaly due

to primary illumination &

neglecting the presence of

rebar (ex. Fig. 8b)

Add Background

Signal to Scatter

From Rebar Array

Determine geometry & material

properties using Surface

Reflection Method & iterative

forward modeling. Run FDTD.

Locate & replace rebars with

point sources (layered

media). Run FDTD.

Observe difference between

Healthy Deck Signal and

Total Signal (ex. Fig. 7)

Remove Background and

Rebar Array With Anomaly

from Total Signal

Add anomaly to rebar

array model. Run FDTD.Add anomaly to background

model geometry. Run FDTD.

Mean of

Total Signal

(ex. Fig. 1)

Fractional

Error (ex.

Fig. 9)

Figure 11.21: Diagram of Computational Procedure

Diagnosis of bridge deck health benefits from clutter reduction techniques. The model-

based approach to determine the thickness of pavement layers and the depth of the rebar

array from observed GPR responses provides accurate systemic clutter signals which can

then be removed from measured or simulated data to reveal defect target responses.

For each of the cases considered, the B-scan difference between the bridge deck with an

anomaly and the healthy bridge deck with no defect (the clutter response) generated

characteristic hyperbolic contours. These contours have the same general shape, which

indicate the similar transverse and depth position of the anomaly. However, upon closer

examination one observes subtle differences in the form of absolute amplitude and number

of major and minor contours, as well as sign of the peak response.

Examples were presented to show the qualitative differences in the responses due

to different anomalies. In Case D, the nominally uniformly spaced rebar array was

transversely perturbed, resulting in an anomalous scattered signal. As this case

represents a structural irregularity rather than evidence of deterioration, it is important

to distinguish the anomalous response from that of air voids and contamination. Indeed,

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the B-scan difference was quite dissimilar to responses from voids and material variations.

In Case F, debonding between the asphalt and concrete layers was close enough to the

surface that the clutter identification algorithm incorrectly determined the rebar array

depth. The resulting B-scan difference is characteristic of a layered medium located at

the rebar array depth. Appropriate correction in rebar depth with iterative geometry

modification in the FDTD model leads to the correct anomaly response.

The final level of processing involves removing the anomaly interaction with the

secondary scattering field by the rebar array. Removing this component of field incident

on the anomaly simplifies the inversion by minimizing mutual field interactions. Without

the secondary interactions, the only field illuminating the anomaly would be the direct

field transmitted from the GPR source. These corrections to the difference fields generate

B-scans that are quite close to the responses produced from a simple layered geometry

with no strongly scattering rebars. Thus, standard half-space and layered media analysis

and inversion methods would be expected to yield acceptable results when applied to the

corrected responses.

Since the fractional error increases as waves travel through the rebar array, it is

expected (and verified) that the closer to the surface the anomaly is (and therefore the

further from the rebar array while being above the array), the less the fractional error.

The delamination case (Case F) has the least fractional error followed by the air void

above the rebar (Case B), the contamination around the rebar (Case E) and the air

void below the rebar (Case C). The difference between the scattering from an anomaly

in the presence of rebar is demonstrably different than the scattering when there is no

rebar present. This difference (fractional error) must be considered when attempting to

identify and quantify anomalies by the waves scattered from them. For the same layer

thicknesses and properties, changes to the rebar parameters (depth of cover, separation,

number of rebar arrays) will change the fractional error.

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Using forward FDTD modeling, the scattering from an anomaly can be isolated.

The technique of replacing the rebar with hard point source excitations fed by the

incident background field recorded at their locations performs well on synthetic data and

establishes a strategy for improved GPR signal processing in the field, as well providing

information required for the future application of inversion methods.

The procedures presented here could be implemented for practical diagnostics using

the following approach: 1) Obtain B-scan measurements of the bridge deck; 2) If there is

a suspicion of anomaly, determine rebar locations as strong features; 3) Use a database

of stored modeled rebar scattering cases (a limited set, since rebar tends to be close to a

given depth, spacing is standardized, as is asphalt overlay thickness); or 4) Model with

FDTD the closest healthy deck case; and 5) Remove the healthy deck background and

secondary scattering by the rebar; finally process or form image of resulting data as if

rebar were absent to more clearly indicate the position and size of an anomaly.

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Chapter 12

Determination of Simulated

Antenna Excitation

Forward modeling requires an accurate model of the sensor which can often be difficult

because the fully characterized antenna parameters, including antenna dimensions and

material properties are unknown. If the antenna were to be fully modeled, it would be

necessary to do so in three-dimensions to capture the geometry. This chapter describes a

procedure to obtain a virtual sensor model from experimental data. The result will, when

used in the 2D FDTD simulation, produce a wave that propagates within the model like

it would in the field. This model of the antenna is useful when the complete antenna

model is not available or computationally infeasible.

Using the surface equivalence theorem (Huygens’s Principle), an equivalent source can

be found that produces the same fields within a given region as the actual antenna [28]. A

virtual sensor composed of a line of sensor in two-dimensions acts as the boundary of the

closed surface on which the virtual source excitations will be determined experimentally.

Figure 12.1 shows an overview of the experimental set up and the location of the virtual

sensor. The equivalent signal is the integration over the surface, or for the purposes

within this chapter, the summation of signals received along the virtual sensor. It should

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be noted that with respect to the direct signal, the closed surface is not actually closed.

Fields from the antenna may ’escape’ the sides and possibly top of the closed region,

but since these fields are not interacting with the bridge deck they are not of interest.

Therefore the virtual source should not be used to represent the direct signal, as it

does not cross the virtual sensor boundary. However, from the virtual sensor downward

(toward the deck), the propagation is the same, and reflection from the surfaces will be

recorded for the virtual sensor. Therefore, when analyzing data, the direct signal should

be removed from all cases.

“Closed” surface

& Virtual Sensor

Location

Target at example position

GPR Unit

Transmitter

to target

Target to

receiver

Figure 12.1: Experimental Layout & Equivalent Virtual Sensor

Since the GPR used in the experiments is a bi-static antenna, with the transmit

and receive antennas located in a single box, experimental results of signal amplitude

and time delay are separated into those associated with the transmit antenna and those

associated with the receive antenna. The effects associated with the transmit antenna

will be used to create the virtual source, and those associated with the receive antenna

will be used after the simulation to translate the response from the virtual source to the

receiver location. The procedure to establish and implement the virtual sensor model is

summarized as follows:

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I. Perform experiment to obtain amplitude & time delay map at the height of the

virtual source as a function of position

II. Compute the phase centers for the transmit and receive antennas

III. Separate amplitude decays and time delays associated with transmit & receive

antennas

IV. Apply transmit antenna amplitude decay & time delay along virtual source (desired

direct signal)

V. Transform function using modulated narrow width Gaussian simulation to obtain

required virtual source excitation

VI. Input along virtual source

The procedure to utilize the virtual sensor is summarized as follows:

I. Execute 2D FDTD simulation

II. Obtain output along the virtual sensor

III. Apply amplitude decay and time delay associated with the receive antenna

IV. Translate the signals from the virtual source to the receiver

For this investigation, a virtual source is established approximately 33cm under the

bottom of the GPR housing unit. By translating the source, the computational space no

longer needs to incorporate the antenna phase centers which results in a model that can

be approximately 30cm shorter. Additionally, propagation characteristics of the waves

are different in 2D versus 3D (see Chapter 9). As will be discussed, the phase centers

are approximately 40cm above the virtual source; using the virtual source instead of

the phase centers results in a total path traveled that is at least 80cm shorter. The

resulting waveform will appear to have been recorded at the receiver, but due to the

shorter distance to the surface will be less influenced by propagation differences between

the 2D and 3D waves.

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12.1 Antenna Characterization Experiment

In the summer of 2006, Geophysical Survey Systems, Inc. (GSSI) donated a SIR-3000

data acquisition system and a 2.2GHz center frequency horn antenna (model 4105 Horn)

to the Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-

CenSSIS) at Northeastern University. This air-coupled antenna is typically used in

both road and bridge investigations, with a suitable penetration depth of approximately

0.75m [60]. The GPR unit measures 21.0cm x 55.6cm x 49.5cm and contains a bi-static

horn antenna configuration.

An experiment was developed to obtain a two-dimensional mapping of reflection

amplitude and time delay of the response to a target in free space at a fixed height

from the GPR unit. This 2D mapping is necessary in the creation of a virtual source for

the 2D FDTD code.

12.1.1 Experiment Set Up

The experiment took place in the Gordon-CenSSIS soilBED facility at Northeastern

University. Photographs of the setup are shown in Figure 12.2. A 30cm x 90cm piece

of Styrofoam with a 3cm grid was placed on the surface of the sand. A 30.5cm tall

Styrofoam cone topped with a 6.4cm diameter foil-covered Styrofoam ball is placed at

each grid location in turn. The grid locations can be seen in plan view in Figure 12.3

with the approximate location of the GPR unit marked. The cone is placed in a given

square such that the center of the sphere is aligned with the center of the square. The

top of the sphere is about 33cm under the GPR unit. In total, 290 data points were

collected by moving the cone to the next 3cm grid point and measure the radar response.

A background record was also recorded.

190

(a) Foil covered ball & Styrofoam cone onsurface of sand

(b) Styrofoam grid, absorbers on back and left(not seen) walls

Figure 12.2: Experiment Set Up Photographs

Approximate antenna locationBack of SoilBED wall

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Grid number indicates data file. Sphere is located in centered in each 3cm x 3cm square.

Y´

X´

Figure 12.3: Experiment Layout & Data File Mapping

The wavelength of the 2GHz center frequency antenna in free space is 15cm. The

experiment set up results in 5 data points per wavelength. The GSSI antenna has a

bandwidth approximately equal to the center frequency. Collecting data in free space

every 3cm satisfies the Nyquist-Shannon sampling theorem up to almost 5GHz. Note

that this is the spatial sampling of the wavefront passing through the plane of the virtual

source. A high-resolution time history is taken at each spatial sample.

191

The GSSI antenna can record data in distance mode or time mode. Distance mode is

the typical setting for road and bridge surveys since, in this mode, data are only recorded

while the GPR unit is moving forward. In time mode, the GPR constantly sends signals

and records data regardless of the forward motion of the unit. Time mode is used in this

experiment since the antennas remained static. For data recorded with the foil sphere at

a particular location, none of the physical components of the experiment are in motion.

Operating the antenna in time mode results in the collection of several hundred traces

that can be averaged to obtain a mean response. Consistent with typical bridge and road

surveys, 512 sample points were collected in 13ns resulting in a temporal resolution of

approximately 25.4ps.

12.1.2 Experiment Results

The mean response for the background signal is plotted in Figure 12.4 along with

the mean response when the foil sphere is located at grid points 136 and at 225 (see

Figure 12.3). The reflection recorded from the sphere does not correspond to a single

point in time. For example, the time of the reflection from the sphere at grid point 136 is

recorded from approximately 2ns to 4ns. For ease of detection of the reflection, the time

of the reflection is considered to be the major valley. For grid point 136, this corresponds

to approximately 3.2ns. Also, note that the SoilBED Reflection referenced in Figure 12.4

includes reflection from the sand surface and the sides of the SoilBED seen in Figure 12.2.

To determine the antenna’s radiation pattern, a map of the amplitude decay and time

delay of the metal sphere reflections is constructed. This data will be used to determine

the plane in which the phase centers exist, from which the location and height of the

phase centers can be computed. In order to create a map of the amplitude decay and

time delay of the reflection from a foil covered ball as a function of position, the time

192

0 1 2 3 4 5 6 7 8 9

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.3

3.31

3.32x 10

4

Time (ns)

Am

plitu

de

BackgroundGrid Point 136Grid Point 225

Direct Coupling

Metal Sphere Reflections SoilBED Reflection

Figure 12.4: Example of Collected Trace Data

and amplitude must be extracted from the mean traces. The background response is first

removed from the traces and the amplitude and time are noted. The time and amplitude

of the reflection from the sphere is determined to be the minimum valley between the

two peaks. Referring to Figure 12.4, it can be seen that if the background is removed

from the data collected at each grid point, the resulting amplitude for the metal sphere

reflection will be a negative number.

The map of the amplitude of the reflection from the foil sphere as a function of the

position of the sphere can be seen in Figure 12.5(a). The approximate location of the

GPR unit is indicated by the black lines. The map of the time of reflection from the foil

sphere as a function of the position of the sphere is shown in Figure 12.5(b).

193

Y ′

(cm

)

X ′ (cm)0 10 20 30 40 50 60 70 80

0

10

20

30

Amplitude

−220

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

(a) Amplitude of Reflection

Y ′

(cm

)

X ′ (cm)0 10 20 30 40 50 60 70 80

0

10

20

30

Time (ns)

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

(b) Recorded Time of Reflection

Figure 12.5: Map of Amplitude & Time of Reflection from Target

The experimental data were recorded at 3cm intervals. For further investigation, it is

desirable to have data at a finer interval. To that end, data contained in the amplitude

and time maps of Figure 12.5 are interpolated using 2-D cubic spline interpolation to

provide measurements every 1mm. The X ′ and Y ′ axes are translated to X and Y axes

on which the origin is located at the point with both the highest magnitude amplitude

reflection and the shortest arrival time. The amplitude and time delay maps using

interpolated data and the translated axes are shown in Figure 12.6.

Y (

cm)

X (cm)−30 −20 −10 0 10 20 30 40

−10

−5

0

5

10

Amplitude

−220

−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

(a) Amplitude of Reflection

Y (

cm)

X (cm)−30 −20 −10 0 10 20 30 40

−10

−5

0

5

10

Time (ns)

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

(b) Recorded Time of Reflection

Figure 12.6: Map of Interpolated Amplitude & Time of Reflection from Target

194

Recall that the GPR unit used in the experiment contains a bi-static horn antenna.

It is assumed that the transmit and receive horn antennas are at the same height and

in the same Y plane. They are located at a distance apart along the X axis called the

bi-static separation. The Y coordinate location of the antennas is along the line resulting

in the highest amplitude and shortest travel time of reflection. This occurs along Y = 0.

For a constant antenna height, the shortest path from the transmit antenna to a target

and then to the receive antenna occurs when the target is at the mid-point between the

two antennas. Compared to targets at the same height, as is the case for the experiment,

the reflections from the target at the origin will have the highest amplitude and shortest

travel time. Therefore, the mid-point between the transmit and receive antennas is at

X = 0.

Since the objective is to incorporate the attenuation and time delay into a 2D FDTD

model, data along Y = 0 are the primary focus. Amplitude and time delay data along

this line are plotted in Figure 12.7. The amplitude data have been normalized to unity

at the peak value to obtain the decay factor. For example, the received reflection of

the sphere at X = 20cm is roughly 70% that of the received reflection of the target at

X = 0cm. Note that this is overall amplitude and delay due to wave propagation from

the transmit antenna to the target to the receive antenna. The time delays have been

set to zero at X = 0. This adjustment removes any constant time effects leaving only

the time delay relative to the reflection from a target at X = 0. For example, when the

target is at X = 20cm the reflection from the target is recorded approximately 0.2ns

later than if the target was at X = 0cm.

195

−30 −20 −10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Target Location X when Y=0 (cm)

Amplitude Decay FactorTime Delay (ns)

Figure 12.7: Amplitude & Time Factors as a Function of Target Location

12.2 Determination of Relevant Antenna

Characteristics

12.2.1 Phase Center Height & Separation

To implement the virtual source into the 2D FDTD code, it is necessary to separate

the effects of the transmitting and receiving antennas. The apparent phase center is

the point from which the electromagnetic wave spreads spherically. For horn antennas,

this point is not the physical location from which the wave is transmitted. Locating

the phase centers in the bi-static horn antenna proves to be challenging because their

separation and height are unknown, and the received signal contains the combined affects

of the transmitting and receiving antennas. The results of the experiment presented in

Section 12.1 are used to determine the height of the phase centers above the target (h)

and their separation (∆x).

Consider an ellipse with the foci corresponding to the transmit (Tx) and receive (Rx)

phase centers separated by ∆x, bisected by the Z axis as shown in Figure 12.8.

196

xi

ai

-hbi

X

Z

Tx Rx

x/2 x/2

d1i

d2i

Figure 12.8: Example Ellipse Geometry

The targets from the experiment are located on a line at a distance, h, under the

phase centers. Each target has a position along the X axis and a constant offset below

the X axis of −h. Any given target (i) has the (X, Z) coordinates (xi,−h). Therefore,

each target has an associated ellipse with constant foci (Tx and Rx) and varying major

and minor axes, ai and bi respectively. Additionally, a property of ellipses is that the

distance from one focus to anywhere on the surface of the ellipse to the other focus is

constant, with a value of twice the major axis. Referring to Figure 12.8, this distance for

the ellipse for a particular target is given as d1i + d2i = 2ai.

The equation for an ellipse with foci along the X axis, bisected by the Z axis is:

x2

a2+

z2

b2= 1 (12.1)

The semi-latus rectum (l) of an ellipse is the distance from the foci to the ellipse along

a line perpendicular to the minor axis. It is given by: l = b2/a. For the ellipse in

Figure 12.8 this occurs at x = ±∆x/2:

∆x2

4a2+

l2

b2= 1 =⇒ l2 = b2

(

1 − ∆x2

4a2

)

(12.2)

197

Equation (12.2) can then be set to the definition of the length of the semi-latus rectum

squared to solve for the minor axis (b) in terms of the major axis (a):

b2

(

1 − ∆x2

4a2

)

=b4

a2=⇒ b =

1

2

√4a2 − ∆x2 (12.3)

Using Equation (12.3), Equation (12.1) becomes

x2

a2+

4h2

4a2 − ∆x2= 1 (12.4)

Expanding Equation 12.4 we obtain:

4a4 − (∆x2 + 4x2 + 4h2)a2 + ∆x2x2 = 0 (12.5)

By letting c = a2, we can solve Equation (12.5) using the quadratic equation:

c =(∆x2 + 4x2 + 4h2) ±

√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.6)

Considering that a =√

c, Equation (12.6) results in four solutions of a:

aa =

√

(∆x2 + 4x2 + 4h2) +√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.7)

ab =

√

(∆x2 + 4x2 + 4h2) −√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.8)

ac = −

√

(∆x2 + 4x2 + 4h2) +√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.9)

ad = −

√

(∆x2 + 4x2 + 4h2) −√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.10)

198

Since the value of a should be positive and non-zero (including at x = 0), we choose

a = aa:

a(x) =

√

(∆x2 + 4x2 + 4h2) +√

(∆x2 + 4x2 + 4h2)2 − 16∆x2x2

8(12.11)

Assume that a(x) is approximated by a second order even function given by:

a(x) = a0 + a2s2 (12.12)

The value of a0 can be obtained by setting x = 0 in Equation (12.11):

a0 =

√

(∆x2 + 4h2) +√

(∆x2 + 4h2)2

8=

1

2

√∆x2 + 4h2 (12.13)

To determine the a2 term, take a Taylor series expansion about x = 0:

a(x) ≈ a(0) +da(0)

dxx +

d2a(0)

dx2

x2

2!+ O(x3) (12.14)

In order to collect the x2 term, we need the second derivative of a (from Equation 12.11)

evaluated at x = 0:

a2 =d2a(0)

ds2

1

2!

=1

16

(

16 +64 h2 − 16 ∆x2

√

16 h4 + ∆x 4 + 8 h2∆x 2

)

1√

2 ∆x 2 + 8 h2 + 2√

16 h4 + ∆x 4 + 8 h2∆x 2

(12.15)

After simplifying, Equation (12.15) becomes:

a2 =4h2

(√

4h2 + ∆x2)3(12.16)

199

The equation a(x) represents the change in length of the major axis of an ellipse for

a particular target location. From the point of view of the experiment, this corresponds

to the increase in the time the reflection from the target is recorded as the distance from

X = 0 increases. The time presented for the experimental data is given as a delta from

the direct coupling signal, instead of the entire actual transmit time. All the recorded

times are considered an increase in the recorded time of reflection from the target at a

particular location over that recorded when the target position was at X = 0. When the

time delay at X = 0 is removed from the data, as in Figure 12.7, the a0 term falls out of

Equation (12.13). The time delay presented in Figure 12.7 is the increase in time to the

target reflection for the overall path of the wave. Since the velocity of the wave through

free space is known, these times can be translated into distance through free space. They

correspond to twice the major axis of the ellipse 2a(x).

The experimental time delay data along Y = 0, translated to distance, are plotted in

Figure 12.9 along with the best fit second order polynomial. The best fit polynomial p

has a coefficient p2 for the x2 term such that

p(x) ≈ p2x2 (12.17)

The value of p2 extracted from the curve shown in Figure 12.9 is found to be

approximately 0.0135. The representation of the experimental data, p(x), is equivalent

to 2a(x): the increase in the reflection time from the transmit phase center to the target

and to the receive phase center, over the reflection time when the target is at X = 0.

Therefore, 2a2 = p2 and using Equation (12.16)

2a2 =8h2

(√

4h2 + ∆x2)3(12.18)

200

−30 −20 −10 0 10 20 30 400

5

10

15

20

25

Target Location X when Y=0 (cm)

Incr

ease

in T

otal

Dis

tanc

e (c

m)

Experimental Data2nd Order Approximation

Figure 12.9: Best Fit Polynomial to Increased Target Distance Along Y = 0

In order to solve for the two unknowns in Equation (12.18), h and ∆x, a second

equation is required. Consider data collected along X = 0 in Figure 12.6(b). Since the

phase centers are bisected by the X axis, for every target location along the Y axis at

X = 0 the distance from the transmitter to the target and from the target to the receiver

are equal. Using the Pythagorean theorem, the total distance, b(y), from the transmitter

to target to receiver along X = 0 is given by:

b(y) = 2

(

1

2

√

∆x2 + 4y2 + 4h2

)

=√

∆x2 + 4y2 + 4h2 (12.19)

where b should not be confused with the minor axis in Figure 12.8. As with the

previous analysis, the time delay at the origin is removed from the data resulting in

only requiring the y2 term such that b(y) ≈ b2y2. This polynomial approximation is

shown in Figure 12.10 compared to the time delay data collected from the experiment

converted to distance through free space. The b2 term can be found from the Taylor

series expansion of Equation (12.19) about y = 0:

b2 =d2b(0)

dy2

1

2!=

2√∆x2 + 4h2

(12.20)

201

−10 −5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

Target Location Y when X=0 (cm)

Incr

ease

in T

otal

Dis

tanc

e (c

m)

Experimental Data2nd Order Approximation

Figure 12.10: Best Fit Polynomial to Increased Target Distance Along X = 0

The coefficient of y2 in the best fit polynomial is extracted from the data in

Figure 12.10. It was found to be approximately 0.0202.

Equations (12.18) and (12.20) can be solved for ∆x and h and the values of 2a2 =

0.0135 and b2 = 0.0202 from the experimental data can be substituted in to obtain

∆x = 2

√

b2(b2 − a2)

b22

≈ 57.20cm (12.21)

h =

√a2b2

b22

≈ 40.53cm (12.22)

Using experimental data, referring to Figure 12.8, the transmit and receive antenna

phase centers are located approximately 40.53cm above the target. Since the targets

are 33.0cm below the bottom of the GPR housing unit, the phase centers are

located approximately 7.5cm above the bottom of the unit and located horizontally

at approximately x = ±∆x/2 = ±28.6cm. This information can be used to separate

the effects from the transmit and receive antennas, a necessary step in determining the

excitation of the virtual source.

202

12.2.2 Isolation of Transmit Antenna Amplitude Factors &

Time Delays

With computation of the phase center separation and height, the distance or time delay

from the transmit antenna to the target can be separated from that of the target to

the receive antenna. Referring to Figure 12.8, the distances from the transmitter to the

target (d1) and the target to the receiver (d2) can be computed using the Pythagorean

theorem:

d1 =

√

(

∆x

2+ x

)2

+ h2 (12.23)

d2 =

√

(

∆x

2− x

)2

+ h2 (12.24)

Recall that the total distance traveled (d1 + d2) equals 2a. From Equations (12.12) and

(12.13)

a(x) = a0 + a2x2 =

1

2

√∆x2 + 4h2 + a2x

2 (12.25)

The experimental data plotted in Figure 12.9 is analogous to 2a2x2. The total distance

traveled is equal to twice the major axis of the ellipses and, for comparative reasons, 2a2

is solved:

d1 + d2 =√

∆x2 + 4h2 + 2a2x2 (12.26)

2a2x2 = d1 + d2 −

√∆x2 + 4h2 (12.27)

These two quantities are plotted in Figure 12.11. The quantity on the left hand side of

Equation (12.27) is the experimental data, and the quantity on the right hand side is the

computed distance.

203

−30 −20 −10 0 10 20 30 400

5

10

15

20

25

Target Location X when Y=0 (cm)

Incr

ease

in T

otal

Dis

tanc

e (c

m)

Experimental DataComputed Distance

Figure 12.11: Comparison of Experimental and Computed Distance to Target

Assured that the total distances closely represent the experimental data, d1 and d2

can be examined separately from each other. The distance from the transmitter to the

target (d1) and from the target to the receiver (d2) are shown in Figure 12.12(a). Unlike

the data in Figure 12.11 and Equation (12.27), this data does not have the distance to

the target at X = 0 removed, per Equations (12.23) and (12.24). The transmitter is

located at x = −∆x/2 = −28.6cm. The distance from the transmit antenna to the

target is shortest at this location and corresponds to the height of the phase center which

is 40.53cm. From this location, a distance of approximately 70cm must be traveled to

reach the receiver phase center. At the location of X = 0, the target is halfway between

the transmit and receive phase centers so the transmit to target and receive to target

distances are equal, approximately 50cm each or the value of a0 from Equation (12.13).

The distances in Figure 12.12(a) can be converted to time delay in free space. These

values are shown in Figure 12.12(b).

The final data to extract from the experiment are the amplitude decays as a function

of position (Figure 12.7) which need to have the effects of the transmit and receive

antennas separated.

204

−30 −20 −10 0 10 20 30 40

45

50

55

60

65

70

75

80

Target Location X when Y=0 (cm)

Dis

tanc

e (c

m)

Transmit to TargetTarget to Receive

(a) Separated Travel Distances

−30 −20 −10 0 10 20 30 40

1.4

1.6

1.8

2

2.2

2.4

2.6

Target Location X when Y=0 (cm)

Tim

e D

elay

(ns

)

Transmit to TargetTarget to Receive

(b) Separated Time Delays

Figure 12.12: Separation Transmit & Receive Distance & Time Delay to Target

Assume that the amplitude decays experienced by the transmit (gt) and receive (gr)

antennas have the same shape. Since the amplitude decay factors in Figure 12.7 have

a points of inflection left and right of center, assume they can be approximated with a

cosine function. The peak of the cosine function for the transmitter occurs at x = −∆x/2

and for the receiver at x = ∆x/2. The equations for the decay factors are given as:

gt = cos (c1(x + ∆x/2)) + c2

gr = cos (c1(x − ∆x/2)) + c2 (12.28)

Since the total amplitude factors obtained from the experiment have been normalized

to unity at x = 0 and the total amplitude decay is the product of the transmit and receive

amplitude decays, c2 can be determined by setting Equation (12.28) equal to 1 at x = 0:

1 = cos (c1∆x/2) + c2 =⇒ c2 = 1 − cos (c1∆x/2) (12.29)

205

and Equation (12.28) becomes

gt = cos (c1(x + ∆x/2)) + 1 − cos (c1∆x/2)

gr = cos (c1(x − ∆x/2)) + 1 − cos (c1∆x/2) (12.30)

The amplitude factors from the transmit and receive antennas are multiplied to obtain

the total amplitude, which is compared to the measured data (M) shown in Figure 12.7:

G = gt gr =2 cos2 (c1∆x/2) + cos2 (c1x) + 2 cos (c1x) cos (c1∆x/2)

− 2 cos (c1x) cos2 (c1∆x/2) − 2 cos (c1∆x/2) (12.31)

An explicit solution for G = M does not exist, so c1 is solved for using a non-linear least

squares minimization of the error between the measured and computed data (E = M−G).

The optimization procedure results in a value of c1 = 1.3043. A comparison of the total

computed and measured decay is shown in Figure 12.13.

−30 −20 −10 0 10 20 30 40

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Target Location X when Y=0 (cm)

Am

plitu

de D

ecay

Fac

tor

Rel

ativ

e to

x=

0

Experimental DataComputed Total Decay

Figure 12.13: Amplitude Comparison of Measured & Computed Data

206

Using Equation (12.30), the amplitude factors for the separated transmit and receive

antennas can be computed. The resulting factors are plotted in Figure 12.14. Using the

separate transmit time delays in Figure 12.12(b) and amplitude factors in Figure 12.14,

the desired excitation along a line at the height of the targets can be assembled.

−30 −20 −10 0 10 20 30 40

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Target Location X when Y=0 (cm)

Am

plitu

de D

ecay

Fac

tor

Rel

ativ

e to

x=

0

Transmit to TargetTarget to Receive

Figure 12.14: Amplitudes from Transmit & Receive Antennas

12.3 Assembling the Desired Direct Signal

The amplitude decays and time delays associated with the transmit antenna are used to

created a desired direct signal. This is the signal that should propagate from the virtual

sensor to mimic the actual antenna system. Consider the experiment discussed in 12.1.

If the signal was recorded along the line of targets, a good image of how the wave was

propagating, considering directivity, would be obtained. However, the data are collected

at the receive antenna located within the GPR unit, and the signal is therefore subject

to effects from both the transmit and receive antennas.

A virtual source for the 2D FDTD is constructed at the height of the targets. The

signal that would be recorded at the target locations is only affected by the transmit

antenna. The virtual source is assembled using the time delays and amplitude factors

207

separated out in Figures 12.12(b) and 12.14. Assume that the 2D FDTD simulation will

utilize a spatial resolution (∆) of 1mm. To satisfy Courant’s condition, a time step (∆t)

of 2.25ps is used.

In order to apply the previously computed time delays and amplitude decays, a pulse

shape is obtained. The direct signal, isolated in experimental data, is time gated and

normalized to a peak amplitude of unity. The signal is interpolated from the sampling

time that the data was collected at (approximately 25.4ps) to the sampling time that

the simulation will be conducted at (2.25ps). The resulting target trace, shown in Figure

12.15 is the base line waveform shape to construct the desired direct signal.

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Time (ns)

Am

plitu

de

Figure 12.15: Desired Direct Signal

The width of the virtual source is found by considering the amplitude factors as

computed using Equation (12.30). Figure 12.16 shows amplitude decay factors over a 3

meter range of x values. The virtual source should be wide enough that the amplitude

decays associated with both the transmit and receive antennas are zero at the extents.

Given this constraint, the virtual source is defined over the range x = ±125cm.

208

−150 −100 −50 0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

Virtual Source Location X (cm)

Tot

al A

mpl

itude

Dec

ay F

acto

r R

elat

ive

to x

=0

Transmit to TargetTarget to Receive

Figure 12.16: Amplitude Decay Functions

The phase center height (h ≈ 40.53cm) and separation (∆x ≈ 57.20cm) are used as

previously computed. The time delay associated with the transmit antenna is computed

over the width of the virtual source using the result of the distance from Equation 12.23

multiplied by the speed of light in free space. The time delay at the transmitter is

subtracted from the time delays presented in Figure 12.12(b) to obtain a time delay

relative to the transmit location. This corresponds to having no time delay at the target

located directly beneath the transmitter. Additionally, the time delay is converted from

nanoseconds to an integer representing the time delay in time steps. This time step delay

is the nearest integer resulting from the division of the time delay (ns) by the simulation

time step (∆t). The resulting time step delay, as a function of position along the virtual

source, is shown in Figure 12.17(a).

209

−100 −50 0 50 1000

200

400

600

800

1000

1200

1400

1600

Virtual Source Location X (cm)

Del

ay in

Tim

e S

teps

(a) Signal Delay in Time Steps

−100 −50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

Virtual Source Location X (cm)

Am

plitu

de D

ecay

Fac

tor

(b) Amplitude Decay Factor

Figure 12.17: Delay & Decay Associated with Transmit Antenna

The corresponding amplitude factors can be computed along the virtual source using

Equation (12.28). Since the amplitude decays are a cosine function, depending on the

range of x values the amplitude factors may become negative in value, and as a result,

inappropriately negate and amplify the excitation signal. Therefore, amplitude factors

less than zero, are set to zero. The amplitude factors along the virtual source for the

transmit antenna are shown in Figure 12.17(b). Note that for values of approximately

x > 50cm, the amplitude factor would be negative and therefore, are set to zero.

For each location along the virtual source, the target trace (Figure 12.15) is multiplied

by the corresponding amplitude factor (Figure 12.17(b)) and delayed by the appropriate

number of time steps (Figure 12.17(a)). This results in the desired direct signal matrix

shown in Figure 12.18. This matrix corresponds to the assumed signals that would

be recorded if there were receivers at the target locations from the experiment. The

excitations along the virtual source are computed to produce the desired direct signals.

The resulting sensor model will propagate through the 2D FDTD model similarly to

how the actual GPR transmit signal propagates through free space. The sensor model

captures the directivity of the antenna without requiring modeling of the actual antenna,

which can only be done in 3-dimensions with detailed specifications.

210

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Figure 12.18: Desired Direct Signal Matrix

12.4 Implementation into the 2D FDTD

When a sensor is a hard source, the excitation replaces the electric field at the sensor

location at each time step. However, the virtual source is comprised of soft sources with

the excitation signal added to the computed electric field at each time step. The resulting

waveform is not the same as the excitation signal. The desired response collected along

the virtual source shown in Figure 12.18 is the desired direct signal; the excitation for

each sensor in the virtual source needs to be determined.

The excitation matrix for the virtual source is determined by creating a filter for

a modulated Gaussian pulse. The initial simulation uses a modulated Gaussian pulse

with center frequency and bandwidth of 2GHz, similar to that of the GSSI GPR.

This waveform is subjected to the amplitude adjustment and time delay present in

Figure 12.17. The resulting excitation matrix is shown in Figure 12.19.

211

Virtual Source Location X (cm)T

ime

(ns)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Figure 12.19: Modulated Gaussian Excitation Matrix

A simulation is then run in free space to obtain the resulting direct signal. The

simulation uses a spatial resolution of 1mm and a temporal resolution of 2.25ps. The

computational model is shown in Figure 12.20(a). The virtual source is located along

Z = 501. Since there are sensors located at each adjacent grid point, the individual soft

sensors, denoted by a ’+’, appear as a solid line. The results of the simulation are shown

in Figure 12.20(b). The response matrix, represented as an image, shows the recorded

results at each static sensor location in the virtual source.

X (grid coordinates)

Z (

grid

coo

rdin

ates

)

500 1000 1500 2000 2500

200

400

600

800

Virtual Source

Freespace

Freespace

(a) Free Space Physical Model

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

(b) Direct Response to Gaussian ExcitationMatrix

Figure 12.20: Free Space Model and Response to Gaussian Excitation

212

The input of a linear time-invariant system can be related to the output with a

transfer function. If the impulse response of the system and the input are known, the

output can be computed using convolution in the time domain or multiplication in the

frequency domain. Assume that the excitation in Figure 12.19 is the input x(t) with a

Fourier transform Fx(t) = X(ω), and the result in Figure 12.20(b) is the output y(t)

with a Fourier transform Fy(t) = Y (ω). The frequency response of the system, H(ω),

is the Fourier transform of the impulse response, h(t), and can be computed as follows:

X(ω)H(ω) = Y (ω) =⇒ H(ω) = Y (ω)/X(ω) (12.32)

The same physical model will have the same frequency response. If the desired output

in Figure 12.18 is y′(t), with a Fourier transform Y ′(ω), the required excitation x′(t) can

be computed as follows:

X ′(ω)H(ω) = Y ′(ω) (12.33)

X ′(ω)Y (ω)

X(ω)= Y ′(ω) (12.34)

X ′(ω) = Y ′(ω)X(ω)

Y (ω)(12.35)

x′(t) = F−1X ′(ω) (12.36)

The resulting excitation for the virtual source is shown in Figure 12.21(a). Running

the simulation of Figure 12.20(a) with the new excitation results in the direct signal

shown in Figure 12.21(b). The direct response matches well with the desired response in

Figure 12.18.

213

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

(a) Filtered Virtual Source Excitation

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

(b) Direct Signal from Virtual Source

Figure 12.21: Virtual Source Excitation and Direct Signal

12.5 Translating from Virtual Sensor to Receiver

Consider the simulation of the virtual source reflected from a PEC plate represented

in Figure 12.22(a). The resulting signals recorded along the virtual sensor are shown in

Figure 12.22(b) with the direct signal removed. The computed time delays and amplitude

decays associated with the receive antenna are shown in Figure 12.23. The delays and

decay factors are applied to the simulation output at each location along the virtual

sensor resulting in the matrix shown in Figure 12.24(a).

X (grid coordinates)

Z (

grid

coo

rdin

ates

)

500 1000 1500 2000 2500

200

400

600

800

PEC plate

Virtual Source

(a) PEC Physical Model

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−0.2

0

0.2

0.4

0.6

0.8

(b) Virtual Sensor Output Minus Direct Signals

Figure 12.22: PEC Simulation Model and Raw Output

214

−100 −50 0 50 100600

800

1000

1200

1400

1600

1800

2000

2200

Virtual Source Location X (cm)

Del

ay in

Tim

e S

teps

(a) Signal Delay in Time Steps

−100 −50 0 50 1000

0.2

0.4

0.6

0.8

1

1.2

Virtual Source Location X (cm)

Am

plitu

de D

ecay

Fac

tor

(b) Amplitude Decay Factor

Figure 12.23: Delay & Decay Associated with Receive Antenna

Huygens’s Principle states that each point of an advancing wave can be considered

a new excitation, and the matrix of signals is equivalent to these new excitations.

The summation of these signals translates the excitations to the received phase center.

The resulting signal, equivalent to the signal recorded at the receiver is shown in

Figure 12.24(b).

Virtual Source Location X (cm)

Tim

e (n

s)

−100 −50 0 50 100

0

2

4

6

8

10

12

Amplitude

−0.2

0

0.2

0.4

0.6

0.8

(a) Adjusted Output from Virtual Sensor

0 2 4 6 8 10 12

−50

0

50

100

150

200

250

300

Time (ns)

Am

plitu

de

(b) Final Received Signal

Figure 12.24: Translation of PEC Reflection from Virtual Sensor to Receiver

As a comparison, the PEC simulation was run using the transmit and receive phase

centers instead of the virtual source. The same desired signal (Figure 12.15) was used

to obtain a soft source excitation in the same manner presented in Section 12.4. The

215

resulting excitation is applied at a single grid point, corresponding to the computed

location of the transmit phase center. The excitation is shown in Figure 12.25(a). The

physical model of the PEC simulation using the computed transmit and receive phase

centers is shown in Figure 12.25(b).

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Time (ns)

Rel

ativ

e A

mpl

itude

(a) Filtered Phase Excite

X (grid coordinates)

Z (

grid

coo

rdin

ates

)

500 1000 1500 2000 2500

200

400

600

800

PEC plate

Transmit Phase Center

Receive Phase Center

(b) PEC Physical Model Using Phase CenterExcitation

Figure 12.25: Excitation & Physical Model for Phase Center Simulation

The comparison of the PEC reflection obtained from the virtual sensor model and

the phase center model is shown in Figure 12.26. The signals have been normalized and

aligned in time, but match well. A contributing factor to the difference between the

signals is the propagation differences in 2- and 3-dimensions. The wave in the phase

center simulations propagates at least 80cm further than in the virtual sensor model

and is thereby more influenced by propagation differences. The virtual sensor accounts

for signal attenuation based on experimental values, in 3-dimensions. The wave only

propagates from the virtual sensor to the PEC and back - approximately 40cm total

instead of 120cm (measured vertically).

216

0 1 2 3 4 5 6 7 8 9 10

−0.2

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e A

mpl

itude

Time (ns)

Phase Center ReceiverVirtual Source

Figure 12.26: PEC Reflection Comparison Between Phase Center & Virtual SourceModels

12.6 Example Bridge Deck Simulation

The physical model of a bridge deck presented in 7.3.1 and shown in Figure 12.27(a) is

subjected to excitation by the virtual sensor. Two examples of the magnitude of the

electric field throughout the computational model are shown in Figure 12.27(b). At

2.25ns, the shape of the wavefront is becoming evident along the virtual source. At

4.50ns, the wave has propagated through the deck and is scattering from the rebars.

Figure 12.28 shows the final received signal from the virtual sensor, which is analogous

to the signal recorded at the receiver.

12.7 Summary & Conclusions

The implementation of a virtual sensor presented in this chapter compares well with the

more traditional implementation of the phase center sources as evidenced in Figure 12.26.

This approach also has the added benefits of a smaller computational model, and due

to the decreased distance between the sensor and the surface, less susceptibility to the

differences in propagation between two- and three-dimensions.

217

X (grid coordinates)

Z (

grid

coo

rdin

ates

)

500 1000 1500 2000 2500

200

400

600

(a) Physical Model of Bridge Deck

X (grid cooridnates)

Z (

grid

coo

ridna

tes)

T = 2.25ns

500 1000 1500 2000 2500

200

400

600

Amplitude

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

X (grid cooridnates)

Z (

grid

coo

ridna

tes)

T = 4.50ns

500 1000 1500 2000 2500

200

400

600

Amplitude

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

(b) Magnitude of Electric Field ThroughoutModel

Figure 12.27: Bridge Deck Model and Magnitude of Electric Field

0 2 4 6 8 10 12

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (ns)

Nor

mal

ized

Am

plitu

de

Surface of Asphalt

Asphalt/Concrete Interface

Rebar

Bottom of Deck

Figure 12.28: Simulation Results from Bridge Deck Model

The approach presented in this section will be improved through two main areas

of future work. The first will be an expansion to the excitation of the virtual

sensor. Currently, only the electric field along the virtual sensor is excited. Future

implementation of the virtual sensor model will include the additional excitation of the

magnetic fields. The second area of future work will be to validate the computational

model. This will be accomplished through experimentation over a well characterized

delaminated bridge deck section donated to Northeastern University by Infrasense, Inc.

218

Chapter 13

Summary and Contributions

13.1 Summary

Despite differences in propagation due to a point source excitation in 2D rather than

3D, as discussed in Chapter 9, two-dimensional modeling of GPR adequately predicts

how waves scatter from bridge deck components. The difference in propagation is more

pronounced deeper into the deck. Two-dimensional simulation of geometries that are

invariant in the plane perpendicular to the simulation plan, such as layered media and

transverse rebar, match very well with 3D simulations. As more geometric variance is

introduced, the deviation increases. Scattering from objects below strong scatters not

present in the 2D geometry, such as longitudinal rebars, is most different when modeled

in 2D instead of 3D. However, in 3D or 2D, removal of a healthy bridge deck model

from delaminated models highlights scattering from the delamination and its subsequent

interactions with the deck structure. Since GPR data collected in the field represents

wave propagation in three dimensions, it is important to validate that the 2D model and

geometry can adequately capture the subsurface information. While there are differences

between 2D and 3D simulations, the isolation of anomalies in 3D data by removing the

2D healthy deck data is evident.

219

The scattering of electromagnetic waves from rebar produces strong clutter when

compared to scattering from air filled voids. Multiple interactions of the array itself and

the bridge deck layers strengthen this clutter and further obscure the target anomaly.

As indicated in Chapter 11, in order to accurately remove the scattering from the rebar,

the secondary scattered fields must be considered. This is not possible to do using field-

collected data because it is essential to fully characterize the bridge deck. However, using

forward FDTD modeling, the scattering from an anomaly can be isolated. It has been

shown that scattering from an anomaly may be significantly influenced by the presence

of rebar. These influences, sometimes constructive, sometimes destructive, can give a

false representation of the extent of the anomaly and its electromagnetic properties of

the anomaly.

13.2 Contributions

The ISMME tool presented in Chapters 6-8 enables the representation of complicated

civil infrastructure geometry such as random rough interfaces, arrays of rebar, and

randomly sized and located inclusions (aggregate). Model geometry, electromagnetic

properties of model components, and simulation parameters are easily varied. The object-

oriented ISMME can be easily extended to applications other than civil infrastructure

and to sensing modalities other than GPR. While making enhancements to ISMME

may require some familiarity with object-oriented MATLAB, the use of ISMME does

not. The code has been successfully used in the Gordon-CenSSIS Research Experience

for Undergraduates program and in a Civil & Environmental Engineering Independent

Study project.

The model-based assessment technique in Chapter 10 results in the creation of the

bridge deck geometry, assuming a healthy condition. This model varies conventionally

obtained approximate parameters to find values that best fit the field-collected data.

220

While the scattering from anomalies may not be distinguishable from scattering from

rebar or other deck features in the raw GPR data, comparing the computed healthy deck

response and field-collected response will help identify areas of suspect condition.

Many inversion techniques deal well with field sources, but are complicated by strong

scatterers such as rebar. As discussed in Chapter 11, replacing rebar with a source that

has a propagating wave equivalent to the scattering from the rebar performs well on

synthetic data and makes the bridge deck problem better suited for inversion.

In Chapter 12, experimentally obtained data was used to characterize a bi-static

GPR horn antenna. This information was used to compute a virtual sensor implemented

into the 2D FDTD code. The waves from the virtual sensor propagate as those from

the GPR antenna. It has been shown that the resulting waveform matches well with

field-collected data. Furthermore, using the virtual sensor in lieu of the more traditional

phase center excitation significantly reduces the size of the computational area. The

virtual source incorporates 3-dimensional propagation characteristics between transmit

and receive phase centers and the virtual source located approximately 40cm below.

Since the wave does not have to propagate through this region, the recorded signal is less

subject to differences in propagation between the 2D and 3D excitations, and therefore

more representative of the actual interactions.

13.3 Future Work

Differences in 2D and 3D propagation were noted in Chapter 9. It is possible that the

implementation of a time dependent scaling factor to the 2D results may remove some

of the discrepancies between the 2D and 3D B-scans. Exploration of this hypothesis is

an area of future work.

Currently, the excitation source for the virtual sensor presented in Chapter 12 includes

only an electric field component. As an area of future work, the corresponding magnetic

221

fields will be computed and applied as additional excitation along the virtual sensor.

Infrasense, Inc. donated a 4-foot by 8-foot reinforced concrete slab to Northeastern

University. The test slab was built to Massachusetts Highway Department specifications

with the notable exception of well characterized subsurface inclusions to simulate air

voids. In future work the test slab will be used to validate the virtual sensor presented in

Chapter 12. Data collected on the slab will serve as the field-collected response for the

model based assessment technique discussed in Chapter 10, which will utilize the more

realistic virtual sensor model.

222

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