Electric characterization of construction materials through radar data inversion

Electric characterization

of construction materials through radar

data inversion

Electric characterization

of construction materials through radar

data inversion

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op maandag 9 september 2013 om 10:00 uur

door

Claudio PATRIARCA

Master of Science in Geology Applied to the Engineering and Land

Planning

Sapienza, Universit`

a di Roma

geboren te Sora, Italy

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. E.C., Slob, Technische Universiteit Delft, promotor Prof. dr. ir. T.J, Heimovaara, Technische Universiteit Delft

Prof. dr. A., Yarovoy, Technische Universiteit Delft Prof. dr.ir. A., Scarpas, Technische Universiteit Delft Prof. dr. ir. S. Lambot, Universit´e catholique de Louvain Dr. ir. D.J.M., Ngan-Tillard, Technische Universiteit Delft Dr. ir. V., Kovalenko, Fugro GeoServices

ISBN 978-94-6203-394-8

Copyright c 2013, by C. Patriarca, Section of Applied Geophysics and Petrophysics, De-partment of Geoscience & Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands.

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic, mechanical, photocopying, recording or by any information storage and retrieval system without the prior written permission of the author.

SUPPORT

The research leading to these results has received funding from the European Community’s Seventh Framework Program [FP7/2007-2013] under grant agreement no. 213651. Typesetting system: LA_{TEX2e Documentation System.}

Contents

1 Introduction 1

1.1 An overview of non-destructive methods . . . 1

1.1.1 Acoustic methods . . . 2

1.1.2 Optical methods . . . 2

1.1.3 Electromagnetic methods . . . 3

1.2 Non-destructive evaluation role in management . . . 3

1.3 Ground Penetrating Radar and Transmission Line: EM methods and techniques . . . 4

1.4 Statement of the Problem . . . 7

1.5 Thesis Outline . . . 8

2 Determining Electromagnetic Properties 9 2.1 Introduction . . . 9

2.2 Techniques . . . 10

2.2.1 Instruments and test methods . . . 10

2.2.2 Materials and experimental procedure . . . 12

2.3 Results . . . 14

2.3.1 Transmission Line Experimental Results . . . 15

2.3.2 GPR Results . . . 26

2.4 Conclusion . . . 27

3 Data Inversion and Interpretation 29 3.1 Introduction . . . 29

3.2 Materials and Methods . . . 31

3.2.1 Forward and Inverse Modeling . . . 31

3.3 Numerical Experiments . . . 35

3.3.1 A: Single Slab . . . 36

3.3.2 Configuration B: Overlapped Slabs . . . 37

3.4 Laboratory Experiments . . . 39

3.4.1 Configuration A: Single Slabs Results and Discussion . . . 42

3.4.2 Configuration B: Overlapped Slabs Results and Discussion . . 45

4 Acquisition and Inversion Uncertainty Quantification 53

4.1 Introduction . . . 53

4.1.1 GPR equipment . . . 55

4.1.2 Antenna characteristic coefficients determination . . . 55

4.2 Model Accuracy . . . 56

4.2.1 Uncertainties in the calibration method . . . 56

4.2.2 Measured and computed Green’s functions . . . 58

4.2.3 Inversion experiments . . . 60

4.2.4 Data Precision . . . 61

4.3 Results and Discussion . . . 62

4.3.1 Uncertainties in the calibration method . . . 64

4.3.2 Measured and computed Green’s functions . . . 65

4.3.3 Inversion results . . . 69

4.3.4 Data Precision . . . 71

4.4 Conclusion . . . 71

5 Conclusions and Recommendations 75 5.1 Conclusions . . . 75 5.2 Recommendations . . . 77 Appendices 79 A Appendix 81 A.1 Introduction . . . 81 A.2 Objective . . . 82

A.3 Theoretical background . . . 83

A.3.1 Time-domain signal picking . . . 83

A.3.2 Full-waveform inversion technique . . . 83

A.3.3 Rayleigh scattering technique . . . 83

A.4 Experimental framework . . . 84

A.4.1 Experimental design . . . 84

A.4.2 Test devices and equipment . . . 85

A.4.3 Materials and laboratory testing . . . 85

A.5 Results and discussion . . . 87

A.5.1 Clay content estimation – Rayleigh scattering method . . . . 87

A.5.2 Clay content estimation from the full-waveform inversion and the time-domain signal picking techniques . . . 93

A.6 Conclusion . . . 97

Bibliography 100

Summary 113

CONTENTS iii

About the Author 120

Chapter 1

Introduction

The object of this thesis is the evaluation of electromagnetic methods as non-destructive assessment and monitoring tools in a variety of applications. These range from the support of refurbishment works on conventional and heritage buildings, to pavement and road subgrade inspection. In this Chapter an overview of the most efficient non-destructive techniques is given for the applications of interest, focusing on electromagnetic methods. These methods find applications outside the domain discussed in the thesis, therefore a brief overview of other possible applications is also discussed. This work mainly aims at the ability to retrieve electrical properties from the media investigated; from this information it is possible, to some extent, to assess the integrity of structures, to individuate defects and their entity. We consider here known methods for the quantitative reconstruction of the electric properties that are still in the research phase, as well as qualitative methods suitable for subsur-face imaging. In both cases an unknown electromagnetic profile is reconstructed, in terms of absolute values or relative contrasts, respectively. The basics of the considered electromagnetic techniques are presented and a prospect in the field of non-destructive testing is examined. Lastly, the outline and the research objectives of this thesis are provided.

1.1

An overview of non-destructive methods

The aim of non-destructive testing or evaluation (NDT/NDE) methods is, gen-erally speaking, to detect and classify defects and damages in the investigated media [Maierhofer, 2010]; quantification is carried out when possible. The aim of non-destructive investigation includes the determination of position and dimensions of heterogeneities, intended as changes in physical properties inside a homogeneous do-main. These methods are now being used mostly to plan repair and maintenance of constructions. Non-destructive techniques find applications in a broad field of inves-tigations. Amongst the numerous applications, the most common include concrete

testing [Beutel et al., 2008; D´erobert et al., 2008; D´erobert and Berenger, 2010], inspection for land use purposes [Fenning and McCannt, 1995; Minet et al., 2011], for ancient building preservation [Ranalli et al., 2004; Solla et al., 2011] and modern infrastructures safety, characterization of stone masonries [Binda et al., 1998, 2005; Diamanti and Redman, 2012; Helmerich et al., 2012], and bridge decks inspection [Hugenschmidt and Mastrangelo, 2006; Saarenketo and Soderqvist, 1994]. In pave-ment engineering, driving safety is strictly related to pavepave-ment surface conditions [Tighe et al., 2000]. Cracks, potholes, and surface deformations generate sudden vertical accelerations on the vehicle tires, decreasing the effective friction between tires and pavement. Such road damages are mostly related to a reduction of the bearing ratio of the sub-base and sub-grade courses [Diefenderfer et al., 2005]. Pre-vious research has demonstrated that structural damage in road pavement depends on the moisture percentage in sub-asphalt courses [Benedetto and Pensa, 2007]. In particular, in the case of clayey unbound courses, the infiltration of water through the pavement causes the transport of plastic materials. In pavement engineering, this type of damage is classified as pumping and it is caused by several factors, such as inadequate compaction during construction, poor mix design, high water table, and poor drainage [Al-Qadi et al., 2004; Tosti and Benedetto, 2012].

NDT methods can be broadly classified into acoustic, electromagnetic, and op-tical; a brief description of each group is provided in this section. A good overview of the most common testing problems and how these can be analyzed with different NDT methods, is given in Maierhofer et al. [2010].

1.1.1

Acoustic methods

The propagation of sound waves is the basic principle of acoustic methods; the frequency bandwidth of the emitted and detected signals can be chosen, and this choice determines the variation in spatial resolution. The propagation time of im-pulses is measured in most of the applications; in some methods, the amplitude of the signal is analyzed. The acoustic methods include, for instance, acoustic/ultrasonic emissions and impact echo [Beutel et al., 2008; Sidney, 2003].

1.1.2

Optical methods

Optical methods have long been used in civil engineering for topographical map-ping, for the quantitative measurement of deformations, e.g., using photogrammetry, laser scanner, laser vibrometer, speckle interferometry and stereography [Maierhofer et al., 2010]. More recent methods record direct images of the surface, e.g., digi-tal photography, videoscopy, thermography [Gary, 2003]; other methods involve a spectral analysis of the object under investigation.

1.2 Non-destructive evaluation role in management 3

1.1.3

Electromagnetic methods

Electromagnetic methods normally used in civil engineering are based on the propagation velocity of electromagnetic waves in materials. Most of these meth-ods are based on the transmission and reflection of very short electromagnetic im-pulses, which are emitted and detected by antennas. Configurations in reflection and transmission are possible, similarly to what is done in acoustic or ultrasonic techniques [Sassen and Everett, 2009; Tsoflias and Hoch, 2006]. The propagation velocity is controlled by the electric permittivity, and therefore, it is highly influ-enced by the moisture content, while the attenuation of the electromagnetic wave energy depends on the conductivity and dielectric losses (which are significant above 1 GHz when water is present). Electromagnetic methods broadly include radar, ca-pacimetry, electrical resistivity, optical, thermographic, potential field methods, and other microwave methods. Other areas of application different from civil engineering where NDT are established as routine techniques include the inspection of aircrafts, nuclear facilities, chemical plants, transport systems, electronic devices and other safety-critical installations.

The application of NDT methods is normally most useful at the beginning of a building assessment; for obvious reasons this is not always the standard procedure, and an assessment might be required on structures with different conditions and age, and at different stages (during or after construction, during service or regular maintenance, after damage and deterioration, before during and after repair). With the application of radar or active thermography, large areas can be covered in a short time enabling a fast overview on the position of the main defects, like the presence of delaminations and voids, the presence and distribution of moisture, and geometrical parameters. Selected smaller areas can be investigated successively with enhanced accuracy with further NDT, or with minor destructive or destructive techniques. In addition, NDT methods can be applied repeatedly over longer periods for monitoring purposes. The application of NDT methods is possible in all areas of a construction that are accessible from the surface. At the present stage of NDT methods devel-opment, conventional destructive methods cannot be replaced completely, but the number of core samples taken can be reduced considerably [Maierhofer et al., 2010].

1.2

Non-destructive evaluation role in management

Despite the techniques mentioned so far can be somehow similar to our prob-lems, a deep difference exists in the monuments conservation approach and in the motivations behind the intervention. Moreover, a clear distinction needs to be made between the assessment before, during and after the intervention in heritage build-ings and in the modern structures. The distinction is based on the fact that ethical questions outside the scope of this work need to be considered when planning an intervention on ancient buildings; on the contrary, these do not normally arise when modern infrastructures are considered. Conservation of ancient buildings involves an

understanding, interpreting and managing the architectural heritage to deliver it to posterity [D’Ayala and Forsyth, 2007]. The institutional bodies and the community are responsible for the care of historic buildings for ensuring utility with minimum loss of fabric and value.

Besides quality assurance during and after the construction of new structures or after reconstruction, the characterization of material properties and damage as a function of time and environmental influences has become a serious concern [Jernberg et al., 2004]. For this reason, non-destructive testing methods have an increasing po-tential to be part of a management system for the infrastructures [Maierhofer et al., 2004]. Even modern infrastructures inevitably deteriorate, requiring monitoring and maintenance. The deterioration can be prevented using the available resources effec-tively through good management strategies. Proper management actions consist in the optimization of maintenance to maximize the benefits and reducing the costs on the long run [Hudson et al., 1993]. Advanced non-destructive evaluation methods are being used to support maintenance, despite a number of applications described in the literature are still under development. However, there is clear evidence that these techniques are already playing a major role in maintenance programs, as demon-strated by the evolving research and applications [Abudayyeh et al., 2004].

The goal of a good maintenance program is to achieve the maximum benefit at the least cost to the public. Management decisions must include the assessment of risk factors, such as roads deterioration due to changing climatic conditions [Mills and Andrey, 2002], over the entire life of an infrastructure. Among the benefits of a successful programme we can include the public safety, the reduction or elimination of risky factors, the correction of deficiencies in a relatively short amount of time, the efficient allocation of funds, choosing the correct maintenance measures, avoid costly repairs through preventive maintenance, and generally, minimize life-cycle costs [Abudayyeh et al., 2004].

Non-destructive testing methods in civil engineering are still not established for regular inspections, and only a few standardized procedures exist worldwide. Guidelines for NDT are currently applied only in special cases, mainly for damage assessment. In recent years, rapid, high-level progress was achieved in the develop-ment of technology, data analysis and reconstruction, automation and measuredevelop-ment strategies [Maierhofer et al., 2010]. However, extra efforts are required to establish these technologies for regular building check-ups and towards the standardization of techniques.

1.3

Ground Penetrating Radar and Transmission

Line: EM methods and techniques

Refurbishment and conservation activities always start with material imaging and characterization to assess the type and extent of damage. The assessment should be carried out in a non-destructive manner, for the obvious reason that

con-1.3 Ground Penetrating Radar and Transmission Line: EM methods and techniques 5

solidation is done with the intention to increase the integrity of stones. Damage assessment usually starts with visual inspection and it is sometimes combined with on-site measurements. This happens if suitable methods are available, and labora-tory investigations on samples may follow. NDE methods are the best candidates for on-site measurements and evaluations, and can be found in different stages of de-velopment. There are standardized methods that have been thoroughly employed in the field, and methods that have been already applied in the field but research efforts are required for clearer interpretation and more successful applications. Developing a NDE method involves three distinct parts:

• testing of the theory which is believed to provide the desired information to be extracted from the measurements;

• testing the equipment used to carry out the measurements. In some cases this is standardized, in some other, an experimental method is used to generate and record the data;

• data analysis, which links the data acquired and the theory in a more or less complex way to extract the physical properties of the tested objects.

Ground penetrating radar (GPR) qualitative imaging can be considered as a fairly developed method. Stepped-frequency continuous-wave (SFCW) radar sys-tems, on the contrary, are still under development and can be considered as local methods, being applicable only to a small area of the object being examined. In order to achieve larger sections a scanning must be employed, which is the appli-cation of sequential local observations in contiguous areas. As a general approach, visual inspection usually precedes the use of GPR to identify the location of obvious problems, and therefore, guides testing to individuate subtle ones. It would not be practical to apply blindly NDE methods to an entire infrastructure, but an input of the area prone to failure should be given in determining where tests need to be conducted. In such way, NDE can help individuate where problems lie.

The problem of repeating measurements to cover a larger area is particularly delicate if acquisition time is long and data processing cannot be done in real time. This is the case for the SFCW approach used in this thesis for quantitative data inversion.

SFCW Radar

Ground penetrating radar is used in earth science and civil engineering for a large variety of applications and in many different disciplines [Daniels, 2004; Jol, 2009; Slob et al., 2010]. The possibility of non-destructive testing has generated growing interest in improving electromagnetic methods, whose prime goal is to char-acterize materials. Non-destructive testing methods can quickly provide qualitative images at reasonable cost; thorough physical characterization is not established as the routine yet, despite the feasibility has been convincingly demonstrated [Lambot et al., 2004b; Minet et al., 2009; Patriarca et al., 2011].

Ground penetrating radar exploits the wave character of EM fields, and it is effective in low-loss materials, where energy dissipation is small compared to energy storage [Jol, 2009]. GPR techniques depend on the detection of scattered signal. Radar systems use transmitting and receiving antennas, which transform electrical signals to and from vector EM fields. The antennas might be coincident (mono-static configuration), two separate features (bi-static configuration) or antenna arrays (sev-eral combinations of sources and receivers are possible). Reflection and transmission coefficients allow to quantify variations in the EM fields across interfaces between materials [Annan, 2009]. The impedance and admittance contrasts between materi-als are responsible for an EM impedance contrast that generates a response.

The most common radar systems operate in the time domain while stepped fre-quency continuous-wave radar operates in the frefre-quency domain. A frefre-quency sweep is performed through linear increments over a chosen bandwidth, with specific start and stop frequencies. The reflected energy is received as a function of frequency and indicates the amplitude and phase of energy scattered from subsurface objects. The received-to-emitted signal is sampled during the sweep and recorded as a function of frequency, i.e., at each discrete frequency [Koppenjan, 2009].

Koppenjan et al. [2000] present several advantages of the stepped-frequency GPR, namely: the controlled transmission frequencies, efficient use of power, and efficient sampling of ultra-wideband signals. These systems permit the collection of complex reflection functions, which allow complex processing algorithms, and more importantly, to control the amplitude information. Disadvantages of SFCW include the complex and relatively expensive electronics, and, for non-specialized users, the need for digital signal processing since commercial software are not developed.

SFCW radars have extensively been used in the detection of landmines [Lang-man and Inggs, 1998; Lopera et al., 2007b; Nicolaescu and Van Genderen, 2008; Soldovieri et al., 2011], for through wall imaging and motion detection [Biying et al., 2011], and for other applications requiring the accurate characterization of subsur-face electric properties [Lambot et al., 2004b]. B¨oniger and Tronicke [2012] used a spectral decomposition method to study thin-bed responses, and showed how it is possible to effectively improve the resolution or compensate for potential frequency attenuation effects at late arrivals. Leckebusch [2011] compared SFCW and pulsed systems using the same frequency bandwidth. They found advantages in using a lim-ited frequency range, as it is usually done in conventional pulsed systems, obtaining better results at a faster recording speed. A signal processing method allowing for the signal phase structure to be used for extracting additional information concerning the physical properties of buried inhomogeneities is presented by Sugak and Sugak [2010]. They consider the relationship between the signal phase spectrum and the electrical properties of a soil and buried objects through experimental measurements. Relationships of random step frequency radar are compared with frequency modu-lated continuous wave noise radar and the statistical characteristics of the ambiguity function and the sidelobe noise floor are analyzed by Axelsson [2007].

1.4 Statement of the Problem 7

1.4

Statement of the Problem

The response of materials to an applied EM field is described by Maxwell’s equa-tion through constitutive parameters [Maxwell, 1865]. These parameters describe the macroscopic electromagnetic properties, namely the electrical permittivity and conductivity, and the magnetic permeability. In geophysics often we are interested in the imaging capabilities of electromagnetic waves for a given material. For this purpose an EM signal is emitted by the antenna, trasmitted into the subsurface and an image is produced from the retrieved response. It is therefore interesting to know how the materials commonly found in the subsurface or below the surface of complex infrastructures react to applied EM fields.

Many models exist that macroscopically describe the EM behavior of materi-als. To investigate the response of the materials we are interested in, laboratory measurements are carried out considering soil mixtures of growing complexity. The materials considered in this thesis are multi-phase mixtures composed by several constituents (solid, liquid and gas). This complexity has an impact on the choice of the model we adopt to represent the subsurface, irrespectively of the environment. For this reason the first part of the thesis is devoted to measuring the response of materials to an applied EM field, and to accurately reconstruct their EM properties on a wide bandwidth. We restricted our study to the permittivity and conductivity because most materials of our interest are non-magnetic.

The most common modelling approach used in EM exploration considers the subsurface as being multilayered. The EM parameters assigned to each layer might be constant (piecewise constant model) or varying as a function of depth, frequency, or both. However, how exactly commonly encountered materials react to an ap-plied field remains unclear, therefore the need exists to study several multi-phase compounds in growing order of complexity and in the frequency range of interest.

Moreover, mixing models allow to compute electrical permittivity from the rel-ative abundances and permittivities of the mix constituents. This is usually done in a narrow frequency spectrum or for a single frequency, but for UWB applications a clear description of the reaction is still needed. From the measured response it is evident that there is not a single model performing better than others, which leads to a site- and application-specific approach, using as starting models the existing ones with due corrections. Experiments are used to test the ability of material char-acterization and of defects detection having no or little a-priori information on the electromagnetic and geometrical parameters.

In inverse problems the quantification of uncertainty should be always accounted for, together with the error propagation in the parameters estimation inversion pro-cess. A statistical description should always be linked to inversion results. This information can help the geophysicist accept the inversion results, which are neces-sarily carried out at a practical computational cost. Realistic results that suggest a close match to field surveys are achieved keeping in mind the limitations of our models, assumptions and instrumentation.

1.5

Thesis Outline

Having introduced the GPR sensitivity to electrical properties of the investi-gated media, the research objective of this thesis is to examine the ability of one par-ticular radar technology to provide quantitative information in damage assessment and monitoring capabilities in the framework of inspection activities. In particular, an off-ground radar system is tested through numerical and laboratory studies to investigate its usage in the built environment.

The thesis is structured as follows:

• Chapter 2 discusses the electromagnetic properties of multiphase aggregate mixtures as a function of water and clay contents. The frequency dependent electrical properties of materials are investigated; the resulting models are the base for the data inversions performed in Chapters 3 and 4. Mixing models are also evaluated to assess the capacity of indirectly retrieving the volumetric water content for a given EM profile and for known sand-clay mixtures. The results shown here are of interest for environmental applications as much as for NDT purposes.

• Chapter 3 describes numerical and laboratory experiments conducted using a radar system with the aim of reconstructing material electrical properties and, eventually, delaminations using full-waveform inverse modeling. Laboratory experiments were conducted on non-reinforced concrete and plaster slabs ly-ing in different configurations. The results showed the good potential of this method: (1) to provide a thorough fracture response model in buildings or artworks and (2) to non-invasively characterize the samples in terms of their electromagnetic properties.

• Chapter 4 discusses the uncertainty quantification in the EM and geometri-cal properties; this uncertainty mainly originates from the data and geometri- calibra-tion measurements acquisicalibra-tion, propagates along the inversion procedure and results in bounding full-waveform inversion results or limiting the ability to retrieve the correct Green’s function. The results obtained in Chapter 3 are therefore given the right relevance in terms of expected accuracy and relative error.

• Chapter 5 is a summary of the main findings, and it concludes the present research giving recommendations for future developments.

Appendix A contains a detailed description of radar experiments performed with different Ground Penetrating Radar tools and techniques used to non-destructively investigate the clay content in sub-asphalt compacted soils. The experiment is aimed at evaluating the electrical properties of typical materials employed for road bearing courses construction. Signals are processed in both time and frequency domains, and the consistency of results is validated by the Rayleigh scattering method, the full-waveform inversion and the time-domain signal picking.

Chapter 2

Determining Electromagnetic

Properties

1

2.1

Introduction

Non-destructive characterization of road pavement layers is crucial for the life, safety, and maintenance of infrastructures [Tighe et al., 2000]. Road maintenance can incur considerable direct and indirect costs [Benedetto and Pensa, 2007], which has led to comprehensive cost-benefit assessments. Geophysics has been used to evaluate pavement conditions and to assess potential or existing damages. In particular, it is widely recognized that ground-penetrating radar (GPR) is a valuable tool for non-destructive road inspection and condition monitoring [Jol, 2009; Loizos and Plati, 2007].

Considerable efforts went into the evaluation of concrete and asphalt thickness [Saarenketo and Scullion, 2000; Spagnolini and Rampa, 1999], reinforced concrete bridge decks deterioration [Hugenschmidt and Mastrangelo, 2006], and subsurface soil investigations. Recent research focused on the detection of damages in the as-phalt layers, recognizing that the origin of instabilities might be due to the subgrade soil behavior [Diamanti and Redman, 2012]. Research was conducted to accurately determine and model the electric properties of soil multi-phase mixtures in the mi-crowave region [Dobson et al., 1985; Hallikainen et al., 1985]. A review of the existing petrophysical models for GPR frequencies is given by Steelman and Endres [2011]. Saarenketo [1998] measured the electric properties of soils at different densities and water contents at given GPR frequencies; they classified water in soils according to the electrical behavior, as being adsorbed, viscous, or in a free state. The frequency dependency of adsorption and dispersion in lossy media was studied by Bano [2004]

1_{This Chapter has been published as journal paper in the Journal of Applied Geophysics, In} Press (Patriarca et al. [2013b]). Note that minor changes have been introduced to make the text consistent with the other chapters of this thesis.

using the quality factor Q and a complex power function with the aim of modeling the electric behavior of materials. Johnson and Poeter [2005] used the Bruggeman-Hanai-Sen (BHS) mixing model on a three-phase system composed of water, sand, and dense nonaqueous-phase liquid to develop an iterative new weighted BHS model. Despite the accuracy of the results shown in literature for multi-phase mixtures, little attention has been given to the ability to detect clay independently from water content in subgrade soils. A study that combines GPR full-waveform inversion and dielectric mixing models was carried out by Tran et al. [2012]. However, full-waveform inversion results have not yet been compared with reliable ground-truth measurements for sand-clay mixtures. The frequency dependency of the electric parameters has been thoroughly investigated [Gregoire and Hollender, 2004; Lambot et al., 2005], but exact measurements and comparisons in the GPR frequency range for multi-phase mixtures are still missing.

In this chapter, the influence of clay on the electric properties frequency de-pendency, usually neglected or considered in a relatively narrow-band, is properly highlighted for different water content mixtures. Topp’s model [Topp et al., 1980] is used to assess whether the presence of clay limits the capability to determine water content. A volumetric mixing model [Birchak et al., 1974] is tested and extended from two to three-phase simple cases to four-phase mixtures. Measurements are performed using a stepped-frequency GPR on natural size samples, in an effort to extend this approach to field-scale measurements. Results show the possibility to relate water and clay contents to the electric properties, eventually via the quality factor Q. The effect of bound water can be hypothesized both from the transmission line and GPR results.

2.2

Techniques

The frequency dependent complex electric permittivity ε∗

r = ℜ(εr) + ℑ(εr) is

determined using two different setups described in Section 2.2.1. The relative electric permittivity εr = ε/ε0 is the ratio between the electric permittivity ε and the free

space electric permittivity ε0= 8.854 × 10−12 Fm−1. Different samples’ preparation

techniques are used, as explained in Section 2.2.2.

2.2.1

Instruments and test methods

Two setups are used, as described in this Section. For both setups an HP Vector Network Analyzer (VNA) 8753C is used together with an HP S-Parameter test set 85046A connected to a horn antenna with a 50 Ω impedance coaxial cable in the case of GPR, or to the transmission line end points. The frequency bandwidth used in the transmission line setup and in the GPR are 300 kHz – 3 GHz, and 500 MHz – 3 GHz, respectively. The antenna used in the GPR setup is a double-ridged broadband horn (BBHA9120A, Schwarzbeck Mess-Elektronik). The Open-Short-Match-Through calibration kit HP85033C is used to calibrate the VNA at the

2.2 Techniques 11

cable terminations. The frequency dependent scattering parameters for reflection and transmission are collected. The signal was sampled at 1601 frequencies over both bandwidth, 0 dBm transmission power and an averaging factor of 10.

Coaxial Transmission Line

The complex relative electrical permittivity is computed from an explicit ex-pression that relies on the measured scattering parameters using the propagation matrices method [Thomson, 1950]; the adaptation of this method to the EM case is based on an analytical solution and is described in details in Gorriti and Slob [2005a].

The measured S-parameters of a transmission line, if only one port is considered as emitter, can be expressed as

" 1 S11 # = LSR " S12 0 # , (2.2.1)

where S represents the propagation inside the sample holder, while L and R stand for the propagation at the left and right of the sample holder. Equation (2.2.1) can be rewritten such that _"

A B # = S " C D # , (2.2.2) where " A B # = L−1 " 1 S11 # and " C D # = R " S12 0 # . (2.2.3)

Considering a transmission line of length ds made of perfectly conducting material

with P and r as outer and inner radii, respectively, the complex relative electric permittivity of the sample can be related to the complex impedance as:

Zs=Z0 εr , whereZ0= ln(P/r) 2π r µ0 ε0 . (2.2.4)

Moreover, the complex propagation factor γs is also expressed as a function of the

permittivity and frequency:

γs=

ω c0

√_ε

r. (2.2.5)

The propagation in the sample holder is given by: S = 1 2 " eγsds_{+ e}−γsds _Z p(eγsds− e−γsds) 1 Zs(e γsds_{− e}−γsds_{) e}γsds_{+ e}−γsds # . (2.2.6)

Table 2.1: Physical properties of the materials used. Ranges of values are given for param-eters determined over different water content samples. The symbol γ indicates the specific weight in natural conditions (subscript n), in dry conditions (dry) or in saturated conditions (sat); εr indicates the real electric permittivity and θv% the volumetric water content.

γn(g cm−3) γdry(g cm−3) γsat(g cm−3) Porosity εr θv% Saturation %

Sand A2 1.58–1.94 1.45–1.54 2.66 0.40 2.6–2.9 0–0.39 0–94

Sand A3 1.39–1.78 1.33–1.39 2.66 0.36 2.5–2.8 0–0.39 0–86

Clay – 1.71 – 2.41 – 2.98–10.0 – –

Substitution of Equation (2.2.6) into (2.2.1) allows to find a solution for the expo-nents as:

eγsds_{+ e}−γsds_{= 2}AB + CD

AD + BC, (2.2.7)

that can be solved for the propagation factor γs.

γs=

arccoshAB+CD_AD+BC ds

. (2.2.8)

Ground-Penetrating Radar

For a single transverse electromagnetic (TEM) horn antenna connected to a VNA and illuminating the earth in the far field, an effective radar antenna mathe-matical model [Lambot et al., 2004b] allows to retrieve the earth impulse response G↑

xx(ω) from the measured data S11(ω) as

G↑ xx(ω) =

S11(ω) − Hi(ω)

H(ω) + Hf(ω)[S11(ω) − Hi(ω)]

. (2.2.9)

The Green’s function is a solution to Maxwell’s equations that can be computed analytically if the geometrical and electric model parameters are known. A modeled Green’s function is therefore iteratively inverted to match the measured Green’s function by mismatch minimization.

2.2.2

Materials and experimental procedure

The experiments described are conducted in a temperature-controlled labora-tory environment, at 20◦_C±2◦_{C. Two different sand grain sizes are used: a 1.00–2.00}

mm and a 0.125–0.250 mm, hereafter named, respectively, A2 and A3; a distinction is made only based on the particle-size distribution according to the AASHTO soil classification [AASHTO, 2011]. Some physical properties of the material used are given in Table 2.1.

2.2 Techniques 13

Table 2.2: The natural specific weight γn expressed in (g cm−3_{) is shown for sand A2 for} increasing clay and water contents.

Water content %w 0 5 10 12 14 16 18 20 22 24 26 28 30 C la y co n ten t 0% 1.58 1.52 1.58 1.62 1.67 1.73 1.79 1.77 1.89 1.86 1.93 1.93 1.94 5% 1.65 1.52 1.57 1.63 1.64 1.67 1.74 1.73 1.79 1.82 1.87 1.94 1.89 10% 1.77 1.59 1.60 1.66 1.68 1.70 1.75 1.82 1.79 1.84 1.87 1.90 1.91 15% 1.73 1.63 1.67 1.68 1.71 1.71 1.76 1.77 1.82 1.85 1.86 1.90 1.91 20% 1.71 1.56 1.60 1.66 1.70 1.75 1.80 1.80 1.83 1.85 1.90 1.,89 1.91 25% 1.78 1.59 1.70 1.73 1.78 1.78 1.86 1.86 1.89 1.90 1.92 1.90 1.87 Transmission Line

The measured electric parameters are functions of the relative abundance of the constituents present in the mixtures and of their relative electric properties. The analyzed components are two different mineralogy sands, bentonite clay, water and air. To evaluate the effect of water and clay content on EM signals the following variables are considered:

• clay content: from 0%w to 25%w using steps of 5% increase in weight in dry

conditions;

• water saturation: from oven-dry to 30%w water content (above 90%

satura-tion) using non-linearly increasing steps for a given mixture.

A total of 78 measurements of the electric properties were performed for each sand as a function of frequency. Fully-saturated samples are prepared for the two sands A2 and A3.

Calculated amounts of water with a conductivity of 508 µS/cm were used to reach the desired moisture; manual mixing followed until a homogeneous mix was obtained. The volumetric content of all phases is known since the densities of the single phases were measured with a gas pycnometer (Tosti et al. [2013]); the trans-mission line sample holder volume is known, as well as the GPR sample containers. The actual water content is determined gravimetrically using the entire sample after measurements. The samples’ natural volume weight, indicated as γn (g/cm3), is

de-termined before and after measurements; the difference was always less than 1.5%, confirming a limited loss of material and water evaporation during measurements (15 minutes).

Samples’ composition, water content and natural specific density γnare reported

in Tables 2.2 and 2.3. The dry sample often shows a higher density compared to the 5%w water content sample of the same mix. This is explained by clay and water

coating sand grains, reducing the amount of grained clay between coated particles and increasing the void ratio.

The dry specific weight γdry (g cm−3) refers to the volume weight of the dry

Table 2.3: The natural specific weight γn expressed in (g cm−3_{) is shown for sand A3 for} increasing clay and water contents.

Water content %w 0 5 10 12 14 16 18 20 22 24 26 28 30 C la y co n ten t 0% – 1.40 1.41 1.50 1.49 1.53 1.58 1.65 1.68 1.76 1.75 1.79 1.79 5% – 1.44 1.50 1.52 1.61 1.59 1.63 1.67 1.67 1.73 1.76 1.80 1.81 10% – 1.50 1.56 1.59 1.62 1.67 1.68 1.74 1.76 1.76 1.80 1.83 1.85 15% 1.57 1.49 1.54 1.58 1.62 1.64 1.70 1.73 1.74 1.78 1.81 1.82 1.84 20% – 1.51 1.59 1.61 1.65 1.67 1.70 1.74 1.73 1.73 1.78 1.79 1.80 25% 160 1.52 1.54 1.63 1.66 1.67 1.71 1.73 1.75 1.76 1.79 2.09 1.75

independently of the level of saturation. However, the γdryincreases with increasing

saturation, showing that hygroscopic water strongly bounds to clay minerals, and that standard drying techniques are not successful in removing it from the mineral lattices [Wang and Schmugge, 1980].

Ground-Penetrating Radar

In GPR measurements the end-member conditions for maximum clay content was varied:

• saturation: from oven-dry to 25%wwater content using non-linearly increasing

steps.

Soil mixtures are placed into a dedicated container for measurements; the GPR mea-surements are a larger scale analogue of the experiments described in Section 2.2.2. The container is PVC made, 10.5 × 40 × 47 cm3 _{in size; this lays on a 1.5 × 1.5}

m2 _{perfect electric conductor (PEC) plate used to control electric bottom boundary}

conditions. The size of the box is chosen such that the penetration depth is satis-factory and is comparable to what is realistically achievable on the site. Necessary assumptions require the samples be homogeneous and laterally infinite extended lay-ers. A detailed description of the sample preparation for this technique is found in Tosti et al. [2013].

2.3

Results

The physical state of water present in the system is to be considered when evaluating the effective permittivity of the mix. Water can be tightly or loosely bound to soil particle surfaces, or unbound free-water [Saarenketo, 1998; Wang and Schmugge, 1980]. The relaxation frequency and the ability to be polarized is different in those states. A quantification of the amount of water necessary to transit from the bound to the free water state is possible. From our measured data only loosely or tightly bound water can be identified. In fact, the humidity step used in the low-end is larger than the ones adopted in dedicated research [Saarenketo, 1998]. The tightly

2.3 Results 15

0.5

1

1.5

2

2.5

3

5

10

frequency (GHz)

ℜ

(

ε

r

)

0.5

1

1.5

2

2.5

3

0

5

frequency (GHz)

ℑ

(

ε

r

)

Figure 2.1: Real and imaginary part of the εr measured for A2, A3 sands and clay over the entire frequency range.

bound mono-molecular water layer was observed by Wang and Schmugge [1980] at volumetric water contents below 0.1 θv. However, we can presume that our first

moisture steps fall into the bound water behavior, being between 0 and 10%w.

2.3.1

Transmission Line Experimental Results

The electric behavior of the single components is discussed in this section. Fig-ure 2.1 shows the real and imaginary parts of the permittivity for dry sands and clay. Despite the grain size and the mineralogical differences, little electrical difference is observed between the sands: they have a constant ℜ(εr) over the entire frequency

range, while ℑ(εr) tends to zero. The clay has a frequency dependent behavior

in the real and imaginary part of the permittivity, showing considerable losses at lower frequencies. Figures 2.2 and 2.3 illustrate the real value of the relative per-mittivity and the electrical conductivity over the entire bandwidth for the measured mixtures. Data are grouped per clay content and different colors represent different water contents.

Undesired layering effects are responsible for the spikes in Figure 2.2(a) around 500 MHz and 1.2 GHz, which also cause instabilities in Q. When coarse sand and water are mixed, water flows at the bottom of the sample holder because of the vertical positioning.

The clay-free data represented in Figure 2.2(a) and 2.3(a) show that for in-creasing water content the permittivity remains independent of frequency. With the presence of clay, even in minimal quantities, a frequency dependent behavior is shown. This behavior is already visible in low-clay and dry or low-water content data, but it becomes more evident for high-water contents below 1 GHz. A higher increase in the electric permittivity was expected as clay and water volume frac-tions increase; however, the observed slight decrease of the electric permittivity is explained by the lower density of clay with respect to sand, which reflects in a lower bulk density of the clay-rich samples. In the clay rich samples the porosity is high, therefore the permittivity tends to decrease when clay exceeds 15%w.

The electric conductivity constantly increases with frequency in clay-free sam-ples. In clay-rich materials, the conductivity is non-linear in the 300 kHz–1 GHz frequency range, while it can be approximated to linear for frequencies higher than 1 GHz. The model of increasing apparent conductivity with frequency sometimes adopted in literature [Lambot et al., 2006; Patriarca et al., 2011] is justified for clay-free samples, and above 1 GHz also for clay-rich sands. An increase in conductivity is observed for increasing water quantities and, as expected, higher conductivity values are registered for higher clay contents.

The quality factor Q is represented in Figures 2.4 and 2.5. Very low losses in the low water and clay contents (Figures 2.4(a)-(b) and 2.5(a)-(b) lead to high values of Q. Those effects are moderated by the dispersive nature of clay. Generally, Q increases linearly with frequency in the 300 kHz–1.5 GHz region, approaching a constant value in the 1.5 GHz–3.0 GHz.

Apart from the instabilities due to very low dispersions shown in Figures 2.4(a)-(b) and 2.5(a)-2.4(a)-(b), the quality factor decreases in samples containing higher amounts of clay. For the same clay content, the quality factor does not vary significantly with different water contents, except for the dry samples.

Empirical Relationship

The volumetric soil water content θvcan be obtained from the bulk permittivity

using empirical relationships. We use Topp’s equation to estimate soil water content [Topp et al., 1980]; an average over frequency is used to determine a single water content value. Topp’s equation is applied for each single frequency, which allows to estimate the average value and the error bounds associated with it (Figure 2.6). Volumetric water content values are compared with the Topp predicted values for each saturation step.

A high correlation is observed between the gravimetrically determined and pre-dicted water content values for clay poor samples (Figure 2.6b), and especially for the 0% clay content (Figure 2.6a). In the dry samples, Topp’s model always overes-timates the water content. This error increases as clay quantities increase, because the high permittivity of clay is interpreted as water, and because clay contains chemically bound water in its mineral grains. Figure 2.6 shows that Topp’s model underestimates water contents for high clay contents. Part of the water becomes

2.3 Results 17 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 0% w0 w5 w10 w12 w14 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 5% w16 w18 w20 w22 (a) (b) 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 10% w24 w26 w28 w30 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 15% (c) (d) 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 20% 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 25% (e) (f)

Figure 2.2: The electric permittivity and the electric conductivity of the A2 sand-clay mix-tures are shown in the 300 kHz–3 GHz bandwidth for the coaxial transmission line measure-ments. Clay contents are shown in the conductivity plot and the legend indicating different water content curves is common to all graphs.

0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 0% Clay 0% w0 w5 w10 w12 w14 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 5% Clay 0% w16 w18 w20 w22 (a) (b) 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 10% Clay 0% w24 w26 w28 w30 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 15% (c) (d) 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 20% 0.5 1 1.5 2 2.5 3 10 20 30 ℜ ( εr ) 0.5 1 1.5 2 2.5 3 0 0.5 1 σ Sm −1 frequency (GHz) Clay 25% (e) (f)

Figure 2.3: The electric permittivity and the electric conductivity of the A3sand-clay mix-tures are shown in the 300 kHz–3 GHz bandwidth for the coaxial transmission line measure-ments. Clay contents are shown in the conductivity plot and the legend indicating different water content curves is common to all graphs.

2.3 Results 19 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 frequency (GHz) Q Clay 0% 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 frequency (GHz) Q Clay 5% 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 10% w0 w5 w10 w12 w14 (a) (b) (c) 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 15% w16 w18 w20 w22 w24 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 20% w26 w28 w30 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 25% (d) (e) (f)

Figure 2.4: The quality factor Q = ℜεr/ℑεr is shown in the 300 kHz–3 GHz bandwidth for the A2-clay mixes. Clay contents are shown in the plots and the legend indicating different water content curves is common to all graphs. To be noted is the change of scale for Q in Figs. 2.4 (a)-(b) with respect to the rest of the Figures.

chemically bound to clay minerals; loosely or tightly bound water is not able to po-larize and behaves as a solid, therefore the expected increase in permittivity is not shown. However, the offset is fairly constant for all water contents, which suggests that a correction of the model taking clay into account is possible. Higher quantities of clay are responsible for larger error bounds in the water content estimates, while negligible levels of uncertainty are observed in the clay-free data. The uncertainty increases for increasing water contents.

An overall good correlation exists between the experimental values and Topp’s prediction, therefore Topp’s model can be used to predict humidity values very close to the observed ones even in the presence of clay. Where the error is larger and the uncertainty higher, the error bounds intercept the diagonal of the expected values. Volumetric Mixing

A volumetric mixing approach is tested that allows to reconstruct the bulk permittivity εb

rof materials by knowing the volume fractions fiof their n constituents

0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 frequency (GHz) Q Clay 0% 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 frequency (GHz) Q Clay 5% 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 10% w0 w5 w10 w12 w14 (a) (b) (c) 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 15% w16 w18 w20 w22 w24 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 20% w26 w28 w30 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 frequency (GHz) Q Clay 25% (d) (e) (f)

Figure 2.5: The quality factor Q = ℜεr/ℑεr is shown in the 300 kHz–3 GHz bandwidth for the A3-clay mixes. Clay contents are shown in the plots and the legend indicating different water content curves is common to all graphs. To be noted is the change of scale for Q in Figs. 2.5 (a)-(b) with respect to the rest of the Figures.

mixing formula is generally written as [Birchak et al., 1974] (εbr)α=

n

X

i=1

fi(εr,i)α, (2.3.10)

where the exponent α is a geometrical factor. When the exponent α takes the value of 0.5 the model is also known as the Complex Refractive Index Measurement (CRIM) [Birchak et al., 1974; Dobson et al., 1985; Gorriti and Slob, 2005b].

Electric properties of the single components are calculated directly from mea-surements; this approach is only possible for water, and if the porosity of two-phase mixture is known. The disadvantage of the volumetric mixing formulae is that the soil porosity, specific densities of the components, their relative abundance and elec-tric permittivity need to be known. The measured permittivity of water is used to model the electric behavior of multi-phase mixtures.

In a two-phase case like air-dry and water-saturated conditions, knowing the porosity φ it is possible to model the matrix permittivity (εmod1

r and εmodr 2) by

2.3 Results 21 0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Estimated θv cm 3 cm −3 Clay 0% Topp A3 θ

v with error bounds Topp A2 θ

v with error bounds

0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Clay 5% Topp A3 θ

v with error bounds Topp A2 θ_v with error bounds

(a) (b) 0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Estimated θv cm 3 cm −3 Clay 10%

Topp A3 θ_v with error bounds Topp A2 θ_v with error bounds

0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Clay 15%

Topp A3 θ_v with error bounds Topp A2 θ_v with error bounds

(c) (d) 0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Gravimetrically Determined θ v cm 3 cm−3 Estimated θv cm 3 cm −3 Clay 20% Topp A3 θ

v with error bounds Topp A2 θ

v with error bounds

0 0.1 0.2 0.3 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Gravimetrically Determined θ v cm 3 cm−3 Clay 25%

Topp A3 θ_v with error bounds Topp A2 θ

v with error bounds

(e) (f)

Figure 2.6: Gravimetrically determined water content and the measured water content de-rived from Topp’s equation. Error bounds extend on the estimated θv axis as a confidence interval for the Topp’s relationship.

A2 A3 Frequency GHz |ε_matdry_−ε mat sat_| 0.2 0.4 0.6 0.8 1 0 1 2 3 0 5 10 15 Frequency GHz α 0.2 0.4 0.6 0.8 1 0 1 2 3 A2 A3 Frequency GHz

|ε_{clay x%}An sat−ε_{clay y%}An sat|

0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 0 20 40 60 80 Frequency GHz α 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 (a) (b)

Figure 2.7: (a) The optimal α value is searched in the 10−2_{≤ α ≤ 1 range for the two sands} A2 (top) and A3 (bottom). The difference between the modeled sand grain matrix obtained from and dry and saturated measurements is shown as a function of frequency. (b) The optimal α value is searched for A2 (top) and A3 (bottom) saturated sand-clay mixtures in the 0.25 ≤ α ≤ 0.85 range. Clay contents of 10 and 15% are used for A3 sands; 20 and 25% are used for A2.

permittivity of the dry phase εdry

r , and for the saturated phase εsatr :

     εmod1 rmatrix = h (εdry r )α−φ 1−φ i1/α εmod2 rmatrix = h (εsat r )α−φεαw 1−φ i1/α (2.3.11)

The parameter α is varied such that it can take values 0.01 ≤ α ≤ 1, and the difference |εmod1

rmatrix− ε mod₂

rmatrix| is plotted versus α in Figure 2.7(a). Alpha does not

depend on frequency and it has a an optimal value corresponding to 0.60 for A2 and 0.59 for A3. Choosing the optimal α value that minimizes the differences between the two εrmatrix modeled, corresponds to a calibration of the volumetric mixing

model on the two end-member conditions.

In Figure 2.8(a) the normalized differences between the predicted and measured permittivities are shown for each water content. Errors below 6% are achieved for the two sands in the end-member cases, except for the dry A3. Despite the optimal α value is used, the error increases significantly with increasing water contents. The procedure of selecting the best α is therefore helpful only if the system remains in a two-phase state, while adding a third phase to the previously calibrated α results in significant errors up to 35%.

A similar procedure was adopted for the three-phase case with sand-clay mix-tures in water saturated conditions. The clay matrix permittivity is modeled by

2.3 Results 23 0 10 20 30 40 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 A3 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 clay 5% A3 clay 5% (a) (b) 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 clay 10% A3 clay 10% 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 clay 15% A3 clay 15% (c) (d) 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 clay 20% A3 clay 20% 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Expected θ % w |εr − ε m od r |/ |εr | A2 clay 25% A3 clay 25% (e) (f)

Figure 2.8: The normalized error on the estimate of εr using the optimal α value for three-phase system (sand, air and water) (a) and for the four-three-phase system (sand, clay, air and water) (b) to (f ). The error is calculated in the 2–3 GHz range for the two sand-clay multiphase systems.

using saturated measurements: εmod r,clay = h_(εsat r ) α_−f

wεαwater−fsandεαsand

fclay

i1/α

. (2.3.12)

The parameter α is varied between 0.25 ≤ α ≤ 0.85, and the difference |εmod1

r,clay −

εmod2

r,clay| is plotted versus α in Figure 2.7(b). The optimal α value depends on

fre-quency, and for high frequencies (between 2–3 GHz, where we limit our analysis) an optimal value is considered for each sand.

In Figure 2.8(b)-(f) the normalized differences between the predicted and mea-sured permittivities are shown for each water content for the four phase mixtures. Errors below 10% are achieved for the two sands in the dry end-member cases (Fig. 2.8(b)) with 5% clay content. The error significantly increases with increasing water contents despite using the optimal α value. The value of this error tends to increase with increasing clay quantities. The procedure of selecting the best α is therefore relatively helpful if the system is in a three-phase state, but adding the fourth phase generates errors up to 50%.

Methods Comparison and Polynomial Fitting

In this section the different methods for indirect εr or θv determination are

compared, and a third-order polynomial equation is used to suggest a correction for Topp’s equation.

In Sections 2.3.1 and 2.3.1 it was shown that the two models used can reliably describe the data when the clay content is zero, and when the variables in the volumetric equation are specifically calibrated for. Topp’s equation was derived for a variety of soils and sand grainsizes, but not to specifically describe clayey soils. The volumetric model limitations encountered in presence of clay are due to its swelling properties and therefore to the lack of control over the voids-clay volume fractions. A comparison between the petrophysical relationship performances is plotted in Figure 2.9. When clay is not present the volumetric model and Topp’s model match the data fairly well; error bounds are given which encompass all the data considering θv equal to ±0.025 (Fig. 2.9(a)). A good agreement is shown by the

volumetric prediction for the A2 sand, and for A3 until 24% water content; after that, the volumetric model severely underestimates the moisture. Measurement values are inside the error bounds of Topp’s equation. For low-clay contents, Topp’s model performs fairly well below 30% water content (Figure 2.9(b)). However, for high-clay and water contents the two models deviate from the measured data. The third-order polynomial fit is also shown in the Figure 2.9(b)-(f), and the relative coefficients are listed in Tables 2.4. The equation used to fit our experimental data can be written as:

εr= 3

X

n=0

λnθn, (2.3.13)

where λ stands for the coefficient; the values of θ used are the gravimetrically de-termined ones. For the sake of comparison, also Topp’s coefficients are shown in

2.3 Results 25 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 θ %_w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 Clay 5% θ %_w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured A2 fit A3 fit (a) (b) 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 Clay 10% θ %_w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured A2 fit A3 fit 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 Clay 15% θ %_w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured A2 fit A3 fit (c) (d) 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 Clay 20% θ % w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured A2 fit A3 fit 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 Clay 25% θ % w εr A2 volumetric A2 measured Topp A3 Volumetric A3 measured A2 fit A3 fit (e) (f)

Figure 2.9: Measured and estimated electric permittivity with different methods for the sands (a) and sand-clay mixtures (b) to (f ); values are averaged between the 2–3 GHz. The red solid curve indicates our proposed polynomial for the A2 sand-clay mixture, the dashed red line indicates the best fit for the A3 sand-clay mixture. Error bounds for θv = ±0.025 are given for Topp’s estimate as dotted lines.

Table 2.4: The third-order polynomial coefficients and the ℓ2 _{norm are given for our} poly-nomial fit and for Topp’s equation for sand-clay mixtures.

Clay %

λ

λ

λ

λ

ℓ

poly

ℓ

Topp

S

an

d

A

2

5%

-1.64e2

2.11e2

0.69

2.89

2.01

2.94

10%

-7.29

1.34e2

-1.41

3.61

1.23

5.52

15%

-61.59

1.56e2

-3.77

3.51

0.87

6.67

20%

-84.91

1.53e2

-1.39

3.31

0.82

8.38

25%

-2.06e2

2.29e2

-13.75

3.84

0.51

8.42

Topp

-76.7

1.46e2

9.3

3.3

–

–

S

an

d

A

3

5%

70.58

60.49

16.34

2.52

1.55

6.26

10%

-2.50

95.82

8.63

2.77

0.77

9.82

15%

-55.200

1.35e2

-0.36

2.94

0.88

11.66

20%

-78.83

1.51e2

-4.36

3.12

0.85

12.49

25%

-44.94

1.34e2

-3.91

3.25

0.68

12.11

Topp

-76.7

1.46e2

9.3

3.3

–

–

Table 2.4; the ℓ2 norm of the residuals is included as an estimate of the error. A significant feature is that the residual error tends to decrease with increasing clay contents (Table 2.4). Topp’s model predicts the volumetric model quite well from low to high clay contents. The Topp’s and volumetric curves for clayey soils are essentially the same: they both underestimate water content of quantities below the 2.5% error interval.

2.3.2

GPR Results

A low-cut filter was applied to the GPR data for frequencies lower than 1 GHz, where the antenna directionality is reduced. Measurements are performed for the A2-clay mix from dry until 18% water content, and from dry until 25% water content for the A3-clay mix. Higher water contents could not be achieved because of the difficulty in samples preparation.

A sudden change occurs in the electric permittivity after 10% water content (Figure 2.10(a)) for the A2-clay mixture, and after 20% water content for the A3-clay mixture. The non-linear increase in the intermediate water contents is explained by the complicated interactions between the water-sand-clay mixtures. For those in-termediate values, water zonation was observed as a relatively big aggregates of clay-water, while virtually no water was present in the rest of the mix. Under these conditions the hypothesis of homogeneity is violated. The hypothesis of a homoge-neous layer with a uniform water content is restored for the high and for the very

2.4 Conclusion 27 0 5 10 15 20 25 30 5 10 15 20 25 30 Gravimetrically Determined θ v ℜ ( εr ) A2c25 A3c25 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 Gravimetrically Determined θ v σ (Sm −1 ) A2c25 A3c25 (a) (b)

Figure 2.10: Relative permittivity and conductivity as a function of water content for the sands A2 and A3 containing 25% clay by weight. The linear increase of the conductivity was observed in the transmission line data; the permittivity suddenly increases for water contents of about 12% and 22% for A2 and A3, respectively.

low water content conditions. Sharp rising values of the electrical conductivity are not recorded, but conductivity increases linearly with water content, in agreement with what was observed in the transmission line measurements.

2.4

Conclusion

The present study is motivated by the need to understand the electric behavior of clayey road subgrades. GPR and transmission line data are analyzed for two distinct sand grainsizes for moisture conditions that cover the range of possibilities encountered in infrastructures for the common GPR frequency range. The same amounts of clay are used for each soil to modify the samples under different water contents. Two mixing models were tested for known values of the water content and for known soil physical properties.

For a given mixture, the electric permittivity increases as a function of water content, exhibiting constant values for frequencies higher than 1.5 GHz; the elec-trical conductivity increases linearly with frequency for increasing clay and water contents. The presence of clay is responsible for the non-linear behavior of the elec-tric permittivity in the low-frequency range, which is independent of the amount of water. The presence of clay, even in small quantities such as 5%w, can be detected

independently of the quantities of water; however, water facilitates the task of clay individuation. In dry soils the frequency dependency of the electric permittivity with low amounts of clay is very gentle and difficult to recognize under field conditions, having still a distinct behavior under laboratory conditions.

The ability to individuate clay is strongly reduced for frequencies above 1 GHz. If the aim of GPR road inspection is to individuate clay, relatively low frequencies

should be used. This conflicts with the need of high-resolution for the localization of imperfections. The seasonal change in water content cannot constitute a distur-bance because the relatively low frequencies experience little attenuation in the first decimeters of soil.

A volumetric mixing model performs well in the prediction of the electric per-mittivity if the constituents volume is known and the electrical properties of the matrix can be inferred. Extending the model from two to three phases, however, significantly increases the error in the predicted electric permittivity. The applica-tion of a mixing model may fail for clayey soil in the presence of water, because of the swelling properties of clay. Using the volumetric mixing model with clay-rich soils leads to an under-prediction in the correct electrical properties. A third-order polynomial model can be used for clayey soils as a function of water saturation, but the coefficients need to be determined for each clay content.

Chapter 3

Data Inversion and

Interpretation

1

3.1

Introduction

Extensive research has been carried out by many agencies and institutions on the applications concerning radar non-destructive evaluation of the built environment. Applications include electromagnetic modeling for detecting cracks in cement-based materials inspection [Malhotra and Carino, 2003; Nadakuduti et al., 2006; Carino, 2008], detecting media delamination and voids in road and bridge pavements [Belli et al., 2009], characterizing materials in terms of electromagnetic properties [Robert, 1998; Soutsos et al., 2001; Lambot et al., 2004b], detecting damages in concrete columns [Buyukozturk and Yu, 2009], surveying and monitoring cultural heritage constructions [Brown et al., 2009] and assessing masonry damages [Stone, 1997; Binda et al., 2005]. Maierhofer and Leipold [2001] adopted a multi-polarization approach to determine moisture content distribution and to detect full and empty joints in masonry structures through semi-quantitative analysis. Tsoflias and Hoch [2006] successfully investigated the response of GPR wave transmission through thin layers using multi-polarization for non-contact characterization of fracture aperture and fluid-fill. Their experiment, however, involved transmission of GPR signals at various angles; this implies two sided access to the masonry, which is not always possible in real cases. Orlando and Slob [2009] used a multicomponent pulsed GPR system to produce 2.5D vector migration images of cracks in the floor of a historical building. With their approach, the structures underneath an ancient building floor could be accurately reconstructed and different orientation cracks could be detected in qualitative images. D´erobert et al. [2008] used GPR combined with a new

capaci-1_{This Chapter has been published as journal paper in the Journal of Applied Geophysics, 74} 26-37 (Patriarca et al. [2011]). Note that minor changes have been introduced to make the text consistent with the other chapters of this thesis.