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## Notions of electrodynamics in material media

This section introduces the basic concepts of electromagnetic theory and some speci c points involved later in the numerical implementations. In particular, I will explain the implications of some assumptions which are commonly used but rarely fully commented in the literature. This presentation is mainly based on the books by Chew (1995) and Ta ove and Hagness (2005), as well as on readings in Landau and Lifchitz (1969) and Feynman et al. (1979).

The equations of electrodynamics and their interpretation are rst presented in the time domain, as it is our physical space. I then move to the frequency domain for the rest of the manuscript because, following Yu a and Scales (2012), it is in the « non-physical, but highly useful, Fourier domain » that I will perform my simulations and inversions. The electromagnetic properties of natural media will be discussed, since they are those I want to quantify by inverting GPR data. I thus detail the dielectric response models that provide parameterisations for these properties. In particular, I comment the so-called universal response of Jonscher (1977), which is widely used in geophysical applications but in a truncated form which is rarely discussed. I also comment the consequences of assuming a simpler parameterisation, which will be used further for the inversion.

**Maxwell’s equations and constitutive relations**

The behaviour of electromagnetic waves has been completely and coherently described by Maxwell (1873), based on the previous work of Faraday, Ampere and Gauss. Maxwell’s equa-tions can be expressed in a di erential form as r E(r; t) = @B(r; t)

; Maxwell-Faraday’s equation, (1.1) @t r H(r; t) = @D(r; t) + J(r; t); Maxwell-Ampere’s equation, (1.2) @t

r D(r; t) = q(r; t); Maxwell-Gauss’ equations, (1.3)

r B(r; t) = 0; (1.4)

where E is the electric eld with an amplitude in V/m, H is the magnetic eld in A/m, D is the electric induction (or electric displacement) in C/m2, B is the magnetic induction in T, J is the conduction current density in A/m2, and q denotes the electric charge density in C/m3. The variable r is the position vector (with coordinates in m) and t denotes time (in s). Following Yu a and Scales (2012), I stress that all elds and variables are real quantities here.

According to the so-called right-hand rule, Faraday’s law indicates that a time-varying magnetic ux B generates an electric eld E which rotates around B. Similarly, Ampere’s equation indicates that a current J or a time-varying electric ux D generates a rotating magnetic eld H. The physical mean of Maxwell-Gauss’ equations is that an electric charge density q is the source of an electric ux D, whereas an equivalent source of magnetic ux does not exist. The reader can refer to Chew (1995, x1.1.2, p. 3) for an integral form of Maxwell’s equations and their interpretation.

**Time-domain constitutive relations**

Maxwell’s equations system is very general but under-determined, because its equations are not linearly independent. For instance, assuming elds without constant components, we can derive the second Gauss’ law r B = 0 (1.4) by taking the divergence of Faraday’s equation (1.1), because r r = 0. Similarly, taking the divergence of Ampere’s equation (1.2), and using the conservation law for the electrical charge r J(r; t) + @q(r; t) = 0; (1.5) @t eturns the rst Gauss’ law r D = q (1.3). In the following, I thus consider only Faraday’s and Ampere’s equations (1.1) and (1.2), and I omit the later Gauss’ equations (1.3) and (1.4) which can be deduced from the former.

Mathematically, we need additional relations to determine the electromagnetic elds in-volved in Maxwell’s equations. On a physical point of view, we can also notice that Maxwell’s equations do not implicate explicitly the properties of the material media we want to investigate with GPR measurements. To represent the response of natural media and solve these equa-tions, the induction vectors are related to the elds vectors through the constitutive equations. In vacuum, these relations are simply

D(r; t) = « oE(r; t); (1.6)

B(r; t) = oH(r; t); (1.7)

where « o ’ 8:85 1012 F/m is the dielectric permittivity of vacuum and o = 4 107 H/m is its magnetic permeability.

In dielectric material media such as rocks and soils, the electromagnetic response is more complex since imposing an electric eld E to the material induces a polarisation, i.e. the orientation of the electrical moments of bounded charges (e.g. electrons linked to their atoms or dipolar molecules such as water) in a given direction. Similarly, applying a magnetic eld induces a magnetisation, i.e. the orientation of the magnetic moments of magnetic particles. In linear, isotropic media, polarisation P and magnetisation M can be described as the moment vectors

P(r; t) = « o e(r; t) E(r; t); (1.8)

M(r; t) = o m(r; t) H(r; t); (1.9)

where denotes time convolution, e is the dielectric susceptibility of the medium, which describes the capability of molecules to get an electrical polarisation when they are embedded in an electric eld E, and m is the magnetic susceptibility, which is the capacity of the particles to get a magnetisation under a magnetic eld H. Induction vectors thus results from the response of vacuum (eqs 1.6 and 1.7), plus the induced polarisation and magnetisation of the medium, which can be summarised in the following constitutive equations:

D(r; t) = « (r; t) E(r; t); (1.10)

B(r; t) = (r; t) H(r; t); (1.11)

where » = « o(1 + e) is the dielectric permittivity of the medium (in F/m) and = o(1 + m) is its magnetic permeability (in H/m). In a conductive medium, an additional relation comes from Ohm’s law which relates the conductive currents Jc to the electric eld E via Jc(r; t) = (r; t) E(r; t); (1.12) where is the electrical conductivity (in S/m). The total current J appearing in Ampere’s equation (1.2) is then the sum of the conductive currents Jc generated by the electric eld and of the source current Js injected in the GPR antenna: J(r; t) = Jc(r; t) + Js(r; t): (1.13)

In practice, Js(r; t) is non-zero only at the transmitting antenna position and during the emis-sion of the GPR pulse.

The constitutive equations and Ohm’s law constitute the material’s relations which describe the response of the medium to the applied electric and magnetic elds. The electromagnetic parameters « (r; t), (r; t) and (r; t) are thus the response functions of the medium along time, corresponding to dielectric, magnetic, and conductive mechanisms, respectively. I already mention that I consider linear, isotropic media. More fundamentally, I also implicitly assume that the polarisation P does not depend on the applied magnetic eld H but only re ects the response of the material to the electric eld E. Such a dependence of polarisation on the magnetic eld can be encountered in the optical frequency range where it implies magneto-optic e ects but we will not consider them here. Moreover, I also assume that the polarisation P is locally related to the electric eld E and does not su er from spatial dispersion. In optics, spatial dispersion can be encountered in materials that are said to be optically active, such as some chiral molecules or crystals. Again, I disregard this possibility. Linearity and isotropy, however, are important assumptions about the properties of the medium and I shall now detail their implications.

Linearity In the constitutive relations (1.10) to (1.12), it is assumed that the medium res-ponds linearly to the applied electric and magnetic elds. Mathematically, it is formalised by the fact that the parameters « , and do not depend on the imposed elds E and H. Phys-ically, it means that the polarisation (/magnetisation) of the particles are proportional to the applied electric (/magnetic) eld: polarisation and current vectors are co-linear to the electric eld, while magnetic moments are co-linear to the magnetic eld. The linearity of the response with respect to the excitation is a property of prior importance and I will often use it in my developments of simulation and inversion algorithms. Isotropy The constitutive relations also consider an isotropic medium: electromagnetic pa-rameters are scalar values and not tensors, so that the polarisation and magnetisation capability of the particles, as well as their conductivity, does not depend on the direction from which the applied elds are coming. Except in the work of Carcione (1996) and Carcione and Schoenberg (2000), who seek for generality, anisotropy is generally not considered in GPR applications. Indeed, the notion of isotropy depends on the scale at which the medium is described: in this sense, we may distinguish between an intrinsic anisotropy, which is described by a tensor, and a structural anisotropy, that results from the anisotropic arrangement of isotropic materials described by the relations (1.10) to (1.12).

Heterogeneity Electromagnetic parameters « , and depend on the position in space r, which simply means that the medium is heterogeneous. It is an obvious but nonetheless impor-tant feature to describe natural media, which can be very heterogeneous at the scale of GPR investigations.

Time-dependence These parameters also vary with time: natural media are non-perfect dielectrics, conductors and magnets. As a consequence, they present a transient response to the applied electric and magnetic elds. If we consider the polarisation phenomenon, it means that the individual electrical moments of charged particles do not align instantaneously with the vector E but present some inertia (relaxation process). The resulting macroscopic polarisation P is well aligned with the electric eld, but its intensity varies with time until the orientations of all microscopic moments stabilise. To respect physical causality, i.e. the fact that the response (e.g. polarisation) can not precede the cause (the imposed electric eld), the dielectric response function « (t) must be zero before the time to when the electric eld is applied. The magnetic permeability and the electrical conductivity follow the same rule for the establishment of magnetisation and of conductive currents, respectively. The ow of free charges represented by the current Jc(r; t) is thus proportional to the applied electric eld, but its intensity can vary with time. In the case of electrolytic conduction such as occurring in natural media, a delay in the establishment of the currents can be due to the viscosity of the interstitial uid carrying the ions.

Arrived to this point, I may specify that my work will focus on the electrical properties of the subsurface, i.e. on the dielectric permittivity » and on the electrical conductivity , since natural media are generally non-magnetic. Formally, the magnetic permeability (r; t) will depend on space and time in the equations and will be allowed to vary in my numerical implementations. But the reader can keep in mind that in my applications, permeability takes the constant value of vacuum o = 4 : 107 H/m. By convention, one commonly refers to the relative permeability r = = o = 1. Similarly, it is usual to speak about the relative permittivity « r = « = »o.

**Time-domain wave equation**

In order to show the wave nature of electromagnetic elds, Faraday’s equation (1.1) and Am-pere’s equation (1.2) can be combined with the constitutive relations (1.10) to (1.12). In the simple case of a homogeneous, time-invariant and source-free medium, it yields the following damped wave equation

r2u(r; t) » @2u(r; t)

propagation

@u(r; t) = 0; (1.14)

@t

| {z }

di usion

with the eld u being either E or H. This equation rst shows that the electric and magnetic elds are waves propagating at the same velocity v = 1=p » and undergoing the same di usion e ects, with a di usivity = 1=( ). Considering plane waves of the form u e{k r and recalling Faraday’s equation, it is also possible to retrieve the well-known rule that the elds E, H and the wavenumber vector k are mutually orthogonal and form a right-handed system (Chew, 1995, x1.2.3, p. 13). More importantly, equation (1.14) gives us a rst insight on the e ects of permittivity and conductivity on the electromagnetic waves: permittivity controls the wave propagation, whereas conductivity appears in the di usion term.

To obtain the wave equation (1.14), I assumed a homogeneous, time-invariant medium. It is also possible to derive a wave equation for inhomogeneous media (see Chew, 1995, x1.2.1, p. 11) but the assumption of time-invariance can not be dropped o . In time-varying media, it is not possible to de ne a constant velocity v, which is generally di cult to account for in the time-domain. Mathematically, this di culty comes from the time-convolution products involved in the constitutive relations, which make the material response non-local in time. To consider the e ect of the transient electrical response on the electromagnetic waves more easily, it is convenient to switch to the frequency domain.

**Frequency domain**

Assuming time-harmonic wave elds, or performing a Fourier transformation of Maxwell’s sys-tem with respect to time1, enables to express equations (1.1) to (1.4) in the frequency domain:

r E(r; !) = {!B(r; !); (1.15)

r H(r; !) = {! D(r; !) + J(r; !); (1.16)

r D(r; !) = q(r; !); (1.17)

r B(r; !) = 0; (1.18)

where the elds are now complex quantities depending on the angular frequency, or pulsa-tion, ! (in rad/s). By convention, I assume here a time-harmonic dependency in e{!t . The corresponding Fourier transformations to pass from the time to the frequency domains are Z +1

f(!) = 2 Z f(t)e{!tdt; (1.19)

and f(t) = + 1 f(!)e{!t d!; (1.20)

where f denotes the function of interest. Please note that I use the same notation for time-domain and frequency-domain quantities. I shall also underline that this Fourier convention leads to opposite signs for the imaginary parts of complex quantities compared to the notations used, for example, by Hollender and Tillard (1998) and Ta ove and Hagness (2005). It is the same convention as in Press et al. (1992), Chew (1995) and Virieux (1996).

In the frequency domain, the constitutive equations and Ohm’s law read

D(r; !) = « (r; !) E(r; !); (1.21)

B(r; !) = (r; !) H(r; !); (1.22)

Jc(r; !) = (r; !) E(r; !); (1.23)

where time-convolution products have been replaced by multiplication, which greatly simplify the consideration of the transient response. Since they are the Fourier coe cients of the cor-responding real-valued response functions in the time-domain, the electromagnetic properties « (r; !), (r; !) and (r; !) are now frequency-dependent complex quantities. Their imaginary parts account for the energy dissipation occurring during the transient response. Due to their frequency dependency, the medium is said to be dispersive. Dispersion and dissipation are thus indivisible frequency-domain mechanisms describing the time-dependent response of the medium (Toll, 1956; Yu a and Scales, 2012).

To complete the overview, I have to mention that the real and imaginary parts of the frequency-domain parameters should be linked through the Kramers-Kronig relations in order to verify physical causality, just like time-domain response functions must be zero before the application of any imposed elds (see e.g. Sohl, 2008; Yu a and Scales, 2012). Strictly speaking, these relations have two consequences. First, it means that the knowledge of one of the two parts (real or imaginary) over the whole frequency spectrum is su cient to derive the other one from the Kramers-Kronig relations. In practice, however, measurements are limited by the frequency bandwidth of the instrument and this derivation is rarely possible. Secondly, it implies that electromagnetic parameters can not be constant over the whole frequency spectrum, since they are the Fourier transforms of the non-constant time-domain response functions, which have to be zero before to to respect causality, and non-zero during the response (except in vacuum). Again, GPR applications generally forget about this physical consideration, due to the limited frequency-bandwidth of the measurements. We shall see in Section 1.1.2.2 in what extent the approximation of constant parameters might be valid in the GPR frequency range.

In the frequency-domain, the propagation and di usion terms of the wave equation can be gathered into a unique dielectric response, re-writing equation (1.14) under the form of the Helmholtz equation r2u(r; !) + « e(!) (!)!2u(r; !) = 0; (1.24) where « e is an e ective permittivity which gathers both permittivity and conductivity: « e(!) = « (!) + { (!) : (1.25)

Note that, in the frequency domain, the derivation of Helmholtz equation does not require the assumption of a non-dispersive medium as it was the case for the time-domain wave equa-tion (1.14). Frequency-domain wave propagation modelling thus enables to consider dispersion in a straightforward manner, solving Helmholtz equation frequency per frequency. For simpli-city, I considered a homogeneous medium to derive equation (1.24), which is in fact a scalar wave equation: In homogeneous media, the eld components ux, uy and uz can be conside-red as independent scalar values and the corresponding Helmholtz equations can be decoupled and solved independently. In heterogeneous media, we can still derive wave equations for the electric and magnetic elds E and H, but we end up with vector wave equations where eld components are not independent anymore (see Chew, 1995, x1.3, p. 17).

Since I am mainly interested in the permittivity and conductivity parameters, and less in the permeability, the question is now how to describe the e ective permittivity. In the next section, I develop some models which have been proposed for this purpose. Working in the frequency domain a priori enables to nely describe the dielectric response of natural media in the simulations. I should already mention, however, that I will consider frequency-independent, real-valued permittivity and conductivity parameters in the inversion, as done by Meles et al. (2010) in the time domain. Indeed, we shall see in Chapter 2 that the 2D multiparameter imaging of frequency-independent, real-valued permittivity and conductivity distributions is already an ill-posed problem whose resolution is challenging. Accounting for dispersion still increases the number of degrees of freedom and the ill-posedness of the problem. Moreover, GPR data are often mainly sensitive to permittivity, in less extent to conductivity, and in minor degrees to dispersion. Nevertheless, it is important to understand the general behaviour of dielectric materials to realise the implications of further simpli cations.

### The electrical response of dielectric solid materials

**Dielectric response models**

As already mentioned, the frequency-domain permittivity and conductivity can be regarded in the case of a linear, isotropic, heterogeneous, dispersive medium as space- and frequency-dependent complex quantities. Thus, we can write them as functions of real quantities as

« (r; !) = « 0(r; !) + {« 00(r; !); (1.26)

and (r; !) = 0(r; !) { 00(r; !); (1.27)

where the imaginary parts « 00 and 00 re ect the energy dissipation occurring during the transient response of the medium to the applied electric eld. The opposite signs before imaginary parts come from our convention for Fourier transformation (eqs 1.19 and 1.20) and have been chosen such that imaginary parts are positive (« 0; « 00; 0; 00 2 R+). In the complex plane, dissipation manifests itself by the fact that the response of the medium, e.g. the induction vector D, is not in phase with the applied electric eld E.

The e ective permittivity can be re-written in terms of the real and imaginary parts of permittivity and conductivity as « e = « 0(!) + 00! +{ « 00(!) + 0 ! : (1.28)

| {ze0 } | {ze }

The total energy losses are then quanti ed by the loss tangent tan = « 00e= » 0e. If we neglect the conductive losses for simplicity ( 00 = 0, I shall justify this hypothesis later on), we have « 00(!)0(!) tan (!) = « 0(!) + ! »0(!) ; (1.29)

where the rst term corresponds to dielectric losses due to the relaxation of bounded charges and the second term to conductive losses due to the displacement of free charges.

Expression (1.28) has two consequences. First, we can note that the imaginary part of the permittivity plays a similar dissipative role as the real part of the conductivity. Conversely, the in uence of the imaginary part of conductivity might be interpreted as a propagative e ect. Secondly, we can anticipate that this type of parameterisation is largely not optimal in view of the inversion as it would require to estimate four values per frequency (for « 0(!), « 00(!), 0(!) and 00(!)). Moreover, inside the in-phase (« 0e) and out-of-phase (« 00e) parts of the dielectric response, we can not discriminate between the contributions of the real and imaginary parts of permittivity and conductivity (« 0(!) vs. 00(!)=!, and 0(!)=! vs. « 00(!)). Indeed, even in a modelling point of view, expression (1.28) is not very adequate. It is certainly exhaustive since all theoretical terms appear explicitly, and therefore it can t all possible measurements of dielectric relaxation. However, it is not very useful to explain the behaviour of the materials in terms of general processes (Jonscher, 1999).

For these reasons, other models have been proposed for the e ective permittivity, involving a limited number of parameters. These models are based on empirical laws derived from dielectric measurements (Jonscher, 1999). Some of them also rely on theoretical assumptions about the underlying relaxation processes (Jonscher, 1981). These dielectric response models take the generic form of « e(!) = « 1 + « o (!) + { DC ; (1.30) where « 1 is an asymptotic limit for permittivity at high frequencies (considered to be real), DC is the (real) static conductivity, and (!) is a complex electrical susceptibility which char-acterises dispersion and dissipation due to both dielectric and conductive frequency-dependent phenomena. Note that this frequency-domain susceptibility does not strictly correspond to the Fourier transform of the time-domain response e. On one hand, does not include the constant asymptotic permittivity « 1 (6= »o). On the other hand, it includes all frequency-dependent parts of the conductivity, because these e ects can not be distinguished from the dielectric ones1.

The di erent models I will now describe only vary by their de nition of the electrical susceptibility , according to the assumptions done on the underlying polarisation mechanisms. Polarisation in solid materials is generally described as a relaxation process but there are many possible models to formalise it. The following explanations are mostly inspired by the enlightening reviews of Jonscher (1981, 1999).

**Debye-type models**

A rst class of dielectric models assume that the electrical polarisation can be explained by a Debye-type relaxation, with an electrical susceptibility of the form (!) = 1 « s « 1 ; (1.31) where « s is a (real) static permittivity valid at low frequency and is the characteristic rela-xation time for the considered mechanism. The Debye model is one of the rst and simplest attempts to characterise relaxation (Debye, 1929) and also nd applications in the mecha-nics of viscous media. Debye’s model considers the individual electrical moments as identical, non-interacting dipoles, having a loss of energy proportional to frequency2. Debye’s model is particularly suitable for describing the polarisation of dipolar molecules such as water, or the polarisation of ions at interfaces (e.g. between soil particles and water), inducing an interfacial capacitance (Maxwell-Wagner phenomena).

Fig. 1.1 shows the evolution of the real and imaginary parts of the e ective permittivity with respect to frequency, for the Debye model of pure water at 25 C (Cassidy, 2009a). The imaginary part of the permittivity « 00 exhibits a clear peak of dissipation at the relaxation frequency !p = 1= corresponding to the characteristic time (!p ’ 19 GHz in this case), whereas the real part « 0 drops down to its high frequency limit « 1r = 5:6.

According to the considered frequency range, diﬀerent relaxation mechanisms can be in-volved (see Fig. 1.2). In the GPR frequency range, dielectric losses at high frequencies are mainly due to dipolar relaxation: it is one of the reasons why GPR data are sensitive to the presence of water in the investigated material (the primary reason being the value εsr = 81 itself, much larger than in any other natural media, see Table 1.1). In a smaller extent, we may observe dielectric losses at low frequencies due to Maxwell-Wagner interfacial phenomena (indicated as ionic processes in Fig. 1.2) but these eﬀects are generally dominated by the con-ductive losses due to the static electrical conductivity of the medium. Actually, most of the GPR frequency band lies outside any Debye-type relaxation peaks (see Fig. 1.2). And it turns out that, apart from this particular loss peaks, pure Debye’s models fail to accurately describe the dielectric behaviour in most of real materials (Jonscher, 1999).

Some authors proposed ner descriptions of the dielectric properties at GPR frequencies involving variants of Debye’s model, for instance the Cole-Cole model (Cole and Cole, 1941) or the Cole-Davidson model (Davidson and Cole, 1951). These parameterisations introduce additional exponent factors in the denominator of expression (1.31) that make the dispersion frequency range wider and roughly correspond to giving a memory to the relaxation process (Hill and Dissado, 1985). Other authors promote the superposition of several Debye-type peaks (Xu and McMechan, 1997) or the combination of Debye-type models with other relaxation mechanisms (e.g., the Kelvin-Voigt mechanical model, see Carcione, 1996). I will not detail the variety of possible models because the variations between them are not signi cant for our purposes. As for every empirical law, the preference for a model or another is case-dependent. On a theoretical point of view, Jonscher (1999) further argues that designing sophisticated variants of the Debye model and invoking superposition or combination of di erent relaxation mechanisms is not satisfactory: of course, it enables to better explain the observed dielectric behaviours (by adjusting more and more tuning parameters) but it fails to provide a uni ed explanation of the underlying physical processes.

Alternatively, Jonscher (1977) proposed a universal response model which is claimed to explain all the observed electrical behaviours with a uni ed physical mechanism, involving a limited number of independent parameters. Jonscher’s model has been promoted by Hollender and Tillard (1998) for geophysical investigations in the GPR frequency band. Since then, it has been used to describe the dielectric properties of rocks (Hollender and Tillard, 1998), sands (Gregoire and Hollender, 2004) or concrete (Bourdi et al., 2008; Ihamouten et al., 2011). Deparis and Garambois (2009) successfully applied it for the characterisation of thin layers in fractured limestones. In the next section, I detail this model, starting from the general form of Jonscher (1981) to understand its meaning and limitations.

We can notice that Jonscher’s model does not evacuate Debye’s model but integrates it (eq. 1.33). Jonscher’s model implies a Debye’s mechanism to describe the dielectric response over the whole spectrum since the power laws (1.32) and (1.34) alone can not reproduce the relaxation peak of dipolar molecules or interfacial polarisation, which are best explained by Debye’s model. The improvement of Jonscher’s model concern the description of the electrical response outside the frequency range of this Debye-type relaxation, where the power laws of eqs (1.32) and (1.34) have been shown to better t the observed dielectric measurements than the at response of Debye’s model (Jonscher, 1981, 1999).

It is worth noting that, in their adaptation of Jonscher’s model to GPR measurements, Hollender and Tillard (1998) only keep the high frequency regime of the general Jonscher’s model. They end up with an expression which veri es equation (1.34), of the form (!) = r ! n1 1 + {cotan n ; (1.35) !r 2

where r is this time a real and constant susceptibility parameter and !r is a reference frequency which can be de ned arbitrarily (it only aims at making the ratio !=!r dimensionless). For GPR applications, it is usual to consider !r = 2 100 MHz.

Since Hollender and Tillard (1998) discarded the low-frequency part of the law, as well as the Debye-type relaxation peak, it should be stressed that expression (1.35) is only valid in the frequency regions above any Debye-type loss peaks. In particular, it can not describe the polarisation of water molecules. However, as most of the GPR frequency band lies out-side any Debye-type peaks (see Fig. 1.2), expression (1.35) provides a good agreement with dielectric measurements at intermediate GPR frequencies, typically between 50 and 300 MHz, in particular in dry rocks (Hollender and Tillard, 1998). But it should be manipulated with cautious when considering conductive materials such as clay at low frequencies (close to the Wagner-Maxwell relaxation peak) or wet rocks and soils at high frequencies (nearby the Debye peak of water relaxation).

Finally, some authors suggest that, using expression (1.35), the static conductivity DC may also be included in the electrical susceptibility (e.g. Deparis, 2007; Lopes, 2009). This is a pragmatic assumption that aims at decreasing the number of parameters required for the description of the e ective permittivity. However, neither Jonscher (1977, 1981, 1999) nor Hollender and Tillard (1998) make this assumption. Hollender and Tillard (1998) simply neglect the contribution of the static conductivity for their applications at GPR frequencies. As for Jonscher (1999, x4.6), he clearly distinguishes between DC conduction and low frequency dispersion. The latter implies a reversible storage of charge in the material and can indeed be described by the universal power-law. Static conductivity, however, consists in owing charges, without any storage. In the generic expression of the e ective permittivity (1.30), I will thus keep the static conductivity decoupled from the frequency-dependent permittivity term: « e(!) = » 1 + « o (! ) +{ DC ; (1.30 again) where I remind that the quantities « 1 and DC are real-valued constants, while the suscep-tibility (!) is a complex variable. Following Jonscher (1981), I then conceive the general, frequency-dependent conductivity as

(!) = DC + ! »00(!); (1.36)

= DC + ! »o 00 (!); (1.37)

which is now a real quantity (justifying the hypothesis 00 = 0 made p. 34), with 00(!) following the universal response and DC being an independent parameter. Again, I shall stress that this decoupling is mainly a matter of interpretation. It only means that I distinguish the ow of (totally) free charges described by the direct current conductivity DC , from the complex, frequency-dependent electrical response « (!) of (more or less) bounded charges that can give rise to an alternating current (AC). The loss tangent is then simply tan (!) = « e00(!) = (!) : (1.38)

Now that I have explained what is exactly an electrical response, i.e. what permittivity « (!) and conductivity (!) represent, we can have a look at the values of these parameters in natural media. In particular, the values of the dispersive parameters r and n in expression (1.35) will tell us if the assumption of non-dispersive materials is reasonable for considering natural media.

#### Electrical properties of natural media

Table 1.1 presents the electrical properties of some materials, compiled from the PhD thesis of Saintenoy (1998), Girard (2002), Jeannin (2005) and Loe er (2005). More exhaustive lists of these parameters can be found in these studies, as well as on the internet (e.g. Wikipedia, 2014; Clipper Controls®, 2014). These references give either rough orders of magnitude for the real part of the relative permittivity « 0r and for conductivity in the GPR frequency range, or the corresponding Jonscher parameters. When the Jonscher parameters are given, I derive the corresponding values for « 0r(!) and (!) for two characteristic frequencies of GPR investi-gations, namely 100 and 200 MHz. I also derive the ratio DC =(« 0!) that gives an indication of the proportion of di usive vs. propagative e ects in the behaviour of electromagnetic waves (see eqs 1.14 and 1.29). Small values for this ratio justify to neglect the static conductivity in the description of the medium at GPR frequencies, as argued by some authors (Hollender and Tillard, 1998; Jeannin, 2005). We shall see wether or not this assumption is reasonable.

A rst look at Table 1.1 enables to distinguish three classes of materials:

1. low-loss media, such as snow, ice, fresh water, quartzite or dry sand. In these media, electromagnetic waves can propagate over very long distances quasi without attenua-tion other than geometrical spreading (GPR surveys can thus achieve deep penetration depths). It is then possible to ignore the e ect of conductivity. The extreme case being air, that behaves like vacuum.

2. lossy media, such as ion-carrying water, clay, and all rocks and soils containing a signi-cant proportion of interstitial water or clay. In these media, electromagnetic waves are strongly attenuated and it is not possible to neglect the in uence of the electrical con-ductivity. Some of these media also exhibit a strong dispersion, which a ects strongly conductivity values, and in a smaller extent permittivity (except for clay).

3. intermediate media, which are weakly dispersive and where the attenuating e ect of conductivity is not dramatic but not totally negligible ( 0:1). In this category, we nd in particular samples of limestone, which is the material that I investigate in Section 3.2.

Note that volcanic rocks can present a non-zero magnetic susceptibility, as well as sands and sandstones depending on their detrital origin. I do not indicate these magnetic properties in Table 1.1 because I do not consider magnetic materials in the following.

**Table of contents :**

**General introduction **

Near surface geophysical imaging and characterisation

Ground-Penetrating Radar (GPR)

Principles of GPR measurements

GPR applications

GPR processing and imaging

Full Waveform Inversion (FWI)

Principles, history, and challenges

FWI of GPR data: state of the art

Objectives of the thesis and outline of the manuscript

**1 The forward problem **

1.1 Notions of electrodynamics in material media

1.1.1 Maxwell’s equations and constitutive relations

1.1.2 The electrical response of dielectric solid materials

1.1.2.1 Dielectric response models

1.1.2.2 Electrical properties of natural media

1.1.3 Wave propagation in two dimensions

1.1.3.1 TE-TM modes and analogy with the acoustic system

1.1.3.2 Wave equations and analytical solutions in homogeneous media

1.2 Numerical modelling of electromagnetic waves propagation in 2D heterogeneous media

1.2.1 Introduction: Choice of the numerical method

1.2.2 The frequency-domain nite-dierence mixed-grid stencil

1.2.3 Validation in a homogeneous medium

**2 The inverse problem **

2.1 Introduction to inverse problems, optimisation and FWI

2.1.1 Denition, properties, and resolution of inverse problems in general, and of FWI in particular

2.1.2 Local descent optimisation algorithms

2.2 A strategy for multiparameter FWI (Lavoue et al., 2014)

2.2.1 Forward and inverse problems formulation in the frequency domain

2.2.1.1 Forward problem

2.2.1.2 Inverse problem

2.2.2 Multiparameter imaging of permittivity and conductivity

2.2.2.1 Parameter sensitivity and trade-o

2.2.2.2 Parameter scaling

2.2.2.3 Behaviour of the inversion with respect to parameter scaling and frequency sampling

2.2.3 A realistic synthetic test

2.2.3.1 Benchmark design

2.2.3.2 Inversion of noise-free data

2.2.3.3 Inversion of noisy data

2.2.4 Discussion

2.3 Further methodological details

2.3.1 Computation and interpretation of the gradient

2.3.2 Validation of the computed gradients

2.3.3 Derivation and validation of the two-parameter problem

2.3.4 Sensitivity to dispersive parameters

**3 Application to real data inversion **

3.1 Validation of the imaging algorithm against experimental laboratory data (Lavoue et al., 2015)

3.1.1 Presentation of the data (Institut Fresnel, Marseille, France)

3.1.2 Forward problem

3.1.2.1 Numerical strategy

3.1.2.2 Data pre-processing

3.1.2.3 Simulation of synthetic data

3.1.3 Data inversion

3.1.3.1 Inverse problem formulation

3.1.3.2 Monoparameter inversion of the dataset FoamDielExt

3.1.3.3 Multiparameter inversion of the dataset FoamMetExt

3.2 Imaging a limestone reservoir from on-ground GPR data at the LSBB (Rustrel, France)

3.2.1 Introduction: Context and aim of the study

3.2.2 Classical processing: velocity analysis, migration, forward modelling

3.2.2.1 Semblance analysis, direct wave and hyperbola picking

3.2.2.2 NMO correction and stack, reverse-time migration

3.2.2.3 Forward modelling in a blocky model

3.2.3 Pre-processing steps towards FWI

3.2.3.1 Data pre-processing: Mute and 3D-to-2D conversion

3.2.3.2 Estimation of a source wavelet

3.2.3.3 Frequency-domain analysis

3.2.4 Preliminary FWI results

**Conclusions and perspectives **

Conclusive sum-up

Perspectives

Forward perspectives: 3D modelling, antenna radiation pattern

Inverse perspectives: optimisation issues, other parameters, other data

Applicative perspectives: starting model, source estimation, acquisition

Discussion: Time-domain vs. frequency-domain FWI of GPR data

Appendix A Adjoint state method using a Lagrangian formulation

Appendix B Complete LSBB data set

B.1 Raw common-oset sections

B.2 Processed common-oset sections

B.3 Filtered common-oset sections

B.4 Frequency-domain data

B.5 Time-domain data t

**Bibliography **

List of notations