Guy, Zoé Lora Ludovic

Merci À François

Ingenieros Del Perú Jafet

Williams Estos

Je commencerai par remercier les membres de mon jury de thèse, et en premier lieu Sylvain Bouley, qui a accepté de le présider, après avoir été mon suiveur de thèse pendant 3 ans. Ensuite, merci beaucoup à mes rapporteurs Alain Reineix et David Stillman. Alain pour son rapport détaillé de cette thèse, qui par son exhaustivité ne laisse aucun doute quant à la rigueur de sa relecture.

Table of contents

Introduction

The Apollo Lunar missions (1969)(1970)(1971)(1972) constituted the first human exploration of another planetary body, as well as the first return of samples collected outside the Earth. They also allowed the first extra-terrestrial application of a new kind of planetary exploration instrument: sounding radars. With the transmission of electromagnetic signals towards the surface, such instruments can remotely reveal the structure of the subsurface as they move across the surface, in a non-invasive way. Even though the Apollo sounding radar was successful in exploring the Lunar subsurface, probably due to the difficulty to store and analyse large quantities of acquisitions in the 1970s, nearly 20 years passed before the selection of any new sounding radar for planetary missions. The developments in electronics and digital technologies led to a renewed interest in radar sounders, now easier to integrate to planetary missions, and acquisitions easier to process. In the last two decades, 9 extra-terrestrial sounding radars have been operated in the solar system, 4 on the Moon, 4 on Mars, and 1 on the nucleus of comet 67/P. And 4 more have been selected for future missions on Mars, Jupiter's icy moons and Venus in the next two decades. The first sounding radars were operated from orbit, but the possibility to carry instruments to the surface of Mars or the Moon, with remotely controlled robots named "rovers", also offered an interesting new type of platform for radar soundings on a local scale. In the last 10 years, 2 radar sounders mounted on rovers explored the subsurface of the Moon, and 2 radars are currently exploring the subsurface of Mars. This is in this context of rising interest for this technology that the WISDOM (Water Ice Subsurface Deposits Observations on Mars) Ground Penetrating Radar, onboard the Rosalind Franklin rover of the ExoMars 2022 mission, will explore the subsurface of Mars in June 2023.

The WISDOM radar is also a first, as it aims at shallower penetration depths (of a few meters), but a better range resolution (of a few centimetres) than all other radar sounders operated to this day on Mars. This peculiar design is the result of the ExoMars 2022 mission objectives: searching for traces of life in the Martian shallow subsurface, where biomarkers would potentially be preserved. In order to collect a sample to be analysed, Rosalind Franklin will be equipped with a 2-m drill. Revealing the nature of the subsurface below the rover, WISDOM will participate in the drilling operations guidance, and the reconstruction of the landing site geological history. In order to prepare such Martian operations, the instrument teams of the LATMOS (Laboratoire Atmosphères, Milieux, Observations Spatiales) and TUD (Technische Universität Dresden), have developed processing and interpretation tools adapted to the WISDOM data, and validated them on both synthetic and experimental acquisitions. Many of these experimental acquisitions were performed on natural sites, selected for their properties analogous to the ones expected on Mars.

After processing, the data acquired by WISDOM consist in "radargrams", displaying the envelope of the received signals (after reflection/scattering on subsurface structures) as a function of their time delays. The present thesis was dedicated to the development and validation of new software tools to be implemented into the WISDOM processing / interpretation chain, with the objective of enhancing the vertical resolution of the obtained radargrams, and automatically infer the dielectric properties of the subsurface from them. This manuscript is divided in 5 chapters. The first one introduces the basic principles and design of a sounding radar, and continues with an overview of sounding radars selected for planetary exploration missions, in order to provide a context for the arrival on Mars of the WISDOM instrument. In the second chapter, the objectives of the ExoMars 2022 mission, the design of WISDOM, and the processing chain developed to generates interpretable data products from WISDOM soundings are detailed. The experimental and synthetic datasets of the instrument, used to prepare future Martian operations, are also presented in this chapter. The third chapter is dedicated to two techniques aiming at a more accurate estimation of the dielectric constant of the Martian subsurface. One is based on the amplitude of the surface echo, and the other on the shape of patterns formed by echoes from point reflectors ("diffraction curves") in WISDOM radargrams. This last process has been automated and corrected to account for the effect of refraction at the surface. After validation, an application of both methods to acquisitions obtained during a field campaign run in the Desert of Atacama during this thesis, is proposed. Chapters 4 ad 5 introduce and validate a number of "Bandwidth Extrapolation" techniques, that are applied in the frequency domain to enhance the final range resolution of the WISDOM soundings, acquired in frequency domain. In the fourth chapter, the classic Bandwidth Extrapolation technique, already applied on other planetary radars acquisitions, is implemented into the WISDOM processing chain, and after validation, applied to radargrams acquired in 3 natural environments. Preliminary validations of potential improvements to the technique (based on more sophisticated signal models) are the subject of the fifth and last chapter. The vertical resolution of WISDOM radargrams could be improved by a factor of 3 with these techniques, revealing more details of the subsurface structure, and thus allowing more accurate interpretations. The implementation of all these new and sophisticated tools in the processing / interpretation chain of the WISDOM data will enhance the quality and readability of future Martian observations. Eventually, this manuscript ends on the conclusion and perspectives of this thesis.

This thesis was conducted in the PLANETO department of the LATMOS. It was funded by the UVSQ (Université de Versailles-Saint-Quentin) and the CNES (Centre National d'Etudes Spatiales), and benefitted from both official and un-official collaborations, with the TUD, the ExoFit simulation of operations teams (Airbus, ESA, CLUPI instrument team, PANCAM instrument team, RAMAN-LS instrument team), and with M. Mastrogiuseppe from La Sapienza Rome university.

Fundamentals of radio waves -matter interactions

A sounding radar is an active and non-invasive instrument, transmitting electromagnetic waves towards a planetary surface, and measuring the portion of these electromagnetic waves reflected or scattered by the surface and the subsurface in the direction of its receiving antenna (generally the backscattering direction). It thus provides information on the structure and composition of the observed surface and subsurface.

The radar remote sensing technique is therefore based on the interactions of the transmitted electromagnetic waves with the different materials of the planetary surface and subsurface. These interactions depend on the structure and intrinsic electromagnetic properties of the materials encountered by the waves.

Intrinsic properties of matter

Three intrinsic characteristics have an impact on the propagation of electromagnetic waves in a given medium: the dielectric permittivity ε, the magnetic permeability and the electrical conductivity .

The dielectric permittivity characterizes the ability of a medium to modify the equilibrium positions of its charged particles when an electric field is applied to it, to compensate for the effect of this field. This redistribution of elements can occur on the scale of electrons, atoms, dipoles and ions. It is expressed in F/m, in harmonic regime, when the complex representation of the fields is used (with the exp 2 convention), the dielectric permittivity is represented by its Fourier transform which is a complex function of the frequency, noted in the following for the sake of simplicity:

= + " ( 1.1 )
The real part of the permittivity , or "dispersion term", corresponds to the energy accumulated in the material when its elements change their equilibrium positions in the presence of an electric field. The imaginary part " , or "absorption term", corresponds to the energy dissipated in the material when its elements move in the presence of an electric field.

A dimension-less value called "relative permittivity", or "dielectric constant" is often defined to characterize the dielectric properties of a material, such as:

= ( 1.2 )
where = 8.854.10 -12 F/m is the dielectric permittivity of vacuum.

CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 5

The magnetic permeability characterizes the capacity of a material to change magnetic flux lines when a magnetic excitation is applied to it. It is expressed in H/m, in harmonic regime, its Fourier transform is a complex function of the frequency, noted: = + " ( 1.3 ) Similar to the dielectric constant, a dimension-less value named "relative permeability" is defined as:

= ( 1.4 )
where = 4 10 H/m is the magnetic permeability of vacuum. In paramagnetic ( ≥ 1) and diamagnetic materials ( ≤ 1), not very sensitive to magnetic excitation, it is often considered that = and therefore = 1. This approximation is not valid for ferromagnetic materials, for which ≫ 1.

The electric conductivity characterizes the capacity of a material to let free charges (electrons or ions) circulate through it. It is expressed in S/m, and in harmonic regime, its Fourier transform is also complex:

= + " ( 1.5 )
It can be noted that the conductivity is the inverse of the resistivity, expressed in Ω/m. The imaginary part " of the conductivity characterizes the movements of free charges in the material leading to its polarization, while its real part corresponds to the accumulation of energy by the movement of these charges. It can thus already be guessed that, even if the two describe different phenomena, the quantities and " will have a combined effect on the accumulation of energy in the material, while " and will have a combined effect of the losses of energy when an electric field is applied. The effects of the permittivity and the conductivity cannot be disentangled by a simple measurement. An effective permittivity, that combines the effects of both parameters, will be introduced later on.

Given the frequency ranges used by radars and, most of all, their limited frequency bandwidth, the frequency dependent character of these three parameters can be in a first approximation neglected.

In the next session, we will show how these three intrinsic properties of a material drive the propagation of electromagnetic waves through it.

Propagation of electromagnetic waves in a material

An electromagnetic wave is a combination of an electric field and a magnetic field. The Maxwell equations express the fact that the spatial variations of one field are related to the temporal variation of CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 6 the other field. But in order to describe the interaction of these fields with matter and find solutions to these equations they have to be combined with the constitutive relations of the considered medium which relate the dielectric displacement to the electric field and the magnetic field to the magnetic field. Here, all considered propagation media are linear, isotropic, homogeneous, spatially local and time-invariant.

For such media, the constitutive equations are convolutional relationships in time domain. For harmonic waves, the constitutive relations in frequency domain simplify to simple products and the wave equations for the electric and magnetic fields can be obtained from the Maxwell equations. In absence of sources (currents and charges), the wave equations are given by formulae (1.6) and (1.7).

, " ⃗ $$$$$$$$$$$$$⃗ = , " ⃗ $$$$$$$$$$$$$⃗ % - , " ⃗ $$$$$$$$$$$$$⃗ % ( 1.6 ) ' , " ⃗ $$$$$$$$$$$$$$⃗ = ' , " ⃗ $$$$$$$$$$$$$$⃗ % - ' , " ⃗ $$$$$$$$$$$$$$⃗ % ( 1.7 )
where , " ⃗ $$$$$$$$$$$$$⃗ is the complex electric field (V/m), ' , " ⃗ $$$$$$$$$$$$$$⃗ is the complex magnetic field (A/m), " ⃗ describes the position where the fields are considered and % = 2 is the angular frequency. In the absence of a source of charges in the volume considered, equation becomes:

$⃗ = -% ( -% ) $⃗ ( 1.8 )
The quantity ( -* +

) combines as already mentioned before the permittivity and conductivity of the medium. In the following, for the sake of simplicity, this complex quantity will be designed as the permittivity , with a real part + *" + (named in the following) and an imaginary part " + * + (named " in the following). In consequence, the same mathematical model will be used for lossy dielectric and conductive materials.

Using the effective permittivity introduced above, the factor , defined by , = % + " is referred to as the "wavenumber". It is often complex and decomposed as , = --. with α called the attenuation factor, and β the phase factor.

As an illustration of the effect of factors α and β, we consider a theoretical harmonic (frequency ) electromagnetic wave, which is a function of the only spatial coordinate / and propagates in the direction of increasing /. We also consider that the direction of the electric field is perpendicular to the propagation direction and is the same whatever the location considered and whatever the time.

The considered wave is an harmonic (temporal behaviour), plane wave (spatial behaviour), which is linearly polarized (direction of vector $⃗ constant). It is a particular solution of the wave equation, which is compatible with the Maxwell equation and the constitutive equations. The value of the electric field along its polarization direction is /, = 0 12 0 3 +4 52 ( 1.9 ) where is the amplitude of the oscillations, and the two factors α andare:

. = %6 7 1 2 8 9 1 + : ′′ ′ < -1=

( 1.10 ) -= %6 7 1 2 8 9 1 + : ′′ ′ < + 1=

( 1.11 )

The attenuation and phase factors characterize the effect of dielectric losses in the material on the amplitude and the phase of the transmitted electromagnetic wave, respectively. The ratio > > between the real and the imaginary part of the permittivity is named the "loss-tangent", and is often written ?@ A . Theoretically ε is frequency dependent, thus ?@ A is also frequency dependent.

Nevertheless, as mentioned above, for radar applications the frequency dependence of the permittivity and the permeability can often be neglected over the frequency bandwidth of the instrument. As a consequence, both . andare proportional to the frequency of the electromagnetic wave, and thus that attenuation and phase shift increase in a given material with frequency. We can introduce a derived parameter B, defined by . = B ( 1.12 ) to highlight this frequency dependence.

In practice, natural materials are often non-magnetic, and = , where the magnetic permeability of vacuum.

Many sounded materials being poorly conductive (low dielectric losses), it is also often considered that ≪ ′. The following simplifications for the expressions of α and β are thus common in sounding radars theory: . ≈ . In this thesis, unless mentioned otherwise, the hypotheses of non-magnetic materials and low dielectric losses will be considered.

As previously mentioned, a sounding radar instrument measures the portion of its emitted electromagnetic signal reflected or scattered at interfaces between materials of different properties towards its receiver part. The following section presents two models describing the interactions occurring when an electromagnetic wave encounters an interface between two materials.

Interactions at an interface between two materials

When an electromagnetic wave such emitted by a radar encounters an interface between two media of different dielectric properties, only a portion of its energy will be sent back to the receiver of the radar.

Other portions of this energy can be reflected or scattered in other directions, absorbed or transmitted in the second medium. As a first approximation, with the simple hypothesis of a plane wave, and an infinite, smooth and flat interface between the two media, the Snell-Descartes laws allow an estimation of the reflected and transmitted portions of the signal. The contrast of dielectric constant generating a change of speed of the electromagnetic wave at the interface, its specular direction of propagation will be affected if it is not perpendicular to the surface. This is the refraction effect, and in the approximation of G = E √> for the propagation speed in both materials, the angle I J between the incident ray and the normal to the plan of the interface @ $⃗ is related to the angle I between the refracted ray and @ $⃗ as:

6 K LM@ I J = 6 LM@ I ( 1.13 ) with K and the dielectric constants of the two materials. The value √ is referred to as the "refraction index" of the material. Another portion of the wave is reflected at the surface in the specular CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 9 direction, the direction symmetric to the incidence direction relative to the normal to the surface @ $⃗. This situation is illustrated in Figure 1.1.

The direction vector of the refracted ray N OP Q $$$$$$$$$$$$⃗ and the direction vector of the reflected ray N OPROE $$$$$$$$$$$$$⃗ are according to Snell-Descartes laws:

⎩ ⎪ ⎨ ⎪ ⎧ N OP Q $$$$$$$$$$$$⃗ = : √ K √ < N WXEW $$$$$$$$⃗ + : √ K √ YZL I J -YZL I < @ $⃗ M[ YZL I J ≥ 0 N OP Q $$$$$$$$$$$$⃗ = : √ K √ < N WXEW $$$$$$$$⃗ + : √ K √ YZL I J + YZL I < @ $⃗ M[ YZL I J ≤ 0 N OPROE $$$$$$$$$$$$$⃗ = N WX\W $$$$$$$$$⃗ + 2 YZL I J @ $⃗ ( 1.14 )
The refracted ray exists only if

( √ > ]√ > ]_
) LM@ I J is between -1 and 1. If not, the reflection is "total": the electromagnetic wave is totally reflected.

From these equations, we can deduce the Fresnel coefficients, describing the amplitudes and phases of the reflected and refracted signals rays. Two particular cases are considered: When the electric field of the incident ray is in the incident plane (defined by @ $⃗ and N WXEW $$$$$$$$⃗ ), the polarization of the incident wave is noted V. When the electric field of the incident ray is polarized in the perpendicular direction to the incident plane, the polarization is noted H. For a polarization of the incident field which is neither H nor V, we use the fact that the Maxwell equations are linear with respect to the fields and the electric field is simply decomposed in a sum of a H polarized and V polarized fields.

For the refracted ray, the Fresnel coefficients that give the ratio between the refracted amplitude and the incident amplitude are:

⎩ ⎪ ⎨ ⎪ ⎧ " `=
√ YZL I -√ K YZL I J √ YZL I + √ K YZL I J " a = √ K YZL I -√ YZL I J √ K YZL I + √ YZL I J ( 1.15 ) And for the reflected ray, the Fresnel coefficients are:

⎩ ⎪ ⎨ ⎪ ⎧ `= 2 √ YZL I √ YZL I + √ K YZL I J a =
2 √ K YZL I √ K YZL I + √ YZL I J ( 1.16 ) When the permittivity values are complex values, the Fresnel coefficients are complex values too and model the change in real amplitude (modulus) and the phase shift (argument) at the interface. If the CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 10 imaginary part of the permittivity is negligible, the coefficients are real but " ` can be negative, which corresponds to a phase shift equal to π at the reflection.

This simple model can be used in the case of a large smooth and flat surface in the far field of the antenna of the sounding radar, to describe the signal reflected at the surface and the refraction of the transmitted signal in the subsurface. However, a more complex model is required to accurately estimate the amplitude and the phase of the signal received by a radar from a buried reflector.

The fundamentals presented in this section are the basis to the design of sounding radars, and the interpretation of their acquired signals. In the following section, the general concept of a radar instrument, and the specific design of a radar sounder will be introduced.

Fundamentals of sounding radars

Principle of a radar

Radar is the acronym for "RAdio Detection And Ranging", and originally designated an instrument used to measure the distance to a target from the time-of-flight of a signal transmitted by the device and reflected by the target towards the device (First patent by Christian Hülsmeyer in 1904). The modern definition of a radar is an instrument with a transmitter illuminating a surface, volume or object with electromagnetic waves, and a receiver collecting data which are linked to the reflected/scattered part of these waves towards it and eventually from which information about the surrounded environment are obtained.

The main elements constituting a radar instrument is a signal generator and amplifier connected to a transmitting antenna, and a receiving antenna connected to an amplifier and a signal processor (that usually down-converts and filters the signal). For some types of radar, with homodyne reception, the receiver processor uses the signal generated in the transmitter. Since the development of digital electronics, most radars use an Analog to Digital Converter (ADC) in order to transform the obtained signal into data that can be stored. A schematic illustrating the basic elements of a radar is presented in Different architectures for the transmission/reception chains are possible depending on the desired performances and the constraints of the mission.

Types of radar and their applications

The operating modes of a radar and the antennas configuration are defined according to the investigation aimed at (see Figure 1.3 for an illustration)..

Mono-static, bi-static and multi-static radars

A mono-static radar has only a single antenna, used for both transmission and reception of the signal. In this configuration, a switch in the electronic is required to alternatively connect the antenna to the transmission and reception channels. Such a radar can obviously not transmit and receive a signal at the same time, which can be required by some radar designs. If a radar has a transmission and a reception antenna separated by a distance much shorter than the distance of the targets illuminated by the radar, it will be called "quasi-monostatic". Depending on the interpretation process to be applied to such a radar data, it can be approximated as being "mono-static". In planetary missions, most of the radars used to this day were mono-static or quasi-monostatic, as this is the easiest configurations (which is the case for the WISDOM radar on ExoMars, see Section 4.4).

A bi-static radar has its transmission and reception antennas at a distance much larger than the distance between each antenna and the illuminated target. With such a radar, echoes from other directions than the incident one can be accessible, or the signal passing through a target can be acquired. As a different accommodation is required for the transmission and reception chains, bi-static radars are challenging to implement in the scope of planetary missions. Only one real bi-static radar has been used in planetary missions to this day the CONSERT radar of the Rosetta mission (see Section 3.3).

If a radar as more than two transmission/reception antennas in total, it is named "multi-static". Such a configuration allows the study of the illuminated target through different angles without having to move the transmission or reception antennas, the LRPR radar of the Chang'E 5 mission (see Section 4.1.3).

Only one planetary mission to this day used a multi-static radar, but with relatively short distances between the antennas. Projects involving the operation of several cubesats have been studied that relies on the use of one transmitting cubesat and a few receiving ones.

Imaging, non-imaging and sounding radars

In planetary sciences, radars (operating either from orbit or on the surface) are used for a variety of purposes. They can be roughly classified in three main groups: imaging, non-imaging and sounding radars. In this sub-section will be briefly introduced the radars of these three groups used in planetary missions.

An imaging radar creates a 2D image of a surface from orbit, illuminating a portion of this surface named the "footprint" of the radar, moving as the radar moves around its orbit. Covering the surface with this footprint, the energy of the signal reflected by the surface in the direction of the receiving antenna is represented on the generated surface image. To avoid ambiguities between equidistant points in two different directions, imaging radars are side-looking radars. Imagins radars are often Synthetic Aperture Radars (SAR) as opposed to Real Aperture Radars (RAR). They provide a way to improve the along track resolution using Doppler shift (azimuth resolution) and across track (range resolution). The idea behind this technique is that the subsequent footprints will partly superimpose, and thus a single surface target will produce echoes in subsequent acquisitions (see Figure 1.4,right). With the range of echoes and the Doppler information, echoes from a same target can be identified, and their energy focused in the corresponding position in the image. The generated radar image of the surface will thus have a better resolution than the radar footprint would in theory allow with a RAR (see Figure 1.4,left). imagers.

Among the non-imaging planetary radars, the most commons are altimeters. Altimeters are nadirlooking radars, measuring the distance between the instrument and a target on the surface from the travel time of the corresponding echo. They are the only way to obtain a digital elevation model for bodies with a very thick atmosphere (like Venus or Titan for instance) where optical instruments are unable to see the surface. As the echoes measured for one position of the radar can originate from targets located anywhere in the radar footprint, and as the first measure echo is not necessarily originating from the center of the footprint depending on the surface topography, two types of radar altimetry techniques are proposed: beam-limitation or pulse-limitation.

The last main type of radars used in planetary exploration is the sounding radar. The idea is to acquire not only signals reflected by the surface, but to use signals able to penetrate into the subsurface and be reflected by buried structures towards the receiving antenna. In the following of this chapter, we will focus on this type of planetary radar, as the WISDOM instrument is one of them.

Design of a sounding radar

A sounding radar is a specific type of radar, designed to transmit an electromagnetic signal towards the surface, and receive the signal reflected/scattered by the surface and subsurface according to the principles described in the first section of this chapter. Such a radar is designed according to the desired penetration depth in the sounded subsurface, spatial resolution and sensitivity, and is always the result of trade-offs. In the context of planetary exploration missions, additional constraints also drive the design of sounding radars, such as the radiative environment, the allocated mass and dimension budgets, the available power or data volume and storage.

In this section, three elements of this design will be introduced, as they are key in understanding the design choices of the different planetary radar sounders presented later in this chapter.

The central frequency and its implications

The "central frequency" of a radar sounder is commonly defined as the median frequency in the spectrum of the signal transmitted by the instrument. For a radar that operates on a relatively narrow frequency bandwidth, studying the propagation of a harmonic wave at a central frequency is sufficient to model the propagation of the signal actually transmitted by the radar. In this case, the central frequency is directly related to the penetration depth of the instrument, as it is directly related to the absorption and scattering losses in the materials the radar signal propagates through.

The penetration depth is commonly defined in a homogeneous medium, is often defined by the electrical skin depth, defined as the depth at which an electromagnetic wave power decreases by a factor of 1/0.

In low losses and a homogeneous material ( " ≪ ′), and considering only absorption losses, the electrical skin depth A OROE can be expressed:

A OROE ≈ Y 2 √ ′ ?@ A ( 1.17 )
At first order, the penetration depth can be considered to be half the skin depth.

For a given material, if the frequency dependence of the permittivity can be neglected over the frequency bandwidth considered, then the electrical skin depth is inversely proportional to the frequency of the electromagnetic wave.

As a consequence, when designing a sounding radar, one should select a central frequency as low as possible, still remaining in a propagating mode, to penetrate as deep as possible in the subsurface.

However, this choice is limited by other constraints. 1) The central frequency drives the dimensions of the transmission and reception antennas, which must be in the order of the corresponding wavelength or its first harmonics (typically c, d or d e

) to be efficient. The dimensions of the antennas and their mass being limited in the context of planetary exploration, typically the central frequency is in the order of a few MHz for radars operating from orbit (ALSE, LRS, MARSIS or SHARAD for instance, see Section

3) with antennas of a few tens of meters' maximum, and between a few hundred MHz and a few GHz for radars operating on the surface (LPR, RoSPR, RIMFAX or WISDOM for instance, see Section 4), with antennas of a few tens of centimetres maximum. 2) A very low central frequency limits the frequency bandwidth of the instrument, and thus its maximum achievable spatial resolution (see the next section).

3) The radio environment (mainly transmission from the sun or Jupiter) and the impact of a planet's ionosphere (if any) must be considered when choosing the central frequency of the instrument.

Additional losses due to scattering at the surface (related to the roughness of the surface), and to volume scattering in the subsurface (related to the heterogeneity of the subsurface) both reduce the actual penetration depth of the radar signal. To evaluate such losses, different scattering models (either numerical or analytical) can be used depending on the spatial dimensions of the surface roughness (see [START_REF] Chimi-Chiadjeu | Soil surface roughness modeling: limit of global characterization in remote sensing[END_REF] for an analytical model) or subsurface heterogeneities (see Ciarletti et al. (2017b) for a numerical model) compared to the wavelength of the instrument.

The choice of the central frequency of a sounding radar is therefore a compromise between the desired penetration depth (considering absorption and scattering losses), the desired resolution (considering the maximum possible bandwidth, see section 2.3.2), and the mission constraints (electromagnetic environment, mass budget, energy budget, accommodation, etc.).

The bandwidth and the signal modulation

The spatial resolution of a sounding radar corresponds to the minimum distance between two reflectors which generate echoes that can be separated in time-domain by the instrument. This value is proportional to fg , with G the speed of light between the two interfaces, and h the duration of the transmitted signal.

In frequency domain, the spatial resolution is proportional to f i with B the frequency "bandwidth" of the instrument. The bandwidth of a radar corresponds to the spectral span of the transmitted signal.

Since the spatial resolution of the instrument is driven by the speed of the waves imposed by the materials in the subsurface and the signal duration, one should apparently work with a signal as short as possible in time domain (or having an as large as possible frequency bandwidth) to obtain the best resolution.

However, the shortness of the transmitted signal in time domain is limited by two main factors. 1) The chosen central frequency that limits the maximum possible bandwidth of the instrument, and thus the minimum signal duration, as explained in the previous sub-section.

2) The energy of the transmitted signal being the product of the transmission power by the signal duration, for a same desired amount of energy a higher transmitted power is necessary to generate a signal of a shorter duration. In the context of space missions, the limited allocated power thus limits the minimum possible signal duration. This last limitation can be lower down with different modulation techniques, three of them being presented in the following.

Amplitude Modulated Continuous Wave radars (AMCW), sometimes named "impulse radars", is the commonest for Earth-based subsurface penetrating radars. The emitted signal is a very short pulse (often a sino-gaussian wave), generated by Radio Frequency switches. Such a radar works in the time-domain, and this is its main asset, the reception channel directly measuring the impulse response of the subsurface (see Figure 1.5,left). The AMCW technique however requires RF switches working on a wide frequency band, a high transmitted power, and is sensitive to noise.

Frequency Modulated Continuous Wave radars (FMCW), is a "synthetic impulse" radar, as it does not generate impulses but harmonic signals of linearly increasing frequencies (see Figure 1.5,middle), and the time-domain impulse response from the subsurface is obtained after signal processing. The emitted signal thus consists in a harmonic signal of linearly increasing frequency over the bandwidth B of the instrument, during a time T. Such a signal is often named a "chirp". The spectrum obtained by the product between the emitted and the received signal is then transformed in time-domain to obtain the impulse response, with a resolution inversely proportional to B. This technique is less sensitive to noise than the AMCW, and the required power for the same range of the instrument is reduced. However, the electronics required is more complicated, and a coupling between the transmission and reception antennas impair the dynamic of the instrument. Stepped Frequency Continuous Wave radars (SFCW), is also a "synthetic impulse" radar. The emitted signal also consists in harmonic waves of increasing frequencies during a time T, but by steps instead of a linear increase (see Figure 1.5,right). From the amplitude and phase of the transmitted (known) and received (measured) signals for each frequency, the complex spectrum of the impulse response is obtained. As this technique has been selected for the design of the WISDOM instrument, subject of this thesis, it will be detailed in Chapter 2. The SFCW allows a high transmitted energy for a low transmission power, and the short band reception for each frequency reduces the effect of noise. However, as for the FMCW technique, this technique requires complex electronics and the coupling between the transmission and reception antennas impact the dynamic of the instrument.

All radar sounders presented in the following used one of these three modulation techniques, depending on their mission objectives in terms of desired resolution and constraints.

2.3.3. The received power and the link budget

The transmitted power, the gains of the antennas, the attenuators in the RF electronics and in general the dynamic of the instrument will drive the capacity of a sounding radar to detect a target at a given depth.

In order to determine the order of magnitude for the values of these different key parameters, a simple model is used to estimate the "link budget" for a variety of configurations. This is the count of all gains and losses encountered by an electromagnetic signal between its transmission and reception. In this subsection, a link budget at a given frequency, and accounting for the main sources of gains and losses is presented for a homogeneous medium.

In radar theory, the portion of the transmitted signal reflected towards the radar is characterized in terms of received power per unit of surface received compared to the incident power per unit of surface S k [START_REF] Campbell | Radar remote sensing of planetary surfaces[END_REF]. The power density radiated by the radar at a distance l is: 1.18 ) where n 4 is the power of transmitted by the instrument (in W), o 4 the directive gain of its transmitting antenna (as a ratio compared to an isotropic antenna) in the direction of the reflector, and m J is expressed in W/m 2 . The power of the reflected by the target and eventually received by the receiving antenna is: 1.19 ) where S is the backscatter cross section (in m 2 ) of the reflector, p OPP is the effective surface of the receiving antenna (for the frequency and the direction considered). This effective surface can be expressed:

m J = n 4 o 4 4 l (
n = n 4 o 4 m p OPP 16 l e (
p OPP = d _ r ] e s
where c is the wavelength of the transmitted electromagnetic wave and o the directive gain of the receiving antenna. The power received by the instrument is therefore:

n = n 4 o 4 o c 64 l e t m
( 1.20 )

The ratio n n 4 u is commonly expressed in dB. The effect of attenuation and scattering losses can be accounted for with two factors p Q44 et p \JPP : n = n 4 o 4 o c 64 l e t m p Q44 p \JPP ( 1.21 ) The power loss, corresponding to the attenuation factor presented in the previous section on the amplitude, can be expressed (Daniels, 2005):

p Q44 = 0 e 1 v ( 1.22 )
The scattering losses can be expressed introducing a scattering extinction power coefficient w, which can be modelled depending on the size of subsurface heterogeneities compared to the wavelength: 1.23 ) In order to adapt the dynamic of the instrument to the subsurface being sounded, most radars are equipped with tuneable amplifiers and attenuators in transmission and/or reception channels. A gain corresponding to each amplifier and attenuator must be added to the link budget. "Gating" can also be used to delay the reception and avoid the first strongest echoes, and adapt the dynamic range to weaker echoes coming from deeper interfaces.

A yzz = e { | (
More than the level of power received by the radar (as long as it is above the sensitivity of the receiver), what is relevant is the ratio between the power of the signal and the power of the noise n X , named Signal to Noise Ratio (SNR) computed at a given frequency:

SNR = P € P • ( 1.24 )
As previously mentioned, different sources of noise can affect the radar measurements: the electromagnetic environment (mainly the sun and Jupiter for planetary missions), the thermal noise of the instrument, and the noise induced by the different components in the electronics and the antennas.

Maximum unambiguous range and blind zone

Even with a favourable link budget, the range of a sounding radar is also limited by design, by its "maximum unambiguous range" and its "blind zone".

The maximum unambiguous range corresponds to the maximum distance a radar signal can travel back and forth between a target and the antennas between two consecutive transmissions of the radar pulse.

If an echo from a target located farther from the radar than the maximum unambiguous range is received, this echo would be considered as a response to the current radar pulse, while it is the response to a precedent one, hence the ambiguity. The Maximum unambiguous range in free-space can thus be expressed, in the case of an impulse radar, as:

R y'ƒ = c T -τ 2
( 1.25 )

with T the repetition time between two pulses, and h the pulse duration.

The blind zone of a radar instrument corresponds to the minimum distance at which the radar is able to properly detect a target. To avoid saturation of the receiver, the receiving channel is commonly turned OFF during the time necessary for the signal to be transmitted. In this case, for a pulse duration h, the blind zone in vacuum can thus be expressed as:

R ƒ ‡k•ˆ= c τ 2 ( 1.26 )
Therefore, the signal reflected by a target towards the reception antenna can be detected without ambiguities only if the distance between the target and the antennas is between l ‰RJX\ and l QŠ‰ . Note that for a bi-static radar, the design of the radar can reduce l ‰RJX\ to zero in the case of a SFCW or a FMCW modulation.

The subsurface of 4 planetary bodies other than the Earth have been revealed by sounding radars to this day: The Moon, Mars, the nucleus of the 67P comet, and the lakes of Titan. And the subsurface of additional bodies will be sounded in the next two decades: the satellites of Jupiter Europe, Callisto, Ganymede, and Venus. Most of these radars were mounted on orbiters. However, since the selection in 2003 for the ExoMars mission of the WISDOM radar to be operated directly on the surface of Mars, similar in-situ sounding radars have been selected to study the Martian and lunar shallow subsurface, and three of them have already been operated on the Moon (the two LPR and the LRPR instruments of the Chang'E missions), and two on Mars (RIMFAX of the Mars 2020 mission, and RoSPR of the Tianwen-1 mission). Section 3 and 4 constitute an overview of this variety of planetary radar sounders, and eventually concludes with the WISDOM instrument, main topic of this thesis.

Exploring the subsurface of planetary bodies with orbiting radars

The first radar sounders to be selected for planetary missions were in-orbit instruments, mounted on spacecraft. Such configuration of operation is an efficient first approach in studying the subsurface of planetary bodies, as soundings are possible on a large scale, sometimes encompassing the entire studied CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 20 object. Since the operations of the first orbiting radar sounder in 1972 on the Moon, other instruments of this type operated or have been selected to operate on very different objects of the solar system: The Moon, Mars, the nucleus of the 67P comet and in the future the icy moons of Jupiter. In addition, the Cassini radar also proved able to sound the lakes of Titan, even if not intended as a radar sounder.

This section proposes a review of the different planetary radars mounted on orbiters, briefly presenting their mission, design and results. Even if this thesis will focus on a sounding radar operated from the surface of Mars, in-orbit planetary radars historically first demonstrated the ability of sounding radars to reveal the subsurface structure and composition of planetary bodies. Furthermore, some elements of their design are driven by the same physical constraints related to their design performances (targeted resolution, penetration depth, etc.), characteristics of the targeted subsurface (subsurface dielectric properties, loss tangents, surface roughness, scattering losses, etc.) and mission requirements (mass and power budgets, maximum dimensions, etc.). This is why a presentation of the different in-orbit radar sounders was important to provide the context of the emergence of on-surface radar sounders.

Sounding the subsurface of the Moon

The Moon was the first planetary object other than the Earth to have its subsurface explored by a sounding radar (see Figure 1.6.a). To this day, two radar sounders operated in orbit around the Moon:

the ALSE instrument in 1974, and the LRS instrument in 2007.

The ALSE instrument of the Apollo 17 mission

The very first sounding radar ever to operate on another planetary object than the Earth was the Apollo Lunar Sounder Experiment (ALSE), onboard the mission Apollo 17 [START_REF] Porcello | The Apollo Lunar Sounder Radar System[END_REF]. If in-orbit subsurface sounding radars did not reveal much of the Earth subsurface, except in polar regions, due to the presence of underground liquid water underground, the low loss tangents (for frequencies below 300 The two main objectives of the ALSE radar were the detection of subsurface discontinuities (expected from different lava flows, regolith layers or crater ejectas), and the generation of lunar surface profile (the previous Apollo missions were only equipped with laser altimeters, with measurement sampling between 30 and 43 km) [START_REF] Porcello | The Apollo Lunar Sounder Radar System[END_REF]. In agreement with loss tangents from Apollo lunar samples, three frequency channels were selected, 5, 15 and 150 MHz, respectively named HF1, HF2 and VHF, with a modulated "chirped" signal. The corresponding bandwidth for each mode is a 10 th of its central frequency (0.533 MHz, 1.6 MHz and 16 MHz), leading to free-space resolutions of 300, 100 and 10 m. The estimated penetration depths from the analysed samples were 1300 m, 800 m and 160 m. The antennas of the radar were separated in two sub-systems; the HF1 and HF2 channels were connected to dipole elements of 10.4 m, parallel to the lunar surface and perpendicular to the orbiter surface track while the VHF channel was connected to a 12 elements Yagi antenna, oriented 20° off-nadir to the North, in a plane perpendicular to the orbiter surface track. A prototype of the ALSE instrument was tested on Earth, over Southwest American deserts, and over Greenland. In the case of Greenland, subsurface echoes could clearly be identified [START_REF] Porcello | The Apollo Lunar Sounder Radar System[END_REF].

The Apollo 17 mission took place in December 1972, the reception of ALSE acquisitions began in January 1973, but the interpretation of all surveys in the three modes took more than a year [START_REF] Porcello | The Apollo Lunar Sounder Radar System[END_REF]. Despite a reflectivity of the lunar surface higher than expected based on the lunar samples, subsurface echoes were detected in some of the processed radargrams.

The original acquisitions were stored with an optical recorder, having a memory capacity of 6 Moon orbits of soundings, and a maximum bandwidth of 16 MHz (digital recording of large amount of data in the order of gigabytes was indeed impossible in 1972). Three main types of data storage were then applied to the original films from the optical recorder. First photographic imagery, the easiest but at the expense of dynamic range and phase information (see After the original interpretation of ALSE radargrams by [START_REF] Porcello | The Apollo Lunar Sounder Radar System[END_REF], later studies were conducted to give additional interpretations to the detected subsurface echoes. Comparing radargrams in holographic and digital formats, and applying a "Multi-Orbit Correlation" technique to confidently identify which observed echoes were from the subsurface, [START_REF] Peeples | Orbital radar evidence for lunar subsurface layering in Maria Serenitatis and Crisium[END_REF] reported the detection of underground reflectors in Mare Serenitatis (0.9 and 1.6 km under the surface, see Figure 1.8) and in Mare Crisium (1.4 km below the surface). Loss tangents estimated from these echoes were consistent with values measured on lunar samples. The two subsurface layers detected in Mare Serenitatis were correlated to the different volcanic events and regolith formations in the region [START_REF] Sharpton | Stratigraphy and structural evolution of Southern Mare Serenitatis: A reinterpretation based on Apollo Lunar Sounder Experiment data[END_REF]. In addition, nearly 20 years after the mission, [START_REF] Cooper | New evidence for graben origin of Oceanus Procellarum from lunar sounder optical imagery[END_REF] re-created an interpretation process for the radargrams in image format (the holographic viewer being difficult to re-create), also with Multi-Orbit Correlation, to verify the plausibility of two formation scenarios for the Oceanus Porcellanum region.

The LRS instrument of the SELENE (Kaguya) mission

Inspired by the ALSE radar, the Japanese space agency JAXA selected a sounding radar for its mission SELENE, also named Kaguya, which arrived in lunar orbit in October 2007. This orbiter operated around the Moon first at a polar orbit with a maximum altitude of 100 km, and then reduced its altitude down to 50 km in February 2009, and down to 10 km in April 2009, until it was voluntarily crashed on the lunar surface. The selected radar, named Lunar Radar Sounder (LRS), was designed to operate at the same frequency as the ALSE instrument to guarantee a significant subsurface penetration, but with an increased transmitted power and a better resolution [START_REF] Ono | Lunar Radar Sounder (LRS) experiment on-board the SELENE spacecraft[END_REF]. The LRS instrument had three main operating modes: a passive radio acquisition mode from 100 Hz to 1 MHz, a natural plasma wave receiver mode for acquisition between 20 kHz and 30 MHz, and a sounding radar mode. Its objectives of the LRS in the SELENE mission were to perform the same sounding operations as ALSE, with a noise level lower than the galactic noise to penetrate several kilometres into the lunar subsurface, and to measure the interaction between the Moon surface and the solar wind at micrometric scale [START_REF] Ono | The Lunar Radar Sounder (LRS) onboard the KAGUYA (SELENE) spacecraft[END_REF][START_REF] Ono | Lunar Radar Sounder (LRS) experiment on-board the SELENE spacecraft[END_REF].

The instrument was a Frequency Modulated Continuous Wave (FMCW) radar, with a nominal central frequency of the radar was 5 MHz, and the bandwidth 2 MHz (with 2048 samples), resulting in a freespace spatial resolution of 75 m. The LRS also had the option to sound at 15 MHz with a 2 MHz bandwidth, or at 1 MHz with a 2 MHz bandwidth, in case the attenuations in the lunar subsurface were lower or higher than expected. Its nominal frequency of 5 MHz required a large 30 m tip-to-tip crossdipole antenna (see Figure 1.9). As a radar working in the frequency-domain, an Inverse Fourier transform was applied to the acquired data to obtain time-domain soundings, as well as a running average to the radargrams to increase the SNR, and reduce the effect of off-nadir echoes, which could be wrongly interpreted as subsurface scatters [START_REF] Ono | The Lunar Radar Sounder (LRS) onboard the KAGUYA (SELENE) spacecraft[END_REF]. SAR processing can also be applied to focus echoes from subsurface scatters and ease radargram interpretation, as performed by Kobayashi et al. (2012a) and illustrated in Figure 1.10. The radargrams from the LRS showed that most of the near-side lunar Maria have subsurface stratifications, as shown in Figure 1.11, at apparent depths of several hundreds of meters [START_REF] Ono | Lunar radar sounder observations of subsurface layers under the nearside maria of the moon[END_REF]. The deepest detected interface reached ~1.05 km of apparent depth, in Mare Imbrium. The acquisition of soundings on spatially close tracks ruled-out the possibility of these detected echoes being off-nadir surface reflections. With a simple Fresnel reflection model, the dielectric constants and loss tangents associated with each detected interface were estimated, and proved to be consistent with previous estimations from the ALSE soundings. The results are presented in Figure 1.12.

If subsurface stratifications are visible in all lunar Maria, they are not present in every sounding performed in these Maria. The distribution of subsurface stratifications detected by LRS have been mapped by [START_REF] Oshigami | Distribution of the subsurface reflectors of the western nearside maria observed from Kaguya with Lunar Radar Sounder[END_REF] in the western part of the lunar near-side, and on a global scale by [START_REF] Pommerol | Detectability of subsurface interfaces in lunar maria by the LRS/SELENE sounding radar: Influence of mineralogical composition[END_REF]. The former observed a correlation between this distribution and the estimated geological age of the region, illustrated in Figure 1.13, concluding in a formation age of more than 3.4

CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 26 billion years for these formations. The later one found a negative correlation between the distribution of the observed subsurface layers and the abundance of TiO2 and FeO as recorded during the Clementine mission. This anti-correlation is interpreted as due to the strong absorption at low frequencies by lunar ilmenite (FeTiO3), implying that the non-detection of subsurface layering in some Maria regions by LRS does not necessarily mean that such layers are not present. The detected interfaces in Mare Serenitatis were compared to the stratifications observed on crater walls in the same region, and showed that these interfaces could be separating materials of the same composition but different porosity, suggesting a layer of regolith [START_REF] Oshigami | The layered structure of lunar maria: Identification of the HF-radar reflector in Mare Serenitatis using multiband optical images[END_REF]. The estimated depths of the detected interfaces have even been used to estimate the lower limits of the volume of the corresponding geologic units [START_REF] Oshigami | Mare volcanism: Reinterpretation based on Kaguya Lunar Radar Sounder data[END_REF]. After the SELENE mission, the following lunar sounding radars were all operated from the surface, by the Chinese space agency (CNSA). The results of the LRS soundings were proposed to guide the selection of the first on-surface lunar radar sounder (Kobayashi et al., 2012b), presented later in this chapter. penetrators, all instrumented [START_REF] Zakharov | The Mars-94 and Mars-96 missions[END_REF]. The United States, India and different European countries, among which France, provided instruments for the mission. In addition to these different elements of the mission, a 5500 m 3 helium balloon carrying a 15 kg instrumented payload was also planned. The balloon project was later postponed, to be integrated to the Mars 98 mission, and definitively cancelled in 1995. One of the instruments in the payload was the sounding radar PRISM (Permafrost Radar Investigation of the Subsurface of Mars).

The PRISM radar was intended to measure the thickness of the Martian permafrost, and determining the dielectric properties of the Martian subsurface [START_REF] Barbin | Mars 96 GPR program[END_REF]. This design required a 12 m long dipole antenna. To create drag and thus stabilize the lift of the balloon, a 6 m long guide rope was to be attached to the payload. The balloon was expected to fly lower during the night, the guide rope touching the ground [START_REF] Barbin | Mars 96 GPR program[END_REF]. The guide rope would be dragged by the balloon on the surface of Mars at a speed of a few meters per second [START_REF] Laplace | The guiderope for the Soviet-French Mars 96 Balloon Mission[END_REF]. The guide rope therefore constituted a unique opportunity to allocate the PRISM radar into. The selected solution consisted into a flexible antenna, half of which was the guide rope of the balloon, composed of a series of 18 metallic cylinder, and the other a metallic cable named the "tail" (see the schematic in In order to test this design, in 1992 two measurement campaigns have been organized with a prototype of PRISM in natural sites: The Pyla dune and the Mont-de-Lans glacier in France. The radargram acquired over the Pyla dune allowed the detection of the water table at around 50 m of depth, expected as sea water was known to be present at the bottom of the dune [START_REF] Barbin | Mars 96 GPR program[END_REF]. With the radargram acquired over the Mont-de-Lans glacier, the ice thickness could be estimated between 30 and 120 m, coherent with the expectations from previous studies on this glacier [START_REF] Mustard | An HF Bi-phase shift keying radar: Application to ice sounding in western alps and spitsbergen glaciers[END_REF]. Later, in 1993, a measurement campaign at the Dumont d'Urville station in Antarctica. The radargrams acquired over the glacier and presented in Figure 1.15 showed reflections probably originating from a bedrock (even if liquid water cannot be totally ruled out), and weaker echoes possibly from a mixture of ice and rocks [START_REF] Herique | Determination of the ice dielectric permittivity using the data of the test in Antarctica of the ground-penetrating radar for Mars'98 mission[END_REF]. The primary objective of the MARSIS instrument is the mapping of the distribution of water on Mars (liquid or solid). This is also one of the primary objectives of the Mars Express mission. The secondary objectives of the mission are the geological study of the Martian subsurface in general, the characterization of the surface, and the sounding of the ionosphere [START_REF] Picardi | Mars advanced radar for subsurface and ionosphere sounding (MARSIS): concept and performance[END_REF].

Sounding the subsurface of Mars is more complex than sounding the Moon, because of its ionosphere.

This later indeed stop the radar waves of low frequency (below 3 MHz) on dayside, and even radar waves able to propagate through are affected by dispersion effects. To take into account this effect, as well as expected dielectric properties of the Martian subsurface, the MARSIS instrument is adaptable and can generate 4 quasi-simultaneous chirp modulated signals, in 1 or 2 different modes among the 2 dayside modes and 3 nightside modes [START_REF] Picardi | Planetary science: Radar soundings of the subsurface of Mars[END_REF][START_REF] Picardi | MARSIS: Mars Advanced Radar for Subsurface and Ionosphere Sounding[END_REF]. Each mode has a bandwidth of 1 MHz, leading to a free-space resolution of 150 m. The MARSIS instrument is also able to sound the Martian ionosphere, with a stepped chirp of 20 kHz steps. The working frequencies of the different modes of MARSIS are summarized in Table 1.1: The antenna system is divided into two main parts. The main antenna is a 40 m long dipole, with a matching network, made of two 20-m long elements. Its deployment was a critical process, which ended only in June 2005. Then, a secondary 7 m monopole receiving antenna with a null in the nadir direction was deployed, to detect off-nadir echoes so that the subsurface echoes could be easily differentiated from off-nadir surface echoes [START_REF] Picardi | MARSIS: Mars Advanced Radar for Subsurface and Ionosphere Sounding[END_REF].

The MARSIS on-board processor uses adaptive range compression to reduce ionosphere and hardware distortion. It is also able to perform Doppler processing. The footprint of the instrument on the Martian surface is ~5x10 km at ~800 km, and measurements are performed every ~5 km.

MARSIS fulfilled its objectives of giving an insight into the distribution of subsurface water (solid and liquid), and helping geologic interpretations of different regions, through many different studies. A few of them are presented below to illustrate the impact of such an instrument on the exploration of Mars.

Regarding water ice, MARSIS estimated the thickness of the south polar layered deposits (SPLD); the signal penetrating down to 3.7 km thanks to very low attenuation rates (Plaut et al., 2007). This estimation was possible because the echo from the interface between the ice and the rocky substrate is clearly visible in MARSIS radargrams for most of the SPLD region. Thickness variations were observed, as visible in Figure 1.17, and a total volume of ice ~1.6.10 6 km 3 was obtained. The low attenuation levels allowing such deep soundings are compatible with nearly pure water ice, with a maximum of 10 % of dust inclusions if any. A complete Martian surface echo power map was also generated from MARSIS radar traces by Mouginot et al. (2010), which allowed the estimation of the dielectric constant at 3-5

MHz, taking into account the roughness of the surface from MOLA observations. Assuming a homogeneous ground composition and a flat surface at MARSIS scale, a total ground water ice volume of ~10 6 km 3 can be estimated from this map, a volume equivalent to a Martian polar cap. In the region of Vastitas Borealis, Mouginot et al. (2012) observed dielectric values too low to be consistent with volcanic materials, but plausible for low density sediments or ground ice. This could be the remnants of an ancient ocean. With a simple two-layers model, it was shown that a 11 m thick layer of CO2 ice above water ice could explain the low reflectivity of the south residual ice cap measured by MARSIS, providing a first estimation of the dimensions of this layer (Mouginot et al., 2009). Regarding liquid water, MARSIS detected a sub-glacial lake in 2018 in the region of Ultimi Scopuli, in the Martian south polar layered deposits (SPLD) (Orosei et al., 2018). The bright reflectivity of the region pointed to dielectric constants superior to 15. To confirm this discovery, which is probably hypersaline perchlorate brines, an interpretation software successfully applied on Earth to detect subglacial lakes has been used. This detector is based on three criteria: signal intensity, signal acuity and intensity variations. It has been applied to 134 MARSIS radar profiles (Lauro et al., 2021). The generated map of dielectric constant estimations (see Figure 1.18) even suggests the present of one major subglacial lake, with smaller lakes around it, possibly connected.

Regarding geological interpretations in general, the MARSIS instrument allowed the detection of buried impact basins in the northern lowlands. From the number of detected craters of more than 200 km, a lower limit of the age formation of the southern lowlands crusts could be proposed [START_REF] Watters | MARSIS radar sounder evidence of buried basins in the northern lowlands of Mars[END_REF].

The Medusae Fossae Formation, possibly the region of Mars presenting the youngest deposits, has also been studied by [START_REF] Watters | Radar sounding of the medusae fossae formation mars: Equatorial ice or dry, low-density deposits?[END_REF] with MARSIS radargrams. The composition of the deposits in this region is unknown, the main possibilities being volcanic ash, eolian sediments or ice-rich deposits. Based on the subsurface echoes identified in MARSIS soundings, a dielectric constant of ~2.9 for the deposits layer was evaluated, consistent with a low density material, or water ice with significantly more dust inclusions than the polar deposits. [START_REF] Boisson | Sounding the subsurface of Athabasca Valles using MARSIS radar data: Exploring the volcanic and fluvial hypotheses for the origin of the rafted plate terrain[END_REF] present another study in which MARSIS greatly helped the geological interpretation of a Martian site, namely it showed that a rafted plate terrain identified near Athabasca Valles, which could be interpreted as being a fluvial or volcanic formation, was probably volcanic rather than fluvial. MARSIS radargrams were compared to numerical simulations with the different formation scenarios, and plausible dielectric properties to establish this diagnostic. Eventually, even if the MARSIS radar was originally designed to sound the subsurface and the ionosphere of Mars, the possibility of sounding the Martian natural satellite Phobos (which will be the target of future missions such as MMX) was studied through simulations (Plettemeier et al., 2009) before being done. Soundings were acquired, with a special mode of the instrument to override the minimum time limit between transmission and reception while preserving its safety [START_REF] Cicchetti | Observations of Phobos by the Mars Express radar MARSIS: Description of the detection techniques and preliminary results[END_REF]. To this day, no subsurface echoes have been detected. Possible explanations include the ideas that the expected spatial resolution of the instrument in Phobos materials is higher than the thickness of the first regolith layer, or that losses in the Phobos subsurface are higher than expected.

For more results of the MARSIS instrument, see [START_REF] Orosei | Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS) after nine years of operation: A summary[END_REF] which propose a summary of its first nine years of activity.

The SHARAD instrument of the MRO mission

The Shallow Radar (SHARAD) instrument of the Mars Reconnaissance Orbiter (MRO) mission of the American space agency NASA (shown in Figure 1.19) has been launched in 2004, and arrived around In the south polar plateau, the MARSIS instrument did reveal the presence of a rocky substrate below the SPLD, but not the expected layering of the ice sheet itself. In Promethei Lingula, SHARAD did observe this layering in 4 close tracks, with dozens of subsurface reflections, some of them seen over 150 km. The layering is interpreted to be due to differences in dust inclusions or ice density [START_REF] Seu | Accumulation and erosion of mars' south polar layered deposits[END_REF]. The stratigraphy of the North Polar Layered Deposits (NPLD) was also revealed by the SHARAD CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 35 instrument, with subsurface echoes from interfaces in the ice with distances of ~500 m between them, and a Basal Unit (also named BU) substrate at ~2 km below the surface (see Figure 1.20). The interfaces are also interpreted as limits between regions of different ice density and dust inclusions. The layers being continuous over 1000 km, a homogeneous mechanism, probably climate forcing from the periodic variations of the Martian orbit, must be generating these interfaces. An alternation of periods with dust storms and ice deposition, and periods with sublimation compacting the dust layers are proposed to explain the observed layering [START_REF] Phillips | Mars north polar deposits: stratigraphy, age, and geodynamical response[END_REF]. Layering of the Basal Unit can also be observed in SHARAD radargrams. With the accumulating SHARAD soundings over the northern polar cap, the four main layers of ice deposits observed in the NPLD have later been studied in 3D, to delimitate them and evaluate their volume [START_REF] Foss | 3D imaging of Mars' polar ice caps using orbital radar data[END_REF][START_REF] Putzig | Subsurface structure of Planum Boreum from Mars Reconnaissance Orbiter Shallow Radar soundings[END_REF]. Additionally, this 3D mapping of the NPLD allowed the detection of 21 potential impact craters buried inside [START_REF] Putzig | Threedimensional radar imaging of structures and craters in the Martian polar caps[END_REF]. An important discovery as crater-counting remains one of the main techniques to date a geological site in planetary science. Subsurface echoes from mid-latitude geologic formations have also been uncovered by the SHARAD instrument. In the Martian plain of Amazonis Planitia, where different episodes of sedimentary and volcanic activities occurred, a subsurface interface was detected, with an increasing depth towards the north (maximum depth of 100-170 m). This interface is likely the limit between Vastitas Borealis sediments and Hesperian volcanic plains [START_REF] Campbell | SHARAD radar sounding of the Vastitas Borealis Formation in Amazonis Planitia[END_REF]. The loss tangent of the layer of sediments was estimated from the detected echoes. In the eastern Hellas impact basin, buttes massifs and escarpments named "Lobate Debris Aprons" (or LDA) were sounded by the SHARAD instrument, to determine their internal structure and composition. The only echoes not matching simulations being found when SHARAD is over these LDA, they were identified as subsurface echoes. The lack of observed volume scattering indicated a homogeneous material, and low attenuation levels suggested a buried water ice glacier below a shallow layer of debris [START_REF] Holt | Radar sounding evidence for buried glaciers in the southern mid-latitudes of Mars[END_REF]. The Medusae Fossae Formation, already scrutinized by MARSIS radargrams, was also sounded by SHARAD. Clear interfaces are visible in only two locations, as shown in Figure 1.21.a, probably due to the surface roughness inducing a strong scattering effect. Nevertheless, the high porosity of the material determined by MARSIS was confirmed by SHARAD, and in the regions where the interface is detected, no clear layering is visible (Carter et al., 2009b). The absence of interfaces between layers of deposits indicates that the Medusae Fossae Formation is not a mid-latitude equivalent of the polar deposits.

Lava flows in Martian volcanic terrains were also successfully sounded by SHARAD, with two subsurface interfaces visible below lava flows in the northwest of the Ascraeus Mons, a volcano of the Tharsis region. The average dielectric constants of ~12.2 and 9.8 estimated for both interfaces, and the loss tangents between 0.01 and 0.03 are compatible with Earth or lunar basalts (Carter et al., 2009a).

This study represented the dielectric measurement of an entire Martian volcanic terrain, and allowed the estimation of the lava flow density. Additional dielectric studies were later conducted with SHARAD in the same region [START_REF] Simon | Studies of lava flows in the Tharsis region of Mars using SHARAD[END_REF], corresponding to the radargram in Figure 1.21.b. The peculiarity of the MOSIR instrument is its ability to perform soundings in two perpendicular polarizations, with orthogonal dipoles antennas of 10 and 9 m tip-to-tip. This asset of the instrument will allow an improved characterization of the detected surface and subsurface reflectors. As consequence, the lander could not recharge its batteries as planned with the sun light and surface operations were reduced and conducted in only a few days (from 12 th to 14 th November). However, ESA announced that around 80% of the Philae scientific objectives were completed in this short time. Based on previous missions to different comets, and Earth observations, a comet nucleus was expected to be quite transparent to radio frequencies, and the duo Rosetta-Philae offered a unique opportunity to sound 67P with a bistatic tomographic radar. The idea proposed by [START_REF] Barbin | The consert instrument for the Rosetta mission[END_REF] and Kofman et al. (1998) is the transmission of a radar signal by the Rosetta orbiter, the reception of the signal by the Philae lander, and a re-transmission towards the orbiter, allowing the sounding (or tomography) of the interior of the 67P nucleus through different angles in bistatic configurations as Rosetta moves around the nucleus. This instrument, named CONSERT (Comet Nucleus Sounding Experiment by Radiowave Transmission), was selected by ESA in 1996 to be part of the Rosetta mission with the following objectives: to measure the mean dielectric properties of the comet nucleus to constraint its composition and porosity, to detect large scale buried structures or layers, and to detect small scale irregularities [START_REF] Barbin | The consert instrument for the Rosetta mission[END_REF].

CONSERT was designed with an estimated budget link, taking into account the expected geometry of the comet nucleus, the bistatic configuration with the planned trajectories, and the expected comet dielectric properties. The selected carrier frequency was 90 MHz (corresponding to a wavelength ~3 m in vacuum), found to be a good compromise in terms of budget link [START_REF] Barbin | The consert instrument for the Rosetta mission[END_REF]. With the expected properties of the comet nucleus, a bandwidth of 10 MHz was chosen to obtain a resolution of 20 m (or 30 m in vacuum). The two main challenges were the synchronization of clocks on Rosetta and Philae, and the acquisition of a free-space measurement between the two units, which would have served as a reference and allowed a direct estimation of the mean dielectric constant of the comet nucleus. Unfortunately, the unexpected final landing site of Philae did not permit any calibration, complicating the estimation of the nucleus mean dielectric constant. During the time scope between the 12 th and the 13 th November, named the "First Science Sequence", the CONSERT instrument acquired soundings with a high SNR (Signal to Noise Ratio) for the first 30 min of the 9 hours sequence, and the last 80 min. Between the two sets of data, a SNR lower than expected was measured or a total lack of reception which can be ascribed to: The wrongly adapted orbit in regard of the unexpected landing site, the wrong orientation of the Philae antennas, and/or a possible absorption stronger than expected in the nucleus.

The data acquired by CONSERT has been divided into four different categories by the science team (see Figure 1.24.a): 1 for strong reception and available synchronization, 2 for acceptable SNR but no synchronization, 3 for low SNR and no synchronization, and 4 for the total absence of signal [START_REF] Kofman | Properties of the 67P/Churyumov-Gerasimenko interior revealed by CONSERT radar[END_REF].

Three additional measurements of 15 min have been performed afterwards to retrieve the position of the Philae lander on the surface of 67P by triangulation (see Figure 1.24.b). CONSERT located the lander in a 150 x 15 m zone, with an accuracy of 10-20 m [START_REF] Herique | Philae localization from CONSERT/Rosetta measurement[END_REF].

The first observations of the interpretable soundings showed few scattering effects (in surface or volume), a sign of low dielectric contrasts or of a homogeneous material at the wavelength of CONSERT. Example of soundings are presented in Figure 1.25. To compensate for the lack of calibration measurement, different scenarios were modelled with possible estimated positions of the lander within the triangulated window, and different mean dielectric constants of the nucleus assuming a value below 2 and negligible loss tangent (in agreement with composition and structure analysis from other instruments). The results of these models were compared to the delays of the echoes detected in the different soundings. The best fits were obtained in a window of 21 x 34 m, and for a mean dielectric constant ~1.27 [START_REF] Kofman | Properties of the 67P/Churyumov-Gerasimenko interior revealed by CONSERT radar[END_REF][START_REF] Herique | Cosmochemical implications of CONSERT permittivity characterization of 67P/CG[END_REF]. A potential near-surface gradient in dielectric constant of the nuclei was suggested using the high-SNR soundings performed at grazing angles [START_REF] Ciarletti | CONSERT suggests a change in local properties of 67P/Churyumov-Gerasimenko's nucleus at depth[END_REF]. Ray-tracing simulations with a nucleus model determined from the Osiris camera observations, the position of Philae estimated by triangulation, the estimated density of the nucleus (~470 kg/m 3 ) and porosity (70-80%) were generated with a half-cosine dielectric gradient model. The idea was to test different dielectric gradients and distances through the nucleus, to exclude values of both parameters incompatible with the experimental reception of the signal. A dielectric constant lower at the surface than inside the 67P nucleus was found to be impossible, and the most likely scenario would be variations from 1.7 at the surface to 1.3 inside the nucleus along a distance of 150 m [START_REF] Ciarletti | CONSERT suggests a change in local properties of 67P/Churyumov-Gerasimenko's nucleus at depth[END_REF].

Other dielectric measurements on the 67P comet nucleus were conducted by the permittivity probe SESAME-PP. These measurements were difficult to interpret due to the unknown exact attitude of the Philae lander. However, with the data from the descent (measurement in vacuum), different possible attitudes of the lander, and the measurements performed on the surface, an estimation of 2.45 for the dielectric constant of the first meter of the nucleus was found by [START_REF] Lethuillier | Electrical properties and porosity of the first meter of the nucleus of 67P/Churyumov-Gerasimenko: As constrained by the Permittivity Probe SESAME-PP/Philae/Rosetta[END_REF]. This measurement tends to confirm a higher dielectric constant for the surface of the nucleus, even if it must be noted that these measurements were conducted for extremely low frequencies of 409-758 Hz.

However, at very low temperature the effect of frequency on the dielectric constant is expected to be low. In early 2016, Philae was detected in images of the surface of 67P (see Figure 1.26), in a location later named Abydos, allowing new models for interpretations. Information on the nucleus structure and composition were for instance inferred from the variation of the pulse width of echoes measured by CONSERT compared to calibration measurements performed on Earth in 2002. Only a very slight broadening of the -3 dB and -6 dB pulse widths was observed, but not significant, confirming the absence of significant scattering (results presented in Figure 1.27). The measured pulse widths were then compared to numerical simulations with two different models of nucleus formation, with spherical inclusions of fractal structures. The conclusion was that no structure with a characteristic size lower than 1 m would impact the pulse width of CONSERT echoes, but 3 m size structures would be compatible with observations if the dielectric contrast stays below 0.25 (Ciarletti et al., 2017b). Despite the unexpected conditions in which the CONSERT soundings were performed, and the absence of signal from Philae in most of the acquisitions, estimations of the 67P nucleus dielectric and structural properties and structure was possible, and gave insights into its possible composition, porosity and formation history. This instrument remains to date the only sounding radar ever to operate on a comet nucleus.

Bathymetry of Titan's lakes

With a diameter of 5150 km, Titan is the largest natural satellite of Saturn, and the second largest in the whole solar system (see Figure 1.28.a). It is even larger than the planet Mercury. Titan is also the only known satellite in the solar system with a thick atmosphere, the atmospheric pressure at its surface being around 1.5 the pressure on the surface of the Earth. Another peculiarity of Titan is the presence of large quantities of liquid on its surface, a singularity it shares only with the Earth in the solar system. Unlike the Earth, liquids on the surface of Titan are not composed of water, but of hydrocarbons (illustration in Figure 1.28.c). These Titanian seas and lakes have been the subject of radar soundings in the last decade, leading to the first ever extra-terrestrial bathymetry measurements.

If their existence was already guessed since the 1970s, Earth-based, Pioneer 11 and Voyager 1 and 2 observations did not allow any confirmation on the presence of liquid surfaces on Titan, the atmosphere of the satellite absorbing a large band of sunlight. Only in 2007, with SAR imaging from the Cassini radar (see Figure 1.28.b), dark patches of various size and shapes in Titan's Northern hemisphere have been clearly identified as lakes [START_REF] Stofan | The lakes of Titan[END_REF]. It must be noted that even if it was not confidently identified as a lake at the time, Ontario Lacus was seen in ISS (Imaging Science Subsystem) imaging as early as 2005[START_REF] Mcewen | Mapping and Monitoring the Surface of Titan[END_REF]. The Cassini radar was in fact a multimode instrument [START_REF] Elachi | Cassini Titan Radar Mapper[END_REF] working at a central frequency of 13.78 GHz (corresponding to a wavelength of ~2. Hydrocarbons lakes on Titan from surface radar images.

The Cassini radar has therefore not been designed as a radar sounder. However, liquid hydrocarbons on the surface of Titan were expected to have a low loss tangent [START_REF] Gelsthorpe | Some measurements on the temperature variation of loss angle of a group of low-loss liquid dielectrics at microwave frequencies[END_REF], and the Cassini radar in altimeter mode (when observing at nadir) could potentially penetrate down to the bottom of Titan seas (with expected penetration depths of a few hundred meters). Nevertheless, the 4.25 MHz bandwidth of the radar altimeter would only allow a 35 m resolution in free-space. The side-lobes from the surface echo could also be stronger than the potential echoes from Titanian seabeds. For these reasons a true bathymetry of Titan's seas and lakes was thought impossible until 2014.

That year, Mastrogiuseppe et al. (2014) proposed a new processing chain for the Cassini radar in altimeter mode, allowing the first bathymetry of a Titan sea, Ligeia Mare. To improve the resolution of Cassini radar soundings, a super-resolution technique known as the "Bandwidth Extrapolation" (BWE),

proposed by Cuomo (1992), has been applied in frequency domain. This technique can enhance the resolution of radar soundings by a factor of 3, leading to a resolution of ~12 m in free-space for the Cassini radar altimeter (corresponding to a new bandwidth of 9 MHz, see Chapter 4 for more details).

Then, to reduce the side-lobes from the surface echo at least 10 dB below the level of subsurface echoes, an adapted Kaiser-Bessel window was applied to the soundings before Inverse Fourier Transform.

Eventually, to improve the SNR of echoes, an average of the different bursts in range-Doppler domain was performed.

This processing chain was applied to the T91 Titan flyby altimetry radar data, a 300 km track at an altitude of 1600 km acquired in nadir direction over Ligeia Mare on 23 rd May 2013. Ligeia Mare is the second largest liquid surface on Titan after Kraken Mare, with 126.000 km 2 . The obtained bathymetric profile shows a maximum depth of ~160m, and slope steeper at the beginning of the traverse than at the end. From the newly detected echoes corresponding to the seabed, the loss-tangent of the liquid forming the sea has been estimated with two different techniques. First, with the shape of the echo pulses, compared to a Cassini radar signal model. Then, with the magnitude of the echoes. A maximum loss tangent of 8.10 -5 was found, consistent with expected values from liquid hydrocarbon. The soundings after the application of the processing chain, as well as the obtained bathymetric profile are presented in To improve the inversion from this new bathymetric soundings, a Monte-Carlo approach has been proposed in [START_REF] Mastrogiuseppe | Radar Sounding Using the Cassini Altimeter: Waveform Modeling and Monte Carlo Approach for Data Inversion of Observations of Titan's Seas[END_REF]. The inversion from the Cassini radar altimetry data was indeed complicated, due to the presence of thermal noise, speckle, ADC, block adaptative quantization and possible saturation of the receiver. All these elements have been modelled, and their effects applied to synthetic Cassini waveforms generated for different depths, surface/subsurface echo magnitude ratios and roughness of the seabed. Synthetic radar signals were generated for randomly selected parameters in the range of what is expected for a Titan lake sounding, and the best fit was found by a Monte-Carlo approach. The depth, surface/subsurface magnitude ratio and roughness of the subsurface corresponding to the best fit were then extracted, leading to the estimation of the loss-tangent in the liquid. A new estimation of 4.4 10 OE ± 0.9 10 OE was found for Ligeia Mare. This new inversion technique, summarized in Figure 1.30, also allowed the determination of the loss tangent in the Ontario Lacus, another lake in Titan's southern hemisphere, where an unexpected strong specular reflection caused a saturation of the receiver and made the interpretation of the soundings very challenging (Mastrogiuseppe et al., 2018a). In addition to Ontario Lacus, the loss tangent of Punga Mare (Mastrogiuseppe et al., 2018b) and Northern lakes [START_REF] Mastrogiuseppe | Deep and methane-rich lakes on Titan[END_REF] have been determined with the same processing.

The estimation of the loss tangent in Titan's lakes and seas is a precious information to determine their composition. The proportion of methane, ethane and nitrogen in the liquid can indeed be inferred from its dielectric properties. For instance, it showed that the concentration of methane in lakes and seas is higher with latitude, and that the South lakes are richer in ethane than the North lakes. All results are presented in Figure 1.31.

A bathymetry of Kraken Mare, Titan's largest liquid body, has yet to be obtained. No echoes from the seafloor are visible in the traverses performed over the centre of the sea. This lack of detection could be either due to a larger loss tangent than other Titan seas, or a deeper seabed. However, a bathymetric profile has been obtained on Moray Sinus [START_REF] Poggiali | The Bathymetry of Moray Sinus at Titan's Kraken Mare[END_REF], an estuary in the Northern part of Kraken Mare, and shows a maximum depth of ~85 m, and a loss tangent of ~4.6 10 -5 . Assuming a homogeneous loss tangent across the lake, the depth of the lake would be greater than 100 m, most likely >300 m. Such an estimation is consistent with radiometry measurements performed over Kraken Mare. The bathymetric profiles obtained with the Cassini radar were used in different studies as an entry to various models of Titan seas [START_REF] Lorenz | A radar map of Titan Seas: Tidal dissipation and ocean mixing through the throat of Kraken[END_REF][START_REF] Rafkin | Air-sea interactions on titan: Lake evaporation, atmospheric circulation, and cloud formation[END_REF][START_REF] Vincent | Normal modes and resonance in Ontario Lacus: a hydrocarbon lake of Titan[END_REF], and even to compare the dielectric properties inferred from Cassini radar soundings to the values estimated from Cassini radiometric acquisitions (Le [START_REF] Gall | Composition, seasonal change, and bathymetry of Ligeia Mare, Titan, derived from its microwave thermal emission[END_REF].

Future orbiting radars for Jovian moons exploration

Jupiter, the biggest planet in the solar system, has four main natural satellites, also called "Galilean moons" in honour of their discoverer: Io, Europa, Ganymede and Callisto. With respective diameters of approximately 3660, 3120, 5270 and 4820 km, these satellites are planetary objects of importance in our solar system, Ganymede being even larger than the planet Mercury. Europa, Ganymede and Callisto (see Figure 1.32) are of particular interest in the search of habitable zone in the solar system, as these three satellites are suspected to host liquid water below their ice shell, as deep oceans or shallow reservoirs.

Direct access to such bodies of liquid water is very challenging, if not impossible, depending on the actual thickness of the satellites ice shells. Solid water being a very favourable medium for radio penetration, and planetary sounding radars having proven their capacity to reveal water ice structures in the Martian polar caps with the MARSIS and SHARAD radars, logically two radar sounders have been selected by the ESA (European Space Agency) and the NASA (National Aeronautics and Space Administration) for their respective missions JUICE (Jupiter ICy moons Explorer) and Europa Clipper (illustrations of the two orbiters in Figure 1.33). to study the early history of the Jovian system on Callisto. The JUICE orbiter will enter the Jovian system, use elliptical orbits around Jupiter to fly-by the three satellites, and finally stabilize at a circular orbit around Ganymede. The RIME instrument (Radar for Icy Moon Exploration), a penetration radar funded by ASI (the Italian Space Agency), able to sound the ice shell of the Jupiter satellites down to 9 km of depth [START_REF] Bruzzone | RIME: Radar for Icy moon Exploration[END_REF]. Different structure and composition characteristics of their subsurface are expected to be revealed by the instrument: interactions with an ocean or a pocket of water, impurities and dust from meteoritic impacts, convection layers, different types of ice crystals, connections between potential pockets of water and ice salinity to cite a few. If the design of this radar is inspired by the design of previous successful in-orbit planetary radars MARSIS and SHARAD, the Jovian systems added new constraints, and thus induced design adaptations [START_REF] Bruzzone | Jupiter ICY moon explorer (JUICE): Advances in the design of the radar for Icy Moons (RIME)[END_REF][START_REF] Bruzzone | RIME: Radar for Icy moon Exploration[END_REF][START_REF] Bruzzone | Radar for icy moon exploration (RIME)[END_REF]. For instance, the surface of the icy moons of

Jupiter has a higher level of roughness than the surface of Mars at the wavelength of MARSIS and SHARAD signals. Therefore, to reduce the scattering losses on the surface, and achieve the desired penetration depth, a central frequency of 9 MHz (lower than SHARAD but higher than MARSIS) was selected. A second constraint is the radiative environment of the Jovian system. Radio noise from Jupiter is indeed expected to be an issue for radar soundings below 40 MHz. It has therefore been decided to perform radar soundings only on the anti-Jovian side of its satellites. A third constraint is related to the distance between the Earth and Jupiter, implying a reduction of data volumes to be transmitted by JUICE.

The RIME instrument will have two modes of operations, a "high-resolution" mode with its full bandwidth, and a "low-resolution" with a third of its bandwidth, high-resolution soundings being favoured for the Europa and Callisto fly-bys, as there will be only a limited number of them. The RIME instrument is also able to reduce or enlarge its receiving time, whether the objective is to reduce data volume or deeply sound the subsurface of the different satellites. The instrument RIME radar signal is a chirp, with a central frequency of 9 MHz, and a maximum bandwidth of 3 MHz. This relatively large bandwidth for this type of radar will lead to a free-space (worst case) resolution of 50 m [START_REF] Bruzzone | Jupiter ICY moon explorer (JUICE): Advances in the design of the radar for Icy Moons (RIME)[END_REF][START_REF] Bruzzone | RIME: Radar for Icy moon Exploration[END_REF]. The cross-track resolution will be between 2 and 10 km, and the along-track resolution between 0.3 and 1 km, depending on the fly-by. If the targeted penetration depth is 9 km, with the maximum reception time, echoes should be detected down to 20 km in water ice.

The instrument is composed of an electronic module, inside JUICE, and an antenna system outside the orbiter, for a total mass of approximately 11.6 kg [START_REF] Bruzzone | Radar for icy moon exploration (RIME)[END_REF]. The electronic module is divided into three main elements: The Receiver and Digital Subsystem (RDS), controlling the instrument in general, generating the signal, filtering and digitizing the received signal; The Transmitter, performing the amplification of the signal and the transmission/reception duplexing; The Matching Network, responsible of the impedance matching with the antennas. The RDS and Transmitter elements are inside a radiation-protected zone inside the JUICE orbiter [START_REF] Bruzzone | Radar for icy moon exploration (RIME)[END_REF]. The selected antennas design is a 16 m dipole to be unfolded when Juice arrives in the Jovian system. The radiation pattern of the antennas, the effect of solar panels or of an incorrect deployment have been studied [START_REF] Hahnel | RIME-Dipole Accommodation aboard JUICE[END_REF]. The instrument as well as trials on different models are presented in Figure 1.34.

To prepare the arrival of the RIME instrument in the Jovian system, numerical electromagnetic simulations with subsurface models of Europa, Ganymede and Callisto have been generated, with different subsurface structures and composition variations scenarios such as fractures in the ice shells or brines aquifers [START_REF] Heggy | Radar probing of Jovian icy moons: Understanding subsurface water and structure detectability in the JUICE and Europa missions[END_REF]. Other approaches for the simulation of future RIME radargrams are also investigated [START_REF] Thakur | An Approach to the Simulation of Radar Sounder Radargrams Based on Geological Analogs[END_REF]. JUICE will be launched in June 2022, and will arrive around Jupiter in January 2030.

The REASON instrument of the Europa Clipper mission

The Europa Clipper mission was selected by the NASA in 2015, with scientific objectives mainly focused on Europa, hence its name. The orbiter will nominally perform 45 flybys of Europa, with the gravitational assistance of Ganymede. The mission will study the habitability of Europa, dividing the research into three main topics: the detection of liquid water below the ice-shell, the study of the chemical processes inside pockets of water or a subsurface ocean, and the detection of energy sources for potential lifeforms [START_REF] Bayer | Europa Clipper Mission: Preliminary Design Report[END_REF]. The REASON radar instrument (Radar for Europa Assessment and Sounding Ocean to Near-surface) (JPL) was selected, for four different applications:

Altimetry, reflectometry, plasma/particles study, and subsurface soundings [START_REF] Blankenship | REASON for Europa[END_REF].

The design selected for the sounding radar mode is a dual-frequency instrument [START_REF] Blankenship | Radar for Europa Assessment and Sounding: Ocean to Near-surface (REASON)[END_REF]. A

VHF mode with a central frequency of 60 MHz will be used for shallow soundings, with a penetration depth larger than 4.5 km in water ice, and an adjustable resolution down to 15 m in free-space (corresponding to a bandwidth of 10 MHz). A HF mode with a central frequency of 9 MHz will be used for deep soundings, with a penetration depth down to 30 km, and a free-space resolution of 150 m (corresponding to a bandwidth of 1 MHz). The constraints in terms of radiations being the same as for the RIME instrument, only soundings in anti-Jovian configuration will be possible with the HF mode.

In August 2018, a preliminary design review of the Europa Clipper mission has been presented. The HF dipole antenna will measure 16 m, just as RIME, and the VHF antennas may be mounted on the solar The Europa Clipper will be launched in October 2024, and arrive in the Jovian system in April 2030.

JUICE and Europa Clipper missions should therefore take place simultaneously, and both instruments perform soundings of Jupiter's icy moons during the same period, unveiling their icy shells and subsurface oceans.

Sounding the subsurface of Venus with the SRS instrument of the selected EnVision mission

Venus is the second planet from the sun, and the second telluric planet in size just before the Earth, with a diameter of ~12100 km. It is renowned for its very dense atmosphere, with a surface pressure of ~93 times that of the Earth, and mainly composed of carbon dioxide, leading to a strong greenhouse effect.

Due to this effect, the mean surface temperature is ~735 K, higher than the surface temperature of (2020). Among the questions the SRS radar will try to answer are the different volcanic episodes in the history of Venus and the types of volcanism involved, the origin of the Venus "tessera" highlands and plains, and the geological age of certain regions by the detection of buried impact craters. et al., 2020). At a bandwidth of 6 MHz, the expected resolution in a volcanic material of dielectric constant 6 is 16 m. The estimated penetration depths from a preliminary performance evaluation accounting for the dielectric permittivity of the expected subsurface materials, the roughness of the surface, the background noise level, and the effect of the ionosphere are between 100 and 1300 m. The antenna will be a dipole, and lowest considered central frequency being 9 MHz, its maximum length will be 16 m.

Two modes of operations are planned for the EnVision mission: A high density mode with raw data, ground tracks separated by 10 km, covering 20 % of the surface of Venus; And a low density mode with compressed data, ground tracks separated by 50 km, covering 80 % of the planet surface. Night-side observations will be preferred to reduce the effect of the ionosphere and the noise induced by solar activity.

Three launch windows are possible for the EnVision mission, 2031, 2032 and 2033. The transit to Venus will last 15 months, and the scientific activities would begin after 16 months of aerobraking. Further studies are ongoing to constraint the effect of high temperatures on the loss tangents of Venusian materials, as well as the exact noise levels induced by the ionosphere and the solar activity. The final design of the instrument will depend on these studies. SRS is the last planetary sounding radar selected by a space agency to date, and will be the very first to sound the subsurface of Venus.

A perspective

This section was dedicated to all in-orbit planetary radar sounders launched or selected for a future launch to date. Their different targeted planetary bodies and missions implied different designs, in particular regarding their selected working frequencies and bandwidth. The working frequencies indeed drives the penetration depth, and are selected considering surface roughness, subsurface scattering and absorptions. They are also limited by the maximum possible size of the appropriate antennas. The bandwidth drives the resolution of the soundings in time-domain, and must be selected according to the chosen working frequencies and the scale of the structures to be revealed by the radar. Despite these different designs, similarities can also be noted.

All subsurface sounding radars listed in

Sounding radars operating on the surface

In the literature, a radar sounder operating from the surface of a planet is often referred to as a "Ground Penetrating Radar" (GPR). On Earth, such radars are used in a wide variety of applications: geophysics, archaeology, civil engineering and mine detection to cite a few (Daniels, 2005). The penetration depth of GPR is in the order of a few meters to tens of meters, the separation between soundings are in the order of ten to a few tens of centimetres, and the vertical resolution is in the order of a few centimetres to tens of centimetres. Soundings are usually performed on straight lines of a few meters to tens of meters. Compared to in-orbit radar sounders, GPR therefore provide an insight on the subsurface of a planet on a local scale. Nevertheless, in a non-destructive manner, they unlock a third dimension in the interpretation of planetary sites explored in-situ. The pioneering Surface Electrical Properties (SEP) experiment of the Apollo 17 was the first instrument sounding the lunar subsurface directly on the surface with electromagnetic waves [START_REF] Simmons | Surface Electrical Properties Experiment Final Report[END_REF].

Even if this electromagnetic interferometer would not be considered as a GPR in the modern sense of the word, it is a precursor of electromagnetic subsurface soundings from planetary surfaces.

Since the selection of the WISDOM instrument as the GPR of the ExoMars mission (ESA) in 2003, this type of instrument has been selected for various Lunar and Martian exploration missions, by various space agencies (Chinese, American and European). This illustrates the rising interest for this technology to explore the shallow subsurface of planetary bodies. While on Earth, GPR measurements are usually performed mounted on a cart or a vehicle driven by a human being, on the Moon and on Mars the radars are and will be mounted on robotic platform known as "rovers".

This section briefly presents the various GPR developed for planetary exploration, their mission, design and results will be briefly presented. The Lunar Penetrating Radar (LPR), integrated to the Yutu rover, had the mission to investigate the shallow lunar crust, the thickness and structure of the lunar regolith in its landing site, and to provide information on the formation of the Moon, and to choose the location of a potential future Chinese Moon base station (G. Y. Fang et al., 2014).

The LPR had two distinct transmission / reception channels, named CH1 and CH2, both working simultaneously in time-domain. The instrument is indeed an impulse radar, generating impulses of two different durations for the CH1 and CH2 channels, with an avalanche transistor system. The duration of the impulse generated by the CH1 channel is 1.6 ns, with working frequencies between 40 and 80 MHz.

For the CH2 channel, the duration is 0.8 ns, with working frequencies ranging between 250 and 750

MHz. The CH1 is therefore a deep penetration, low resolution channel, with a resolution of several meters, while the CH2 is a shallow penetration and high resolution channel, with a free-space resolution For the CH1 channel, 2 monopole antennas, with a Wu-King impedance loading (King, R. Wu., 1965) to obtain a wide bandwidth have been selected to transmit the signal. They are located on the rear of the rover, and have been unfolded after the landing. For the CH2 channel, a set of 3 bow-tie antennas, at the bottom of the rover and 30 cm above the surface, have been selected. One of the antenna is for the transmission, the other two for the reception. The antennas and their allocation in the rover are illustrated in Figure 1.39. Before the launch of the Chang'E 3 mission, a field trial of the instrument was performed in a natural environment, on a glacier in the Gansu province, in China (Fang et al., 2014). Once on the Moon, the LPR acquired an equivalent distance of 45 m of GPR traverses, shown in Figure 1.40 for the CH2 channel, with an adaptation of the acquisition parameters between each rover traverse [START_REF] Su | Data processing and initial results of Chang'e-3 lunar penetrating radar[END_REF]. The soundings performed by the LPR during the Chang'E 3 mission allowed the study of the stratigraphy of the subsurface, with the dielectric properties of each layer estimated by fitting the hyperbolic diffraction curves probably generated by buried boulders (Lai et al., 2016), and the dielectric constant of the first layer estimated from the surface echo [START_REF] Dong | Parameters and structure of lunar regolith in Chang'E-3 landing area from lunar penetrating radar (LPR) data[END_REF]. The obtained results have been compared to formation models for the landing site, and in particular to models of the lunar regolith structure in order to estimate its porosity.

The LPR instrument of the Chang'E 4 mission

In January 2019, the Chang'E 4 mission became the first mission to land a rover on the far-side of the moon, in the Von Karman crater. Based on orbital images, this crater was expected to be shaped by different ejecta materials and mare basalt units. As the main scientific objectives of the mission were the geological study of the landing site surface materials and subsurface structure, to confirm the geological history of the crater, the Yutu-2 rover of the mission was also equipped with a Lunar Penetrating Radar instrument, the spare model of the Change'E 3 radar [START_REF] Jia | The scientific objectives and payloads of Chang'E-4 mission[END_REF]. This time, the rover was operational during 12 lunar days, and travelled more than 400 m, including 320 m of soundings with the LPR instrument (see Figure 1.41). The SNR of the soundings performed by the LPR during the Chang'E 4 mission was higher than during the Chang'E 3, and clear reflections from subsurface structures were observed both in the CH1 and CH2 channels. The penetration depth was also CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 58 clearly higher, with detectable interfaces down to more than 360 m below the surface [START_REF] Zhang | Stratigraphy of the Von Kármán Crater Based on Chang'E-4 Lunar Penetrating Radar Data[END_REF].

The radargram acquired with the CH1 channel has been interpreted as presenting a first layer of regolith, with different layers of stratified ejecta materials and separated by mare basalt layers below (see Figure 1.42), confirming the succession of impact and basaltic events in the Von Karman crater [START_REF] Lai | Comparison of Dielectric Properties and Structure of Lunar Regolith at Chang'e-3 and Chang'e-4 Landing Sites Revealed by Ground-Penetrating Radar[END_REF][START_REF] Zhang | Stratigraphy of the Von Kármán Crater Based on Chang'E-4 Lunar Penetrating Radar Data[END_REF]. The four basaltic layers are interpreted as four different magmatic events, from the lunar Imbrian period (3.85-3.2 billion years ago), the last two having formed thinner layers.

The high-resolution / shallow-penetration results of the CH2 channel are also impressive, as they show a penetration depth of nearly 35 m, with the layering of the first ejecta unit easily guessed from the detected echoes. The first layer of regolith can also be seen as a zone with few reflections returned to the LPR, surely due to the fine-grain structure of the regolith, far finer than the wavelength of the transmitted signal in this material (see Figure 1.43). These first layers, detected by the CH2 channel of the LPR, are surely made of low loss (with an estimated loss tangent around 5.10 -3 ), and porous material. Hyperbolic diffraction curves, likely produced by buried boulders in these ejecta materials, allowed an estimation of the dielectric constant of the material, and thus of the depths of the detected layers [START_REF] Li | The Moon's farside shallow subsurface structure unveiled by Chang'E-4 Lunar Penetrating Radar[END_REF]. The regolith layer is quite thick, with a maximum depth of 12 m, and relatively homogeneous along the rover track, and is expected to originate from different impact events. The ejecta materials just below the regolith have been separated into two main units: the first one with rough materials and buried boulders, from 12 to 24 m of depth, and the second one composed of a succession of rough and fine materials, from 24 to 40 m of depth. This last unit is interpreted as originating from older impacts than the one above it (interpretation illustrated in Figure 1.44). Oceanus Porcellanum region, near the Louville-ω hill, and return them on Earth for analysis. The lander was equipped with a drill, able to collect a 2 m deep core sample, and therefore required guidance regarding the structure and composition of the subsurface [START_REF] Kramer | Chang'e-5 (China's Lunar Sample Return Mission) / CE-5[END_REF]. This lander, the drill and the capsule containing the sample are shown in Figure 1.45. A situation similar, but in a static configuration, to the Martian mission ExoMars 2022, further presented in the last section of this chapter. To achieve this guiding mission, and provide a geological context to the collected samples, the Lunar Regolith Penetrating Radar (LRPR) instrument was selected to be mounted inside the lander [START_REF] Shen | Lunar Regolith Penetrating Radar on the Lander for Chang'E-5 Mission[END_REF]. More specifically, the 12 Vivaldi antennas are separated into three different arrays, because of the constraints on the available space below the lander. The array "A" is composed of 7 antennas, the array "B" of 4 antennas, and the array "C" of 1 antenna. In each array, the antennas are separated by 12 cm and the antennas are 90 cm above the lunar surface. The electronic unit of the instrument is able to trigger the transmission of the LRPR signal from any of the antennas, and successively receive the surface and subsurface echoes from all the other antennas (Lu et al., 2021b). This configuration leads to 11x12 permutations, and therefore 132 different soundings. The disposition and transmission/reception process is illustrated in Figure 1.47.

In the objective of guiding the drilling operation of the mission, the LRPR required a high resolution,

and not a high penetration depth. For this reason, the impulse radar has a large band of working frequencies: from 0.5 to 4 GHz, the highest to date for a planetary GPR. The impulse emitted by the radar has a duration of 0.2 ns, and the free-space resolution of the instrument is ~4.3 cm [START_REF] Shen | Lunar Regolith Penetrating Radar on the Lander for Chang'E-5 Mission[END_REF][START_REF] Xiao | Ground experiments of Chang'e-5 lunar regolith penetrating radar[END_REF]. The minimum expected penetration depth of the instrument was 2 m, in agreement with the thickness of the core sample to be collected with the lander drill. As in classic ultra-wide band GPR on Earth, a free-space measurement subtraction and a low-pass filter are applied to the acquired soundings to remove the direct coupling, reduce the electronic noise, and DC-bias. A migration algorithm is eventually applied to transform the 132 soundings into a radargram of 1.56 m equivalent horizontal distance [START_REF] Xiao | Ground experiments of Chang'e-5 lunar regolith penetrating radar[END_REF]. An experimental example is shown in Figure 1.48. The three GPR of the Chang'E missions are therefore the first instruments of this type to sound the subsurface of a planetary body. Revealing the geological history of their landing site, and in the case of the LRPR, guiding the collection of samples, they validate the interest for GPR instruments in planetary exploration shown by space agencies. The three next planetary GPR presented in the following section will be operated on Mars, in the frame of missions selected by the Chinese space agency CNSA, the American space agency NASA, and a collaboration between the European space agency ESA and the Russian space agency ROSCOSMOS.

The RoSPR Ground Penetrating Radar of the Tianwen-1 mission

In February 2021, the orbiter of the Tianwen-1 mission arrived around Mars, along with the first Martian rover of the CNSA. The rover, named Zhurong, landed in the region of Utopia Planitia in May 2021 (see Figure 1.49.a), and the orbiter will be used as a relay for communications between the rover and the Earth. As a first mission on the Martian surface for the CNSA, the rover is intended to explore a wide range of planetary topics, such as the general composition and morphology of the surface and subsurface CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 63 of the landing site, the atmosphere and climate of this region, and the magnetic field of the planet [START_REF] Zou | Scientific objectives and payloads of Tianwen-1, China's first Mars exploration mission[END_REF].

Among the instruments selected to be on board the rover, there is a GPR, the RoSPR instrument (Rovermounted Subsurface Penetration Radar). Its main objectives are the evaluation of the thickness of the first layer of lithic materials, the structure of the buried water or carbonic ice underneath, and to study the stratigraphy of the subsurface in general [START_REF] Zhou | The Mars rover subsurface penetrating radar onboard China's Mars 2020 mission[END_REF]. To achieve these objectives, the instrument is divided into main channels, named CH1 and CH2. CH1 is a low-frequency channel, with working frequencies between 15 and 95 MHz. Its penetration depth will be between 10 and 100 m depending on the subsurface composition. The resolution of the soundings acquired in CH1 will be in the order of a few meters. This channel is connected to two monopole antennas at the rover rear, with Wu-King impedance loading to enlarge their bandwidth (Lu et al., 2021a).

CH2 is a high-frequency channel, with working frequencies between 0.45 and 2.15 GHz. The expected penetration depth will be between 3 and 10 m, with a free-space resolution ~8.8 cm. The channel is connected to two sets of Vivaldi antennas accommodated at the rover rear, allowing the transmission and reception in four different polarimetric configurations. A design very similar to the antennas of the WISDOM GPR, presented in the following.

The integration of the two channels antennas is presented in Figure 1.49.a, and virtual models of the antennas are presented in Figure 1.50. Both channels share the same electronic unit, located inside the rover body, with the same modulation technique, but adapted to the working frequencies of each channel. The Martian atmosphere being very favourable to electric discharges at the voltages required in the Lunar Penetrating Radar, the same impulse generation could not be used for RoSPR. A Linear Frequency Modulation (LFM), emitting a waveform better known as a "chirp", has been chosen instead. In the case of the CH1 channel, the signal in time-domain will be obtained by convolution between the received signal and the emitted chirp. For the CH2 channel, the emitted and the received signals are mixed, and an Inverse Fourier Transform is Tests have been performed with RoSPR mounted on a replica of the Zhurong rover, with reflectors buried in a soil trench filled with synthetic volcanic ashes in a laboratory, and in a natural volcanic environment, where reflectors from different volcanic deposits are visible in the acquired soundings [START_REF] Zhou | The Mars rover subsurface penetrating radar onboard China's Mars 2020 mission[END_REF]. See Figure 1.52 for an insight on the campaign and its results.

The RIMFAX Ground Penetrating Radar of the Mars 2020 rover mission

In February 2021, the Perseverance rover of the NASA (National Aeronautics and Space Administration) mission Mars 2020 landed in the Jezero crater, on Mars. With a diameter of 45 km, this crater is part of a wider impact basin, named Isidis. Deposits from an ancient delta are visible on the west side of the Jezero crater (see the geological map in Figure 1.53), the last deposits dating from ~3.8 billion years [START_REF] Stack | Photogeologic Map of the Perseverance Rover Field Site in Jezero Crater Constructed by the Mars[END_REF]. Sedimentary fluvial deposits being representing an interesting target for the search of potential traces of life, the Perseverance rover will collect surface samples on different sites of interest in the crater, which will be returned to Earth for analysis in the frame of the "Mars Sample Return" project [START_REF] Mustard | An HF Bi-phase shift keying radar: Application to ice sounding in western alps and spitsbergen glaciers[END_REF]. Figure 1.54 displays illustrations of the Perseverance rover.

To correlate the sedimentary layers in an identified outcrop with the underground stratigraphy, the RIMFAX instrument (Radar Imager for Mars' Subsurface Exploration) has been selected by the NASA CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 66 in 2014 to be integrated in the Perseverance rover. RIMFAX is a Ground Penetrating Radar, intended to provide soundings of the subsurface of the Jezero crater along the rover movements (or "traverses") or stationary measurements to observe the daily evolutions of the subsurface. It will provide a geological context to the different collected samples, by revealing the structure and dielectric properties of the subsurface [START_REF] Hamran | RIMFAX: A GPR for the Mars 2020 rover mission[END_REF][START_REF] Hamran | Radar Imager for Mars' Subsurface Experiment-RIMFAX[END_REF]. RIMFAX is a Frequency Modulated Continuous Wave radar (FMCW). It consists in transmitting a harmonic signal of linearly increasing frequency over the bandwidth of the instrument. Such a frequency sweep requires a more complex Radio Frequencies electronic unit and has higher sampling requirements than a Stepped Frequency Continuous Wave (SFCW) used by the WISDOM GPR, but it is more energy efficient and does not have an ambiguous range (Øyan et al., 2012). These two points are important for the RIMFAX instrument, as it aims at more important depths than WISDOM. The free-space range of the instrument can be selected between 5.4 m to 1 km, with three different modes: "surface", "shallow" and "deep". Each mode targets echoes at a specific range from the antennas, and the combination of these modes can provide radargrams with a larger dynamic range than one mode alone. The bandwidth of the emitted signal can be selected, with working frequencies between 150 MHz The radar antennas are located under the rover body, 60 cm above the surface, while the electronic unit is inside. As for the other proposed and launched planetary GPR, this "air-coupled" antenna set-up will record an echo from the surface, from which the ground dielectric constant can be estimated in the hypothesis of a smooth surface. Only one antenna will be used for transmission and reception, with a gating to reduce the direct coupling of the transmission on the received signal. A compact bow-tie design has been selected for the antenna. See

The WISDOM Ground Penetrating Radar of the ExoMars 2022 rover mission

The WISDOM radar on the Rosalind Franklin rover of the ExoMars 2022 mission was the first GPR mounted on a planetary exploration vehicle selected by a space agency, namely the European Space Agency as early as 2003.

The Rosalind Franklin rover is part of the ExoMars joint program between the ESA and ROSCOSMOS, subdivided into two main missions, ExoMars 2016 and ExoMars 2022 [START_REF] Vago | ESA ExoMars program: The next step in exploring Mars[END_REF]. The ExoMars 2016 mission consisted in an instrumented Martian satellite, the Trace Gas Orbiter (TGO), and an instrumented lander, Schiaparelli. TGO is equipped with four instruments: NOMAD (Vandaele et al., 2018) and ACS [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF], two spectrometers units to study the composition of the Martian atmosphere, CASSIS [START_REF] Thomas | The Colour and Stereo Surface Imaging System (CaSSIS) for the ExoMars Trace Gas Orbiter[END_REF], an imager with an optical resolution of 5 m, and FREND [START_REF] Mitrofanov | Fine Resolution Epithermal Neutron Detector (FREND) Onboard the ExoMars Trace Gas Orbiter[END_REF], a neutron detector, to evaluate the hydrogen concentration of the Martian shallow subsurface. The orbiter, operational since April 2018, is also intended to be a relay for the telemetry of the Rosalind Franklin rover (see Figure 1.57 for illustrations of the ExoMars mission main The ExoMars 2022 mission, and in particular the Rosalind Franklin rover, will be further described in the following chapter. As for the Jezero crater, the Oxia Planum site presents signs of aqueous alteration from ~3.9 billion years ago, and the sedimentary deposits observed in orbit are interesting places to look for potential traces of ancient life (Quantin-Nataf et al., 2021). The main difference between the Mars 2020 and ExoMars 2022 missions consists in the sample collection and analysis process. The Rosalind Franklin rover will be able to collect better preserved samples down to 2 m in the Martian subsurface, but only an in-situ analysis will be performed on them, inside the rover body. There will be no sample return to Earth. The WISDOM radar, which flight model was integrated to the rover in 2019, was designed to guide the search of potentially interesting samples, and provide an insight on their geological context [START_REF] Ciarletti | WISDOM GPR Designed for shallow and high-resolution sounding of the martian subsurface[END_REF](Ciarletti et al., , 2017a)). As the previously presented GPRs, the interpretation of WISDOM soundings will provide information on the structure and dielectric properties of the subsurface, but with a shallower penetration depth (3 to 10 m) and a thinner theoretical resolution (6 cm in free-space) than the LPR, the RoSPR and the RIMFAX instruments. As the LRPR instrument, such characteristic is directly linked to the requirements of the mission, as samples of a 3 cm length will be collected down to a maximum depth of 2 m. The WISDOM instrument will also be further described in the following chapter. In a nutshell, the WISDOM GPR electronic unit is inside the rover, while its receiving and transmitting antennas are located on the rear of the rover. They are lying at 38 cm above the surface, and will therefore record the surface echo as well as echoes from the subsurface. The WISDOM selected design is a SFCW radar with Vivaldi antennas, able to transmit and receive signal in four polarimetric configurations. Its working frequencies are between 0.5 and 3 GHz, to guarantee the required high-resolution. Eventually, as for the other presented planetary GPR, WISDOM has been tested in various natural environments or laboratory set-ups on Earth, and representative synthetic soundings have been generated in preparation of the instrument for its arrival on Mars (Dorizon et al., 2016;Ciarletti et al., 2017a;Hervé et al., 2020).

The position of the antennas at the rear of the rover, the electronic unit and the Vivaldi antennas are shown in Figure 1.58.

Processing and interpretation tools have been and still are developed to prepare the scientific return of the WISDOM instrument (Hervé et al., 2020;Oudart et al., 2021), as well as the guidance of the sample collection operations. WISDOM, and in particular its processing and interpretation chains are the topic of this thesis. In the next chapter, the WISDOM radar, its design, performances, processing chain and dataset collected on Earth will be detailed.

A recapitulative table of the characteristics of the presented planetary GPR is proposed in Table 1.3, as a comparison with the WISDOM instrument. These GPR being designed according to different mission objectives, none of them should be considered as "better" or "worse" than the others. The purpose of Table 1.3 is rather to emphasize how different mission requirements drive the selection of different GPR designs, or likewise, how similar mission requirements lead to similar GPR characteristics.

Regarding WISDOM, it can be noted that the ExoMars 2022 mission objectives led to a high-resolution (and thus a wide band of working frequencies) and relatively low penetration depth design, and also to a very light and low power-consuming instrument. These choices will be key in the following chapter.

Summary

• A sounding radar is an instrument transmitting electromagnetic waves towards a planetary surface, and measuring the portion of these electromagnetic waves reflected or scattered by the surface and the subsurface in the direction of its receiving antenna. It thus provides information on the structure and composition of the observed surface and subsurface.

• Since the 1970s, soundings radars are applied to the subsurface of other planets than the Earth, revealing their subsurface. A presentation of key elements in the design of planetary radars was proposed, in order to understand the selected characteristic for each of these instruments

• Planetary radar sounders can be operated from orbit, or directly on the surface, providing observations on different scales. An overview of all sounding radars selected for planetary missions was presented in this chapter, providing a context for the selection and design of the WISDOM instrument, main topic of this thesis.

ExoMars 2022 and the subsurface of Oxia Planum

As mentioned in the previous chapter, the objective of the ExoMars 2022 rover mission is finding potential traces of life, past or present, in the subsurface of Oxia Planum. To achieve this objective, the Rosalind Franklin rover will be the first in the robotic exploration of Mars to be equipped with a drill able to collect samples down to 2 m in the Martian subsurface. The samples will be rapidly analyses inside the rover body in order to limit the alterations that would occur if they were exposed for a long time to the Martian conditions at the surface.

The Oxia Planum region has been selected as the landing site of the ExoMars 2022 mission after a five This peculiarity of Oxia Planum suggests that any potential biomarkers in the Oxia Planum sediments could have been shielded from deteriorating conditions on the Martian surface, such as radiations (both UV and ionizing) or oxidation. Furthermore, the clay-bearing unit represents more than 70% of the surface in the landing uncertainty ellipse. On Earth, biosignatures are often encountered in sedimentary material, such as the clay-bearing units detected on Oxia Planum. Therefore any landing location would be close to a site of interest [START_REF] Carter | Oxia Planum, a Clay-Laden Landing Site Proposed for the Exomars Rover Mission: Aqueous Mineralogy and Alteration Scenarios[END_REF]. This site is therefore ideal for a sample collection in the objective of finding preserved traces of life.

The plan for the sample collection with the drill as described in (Vago et al., 2017) For this reason, in the nominal mission scenario, the Rosalind Franklin rover science team will seek out outcrops of sedimentary materials, likely to continue in the subsurface (below a regolith unit) down to 2 m of depth. To detect an outcrop of interest, the rover is equipped with the PANCAM (panoramic camera (Coates et al., 2017)), ISEM (infrared spectrometer [START_REF] Korablev | Infrared Spectrometer for ExoMars: A Mast-Mounted Instrument for the Rover[END_REF]) and CLUPI (close-up imager [START_REF] Josset | The Close-Up Imager Onboard the ESA ExoMars Rover: Objectives, Description, Operations, and Science Validation Activities[END_REF]) instruments. The images from the PANCAM instrument will be processed to produce digital terrain mapping, providing the precise location of the different units of interest. ISEM is co-aligned with PANCAM and can remotely provide the bulk mineralogical composition of an outcrop, and for instance detect the presence of minerals linked to the presence of water in the past. The depositional history of a detected outcrop can be studied from the CLUPI images. An example of interesting texture feature detectable by CLUPI would be traces of biolamination.

Then, the rover science team must have confirmation that the interface between the regolith and the detected clay-bearing unit extends in the subsurface, and determine if and where exactly it reaches the required depth for sample preservation and allowance of drilling operations. To study the structure and composition of the subsurface prior to any drilling operations, Rosalind Franklin is equipped with two instruments: WISDOM (ground penetrating radar (Ciarletti et al., 2017)) and ADRON-RM (neutron spectrometer that detects hydrogen in the top first meter of the subsurface, indicating the presence of hydrated minerals or water, [START_REF] Mitrofanov | The ADRON-RM Instrument Onboard the ExoMars Rover[END_REF]). The chosen drilling spot must be both of scientific interest (a sedimentary material shielded from the surface), in reach of the rover's drill and safe (free of any hazards which could harm the drill such as buried boulders).

Once a suitable drilling spot is selected, drilling operations can begin. In specific locations showing very interesting and promising structures, a full drilling operation, or "vertical survey", will be performed. It consists in five 3 cm long sample collections at depths of 0, 0.5, 1, 1.5 and 2 m in the subsurface. But, since this process has a high cost in mission time and energy, most of the operations planned will consist in one 3 cm long sample collected on the surface (to characterize the outcrop) and one at a depth of probably less than 1 m. In the nominal mission plan, only two "vertical surveys" are planned, against six simpler sample collections. During the drilling, the MA_MISS (visible and near-infrared spectrometer)

instrument accommodated inside the drill (De Sanctis et al., 2017) will provide information on the composition of the different subsurface layers the drill progress through with a vertical resolution of a few millimeters.

Eventually, the collected samples will be transferred to a carrousel, to be shared between three instruments able to analyse them: RAMAN (Raman laser spectrometer [START_REF] Rull | The Raman Laser Spectrometer for the ExoMars Rover Mission to Mars[END_REF]), MICROMEGA (infrared spectrometer [START_REF] Bibring | The MicrOmega Investigation Onboard ExoMars[END_REF]), and MOMA (gas phase chromatograph which will look for biomolecules [START_REF] Goesmann | The Mars Organic Molecule Analyzer (MOMA) Instrument: Characterization of Organic Material in Martian Sediments[END_REF]). It must be noted that the synergy between The entire experiment cycle (outcrop characterization, subsurface soundings, drilling site selection, drilling, sample collection on the surface, at -1.5 m and analysis) will last between 13 and 15 Martian days, while a vertical survey (the same with samples collected at 0, -0.5, -1, -1.5, -2 m) will last around 20 Martian days. Both operations will be separated from one point of interest to the next one by traverses during which PANCAM, ADRON and WISDOM will be operated.

The Rosalind Franklin rover will be launched from Baikonur in September 2022, and will arrive on the Oxia Planum in June 2023. The descent module containing both the rover and its landing platform (see Planum. The time in-between being for data analysis (less than 40 minutes) and decision making.

The information provided by the WISDOM GPR will, as explained, be crucial when selecting a drilling spot in a geological site of interest. Members (scientists and engineers) of the WISDOM team will be at the ROCC during Martian operations, checking the instrument health, processing and interpreting the data. WISDOM has therefore been specifically designed to meet the objectives of the ExoMars 2022 mission in terms of penetration depth commensurate with the drill capacity and vertical resolution commensurate with the sample length. In this chapter we present the instrument, its data processing chain, and the dataset and observations numerically generated or acquired on Earth, to prepare the interpretation of future Martian observations.

Objectives, characteristics and expected performances of WISDOM

Scientific objectives of WISDOM

The Ground Penetrating Radar of the ExoMars 2022 mission WISDOM has three main scientific objectives (Ciarletti et al., 2017):

• To reveal the 3D geological history of Oxia Planum by providing the structure, stratigraphy, physical composition, heterogeneity and electromagnetic properties of its shallow subsurface (thus providing constraints on its composition). On Earth, GPR proved to be useful when a noninvasive analysis of a subsurface is required, in natural environments, in civil engineering and archeology (Daniels, 2005 for an overview). Contrasts in the dielectric properties of a subsurface generates radar echoes with different amplitudes and time-delays in GPR soundings, allowing the detection of changes in structure and composition, as well as estimations of the permittivity of the different subsurface layers. For GPR with polarimetric capabilities such as WISDOM, additional information on the texture and shape of reflectors can even be deduced. • To identify the potential (persistent or transient) existence of liquid water or ground water ice in the shallow subsurface of Oxia Planum. Radars are particularly efficient when it comes to detect liquid water, as the dielectric constant of liquid water is far higher than that of other natural materials. This important difference generates high dielectric contrasts between materials containing liquid water and material deprived of it. On Earth, natural environments with high humidity levels at the surface are therefore unfavourable to GPR soundings, as most of the transmitted signal will be reflected by the surface. Water ice, on the contrary, has dielectric properties similar to high porosity sediments, and is therefore difficult to differentiate from them.

However, if the presence of ground ice is detected by another instrument (ADRON in the frame of ExoMars 2022), GPR can provide information on its distribution in the subsurface.

Measurements performed by WISDOM at the very same location during the day would allow to monitor any change occurring in the very shallow subsurface (few cm) due to the thawing of ice, if any.

• To guide the selection of a scientifically relevant drilling target, with minimum risks for the drill of the rover. When designing a GPR, a compromise between resolution and penetration depth is made according to the constraints and objectives of the mission. This was the case for WISDOM, which must sound the Martian subsurface with adequate vertical resolutions (a few cm) and penetration depths (a few meters) to guide the drilling operations. WISDOM is indeed able to detect boulders buried in the subsurface and use this information to guide the drilling operations to safe areas.

The design of WISDOM has been chosen to account for the mission constraints and objectives, especially in terms of required performances. It will be presented in the following sections.

A stepped-frequency GPR

Various modulation techniques are possible for the design of a Ground Penetrating Radar, each having its advantages and drawbacks (Daniels, 2005). Impulse radars, hence their name, generate impulses of very short duration towards the subsurface, and a time-domain acquisition of the echoes from the impulse are performed. Easier to design for frequencies above 1 GHz, Frequency Modulated Continuous Wave radars operate in the frequency domain. They generate a signal of continuously increasing frequency, over a limited frequency range. The emitted and received signals are mixed together to obtain the phase and amplitude of each target return.

WISDOM is a stepped-frequency radar (also known as "synthetized pulse modulation"). As the FMCW radars (see Section 2.3.2 of Chapter 1), it operates in the frequency domain to overcome the fact that it is technically difficult to generate a very short pulse carrying significant energy given the limited power available on the platform. The idea behind this type of radar is that the frequency of an infinitely narrow impulse signal (or Dirac) is a complex exponential in the frequency domain and has thus a constant value in modulus. An infinite frequency bandwidth is unrealistic, but the wider it is, the narrower the corresponding pulse in time domain.

Hence a stepped-frequency radar generates and transmits a series of harmonic signals of known amplitude and phase, which are distributed over the bandwidth of the radar. After propagation and interaction of the harmonic waves with the sounded medium, the radar measures for each frequency the amplitude and the phase of the returning signals. The spectrum constructed frequency by frequency is then finally used to calculate a synthetized pulse in the time domain. In practice, a discrete series of N harmonic signals of equally spaced frequencies, and known amplitude and phase are generated over a limited frequency range of width B (the instrument's frequency bandwidth), for a duration τ (the repetition time). Hence the name of this modulation: step-frequency.

• Resolution:

The resolution ∆t in time domain of a radar measures its capacity to discriminate two pulses with different propagation delay values (see Section 2.3.2 of Chapter 1), ∆t is the smallest value allowing a clear discrimination of the two signals assuming they have the same amplitude.

Even if formulae may differ according to the actual waveform of the pulses (see the discussion in Section 2.2 of Chapter 4), it is inversely proportional to the system frequency bandwidth and commonly defined as

∆ = 1 ( 2.1 )
with B the total frequency bandwidth of the radar.

• Unambiguous delay:

The sampling step in frequency domain Δ needs to be smaller than twice the maximum delay investigated (to avoid aliasing in time domain). Hence, the unambiguous delay in time domain is

= 1 2 = -1 2 ( 2.2 )
In the case of WISDOM, typical values for the parameters are N = 1001 for the number of harmonic steps, ranging between 0.5 and 3 GHz (B = 2.5 GHz and Δ = 2.5 MHz). Thus, resolution of WISDOM in time domain is 0.4 ns and its maximum time range is 200 ns.

In terms of distance travelled in vacuum, the spatial resolution of WISDOM (considering round-trip) would be 6 cm and the maximum non-ambiguous distance 30 m.

• WISDOM data:

For each transmitted frequency the received signal results from interferences between different echoes coming from the surface and subsurface. It is also a harmonic signal of frequency , but with a modified amplitude and phase relative to the emitted signal. The ratio between the amplitude of the received and the emitted signal A( ) and the phase difference between both signals ϕ( ) corresponds to the amplitude and phase of the WISDOM spectrum at frequency .

In order to measure A( ) and ϕ( ), the emitted and received signals are multiplied in the receiver, to obtain a new signal . This signal is low-pass filtered to retrieve only . It is then separated into its real part (In-phase component) = cos!" # and imaginary part (Quadrature-phase component) $ = sin " . This process is illustrated with an example on 5 frequencies in Figure 2.6.

In the case of WISDOM, in order to fit into the mission's constraints, only the In-phase component of the spectrum is measured. Measuring the Quadrature-phase component would indeed imply to double a portion of the RF (Radio Frequency, see section 2.3) circuit, to increase the total mass of the WISDOM Electronic Unit (WEU), and also double the data volume. The imaginary part of the signal is retrieved using the Hilbert transform of I. This point will be detailed later in this chapter.

The WISDOM Electronic Unit (WEU)

The WISDOM Electronic Unit is a pack of three electronic modules inside the rover body. These The three modules of the WISDOM Electronic Unit are the following ones:

• The Radio Frequency (RF) board (see Figure 2.8 for an illustration) consists in the emission and reception chains, with a mixer to multiply and low-pass the transmitted and received signals, as explained in the previous section. A frequency generator able to generate frequencies between 0.5 and 3 GHz is the source of the emission chain. This signal is amplified, a switch directs it to one of the emission antennas, depending on the required polarization channel. The signal is eventually directed towards the transmission antenna through a coaxial cable. The reception chain is the exact opposite: the signal from the reception antenna is directed to the WEU through a coaxial cable, gating can be applied to this signal to suppress the strong first echoes (antennas coupling and surface echo) thus improving the dynamic of the instrument, a polar switch allows only the required polarisation channel to be acquired, and the signal is eventually amplified. The original signal generated and the processed received signal are compared with the mixer, and sent to the Data Processing Unit (DPU) board. • The Data Processing Unit (DPU) board (see Figure 2.9 for an illustration) consists in a digitization of the final signal from the RF module, a stable reference clock, the control of the RF module, the transfer of the measurements and the reception of the commands. The signal from the RF module is amplified, digitized to 16 bits by an ADC (Analog to Digital Converter)

and transferred to a FPGA (Field Programmable Gate Array, a type of programmable logic device). The FPGA stores and then returns the measured values via a CAN bus. Through this bus, it can also receive commands, and control accordingly the polarization switch and gating of the RF module. "Housekeeping" data, such as the temperature of the different electronic boards and the voltages delivered to the different modules, are also returned by the FPGA through the CAN bus. With a 16 bits ADC, the theoretical dynamic of the instrument is 96 dB (= 20 ()* +, 2 +-). In practice, accounting for the actual performances of the components, this dynamic is shorter, typically 88 dB, 83 dB in the worst case. And finally, controlled attenuators (31 dB both in emission and reception) can be tuned to adjust the dynamic of the instrument to avoid saturation from the strongest echoes and detect echoes as weak as possible. • The DC/DC converter board (see Figure 2.10 for an illustration) is connected to a 28 V primary power source from the rover, and delivers secondary voltages adapted to the RF and FPGA modules (from -5 to 15 V). The primary power source has a redundant connection, the FPGA module is always alimented, and the RF module can be switched ON and OFF to save power between consecutive soundings.

The WISDOM Antennas Assembly (WAA)

The WISDOM Antennas Assembly (WAA) is the name of the different units composing the antennas of the instrument, located outside the rover body, and connected to the WEU via coaxial cables. The WISDOM antennas are divided in two main parts, the transmission and the reception antennas, both directed towards the subsurface. The WAA was designed, built, qualified and characterized by the TU Dresden in Germany. towards the subsurface, with as few radiation lobes as possible in the direction of the rover structure (especially the wheels). The WAA also had to be light (with a maximum mass budget of 400 g). These are the reasons why Vivaldi antennas have been selected (Plettemeier et al., 2009). and two "cross-polar" (the emission and reception antennas having orthogonal linear orientations). The measured polarization ratio of the WAA is in the order of 20-30 dB. The orientations of the four antennas are at a ±45° angle with respect to the rover path, to reduce the volume of the whole WAA. Such orientations give an insight on the position of a reflector relative to the rover (such reflector will not have the same reflection coefficients on the co and cross polar measurements). Having different polarimetric channels also allows the classification of scatters structures with the H-alpha method [START_REF] Plettemeier | Analysis Methods for the Polarimetric WISDOM Radar aboard the ExoMars Rover To cite this version[END_REF], and the evaluation of the surface roughness. The H-alpha, or "Entropy-alpha" method consists in the classification of radar echoes according to the entropy H (in the data processing sense) or "randomness" of the scattering process, and the scattering process angle α (indicating surface scattering, volume scattering or multiple scattering processes).

The original design of the antennas consisted in two Vivaldi "horns", the emission and reception horns containing two perpendicular Vivaldi antennas. However, with such configuration, the height of the WAA had to be 28 cm, 8 cm above the mission constraint on the height of the antennas. A new design, consisting in in-phase dual Vivaldi antennas for each polarization configuration was thus selected. To feed both Vivaldi antennas of each dual slot, a compact Wilkinson divider was proposed, with power losses as low as possible [START_REF] Benedix | Advance of WISDOM GPR antenna for ExoMars 2018 mission[END_REF]. The final dimensions of the emission and reception elements of the WAA are 20x20x16 cm, 16 cm being the height. Both elements are separated by 1 cm, in order to have footprints on the surface that overlap as much as possible. The total dimensions of the WAA, including the baseplate (see below), are 41x20x18 cm.

The antennas are mounted on a baseplate, fixing the WAA to the rover body with a slight tilt of 7° (see Figure 2.12). This tilt, in the opposite direction to the rover body, results from a compromise between reducing the influence of the rover on the measurements, and pointing the antennas towards the subsurface as directly as possible. For the safety of the WAA and because of engineering constraints (otherwise a system would have been required to lift the antennas from the surface each time the rover moves and put them in contact with the surface to perform a sounding), the bottom of the WAA is 38 cm above the Martian surface, and thus not directly in contact with the ground unlike typical GPRs, due to engineering constraints (otherwise a system would be required to lift the antennas when the rover moves). It will be explained later (see chapter 3) that this "air-coupled" configuration is both a drawback and an opportunity for the scientific return from WISDOM soundings.

Expected performances of WISDOM

The WISDOM GPR is, as presented, a stepped-frequency radar. The different characteristics of the radar are summarized in Table 2.1 (Ciarletti et al., 2017):

As previously mentioned, the design of a GPR is guided by the targeted performance, but is always the result of a compromise between the vertical resolution of the instrument and its maximum penetration depth, that also needs to take into account the numerous constraints of the mission. Both resolution and penetration depth values depend not only on the parameter's values presented in Table 2.1 (which have been obtained on the WISDOM Flight Model) but also on the properties of the subsurface. Indeed, the main properties having an impact on the resolution and the penetration depth are, for the electrical properties: the dielectric constant and the loss tangent of the materials; and for the structural properties:

the roughness and dielectric contrast at the interface to be detected, the heterogeneity of the layers and the stratigraphy. To estimate the maximum depth at which the interface between two layers can be detected and the corresponding vertical resolution, we considered a simple two-layers subsurface model and assumed the fact that the interface is detected at a given frequency as soon as the corresponding Signal to Noise Ratio is larger than 3 dB. The homogeneous case has already been studied (Hervé et al., 2020), where the power of the signal reflected by the buried interface was estimated by an analytical formula and the following characteristics for the subsurface were considered: a typical permittivity value of 4 for the first layer and of 8 for the second layer, separated by a smooth or a rough interface, with different levels of absorption losses (." = 0.05, 0.10 and 0.25).

In order to refine this simplified model and account for the effect of volume scattering induced by the presence of heterogeneities in the top layer, we used an electromagnetic numerical code based on Finite Differences in Time Domain (see Section 5.2) to simulate the propagation of the EM waves transmitted by the WISDOM antenna through a non-homogeneous top layer and estimate the resulting additional losses. We generated synthetic WIDOM soundings for fractal heterogeneous subsurface models of different dielectric contrasts ∆. (0.5, 1 and 1.5) and different characteristic sizes 0 of heterogeneities (2.5, 4.5 and 8.5 cm). The thickness of the top layer was set to 2.5 m, with a flat and smooth perfect reflector at the bottom to generate a clear reflected signal. The amplitude of the reflected signal was eventually compared to the amplitude of the reference level obtained in the case of a homogeneous layer to estimate the additional scattering losses due to volume scattering. As a first approach, a mean attenuation over the WISDOM frequency bandwidth is considered here.

The mean measured scattering losses are presented in Table 2.2, for the 3 ∆. values, and the 3 0 values considered. As expected, the mean scattering loss increase with the value of ∆. for any size of heterogeneities, the attenuation of the returned signal increasing with the dielectric contrast. For the same value of ∆., the level of scattering losses varies with 0. This was also expected: if the size of heterogeneities is too small compared to the wavelength of the transmitted signal, the situation is close to the propagation in a homogeneous subsurface of the same mean dielectric constant; if the size of heterogeneities is too large compared to the wavelength of the transmitted signal, the interaction is close to reflections on subsurface structures; and if the size of heterogeneities is comparable to the wavelength of the signal, high scattering losses are expected. For the tested subsurface models, it can be seen that a characteristic size of heterogeneities between 4.5-8.5 cm corresponds to the highest mean measured scattering losses (to be compared with the wavelength of the WISDOM signal in a subsurface of dielectric constant 4, ranging between 5 and 30 cm, ~8.6 cm for the central wavelength). To provide an order of magnitude of the penetration depths and resolutions achievable with the WISDOM instrument in the case of a heterogeneous subsurface, the worst-case of mean scattering losses measured for a given dielectric contrast has been used to generate Figure 2.13. For a homogeneous layer, in the most favourable case (a smooth interface, and low losses), the interface can be detected down to a depth of ~8.5 m, with a resolution of 3 cm in the best case. In the worst case, a rough interface and high losses, the interface can be detected down to ~1.5 m with the best possible resolution of 3 cm. At 2 m (the maximum depth reachable by the ExoMars 2022 rover drill), the resolution would be ~4.5 cm.

For a heterogeneous layer with ∆. = 1.5, and a resolution of 3 cm, the interface can be detected down to 6 m in the most favourable case. In the worst case, at 2 m of depth, the resolution would still be ~5 cm. As an instance, a dielectric contrast of 1.5 is consistent with the contrast expected between dry sand deposits (with dielectric constants between 2 and 3) and dry limestone (with dielectric constants between 3 and 5) (Ciarletti et al., 2017).

The vertical resolutions and penetration depths estimated with a simple but representative model of the WISDOM transfer function shows that WISDOM can provide an accurate guiding of the ExoMars 2022 drilling operations, where samples can be collected down to 2 m of depth, with a resolution of ~5.2 cm in the worst tested case of scattering and absorption losses, to be compared with the size of the collected sample, around 3 cm (Vago et al., 2017).

We will see later in Chapter 4 and 5 that this vertical resolution can be further improved by specific postprocessing techniques.

WISDOM data processing chain

The data processing of the WISDOM soundings will be performed on Earth, after reception of the incoming telemetry from the Rosalind Franklin rover by the Rover Operation Control Center. The idea is to transform the frequency-domain soundings acquired by WISDOM into an image displaying these soundings, processed and in time-domain, to ease their interpretation.

The WISDOM processing chain employed for this thesis is based on the one developed by (Hervé, 2018;Hervé et al., 2020), with a few additions required by the new interpretation tools presented in chapters 4 and 5. It is illustrated by the schematic in Figure 2.14.

In a nutshell, to the 1001 real frequency-domain samples of each WISDOM sounding are applied the following processes: a correction for the effect of temperature, a Hilbert transform to obtain complex frequency-domain samples, a "free-space" removal to reduce the direct coupling between the antennas, a "whitening" to compensate the non-linear effects on the signal, windowing and Inverse Fast Fourier Transform to obtain samples in time-domain, a horizontal stacking of all soundings to obtain an "radargram", and optionally time-domain filtering and horizontal interpolation to ease the reading of the radargram.

In this section will be detailed these different steps of the WISDOM data processing chain used during this thesis, as well as the additional "free-space" and "calibration" measurements required for some of the processing operations. To characterize the effect of temperature on WISDOM acquisitions, soundings have been performed for temperatures of -54°C, -49°C, -30°C, -5°C, 40°C and 55°C. The temperature has no effect on the timedelays of the echoes, but it causes a decrease in their amplitudes.

This effect cannot be neglected, because the free-space and calibration measurements (see section 3.5) with the instrument mounted on the rover have been performed at ambient temperatures, which can be very different from the ones expected on the surface of Mars (for instance, diurnal variations between -89°C and -31°C were measured during the Viking 1 mission). Not correcting the effect of temperature on the amplitudes in WISDOM soundings would thus impair the suppression of the direct coupling between antennas, and any estimate of dielectric properties from the surface and subsurface echoes.

The temperature of the RF-board being one of the "health-check" measurements returned by WISDOM for every sounding, once the effect of temperature on the amplitude is well characterized, a correction is straightforward to implement. Relying on the characterization of the temperature effect on WISDOM measurements performed in the Mars chamber, the following correction has been proposed by (Hervé et al., 2020):

= >?@@ ! @AB # = = CD 10

,.,EFE G HI G JKL M, ( 2.3 ) where U is the amplitude in Volts, CD is the "health-check" measured temperature of the RF-board, and @AB = 33°P is an arbitrary reference for temperature. As the temperature of the WISDOM RF-board can significantly vary even during a same series of soundings performed during a measurement campaign, the temperature correction will be applied for every experimental results presented in this thesis.

Hilbert transform

Raw WISDOM acquisitions are frequency-domain spectra of 1001 samples between 0.5 and 3 GHz, with a step of 2.5 GHz. As previously mentioned (see section 2.2), due to engineering constraints, the WEU only acquires the real part of the reflected signal received from the subsurface. This means that the half of the information, the imaginary part of the spectrum, required to reconstruct the amplitude and phase of the different echoes of the soundings in time-domain is in theory missing.

In order to reconstruct the imaginary part of the spectrum, we use the fact that the spectrum of the signal, real and causal in time-domain, has an imaginary part which is the Hilbert transform of its real part. The Hilbert transform of the In-phase signal I( ) is defined as:

ℋR S = 1 T UV WX -′ ′ Z[ [ \ ′] ( 2.4 )
where VP is the Cauchy Principal Value. Therefore, the complete spectrum ^ is therefore:

^ = + ` ℋR S ( 2.5 )
Equation (2.5) shows that this requires the knowledge of for all possible a values, hence over any infinite bandwidth. However, as previously mentioned, the WISDOM bandwidth, though broad is limited to 2.5 GHz, and thus errors are expected in the reconstruction of the complex spectrum, that increase as the frequency approaches the sides of the bandwidth. We will see later, in section 3.3, that the window functions used to convert by Inverse Fourier Transform (IFT) the signal in time domain will lessen this effect and, in Chapter 4, that the removal of 5% of samples on both edges of the spectrum needed to apply properly parametric spectral estimation techniques also reduces the problem. To validate the ability of the Hilbert transform to provide a correct estimate of the complete complex spectrum, soundings have been acquired during WISDOM-like operations on an experimental soil trench, with a representative model of the WAA, and a Vector Network Analyser (VNA) instead of the WEU (see Figure 2.16).

The VNA used for this experiment has the ability to measure both the In-phase and Quadrature-phase part of the signal on the same range of frequencies as WISDOM. The two signals have been acquired

for various WISDOM traverses over the soil trench, with various buried reflectors. The idea was to compare the complete complex acquisitions (with a VNA) to the signal reconstructed by Hilbert transform with only its real part (as WISDOM would do). A direct comparison between the complex spectra acquired with the VNA and the complex WISDOM spectra reconstructed by Hilbert transform is indeed impossible as the differences between the two will mainly be caused by the differences between their RF electronics.

As a first result, the error on the modulus (or "envelope") reconstruction of the WISDOM frequency spectra using the Hilbert transform acquired during a traverse over reflectors placed in a hole dug in the soil trench has been measured. Figure 2.17 shows the mean measured error and the corresponding standard deviation versus frequency computed on the 70 available soundings. As anticipated, the largest error levels are observed on the edges of the spectra. As an indication, the portion of WISDOM spectra kept when applying parametric spectral estimation techniques described in Chapter 4 and 5 is shown in green in Figure 2.17. On this portion, the mean relative error on the reconstructed amplitude stays below 3%.

The reconstruction of the complex spectrum in frequency-domain seems thus satisfactory, but the effect in time domain still has to be investigated. For this reason, both the original complex soundings and the one obtained by Hilbert transform have been transformed in time-domain using a classic method (an IFFT after application of a Hann window, see section 3.3 for more details). For echoes of calibrated amplitudes superior to -40 dB, the relative error on the amplitude is always inferior to 1%, and inferior to 3% for most of the echoes of amplitudes superior to -60 dB. This low level of error is very satisfying, and illustrates the ability of the Hilbert transform to accurately estimate the envelope of the WISDOM signals in time domain despite the fact that the imaginary part of its spectrum is missing. This point is of crucial importance, as the amplitude of echoes is commonly used in WISDOM soundings to estimate the soil dielectric properties (see Chapter 3).

Inverse Fourier Transform

The WISDOM GPR is operated in the frequency-domain, but the interpretation of the data is done in time domain that allows to detect the different echoes, their amplitude and time delays. For a signal in frequency domain S( ), the Inverse Fourier Transform of the signal is defined as: 2.6 ) In practice, the Inverse Fourier Transform of a series of sampled data is estimated numerically, with the Inverse Discrete Fourier Transform. For a series of discrete frequency domain samples d e of length N, the elements of the Inverse Discrete Fourier Transform (IDFT) of this series b f (with 0 ≤ h ≤ ) are calculated as:

b = ℱ + Rd S = 1 2 T X d M \ Z[ [ (
b f = 1 i d e
M f e j j + ek, ( 2.7 ) The algorithm used to compute the IDFT of WISDOM acquisitions is the Inverse Fast Fourier Transform (IFFT). This algorithm, popularized by [START_REF] Cooley | An algorithm for the machine calculation of complex Fourier series[END_REF], is named "Fast" because it has a lower complexity than the "naïve" algorithm to calculate the IDFT.

The IFFT is time-effective and robust even when applied to experimentally acquired soundings.

However, with the IFFT, the resolution of the signal in time-domain is equal to the inverse of its frequency bandwidth (0.4 ns in the case of WISDOM, corresponding to 6 cm in free-space). A limitation that the implementation of the Bandwidth Extrapolation will improve by a factor of 3 (see chapter 4).

Another drawback of the IFFT is the spectral leakage effect, creating "side-lobes" for each echo in timedomain. This effect, due to the limited frequency bandwidth of the considered signal, can be mitigated by applying a window function prior to IDFT.

A window function (or apodization function) is a function defined on the same frequency band as the spectrum it is multiplied to. It usually tapers the signal on the sides of the spectrum, while preserving it in the middle frequencies. This smoothing of the discontinuities at the ends of the signal reduces the amplitude of side-lobes in time-domain. However, this improvement has a cost, it reduces the signal bandwidth and therefore the spatial resolution of soundings in time-domain. Choosing a window function must result from a compromise between preserving the resolution and reducing side-lobes, and depends on the purpose of its application (detecting faint echoes, separating closely-spaced echoes, ...).

If no window is applied to the signal prior to IFFT, it is equivalent to having multiplied the signal by a rectangular window.

In Figure 2.20 is presented a few examples of commonly used windows, and their application to a synthetic WISDOM sounding with only one echo. The effect on the resolution and the side-lobes of the echo can be appreciated. The characteristics of each presented window are indicated in Table 2.3 (National Instruments Inc., 2009). In the frame of this thesis, techniques to improve the resolution of soundings transformed in time-domain have been applied and validated on WISDOM data. Preserving the resolution of soundings was therefore considered more important than reducing the amplitudes of side-lobes. For this reason, the Hamming window has been selected, and is used to generate the time-domain signals presented in this thesis, where the preservation of the resolution is key. The Hamming window produces the narrowest pulse of all the presented windows (except for the rectangular one), while reducing the maximum side-lobe height by 30 dB. This window is recommended by (National Instruments Inc., 2009) to separate closely-spaced sine-waves (equivalent to echoes in time-domain). Directly applying the IFFT after windowing a WISDOM signal would lead to a sampling rate of 0.4 ns in time-domain. With such a sampling, the sounding would be hard to interpret. For this reason, the "zero-padding" technique is used with the IFFT. It consists in adding zeros at the end of the signal, which is equivalent to an interpolation in time-domain after application of the IFFT. The signal returned by the IFFT algorithm will have the same length as the input frequency-domain signal, meaning that the more zeros are added to the signal in frequency domain, the higher is the level of interpolation in time-domain.

Obviously, this process being equivalent to an interpolation, it improves the interpretation of WISDOM soundings, but does not "create" any new information. The use of the zero-padding technique to interpolate WISDOM signals in time-domain has been validated by (Hervé et al., 2020). The echo from a metallic plate has been reconstructed with acquisitions at various distances from the antennas, corrected in amplitude to compensate dispersion losses. The resulting echo with this "stroboscopic" method has been found to match the samples interpolated with zero-padding for one of the distances (see Figure 2.21).

An example of application of the IFFT to a processed experimental WISDOM sounding, with a

Hamming windowing and zero-padding interpolation is presented in Figure 2.22. In this sounding, only one echo from a metallic plate and its multiple reflections can be seen, as acquired in laboratory, at the TU Dresden facilities. Hamming window and zero-padding.

Radargrams

A WISDOM sounding in time-domain displays the time delays and amplitudes of echoes from reflectors whithin the area illuminated by the antenna radiation pattern. To localize reflectors and provide the 2D variations of the subsurface, "GPR traverses" are performed with the WISDOM instrument.

In the nominal mission plan, WISDOM will perform soundings every 10 cm along a several meters' straight line called a "traverse", the rover moving between two consecutive soundings. The combination of these soundings may reveal variations of the subsurface structure and composition along the traverse direction, with echoes changing in amplitude and time-delays, or even new echoes appearing. Giving the wide footprint of WISDOM antennas on the ground, and their polarimetric capabilities, objects offtrack can be detected too, adding 3D information to the soundings.

To ease the interpretation of all the soundings in the traverse, a graphical representation called "radargram" is commonly used in GPR studies. Every sounding, usually referred to as a "A-scan" (see While this representation of the GPR radargrams ease their interpretation, it must be read with caution.

In particular, as aforementioned, the wide WISDOM radiation pattern implies that echoes from objects off-track can be captured in the radargram. The conversion of time-delays into depth is therefore not straightforward, it requires an estimate of the permittivity value for the conversion in distance and also a way to locate the reflectors in 3D to transform the distance in depth and identify the reflectors that are Migration processing allows to refocus such a signature on a single point which corresponds to the location of the scatterer [START_REF] Berkhout | Wave field extrapolation techniques in seismic migration, a tutorial[END_REF][START_REF] Hagedoorn | A PROCESS OF SEISMIC REFLECTION INTERPRETATION[END_REF]. In the mission plan, "WISDOM grids" are also planned to provide further 3D information on the sounded subsurface. This type of operations consists in performing 3 or 5 parallel WISDOM traverses, to obtain what is known as a "C-scan", as illustrated in Figure 2.24. These measurements allow the reconstruction of 3D shape of subsurface reflectors, as successfully demonstrated by (Dorizon et al., 2016). WISDOM grids are planned to consist of either 3 traverses of 10 m, or 5 traverses of 5 m, separated by approximately 1 m (Ciarletti et al., 2017).

Free-space and calibration measurements

When processing and interpreting WISDOM soundings and radargrams, two main types of measurements are required in order to reduce parasitic signals, calibrate the amplitudes of the echoes, and perform a "whitening" of the signal: a free-space and a calibration measurement. First, within the instrument itself, the internal coupling between the Rx and Tx channels of the RF-board in the WEU appear at the very beginning of WISDOM soundings in time-domain. Second, small differences in impedance between the WEU, the WAA and the coaxial cables can trigger reflections.

Third, when the emitted signal reaches the Tx antenna, the direct coupling between the Tx and Rx antennas also generates a strong echo, which can interfere with the surface echo. Its repetitions can even interfere and mask some subsurface echoes. Other parasitic echoes originate from reflectors not located in the subsurface: 1) Reflections of the transmitted signal arising from the rover body, even if the slight tilt of the WAA reduces the presence of such echoes. 2) Reflections on the surface then bounce between the WAA and the surface, generating multiple repetitions of the surface echo. 3) Reflections on the rover body, then reflected again on the surface and returned to the antennas, or the opposite. Eventually, parasitic echoes can also be generated by reflections on the metallic structure of the rear wheels of the rover which can lie in the footprint of the antennas. If the rover body do not move relative to the WAA, the wheels rotate and reflections coming from their grousers change periodically along a traverse, making these parasitic echoes identifiable. All these sources of parasitic signals are summarized in Fortunately, the internal and direct couplings, the reflections in the coaxial cables and on the rover body are expected to be constant for one mode of the instrument (for a given polarization, set of attenuations, gating, etc.), they can therefore be readily removed provided a prior correction of the temperature effect.

In order to remove these parasitic echoes from the signal, one can subtract to all performed soundings an acquisition performed without any surface or subsurface echoes but that captures the internal couplings, the direct coupling between the transmitting and receiving antennnas as well as any interaction between the antennas and the rover structure. Such an acquisition is called a "free-space measurement". Free-space measurements have been performed during most of WISDOM field trial, since this acquisition depends on the mode of the instrument, but also on the model of the WEU and the WAA (prototype, qualification model, etc.). During a field trial in a natural environment with the instrument on a GPR cart, when no other option is possible, a free-space measurement can be performed by pointing the WAA towards the sky. However, an ideal free-space measurement should be carried out in an anechoic chamber, the antennas pointing towards absorbent material only. Several free-space measurements have been performed in the TU Dresden anechoic chamber, and other measurements have been acquired in Airbus facilities, with WISDOM mounted on the Rosalind Franklin rover.

If not really free-space measurements, additional acquisitions in the Airbus facilities were performed to well characterize the effect of the rover body and of the rotating rear wheels on WISDOM soundings (see Figure 2.28).

Note that the subtraction of a free-space measurement does not remove echoes arising from multiple bounces between the surface, the rover and the antennas because these echoes depend on the environment to be characterized. However, filtering in the time-domain can reduce their presence (see section 3.6).

Calibration of time-domain echoes amplitudes and whitening of the frequency spectra

When interpreting WISDOM soundings or radargrams, the amplitudes of echoes provide an important clue on the dielectric contrast between buried interfaces (see chapter 3), and on their losses. The amplitude of subsurface echoes can be compared with the amplitude of previous echoes, and eventually to the amplitude of the surface echo. A reference is thus required to compare the surface echo to.

The ideal reference would be the amplitude of an echo arising from a perfect reflector detected for the same distance between the WAA and the ground surface. As a perfect reflector reflects all the electromagnetic signal it receives, the amplitude of the echo it returns is the highest possible. To approximate such an echo, a perfect reflector of finite dimensions, namely a metallic plate, is used. The typical dimensions of the plates are between 1 and 2 m of length and width. This type measurement is referred to in the following as a "calibration measurement".

Like for free-space measurements, calibration measurements are specific to a mode and a model of the WISDOM instrument. Therefore, calibration measurements have been performed during different field trials, with a metallic plate brought on the field. They have also been carried out in anechoic chamber at the TU Dresden, and in Airbus facilities (see Figure 2.29).

In addition to giving a reference to compare the surface echo amplitude to, calibration measurements also allow the retrieval of the equivalent time-domain impulse of WISDOM taking into account the transmitting and receiving channels of the WEU and WAA. For instance, this technique has been used in (Hervé et al., 2020) to test the validity of the zero-padding time-domain interpolation to reconstruct echoes. In this thesis, the equivalent impulse of the WISDOM radar retrieved from calibration measurement is used to perform a "whitening" of the original frequency-domain soundings. Calibration measurement performed during a simulation of operations in the Desert of Atacama.

The power generated by the WEU and the gain of the antennas is not constant with frequency, and the selected attenuation mode in the case of the WEU (Hervé et al., 2020). Such variations can be an issue when applying parametric spectral estimation techniques to obtain soundings in time-domain (see chapter 4 for more details), as these techniques assume that the signal emitted by the radar has a frequency spectrum of constant amplitude all across its bandwidth. For this reason, the amplitudes in the frequency spectrum of a WISDOM equivalent impulse can be used to inverse the gain applied to the signal by the WEU and the WAA. This process is called the "whitening" of the signal. it is not surprising to observe defaults on the edges of the frequency spectrum of the impulse. These defaults will not affect the parametric spectral estimation techniques presented in this thesis, as a short portion of frequencies will be removed on each side of WISDOM spectra prior to their application (see section 3.2). In order to validate the ability of our whitening process to correct the spectrum from its frequency variations in the modulus and obtain a spectrum modulus as flat as possible over the whole frequency bandwidth, the whitening algorithm has been applied on synthetic WISDOM data.

As an instance, Figure 2.31 presents the result of this whitening process applied to a synthetic WISDOM sounding (generated with the technique presented in section 5.1 of this chapter), for one echo from a perfect reflector and polluted by an additive white Gaussian noise (SNR = 30 dB). The expected frequency spectrum of such a signal would be a flat line in modulus, except for the presence of noise. A gain has then been applied to the spectrum, generated with a model of a WISDOM impulse determined by (Hervé, 2018) on the prototype version of the instrument. After selection of the impulse in timedomain, and inversion of the estimated gain, the "whitened" signal can be compared to the original synthetic sounding (that is before application of the gain).

As can be observed, the whitening process is able to compensate accurately the unwanted variations of the spectrum modulus. The largest errors are, as expected, on the edges of the signal. Nevertheless, this is not major problem as these parts of the spectrum (5% of samples on each end of the signal, as shown in Figure ) will be removed before the application of a parametric spectral estimation technique. white Gaussian additive noise (SNR=30dB), to which a known "gain" was applied (the envelope of the spectra is displayed).

Time-domain filtering and horizontal interpolation

Once a time-domain radargram is obtained after IFFT, some parasitic signals not eliminated by the freespace removal (multiple reflections between the surface and the antennas for instance) may still impair its interpretation. The attenuations in the subsurface also tend to reduce the amplitude of signals from

deep reflectors, making their detection difficult to the human eye. Additionally, having soundings every 10 cm for traverses a few meters long can give a "pixelated" aspect to the radargram.

Different solutions, some of them commonly used in GPR radargrams processing, have been proposed by (Hervé, 2018) to solve these problems. Such processing is applied directly to the generated radargram image, and can therefore be classified as "time-domain filtering" or simply "image processing" technique.

In order to reduce parasitic signals the free-space removal would not have eliminated, a first simple This technique is a good first approach only if the surface on which the soundings are performed in relatively flat. Otherwise, the parasitic signals are not expected to be constant over the entire horizontal axis of the radargram, and more complex techniques are therefore required. A second approach would be to remove a moving average column vector, to account for variations along the horizontal axis of the parasitic signals to be reduced. This approach can also reduce signals from relatively flat subsurface layers, and the length of the moving average window must therefore be chosen carefully.

More sophisticated approaches have been proposed by (Hervé, 2018) Another time-domain filtering process that can be applied to WISDOM soundings is a gain to compensate for the different losses the radar signal undergoes when spreading in the subsurface (dispersion, absorption or diffusion). This time-domain "gain" is a function increasing with time delays, multiplied to each sounding to amplify the amplitude of soundings with longer time delays. This technique can reveal deep subsurface structures, but at the expense of an increased noise level. In the case of WISDOM, an exponential gain with a tuned coefficient has been selected by (Hervé, 2018). Any interpretation on the amplitudes of echoes in WISDOM radargrams must of course be performed prior to the application of this gain.

To enhance the readability of the radargram, a Spline horizontal interpolation has been selected. For nominal WISDOM Martian soundings, a horizontal sampling of 10 cm between two soundings is planned. To see the benefits of the horizontal interpolation, the radargram presented in Figure 2.33 has been acquired with a horizontal sampling of 5 cm, and down-sampled to 10 cm before applying the interpolation by a factor of 2.

Comparing these three radargrams, it is clear that the horizontal interpolation, if improving the aesthetic quality of the nominal 10 cm sampled radargram, does not add any new details observed in the 5 cm sampled radargram. The interpolated radargram is however easier to read than the nominal 10 cm sampled one, the global scattering signature of the 6 main reflectors appearing enhanced. The time-domain filtering and horizontal interpolation processes presented in this section are optional in the WISDOM processing chain, but they can improve the readability of the generated radargrams, and therefore ease their interpretation. Results from this measurement campaigns were processed and interpreted in (Hervé, 2018), and also compared to FDTD simulations performed with a model of the sandpit. Additional measurements were performed with a commercial GPR, to prepare the Colorado Provençal campaign, presented in section 4.2.5. Then, several traverses have been performed with spherical reflectors, metallic or not, on the surface or buried in the subsurface. The humidity of the subsurface was one of the uncontrollable factors in this measurement campaign, and it appeared that the losses due to subsurface moisture were high enough to make most of the non-metallic reflectors undetectable. For this reason, most of the reflectors buried for the rest of this campaign were metallic.

A metallic staircase, composed of concrete blocks wrapped in aluminium foils to make its faces totally reflective to radar soundings, have been buried in the soil trench (see Figure 2.39). WISDOM traverse's acquisitions have been performed before and after burying the staircase. With a height of 10 cm, the different steps were expected to be resolved in WISDOM radargrams once buried, but under-resolved in the air. Two metallic plates buried with a depth shift of 5.4 cm have also been placed in the soil trench subsurface (see Figure 2.40). This distance is inferior to the resolution of the instrument in the air, and measurements have been performed before and after burying both metallic plates.

Eventually, to test the effect of a non-flat surface on the WISDOM acquisition, ridges were carved on the surface of the soil trench with a depth of 5 cm, below the resolution of the instrument in the air, and a distance of 15 cm between two peaks, slightly above the distance between two WISDOM soundings in a traverse (see Figure 2.41). Some of the results from this measurement campaign have been published in (Oudart et al., 2021).

The presented campaigns in semi-controlled environments were performed with reflectors in a relatively homogeneous material, not representative of the complex structure and variations of dielectric properties which can be expected in a natural environment. WISDOM campaign have thus also been organized in natural sites presenting analogous composition or structure properties as the expected Oxia Planum subsurface.

Acquisitions in natural environments

The Etna campaign

The first field trial of the WISDOM instrument in a natural environment outside of France took place in October 2010, in the region of the Etna Mt., in Sicily (see Figure 2.42). Soundings have been acquired with a prototype of the WISDOM instrument in two main locations in the Etna park, and layers from at least two main pyroclastic events are visible in one of them (Ciarletti et al., 2017). This result has been obtained despite an unfavourable wet weather (it is thus assumed that most of the water in the subsurface escaped rapidly thanks to the porosity of the volcanic material).

As volcanic terrains are expected on Oxia Planum, even if not the prior target of the ExoMars 2022 mission, the radargrams acquired during this campaign are very useful to assess the capacities of the WISDOM processing/interpretation chains on potential Martian analog environment. Results from this campaign have been published in (Ciarletti et al., 2017). Acquisitions from this campaign have been re-processed with the "Bandwidth Extrapolation

Technique", a spectral estimation technique implemented into the WISDOM processing chain during this thesis, and will therefore be further described in chapter 4.

The Chamonix campaign

A few months after the Etna campaign, in January 2011, another field trial has been organized in the region of Chamonix, in the French Alps, with a prototype of the WISDOM instrument mounted on a sledge (see Figure 2.43). The snow and the ice being very favourable to radar soundings, reflectors are seen at estimated depths of more than 12 m in some of the radargrams acquired during this campaign. If not really a Martian analog, the layered structure of the sounded subsurface remains very interesting to train WISDOM interpretation chains, as environments presenting a layered subsurface (but of clay-rich materials) are expected on Oxia Planum as a prior target.

As for the Etna campaign, acquisitions from the Chamonix campaign have been re-processed with the "Bandwidth Extrapolation Technique" (a spectral estimation technique implemented into the WISDOM processing chain during this thesis), and will therefore be further described in chapter 4. During this campaign, the WISDOM instrument has been integrated to three different platforms: the classical GPR cart, the "cliffbot" (from the "Planète Mars" association), a small rover able to move on a slope, and the Magma rover (from the Polish Mars Society), winner of the University Rover Challenge.

Traverses and grids have been performed over ice ponds, with a bedrock and trapped boulders visible in the radargrams. The layered nature of the ice, guessed from ice outcrops but unseen in GPR soundings performed in the cave until this campaign, is also clear in WISDOM radargrams. Results from this campaign have been published in (Dorizon et al., 2016).

As for the Etna and the Chamonix campaigns, acquisitions from the Dachstein Giant Ice Cave campaign have been re-processed with the "Bandwidth Extrapolation Technique" (a spectral estimation technique implemented into the WISDOM processing chain during this thesis), and will therefore be further described in chapter 4. The selected site presented a buried channel bed, exposed on the surface, and WISDOM allowed the detection of a ~50 cm thick layer of sediments.

In the location where the WISDOM grid was performed, a ~70 cm deep interface corresponding to layered sediments was detected in the acquired parallel radargrams. Its presence was later confirmed by digging in the same site.

This campaign represented the first simulation of ExoMars rover operations ever organized, followed in 2019 by two other campaigns in Andalusia and another site in the Desert of Atacama. Results from this campaign have been published in (Ciarletti et al., 2017).

The Colorado Provençal campaign

In July 2017, the LATMOS WISDOM team organized a measurement campaign with a prototype of the instrument in the "Colorado Provençal" geological site (see Figure 2.46), in the vicinity of Rustel (France). WISDOM measurements being impaired by the presence of water in the subsurface, finding a lithic natural environment in France with a dry subsurface, and with outcrops from semi-buried geological units is difficult. The Colorado Provençal site has been selected because it filled both criteria.

In addition to the traverses performed with the WISDOM prototype, a commercial GPR with different working frequencies in the WISDOM Bandwidth has been used to perform the same soundings, as a comparison.

In one of the acquired radargrams in particular, an outcrop continuing underground, below a stratified layer of white sand is clearly visible. A situation allowing the simulation of an ExoMars-like drilling site selection process with this radargram (Hervé, 2018). A free-space measurement has been performed with the antennas pointed towards the sky, and a calibration measurement with a metallic safety blanket (easier to carry in the desert than a metallic plate).

If no clear subsurface structures have been identified in the acquired WISDOM radargrams from this campaign, the dielectric constant of the material has been estimated from the surface echo, a few hyperbolic patterns from point scatters are visible, and a study of the heterogeneities in the subsurface from the texture of the radargrams is ongoing.

Analytical and numerical simulation of WISDOM data

In order to test the processing and interpretation chains prepared for the WISDOM instruments, an apriori knowledge of the studied reflectors is necessary. With experimental soundings, even performed in "controlled" environments, a part of the environment always remains uncontrollable (undesired echoes from parasitic reflectors, physical properties of the selected soil, etc.). For this reason, to validate the different developed chains, they are first applied to synthetic WISDOM soundings, generated with models of the instrument signals and configuration, as well as models of the subsurface.

During this thesis, two different types of synthetic soundings have been generated, depending on the technique to be validated.

Analytical synthetic WISDOM soundings

For the frequency range of the WISDOM instrument (0.5-3 GHz), the response signal S( ) acquired in frequency-domain by a radar can be modelled as a sum of complex sinusoids coming from K point scatters:

d = i l f mn -` 2 T o f mn -p q f + r s fk+ ( 2.8 )
where l f are the amplitude coefficients of the h t point scatter, o f is its associated time delay, p the attenuation coefficient in the medium, q f the equivalent distance in the subsurface of the point scatter, and r a white Gaussian noise.

With this model, a complex spectrum can be generated for frequencies between 0. However, to be representative of experimental WISDOM soundings, the white Gaussian noise must be added in frequency-domain. To know which level of SNR in frequency-domain would correspond to the SNR measured in experimental WISDOM soundings, synthetic soundings in a similar "two reflectors" scenario have been generated for different levels of SNR in frequency-domain. For each level, a 1000 cases of noise have been considered (Monte-Carlo approach).

From the results presented in WISDOM measuring only the In-phase part of the returned signal, only the real part of the generated spectrum is generated, the complex representation of the signal being obtained via Hilbert transform, as explained in Section 3.2 of this chapter.

In Figure 2.52 is presented an example of synthetic WISDOM spectrum generated for two reflectors in free-space, separated by 15 cm. The real signal and the modulus obtained by Hilbert transform are shown.

As already mentioned, errors of the estimation are clearly observed at the extremities of the spectrum.

This type of synthetic WISDOM soundings is useful to simulate rapidly the radargrams corresponding to simple scenarios, such as the "two reflectors" experiments that will be presented in the two last chapters. However, this model does not take into account the geometry of the WISDOM antennas, the effects of dispersion and diffusion in the different materials, and complex geometries of the reflectors are hard to model. Electromagnetic simulations, if much longer to run, can account for all these effects.

Finite Differences in Time Domain (FDTD) simulations

The "Finite Differences in Time Domain" is an electromagnetic simulation technique introduced by [START_REF] Yee | Numerical Solution of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media[END_REF], using discrete versions of the Maxwell equations to simulate the propagation of electromagnetic waves through a modelled materials or vacuum. The modelled space is divided into a pre-defined number of cells (also known as "Yee cells", see Figure 2.53), of sizes Δx, Δy and Δz. To each cell is associated values of the dielectric constant, the conductivity and the magnetic permeability, pre-defined according to the materials being modelled. The components (Ex,Ey,Ez) of the electric field are associated to the edges of each cell, while the components (Hx,Hy,Hz) of the magnetic field are associated to each cell. At each time (n+1/2)Δt, Δt being the time step of the simulation, the components of the magnetic field are updated for each cell. At each time nΔt, the components of the electric field are updated. This technique can easily handle more complex models than analytic methods, however the computation time for each time step is driven by the number of cell, and can be relatively large (even if the algorithm is easily parallelizable).

In order to avoid the numerical instability of the simulation, the time step duration Δt must be chosen as:

∆ < 1 }~1 ∆m M + 1 ∆• M + 1 ∆q M ( 2.10 )
with } the speed of light in vacuum, corresponding to the highest speed of electromagnetic waves in the simulations for WISDOM. The time step required for a given FDTD simulation will therefore decrease with the size of cells, leading to a longer computation time for a same duration of simulation.

In the case of WISDOM FDTD simulations presented in this thesis, the selected step time is

∆ = 0.95 }~1 ∆m M + 1 ∆• M + 1 ∆q M ( 2.11 )
The simulation volume being finite, the simulated electromagnetic waves can be reflected on the borders of the volume, generating undesired echoes. A classic solution to reduce the impact of such reflections is to add "Perfectly Matched Layers" (PML) on the borders of volume. Introduced by [START_REF] Berenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF],

PML are imaginary materials adapted to absorb electromagnetic waves from a specific direction, and for a specific dielectric constant of the material they are in contact with. The thickness of the PML is a compromise between the efficiency of the absorption, and the increase of computation time induced by adding PML cells to the simulation volume. In the case of WISDOM FDTD simulations presented in this thesis, PML have a thickness of 10 cells. To generate subsurface models and FDTD simulations, the TEMSI-FD software of the XLIM laboratory.

In order to obtain representative simulations of WISDOM radargrams, the WISDOM antennas and the time-domain equivalent impulse of the WISDOM instrument have both been modelled. A complete model of Vivaldi antennas elements would require Yee cells with a size of 1 mm, due to the thickness of these elements. For a subsurface model having dimensions in the order of a few meters, this would lead to enormous computation times. For this reason, the slots of the antennas were represented by oblique wires [START_REF] Guiffaut | New oblique thin wire formalism in the FDTD method with multiwire junctions[END_REF][START_REF] Holland | Finite-Difference Analysis of EMP Coupling to Thin Struts and Wires[END_REF], with a resistive profile to reduce reflexions at the tips of the antennas (King, R. Wu., 1965). The emission antenna is fed by a signal generator, and the output of the simulation is the signal measured in the reception antenna. This model (Hervé, 2018) allows the representation of the polarimetric disposition of the antennas, the distance between the emission and reception antennas (20 cm), and the tilt of the antennas (7°). Both antennas are in the air, 38 cm above the surface of the modelled soil. The signal used to feed the emission antenna is a sino-Gaussian model of the WISDOM time-domain equivalent impulse, experimentally determined by (Hervé, 2018). An example of subsurface model with the modelled WISDOM antennas is shown in In the case of TEMSI-FD, the Nit can have values between 1 and 7, leading to 7 different scales for the heterogeneities (see Figure 2.55 for an example of subsurface with heterogeneities of all different scales)

with the Δl = 0.5 cm selected for the subsurface models in this thesis. All 7 scales are summarized in When complex subsurface models or the effect of the antennas geometry is required in order to validate a processing or interpretation tool in this thesis, FDTD simulations will be employed.

In addition to the radargrams simulated by FDTD for this thesis, "free-space" and "calibration"

simulations have also been generated to remove the direct coupling between the modelled antennas, estimate the dielectric constant from the surface echo, and perform a whitening of the signal. The "freespace" sounding is generated with a simulation volume containing only vacuum, and the "calibration" sounding with a simulation containing a metallic surface at 38 cm from the bottom of the antennas.

Summary

• The Rosalind Franklin rover of the ExoMars 2022 mission will seek out potential traces of life (past or present) in the shallow subsurface of Oxia Planum, on Mars, where such traces could have been preserved. To achieve this mission, the rover is equipped with a 2m drill, and 9 instruments, among which a Ground Penetrating Radar: WISDOM.

• WISDOM will reveal the structure and provide clues on the composition of the subsurface prior to any drilling operations, with a resolution of a few centimetres, down to a few meters of depth. It is a stepped-frequency radar, working on the 0.5-3 GHz frequency band, and is divided into an electronic unit (WEU), and an antennas assembly (WAA).

• A processing chain has been developed and validated by the instrument team, in order to produce a final product easing the interpretation, named a "radargram". This thesis contributed to this processing chain.

• In order to validate the processing and interpretation tools developed in preparation of future Martian operations, a dataset of experimental WISDOM soundings (acquired in laboratory and in natural environments) has been constituted in the last decade.

Preliminary validations on synthetic WISDOM soundings generated with analytical models and FDTD simulations are also possible.

Introduction

The signals measured by a radar sounder consist, in time-domain, in a series of echoes, with associated time-delays. To determine the structure of the subsurface, these time-delays must be converted into distances between the antennas (or the surface) and the buried interfaces/scatterers responsible for the different echoes. A knowledge of the speed of light in the different subsurface materials is thus required.

In the hypothesis of non-magnetic subsurface materials with low loss tangents (much smaller than one), the speed of light in these materials is only driven by their dielectric constant that is the real part of their relative permittivity:

= √ ( 3.1 )
For the sake of simplicity, in this chapter the dielectric constant will be noted .

Determining the dielectric constant of the subsurface is therefore of great importance in the interpretation of planetary radar soundings. It also provides a clue on the nature of the sounded materials, as their dielectric constant will depend on their composition, their porosity, their temperature, and their water content to cite a few.

In planetary sciences, where an in-situ permittivity analysis of samples is rarely possible, so the dielectric constant of the shallow subsurface is often estimated from the reflectivity of the planetary surface, measured by radar [START_REF] Campbell | Merging Magellan Emissivity and SAR Data for Analysis of Venus Surface Dielectric Properties[END_REF]Lauro et al., 2021;Mouginot et al., 2012Mouginot et al., , 2010Mouginot et al., , 2009;;Orosei et al., 2018), or its emissivity, measured by radiometry [START_REF] Campbell | Merging Magellan Emissivity and SAR Data for Analysis of Venus Surface Dielectric Properties[END_REF][START_REF] Janssen | Titan's surface at 2.2-cm wavelength imaged by the Cassini RADAR radiometer: Calibration and first results[END_REF]). An estimation of the dielectric constant can also be inferred from an a-priori knowledge on the material observed on the surface, by comparison to a catalogue of expected dielectric constants previously experimentally determined on Earth [START_REF] Dh | Dielectric properties of Apollo 11 lunar samples and their comparison with earth materials[END_REF][START_REF] Paillou | Microwave dielectric constant of Titan-relevant materials[END_REF][START_REF] Robinson | The Dielectric Response of the Tropical Hawaiian Mars Soil Simulant JSC Mars-1[END_REF].

Derivation of the surface dielectric constant from WISDOM surface echo 2.1 The amplitude of the surface echo

The air-coupling set-up of the WISDOM antennas, due to engineering/mission constraints, can be a drawback because: 1) it introduces losses by dispersion in the air above the surface, 2) the strength of the surface echo reduces the dynamic of the instrument, 3) multiple reflections of the surface echo may mask potential faint subsurface echoes, 4) it induces refraction of the transmitted electromagnetic waves on the surface. Nevertheless, this a-priori disadvantage can also offer the opportunity of estimating the dielectric constant of the shallow subsurface from the amplitude of the surface echo.

As explained in Chapter 1, if there is a smooth, flat and infinite interface between two homogeneous materials of dielectric constants and , the ratio Γ between the amplitude of the reflected and the amplitude of the incident plane waves at the interface is given by the absolute value of the Fresnel coefficients. In case of an incident angle equal to zero, the coefficient value is the same whatever the orientation of the electrical field, and is equal to:

= √ -√ √ + √ ( 3.2 )
In the case of a WISDOM signal reaching the surface, the dielectric constant of the air around the WAA can be considered as equal to one. Therefore, the amplitude ratio of the surface echo is:

= √ -1 √ + 1 > 0 ( 3.3 )
where ε is the dielectric constant of the near surface.

If the surface was metallic, the ratio would be = 1. The amplitude of the surface echo acquired during a field trial or once on Mars can therefore be compared to the amplitude measured in calibration soundings performed on a metallic plate, in order to obtain an estimation of the dielectric constant ε as follows:

= 1 + 1 - ( 3.4 )
This method has already been applied to WISDOM data acquired during field campaigns, for instance in the Dachstein or the Desert of Atacama (see Section 4 of Chapter 2). Further, the robustness of this derivation and, more specifically, the impact of the surface roughness on the dielectric constant estimation have been explored in Hervé (2018) on both synthetic and experimental soundings.

Equation (3.4) is obviously only valid if the calibration measurement was performed on a metallic plate at the same distance to the antennas as the surface. However, variations of the topography can alter the distance antennas-surface (that should be equal to 38 cm in a flat and hard terrain with the actual value of the rear wheels' diameter), and a calibration measurement with a metallic plate at exactly 38 cm to the antennas is not always suitable. To account (and correct) for the variations of the antennas-surface distances in WISDOM radargrams, the decrease of signal amplitude with distance has to be accurately measured in laboratory and modelled.

Such measurements have been performed with the WISDOM in the TU Dresden anechoic chamber, with a metallic plate placed at more than 90 different distances (between ~15 and 105 cm with steps of ~1 cm) from the WISDOM antennas. The decrease in amplitude was modelled from these measurements.

Even if the tested distances do not correspond to far-field situations for all WISDOM frequencies (the far-field domain starting at ~1.6 m from the antennas in the worst case), Hervé et al. (2020) found that the measured amplitudes are well fitted by two models of the following form:

= + ( 3.5 )
with the distance between the reflector and the antennas, and and two parameters to be determined. The amplitude is thus inversely proportional to the distance , with an offset .

The first and second models are valid for distances of the reflector below and above = 40 cm, respectively. A similar couple of models has been found to fit the amplitudes measured in the TUD anechoic chamber as presented in Figure 3.1, with transition for = 45 cm. In the following, a correction based on the models displayed in Figure 3.1 will be applied to account for variations in the antenna-surface distance before comparing any amplitude value.

In particular, this correction has been applied during this thesis to the data collected in the frame of the TU Dresden campaign and the ExoFit simulation of operations (see Section 4 of Chapter 2). The following sub-section presents the results from the TU Dresden campaign.

Application of the method to the TU Dresden campaign

As presented in Chapter 2, a measurement campaign with WISDOM has been organized at the TU Dresden in June 2019. A spare model of the WISDOM flight instrument was tested above a 70 cm deep soil trench from the TUD agrarian laboratory where a variety of targets, with simple geometries were buried. The following section describes the application of the estimation of the soil dielectric constant from the surface echo using the TUD campaign WISDOM dataset. Before any analysis of the surface echo, it must be noted that the three radargrams have been acquired

during three consecutive days, and thus a variations of the moisture level of the soil trench close to the surface cannot be ruled out. Additionally, the soil having been mixed in the process of digging holes to bury reflectors and re-filling them, the humidity gradient naturally present in the subsurface is expected to disappear, and the dielectric constant estimated from the surface echo to be after target burial.

Eventually, the radargram acquired for the original soil trench was performed with 10 cm between each sounding (a nominal WISDOM traverse), while the other two radargrams were performed with 5 cm between each sounding.

In order to verify that the "smooth surface" hypothesis was valid for the TU Dresden soil trench, the difference in the amplitudes of the surface echo measured in cross-polar channels (01 and 10) were compared to its amplitudes in co-polar channels (00 and 11). This result was then compared to the difference observed with the echo on a metallic plate (the calibration measurement). Similar results were obtained for the surface of the soil trench and the metallic, indicating few depolarization effect, and thus a smooth surface. In the case where two metallic plates are buried at horizontal distances > ~1.25 m, the inferred dielectric constant also reaches values higher than what was measured on the original soil trench. Figure 3.5 (bottom) indeed displays peaks of the dielectric constant > 5.5 that correlate well with the location of the metallic plates. Like for the metallic staircase case, the burial of the reflectors probably changed the humidity and the porosity of the very shallow subsurface, causing such high values. The presence of the metallic plates below the surface could also cause interferences, altering the surface echo.

Figure 3.6 combines all results obtained for the 3 cases of study; the similarities before ~1.25 m of horizontal distance, and the differences after can be appreciated.

The radargram acquired with two buried metallic plates is particularly interesting, as the depth at which each metallic plate has been buried can be inferred from the radargram acquired prior to their burial comparing the time delays of echoes from the metallic plates before and after burial (~14 and 19 cm of depth for the first and second plates respectively) as illustrated in Figure 3.7. The mean estimated dielectric constant for echoes from the first plate is 4.73, with standard deviation of 0.27, and the mean estimated dielectric constant for the echoes from the second plate is 4.54 with a standard deviation of 0.23. From the surface echo between 1.4 and 3.6 m of horizontal distance (the region where the metallic plates were buried), the mean estimated dielectric constant is lower, equal to 3.96 with a higher standard deviation of 1.01.

Conversely, the dielectric constants estimated from the time delays of echoes from the metallic plates can be compared with the estimations from the intensity of the surface echo as in Figure 3.8. Both techniques do not measure exactly the same thing, as the values measured from the time delays of echoes from the metallic plates provides the mean dielectric constant over the first ~15 or ~20 cm of depth. In Figure 3.8 it can be noted that, except for the two peaks at ~1.5 and ~3 m of horizontal distance, the dielectric constant measured from the time delays of the metallic plates is in average ~1 unit higher than the estimations from the surface echoes, even if for some points both values are very close. This could be explained by a plausible increase in the humidity level and compaction of the soil with depth, over even by interferences between surface and subsurface echoes. Nevertheless, the dielectric constant inferred from the time delays remains a good proxy for the surface dielectric constant as the depth of the metallic plates remains relatively low. In addition, a sample of the soil trench was analysed with a permittivity probe at the TU Dresden. With a Vector Network Analyser, the probe is able to measure the dielectric constant of a sample for frequencies between 200 MHz and 20 GHz (see the set-up in Figure 3.9.a). To achieve a reliable measurement, the sample must fit in a cylinder with a minimum diameter of 20 mm, and a minimum thickness of 20 mm. 7 consecutive measurements have been performed on the soil trench sample, all returning relatively high dielectric constants between ~4.75 and ~6.00 for frequencies in the WISDOM bandwidth (results in Figure 3.9.b).

The dielectric constants measured by the permittivity probe are compatible with the estimations from the time delays of subsurface echoes in the radargrams acquired for two buried metallic plates. They are also consistent with values inferred from the surface echo in some regions of the soil trench, even if generally higher. However, these results must be considered as an indication and not an exact measurement of the soil dielectric constant, as the porosity and humidity of the sample may be altered when collecting and preparing the sample for its study.

In the cases where the surface of a site explored by WISDOM cannot be considered smooth for the wavelengths of the instrument bandwidth, or in complement to the dielectric constant estimations performed using the surface echo, the dielectric constant of the first subsurface layer can also be inferred from the diffraction curves that may be present in the radargram. This will be the topic of the following section.

Derivation of the subsurface dielectric constant from diffraction curves in WISDOM radargrams

As previously mentioned, one of the main objectives of the WISDOM radar is guiding the drilling operations of the ExoMars 2022 Rosalind Franklin rover (Vago et al., 2017). WISDOM radargrams will help identifying interesting geological structures in the subsurface as well as buried boulders which could be a potential threat for the drill. In particular, WISDOM dataset will localize such scatters by providing their distance to the antennas. A knowledge on the speed of electromagnetic waves at the WISDOM working frequencies in the sounded subsurface is thus required to convert the measured time delay for each received echo into distance.

Here we propose to estimate the near-surface dielectric constant using patterns named "diffraction curves" that may be present in WISDOM radargrams. For a GPR, diffraction curve corresponds to the pattern formed by echoes from a same point reflector in consecutive soundings. More specifically, we use a Hough transform-based technique to automatically detect diffraction curves in radargrams.

The "Generalized Hough transform", simply referred to as "Hough transform" in this section, was first introduced by [START_REF] Duda | Use of the Hough Transformation to Detect Lines and Curves in Pictures[END_REF] as a shape detection tool in image processing. Its most common applications are for the detection of lines and circles in optical images. However, it can also be adapted to detect hyperbolic diffraction curves in a radargram, as done by Capineri et al. (1998). This technique has been successfully implemented in GPR processing softwares for a variety of applications, such as detecting land mines [START_REF] Carlotto | Detecting buried mines in ground-penetrating radar using a Hough transform approach[END_REF], detecting and determining the diameter of buried pipes [START_REF] Windsor | The estimation of buried pipe diameters by generalized hough transform of radar data[END_REF] or of tree-roots [START_REF] Li | Tree root automatic recognition in Ground penetrating radar profiles based on randomized Hough transform[END_REF]. In the case of WISDOM, the classical algorithm had to be modified in order to account for the refraction of electromagnetic waves at the surface. Indeed, WISDOM antennas are mounted 38 cm above the ground (Ciarletti et al., 2017) which diverts the shape of the diffraction curves from perfect hyperbolas.

This section describes the implemented modified Hough transform algorithm. First, the effect of refraction on the results of the "classical" Hough transform was characterized to demonstrate that a correction was necessary. Second, the new automated interpretation tool was validated on both synthetic and experimental radargrams.

Diffraction curves for a ground-coupled radar

The subsurface model tested in this section includes a reflector small enough with respect to the instrument wavelength in the subsurface to be considered as a point scatter embedded in a homogeneous medium. The GPR antennas are directly on the surface, and a traverse is performed. Such a case where the GPR has its antennas directly touching the ground, the set-up is referred to as "ground-coupled GPR". In this situation, depicted in Figure 3.10, the calculation of time delays expected for each sounding performed over a buried reflector is straightforward:

= 2 √ - + ! ( 3.6 )
where x is the horizontal position of the antennas, X is the horizontal position of the reflector, Z is the depth at which the reflector is buried, and

" √#
is the speed of electromagnetic waves in the homogeneous medium of dielectric constant ε.

Equation (3.6) can also be written in the following form:

= $ + % -& ( 3.7 )
where

⎩ ⎪ ⎨ ⎪ ⎧ $ = 2√ ! = % = 4 & = ( 3.8 )
We recognize the equation of a mathematic hyperbola. The parameters α, β and γ represents the timedelay (related to the depth of the reflector), the shape of the diffraction curve, and the horizontal position of the reflector, respectively.

Therefore, detecting and characterizing hyperbolas in GPR radargrams allows for the retrieval of the horizontal and vertical positions of the associated reflectors, as well as for the dielectric constant of the medium.

Diffraction curves for an air-coupled radar

We now consider a subsurface model similar as in the previous section, but with the antennas at a given height above the surface, in order to mimic the configuration of WISDOM operations. Such a set-up is referred to as "air-coupled GPR". In this situation, depicted in Figure 3.11, the electromagnetic waves are refracted at the surface, and the time-delays expected for each sounding performed over the buried reflector are the sum of a travel time in the air and in the subsurface:

= 2 , -- + . + 2√ , -- + ! ( 3.9 )
where H is the height of the antennas above the surface, and -is the horizontal position of the refraction point (see Figure 3.11).

Equation (3.9) no longer corresponds to a mathematic hyperbola. To calculate the time delay associated with every horizontal position of the antennas, it is necessary to know the horizontal position of the The size of the reflector is exaggerated.

Formula (3.10) can be rewritten as

3 --3 -- + . = √ 3 --3 -- + ! ( 3.11 )
In order to calculate the refraction point -corresponding to a given horizontal position x, this equation has been solved in Persico et al. (2015), where it was developed:

- It means that when the reflector is shallowly buried, the diffraction curve tends to be very close to a hyperbola with ε = 1, and when the reflector is deeply buried, the diffraction curve tends to be close to a hyperbola with ε being the dielectric constant of the subsurface, and the addition of a time delay 6 .

Due to the increase of the refraction angle with the distance | -|, the higher is the number of points in the detected diffraction curve, the more the diffraction curve differs from a perfect hyperbola. The number of points in a diffraction curve is therefore expected to have an impact on our ability to estimate the dielectric constant of the subsurface. For this reason, we define the parameter "Field of View" (FoV)

of the antennas as twice the maximum angle between the vertical direction of the antennas and the direction of arrival of the furthest side echo that can be detected by the antennas, as illustrated in Figure 3.12. This parameter is only a way to characterize the size of a diffraction curve, as the ability of the WISDOM instrument to detect a reflector at a given position will depend on the aperture of its antennas radiation pattern, which is frequency-dependent, and on Mars, will be affected by the vicinity of the rover body and wheels.

Figure 3.12: Illustration of the WISDOM antennas Field of View (FoV).

From the radiation pattern of the antennas in both E and H planes, at -10 dB, and for a frequency of 2

GHz (close to the WISDOM central frequency) the main lobe has an aperture of ~75° for the E plane, and ~140° for the H plane. For this reason, FoV between 60° and 120° will be considered when analytically generating diffraction curves in the following.

It must be noted that in this chapter, we assume that the antennas are pointed directly toward the subsurface while, in practice, they will be tilted by 7° to reduce the reflections coming from the rover body and wheels. This inclination can be even larger depending on the topography of the terrain.

The generalized Hough transform applied to hyperbolas

The original Hough transform as patented by Hough was created to detect lines in an image, and retrieve their position and slope. We here describe its principle.

First, a line in a 2D Euclidian space can be represented by the Hesse normal form with the equation

J = K0 / + L 012 / ( 3.15 )
where x and y are the two variables in the Euclidian space, ρ is the shortest distance of the line from the origin of the Euclidian space, and θ is the angle between the shortest line connecting the origin of the Euclidian space to the line and the horizontal axis (corresponding to the x variable). This representation is chosen because it allows for the characterization of vertical lines. The idea of the Hough transform is, after the application of an edge detection algorithm to the image, to calculate the parameters ρ and θ of the line passing through every couple of points in the detected edges.

A 2D accumulator is incremented by one at each calculated coordinates (ρ, θ). The resulting accumulator is a Hough transform of the image (see Figure 3.13 for an illustration of the process with one couple of points).

The detailed algorithm of the Hough transform for the detection of lines is the following one:

1) Apply an edge detection to the image to retrieve the position of pixels on edges.

2) Initialize a 2D accumulator with zeros, one axis corresponding to ρ values and the other to θ values, with chosen boundaries and steps depending on the size of the image and the precision required on the line detection.

3) For each couple of pixel position in the edge detection (1 M , O M ) and (1 P , O P ), the parameters J MP and / MP of the line passing through are determined, and the accumulator is incremented of one at the coordinates corresponding to J MP and / MP (with the precision of the accumulator).

4) Local maxima are then detected in the accumulator. A threshold on the values of the maxima can be applied to select only the most probable detections.

5) The corresponding position and slope of each detected line is eventually returned by the algorithm with the retrieved parameters J and /.

The Hough transform can be adapted to any shape that can be defined by a simple analytical formula, like lines. In the case of radargrams from ground-coupled GPR, it can be adapted to detect hyperbolas (signature of point scatters).

Unlike lines, three parameters α, β and γ are required to define a hyperbola. Thus, three distinct points on a hyperbola are enough to define it. To adapt the Hough transform to the detection of hyperbolas, it is therefore necessary to find how to determine these parameters from the position of three pixels (1 Q , O Q ),

(1 M , O M ) and (1 P , O P ) in the edge detection.

Starting from the equation of a hyperbola, we can write:

R O Q = $ + % 1 Q -& O M = $ + % 1 M -& O P = $ + % 1 P -& ( 3.16 )
The α parameter is therefore related to the other parameters and the positions by

R $ = O Q -% 1 Q -& $ = O M -% 1 M -& $ = O P -% 1 P -& ( 3.17 )
From these equations can be drawn the following equalities:

S O Q -% 1 Q -& = O M -% 1 M -& O Q -% 1 Q -& = O P -% 1 P -& ( 3.18 )
The β parameter is therefore related to the other parameters and the positions by

⎩ ⎪ ⎨ ⎪ ⎧ % = O Q -O M 1 Q -& -1 M -& % = O Q -O P 1 Q -& -1 P -& ( 3.19 ) Hence: O Q -O M 1 Q -& -1 M -& = O Q -O P 1 Q -& -1 P -& ( 3.20 )
Finally, the γ parameter is related to the positions by

& = O Q -O M 1 P -O Q -O M 1 Q + O Q -O P 1 Q -O Q -O P 1 M 2 TUO Q -O M V1 P -UO Q -O M V1 Q + UO Q -O P V1 Q -UO Q -O P V1 M W ( 3.21 )
Therefore, the parameters can be determined with the positions of the pixels by first calculating γ, then calculating β with the calculated value of γ, and then calculating α with the calculated values of β and γ with the following formulas:

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ & = O Q -O M 1 P -O Q -O M 1 Q + O Q -O P 1 Q -O Q -O P 1 M 2 TUO Q -O M V1 P -UO Q -O M V1 Q + UO Q -O P V1 Q -UO Q -O P V1 M W % = O Q -O M 1 Q -& -1 M -& $ = O Q -% 1 Q -& ( 3.22 )
The principle of the Hough transform adapted to the detection of hyperbolas is thus, after application of an edge detection algorithm to the image, to determine the hyperbolas passing through every triplet of points in the detected edges, and increment a 3D accumulator accordingly, as presented in Figure 3.14 for one triplet of points.

The detailed algorithm of the Hough transform for the detection of hyperbolas is the following one:

1) Apply an edge detection to the image to retrieve the position of pixels on edges.

2) Initialize a 3D accumulator with zeros, the first axis corresponding to the α parameter, the second axis to β and the third to γ, with chosen boundaries and steps depending on the size of the image and the precision required on the hyperbolas positions and shapes.

3) For every possible triplet of pixel position in the edge detection (1 Q , O Q ), (1 M , O M ) and (1 P , O P ), the parameters $ QMP , % QMP and & QMP of the hyperbola passing through are determined, and the accumulator is incremented of one at the coordinates corresponding to $ QMP , % QMP and & QMP (with the precision of the accumulator). The Hough transform adapted to the detection of hyperbolas can be long to run (~25 s for a million iterations), as every triplet of pixels in the edges of the image must be tested. It is therefore often applied in a "randomized" version: instead of testing every triplet of pixels in the detected edges, a large number of randomly selected triplets are tested among all possible. However, a criterion is required to define what is a "large" number of pixel in this application. [START_REF] Xu | Randomized Hough Transform (RHT): Basic Mechanisms, Algorithms, and Computational Complexities[END_REF] propose such a criterion, and the number of triplets to be tested for hyperbola detection is: 3.23 ) where N is the total number of pixels identified after edge detection, and coeff a coefficient proposed to be between 10 and 100.

X = KYZZ [ 5 27
(
A randomized version of the Hough transform adapted to the detection of hyperbolas has been implemented to detect point scatters signatures in WISDOM radargrams. However, as previously explained, WISDOM is an air-coupled GPR, and point scatters do not present perfect hyperbolic signatures. The actual diffraction curves expected with an air-coupled GPR are defined by an analytic formula which do not allow a simple inversion as with hyperbolas. For this reason, it is important to assess the impact of the refraction at the surface on the detection by Hough transform, and correct this impact if necessary.

The effect of refraction at the surface on diffraction curve detection

As shown in section 3.2, a diffraction curve from a subsurface reflector and in an air-coupled situation is not a perfect hyperbola, but tends to be a hyperbola when the depth of the reflector tends to the infinity.

In practice, when guiding the drilling operations of the ExoMars 2022 mission, the WISDOM science team will be especially interested in reflectors within the reach of the rover drill, that is between 0 and 2 m of depth. Such distances are in the same order of magnitude as the height of the antennas above the surface; any existing diffraction curves in the first two meters of the subsurface are therefore expected to be significantly impacted by the refraction at the surface which will, in consequence, impair the derivation of the dielectric constant. To characterize this effect, time-delays of echoes from a reflector in at different depths, and in homogeneous media of different dielectric constants have been estimated using the model presented in section 3.2. Depths between 0 and 3 m with a step of 5 cm, dielectric constants between 1 and 7 with a step of 1, and Field of Views of between 60° and 120° with a step of 20° have been tested. The horizontal distance steps for the diffraction curve is 10 cm, similar to WISDOM traverses.

The resulting time-delays obtained for different horizontal positions of the antennas relative to the reflector have then been set as the input of the Hough transform algorithm, as if they were the detected edges in a radargram. All the points given as an entry to the algorithm here are thus part of one diffraction curve, unlike in WISDOM radargrams where the detected edges will come from echoes of various origins. Figure 3.15 displays examples of generated diffraction curves.

For every tested diffraction curve, the calculated parameters α, β and γ for all possible triplets of points in the diffraction curve are stored.

In Figure 3.16.a is presented an example of Hough accumulator obtained for one of the generated diffraction curves, corresponding to a reflector at a depth of 1 m in a homogeneous subsurface of dielectric constant 4, and for a FoV of 100° of the GPR. As a 3D accumulator cannot be displayed in a readable way, three projections along each of the 3 axis are proposed here, as 2D histograms. The selected precisions of the accumulator are 5 cm for horizontal distances, 0.1 ns for time delays, and 0.1 for dielectric constants. For each of the 3 projections, one main peak is visible, which value corresponds to the expected horizontal distance of 0 m and time delay of 13.3 ns in the Hough accumulator, but not the dielectric constant 4. This error on the dielectric constant was expected as equations (3.22) are valid for hyperbolas and the generated diffraction curves are not perfect hyperbolas. It can also be observed that a few detections are outside the mains peak, explaining the presence of standard deviations in the results of Figure 3.16.b. It can also be noted in Figure 3.16 that the standard deviation of the estimated dielectric constant increases with depth and with the FoV. This is because the effect of refraction at the surface increases with the number of points in the diffraction curve, this number rising with the depth of the reflector, and the FoV of the GPR.

This demonstrates the need for a correction of the Hough transform usually employed to detect diffraction curves in GPR radargrams. It can also be foreseen that the estimation of the dielectric constant of the subsurface will be difficult for very shallowly buried reflectors, for which the shape of the diffraction curve is close to that of a reflector in the air or at the surface. The discrimination between shallowly buried targets and targets on the surface will thus require the synergy with the PANCAM instrument (see Section 1 of Chapter 2), to confirm if a surface reflector at the right distance from the antennas is in sight.

The correction for the refraction effect

Equation (3.9) is not easy to inverse contrarily to equation (3.6). Therefore, estimating the diffraction curve passing through a triplet of points cannot be done using a simple set of formulas. Instead, another technique had to be found to account for the layer of air between the WISDOM antennas and the surface when detecting and characterizing hyperbolas in radargrams.

An equivalent dielectric constant as a first approach

A first approach consists in using an equivalent dielectric constant for the medium between the antennas (at a height . above the ground) and the buried reflector (at a depth !):

̅ = . + !√ . + ! ( 3.24 )
̅ is the mean dielectric constant of the air and subsurface layers, weighted by their thickness. The resulting hyperbola is expected to be close to the actual diffraction curve in cases where the refraction effect is not too strong, for instance for low dielectric constants and/or narrow FoV. After fitting the diffraction curve with the hyperbola of dielectric constant ̅ , the actual dielectric constant of the subsurface ε can be readily calculated from: of Chapter 1). Indeed, the LPR is also an "air-coupled" GPR, with the antennas at approximately 31 cm from the surface.

In the case of WISDOM, with H = 38 cm, the mean dielectric constant of the layers of air and homogeneous subsurface above the reflector can be calculated for different dielectric constants and reflector depths as shown in Figure 3.17.

As expected from the model defined in equation (3.9), the obtained mean dielectric constant begins from one and tends to the actual dielectric constant of the subsurface. To observe the ability of this correction to improve the estimation of the dielectric constant, the "corrected" Hough transform has been applied

to the exact same dataset of synthetic diffraction curves as the original Hough transform in section 3.4.

The results are presented in Figure 3.18. As anticipated, the correction is efficient for low dielectric constants, but much less efficient for dielectric constants above ~4, for which the refraction effect is high. In terms of subsurface materials, this would translate as a correct estimation of the dielectric constant for low density of high-porosity materials, but an insufficient correction for high-density / low-porosity materials. As for Figure 3.16, the correction not accounting for the refraction at the surface, the standard deviation of the dielectric constant estimation increases with the number of point, and thus with the depth and FoV.

A new Nelder-Mead optimization approach

In order to accurately correct the bias in the estimation of the dielectric constant in the Hough transform, when fitting triplets of points with a theoretical diffraction curve presented in Section 3.2, we decided to inverse the problem posed by equations (3.6) and (3.9) with an optimization method.

The diffraction curve passing through a given triplet of points (1 Q , O Q ), (1 M , O M ) and (1 P , O P ) is determined with the optimization algorithm known as "Nelder-Mead" [START_REF] Nelder | A Simplex Method for Function Minimization[END_REF][START_REF] Spendley | Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation[END_REF]. This "direct search" approach is sometimes referred to as the "downhill simplex method". This algorithm has multiple assets: it is simple and requires few tests per iteration; it always returns the same result for a given situation; it can handle multidimensional problems and even discontinuities in the space of possibilities.

The main known drawbacks of the algorithm are the risk of converging to a local minimum of the error function, and its sensitivity to the initial positions of the simplex points. For this reason, the Nelder-Mead algorithm will be initialized with the estimation of (X,T(X),ε) returned by equation (3.22). T(X)

is the time-delay of the diffraction curve apex, the horizontal position of the apex being X. From T(X) can be calculated the depth of the reflector ! = " √#

. The algorithm is the following one:

1) Initialization: 4 points in the 3D space of (X,T(X),ε) possibilities are considered, forming a 3D simplex: a tetraedron. The 4 points of the simplex (see Figure 3.19 for an illustration) are at different positions (X1,T1,ε1), (X2,T2,ε2), (X3,T3,ε3), and (X4,T4,ε4), with T1 = T(X1), T2 = T(X2), T3 = T(X3) and T4 = T(X4). To initialize the algorithm, one of the points is set at the (γ,α,β) values determined with equations (3.22).

2) The error function: For each of the 4 points of the "pattern", the corresponding diffraction curve is generated with equations (3.9) and (3.11) to calculate T(1 Q ), T(1 M ) and T(1

P ), (1 Q , O Q ), (1 M , O M )
and ( 1 3) Ordering: The positions of the simplex points (X1,T1,ε1), (X2,T2,ε2), (X3,T3,ε3), and (X4,T4,ε4) are ordered according to the values of F(X1,T1,ε1), F(X2,T2,ε2), F(X3,T3,ε3), and F(X4,T4,ε4). The point which coordinates (X,T(X),ε) correspond to the smallest value of F(X,T(X),ε) is named ^_-G . Similarly, the point corresponding to the highest value of F(X,T(X),ε) is named ^`a G , and the point corresponding to the second highest value of F(X,T(X),ε) is ^`a G . The centroid of all points except ^`a G is calculated and named ^"-bG a .

To find the minimum of F in the space of possibilities, different operations can be applied to the simplex: "reflection", "expansion", "contraction" or "shrink".

4) Reflection:

The "reflected point" ^ -is first calculated: ^ -= ^"-bG a + ^"-bG a -^`a G .

If c ^_-G ≤ c ^ -≤ c ^`a G then ^`a G is replaced by ^ -, and the algorithm returns to step 2.

5) Expansion: If F(^ -)< c ^_-G then the "expanded point" ^-ef = ^"-bG a + 2 ^ --^"-bG a is calculated. If F(^ef )< c ^ -then ^`a G is replaced by ^-ef and the algorithm returns to step 2. Else, ^`a G is replaced by ^ -and the algorithm returns to step 2.

6) Contraction: If F(^ -)≥ c ^`a G then the "contracted point" ^"abG = ^"-bG a + 0.5 ^`a G -^"-bG a is calculated. If F(^" abG )< c ^`a G then ^`a G is replaced by ^"abG and the algorithm returns to step 2.

7) Shrink: Else, replace all points except ^_-G by ^= ^_-G + 0.5 ^-^_-G .

8) While the value ^_-G is superior to a defined threshold, steps 2) to 8) are repeated.

The output of the algorithm is thus the position (X,T(X),ε) corresponding to the last ^_-G value.

The main idea behind this algorithm is to replace the "worst" point in the simplex to make the simplex "move" in the direction of the reflection of the point compared to the centroid of the simplex. If the reflection point is not better than the worst point, the simplex can expand or contract relative to the centroid, towards a better point. The shrink case avoids the reflected point to cross a "valley" in the F(X, T(X), ε) values instead of moving towards it.

The diffraction curves detection strategy

The use of an optimization algorithm such as the Nelder-Mead approach in the Hough transform leads to an accurate detection and characterization of the synthetically generated diffraction curves (presented in Section 3. 4), but at the expense of a long computation time. This could be an issue once on Mars where a quick interpretation of WISDOM radargrams will be expected. On the other hand, while the classic Hough transform for mathematic hyperbolas is unable to accurately estimate the dielectric constant of the subsurface from the diffraction curves, it nevertheless accurately detects the position of the apex (horizontal position and time-delay) of diffraction curves on synthetic radargrams. In other words, the classic Hough transform fits well a hyperbola to the diffraction curves, but returns a wrongly estimated dielectric constant.

The proposed detection/characterization strategy is therefore the following one: first applying the classic Hough transform to the entire radargram, to detect the position of diffraction curves apex in a reasonable time, and then applying the Hough transform corrected with the Nelder-Mead approach to points selected in windows around the detected diffraction curves. The precision on the horizontal position, time delay and dielectric constant of the diffraction curves will be equal to the corresponding steps in the 3D

accumulator. Figure 3.20 summarizes this strategy.

In the following, we will refer to the "classic Hough transform" for the algorithm described in section 3.3 for the detection of mathematic hyperbolas, and to the "corrected Hough transform" for the algorithm modified with the Nelder-Mead approach as described in section 3.5. Even if the diffraction curves in the cloud of points are not mathematic hyperbolas, it can be seen in Figure 3.21 that the classic Hough transform well detect their presence and fit a mathematic hyperbola having the right apex position. On the other hand, as expected, the estimated dielectric constant of the subsurface (and therefore the estimation of depth for the buried targets) is wrong. Figure 3.22 displays a 3D Hough accumulator integrated over the ε axis to allow a 2D representation; the 4 peaks corresponding to the 4 detections can be clearly observed, with very low contamination from the parasitic points. The corrected Hough transform with the Nelder-Mead optimization method will then be applied to points selected in windows around the detections to correctly estimate the dielectric constant of the subsurface. In experimental WISDOM radargrams, perfect diffraction curves as defined by equation (3.9) may be distorted due to heterogeneities of the dielectric constant in the subsurface which will affect time delays of echoes. In order to test the ability of the classic Hough transform to detect diffraction curves despite subsurface dielectric constant variations, tests were performed with a randomly selected dielectric constant with a normal distribution of mean parameter 4, and standard deviation 0.05.

Validation on synthetic WISDOM radargrams

As illustrated in Figure 3.24, this modifies the time delays of the diffraction curves. However, the corrected Hough transform algorithm is still able to detect the diffraction curves and to estimate the depth of each corresponding target quite accurately. The estimated depths are indeed 0.45, 0.97, 1.55 and 2.01 to be compared to input values 0.5, 1.0, 1.5 and 2.0 m, respectively.

Validation on FDTD simulations

For further validation, the diffraction curve detector was then applied to synthetic radargrams generated by FDTD simulations (see Section 5.2 of Chapter 2). In such simulations, the WISDOM antennas are modelled with representative radiation patterns and the effects of the orientation, tilt and geometry of the antennas, which can affect the estimation of the dielectric constant, are taken into account.

Reflectors in the air

First the classic Hough transform detection was applied to radargrams generated for reflectors separated from the antennas only by air, in order to test the ability of the method to detect reflectors lying at the surface in future WISDOM Martian radargrams.

The radargram presented in Figure 3.25 has been generated for one spherical reflector with a diameter of 5 cm in the air ( = 1), at a distance of 1 m from the antennas, and at a horizontal distance of 2 m from the beginning of the 4-m long traverse. A free-space measurement has also been generated for this radargram, and subtracted to the other soundings. As expected, a mathematic hyperbola is visible, with a slight dissymmetry due to the tilt of the antennas.

The edge detection applied to the radargram consists in a modal decomposition technique further described in Chapter 5 (Piou et al., 1999). This technique allowed the detection of the two expected diffraction curves, corresponding to the interface between the air and the reflector, and then between the reflector and the air (as the reflector is a sphere with a diameter of 5 cm), despite their interferences in the radargram. In the following, the edge detection applied to the other radargrams is a peak detection with a threshold, for the sake of simplicity.

Then, the classic Hough transform was applied to the detected points, with precisions of 5 cm in horizontal distance, 0.1 ns in time-delays, and 0.1 in dielectric constant (see results in Another case of diffraction curve from a reflector in the air has been tested, with a FDTD generated radargram for a spherical reflector with a diameter of 10 cm lying at the surface. Unlike the previous study case, the reflector is 0.5 cm on one side of the traverse and not just below. The diffraction curve should appear below the surface echo in the generated radargram, as would a reflector buried below the surface. The idea behind this scenario, illustrated in Figure 3.26, is to assert the ability of the classic Hough transform to detect a diffraction curve and deduce that the reflector is on the surface and not in the subsurface.

Figure 3.27 shows the generated synthetic radargram where two diffraction curves appear below the surface echo. The first diffraction curve corresponds to the reflection on the interface between the air and the sphere, the second to the reflection on the interface between the reflector and the air. The first diffraction curve is a perfect hyperbola, for a dielectric constant = 1. The second one, with double path of the electromagnetic signal inside the reflector, is not a hyperbola, even if the Hough transform try to fit a hyperbola to this curve. Both diffraction curves present a branch much larger than the other, as the reflector is off-track the antennas, and only one co-polar channel of WISDOM is simulated here. The detection of echoes was applied after time-domain filtering, hence the near-absence of surface echoes. on the dielectric constant. As expected from the preliminary tests on synthetic diffraction curves, the dielectric constant is always underestimated in absence of correction for the refraction effect, leading to a potentially dangerous overestimation of the reflector depth. In these examples, the overestimation is far from being negligible, from 29% for 2 m of depth, to 48% for 0.5 m of depth. In addition, the geological interpretation of WISDOM's radargrams may be impaired, with a risk of confusion between basaltic materials which typically have a dielectric constant of ~6 with dry sand or/and water ice which dielectric constant estimations are < 4.

On the other hand, Figure 3.31 demonstrates that after application of the corrected Hough transform to the 9 generated diffraction curves, the estimation of the dielectric constant is accurately estimated. The maximum error on the depth of the reflector is only ~3 cm in all examples, which is very satisfactory.

As an additional conclusion, the tests shown in this section suggest that the geometry of the antennas (their relative position and tilt), which were not considered in the model presented in Section 3.2, has only a limited impact on the diffraction curve detection and characterization. This result illustrates one of the limitations of the proposed technique: if a clear interface between two media is detected, dielectric constants inferred from diffraction curves visible after this interface must be interpreted with caution. Overall, any significant dielectric gradient in the subsurface will likely also affect the characterization of the diffraction curves.

Several reflectors in a heterogeneous subsurface

No such thing as a perfectly homogeneous subsurface exists in nature, that is why the corrected Hough transform was also tested on WISDOM synthetic radargrams obtained for a heterogeneous subsurface.

In particular, the shape of diffraction curves is expected to vary as a function of the size of the subsurface heterogeneities with respect to the transmitted wavelengths. Simulations with other heterogeneous subsurface models, varying the dielectric contrast and size of heterogeneities, ought to be tested in order to conclude on the robustness of the technique to heterogeneities. Nevertheless, this first result on a particular case of heterogeneities is encouraging, confirming the implemented Hough transform is able to detect diffraction curves confronted with this subsurface model.

Validation on experimental WISDOM radargrams

Validation on hyperbolas from reflectors in free-space

As a starting point, the Hough transform was tested on experimental WISDOM radargrams including reflectors in free-space. Only the classic form of the Hough transform (see Section 3.3) was applied here since there is no refraction to be taken into account.

During the June 2019 TU Dresden campaign (see Section 4.1, Chapter 2), WISDOM performed a traverse over a soil trench where a metallic staircase was buried. Before burying the staircase, it acquired data with the staircase in the hole dug in the soil trench. The resulting radargram is particularly useful to test the Hough transform as a clear diffraction curve from the first step of the staircase is visible. The diffraction probably originates from the corner of the metallic step, which was in the air during the experiment. The curve is thus expected to be a perfect hyperbola, for a dielectric constant of 1 as confirmed by the application of the classic Hough transform (see Figure 3.40). The derived "depth" is also consistent with the known geometry of the metallic staircase, the last step being at 70 cm from the surface, and each step having a height of 10 cm. Note that the radargram of Figure 3.40 has also been used to test a technique to improve its spatial resolution, namely the "Bandwidth Extrapolation" (BWE) technique that is the subject of Chapter 4.

Even after application of the BWE, the good results of the Hough transform remain unchanged.

Validation on the Dachstein campaign radargrams

Radargrams from the Dachstein campaign (presented in Section 4.2.3 of Chapter 2) constitutes a relevant dataset to validate the corrected Hough transform technique, as they include tens of diffraction curves, many of them originating from reflectors trapped in the water ice slab. As a reminder, pure water ice has a dielectric constant of ~3.1-3.2 at the WISDOM working frequencies [START_REF] Fletcher | Liquid water and freezing[END_REF]. Fits of the diffraction curves have already been performed manually in a previous study (Dorizon et al., 2016), and revealed that most of the scatters were objects embedded in water ice, with dielectric constants between 3 and 4 (see the histogram of the estimated dielectric constants in Figure 3.41). The rest of them were either coming from above the surface, with dielectric constants around 1, or from the bedrock with dielectric constants around 6-7. For this section, we focused on diffraction curves identified as corresponding to reflectors in the ice. With the corrected Hough transform, the estimated dielectric constants are coherent with the results obtained manually for the Dachstein campaign (Dorizon et al., 2016). The diffraction curves around positions (1.5 m, 20 ns) In addition to the 12 days of simulation of rover operations [START_REF] Hall | ExoFiT: ExoMars-Like Field Trials -a Mission Simulation[END_REF], on the 25 th and 27 th of February 2019, 3 parallel traverses have been performed with the WISDOM instrument mounted inside a cardboard support to maintain the antennas 38 cm above the surface. The selected site for these soundings (see Figure 3.45) have later been used to simulate the first operations days, including the descent of the rover from its platform. The surface was strewn with boulders of various sizes ranging from a few 10s of cm to a few meters, separated by wide patches of sand, with small patches of white evaporites identified as salt crusts. The presence of clay deposits (phyllosilicates), especially relevant in the scope of the ExoMars 2022 mission simulation, have also been detected on-site (see Figure 3.46.a). The soundings were performed in full-polar mode, implying that radargrams for the 4 polarimetric channels (00,01,10,11) can be generated for each traverse. In addition, a free-space and a calibration measurement were performed with the antennas pointed perpendicular to the ground towards the sky, and metallic plate in front of the antennas, respectively. Lastly, a shallow subsurface sample was collected and analysed by the RAMAN-LS instrument between the traverses performed on the 25 th and the afternoon of the 27 th of February, providing constraints on the local composition [START_REF] Veneranda | ExoFiT trial at the Atacama Desert (Chile): Raman detection of biomarkers by representative prototypes of the ExoMars/Raman Laser Spectrometer[END_REF]. During the ExoFit simulation, the RAMAN-LS team used a demonstration transportable device named

Rad1. This latter provided rapid in-situ characterization and thus useful information for the simulation campaign. More representative models of the instrument (named RLS in Figure 3.47) as well as a X-ray diffractometer (named XRD in Figure 3.47) later analysed the collected samples in laboratory. The two 30 cm thick core sample were separated into 2 parts, "UP" (Upper Part) from 0 to 15 cm of depth, and "LP" (Lower Part) from 15 to 30 cm of depth. The main components of both UP and LP samples are quartz, K-feldspar and muscovite. This latter mineral is particularly interesting as it a phyllosilicate, one of the targets of the ExoMars 2022 mission. Minor components include calcite in the UP samples only, and potential biomarkers probably preserved in the phyllosilicates.

Table 3.1 reports the expected dielectric constant of a selection of materials detected in the shallow subsurface of the Atacama operation site at WISDOM frequencies [START_REF] Church | Dielectric Properties of Low-loss Minerals[END_REF][START_REF] Zheng | Dielectric properties of hard rock minerals and implications for microwave-assisted rock fracturing[END_REF]. The actual subsurface dielectric constant can differ from these values depending on the exact mixing of the materials, their porosity and level of moisture.

Analysis of the surface echo

The dielectric constant was estimation from the surface echo, as shown in Figure 3.50 (after correction of the antenna-surface distance variations as explained in Section 2.1). Doing so, the inferred values oscillate between 2 and 6, with most of the values observed slightly below 3 which is consistent with dry sand. It can be further noted that the results obtained from the 00 and 11 polarimetric channels are overall very close.

To identify general trends in the evolution of the dielectric constant measured from the surface echo, a moving average can be applied to the presented figures. A window of horizontal distance 1 m is used to generate Figure 3.51. Assuming a dielectric constant of 3, as supported by both surface echo analysis and laboratory measurements, the Atacama radargrams can be displayed as a function of "depth" (more accurately, as a function of distance accounting for the possibility of off-track scatters) instead of time-delay. radargrams. If this reflection arises from a buried structure, this latter would be lying at a depth ~30-40 cm. On the other hand, if it is lying on the surface, its distance from the antennas should be ~90-110 cm.

For sure, the origin of the reflection is not the large rock observed on between the 27/02 afternoon and 25/02 traverses (see Figure 3.46.b) because it is too afar (> 1.7 m). Comparison of the radargrams acquired in different polarization provides further clues on the origin of the strong reflection. This latter is present, at nearly the same horizontal distance, on both the 00 and 11 channels which implies that the reflector must be on-track, or just slightly off-track. A subsurface reflector is therefore a reasonable hypothesis. The significant rise in the dielectric constant observed for the 27/02 morning traverse occurring right after the horizontal position of this reflector, it could be one of the many partly-buried rocks observed on-site, a portion of it rising to the surface (see Figure 3.48.b).

Analysis of a diffraction curve

Unfortunately, the reflector in question did not generate any clear diffraction curve to be analysed by the corrected Hough transform. Nevertheless, a few potential diffraction curves can be exploited, especially on the radargrams acquired on the 25/02. Figure 3.55.a shows one of them which is only visible in one of the co-polar channels thus pointing to an off-track reflector. As a further argument, the left branch of the diffraction curve is longer and stronger in amplitude that the right one; as WISDOM antennas were not tilted during the Atacama campaign, such effect is likely to be caused by an off-track reflector. In order to determine whether this reflector is at the surface or in the subsurface, a fit to the diffraction by the Hough transform was performed (see Figure 3.55.b and c). The estimated dielectric constant is 0.8±0.1, compatible with an off-track reflector lying at the surface. It is probably one of the many surface boulders visible on the operation site, and would be at a minimum distance of ~57 cm from the antennas considering the dielectric constant of free-space. In the context of the ExoMars 2022 mission, the hypothesis of a potential subsurface hazard for the rover drill would therefore have been discarded thanks to this analysis. As a conclusion, the set of 3 WISDOM traverses acquired during the ExoFit simulation of operations are challenging to interpret, with no clear subsurface structures identifiable in the generated radargrams.

Nevertheless, the dielectric constant estimation techniques proposed in Sections 2 and 3 of this chapter provided interesting clues for the interpretation of two visible reflectors. With the estimation of the subsurface dielectric constant from the surface echo, time delays in the radargrams could be converted into distances, constraining the "depth" of the strongest identified reflections, if coming from the subsurface. A neat difference between the dielectric constants estimated for the 27/02 morning traverse and the two others was also observed. Eventually, a diffraction curve was detected by the Hough transform, and associated with a boulder lying on the surface. Only the correction implemented for the Hough transform could not be tested on this dataset, as no diffraction curves corresponding to subsurface reflectors could be identified.

Summary

• For non-magnetic and low loss tangents, the dielectric constant of the subsurface characterizes the propagation speed of electromagnetic waves through it, allowing the conversion of time delays to distances in WISDOM radargrams. It can also be a clue on the subsurface composition.

• In this chapter were presented two techniques to estimate the dielectric constant from WISDOM radargrams: from the amplitude of the surface echo, and from the shape of diffraction curves.

• In the case of a smooth surface, by comparing the amplitude of the reflection from a metallic plate to the amplitude of the surface echo can be inferred the dielectric constant of the shallow subsurface. An application to the TUD campaign was proposed.

• A new automated technique to detect and characterize diffraction curves (the patterns formed by echoes from point reflectors in WISDOM radargrams), based on the Hough transform, was implemented.

• A correction to account for the impact of refraction at the surface on the diffraction curves was added to the Hough transform, allowing an accurate estimation of the subsurface dielectric constant. Validations on synthetic and experimental radargrams were proposed.

• We applied both techniques to 3 radargrams acquired during this thesis in the Desert of Atacama, potential Martian analog environment.

• An accurate estimation of the Martian subsurface dielectric constant from WISDOM radargrams will be key in the reconstruction of the geological history of the ExoMars 2022 landing site, and when guiding precisely the drilling operations.

Introduction

WISDOM has been designed to unveil the structure and dielectric properties of the Martian subsurface below the ExoMars rover down to a likely maximum depth of ~3 m in a lithic environment with a vertical resolution of a few centimeters. In the context of searching for a drilling place to collect 3-cm long samples (Vago et al., 2017), this latter parameter is key. However, the range resolution of a radar is limited by the instrument bandwidth (2.5 GHz in the case of WISDOM), and the Inverse Fourier

Transform processing used to obtain soundings in time-domain. A new processing technique able to improve the resolution of radar soundings in time-domain would therefore provide precious additional information when interpreting WISDOM radargrams acquired on Mars.

A variety of modern spectral estimation techniques have been developed as alternatives to the classic Fourier Transform, limited by the frequency bandwidth and windowing [START_REF] Kay | Spectrum Analysis-A Modern Perspective[END_REF].

These techniques can be divided into two main categories: parametric categories, and non-parametric categories. While non-parametric spectral estimation techniques, such as the Fourier Transform-based algorithms, do not rely on an a-priori model of signal, parametric spectral estimation techniques assume a spectral density function defined by a limited set of parameters. In this case, the parameters can be determined with different algorithms.

The application of a parametric technique called Multiple Signal Classification (MUSIC) on WISDOM soundings has already been studied (Hervé, 2018). MUSIC uses the eigenvectors of the autocovariance matrix of the signal to determine the parameters of the spectral density function [START_REF] Schmidt | MULTIPLE EMITTER LOCATION AND SIGNAL PARAMETER ESTIMATION[END_REF]. If the results of the MUSIC technique to WISDOM synthetic soundings were very promising, no improvement in the resolution of experimental soundings was obtained, probably due to the difficulty of estimating properly the number of sources. It was therefore decided to continue the study of WISDOM's vertical resolution enhancement with other parametric spectral estimation techniques.

In this chapter, we describe how to expand the nominal bandwidth of WISDOM and therefore to improve the instrument resolution, by implementing in the data processing chain a super-resolution method called the Bandwidth Extrapolation (BWE) technique. This spectral estimation technique uses both a parametric method and the Fourier Transform, to obtain soundings with a resolution three times better than with the Fourier Transform alone, but more robust on experimental data than parametric techniques alone.

The BWE super-resolution technique was proposed by the Radar Imaging Technique group of the Lincoln laboratory. The concept was introduced by S. B. Bowling (Bowling, 1977) and then further developed by K. M. Cuomo (Cuomo, 1992) who applied it to radar data. Moore et al. (1997) then applied it to SAR imaging. More recently, the technique has been used to enhance resolution in planetary radar imaging, for example to improve the bathymetric capabilities of the Cassini Radar instrument (Mastrogiuseppe et al., 2014), the stratigraphic analysis of the Martian polar ice sheets using SHARAD/MRO dataset (Raguso et al., 2018), and lastly to the LRS/SELENE lunar soundings [START_REF] Poggiali | SURFACE CLUTTER SIMULATIONS AND SUPER RESOLUTION PROCESSING OF THE SELENE[END_REF].

Following such promising results, we have adapted and applied the BWE technique to WISDOM data.

In this paper, we briefly introduce the instrument and its performances in term of range resolution. Then, we present the BWE technique, and its integration to the WISDOM data processing chain. The method is eventually applied to WISDOM experimental data collected both in controlled, semi-controlled and natural environments.

The content of this chapter has been the object of a peer-reviewed publication in the Planetary and Space Science journal (Oudart et al., 2021b).

The WISDOM vertical resolution

As Only the real part of the return signal is measured by the instrument, and the complex form is derived via Hilbert transform before applying an Inverse Fast Fourier Transform (IFFT) to obtain the response of the sounded volume. In the time domain, each reflector generates an echo which is characterized by a propagation delay (that can be then converted in distance with assumption on the velocity of the wave in the sounded material) and an amplitude. Since WISDOM antennas are looking downwards at the surface and, while echoes can sometime be generated by off-track reflectors, the retrieved distances are commonly and improperly interpreted as depths. In the following, for the sake of simplification, we represent radargrams in the terms of depth and vertical resolution assuming a value of dielectric constant, rather than range and range resolution.

As previously mentioned, WISDOM is in charge of providing information that will help the mission's scientific team to identify the very layer where to take a sample. In order to achieve this goal, the instrument requires a vertical resolution commensurate with the 3-cm length of the sample. The range resolution is defined as the shortest distance between two echoes that can be discriminated.

In a horizontally layered subsurface, it corresponds to the smallest thickness that can be resolved by the instrument. As previously mentioned, WISDOM is operating in the frequency-domain over a frequency bandwidth = 2.5 GHz and an Inverse Fourier Transform (IFT) is applied to the data to reconstruct the response in time-domain. A radar range resolution is limited by the width of the synthetic pulse obtained after IFT, which is inversely proportional to the effective bandwidth of the instrument. In the time domain, two echoes of a same amplitude are commonly considered as resolved if the time delay between them is superior to half the effective pulse width. For a pulse width at -6 dB, in a material having a dielectric constant of , it follows:

= 1.21 2 √ ( 4.2 )
where c is the speed of light in vacuum (National Instruments Inc., 2009).

Commonly, IFT process involves the use of a window to reduce side-lobe contribution, that could be misinterpreted. With the Hamming window used for this study, the resulting range resolution is degraded to ~1.5 . The vertical resolution of WISDOM radargrams in a subsurface with a typical permittivity value of 4 after windowing would therefore become ~5.5 cm which is slightly larger than the desired 3 cm.

Lastly, it is also important to note that the resolution of the radargrams will also be affected by the roughness of the investigated surface and buried interfaces and by scattering due to heterogeneities within the subsurface volume. In addition, because absorption losses often increase with frequency, WISDOM high frequencies (2-3 GHz) are more rapidly attenuated in the medium than smaller ones and the resolution therefore tends to decrease with depth (Hervé et al., 2020).

In the following section, we propose to implement the Bandwidth Extrapolation technique to extrapolate WISDOM spectra to additional frequencies, and therefore obtain a wider bandwidth that would result in an enhanced vertical resolution of the radargrams.

Bandwidth Extrapolation technique applied to WISDOM data

Extrapolation in frequency domain by an autoregressive (AR) linear model

In the frequency domain, a point scatter inside a non-dispersive and lossless sounded volume would generate a sinusoidal signal and the spectrum S resulting from scatterers is expected to be a sum of sinusoids:

( ) = ! (-" 2 # $ ) % &' ( 4.3 )
where is the frequency, $ time delay associated with the k-th scatterer, and the amplitude of the return signal from the k-th scatterer. As previously explained, a WISDOM sounding consists in a set of spectrum values collected for N discrete frequencies ( = ( ) where is given by equation (4.1).

This sum of harmonic signals can be modelled by a linear autoregressive all-pole model [START_REF] Kay | Spectrum Analysis-A Modern Perspective[END_REF]. Indeed, each sample of the frequency spectrum is the weighted sum of the previous samples for a model in the forward direction (see equation ( 4 A demonstration for a sum of two complex sinusoids is proposed in the following box, and a demonstration for a sum of any number of complex sinusoids proposed by (Raguso, 2018).

Predictability demonstration for a sum of two complex sinusoids:

We consider a set of spectrum values ( being a sum of two complex sinusoids corresponding to echoes of time delays $ ' and $ ) :

( = *+ ) , -. / 0 + *+ ) , -. / 1
We will demonstrate that this sequence can be predicted in the forward direction by a linear autoregressive model of order 2.

If this sequence of spectrum values is predictable by such an autoregressive model, any sample ( is therefore equal to a weighted sum of the two previous samples. Naming 2 ' and 2 ) the weights of such a sum:

( = 2 ' ( *' + 2 ) ( *)
And thus the following equation can be established:

*+ ) , -. / 0 + *+ ) , -. / 1 = 2 ' 3 *+ ) , -.40 / 0 + *+ ) , -.40 / 1 5 + 2 ) 3 *+ ) , -.41 / 0 + *+ ) , -.41 / 1 5

According to equation (4.1), ν 7 = ν 897 + (n -1)∆ν. Thus, ν 7*' = ν 7 -∆ν and ν 7*) = ν 7 -2 ∆ν.

The previous equation can therefore be simplified:

*+ ) , -. / 0 + *+ ) , -. / 1 = *+ ) , -. / 0 32 ' + ) , ∆; / 0 + 2 ) + < , ∆; / 0 5

+ *+ ) , -. / 1 32 ' + ) , ∆; / 1 + 2 ) + < , ∆; / 1 5
Thus, ( is predictable by an autoregressive model only if the following system has solutions:

= 2 ' + ) , ∆; / 0 + 2 ) + < , ∆; / 0 = 1 2 ' + ) , ∆; / 1 + 2 ) + < , ∆; / 1 = 1
This system can be expressed in the matrix form:

> + ) , ∆; / 0 + < , ∆; / 0 + ) , ∆; / 1 + < , ∆; / 1 ? > 2 ' 2 ) ? = > 1 1 ?
Assuming the coefficient matrix is non-singular, it is therefore evident that this system has a unique couple solutions 2 ' and 2 ) , QED.

A similar demonstration can be done for the predictability in the backward direction.

If p is the order of the model: where (̂ is the n-th sample of the modelled frequency spectrum (indices f and b corresponding to "forward" and "backward" models), 2 and 2 * the i-th coefficient of the model and its complex conjugate, respectively.

For ! + 1 < < (̂ B = - 2 ( * C &' ( 4.
For a given order ! of the model, the coefficients of the model are determined by minimizing the absolute value of the error between the available samples and their modelled values in the backward and forward directions which are, respectively: Therefore, for a frequency spectrum of samples and a given order ! for the model, the total energy G C of the residual has to be minimized in order to determine the coefficients of the model:

For ! + 1 < < C B ( ) = ( -(̂ B = ( + 2 ( * C &'
G C = 1 2 (H C B ( )H ) + H C D ( )H ) ) I &CF' ( 4.6 )
This process is known as the "Maximum Entropy Method" (Burg, 1967). It can be implemented using several algorithms, but only the Burg algorithm is considered in this study, as it has proved to guaranty the stability of the model, and performs better on experimental data than the others (Raguso, 2018).

In a classical parametric spectral estimation, the signal in time-domain ((J) would be estimated with the determined coefficients:

((J) = ' H'F∑ L M 4N L O P LQ0
H

, with the time-delays of point scatters corresponding to the zeros of the denominator. However, such an estimation lacks the information on the phase of each echo, and has proven to be inaccurate in the presence of noise (Moore et al., 1997). For this reason, the Bandwidth Extrapolation technique proposes to combine the previously obtained autoregressive model and the Inverse Fourier Transform.

The idea of the Bandwidth Extrapolation technique is to use the model obtained from the available data (i.e., the order ! and 2 coefficient values) to extrapolate the spectrum to frequency values outside the initial frequency range, and therefore keep information usually lost when applying windowing and IFT.

The time-domain sounding obtained after Inverse Fourier Transform will thus be improved in resolution compared with the IFT alone, have information on the phase of echoes, and be more robust on experimental data.

Considering an initial frequency spectrum made of N samples, the ( + 1)-th sample can be estimated in the forward direction:

(Î F' B = -∑ 2 ( I* F' C &'
for frequency IF' = + ∆ ( 4.7.a )

The same process can be performed in the backward direction,

(R D = -∑ 2 * ( C &'
for frequency R = -∆ ( 4.7.b ) leading to negative indices for extrapolation to even lower frequency

(̂* ' D = -∑ 2 * ( *C C &'
for frequency *' = -2∆ ( 4.8 )

Further estimates are obtained based on the initial dataset gradually extrapolated.

The Burg method (Burg, 1967) is a recursive algorithm which minimizes the value of G C for increasing values of p. For each order, it calculates the coefficients of the autoregressive linear model with the Levinson-Durbin recursion.

By minimizing the error energy at each iteration, the Burg algorithm ensures that the solution will be stable, all poles of the autoregressive model being within the unit circle. This is an advantage of the Burg method over algorithms such as the Covariance, Modified Covariance and Yule-Walker methods [START_REF] Kay | Spectrum Analysis-A Modern Perspective[END_REF]. As previously mentioned, the order of the returned model (the last iteration of p in the algorithm) is an entry of the Burg method.

For the best model's order, a value of ! = /3 has been experimentally found to yield the best results

for soundings corrupted by white noise (Cuomo, 1992;Raguso, 2018) and is commonly used in literature. We will show in section 4.2 that this also applies to the WISDOM dataset.

Implementation of the Bandwidth Extrapolation technique in the WISDOM data processing chain

The data processing pipeline developed for WISDOM is described in (Hervé et al., 2020). Here we briefly recall its main steps and show how it can be completed by the Bandwidth Extrapolation technique.

First, as previously mentioned, WISDOM acquires soundings composed of 1001 real frequency-domain samples. In order to accurately determine the autoregressive model, a Hilbert transform is applied to the real data to convert the signal in complex form. The result is 1001 complex frequency-domain samples.

Since the Hilbert transform does not produce any information, the obtained complex data are oversampled by a factor of two. Therefore, before applying the BWE, following a process recommended in [START_REF] Kay | Spectrum Analysis-A Modern Perspective[END_REF], the data are down sampled by a factor of two. This and has no effect on the range resolution as the bandwidth remains the same. The input soundings in the processing pipeline are thus composed of = 500 complex frequency samples.

WISDOM data then undergo a temperature correction, followed by a free-space measurement removal to supress constant parasitic signals in radargrams (internal coupling, antenna cross-talk, etc.). A signal whitening or spectrum compensation can be performed by dividing the measured spectra by the instrument transfer function. This transfer function is experimentally obtained from the WISDOM spectrum of an echo obtained on a perfect reflector (a metallic plate in the case of this study). Because there are several models of the WISDOM antennas and electronic systems (see Hervé et al., 2020), the free-space measurement and the echo on a perfect reflector had to be obtained for each model of the instrument. A DC-offset removal can also be applied to further reduce constant parasitic signals (such as reflections on the rover body).

Then, 5% of the frequency samples are removed on each side of the spectrum, and the Bandwidth Extrapolation technique is applied with an order of ! = / 3 for the model (see section 4.2). The removal of 10% of the samples in the spectrum, causing high side-lobes, is part of the Bandwidth Extrapolation technique as described by (Cuomo, 1992). In the case of WISDOM, this effect is caused by the Hilbert transform, which reconstructs better samples near the center than on the edges of the spectrum.

Eventually, windowing is applied to the signal, and a time-domain sounding is obtained after an Inverse Fast Fourier Transform (IFFT) with zero-padding. The signal is extrapolated by a factor of 3, as prescribed by (Cuomo, 1992;Moore et al., 1997). The implementation of the Bandwidth Extrapolation into the WISDOM processing pipeline is illustrated by Additionally, a horizontal interpolation can be employed to ease the interpretation of the radargram. The effect of this last processing is of course purely aesthetic as it does not bring any new information to the radargram.

With this modified processing pipeline, we expect to improve the resolution of WISDOM radargrams by a factor of 3 (the Bandwidth Extrapolation factor). Different tests on synthetic and experimental WISDOM radargrams have been performed to confirm this improvement as further described below.

Validation on synthetic WISDOM data

Before testing the Bandwidth Extrapolation on experimental WISDOM soundings, preliminary tests are performed on synthetic data to assert different expected results of the technique.

Generation of synthetic WISDOM radargrams

Synthetic soundings simulating the presence of two reflectors in vacuum (the worst-case scenario in term of resolution) were generated. For a given distance d between the two reflectors, a spectrum between 0.5 and 3 GHz was generated by the addition of two sinusoids of the same amplitude with a phase difference corresponding to twice the distance d. A radargram was generated, for d values regularly increasing from 0 to 15 cm with a step of 0.25 cm. To some extent, this simulated experiment is similar to what WISDOM should measure over a smooth surface overlying a homogeneous layer getting thicker as the radar moves horizontally above the surface. For sake of realism, an additive white Gaussian noise, resulting in a SNR of 30 dB (in agreement with what is observed on experimental data, see section 5) was applied in the frequency domain. times larger than the original one. As expected, the resolution is improved by a factor 3 therefore reaching ~4 cm (see section 4.3). We will describe in the following how this result has been obtained.

In frequency domain, the accuracy of the BWE applied to this synthetic dataset can also be appreciated.

As an instance, Figure 4.3 presents the application of the BWE to a synthetic sounding generated for d = 15 cm, with the same Bandwidth as WISDOM, and an SNR = 30 dB. As mentioned in section 3.2, first 5% of the samples on each side of the spectrum are removed. Then, the spectrum is extrapolated by a factor of 3. The same sounding has also been generated with three times the Bandwidth of WISDOM but without noise, for the extrapolated samples to be compared to.

Determination of the order of the model

To generate the radargram enhanced by BWE shown in Figure 4.2.b, we have applied the method described in section 3. The first step of the method is to choose the value for the order ! of the autoregressive model. To do that, we limit all the initial spectra to one third of their bandwidth by symmetrically removing frequencies on both sides. Then, we apply the BWE technique to reconstruct the initial dataset for different orders values ! ranging from 1 to N. For each ! value, the reconstructed radargram is compared to the original one to identify the best order for the BWE algorithm. The correlation between both radargrams is computed, which provides a quantitative analysis of the reliability of the extrapolation. This process is repeated for 1000 noise realisations (Monte Carlo approach). The result is shown in Figure 4.4.

As can be observed in Figure 4.4, the correlation coefficient values are high (>0.97) for all ! values (which is due to the fact that we only consider 2 reflectors in the case). The highest correlation is obtained for an order of the autoregressive model close to a third of the bandwidth, consistent with the values prescribed in literature. A similar experiment has also been performed by Raguso (2018) on simulated SHARAD soundings, yielding the same result. In the following sections, the order of the model has been therefore set to ! = / 3.

Effect of the BWE on the resolution

With the same dataset, it is possible, from the time delays measured at the peaks of the two echoes (when resolved), to obtain an estimate of the distance between the two reflectors and to compare this estimate U V to the known distance U in order to test the ability of BWE to improve the resolution of the soundings.

In Figure 4.5, the mean value of the error U V -U, with its standard deviation computed on 1000 realisations (Monte Carlo approach) a randomised phase for the first echo is displayed as a function of U.

Without BWE, the simulated echoes obtained with U < 11 cm cannot be separated, while with BWE this limit is brought to 3.75 cm with an error that remains well below the expected resolution.

It can be also noted that, with or without BWE, the error oscillates, depending on the phase between the two echoes. Further, the error decreases as U increases. For the considered U values, it lies between -0.2±0.15 and 1.6±0.13 cm. The amplitude of the oscillation is higher after the application of the BWE due to the sensitivity of the Burg algorithm to both phase and noise, as already depicted by [START_REF] Kay | Spectrum Analysis-A Modern Perspective[END_REF].

Synthetic radargrams have also been generated for media other than vacuum i.e., with higher dielectric constants; they show that the amplitude of the oscillation decreases as the dielectric constant increases, the error always remaining below the expected resolution.

Effect of the BWE on the precision

Having studied the effect of the BWE on the resolution with the generated synthetic dataset, it is also important to verify that this technique does not alter the positions of the reflectors. In other words, that the precision of the super-resolved radargrams remain barely changed after application of the BWE. For each echo, the error on the estimation of its position is measured for each value of d before and after application of the BWE. A precision of 1 cm on the position of an underground layer is three times lower than the length of the sample to be collected by the mission drill. This result is therefore satisfactory, allowing to aim at a precise underground interface without the risk of missing it.

Effect of the BWE on the echo amplitude

An advantage of the BWE, when compared to other super-resolution techniques (MUSIC, ESPRIT), is the conservation of the amplitude of each echo in the enhanced radargram (Cuomo, 1992). This advantage is essential because, in experimental data, the echo amplitude will ultimately be used to infer information about the reflecting structures (starting with the permittivity value in a case of a smooth interface). The ability to obtain a satisfactory estimate of the amplitude has been tested on the previously described synthetic dataset for which the amplitude of both echoes has been set to 1. When these latter ones are resolved by BWE (i.e., for d > 3.75 cm after BWE), the ratio between the first echo amplitude and the second echo one is computed. Figure 4.7 shows the mean value of this ratio with the corresponding standard deviation as a function of U.

By construction the amplitude ratio between echoes is 1 and, after BWE, its mean value remains within between 0.97 and 1.05 with a standard deviation of about 1.6%, which is very satisfactory. On data collected in a natural environment corresponding to a similar configuration (that is a smooth surface on the top of a layer having an increasing thickness along the radar displacement) the measurement of such a ratio would provide an estimate of the losses in the uppermost layer, thus bringing information on the loss tangent (ratio between the imaginary and real parts of the permittivity) and/or on the heterogeneities embedded within this layer. For example, considering the case of a homogeneous layer with a typical dielectric constant 4 and loss tangent 0.03, over a layer of dielectric constant 6 with smooth interfaces, and soundings for distances ranging from 3 to 15 cm with a step of 0.25 cm, an error with a standard deviation of 5% on the ratio estimate would result in an error on the loss tangent with a standard deviation of 3% (Monte-Carlo with 100000 cases) according to the model proposed by [START_REF] Picardi | Marsis data inversion approach: Preliminary results[END_REF]. The derivation of the loss tangent and, overall, a better understanding of the composition of the investigated terrains require to know the dielectric constant of the uppermost layer. This later, especially if the surface is smooth, can be derived from the amplitude of the surface (first) echo (Hervé et al., 2020) which, as shown below, is also well preserved after application of the BWE. set to 1 and remains close to unity after BWE. We further note that both echoes present similar variations in amplitude, with oscillations that are once again clearly correlated to the phase between the echoes.

For distance values U > 9.5 cm in vacuum (i.e., 4.75 cm in a medium with a typical dielectric constant of 4), the error on the amplitude of the first echo (corresponding to the surface echo) is less than 2.5%

(see Figure 4.8.a). As an illustration, such an error on the surface echo amplitude would lead to an error on the retrieval of the permittivity of the top layer of less than 4% for a permittivity value of 4. This would result in an error in distance estimate of less than 2 cm at a depth of 1 meter. For d > 5 cm (i.e., 2 cm with the same hypothesis as above), the error on the first echo being less than 7%, the error at 1 m depth would be ~5.2 cm.

This same error on the amplitude estimate would also translate in an error on the material's porosity retrieval. In fact, assuming an a priori knowledge on the composition of a Martian soil, its porosity can also be derived from the real part of the permittivity. For instance, according to the model experimentally determined by [START_REF] Brouet | A laboratory-based dielectric model for the radar sounding of the martian subsurface[END_REF], at 0.5 GHz and 200 K the dielectric constant is linked to the porosity ϕ by ≈ (12.03 + 2.1) '*[ -0.043, and at 3 GHz by ≈ (8.87 + 2.05) '*[ -0.043 for pure JSC Mars-1 Martian simulant. In the first case (0.5 GHz), for a measured dielectric constant of 4, the error on the estimation of porosity would be ~5.7%. In the second case (3 GHz), the error would be ~7.2%.

In both cases, a satisfying low level of error.

These different tests validate on synthetic data the ability of BWE to improve by a factor of 3 the vertical resolution of WISDOM radargrams. We have also demonstrated that the distance between reflectors and the amplitude of the echoes are well preserved by this technique and can therefore be used to characterize the surface and subsurface. In the following, the method is applied to experimental WISDOM data collected in controlled environments to further confirm the value of the BWE technique. preserved after BWE although some oscillations appear due to phase differences between the echoes.

Effect of the noise on the BWE performances

In the previous sub-sections, the performances of the BWE in terms of reconstitution of the distance between two echoes, the precision on their position, their amplitudes and the ratio between their amplitudes has been validated on synthetic WISDOM soundings generated for an SNR = 30 dB in frequency domain (additive white Gaussian noise). This level of noise was selected to validate the BWE technique because it is coherent with the SNR measured in laboratory on experimental WISDOM soundings.

Nevertheless As could be expected, a lower value of the SNR increases the values of the error on the distance between the two reflectors, but the values of the error remain between ~-0.6 and ~2.3 cm down to a SNR = 1 dB.

This level of error remains very satisfactory, even if higher than the -0.2 and 1.6 cm found for a SNR = 30 dB. The standard deviation of the error also increases as the SNR becomes very low, with values up to ~0.35 cm.

Figure 4.10 presents the measured amplitude of each echo as a function of the distance between the echoes and the SNR, knowing the amplitude of both echoes was set to 1 when generating the synthetic soundings.

As for the error on the distance between the reflectors, the error on the amplitude of each echo globally reduces when the SNR increases. Even so, the level of error remains satisfactory for d > 5 cm. The reconstruction of the amplitude of the two echoes remains also satisfactory, even for low SNR.

For d > 5 cm and a SNR > 3 dB, the average error on each amplitude stays below 15%, to be compared with the 7% found at SNR = 30 dB. Such an error on the amplitude would translate into ~11.7 cm of error at 1 m of depth for a subsurface of dielectric constant 4. For d >10 cm, this average error stays below 4% for all tested SNR values, translating into an error of ~3.8 cm at 1 m for the same subsurface. The standard deviation is always below 10%, and below 5% for all SNR > 5 dB.

Validation of the vertical resolution improvement on experimental data

Application of the Bandwidth Extrapolation technique to WISDOM "two reflectors" laboratory experiment

In order to validate the gain in resolution brought by the BWE technique on experimental data, we have performed laboratory measurements, at LATMOS, with the spare model of the instrument, in configurations as close as possible to the simulated "two reflectors" case described in Section 4. Two metallic plates were placed side by side in front of WISDOM antennas and separated by a varying distance d (Figure 4.11). More specifically, the first metallic plate remained at the same distance from the antennas (~42.6 cm) while the second one was progressively moved away from 1 to 14 cm by steps of about 1 cm. The resulting WISDOM radargram (Figure 4.12.a to be compared to Figure 4.2.a) shows echoes from the two plates but also undesired reflections (at the corners of the metallic plates for instance) and constructive and destructive interferences due to the superposition of the two echoes from the plates.

This radargram has been processed as described in Section 3.3 with the following exception: no DCoffset removal has been performed, because it would have suppressed the echoes from the first metallic plate (which displays a constant arrival time and amplitude).

The measurement error on distance d is estimated to be around 0.2 cm (measurements were performed with a graduated ruler). In order to reduce the measurement uncertainty some of the soundings were performed with polystyrene foam plates of known thicknesses (instead of air) between the metallic plates (see Figure 4.12.b); this ensures a more accurate positioning of the metallic plates. Polystyrene foam is nearly transparent to radio wavelengths and has a permittivity value close to 1, such a set-up therefore remains close to a "two reflectors" case. In the following, measurements performed with a polystyrene foam support will be highlighted. Comparison between Figure 4.12.a and Figure 4.2.a reveals that simulated and experimental data present the same behaviour except that the second echo is weaker than the first one in experimental data. In particular, we note that the constructive and destructive interferences are very similar in these datasets.

Both original radargrams show that, with conventional processing, it is impossible to confidently separate echoes when the distance between the reflectors is smaller than 11-12 cm. This is consistent with the expected theoretical vertical resolution after windowing (~11 cm in vacuum using equation In Figure 4.13.b, the distance of 10.5 cm is still slightly below the resolution according to the criterion defined in section 2.2. In fact, the presence of two echoes can already be noticed before BWE, but they still interfere strongly. The application of the BWE once again reducing the pulses widths makes the echoes more clearly resolved. The error on the estimated distance between the reflectors is ~0.94cm.

Because of the interference between the two echoes, the difference in amplitude before and after resolution enhancement is not unexpected.

Distance estimation after application of the BWE

A peak detection can easily be applied to the different soundings performed without and with BWE to measure the time delay between the two echoes coming from the metallic plates and therefore the distance between these plates, as performed in Section 4 on synthetic soundings. Figure 4.12 shows the error on the estimated distance as extracted both from the synthetic and the experimental datasets and before and after BWE.

As previously mentioned, for the synthetic data, each sounding has been generated with 1000 cases of On simulated and experimental data, the BWE significantly improves the resolution which is enhanced to 4 cm compared to 11 cm without BWE, i.e. by a factor close to 3. The oscillations in the measured distance error, already noticed on synthetic data (Figure 4.5), are also clearly visible on the experimental data, with and without BWE. As mentioned in section 4, their presence is due the phase difference between the two echoes (Raguso, 2018).

We further note a good match between the errors measured on synthetic and experimental data for distances smaller than 10 cm. For greater distances, while the test on synthetic data predicts a diminution of the oscillations, such diminution is not observed on experimental results and the error is actually slightly smaller without BWE (< 1.55 cm). This difference could come from imperfections in the "whitening" process, which is not simulated in the generated synthetic soundings. However, the maximum absolute error after BWE is ~2 cm which remains below the desired vertical resolution.

As a conclusion, the present experiment demonstrates that an improvement in resolution by a factor of 3 in WISDOM radargrams can be achieved thanks to BWE. The reached resolution is 4 cm in vacuum with an error on the estimated distance < ~2 cm. Both the vertical resolution and error would be even smaller in media with a dielectric constant > 1. From this experiment, unlike on synthetic data in section 4, the error on the amplitude of each echo cannot be drawn. The synthetic data were indeed generated with an amplitude we fixed for each echo while the echo amplitude is not known for experimental data.

The only comment possible here is that for soundings where the two echoes are resolved both before and after BWE, amplitudes are quite close. The differences are likely due to interferences, which are still present for distances near the resolution limit.

The experiment described in this section has also been performed with experimental soundings obtained in an anechoic chamber on a single plate. The soundings have been modified to simulate a delay in time domain and added in order obtain two echoes. The results are very similar to the experimental results

shown in this section.

In the following, the BWE is applied to WISDOM experimental radagrams acquired during field tests performed on different types of environments.

Application of the Bandwidth Extrapolation technique to WISDOM field test data

Between 2010 and 2019, several field campaigns have been organised in order to evaluate WISDOM performances on different environments, some of them potentially analogs to the Martian subsurface.

These field tests have been performed with various models of the instrument, the first ones with prototypes and the most recent ones with models similar to the instrument flight model. The result is a large dataset of radargrams to which BWE can be applied.

We have selected four campaigns, during which structures of interest were detected in the subsurface, and for which the application of the BWE proved to be very helpful for radargram legibility and interpretation. For some campaigns, no free-space or no calibration measurements are available. When no free-space is available, only a DC-offset removal is applied to the signal. When no calibration is available, the surface echo is used for the whitening process (see section 3.3).

One of the experiments performed during this measurement campaign is called the "metallic staircase" experiment. Seven blocks of concrete wrapped in aluminium foils were organized to form a staircase (see Figure 4.15). Regularly spaced WISDOM soundings were performed above the staircase, with a step of 5 cm. The resulting radargram is shown on As expected, the steps can now be clearly separated. The staircase being in the air, the distance can easily be estimated from the time delay. The vertical separation of 10 cm between each step, and their horizontal extension of 20 cm are retrieved. Hyperbolic branches corresponding to diffraction effects at the corners of the steps are revealed (arrow 2), even if, due to the geometry of the metallic staircase, only the left branch of the hyperbolic shapes is visible. These hyperbolic branches appear here because the improvement in resolution reduces the effect of interferences between them, which explains why the hyperbolic branches cannot be seen before BWE.

Being able to detect these hyperbolic shapes is very important because they are commonly used to estimate the averaged permittivity of the medium separating the antennas from the reflecting structures (Daniels, 2005) as explained in Chapter 3. We applied the Hough transform (Capineri et al., 1998) to the region of the deepest hyperbolic shape, and derived a dielectric constant of 1.0±0.1 (0.1 being the precision on the dielectric constant in the Hough accumulator) consistent with a scatter in the air (see

Icy environments: The Chamonix and the Dachstein Giant Ice Cave campaigns

When selecting a field test location for WISDOM, one must favour environments with as little liquid water as possible. Indeed, liquid water tends to increase absorption losses in the subsurface thus causing strong and rapid attenuation of the electromagnetic waves as they propagate. In contrast, water ice is a low-loss material at radar frequencies, and thus a favourable environment for GPR soundings. For these reasons, the two field campaigns presented here were performed in icy environments (Figure 4.19). Even if no evidence of underground water ice at Oxia Planum have been reported so far, and even if the latitude of the landing site is too close to the equator to sustain stable water ice in its shallow subsurface [START_REF] Mellon | Geographic variations in the thermal and diffusive stability of ground ice on Mars[END_REF], remnants of ground water ice [START_REF] Clifford | The stability of ground ice in the equatorial region of Mars[END_REF][START_REF] Forget | Formation of glaciers on Mars by atmospheric precipitation at high obliquity[END_REF] in equatorial regions is a possibility. [START_REF] Wilson | Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data[END_REF] for instance presents observations consistent with the presence of such remnants or, alternatively, hydrated minerals, in other equatorial regions. To obtain an estimation of depth in the radargram from the measured time delays, a dielectric constant of 3.1 (close to the dielectric constant of water ice at GHz frequencies) has been assumed. Close to the surface, in the uppermost subsurface layer, ~10 cm thick isochrone layers of snow can be observed. They likely correspond to recent precipitation (snow fall) events. A snowpack internal layering can be observed around a depth of 1.5 m, likely corresponding to older precipitation events, and possibly to the separation between snow layers and névé layers. The interface, at a depth around 3 m, may indicate the limit between the snowpack and the ice of the glacier. Beneath this last interface, no evidence of structure is visible suggesting that the compacted ice below is homogeneous at spatial scale of the order of a ~5 cm. Such a 3-layer structure of the snowpack is not unusual, and is for instance also visible in radargrams acquired by [START_REF] Gusmeroli | Helicopter-borne radar imaging of snow cover on and around glaciers in Alaska[END_REF] on the snowpack of the Scott Glacier, in Alaska.

The Dachstein Giant Ice Cave campaign

The second WISDOM campaign on an icy environment took place in the Dachstein Giant Ice Cave (Austria), in April 2012 and was part of the Dachstein Mars Simulation campaign organised by the Austrian Space Forum [START_REF] Groemer | Dachstein Mars Simulation 2012[END_REF]. Figure 4.18 displays the radargram acquired in this cave before and after BWE; different buried structures of interest can be identified, such as layering in the ice sheet, the bedrock below and trapped boulders in between. Similar structures will be hunt down in the subsurface of Oxia Planum, on Mars (Vago et al., 2017).

At depths of 0.6 -1.2 m, the ice-sheet exhibits a layered structure (yellow lines in Figure 4.18.c). In a previous study, Dorizon et al. (2016) estimate the thickness of Dachstein ice layers to range from a few cm to 10 cm. The resolution of WISDOM in pure water ice after windowing being ~6 cm, these underground structures are particularly enhanced after application of the BWE and a new layer is even revealed (green arrow in Figure 4.18.b). While segregated ice is not expected in abundance in Oxia Planum, layered structure related to sedimentary deposits may be present in this region which is rich in clay-bearing units [START_REF] Quantin-Nataf | ExoMars at Oxia Planum, Probing the aqueous-related Noachian environments[END_REF].

Hyperbolic patterns corresponding to echoes generated by rocks beneath the ice-sheet can be observed at depths 0.4 -2 m. After application of the BWE, they are more readily located and can be used with more confidence to derive the dielectric constant of their surrounding medium. We recall that, in the frame of the ExoMars mission, hyperbolic patterns indicating the likely presence of buried rocks will be regions to avoid for the safety of the drill.

On the contrary, the underground continuation of surface outcrops will be a favored target for the drill of the Rosalind Franklin rover. In that regard, as another gain of BWE, the echoes from the bedrock beneath the ice-sheet of the Dachstein can be observed in much greater details after application of this technique (blue lines in Figure 4.18.c). They reveal 2 underground rough interfaces that seem to be interrupted at a distance of ~5.5m unless they form a continuous bedrock exhibiting a very steep slope in this area which would make it invisible to WISDOM (Ciarletti et al., 2017). The radargrams from both campaigns in icy environments have shown the value of the BWE technique in terms of resolution and interpretation enhancement. However, as mentioned above, icy environments are very favourable to radio wave propagation and it is important to also demonstrate the ability of the BWE to improve WISDOM scientific return in dry, but more lossy, lithic environments that is, environments likely more analogous to the region that will be explored by the ExoMars rover. In 2010, a prototype model of the WISDOM instrument was brought to a volcanic site in the Etna mountain region (see Figure 4.19), in Sicily (Italy). Some results of this field test are shown in Ciarletti et al. (2017). Such an environment may be a good analog for Oxia Planum, where likely volcanic "dark resistant units" are exposed at the surface [START_REF] Quantin-Nataf | ExoMars at Oxia Planum, Probing the aqueous-related Noachian environments[END_REF]. Though is it not a primary objective of the mission, volcanic terrains will be most likely investigated by WISDOM on Mars. More specifically, 2 highly reflective layers, parallel to the surface, can be observed between 20 and 30 cm of depth (yellow lines in Figure 4.20.c); they probably originate from the same pyroclastic event.

Beneath these two layers, tilted interfaces are detected (in blue). They have been previously interpreted as deposits from an older pyroclastic event (Ciarletti et al., 2017). The material from these two events is superimposed over distances inferior to 4 m.

As a conclusion, the application of the BWE technique to WISDOM radargram proves to be very valuable even on a lithic environment which is promising for future operations Mars.

Summary

• The range resolution of the WISDOM instrument (the vertical resolution of its radargrams)

is inversely proportional to its frequency Bandwidth. The length of the samples to be collected by the rover being of only 3 cm, a thin resolution is necessary for the interpretation of WISDOM radargrams.

• A super-resolution technique based on the "Bandwidth Extrapolation" (BWE) has been implemented into the WISDOM processing chain, allowing an enhancement of the vertical resolution of radargrams by a factor of 3.

• The performances of the technique in terms of resolution enhancement, precision and amplitudes reconstruction were validated on synthetic and experimental WISDOM soundings for 2 reflectors at different distances from each other. Errors on the distance between reflectors ranging from -2.3 to 1.6 cm are experimentally found,

• The technique was applied to experimental acquisitions from 3 natural environments presenting interesting subsurface structures: The Chamonix, Dachstein and Etna campaigns. In all 3 cases, the application of the BWE allowed an improved interpretation of the enhanced radargrams.

• Once on Mars, the improvement in vertical resolution of WISDOM radargrams, and therefore the improved separation of echoes from underground reflectors, will indeed be essential to ensure the drill safety by avoiding buried rocks as well as to investigate the geological history of the ExoMars landing site. Potential improvements to the technique are assessed in the following chapter.

Introduction

Since its invention by (Bowling, 1977) and the first applications to radar data by (Cuomo, 1992) and (Moore et al., 1997), different algorithms have been used to determine the coefficients of the autoregressive (AR) model used for BWE (see Chapter 4). Among them, the Burg algorithm while sensitive to noise and to the phase difference between echoes, has indeed the main advantage of guaranteeing the stability of the returned model, and is therefore the most robust when it comes to modelling experimental signals.

However, the choice of a simple AR model to represent radar spectra has its limitations. First, the white Gaussian noise present in the experimental signal is not considered. As a consequence, the BWE is sensitive to noise and a large required model order (typically a third of the number of samples) is required to obtain a satisfying result. In addition, the effect of the attenuation of the signal in the subsurface with increasing frequency has also been neglected. In fact, the currently used AR model assumes that the amplitudes of the complex sinusoids constituting the received signal in frequency domain are constant over the bandwidth of the radar instrument. As a consequence, the BWE technique is expected to be less efficient in improving the vertical resolution of soundings acquired in high-loss subsurfaces, or deeply buried reflectors.

In order to improve the efficiency of the BWE technique applied to WISDOM soundings, more complex models of the signal's spectrum can be considered. This chapter focuses on two alternative techniques:

The Polarimetric Bandwidth Extrapolation (PBWE) and the State-Space Bandwidth Extrapolation (SSBWE).

The Polarimetric Bandwidth Extrapolation

Theory and application to WISDOM soundings

Because SAR radar imagers often have four polarimetric channels, Suwa and Iwamoto (2007) introduced a multi-channel version of the Burg algorithm designed to take profit of the different channels of polarimetric radars. The principle of this technique consists in extrapolating each channel with a weighted sum of previous/next samples not only using this polarimetric channel, but also the 3 others. This is possible if we consider that a point scatter generates in the four polarimetric channels, four sinusoids of the same period with different complex amplitudes. Hence the multichannel model for the frequency response of K point scatters for time delays τ :

In the forward direction, for + 1 < 7 < 8, * ) is modelled by a linear combination of the previous samples as in Chapter 4, but represented with matrices to account for the 4 polarimetric channels:

* 9 ) : = ; / : 2 . ) ,/ ( 5.4 ) where where

; / F = 0< /,/ F … < /, F 4 2 ( 5.8 ) with < /,= F = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F > = , F ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
( 5.9 ) > = ?@A,B@CDE) F being the coefficients of the backward model.

As for the Burg algorithm in Chapter 4, in order to determine the coefficients of the forward and backward model, the prediction errors between the samples and the model have to be minimised for increasing values of . In the present case, the forward and backward prediction errors are:

For + 1 < 7 < 8

for 7 ∈ [ + 1; 8]
6) The energy of the prediction error for the order p is then computed: ( 5.21 )

R
7) The process is repeated from step 2 until the energy of the prediction error converges (a threshold has to be defined).

Once the multichannel AR model is determined, the signal from each polarimetric channel can be extrapolated as done in simple BWE.

Validation on synthetic WISDOM soundings generated with analytical models

As a first validation step, the PBWE has been applied to synthetic WISDOM soundings, generated with a simple subsurface model, similar to the synthetic soundings generated to validate the BWE technique in Section 4 of Chapter 4.

An independent additive white Gaussian noise corresponding to a SNR = 30 dB is added to each of these soundings, in agreement with what is experimentally measured in WISDOM experimental acquisitions.

The synthetic WISDOM acquisitions generated with this model can be used to test the efficiency of the PBWE, which is expected to perform better than the BWE provided that at least one of the reflectors shows different scattering properties depending on the polarization channel considered.

As a preliminary example, new synthetic soundings have been generated for the two co-polarization channels (00 and 11), with two reflectors having different polarimetric properties. The first reflector is a plane reflector, with a scattering vector U 1 0 0 1 W, and the second reflector behaves like a dihedral corner reflector, with a scattering vector equal to U 1 0 0 -1 W.

Therefore, the relative phase difference between the 00 and 11 channels for the second echo is 180°.

Soundings were generated for distances d between the two reflectors ranging from 0 to 15 cm, with a step of 0.25 cm. Modelling the signal with data from the two polarimetric channels, and extrapolating with samples from the two channels clearly reduces the effect of the interference between the echoes on the spectrum, which has been shown to be amplified by the presence of noise for the BWE (see Section 4.5 of Chapter 4).

For d > 4 cm, the error on the estimation of the distance between the reflectors becomes negligible with the PBWE. the mean error on the amplitude ratio is always inferior to 1.5%, to be compared with 5% observed for the BWE. On the amplitude of each echo, the maximum mean error on the amplitude is 2%. Considering a surface of dielectric constant 4, an error of 2% on the evaluation of the surface echo would translate into an error of 3% on the estimation of the dielectric constant, and thus an error of 2.5 cm on the estimation of the depth of a reflector at 1 m in the subsurface.

Other values of the phase angle in the scattering coefficient of the second echo for channel 11 have been tested, and the performances both in terms of the error on the distance d between the reflectors and the amplitude of the echoes measured. As for the previous test, 20 cases of noise have been tested, for the sake of computation time. When the phase angle of the scattering coefficient of the second echo is equal to 0° for channel 11, and all scattering coefficients for the two echoes being thus equal, the average performances of the PBWE for the estimation of d and amplitudes become nearly the same as with the BWE. This result confirms that a difference in the scattering coefficients in the different polarimetric channels for at least one echo is required for the PBWE to perform better than the BWE.

As mentioned, the PBWE being much longer to run than the BWE, a full test on how the noise level may affect the technique as performed in Chapter 4 was impossible. For this reason, the results shown in Figure 5.6 are based on a series of 20 synthetic soundings that have been generated in the same conditions but for lower SNR values of 20, 10 and 3 dB. The robustness of the PBWE when confronted to low SNR levels can be appreciated in Figure 5.6. Even for a level as low as 3 dB, the average error on the estimation of d remains below 0.5 cm, and the error on the amplitude inferior to 10 % for all tested distances between the reflectors. The standard deviation significantly increases for both the estimation of d and of the amplitudes when the SNR decreases, but remains below 0.5 cm for the error on d, and below 10% for amplitudes measured at X > 5 cm.

The PBWE can therefore be expected to perform better than the BWE even when the SNR of the signal is low, provided that at least of the reflectors to be separated has different scattering coefficients for the considered polarimetric channels.

Validation on synthetic WISDOM soundings generated with FDTD simulations

The PBWE technique was also validated on FDTD (Temsi-FD) simulated radargrams, for two co-polar configurations: 00 and 11. FDTD simulations include a fine modelling of the WISDOM antennas which is of prior importance to validate the PBWE, because as previously mentioned, depending on the position of reflectors relative to the antennas, some echoes can be present in one co-polar channel and not in the other. Improvements between the BWE and the PBWE are expected in cases where reflectors are visible in both polarimetric channels. Freespace and calibration measurements have also been simulated to remove the direct coupling, and to perform a whitening of the signals (see Section 3.5.2 of Chapter 2). These two operations are necessary, as models of the WISDOM antennas are used in FDTD simulations.

The results obtained for the 00 channel, with the original WISDOM bandwidth, after BWE, and after PBWE are presented in Figure 5.8. The results obtained for the 11 channel are presented in Figure 5.9. As expected, it can be noted when comparing the radargrams obtained with the original WISDOM bandwidth for the 00 and 11 channels that the two are not identical at all. For many of the many of the measured echoes, the scattering coefficient are thus different for the 00 and the 11 channels.

Perspectives of application to experimental WISDOM radargrams

Lastly, a preliminary validation of the PBWE technique has been performed on modified experimental WISDOM soundings. In the TU Dresden anechoic chamber, soundings were performed with a metallic plate at different distances in front of the WISDOM antennas. By adding 2 frequency spectra from this dataset, a sounding with two echoes at a distance corresponding to the difference of distance between the metallic plate and the antennas for the 2 spectra is obtained. This process, inspired by an experiment performed by Le [START_REF] Bastard | Apport des techniques de traitement du signal super et haute résolution à l'amélioration des performances du radar-chaussée[END_REF], has been applied to WISDOM soundings for different distances of the metallic plates in order to generate a radargram with two echoes at increasing distances from each other, between 0 and 30 cm (see Figure 5.10 for the experimental set-up and a schematic summarizing the process). The classic BWE technique was already applied to such a radargram. However, as the reflectors in this experimental set-up are metallic plates perpendicular to the WISDOM antennas, and thus having similar scattering coefficients for the 00 and 11 channels, no improvements can be expected from the PBWE with respect to the BWE.

For this reason, two sets of soundings have been generated: one by the addition of couples of soundings, and the other by their subtraction. The first one would be equivalent to the 00 channel, and the second on to the 11 channel for soundings acquired with a reflector having a scattering vector U 1 0 0 1 W, at different distances from a reflector of scattering vector U 1 0 0 -1 W. A situation similar to the validation on synthetic data proposed in Section 2.2. The PBWE technique has been applied to these two generated "polarimetric channels". As this set of WISDOM soundings has been modified, this experiment is not expected to be fully representative of the application of the PBWE to actual experimental WISDOM data. However, the validation of the technique on such a dataset is of interest, as both echoes added or subtracted are experimental WISDOM echoes, to which the different steps of the processing chain have been applied

(temperature correction, free-space removal, Hilbert transform, signal whitening, see Section 3.2 of Chapter 4), and with a level of noise representative of the instrument integrated to the ExoMars 2022 rover (typically 30 dB).

Figure 5.11 displays the radargrams obtained after the application of both the BWE and PBWE techniques to this dataset. The improvement with the PBWE can visually be appreciated. As expected from the application on synthetic WISDOM soundings in the same configuration (a plane and a corner dihedral reflectors at different distances from each other, see Section 2.2), the time delays of the two echoes seem to be better reconstructed with the PBWE, in particular they show much less oscillations of their position compared to what is obtained with the classic BWE technique. However, it can also be noted that it seems difficult for the PBWE technique to accurately extrapolate the signal for the two soundings corresponding to the smallest distance between the reflectors. Another point to be mentioned is that to obtain such results 10% of frequencies had to be removed on each side of the spectrum instead of 5% when applying the PBWE. For only 5%, the extrapolation diverged for some of the soundings in the radargram with the PBWE. The technique seems thus to be more sensitive than the BWE to inaccuracies in the values on the side of the spectrum, which are mainly due to inaccuracies in the gain determined for the whitening process. • It has been shown on analytically generated synthetic WISDOM soundings that the PBWE performs better than the BWE in terms of reconstruction of the distance between reflectors, or amplitudes of their echoes, as long as the echoes have different scattering properties in the different polarimetric channels.

• In free-space (the worst-case for the resolution), the predicted errors on the distance between two echoes are below 0.5 cm in average down to a low SNR level of 3 dB. For distances superior to 5 cm between the echoes, the error on the amplitudes remains below 10% for the same SNR.

• Qualitative comparisons between the BWE and PBWE results were also done with a couple of FDTD generated radargrams for two co-polar channels, and modified experimental WISDOM soundings with 2 reflectors in free-space. On these radargrams, the PBWE also performed better than the BWE, better separating diffraction curves in the first case, and better reconstructing the relative position of echoes in the second.

• After application of the PBWE, WISDOM radargrams are therefore expected to be very accurately reconstructed in amplitudes and time delays, allowing an improved geologic interpretation. If these promising results are confirmed on experimental radargrams in a controlled environment, the technique could be confidently applied to WISDOM acquisitions in natural environments as in Chapter 4 and later on to future Martian acquisitions.

The State-Space Bandwidth Extrapolation

Unlike Section 2, no polarimetric data will be considered in this section, and the notations for the samples to be extrapolated are the same as in Chapter 4, dedicated to simple BWE. The technique presented in this section is indeed based on an improved signal model for only one series of 8 samples ( ) = ( ) ).

Theory and application to WISDOM soundings

In addition to the effect of noise, the model employed in the classical BWE technique has another drawback: it considers the signal as a sum of sinusoids of constant amplitude, each corresponding to a point scatter. As the WISDOM bandwidth is of the same order of magnitude as its central frequency, we expect the amplitudes of the sinusoids to vary with frequency. In particular, high frequencies should be damped more rapidly than low ones with time/depth, as they are preferentially absorbed by the subsurface even if homogeneous (see the definition of the penetration depth in Section 2.3.1 of Chapter 1).

To account for these two effects in the model used to extrapolate the bandwidth of a radar, Piou et al. (1999) proposed to use an autoregressive moving-average (ARMA) model instead of the AR model used in the classical BWE. The ARMA model proved its efficiency on a variety of experimental radar data:

displaying a decay of amplitude with increasing frequency, or obtained from reflectors with complex shapes (Naishadham andPiou, 2008, 2005), or even data used for the estimation of cardio-pulmonary parameters with an ultra-wideband radar [START_REF] Naishadham | Estimation of Cardiopulmonary Parameters from Ultra Wideband Radar Measurements Using the State Space Method[END_REF]. These are the reasons why we decided to adapt this ARMA model for the BWE technique to WISDOM data, and to validate it on both synthetic and experimental soundings.

In the frequency domain, a point scatter inside a non-dispersive (all wavelength having the same propagation speed) sounded volume would generate a sinusoidal signal and the spectrum S resulting from [ scatterers is expected to be a sum of damped sinusoids corrupted by an additive white noise: 5.22 ) where is the frequency, $ the time delay associated with the k-th scatterer and the radar, \ the amplitude coefficient of the return signal from the k-th scatterer (the attenuation, which is frequencydependent is not accounted for in this parameter), ] the decay factor associated with the k-th scatterer (for a homogeneous subsurface, ] would be the same for all scatterers), ^ the equivalent distance in the subsurface associated with the k-th scatterer, and _ a white Gaussian noise. As previously mentioned, a WISDOM sounding consists in a set of spectrum values collected for N discrete frequencies ( = ( ) where is given by equation (4.1).

( ) = \ (-! 2 # $ -] ^ ) + _( ) % & (
The ARMA model proposed by Piou et al. (1999) to simulate the signal, with p and q the orders of model, is the following one:

For max ( , c) + 1 < 7 < 8 5.23 ) where () is the n-th sample of the modelled frequency spectrum e = and f the i-th coefficient of the AR part of the model and the j-th coefficient of the MA part of the model, respectively.

() = e = ( ) = / =& + f _(7 -!) + f _(7) g & (
To determine the coefficients of the model, a State-Space representation of the model is used. This representation of linear systems has been introduced by [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], and can be used to control or extrapolate a linear system. For this reason, the technique studied in this section is called "State-Space BWE" or SSBWE. In the case of our ARMA model the two State-Space equations are: h (7 + 1) = i (7) + K _(7) j(7) = < (7) + _(7) ( 5.24 ) where x is the "state" vector (K × 1), i the "state" matrix (K × K), K the "input" matrix (K × 1), < the "output" matrix (1 × K), and j(7) = ().

With this representation, any sample of the signal can therefore be estimated as:

() = j(7) = < i ) K ( 5.25 )
1) The three matrices i, K and < need to be determined, this is done with the following procedure.

The vector y is filled with the different samples in a WISDOM spectrum:

j = [( ( T … ( H ] ( 5.26 )
2) The Hankel H matrix is calculated with vector y, m = n j(1) ⋯ j(p) ⋮ ⋱ ⋮ j(8 -p + 1) ⋯ j(8) s

( 5.27 ) where p =

3) This matrix H is then decomposed in signal and noise subspaces with a singular value decomposition as follows: 5.28 ) where indices s and n correspond to signal and noise subspaces, respectively.

m = [v w v ) ] x ∑ w 0 0 ∑ ) z { | w * | ) * ~ (
4) To better model the signal, only the Hankel matrix in the signal subspace is considered: 5.29 ) Different criteria will be discussed to separate the signal subspace from the noise in this section. ( 5.32 ) where ; • H is a new controllability matrix defined with N columns.

m • = v w ∑ w | w * (
8) Eventually, the matrices A, B and C being determined, the data vector y can be extrapolated in the forward direction:

For n > N, j : (7) = <i HI) K ( 5.33 ) And in the backward direction, by flipping the data vector y for the sake of simplicity:

For n > N, j F (7) = < ? i ? HI) K ? ( 5.34 ) where i ? , K ? and < ? are determined with the flipped data vector.

It is interesting to note that the estimation of the amplitude coefficient \ , the time-delay $ and the attenuation ] associated with each echo can be obtained from the eigenvalues and eigenvectors of the matrix A for 1<k<K: 5.35 ) where Ž and ‹ are respectively the eigenvalues and eigenvectors associated with the k-th point scatter, ‹OE7• the inverse of the eigenvector matrix, and ' the phase of Ž .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧\ = (< ‹ )(‹OE7• K) Ž • /•• $ = -' 2#' ] ^ = -'" (|Ž |) ' (
The whole process described above has been implemented into the WISDOM processing pipeline we developed with the Bandwidth Extrapolation technique, and has been tested both on synthetic and experimental data.

However, as for the order of the AR model in the classical BWE technique, the number of point scatters in a WISDOM spectrum K has to be estimated with some criterion to generate the State-space model.

This number is required to disentangle signal and noise subspaces when decomposing the Hankel matrix H into singular values. Different criteria can be used to achieve this estimation. • The approximation error Sm -m • S (‖. ‖ being the spectral norm here) can be calculated for any number of elements possible in m • (between 1 and K). It should decrease with an increasing number of elements, but the drop is expected to slow down after a critical number of elements marking what is called the "breaking point" (see an example in Figure 5.12). The idea is thus to calculate the difference between two consecutive values of Sm -m • S, and to detect the breaking point based on a threshold on the difference: when the difference is lower than the threshold, the number of elements in m • is considered as the number of elements in the signal subspace.

However, the main problem of this technique is that a threshold has to be defined.

Another solution to determine the number of elements to be considered as signal in H is to use an information-based criterion, three of them have been considered and compared:

• The most famous being Akaike's Information Criterion (AIC), defined by [START_REF] Akaike | A New Look at the Statistical Model Identification[END_REF]. It aims at finding a trade-off between the likelihood and the simplicity of a model. AIC is calculated for every number of elements between 1 and K, and the number of elements k that minimizes AIC should provide a good estimate of the number of elements in the signal subspace. [START_REF] Cheng | Independent component analysis based source number estimation and its comparison for mechanical systems[END_REF] proposes an adaptation of the AIC formula to the separation of sources in a signal:

i-<(˜) = -28(p -˜)'" (pOE˜ ) + 2˜(2p -˜) ( 5.36 ) where pOE˜ =

∏ š › oe• oe ›ž Ÿ oe• ∑ š › oe ›ž Ÿ
is named the maximum of the "likelihood function" determined by [START_REF] Cheng | Independent component analysis based source number estimation and its comparison for mechanical systems[END_REF] for this application, the Ž = values being the eigenvalues of x ∑ w 0 0 ∑ ) z. The first part of the sum in equation ( 5.36) corresponds to the likelihood of the model, and the second part is a penalty to avoid a too complex model.

• Other information-based criteria exist, such as the Minimum Description Length (MDL), introduced by [START_REF] Rissanen | Modelling by the shortest data description[END_REF]. It is based on the minimum length required to encode the data.

Adapted to our problem, the formula to calculate the MDL is as follows:

Lp(˜) = -8(p -˜)'" (pOE˜ ) + 0.5˜(2p -˜)'" (8) ( 5.37 )

• Lastly, the Bayesian Information Criterion (BIC), introduced by [START_REF] Schwarz | Estimating the Dimension of a Model[END_REF], relies on the Bayesian statistics to select the order of the model. Adapted to the present problem, the formula to calculate the BIC is:

K-<(˜) = -28(p -˜)'" (pOE˜ ) + ˜(2p -˜)'" (8) ( 5.38 ) To select the criterion to be used in order to determine the appropriate State-space model to extrapolate WISDOM spectra, we tested all aforementioned solutions on synthetic data prior to the implementation of the technique. More specifically, synthetic soundings were generated with numbers of scatters from 1 to 60, at random depths down to 3 m, a typical dielectric constant of 4, random amplitudes on a 60 dB range, and a SNR = 30 or 10 dB. For each number of scatters, 1000 soundings were generated in a Monte-Carlo approach, and the order estimate was calculated with every criterion we presented. The results are presented in Figure 5.13, both in terms of estimated number of scatter points and estimation error.

It must be noted that for the application on synthetic WISDOM soundings, the BIC and MDL criteria always return the same value in all the tests we performed. For this reason they are displayed with the same color in Figure 5.13. Then, we observe that the results returned by the AIC and the BIC-MDL criteria are very similar, both in terms of mean values and standard deviation. The error on the estimations by both criteria is relatively low when the number of echoes remains low (~30 for SNR = 30 dB and ~20 for SNR = 30), and for higher numbers of scatterers, the criteria underestimate the number of echoes and the absolute error rises rapidly. The estimation from the AIC is observed to be always higher than the estimation from the BIC, but the difference between the two stays below 1 in average for the two tested configurations. None of these two criteria thus performs significantly better than the other in all situations. As a consequence, unless mentioned otherwise the SSBWE is used with the AIC to estimate the number of scatters. But on the long term, another efficient mean of estimating the number of echoes in a WISDOM sounding should be found in order to use other criteria. This constitutes for now the main limitation in the application of this technique (see Section 3.4 for more details). As previously mentioned, the determination of the SSBWE model provides a characterization of the different point scatters detected in the soundings, in terms of amplitude coefficient, time delay and frequency-domain decay/growth. In this very simple example, as shown in Figure 5.15, two echoes of the same amplitude are expected to be detected, with a linearly increasing time delay between them, and no decay nor growth in frequency-domain for both of them (since ¡" = 0).

Figure 5.15 shows that, as expected, in most of the soundings, the SSBWE, as expected, has detected the two echoes and found similar amplitude, and no decay/growth in frequency-domain for both of them.

The two scatters are well detected down to nearly 1.5 cm of distance between them, to be compared with the expected ~11 cm of resolution in free-space. It must be noted that 3 false detections are visible, corresponding to soundings where the AIC overestimated the number of echoes by 1. The distance d between the two reflectors can be estimated from the time-delay between the two corresponding echoes in time-domain. This estimation X £ can then be compared to the known distance between the reflectors, as shown in Chapter 4.

As expected from the radargrams, and as presented in Figure 5.16, after enhancement by SSBWE, the error oscillates between values much lower than after BWE. For a distance d superior to 4 cm between the reflectors, the error remains between -0.1 and 0.1 cm with the SSBWE, to be compared with an error between -0.4 and 1.7 cm with the BWE. Such an excellent result is obtained thanks to the use of an ARMA-model instead of an AR-model, which takes into account white noise in its modelling.

Figure 5.17.a compares the values of the amplitude coefficients retrieved with the BWE and the SSBWE, the two coefficients being set to 1 when generating the synthetic soundings. The amplitude ratio between the two echoes is even closer to 1 after SSBWE than it was after BWE, with errors of less than 1%. The amplitude coefficient of each echo measured independently, shown in Figure 5.17.b, is also very close to the expected values of 1, with errors inferior to 1% even for distances d < 4 cm. Mars, the robustness to noise of the SSBWE has also been tested by the generation of the same "two reflectors" synthetic soundings for SNR lower than 30 dB. For the sake of computation time, only 100 cases of noise have been tested for each SNR level between 1 and 40 dB, with a step of 1. As for the classic BWE, the error on the amplitude coefficient decreases when the SNR rises, but with lower values of errors and a stronger decrease of the error with the SNR. It can be observed that for both echoes the mean error on the amplitude estimation remains below 2% for a SNR > 10 dB for all distances d > 3.75 cm, and below 5% for SNR > 5 dB. Equally, the standard deviation diminishes rapidly with the SNR, the standard deviation of this error remaining below 10% for SNR > 10 dB.

Synthetic WISDOM soundings with point scatters and absorption losses

In order to add the effect of absorption losses in the subsurface when generating synthetic soundings, the signal model has been modified to account for the attenuation coefficient α in the medium at a given frequency (see Section 1.2 of Chapter 1).

The spectrum S resulting from [ scatterers corrupted with white noise is described in equation (5.22), and ¤( ) the attenuation constant defined as

¤( ) = 2 # ¥¦ ¡′ ¨© Iª «¬¬ «¬ - ® T = ]
in the approximation of a non-magnetic medium, and with values of ¡′ and ¡′′ constants in the bandwidth of the instrument (non-dispersive medium). The amplitude of the echoes is given by \ exp (-] ^ ), the amplitude coefficient \ being a constant.

As a first example, a synthetic radargram with point scatters has been generated using the analytical model of equation (5.22). A first point scatter, corresponding to the surface echo, and couples of point scatters with echoes of the same amplitude coefficient, at depths between 0.5 and 2.5 m, with a step of 0.5 m. In each couple of reflectors, the distance between the two buried reflectors is set between 0 and 15 cm, with a step of 0.25 cm. The relative permittivity of the medium is ¡ ? = 4 + 0.03!, a plausible value for dry sandstones (Ciarletti et al., 2017). With such a permittivity, the electric skin depth would be ~1.82 m at the WISDOM central frequency. The transmission/reflexion coefficients between layers are not accounted for in these synthetic data.

The values of the amplitude coefficients \ of the associated complex sinusoids are all set to 1 to ease the comparison.

The radargram of Figure 5.20 gives an insight into the ability of the SSBWE to separate echoes despite absorption losses. It can be noted that the technique is able to characterize echoes, retrieving the fact that they all have the same amplitude coefficients \ . The SSBWE also finds that echoes have increasing decay parameters ] , the surface echo having no decay in frequency-domain. With the increasing depth, the amplitude of the echoes diminishes even though they all have amplitude coefficients \ = 1, as the high frequencies are attenuated with depth. A few false detections of echoes can be noted, as the AIC was used to estimate the number of point scatters.

The imaginary part of the subsurface permittivity can be inferred from the attenuation coefficient estimated by SSBWE modelling for a point scatter k, with the following: ( 5.39 ) with ] calculated as in equation (5.35). To further assert the capacity of the SSBWE to accurately estimate the imaginary part of the permittivity of a material, a synthetic radargram has been generated

¡ ³³ = ¡ ³ ¨´1 + ] T 8 # T ¦ ¡′ ¶ T - 1
with only one couple of reflectors below the surface in a subsurface of permittivity 4+0.03j. The depth of the first reflector is set to 0.5, 1.5 or 2.5 m, plausible values considering the expected penetration depths for WISDOM. The distance d between the two subsurface reflectors remains between 0 and 15 cm with a step of 0.25 cm.

In a Monte-Carlo approach, the imaginary part of the dielectric constant is estimated for each reflector and each distance between, for a 1000 cases of white Gaussian noise (SNR = 30 dB). The mean value of the estimated imaginary part of the dielectric constant and its standard deviation are displayed for It can be noted from Figure 5.21 that the level of error is very low. For a depth of 1.5 m, estimations of the subsurface imaginary part of the permittivity between 0.028 and 0.032 according to the standard deviation in every case. This corresponds to a maximum relative error below 7% on the estimation of the loss tangent. After d~5 cm, this error becomes inferior to 1%.

In the tested scenario, the imaginary part of the permittivity is relatively high. To be sure that the SSBWE modelling would also be able to accurately estimate the loss tangent of a low-loss material, a radargram for a second scenario has been generated. The permittivity of the subsurface was set to 3+0.0015j, consistent with unconsolidated sand (Ciarletti et al., 2017). Figure 5.22 displays the results of this experiment: the level of error is higher, but only slightly for distances d higher than 5 cm, with a relative error on the loss-tangent inferior to 10 % for a depth of 1.5 m. A low imaginary part of the permittivity thus leads to higher estimation errors, but the level of error is still very satisfactory.

It can be noted that the standard deviation of the estimated imaginary part of the permittivity increases with the depth of the reflector in both scenarios. At 0.5 m, it stays below 2% for d > 5 cm in the first scenario, while it can be up to 30% in the second. The performances of the dielectric losses estimation are thus expected to improve with the depth of the reflector and the imaginary part of the permittivity.

These are of course very simple scenarios, and validations on more complex ones are the next logical steps before applying this process to experimental data. In a homogeneous subsurface of dielectric constant 3, the expected resolution of WISDOM with classical processing techniques is ~6.3 cm (see Chapter 4), hence the selected thicknesses of the first layer. After BWE by a factor of 3, the expected resolution would therefore be ~2.1 cm.

Freespace and calibration measurements have also been simulated to remove the direct coupling, and to perform a whitening of the signals (see Section 3.5.2 of Chapter 2). This last processing is very important here, as numerical models of the WISDOM antennas are used in FDTD simulations that have a non-negligible frequency dependency. The described subsurface model is presented in a schematic in Figure 5.23. Further, the dielectric constant of the first layer can be estimated from the calibrated amplitude of the first echo (the surface echo in this scenario). Figure 5.26 shows, when the first echo is clearly separated from the second one, the value of the dielectric constant derived from amplitudes measured on the original radargram (see Section 2 of Chapter 3), the radargram after BWE and after SSBWE.

After SSBWE, the results are even better than after BWE: for thicknesses superior to 3.5 cm, the derived value of the dielectric constant stays between 2.9 and 3.1 (against 2.5 to 3 with the BWE), very close to the input value of 3. It can also be noted that when the echoes can be separated (X > 7 >‹), the estimation of the dielectric constant from the original radargram is, as expected for homogeneous layers, excellent.

Two heterogeneous subsurface layers model

The first application of the SSBWE technique to radargrams generated by FDTD proved its ability to enhance the spatial resolution, retrieve the thickness of the first layer, and estimate its dielectric constant.

However, in the scenario tested so far both subsurface layers were homogeneous. To further test the technique on more realistic subsurface layers, fractal heterogeneities have been introduced with the "square-diamond" method (see Section 5.2 of Chapter 2).

As illustrated by Figure 5.27, the thickness of the first layer is now set to 5 cm, a value inferior to the theoretical resolution of WISDOM. Its dielectric constant is set to 3 with variations between 2.5 and 3.5 due to heterogeneities having a characteristic size L = 2.5 cm. The dielectric constant of the second layer is set to 6 with variations b between 5.5 and 6.5, due to heterogeneities of a characteristic length of L = 16.6 cm.

Such a scenario could be representative of a layer of sand with heterogeneities smaller than the central wavelength of the instrument, lying over a denser material with heterogeneities larger than the median wavelength. The modelled environment consists in a 2.5 m soil trench with margins, where soundings were performed along a linear traverse, every 5 cm. Lastly, the generated soundings are corrupted by an additive white Gaussian noise to obtain a SNR = 30 dB. The heterogeneities in the dielectric constant changing the speed of light in a medium, an estimation of the distance between the interfaces is not straightforward, neither is the interpretation of the amplitude of the echoes. Nevertheless, the generated radargram can be used to validate the ability of the SSBWE to separate echoes in the presence of heterogeneities. guess the presence of two interfaces. After the application of the BWE to these soundings, the presence of two echoes is much clearer, even if the result is variable depending on the soundings (due to the heterogeneities and the white noise added). There is again a clear improvement when applying the SSBWE instead of the BWE: in most of the soundings, the two echoes appear better separated, with a few exceptions probably due to a wrong estimation of the number of scatter points.

The results are also very satisfactory regarding the estimation of the dielectric constant of the first layer from the surface echo amplitude despite the presence of echoes from the second layer and from embedded heterogeneities. In Figure 5.29 these results are obtained using the entire radargram and deriving the dielectric constant from the surface echo for 1000 cases of noise (Monte-Carlo approach).

The dielectric constant is overestimated using the original radargram, due to interferences between echoes coming from the surface and the buried interface. After BWE, the estimation is closer to the expected dielectric constant of 3, with values between 2.5 and 2.8. And after SSBWE, the mean estimation of the dielectric constant is very close to 3, between 2.8 and 3.2 in most of the cases. For some soundings, the standard deviation can be as high as 0.4, probably because in some cases of noise the estimation of the number of scatter points in the radargram was wrong.

Perspectives of application to experimental WISDOM radargrams

To this day, the SSBWE could not be applied successfully to experimental WISDOM radargrams, due to errors in the order chosen for the ARMA model. Unlike the BWE, the SSBWE technique does not guarantee the stability of the returned model, no matter the chosen order. Additionally, unlike the BWE, the SSBWE does take into account the effect of noise in the model used to extrapolate the signal. Without noise, the order of the model for the BWE and the SSBWE should be directly proportional to the number of complex sine waves in frequencydomain soundings, and thus to the number of echoes in time-domain. When the signal is corrupted by white noise, it has been shown in Chapter 4 and previous studies (Cuomo, 1992;Moore et al., 1997;Raguso, 2018) that an order of a third of the total number of samples is the best compromise for the BWE model. This is not the case for the SSBWE, for which the presence of noise is accounted for, and the choice of order for the model still proportional to the number of echoes in the sounding. As previously explained a criterion has been selected to determine the order of the model for the SSBWE. The SSBWE not guarantying the stability of the model, and the estimation of the order of the signal model being necessary for each sounding, errors in the determination of this order may lead to unstable models and thus diverging extrapolations. This is particularly true as the number of echoes in soundings increases.

In • On analytically generated synthetic WISDOM soundings with two echoes from reflectors in free-space (worst-case for the resolution), the SSBWE has been shown to perform better than the BWE, even for very low SNR values.

• The SSBWE improves the estimation of the distance d between reflectors, with average errors below 0.3 cm for SNR > 8 dB, and below 1 cm for all tested SNR levels. It also improves the estimation of the echoes amplitudes, with average errors below 5% for SNR > 5 dB.

• The SSBWE technique also allows the retrieval of the detected echoes properties: time delays, amplitudes, and decays in frequency domain. An estimation of the absorption losses is thus possible, and was also tested on synthetic WISDOM soundings.

• The validation of the SSBWE continued with FDTD simulations, with subsurface models divided in two homogeneous or heterogeneous subsurface layers. The performances of the SSBWE on these more realistic radargrams were also better than with the BWE, with errors on the distances and amplitudes negligible as long as the layer was thicker than 4 cm.

• The only issue with this technique is the determination of the order for the signal model, as the stability of the model is not guaranteed for a wrong selected order. Before being able to apply the technique to experimental WISDOM radargrams, an efficient criterion to estimate the order providing the best model for the will have to be found. Considering the The challenging requirements, specifically in terms of depth penetration, accuracy and vertical resolution imposed to the WISDOM GPR by the objectives of the ExoMars rover mission motivated the studies and developments that are presented in this manuscript, together with the obtained results.

Conclusion and perspectives

WISDOM will be key in the selection of sample collection sites, as the Rosalind Franklin rover will be the first Martian robot able to collect samples down to 2 m in the subsurface of Mars with a drill. The goal of the WISDOM team will be to provide accurate information on the subsurface structure and clues on its composition prior to drilling operations. This way, it will help to select a promising and safe drilling site, and bring insights into the geological context of the collected samples. To achieve this task, a quantitative analysis of the WISDOM data is needed that requires efficient tools specifically designed to process as automatically as possible and interpret the future WISDOM Martian data. This was the subject of this thesis carried out in close interaction with the WISDOM instrument team at LATMOS (PI's institute) and TUD (Co-PI's institute in Germany), and under CNES funding. The new WISDOM data processing and interpretation tools developed during this thesis (even some previously developed but not fully validated) have been systematically tested and validated on simulated data of gradually increasing complexity (the most elaborate one have been obtained with a numerical FDTD code able to model the WISDOM antennas, the roughness of the interfaces and the heterogeneities inside the subsurface layers). After these first series of tests, the successful approaches have been further validated on experimental data acquired in laboratory and controlled environment, and eventually (when possible) used on experimental data acquired in natural environments. As a result, some new features to the automatic base-line processing chain of the WISDOM data have been introduced, such as the signal whitening step, that proved to be essential for a correct quantitative interpretation of the data. Some processing tools previously developed, as for instance the Hilbert transform process, have also been carefully validated with new experimental acquisitions in the frame of this thesis.

In the context of the ExoMars 2022 mission, decisions on the drilling site locations will have to be taken rapidly, while assuring the safety of the drill and the relevance of the considered site. In order to evaluate precisely the depth of a subsurface target, and to provide constraints on the subsurface composition and porosity prior to drilling, an assessment of the subsurface dielectric properties is required. In the manuscript, we presented the different techniques developed or improved to automatically estimate the dielectric properties of the subsurface volume sounded by WISDOM.

• The estimation of the surface permittivity value from the amplitude of the surface echo (Section 2 of Chapter 3) is a technique that had been already applied to WISDOM radargrams and validated during the thesis of Sophie Dorizon (2016) and Yann Hervé (2018). It is now significantly improved as the new version of the method is accounting for the variations of distance between the surface and the WISDOM antennas due to surface large-scale roughness or topography. It has been successfully

applied to the TU Dresden campaign data. • An algorithm has been developed to automatically detect and characterize the diffraction curves in WISDOM radargrams, that are signatures of small-size subsurface reflectors such as buried rocks.

(Section 3 of Chapter 3). These diffraction curves are of great interest because they allow the detection of rocks that are potentially hazardous for the rover's drill and, also because their shape is directly related to the dielectric constant of the material in which they are embedded. The new detection and characterization technique proposed during this PhD relies on the Hough transform, a famous image processing technique used to detect specific shapes in images. This technique has been modified to account for the refraction of WISDOM signals at the surface. As a result, rocks are automatically detected in radargrams and the average permittivity in the subsurface is estimated (a maximum error of 0.4 (10%) was found on a simulated radargram for a homogeneous subsurface of dielectric constant 4) along the propagation path between the antennas and the scatterer, which can be readily used to retrieve the distances (at a depth of 1.5 m, an error on the distance of 6 cm was found for a simulated homogeneous subsurface of dielectric constant 4).

• Eventually, the application of the two aforementioned techniques to the ExoFit campaign radargrams was proposed at the end of Chapter 3, with a comparison to independent information provided by samples analysis and pictures taken on-site.

In the design of any radar sounder, a compromise has always to be found between the desired range resolution and penetration depth in the sounded subsurface. The range resolution of a sounding radar in a given subsurface is limited by its frequency bandwidth. In order to overcome this limitation and improve the resolution in distance of WISDOM radargrams, the "Bandwidth Extrapolation" technique (BWE) was successfully adapted to WISDOM data and implemented in the processing and interpretation chains of the future Martian acquisitions. The resolution is indeed key in the detection of thin layers in the WISDOM radargrams, as the 3-cm long sample collection must be guided as precisely as possible to the right place in the subsurface. By design the instrument has already a high-resolution thanks to its 2.5 GHz of frequency bandwidth but does not meet the mission requirement of a resolution about 3 cm. Chapter 4 was dedicated to the validation of the BWE on both synthetic and experimental WISDOM soundings, the effects of the technique on the range resolution, range precision and amplitudes of echoes quantified. The technique was then applied to radargrams acquired on natural environments, Martian analogs in terms of structure and/or composition, and the improvement in their interpretation detailed. • Largeur de bande de fréquences, modulation et résolution spatiale :

La largeur de bande de fréquences correspond à la largeur du spectre fréquentiel du signal émis.

Elle contrôle la résolution spatiale de l'instrument, qui est proportionnelle à .

Plus la bande est large, et plus fine est la résolution. Cependant, à énergie égale, un signal plus court (donc plus large bande) devra être émis avec plus de puissance, alors que cette dernière est limitée dans le contexte de missions spatiales. Pour résoudre ce dilemme, différentes techniques de modulation du signal émis (en fréquence ou en amplitude) peuvent être mises en place, afin de réduire la puissance émise tout en garantissant une bonne résolution.

WISDOM, le radar à pénétration de sol de la mission ExoMars 2022

Les objectifs de WISDOM dans le contexte d'ExoMars 2022 : Le rover Rosalind Franklin de la mission ExoMars 2022 (voir Figure 1.a) a pour objectif principal de rechercher des traces de vie passée ou présente dans le proche sous-sol martien (Vago et al., 2017). Ce La chaîne de traitement des données WISDOM : La chaîne de traitement des données WISDOM compte comme étapes principales :

• La correction de l'effet de la température :

La température de la carte électronique RF du WEU influe sur l'amplitude des signaux mesurés dans un sondage. Un correctif déterminé expérimentalement doit être appliqué à chaque sondage en fonction de la température mesurée pour compenser cet effet.

• La transformée de Hilbert : Afin de respecter les budgets de masse et de volume imposés par la mission en économisant une chaîne de réception RF, seule la partie réelle du signal reçu est mesurée par WISDOM. La partie imaginaire du signal est reconstituée à partir de la transformée de Hilbert du signal reçu, ce qui est théoriquement possible car la bande de fréquences de WISDOM est large. Afin de vérifier RÉSUMÉ EN FRANÇAIS 291 l'exactitude de cette reconstitution, durant cette thèse des sondages ont été réalisés avec un analyseur de réseau connecté à la WAA et capable d'envoyer des signaux harmoniques dans la même gamme de fréquences que WISDOM. Il a ainsi été démontré que l'erreur commise sur le module du signal reste inférieure à 3% pour toutes les amplitudes mesurées au-dessus de -60 dB par rapport au pic maximum de la réponse.

• La Transformée de Fourier Inverse (ou IFT) et fenêtrage :

Comme expliqué plus tôt, la modulation à sauts de fréquences de WISDOM fait que les données sont acquises dans le domaine fréquentiel, et doivent être passées dans le domaine temporel pour obtenir la réponse impulsionnelle du sous-sol. Pour ce faire, une Transformée de Fourier Inverse est appliquée. Afin de réduire les effets des lobes secondaires pouvant être confondus avec des échos dans les sondages de WISDOM, un fenêtrage des spectres acquis et nécessaire avant IFT.

Le choix de la fenêtre dépend de l'effet visé ; dans le cadre de cette thèse qui s'est notamment attachée à améliorer la résolution spatiale de l'instrument, une fenêtre de Hamming a été retenue.

• La génération d'un radargramme :

Les sondages de WISDOM sont en général réalisés le long d'une ligne droite, régulièrement espacés (typiquement tous les 10 cm). Les sondages sont ensuite concaténés horizontalement afin de générer une image appelée « radargramme » (voir Figure 2). Un radargramme montre l'amplitude (ou la puissance) du signal reçu en fonction du temps d'arrivée de ce signal (axe vertical) et de la distance parcourue au sol (axe horizontal).

Pour permettre l'interprétation des radargrammes de WISDOM, une donnée dite « en espace libre » ainsi qu'une donnée dite « d'étalonnage » sont souvent utilisées :

• Donnée en espace libre et filtre en domaine temporel :

Les sondages de WISDOM sont parasités par des échos ne provenant ni de la surface, ni de la sous-surface. Tout d'abord, des couplages internes ont lieu au sein même du WEU. Ensuite, il y a un couplage direct entre l'antenne d'émission et l'antenne de réception. Finalement, le corps du rover générera aussi des échos, bien que réduits par l'inclinaison des antennes. Afin de réduire l'effet de ces échos parasites, une mesure dite « en espace libre », sans aucun réflecteur en face des antennes de WISDOM est réalisée puis retirée de tous les autres spectres acquis par l'instrument.

De plus, d'autres échos non désirés pouvant provenir de multiples réflexions entre le sol, le rover et les antennes peuvent être retirés par différentes méthodes de filtrages appliquées directement au radargramme.

• Etalonnage et blanchiment de spectre : Une plaque métallique se comportant comme un réflecteur parfait pour les signaux radar, effectuer une mesure avec une plaque métallique devant les antennes de WISDOM permet à la Estimation de la constante diélectrique à partir de l'amplitude de l'écho de surface :

Pour une surface lisse au regard de WISDOM, la constante diélectrique de la surface peut être estimée à partir de l'intensité de l'écho de surface. En effet, le coefficient de réflexion à la surface, et donc le rapport entre l'amplitude reçue et l'amplitude attendue pour une réflexion totale , dépendent directement de la constante dielectrique ′ de la surface :

= √ ′ -1 √ ′ + 1
′ peut donc facilement être estimée à partir du rapport entre l'amplitude de l'écho de sol et l'amplitude de l'écho d'une plaque métallique (donnée d'étalonnage) disposée à la même distance des antennes.

Si cette technique a déjà été appliquée à des radargrammes WISDOM dans des travaux précédents (Dorizon et al., 2016;Hervé, 2018), un correctif a été mis en place pour pouvoir utiliser une donnée Pour les 9 = 1001 échantillons en fréquentiel notés ; < = / < acquis par WISDOM, le signal est extrapolé dans le sens direct à partir du modèle AR :

;̂> ?8 = -0 @ A ; >BA?8 C A78 avec 4 l'ordre du modèle, et @ A ses coefficients, déterminés avec l'algorithme de Burg (Burg, 1967). Le signal peut aussi être extrapolé en sens indirect de la même façon. Pour fonctionner correctement, cette technique nécessite un blanchiment du spectre à extrapoler, ainsi que le retrait de 5% des échantillons de chaque côté.

Validation sur des données synthétiques :

Les performances de la BWE ont d'abord été validées sur des données synthétiques générées à partir d'un modèle analytique de signal, corrompu par un bruit blanc Gaussien. Dans ces simulations, deux réflecteurs en espace libre, séparés par une distance d, se rapprochent l'un de l'autre jusqu'à une distance inférieure à la résolution théorique de WISDOM (voir les résultats en Figure 6.a et b).

Il a notamment été vérifié que les échos peuvent être séparés jusqu'à une distance d de ~3.75 cm (contre ~10.9 cm sans BWE) et que leur amplitude sont parfaitement reconstituées. Les performances de la BWE ont été vérifiées pour différents niveaux de bruit, montrant ainsi la robustesse de la technique fasse au bruit.

Validation sur des données expérimentales :

La BWE a ensuite été appliquée avec succès à des données expérimentales WISDOM, aussi bien acquises dans des milieux contrôlés au LATMOS et à la TU de Dresde que sur des surfaces naturelles. A partir de ce modèle, une version multivoie de la BWE, nommée « Extrapolation de largeur de bande polarimétrique » ou PBWE (Suwa and Iwamoto, 2007), peut être utilisée pour extrapoler chacun des spectres à partir des 4 spectres polarimétriques.

Cette technique a été appliquée à des données synthétiques obtenues analytiquement pour les 2 vois copolaires (00 et 11), dans une situation de deux réflecteurs se rapprochant l'un de l'autre, mais avec différents déphasages pour les échos selon la voie (exemple de résultat en Figure 8). Les performances de la PBWE sont meilleures que celles de la BWE lorsque les S et S 88 sont différents pour au moins un des échos. La sensibilité de la PBWE face au bruit est aussi plus faible que celle de la BWE. Cette technique a été appliquée à des données synthétiques analytiques simulant 2 échos séparés par une distance variable d en présence d'un bruit blanc Gaussien, puis en ajoutant l'effet des pertes par

  where c is the speed of light in vacuum. The propagation speed of the electromagnetic waves being defined as G

Figure 1 . 1 :

 11 Figure 1.1: Illustration of the Snell-Descartes laws.

Figure 1

 1 Figure 1.2.

Figure 1 . 2 :

 12 Figure 1.2: Schematic representing the general functioning of a radar.

Figure 1 . 3 :

 13 Figure 1.3: Schematic representing different radar configurations.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 13

Figure 1 . 4 :

 14 Figure 1.4: Illustrations of Real Aperture Radar (on the left) and Synthetic Aperture Radar (on the right)

Figure 1 . 5 :

 15 Figure 1.5: Illustration of the signal emitted by a AMCW radar (left), a FMCW radar (middle), and a SFCW radar (right), with the signal amplitude A, and the signal frequency f.

  MHz) measured on lunar samples from previous Apollo missions suggested that subsurface radar soundings could be successful on the Moon. Hence the selection of this instrument by the American space agency NASA. The Apollo 17 Command and Service Module and the ALSE instrument are presented in Figure 1.6 b and c.

Figure 1 . 6 :

 16 Figure 1.6: a. Picture of the Moon taken from the Apollo 17 orbiter, b. Apollo 17 Command and Service Module (CSM), c. ALSE antennas disposition on the CSM.

  Figure 1.7.a). Second, holographic imagery, improving the dynamic range and adding phase information to amplitudes, but required holographic viewers (see Figure 1.7.b). And third, digital recording, but limited to a 55 km track in Mare Serenitatis (see Figure 1.7.c). A FORTRAN software was created to read the acquisitions in imagery format and CHAPTER 1: PLANETARY SUBSURFACE SOUNDING BY RADAR 22 produce an interpreted radargram (see Figure 1.7.d). With such a data storage, the processing and interpretation processes of the ALSE were long and sometimes complicated compared with all the others instruments presented in this chapter.

Figure 1 .

 1 Figure 1.7: a. Photographic imagery of an ALSE radargram, b. Holographic imagery of an ALSE radargram through the holographic viewer, c. Digital imagery of an ALSE radargram, d. Interpreted ALSE radargram in the Montes Cordillera region.

Figure 1 . 8 :

 18 Figure 1.8: Interpreted ALSE radargram showing two subsurface interfaces in Mare Serenitatis.

Figure 1 .

 1 Figure 1.9: a. Disposition of the LRS antennas on the SELENE orbiter, b. The LRS instrument before integration.

Figure 1 . 10 :

 110 Figure 1.10: Example of radargram acquired by the LRS instrument, before (a) and after focusing (b).

Figure 1 . 11 :

 111 Figure 1.11: Detection of subsurface interfaces by the LRS instrument in 5 different lunar Maria regions.

Figure 1 . 12 :

 112 Figure 1.12: Estimated dielectric properties and depths of the interfaces detected by LRS in the subsurface of 5 different lunar Maria regions.

Figure 1 . 3 . 2 .

 132 Figure 1.13: Distribution of the subsurface interfaces detected by LRS, in perspective with the TiO2 contents measured during the Clementine mission.

  It has been designed by the CNRS (Centre National de Recherche Scientifique) in France through the Service d'Aéronomie and the CEPHAG (Centre d'Etude des Phénomènes Aléatoires de Grenoble), and the Riga Aviation University in Latvia.PRISM was an impulse radar, with a centre frequency around 12 MHz, and a bandwidth of 6 MHz, for an expected penetration depth of ~1 km and resolution of 12 m in the Martian permafrost. For each of the 10 nominal days of the mission, 4 profiles were planned with a separation of 5 km. Each profile would have been 400 m long with 10 m between soundings.

Figure 1 .

 1 Figure 1.14).

Figure 1 . 14 : 1 to 2

 11412 Figure 1.14: Simple schematic of the Mars 96 balloon concept (not to scale), with the guide rope and the "tail".

  The use of migration techniques to focus the signal from subsurface scatters have been validated on the Dumont d'Urville radargrams, with the objective of applying this technique to Martian PRISM radargrams. Additionally, numerical simulations with Martian permafrost subsurface models presenting different levels of heterogeneities have been generated to validate the targeted penetration depth of PRISM.

Figure 1 .

 1 Figure 1.15: PRISM radargram acquired during the Dumont d'Urville campaign.

Figure 1 .

 1 Figure 1.16: a. The Martian Northern hemisphere pictured by the Mars Express VMC camera, b. Artist view of the MARSIS orbiter with its antennas, c. Folded MARSIS dipole antenna before integration.

Figure 1 .

 1 Figure 1.17: MARSIS radargrams acquired over the South Polar Layered Deposits, with the interpreted elevations and detections of the interface between the ice and the substrate.

Figure 1 . 18 :

 118 Figure 1.18: Mapping of dielectric constant estimations from MARSIS soundings over the Ultimi Scopuli region, revealing the plausible distribution of subglacial lakes.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 34 Mars in March 2006. Still in activity to this day, the SHARAD sounding radar operated during nearly the same period as the MARSIS radar, complementing subsurface observations at shallower depths but a resolution ten times better.The announced main objective of the SHARAD instrument was indeed to map dielectric interfaces in the subsurface of selected Martian sites with a resolution in the order of ~10 m, a horizontal sampling between 300 and 1000 km, and a penetration depth of a few hundreds of meters. The stratigraphy of rocks, regolith, solid and liquid water in the Martian subsurface was expected to be revealed at such a resolution. To achieve this objective, the chirp modulated signals generated by the SHARAD instrument were between 15 and 25 MHz, corresponding to a bandwidth of 10 MHz. This bandwidth leads to a freespace resolution of 15 m, to be compared with the 150 m free-space resolution of the MARSIS instrument. With these working frequencies, the dipole antenna required for the SHARAD instrument was four times shorter than for MARSIS, namely 10 m. To determine if echoes detected by the instrument are from the subsurface or off-nadir reflections, representative simulations named "cluttergrams" are performed using the MOLA (Mars Orbiter Laser Altimeter instrument of the Mars Global Surveyor mission) elevations digital models. The idea is to generate a synthetic radargram representing what would be obtained if only the Martian surface reflected SHARAD signals.The SHARAD instrument fulfilled its main objective of revealing new stratigraphic details of the Martian subsurface that the MARSIS radar is not able to resolve. In only a few years, identified subsurface echoes allowed new in interpretations of different Martian regions.

Figure 1 .

 1 Figure 1.19: a. Artist view of the MRO orbiter, b. The SHARAD instrument before integration, c. Folded SHARAD antenna before integration.

Figure 1 .

 1 Figure 1.20: SHARAD radargram acquired over the North Polar Layered Deposits, with the different identified interfaces and the basal unit indicated.

Figure 1 .

 1 Figure 1.21: a. SHARAD radargram acquired over the Medusae Fossae Formation with the identified subsurface interface indicated, b. SHARAD radargram acquired over the Tharsis volcanic region with identified subsurface interfaces from lava flows.

Figure 1 . 3 . 3 .

 133 Figure 1.22: a. The MOSIR instrument before intergration, b. Model of the MOSIR instrument mounted on a mock-up Tianwen-1 orbiter for a trial, c. Hot-air balloon trial with the MOSIR instrument on the mock-up Tianwen-1 orbiter.

Figure 1 .

 1 Figure 1.23: a. Artist view of the Rosetta orbiter and its payload, b. Artist view of the Philae lander and its instruments, c. The 67P nucleus pictured from Rosetta.

Figure 1 .

 1 Figure 1.24: a. Representation of the different CONSERT soundings of the 67P nucleus categorized into 4 signal classes, b. Representation of the different tested positions of the Philae lander on the 67P nucleus surface and the goodness of the fit for each option.

Figure 1 .

 1 Figure 1.25: a. Examples of soundings performed by CONSERT, b. Echoes detection in CONSERT soundings with an acceptable SNR (in order of amplitude, the first echo in red, the second echo in blue, and the third echo in green).

Figure 1 . 26 :

 126 Figure 1.26: Images from the Rosetta orbiter which allowed the localization of the Philae lander on the surface of 67P in 2006.

Figure 1 . 27 :

 127 Figure 1.27: Histograms of the measured pulse width at -6 dB of echoes received by CONSERT when sounding the 67P nucleus.

  2 cm in vacuum), with four different modes: scatterometer, imaging, altimeter and radiometer). To transmit and receive signals in the different modes of operation, the instrument used the 4 m diameter high gain communication antenna of the Cassini probe.

Figure 1 .

 1 Figure 1.28: a. The surface of Titan seen in infrared, b. The Cassini orbiter before launch, c.

Figure 1

 1 Figure 1.29.

Figure 1 .

 1 Figure 1.29: a. Radargram acquired over Ligeia Mare with a clear echo from the seabed, b. Examples of soundings from this radargram before and after bandwidth extrapolation process, c. Ligeia Mare bathymetric profile obtained from the radargram.

Figure 1 . 30 :

 130 Figure 1.30: Interpretation chain developped to inverse the bathymetric radar soundings obtained over Titan's seas and lakes.

Figure 1 .

 1 Figure 1.31: Summary of the determined loss tangents and corresponding plausible compositions for Titan's liquid hydrocarbons bodies sounded by the Cassini radar.

Figure 1 . 32 :

 132 Figure 1.32: The icy Galilean moons of Jupiter pictured by the Galileo orbiter.

Figure 1 . 33 :

 133 Figure 1.33: Artist views of the JUICE and Europa Clipper orbiters.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 48

Figure 1 .

 1 Figure 1.34: a. The RIME (ASI) instrument before integration, b. Characterization of the RIME electromagnetic signal with a mock-up JUICE orbiter in an anechoic chamber, c. Trial of the RIME instrument mounted on a high-altitude balloon.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 50 panels of the orbiter[START_REF] Bayer | Europa Clipper Mission: Preliminary Design Report[END_REF]. Preliminary tests of the HF antennas have been conducted in the Jet Propulsion Laboratory facilities. The total mass of the instrument should be around 32 kg. The integration of the antennas to the orbiter, as well as their functional test are presented in Figure1.35.

Figure 1 .

 1 Figure 1.35: a. Disposition of the different instruments of the Europa Clipper orbiter, b. Test of the REASON instrument performed at the JPL facilities.

  Figure 1.37.

Figure 1 .

 1 Figure 1.36: a. Venus as seen from the Mariner 10 mission (NASA) camera, b. Compilation of radar images from the Magellan mission (NASA), c. Images of the surface of Venus in Phoebe Regio from the Venera 13 Sovietic lander.

Figure 1 .

 1 Figure 1.37: a. Artist impression of the EnVision orbiter around Venus, b. Location of the different instruments inside the EnVision orbiter as presented in the mission Assessment Study Report in 2021.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 52 A baseline central frequency of 9 MHz and bandwidth of 6 MHz are proposed for the radar, with possible changings in the range 9-30 MHz for the central frequency and 6-20 MHz for the bandwidth (Bruzzone

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 54

4. 1 .

 1 The Lunar Penetrating Radars of the Chang'E missions 4.1.1. The LPR instrument of the Chang'E 3 mission The CNSA (Chinese Space Agency) landed its first rover, named Yutu, on the surface of the Moon in December 2013 (see Figure 1.38.a). The rover, as part of the Chang'E 3 mission, explored its landing site, in the region of Mare Imbrium, near the Laplace crater. It travelled 114 m, during around two lunar days, and completely ceased operations after three lunar days (see Figure 1.38.b). The mission was a successful demonstration of the Chinese Space Agency capacities of planetary robotic exploration, and one of its instruments was a first on the surface of the Moon: a GPR.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 55 of 30 cm. The received signal, in time-domain, is coded on 8 bits, with sampling rate of 400 MHz for the CH1 channel, and 3.2 GHz for the CH2 channel.

Figure 1 .

 1 Figure 1.38: a. The Yutu rover of the Chang'E 3 mission pictured from its landing platform, b. The Yutu rover path with the different step points of its mission.

Figure 1 .

 1 Figure 1.39: a. Disposition of the CH1 and CH2 antennas of the LPR instrument in the Yutu rover, b. Schematic of a CH1 antenna, c. Schematic of the CH2 antennas.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 56

Figure 1 . 40 :

 140 Figure 1.40: Radargram of all the soundings acquired by the LPR instrument (CH2 channel) during the Chang'E 3 mission with identified subsurface reflectors highlighted.

CHAPTER 1 :Figure 1 . 41 :Figure 1 . 42 :

 1141142 Figure 1.41: Yutu 2 rover path during the Chang'E 4 mission with step points

Figure 1 .

 1 Figure 1.43: Radargram of all the soundings acquired by the LPR (CH2 channel) instrument during the Chang'E 4 mission with geological interpretations.

Figure 1 . 44 :

 144 Figure 1.44: Subsurface model established from the LPR radargrams during the Chang'E 4 mission.

Figure 1 .

 1 Figure 1.45: a. Artist view of the Chang'E 5 lander, b. Schematic of the Chang'E 5 lander sample collecting system, c. Retrieval of the capsule containing lunar samples collected by the Chang'E 5 mission.

Figure 1 .

 1 Figure 1.46: LRPR antenna array (a) and electronics (b) before integration.

Figure 1 .

 1 Figure 1.47: a. Disposition of the LRPR antenna arrays in the Chang'E 5 lander, b. Schematic representing the LRPR signal acquisition process.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 62 Different experimental radargrams have been performed with a model of the instrument (with working frequencies between 1 and 3 GHz), in a soil trench filled with lunar material simulants and buried reflectors in the Institute of Electronics of the Chinese Academy of Sciences(Lu et al., 2021b; Xiao et al., 2019). The Chang'E 5 being recent, no experimental results have been published to this day, but the collected lunar samples have been successfully returned to Earth the 16th December 2020.

Figure 1 .

 1 Figure 1.48: LRPR radargrams acquired in a soil trench in the Chinese Academy of Sciences, (a) the raw radargram, (b) the free-space radargram, (c) the radargram after free-space subtraction, (d) the migrated radargram.

Figure 1 .

 1 Figure 1.49: a. The Zhurong rover in front of its landing platform as seen by an independent camera, b. Disposition of the RoSPR antennas in the Zhurong rover.

Figure 1 .Figure 1 .

 11 Figure 1.50: a. Model of the CH1 antennas folded and unfolded, b. Model of the CH2 antennas

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 65 applied to obtain the signal in time-domain. Figure 1.51 shows examples of soundings acquired for both channels.

Figure 1 . 52 :

 152 Figure 1.52: Picture a field campaign organized in a volcanic terrain with a model of the instrument in a mock-up Tianwen-1 rover (a), and a radargram acquired with the CH2 channel on this site (b).

Figure 1 . 53 :

 153 Figure 1.53: Geological map of the Jezero crater around the Mars 2020 landing site.

Figure 1 .

 1 Figure 1.54: a. Artist view of the Perseverance rover, b. The Perseverance rover pictured during its landing, c. The Perseverance rover pictured a few days after landing.

and 1 . 2

 12 GHz. The high resolution / shallow sounding mode has a central frequency of 675 MHz, the low-resolution / deep sounding mode 375 MHz. The highest resolution possible with the instrument in therefore 20 cm for a relative dielectric constant of 2, and down to 10 cm for a high relative dielectric constant of 9.

Figure 1 .

 1 Figure 1.55: a. Electronics of the RIMFAX instrument before integration, b. The RIMFAX antenna before integration, c. Disposition of the instrument inside the Perseverance rover.

Figure 1 .

 1 55.a and b for illustrations of the RIMFAX instrument units.RIMFAX has been integrated to the Perseverance rover in 2019 (see Figure1.55.c), and the first Martian data from the instrument are currently acquired. To prepare the arrival of the instrument on Mars, numerical simulations of RIMFAX operations on terrain models representative of the Jezero crater sites have been generated, and the processing / interpretation tools tested on them[START_REF] Eide | Ground-Penetrating Radar Modeling across the Jezero Crater Floor[END_REF]. The subsurface model has been created with HiRISE images and assumptions on the composition and heterogeneities of the different exposed layers visible on the images. Experimental soundings have also been acquired in different dry natural environments, such as the Svalbard archipelago (Norway) and the Navajo Desert (USA)[START_REF] Hamran | Results from Field Testing the RIMFAX GPR on Svalbard[END_REF][START_REF] Hamran | Radar Imager for Mars' Subsurface Experiment-RIMFAX[END_REF]. The results from the Svalbard campaign are presented in Figure1.56.

Figure 1 .

 1 Figure 1.56: RIMFAX radargram acquired during a field trial in Midtre Lovenbreen glacier site in Svalbard, with interpretations and enlarged portions.

CHAPTER 1 :

 1 PLANETARY SUBSURFACE SOUNDING BY RADAR 69 elements). The Schiaparelli lander, unfortunately crashed on the surface of Mars in October 2016. The ExoMars 2022, initially planned for 2018, is the second act of the ExoMars program, with an instrumented platform named Kazachok, and the Rosalind Franklin rover(Vago et al., 2017). Both will be launched in September 2022, and will land in June 2023, in the region of Oxia Planum.

Figure 1 . 57 :

 157 Figure 1.57: Artist views of the Trace Gas Orbiter, the Schiaparelli lander, the Kazachok landing platform and the Rosalind Franklin rover of the ExoMars 2022 mission, with the location of the different instruments.

Figure 1 .

 1 Figure 1.58: a. Disposition of the WISDOM antennas in the Rosalind Franklin rover, b. The WISDOM electronics integrated to the rover, c. The WISDOM antennas to be integrated to the rover.

1 . 4 . 5 . 6 .

 1456 ExoMars 2022 and the subsurface of Oxia Planum ................................................................... 2. Objectives, characteristics and expected performances of WISDOM ..................................... 3. WISDOM data processing chain ................................................................................................. WISDOM experimental dataset ................................................................................................ Analytical and numerical simulation of WISDOM data ......................................................... Summary ......................................................................................................................................

  years' selection process initiated by the European Space Agency in 2014. This plain is located between 16° to 19° N and -23° to -28° E, on the edge of the Martian dichotomy, between the outflow channels of Ares Vallis and Mawrth Vallis. It satisfies the objectives of ESA both in terms of safety (assuring a safe landing in the calculated uncertainty landing ellipse and the possibility for the rover to move on the surface), and the scientific return of the mission.

Figure 2 . 1 :

 21 Figure 2.1: Situation of the Oxia Planum site in the Oxia Palus quadrangle region, presented as a geological map (A) and an elevation map (B).

Figure 2 . 2 :

 22 Figure 2.2: Proposed scenario for the geological evolution of the Oxia Planum site.

  is presented in the schematic of Figure2.3. On Earth, biosignatures are often encountered in sedimentary material, such as the clay-bearing units detected on Oxia Planum. However, as explained, a sample collected in clays on the Martian surface would likely be deprived of any traces of ancient life, due to harsh conditions in terms of radiations and oxidation for billions of years. A sample of clay should therefore be collected in the subsurface, beneath shielding layers of materials. Models predict that approximately below 2 m of typical Martian regolith the absorption of the radiations would be enough to preserve potential traces of life.

Figure 2 . 3 :

 23 Figure 2.3: Schematic presenting the ExoMars 2022 rover mission strategy.

  the PANCAM, ISEM, CLUPI, ADRON, WISDOM, and MA_MISS instruments will give a geological context for the different collected and analysed samples. All 9 instruments are shown in Figure 2.4.

Figure 2 . 4 :

 24 Figure 2.4: Pictures of the 9 Rosalind Franklin rover instruments (from left to right, and up to bottom, PANCAM, ISEM, CLUPI, ADRON-RM, MOMA, RAMAN-LS, MICROMEGA and MA_MISS).

Figure 2 .

 2 Figure 2.5.b) requires parachutes of 15 and 35 m of diameter. The later will be the largest parachute ever used on Mars. The rover landing platform, named Kazachok and developed by ROSCOSMOS, is instrumented and will thus be a fixed station on Oxia Planum after the landing. Two ramps will allow the rover to descend the platform with two possibilities (see Figure 2.5.a), in case of hazards in one of the ways. The descent from the platform will occur on the 10 th Martian day after landing. Science operations will begin 5 Martian days afterwards, during which the rover will move away from the platform and perform a blank analysis run.

Figure 2 . 5 :

 25 Figure 2.5: a. Picture of a mock-up Rosalind Franklin rover on a mock-up Kazachok platform during a simulation of operations in the Desert of Atacama, b. The Rosalind Franklin rover integrated on the Kazachok platform in the Thales Alenia Space facilities.

Figure 2 . 6 :

 26 Figure 2.6: Illustration of the stepped-frequency modulation process with 5 frequency steps between 0.5 and 1.5 GHz. From top to bottom, the transmitted frequency steps, the transmitted signal, the received signal, the mix between the transmitted and received signals, the In-phase signal measured by the RF channel, the Quadrature-phase signal not measured by WISDOM.

  modules are accommodated inside a nickelled aluminium isolating box, and connected by internal harnesses (see Figure2.7). The dimensions of the box are 14.5x15.9x5 cm, and its weight is 803g. It is connected to the WISDOM Antennas Assembly (WAA) through four coaxial cables (see the next subsection). The WEU was designed by the LATMOS and the Laboratoire d'Astronomie de Bordeaux (LAB), and its different PCB (Printed Circuit Board) and components fabrication subcontracted to private companies.

Figure 2 . 7 :

 27 Figure 2.7: The WISDOM Electronic Unit of the Qualification Model (on the left) and of the Flight Model integrated to the rover (on the right).

Figure 2 . 8 :

 28 Figure 2.8: Front and Rear views of the Radio Frequency board of the WEU (Qualification model).

Figure 2 . 9 :

 29 Figure 2.9: Front and Rear views of the Data Processing Unit board of the WEU (Qualification model).

Figure 2 .

 2 Figure 2.10: Front view of the DC/DC converter board of the WEU (Qualification model).

Figure 2 .

 2 Figure 2.11: Block diagram of the WISDOM Electronic Unit, presenting the interaction between the different elements of the DC/DC converter, FPGA and RF boards.

CHAPTER 2 :

 2 WISDOM, THE GROUND PENETRATING RADAR OF THE EXOMARS 2022 MISSION 94 WISDOM being an ultra-wideband radar, it required antennas able to send harmonic signals with a smooth gain function on the entire 0.5-3 GHz bandwidth. It also required a radiation pattern directed

Figure 2 .

 2 Figure 2.12: a. Picture of the WISDOM Antennas Assembly integrated to the Rosalind Franklin rover in the Airbus UK facilities, b. Polarimetric set-up, tilt and the main elements of the WAA.

Figure 2 .

 2 Figure 2.13: Estimations of the maximum depth of interfaces detected by WISDOM as a function of its vertical resolution, for different dielectric properties of the subsurface. Plain curves correspond to a smooth interface, and dashed ones to a rough interface. On the left are presented the results for a homogeneous layer, in the middle for a heterogeneous layer of ∆. = 1, and on the right for a heterogeneous layer of ∆. = 1.5.

Figure 2 .

 2 Figure 2.14: Schematic of the WISDOM processing chain used during this thesis.

Figure 2 .

 2 Figure 2.15 displays the model established from the measured power received by the WISDOM instrument for the different temperatures of the RF board in the LATMOS Mars Chamber. The model has been determined by a fit of the power measured for both the 00 and 11 polarimetric channels. The amplitude of the direct coupling, which should be constant in theory, has been measured before and after application of the correction. After correction, the amplitude becomes nearly constant, validating the temperature correction.

Figure 2 .

 2 Figure 2.15: Model of the power received by WISDOM as a function of the RF board temperature established with the experimental acquisitions in the Mars chamber (for the 00 and 11 co-polar channels).

Figure 2 .

 2 Figure 2.16: WISDOM acquisitions performed with a Vector Network Analyser mounted of the WISDOM Antennas Assembly in an anechoic chamber (on the left) and on an experimental soil trench (on the right) at the TU Dresden.

Figure 2 .

 2 Figure 2.17: Relative error on the amplitudes of the modulus measured for 70 experimental spectra acquired with the VNA, reconstructed with the Hilbert transform (mean value and standard deviation), and indications of the length of a WISDOM spectrum after removing 5% of samples on each side.

Figure 2 .

 2 Figure 2.18: Example of experimental sounding acquired with the VNA, a. The modulus of the original complex sounding and the reconstruction obtained by Hilbert transform from the real-part, b. Relative error between the amplitudes of the modulus reconstructed by Hilbert transform and the amplitudes from the modulus of the original complex sounding, c. Modulus of the time-domain transformation of the original complex experimental sounding and the modulus of the time-domain transformation of the sounding reconstructed by Hilbert transform, d. Error between the amplitudes of the modulus in timedomain after reconstruction by Hilbert transform and the amplitudes in time-domain with the original complex sounding.

Figure 2 .

 2 Figure 2.19: 2D histogram of the relative error on the amplitudes of the modulus measured for peaks detected in 70 time-domain soundings acquired with the VNA, reconstructed with the Hilbert transform, and the radargram generated with these 70 soundings.

Figure 2 .

 2 Figure 2.20: Examples of windows commonly used in signal processing (on the left), and the IFFT of their application to a synthetic WISDOM echo at a time-delay of 14 ns (on the right).

Figure 2 .

 2 Figure 2.21: Experimental set-up used to determine the WISDOM "equivalent impulse" in time-domain with a "stroboscopic" technique (on the left), and the results of this experiment compared to the zeropadding time-domain interpolation (on the right).

Figure 2 .

 2 Figure 2.22: Example of application of the IFFT on an experimental WISDOM sounding, with a

Figure 2 .

 2 Figure 2.23.a), is converted into a one-pixel wide image (see Figure 2.23.b), the colour-scale representing amplitudes, and its height the time-delays. By horizontally stacking all the soundings, animage is obtained, with its height representing time-delays, its width the traverse distance on the surface, and its colour variations the amplitudes of the different echoes (seeFigure 2.23.c). This image is the "radargram", also known as "B-scan".

CHAPTER 2 :

 2 WISDOM, THE GROUND PENETRATING RADAR OF THE EXOMARS 2022 MISSION 109 The values represented by the colour-scale of a GPR radargram are usually either the real part of the signal or its envelope, depending on the processing and interpretation techniques applied, or just by convention within a GPR science team. All radargrams processed for this thesis display the signals in envelope, unless explicitely mentioned otherwise.

Figure 2 .

 2 Figure 2.23: Illustration of the construction of a "radargram" from individual soundings, a. An example of WISDOM sounding, b. This sounding transformed into a 1 pixel wide image with a colorscale representing the envelope of the signal, c. The "radargram" obtained by stacking the 1 pixel wide images with the position of the example sounding indicated.

CHAPTER 2 :

 2 WISDOM, THE GROUND PENETRATING RADAR OF THE EXOMARS 2022 MISSION 110 coming from the surface rather than the subsurface. In addition, due to the broad radiation pattern of the antennas, an echo coming from a single point scatter will appear on different consecutive soundings, and create a diffraction curve that looks like a hyperbolic signature (see Chapter 3 for more details).

Figure 2 .

 2 Figure 2.24: Illustration of the WISDOM "traverse" and "grid" operations.

3. 5 . 1 .

 51 Subtraction of a free-space measurement WISDOM soundings displays parasitic signals of different origins which can impair its interpretation.

Figure 2 .

 2 Figure 2.25 Illustration of the different sources for parasitic signals in WISDOM soundings.

Figure 2

 2 Figure 2.25.

Figure 2 .

 2 Figure 2.26 presents an example of WISDOM sounding in time-domain, acquired during a measurement campaign on an experimental soil trench. The internal and direct coupling are clearly visible, with a time delay between them roughly corresponding to the time of flight of the signal in the coaxial cables. The proximity of the direct coupling to the surface echo can also be noticed. Their interference can impair the retrieval of the position and amplitude of the surface echo, which are key information for the WISDOM scientific return.

Figure 2 .

 2 Figure 2.26: Example of WISDOM sounding in time-domain without reducing or removing the effect of parasitic signals.

Figure 2 .

 2 Figure2.27 illustrates the effect of the removal of a free-space measurement on an experimental WISDOM radargram. The free-space subtraction eliminates not only the internal and direct couplings, but also multiple echoes likely to be repetitions of the direct coupling. Once this removal done, the surface echo appears more clearly, and the different reflectors at longer time delays can be much easily separated from repetitions of the direct coupling. On the other hand, as expected, repetitions of the surface echo, due to bounces between the surface and the antennas, are still visible after free-space removal.

Figure 2 .

 2 Figure 2.27: Experimental WISDOM radargram before and after the removal of a free-space measurement.

Figure 2 .

 2 Figure 2.28: a. Experimental set-up to measure the effect of the rover (its wheels in particular) on WISDOM acquisitions, b. Free-space measurement perfomed in Airbus UK facilities with WISDOM integrated to the Rosalind Franklin rover.

Figure 2 .

 2 Figure 2.29: a. Calibration measurment performed in an anechoic chamber at the TU Dresden, b.

Figure 2 .

 2 Figure 2.30 presents an example of calibration measurement transformed in time-domain. Where the impulse from the metallic plate is selected (Figure 2.30.a), and used to determine the gain to be inverted to perform the "whitening" of the signal (Figure 2.30.b). As the technique requires a FFT and an IFFT,

Figure 2 .

 2 Figure 2.30: a. Calibration sounding performed on a metallic plate at the TU Dresden anechoic chamber, b. Frequency spectrum of the impulse selected in the time-domain sounding.

Figure 2 .

 2 Figure 2.31: Application of the whitening process to a synthetic WISDOM sounding generated with a

  Figure 2.32.

  , with promising results on experimental WISDOM radargrams, such as the Singular Values Decomposition. The image is considered as a matrix to be decomposed in singular values, where first singular values are expected to contain the information corresponding to the parasitic signals to be removed.

Figure 2 .

 2 Figure 2.32: Experimental WISDOM radargram acquired on a natural subsurface, before and after mean filtering, the main reduced parasitic echoes being indicated by a red arrow.

Figure 2 . 33 :

 233 Figure 2.33: Radargram with the nominal WISDOM 10 cm horizontal sampling (on the left), radargram with a 5 cm horizontal sampling (in the middle), radargram with the nominal WISDOM horizontal sampling after horizontal interpolation (X2).

  After the validation and characterization of the WISDOM flight model, the instrument has been delivered for integration to the Rosalind Franklin rover in January 2019. Additional functional tests, free-space and calibration measurements have then been performed at the Airbus United Kingdom facilities from August to December 2019, with WISDOM integrated to the rover. Further laboratory experiments have been performed with other models of the instrument, to validate new processing and interpretation tools requiring new experiments.

Figure 2 . 34 :

 234 Figure 2.34 : Summary of the LATMOS (OVSQ) "clean room" functional tests campaign.

Figure 2 . 35 :

 235 Figure 2.35: Summary of the LATMOS (OVSQ) "Mars Yard" campaign.

CHAPTER 2 :

 2 WISDOM, THE GROUND PENETRATING RADAR OF THE EXOMARS 2022 MISSION 123 4.1.3. The TU Dresden campaignIn June 2019, a WISDOM measurement campaign has been organized at the Technische Universtät Dresden facilities (seeFigure 2.36). For this campaign, Flight Spare Models of the WISDOM Electronic Unit, WISDOM Antennas Assembly and coaxial cables have been used. This experimental set-up is therefore very close to the instrument that will sound the Martian subsurface in 2023.The WISDOM Antennas Assembly was mounted on a metallic stick, to face the surface at a similar height as they will be on Mars, with minimum parasitic echoes from the set-up. The antennas were connected to the WISDOM Electronic Unit or alternatively a Vector Network Analyzer (VNA), through the coaxial cables. The VNA being able to measure both the real and the imaginary part of the steppedfrequency radar signal, it allows the reconstruction of the imaginary part by the Hilbert transform to be tested. Finally, the whole set-up was mounted on a motorized rail system, allowing the antennas to move in the straight line above the surface, and thus WISDOM to perform traverses.

Figure 2 . 36 :

 236 Figure 2.36: Summary of the TU Dresden measurement campaign (June 2019).

Figure 2 .

 2 Figure 2.37: Free-space and calibration measurement performed in the TU Dresden anechoic chamber to prepare the future processing of the measurement campaign results.

Figure 2 . 38 :

 238 Figure 2.38: Experimental set-up of the TU Dresden measurement campaign, and second calibration measurement performed on the experimental terrain to assure a distance from the antennas to the metallic plate equal to the distance between the antennas and the surface.

Figure 2 .

 2 Figure 2.39: The TU Dresden "metallic staircase" experimental set-up, a. Pictures of the set-up before the staircase burial, b. Schematic of the set-up with the different dimensions.

Figure 2 .

 2 Figure 2.40: The TU Dresden "metallic plates" experimental set-up, a. Pictures of the set-up before the metallic plates burial, b. Schematic of the set-up with the different dimensions.

Figure 2 .

 2 Figure 2.41: TU Dresden experiment with a "non-flat surface", a. Picture of the ridges carved on the surface, b. Schematic with the dimensions of the ridges.

CHAPTER 2 :

 2 WISDOM, THE GROUND PENETRATING RADAR OF THE EXOMARS 2022 MISSION 127

Figure 2 . 42 :

 242 Figure 2.42: Summary of the Etna Mt. campaign (October 2010).

Figure 2 .

 2 Figure 2.43: Summary of the Chamonix campaign (January 2011).

Figure 2 .

 2 Figure 2.44: Summary of the Dachstein Giant Ice Caves campaign (April 2012).

Figure 2 .

 2 Figure 2.45: Summary of the SAFER simulation of operations in the Desert of Atacama (October 2013).

Figure 2 .

 2 Figure 2.46: Summary of the Colorado Provençal campaign (July 2017).

Figure 2 .

 2 Figure 2.47: Summary of the ExoFit campaign in the Desert of Atacama (February 2019).

Figure 2 .

 2 Figure 2.48: a. Drilling and sample collection operations, b. WISDOM grid operations after descent from the landing platform.

Figure 2 .

 2 Figure 2.49: a. Terrain on which the 3 WISDOM traverses without the mock-up rover have been performed, b. Experimental set-up to perform the 3 traverses without the mock-up rover.

Figure 2 . 50 :

 250 Figure 2.50: Schematic representing the relative position of the 3 long WISDOM traverses and the 2 traverses grid performed during the ExoFit campaign.

5 and 3

 3 GHz with a step of 2.5 MHz, just as WISDOM soundings. The amplitudes, distances, depths and attenuations associated with each point scatter are entries of the model. An additional phase shift can even be added.An additive white Gaussian noise is then applied to the generated signal. To obtain the order of magnitude of the Signal to Noise Ratio (SNR) in an experimental WISDOM sounding, the SNR level of experimental acquisitions in the LATMOS radar laboratory have been measured in time-domain, with:d u = 20 ()* +, v wxy uzd b + M { ( 2.9 )where wxy is the amplitude of the highest considered echo, RMS() the root mean square, b + M the portion of the time-domain signal between times t1 and t2, the limits of the signal where only noise is expected.

Figure 2 .

 2 Figure 2.51: Measured SNR in time-domain as a function of the SNR set in the frequency domain for synthetic WISDOM soundings (Monte-Carlo approach : 1000 cases of noise for each tested SNR level).

Figure 2 .

 2 51, a SNR = 30 dB in frequency domain has been chosen when generating examples of synthetic soundings in this document. When possible, signals with lower SNRvalues have also been generated in order to test the effect of noise of some processing and interpretation tools (seeChapter 4 and 5). It can also be noted that the results presented in this Figure2.are coherent with the 27 dB IFT gain from Table2.1.

Figure 2 . 52 :

 252 Figure 2.52: Example of synthetic WISDOM signal (real part and envelope reconstructed by Hilbert transform) generated with a scattering model and an additive white Gaussian noise (SNR=30dB).

Figure 2 . 53 :

 253 Figure 2.53: Schematic of a 3D Yee cell.

Figure 2 .

 2 Figure 2.54.

Figure 2 . 54 :

 254 Figure 2.54: Example of subsurface and WISDOM antennas models.

Figure 2 .

 2 Figure 2.55: Example of subsurface models with all possible scales of heterogeneities generated with the Diamond-Square algorithm in Temsi-FD.

Figure 3 . 1 :

 31 Figure 3.1: Amplitude of echoes measured for a metallic plate at different distances from WISDOM antennas (normalized by the highest measured amplitude), and models determined for echoes measured for distances below and above ~45 cm (respectively in red and blue).

Figure 3 . 2 :

 32 Figure 3.2: The 3 experimental set-up selected to apply the estimation of the dielectric constant from the surface echo to, a. The original soil trench, b. The metallic staircase before burial, c. The two metallic plates before burial.

Figure 3 . 3 :

 33 Figure 3.3: Radargram acquired on the original soil trench at the TUD, in the 00 polarimetric channel (top), in the 11 polarimetric channel acquisition (middle), and the dielectric constant estimated from the surface echo amplitude for both channels (bottom).

Figure 3 .

 3 Figure 3.3 presents the results obtained with the original soil trench; they show variations of the inferred dielectric constant along the traverse, with values rising from less than 2 for the first 50 cm to punctual values around 4. The especially low values below 2 at the very beginning of the traverse can be explained locally by a higher porosity and/or roughness of the soil near the surface. Furrows made by a prototype tilling machine on the soil trench were indeed clearly visible at the beginning of the path. It can also be observed that the dielectric values inferred from the 00 and 11 polarimetric channels are very similar, which is consistent with a flat surface as the TU Dresden soil trench.

Figure 3 . 4 :

 34 Figure 3.4: Radargram acquired on the buried metallic staircase at the TUD, in the 00 polarimetric channel (top), in the 11 polarimetric channel acquisition (middle), and the dielectric constant estimated from the surface echo amplitude for both channels (bottom).

Figure 3 . 5 :

 35 Figure 3.5: Radargram acquired on the two buried metallic plates at the TUD, in the 00 polarimetric channel (top), in the 11 polarimetric channel acquisition (middle), and the dielectric constant estimated from the surface echo amplitude for both channel (bottom).

Figure 3 . 6 :Figure 3 . 7 :

 3637 Figure 3.6: Comparison of the dielectric constants inferred from the surface echoes for the original soil trench (top), the buried metallic staircase (middle) and the two buried metallic plates (bottom)

Figure 3 . 8 :

 38 Figure 3.8: Dielectric constant of the soil trench inferred from the time delays and known depth of each metallic plate, compared to the dielectric constants estimated with the surface echo for the corresponding traverse.

Figure 3 .

 3 Figure 3.9: a. Experimental set-up at the TUD for the permittivity measurement of soil samples, b. Dielectric constant measurements performed at the TUD for the sample of the soil trench (measurements named from M1 to M7).

Figure 3 .

 3 Figure 3.10: GPR in a ground-coupled set-up, with one small reflector embedded in a homogeneous subsurface. The size of the reflector is exaggerated.

Figure 3 .

 3 Figure 3.11: GPR in an air-coupled set-up, with one reflector embedded in an homogeneous subsurface.

Figure 3 .

 3 Figure 3.13: Illustration of the Hough transform applied to the detection of lines.

4 )

 4 Local maxima are then detected in the accumulator, and the corresponding parameters $, β and & are retrieved. A threshold on the values of the maxima can be applied to select only the most probable detections. 5) The position and slope of each detected hyperbola is eventually returned by the algorithm with the retrieved parameters $, β and &.

  Figure 3.14: Illustration of the Hough transform applied to the detection of hyperbolas.

Figure 3 .

 3 Figure 3.15: WISDOM diffraction curves generated with the air-coupled model (see Section 3.2) for different possible FoV angles of the GPR, subsurface dielectric constants and depths of reflectors.

Figure 3 .

 3 Figure 3.16.b shows the mean estimated dielectric constant (from parameters % returned by the Hough transform) and the associated standard deviation for each generated diffraction curve. Except for reflectors in the air ( = 1) (as rocks on the surface), the classical Hough transform does not result in accurate estimates of the dielectric constant of the subsurface. The inferred dielectric constant is systematically underestimated, leading to an overestimation of the reflector depth. Such an error could be dangerous when selecting a safe drilling target, as the depth of a potential underground hazard would be shallower than expected.

Figure 3 .

 3 Figure 3.16: a. Projections of a 3D Hough accumulator (2D histograms) obtained for a diffraction generated for a reflector at a depth of 1 m, in a homogeneous subsurface of dielectric constant 4, and for a FoV of the GPR of 100° b. Estimations of the dielectric constant of the subsurface by the Hough transform (all possible triplets of points considered) when applied to diffraction curves generated with the previously modified hyperbola model (see Section 3.2). Different possible subsurface dielectric constants, depths of the target and FoV of the GPR are tested.

Figure 3 .

 3 Figure 3.17: The "equivalent" dielectric constant as function of the depth of the target, for different possible dielectric constants of the subsurface.

Figure 3 . 18 :

 318 Figure 3.18: Dielectric constant of the subsurface estimated by the Hough transform (all possible triplets of points considered) with the "mean" dielectric constant correction applied, for different possible depths of the target, dielectric constants of the subsurface and FoV of the GPR.

Figure 3 . 19 :

 319 Figure 3.19: Schematic representing an example of simplex in the space of solution for the Nelder-Mead algorithm adapted to the presented optimization problem.

  P , O P ) being the positions of the considered triplet of points in the detected edges of the radargram. The resulting values of T(1 Q ), T(1 M ) and T(1 P ) for the 4 points are compared in the least squares sense with O Q , O M and O P . This is the error function F(X,T(X),ε).

Figure 3 . 20 :

 320 Figure 3.20: Selected strategy for the detection of diffraction curves in WISDOM radargrams.

3. 6 . 1

 61 Validation on synthetic diffraction curvesA preliminary validation of the Hough transform detector developed for WISDOM has been performed on clouds of points including diffraction curves generated using equation (3.9), among randomly located points. The idea is to prove the ability of the classic Hough transform to detect the diffraction curves in these clouds of points.

Figure 3 . 21 :

 321 Figure 3.21: Generated cloud of points including 4 diffraction curves for a subsurface of dielectric constant 4 and 500 parasitic points (on the left). Result of the classic Hough transform applied to the cloud of points, with the detected diffraction curves represented and their characteristics indicated (on the right).

Figure 3 . 22 :

 322 Figure 3.22: Example of Hough transform accumulator integrated over the dielectric constants axis to allow a 2D-representation. This accumulator was obtained for the cloud of points shown in Figure 3.21.

Figure 3 .

 3 Figure 3.23 shows that, after the application of the corrected Hough transform the inferred dielectric constant from all 4 diffraction curves is correctly estimated, as well as the depth of the corresponding target perfectly estimated. This preliminary test illustrates the ability of the proposed strategy to detect diffraction curves among other edge detections, and correctly characterize diffraction curves affected by refraction at the surface, as long as they follow the model proposed by equation (3.9).

Figure 3 .

 3 Figure 3.23: Result of the corrected Hough transform applied to the generated cloud of points, with the detected diffraction curves represented and their characteristics indicated.

Figure 3 . 24 :

 324 Figure 3.24: Cloud of points generated for 4 diffraction curves with dielectric constant variations (normal distribution of mean parameter 4 and standard deviation 0.05) and 500 parasitic, with the results of the application of the corrected Hough (the detected diffraction curves represented and their characteristics indicated).

Figure 3 .

 3 25). The diffraction curve corresponding to the highest peak in the Hough accumulator is displayed in Figure3.25 (right). The estimated dielectric constant is as close to 1 (0.9±0.1), the estimated horizontal distance is ~1.95 m and the estimated distance to the antennas is ~1.04 m.

Figure 3 .

 3 Figure 3.25: FDTD generated radargram for a spherical reflector (diameter of 10 cm) in free-space at 1 m from the antennas (on the left), detection of echoes in the radargram (in the middle), and results of the classic Hough transform applied to the radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 .

 3 Figure 3.26: Model used to generate a FDTD simulated radargram with a diffraction curve from an off-track spherical reflector (of diameter 10 cm) lying on the surface.

Figure 3 .

 3 Figure 3.27: FDTD simulated radargram with an off-track target lying on the surface (on the left), detection of echoes in the radargram (in the middle), and result of the classic Hough transform applied to the radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .The dielectric constant estimated for the first diffraction curve is = 0.9±0.1, pointing the fact that the reflector is lying on the surface and not embedded in the subsurface. The estimated positions of the

Figure 3 . 28 :

 328 Figure 3.28: Subsurface model used to generate FDTD simulated radargrams with one spherical buried reflector (diameter of 10 cm) embedded in a homogeneous material with 3 different possible dielectric constants.

Figure 3 .

 3 Figure 3.29 shows the resulting radargrams. Each of them displays surface echoes, the main echoes from the subsurface reflector forming a diffraction curve, and secondary reflections from the subsurface reflectors. It can also be noted that due to the tilt of the antennas, the left side of the diffraction curves presents stronger echoes than the right side.

Figure 3 .

 3 Figure 3.29: FDTD generated radargrams for one subsurface reflector embedded in a homogeneous medium of dielectric constant >1.

Figure 3 .

 3 Figure 3.30 presents the results of the classic Hough transform applied to the different radargrams. It shows what would be the estimation of the dielectric constant (and thus of the reflector depth) without any correction for the refraction effect. The Hough transform has been performed with a precision of 0.1

Figure 3 . 30 :

 330 Figure 3.30: Results of the application of the classic Hough transform to the FDTD generated diffraction curves. The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 .

 3 Figure 3.31: Results of the application of the corrected Hough transform to the FDTD generated diffraction curves. The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 . 32 :

 332 Figure 3.32: Subsurface model used to generate a FDTD simulated radargram with 5 spherical buried targets (10 cm diameter) in a homogeneous material (ε = 4).

Figure 3 .

 3 34 (right) shows that after application of the corrected Hough transform, the estimated dielectric constants are much closer to the expected 4. As a consequence, the estimated depths of the different reflectors are also much closer to the model input values with errors of the order of only a few centimetres.Even in the presence of parasitic echoes (in Figure3.35, 500 parasitic echoes at random time delays and horizontal distance have been added to the list of detected echoes), the detection of the diffraction curves by either the classic or corrected Hough transforms is preserved and the derivation of the dielectric constant by this later technique is still very satisfactory. This further confirms the results obtained on analytically generated diffraction curves (see Figure3.23) and the validity of the tool and strategy we propose.

Figure 3 .

 3 Figure 3.33: FDTD generated radargram with 5 buried targets in a homogeneous subsurface (on the left), detection of echoes in the radargram (on the right).

Figure 3 .

 3 Figure 3.34: Results of the classic Hough transform applied to the radargram from Figure 3.33 (on the left), and results of the corrected Hough transform applied to this same radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 . 35 :

 335 Figure 3.35: Detection of echoes in a FDTD generated radargram with 5 buried targets in a homogeneous subsurface after adding 500 parasitic echoes (on the left), results of the classic Hough transform applied to the radargram (in the middle), and results of the corrected Hough transform applied to the radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 .

 3 Figure 3.36: Two-layer subsurface model used to generate a FDTD simulated radargram with 5 spherical buried targets (10 cm diameter) embedded in two homogeneous materials of different dielectric constants ε = 3 and 6.

Figure 3 .

 3 Figure 3.37: FDTD generated radargram with 5 buried targets in a subsurface with two homogeneous layers of different dielectric constant (on the left), detection of echoes in the radargram (in the middle),

Figure 3 .

 3 Figure 3.38 shows a slice of the modelled subsurface: 5 spherical targets are embedded in a fractal matrix of mean dielectric constant equal to 4, with variations between 3.5 and 4.5, generated with the Diamond-Square algorithm (see Section 5.2 of Chapter 2). The characteristic size of the dielectric heterogeneities is 16.6 cm (of the same order as that of the target diameter, and approximately twice the central wavelength of WISDOM for a dielectric constant of 4).Figure 3.39 displays the resulting radargram where the alteration of the shape of the diffraction curves can be observed, especially for the deepest targets (to be compared with Figure 3.33).

Figure 3 . 38 Figure 3 . 38 :

 338338 Figure 3.38

Figure 3 .

 3 Figure 3.39: FDTD generated radargram with 5 buried targets in a heterogeneous subsurface (on the left), detection of echoes in the radargram (in the middle), and results of the corrected Hough transform applied to the radargram (on the right). The precision is 10 cm on X, 0.2 ns on T(X) and 0.25 on .

Figure 3 . 40 :

 340 Figure 3.40: Selected radargram (real part of the signal) from the TU Dresden campaign where a clear hyperbolic shaphe is visible (left), echoes detection applied to this radargram (middle), Classic Hough transform applied to this radargram (right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 . 41 :

 341 Figure 3.41: Histogram of the dielectric constants estimated from manual fits of the different observed diffraction curves in the Dachstein campaign radargrams (extracted from(Dorizon et al., 2016)).

Figure 3 .

 3 Figure 3.42: A selected portion of radargram (real part of the signal) from the Dachstein campaign (on the left), the detection of echoes in this radargram (in the middle), and the result of the corrected Hough transform for this radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

Figure 3 .

 3 Figure 3.43: A selected portion of radargram (real part of the signal) from the Dachstein campaign (on the left), the detection of echoes in this radargram (in the middle), and the result of the corrected Hough transform for this radargram (on the right). The precision is 5 cm on X, 0.1 ns on T(X) and 0.1 on .

  As presented in Section 4.2.6 of Chapter 2, the second half of the ExoFit simulation of operations took place in February 2019 in the Desert of Atacama (Chile). The selected site is located in the region of the Sierra Vicuña Mackenna, at ~12 km from the Cerro Paranal hill where the Very Large Telescope (VLT) observatory is implanted.The Desert of Atacama is the driest non-icy desert on Earth (5-20% air humidity); it features high UV levels (under the same conditions, UV-B irradiance was found 20% more elevated than in the Canary Island by[START_REF] Cordero | The Solar Spectrum in the Atacama Desert[END_REF]) and oxidizing conditions altering its surface where evaporitic materials are exposed. For all these reasons, it is considered as a good geological Martian analogue. According to the official geological map established by the Chilean government (see Figure3.44), 5 main geological units are present in the region where ExoFit operations took place: magmatic rocks (mainly diorites, resulting from the partial melting of silicate materials above the subduction zone) from the Jurassic and the Cretaceous Inferior (~202 to 124 Ma), and alluvial or fluvial sedimentary deposits from the Cenozoic (~23 Ma to this date). This combination of magmatic and sedimentary materials motivated the selection of this site to simulate ExoMars 2022 operations on Oxia Planum.

Figure 3 . 44 :

 344 Figure 3.44: Geological map of the Sierra Vicuña Mackenna region, extracted from the geological map of Chile, with the geological units present around the site of operations (indicated by a white star).

Figure 3 .

 3 Figure 3.45: Satellite images of the site where the ExoFit 2019 simulation of ExoMars operations took place, in the Desert of Atacama. A yellow star to indicates the location where the 12 m WISDOM traverses were performed.

Figure 3 .

 3 Figure 3.46: a. Drone view of the site where 3 WISDOM traverses were performed, with the exact zone (highlighted in yellow), b. Picture of the same zone, with the path of WISDOM indicated for the 3 traverses.

Figure 3 . 47 :

 347 Figure 3.47: Table summary of the elements detected in the collected samples with the RAMAN-LS instrument in two different modes (ExoFit and planned ExoMars conditions), and with additional techniques for a cross-check.

Figure 3 .

 3 Figure 3.48 displays compositional clues on the sounded site, with surface rocks partly buried or entirely exposed at the surface (a, b and d), evaporites at a few millimetres of depth in the subsurface (c), sand grains possibly made of quartz (e), and different rocks collected on-site during the campaign (f). Among

Figure 3 .Figure 3 .

 33 Figure 3.48: a. Surface rocks and outcrops present on the selected ExoFit site, b. Picture taken during the 25/02 traverse showing a standing outcrop (indicated by a white arrow), c. Evaporitic material observed in the first millimetres of the subsurface in various locations on the selected site, d. Large surface boulder visible between the 25/02 and 27/02 afternoon paths, e. Microscope image of a sand sample collected on site, f. Samples collected during the ExoFit campaign named from 1 to 6.

Figure 3 . 49 :

 349 Figure 3.49: Radargrams acquired during the 3 traverses performed with WISDOM in the Desert of Atacama, displaying the real part of the signals (co-polar channels). Blue arrows indicate a region of strong reflection/scattering observed in the two co-polar channels. The red arrow indicates the two successive diffraction curves visually identified.

Figure 3 .

 3 Figure 3.51 further shows that the strongest variations in the general trend of the dielectric constant along the traverse were observed in the morning of 27/02/2019. The moving mean dielectric constant rises up to 4 around 6 m of horizontal distance, with a neat increase beginning at ~3 m and ending at ~9 m. In addition, for this same traverse we note some differences between the values inferred from each polarimetric channel, especially in the first meter of the track and around 6 m. Such difference suggests a dissymmetry at the surface rather than structures in the subsurface. Considering the relative flatness of

Figure 3 . 50 :

 350 Figure 3.50: Estimate of the dielectric constant from the surface echoes for the 3 traverses performed in the Atacama on the 25/02 (middle) and 27/02/2019 (top and bottom) in both polarizations.

Figure 3 .

 3 Figure 3.51: General trend in the estimation of the dielectric constant from the surface echo in the 3 selected radargrams, with a moving average of 1 m large window.

Figure 3 .

 3 Figure 3.52 further illustrates the difference in terms of derived dielectric constant between the traverse performed on the 27/02/2019 morning and the other two traverses. The distribution of derived dielectric values is centred around 2.75 for the 25/02 and 27/02 afternoon traverses, while it is more spread out for the 27/02 morning traverse, with non-negligible amount of estimated dielectric constants above 3. Consistently, the derived mean dielectric constant and its standard deviation are both larger for the 27/02 morning path.

Figure 3 . 52 :

 352 Figure 3.52: Histograms representing the dielectric constants estimated from the surface echoes in the soundings of the 3 selected traverses from the ExoFit Atacama campaign.

Figure 3 .

 3 Figure 3.54 offers then a closer look at the main reflection observed below the surface echo in all 3

Figure 3 .

 3 Figure 3.53: a. Crushed sample 1 collected in the Desert of Atacama and analysed with the TUD permittivity probe, b. Dielectric constant measurements of the crushed sample 1 named from M1 to M5.

Figure 3 . 54 :

 354 Figure 3.54: Zoom on the section of radargrams acquired on the 27/02/2019 morning (polar 00 and 11), where a strong reflector is visible below the surface echo (indicated by white arrows), with depths estimated for a dielectric constant of 3.

Figure 3 . 55 :

 355 Figure 3.55: Application of the Hough transform to a selected section of the 25/02 radargram.

  presented in Chapter 2, WISDOM is a polarimetric Stepped Frequency Continuous Wave (SFCW) radar operating in the UHF (Ultra High Frequency) frequency domain and designed to investigate the first meters (typically 3-10 m below the surface) of the Martian subsurface with a centimetric vertical resolution. When performing a sounding, WISDOM transmits and receives a series of =

CHAPTER 4 :

 4 RANGE RESOLUTION ENHANCEMENT OF WISDOM RADAR SOUNDINGS BY THE BANDWIDTH EXTRAPOLATION TECHNIQUE: VALIDATION AND APPLICATION TO FIELD CAMPAIGN MEASUREMENTS 206

  .4.a)) or of the following samples model for a model in the backward direction (see equation (4.4.b)).

CHAPTER 4 :

 4 RANGE RESOLUTION ENHANCEMENT OF WISDOM RADAR SOUNDINGS BY THE BANDWIDTH EXTRAPOLATION TECHNIQUE: VALIDATION AND APPLICATION TO FIELD CAMPAIGN MEASUREMENTS 209

Figure 4

 4

Figure 4 . 1 :

 41 Figure 4.1: WISDOM processing pipeline including the application of the BWE technique.

Figure 4 . 2 .

 42 a displays the resulting synthetic radargram (obtained with a Hamming window, zero padding but without BWE). It shows constructive and destructive interference features between the two echoes (that are also observed on the corresponding experimental radargram, Figure4.12). The instrument is unable to resolve the echoes when the distance between the reflectors is less than ~11 cm in vacuum, which is consistent with equation (4.2) for = 1.

Figure 4 . 2 :

 42 Figure 4.2: Synthetic WISDOM radargram generated with two echoes and a white Gaussian noise (SNR = 30 dB) a) without BWE b) with BWE.

Figure 4 .

 4 Figure 4.2.b displays the radargram that is obtained with the same dataset after BWE to a bandwidth 3

Figure 4 .

 4 3 shows that both the module and phase are well extrapolated by the BWE, leading to a resolution enhanced by a factor of 3 in time domain.

Figure 4 . 3 :

 43 Figure 4.3: Synthetic spectra (module (a), phase (b) ) and corresponding temporal responses obtained by IFT (c), generated for d = 15 cm. The initial spectrum in red is 3 times the bandwidth of WISDOM (7.5 GHz), and the extrapolation by BWE after reduction of this spectrum to the Bandwidth of WISDOM (2.5 GHz) and addition of a white Gaussian noise (SNR = 30 dB) is displayed in dashed black. 5% of the samples are removed on each side of the spectrum before extrapolation.

Figure 4 . 4 :

 44 Figure 4.4: Correlation between a synthetic WISDOM radargram similar to Figure 4.2, displaying two echoes and white noise (SNR=30 dB) and its reconstruction by BWE after removal of 2/3 of its bandwidth as a function of the order of the autoregressive (AR) model.

Figure 4 . 5 :

 45 Figure 4.5: Error on the estimation of the distance between two reflectors generating synthetic echoes of the same amplitude as a function of the distance between these reflectors. 1000 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered; the first echo has a random phase.

Figure 4 .

 4 Figure 4.6 displays the error on the position of both echoes as a function of d. It can be observed that the absolute error on the position stays below 1 cm as it oscillates. It is even lower than 0.5 cm for d > 5 cm. This worst case result (i.e., in vacuum) demonstrate the ability of the BWE to increase the resolution of the WISDOM soundings without impairing their precision.

Figure 4 . 6 :

 46 Figure 4.6: Error on the position of the synthetic echoes before and after application of the BWE (Figure 4.2) as a function of the distance d between these reflectors. 1000 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered. The first echo has a random phase.

Figure 4 . 7 :

 47 Figure 4.7: Amplitude ratio measured between two synthetic echoes after application of the BWE (Figure 4.2.b) as a function of the distance d between these reflectors. 1000 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered. The first echo has a random phase. The echoes have been initially generated with the same amplitude (Figure 4.2.a) so that the expected amplitude ratio is 1.

Figure 4 .

 4 Figure 4.8 displays amplitude of the first (Figure 4.8.a) and the second echo (Figure 4.8.b) extracted from Figure 4.2.b after application of the BWE. The amplitude of the two synthetic echoes was initially

Figure 4 . 8 :

 48 Figure 4.8: Amplitudes of the first (a) and second (b) synthetic echoes after application of the BWE (Figure 4.2b) as a function of the distance d between the two reflectors. 1000 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered. The first echo has a random phase. Both echoes have been initially generated with an amplitude of 1 (Figure 4.2.a) and this amplitude is well

  , lower SNR values can be expected in soundings performed during Martian WISDOM operations. It is therefore important to evaluate how a lower SNR would affect the previously shown performances of the BWE. For this reason, synthetic WISDOM soundings for two reflectors in freespace separated by distances d have been generated with the same analytical model presented in Section 4.1, and values of SNR between 1 and 40 dB with a step of 1. For each value of SNR, 100 cases of noise have been tested for the sake of computation time.

Figure 4 .

 4 Figure 4.9 displays the measured error U V -U on the distance between the two reflectors estimated from the time delays of the corresponding echoes, as a function of d and the SNR.

Figure 4 . 9 :

 49 Figure 4.9: Error on the estimation of the distance between two reflectors generating synthetic echoes of the same amplitude as a function of the distance between these reflectors. 100 synthetic cases with a Gaussian noise for each SNR between 1 and 40 dB with a step of 1 have been considered; the first echo has a random phase. (a) Mean value of the error, (b) Standard Deviation of the error.

Figure 4 . 10 :

 410 Figure 4.10: Amplitudes of the first (a,b) and second (c,d) synthetic echoes after application of the BWE as a function of the distance d between the two reflectors and the SNR. 100 synthetic cases with a Gaussian noise for a SNR between 1 and 40 dB have been considered. The first echo has a random phase. Both echoes have been initially generated with an amplitude of 1. The mean value of the amplitude (a,c) and the standard deviation (b,d) are displayed.

Figure 4 .

 4 Figure 4.11: Illustrations of the experimental set-up for the "two reflectors" experiment performed at LATMOS. a. Experimental set-up without support between the plates b. Experimental set-up with a support between the plate.

Figure 4 . 12 :

 412 Figure 4.12: Experimental radargram before (a) and after (b) application of the BWE.

( 4 . 1 )

 41 and 1.5 , section 2). After application of BWE (Figure 4.2.b and Figure 4.12.b), the echoes are resolved down to a distance between 3 and 4 cm which translates into a resolution improvement by a factor ~3.

Figure 4 .

 4 Figure 4.13 shows two examples of experimental soundings obtained for the specific cases of d = 6 cm (Figure 4.13.a) and d = 10.5 cm (Figure 4.13.b). In Figure4.13.a, the distance of 6 cm is well below the resolution expected in vacuum with common spectral processing techniques and windowing according to the criterion defined in section 2.2. After application of the BWE, the two echoes are clearly resolved, and their pulse width is significantly reduced which gives a further illustration of the benefit of this technique. The distance between the two reflectors is only slightly underestimated by ~0.37cm. The amplitude of the first echo before BWE is higher than the amplitude of each echo after BWE, which can be explained by the fact that before BWE the two echoes are interfering.

Figure 4 .

 4 Figure 4.13: Experimental WISDOM soundings before and after application of the BWE for distances of 6 cm (a) and 10.5 cm (b) between the two reflectors.

  noise, with a SNR = 30 dB. Mean values of the distance error with associated standard deviations are displayed in Figure4.14 as a function of the distance between the reflectors. For the experimental data, only one measurement is performed for a given distance d which has been measured with an uncertainty of about 0.2 cm. This uncertainty on the distance between the metallic plates leads to the same uncertainty on the error between the distance and its estimation. Soundings performed with a foam support between the two metallic plates are indicated by an arrow.

Figure 4 . 14 :

 414 Figure 4.14: Error on the distance estimation between the reflectors before and after application of the BWE as a function of d for both synthetic (1000 cases of noise with SNR = 30 dB) and experimental sounding.

Figure 4 .

 4 14.a. It displays clear scattering patterns but because each step of the staircase had a height of 10 cm, they are barely resolvable in the air without BWE. First, as a further test of the BWE value, the soundings acquired on the metallic staircase have been reduced in bandwidth: a third of the frequencies were removed on each side of the spectra (Figure 4.14.b) before being reconstructed by BWE (Figure 4.14.c). In the original radargram (Figure 4.16.a) obtained without BWE, even if the echoes from two successive steps are not vertically resolved, they can be located thanks to the horizontal variations as the radar moves at the surface. After reduction of the bandwidth (Figure 4.16.b), the resolution is drastically coarser and identifying the staircase is very challenging. After reconstruction by BWE (Figure 4.16.c), individual step can be located anew. Comparison between (Figure 4.16.a) and (Figure 4.16.c) demonstrates the accuracy of the reconstruction by BWE and that the technique does not introduce artefacts. The SNR after reconstruction by BWE also seems higher, which is expected as the AR models are determined by the Burg algorithm with an order corresponding to a third of the bandwidth, meaning that soundings are extrapolated accounting for a large portion of previous/following samples, and thus reducing the effect of noise from the bandwidth reduced signal in the reconstruction.

Figure 4 .

 4 Figure 4.16: TUD "staircase" radargram as acquired (a), after reduction of the bandwidth by removing one third of the frequencies (b), and after reconstruction by BWE (c).

Figure 4 .

 4 Figure 4.15).

Figure 4 .

 4 Figure 4.17: TUD "staircase" radargram after BWE (a) and its interpretation (b).

Figure 4 .

 4 Figure 4.18: TUD "buried staircase" radargram before (on the left) and after BWE (on the right).

Figure 4 .

 4 Figure 4.19: Illustrations of the icy-environments WISDOM field campaigns a. The Chamonix campaign (France) b. The Dachstein Giant Ice Caves (Austria).

6. 2 . 1 .

 21 The Chamonix campaignThe first WISDOM campaign performed in an icy environment took place in Chamonix (French Alps) in winter 2011. To ease the movements of the instrument in this snowy environment, a prototype version of WISDOM was mounted on a sledge.

Figure 4 . 20 :

 420 Figure 4.20: Radargram from the Chamonix field campaign, before (a) and after (b) application of the BWE. Panel (c) proposes interpretation. The time delays have been converted into distance assuming a dielectric constant of 3.1.

Figure 4 .

 4 20 shows clear improvement in resolution after BWE which leads to a better separation and therefore detection of the layers. New details even appear in some regions of the radargram (see green arrows in Figure 4.20.b).

Figure 4 .

 4 Figure 4.21: Radargram from the Dachstein field campaign before (a) and after (b,c) application of the BWE. In (b) the green arrows point to details of the scattering structures that where not resolved before application of the BWE. In (c) time delays have been converted into distance assuming a dielectric constant of 3.1 and structures of interest are indicated.

6. 3

 3 Figure 4.22: WISDOM on the flank of the Etna volcano.

Figure 4 . 23 :

 423 Figure 4.23: Radargram from the Etna field campaign before (a) and after (b, c) application of the BWE. In (b) the green arrows point to details of the scattering structures that where not resolved before application of the BWE. In (c) time delays have been converted into distance assuming a dielectric constant of 6 (as derived from the surface echo amplitude) and structures of interest are indicated.

  ,B@CDE) : being the coefficients of the forward model. The column indexes correspond to the polarimetric channel the coefficient is associated with, and the row indexes to the polarimetric channel to be extrapolated.Similarly in the backward direction, for + 1 < 7 < 8, * ) is modelled by a linear combination of

Figure 5 .

 5 Figure 5.1 displays the radargrams generated at 00 and 11 polarizations, along with the radargrams obtained after BWE and PBWE. It can first be noted that in both polarization channels, if the two echoes are better resolved after application of the BWE, the PBWE allows an even better separation of the echoes, with the oscillations corresponding to the phase coupling between them disappearing.

Figure 5 . 1 :

 51 Figure 5.1: Synthetic radargrams generated with an analytic signal model, for 2 reflectors having different polarimetric properties, in 00 and 11 channels before and after application of the BWE and the PBWE techniques.

Figure 5 .

 5 Figure 5.2 clearly demonstrates the improvement in the estimation of the distance between the two reflectors after application of the PBWE. While the mean error on the estimate oscillated between -0.2 and 1.6 cm after BWE, the mean error is ranging between less than -0.1 and ~0.2 cm with the PBWE.

Figure 5 . 2 :

 52 Figure 5.2: Error on the estimation of the distance between two reflectors generating synthetic echoes of the same amplitude as a function of the distance between these reflectors. Application of the BWE and the PBWE to the 00 and 11 channels, estimation of the distance with the 00 channel. 20 synthetic cases of noise such as SNR = 30 dB have been considered; the first echo has a random phase.

Figure 5 . 3 :

 53 Figure 5.3: Amplitude ratio measured between two synthetic echoes after application of the BWE and PBWE as a function of the distance d between these reflectors. 20 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered. The echoes have been initially genererated with the same amplitude so that the expected amplitude ratio is 1.

Figure 5 .

 5 Figure 5.3 and Figure 5.4 show how the amplitude ratio between the two echoes, and the amplitude of each echo independently are almost perfectly reconstructed after application of the PBWE. In particular,

Figure 5 .

 5 5 shows that the average performances of the PBWE in the estimation of d and the amplitudes of echoes remain nearly the same even for phase angles of 20° and 10°. Only the standard deviation increase as the angle diminishes, with values of up to more than 0.4 cm for the estimation of d and nearly 7% of the amplitude of the echoes. Nevertheless, the level of error remains very satisfying.

Figure 5 . 4 :

 54 Figure 5.4: Amplitudes of the first (a) and second (b) synthetic echoes after application of the BWE as a function of the distance d between the two reflectors. 20 synthetic cases with a Gaussian noise such as SNR = 30 dB have been considered. The first echo has a random phase. Both echoes have been initially generated with an amplitude of 1.

Figure 5 . 5 :

 55 Figure 5.5: Results obtained with PBWE for the 00 channel of the same model as in Figure 5.1 but for different values of the phase of the scattering coefficient of the second echo in channel 11, a. The error on the estimation of the distance between the two reflectors as a function of this distance, b. Amplitude of the first and second echo as a function of the distance d between the reflectors.

Figure 5 . 6 :

 56 Figure 5.6: Results obtained for the 00 channel of the same model as in Figure 5.1 but for different values of SNR corrupting both channels, a. The error on the estimation of the distance between the two reflectors as a function of this distance, b. Amplitude of the first and second echo as a function of the distance d between the reflectors.

Figure 5 . 7 :

 57 Figure 5.7: Schematic of the subsurface model used to generate a FDTD simulation of a WISDOM traverse over 30 metallic cubes (5x5x5 cm) buried in a homogenenous soil of dielectric constant 3.

Figure 5 . 8 :

 58 Figure 5.8: Synthetic radargram generated by FDTD simulation for the polarimetric channel 00 (on the left), The radargram after application of the BWE (in the middle), and the radargram after application of the PBWE (on the right). The green arrows indicate details enhanced by PBWE.

Figure 5 . 9 :

 59 Figure 5.9: Synthetic radargram generated by FDTD simulation for the polarimetric channel 11 (on the left), The radargram after application of the BWE (in the middle), and the radargram after application of the PBWE (on the right). The green arrows indicate details enhanced by PBWE.

Figure 5 .

 5 Figure 5.10: a. Experimental set-up of the metallic plate at different distances in front of the WISDOM antennas inside the TUD anechoic chamber, b. Schematic representing the addition of 2 soundings performed for the metallic plates at different distances from the WISDOM antennas.

Figure 5 . 11 : 2 . 5 .

 51125 Figure 5.11: Radargrams for the 00 and 11 polarimetric channels generated by the addition and the subtraction of experimental WISDOM soundings with one reflector in front of the antennas, the same radargrams after the application of the classic BWE, and after the application of the PBWE.

5 )

 5 The observability matrix Ω • and controllability matrix ; • are then derived from the Hankel matrix as m • = Ω • ; • . Therefore, and ; • ")… are respectively the controllability matrix without the first and the last columns. 7) Then, the matrices B and C are matched to the data vector y:

Figure 5 . 12 :

 512 Figure 5.12: Approximation error Sm -m • S as a function of the number of elements considered in m • , obtained for a synthetic WISDOM sounding with 10 random point scatters (indicated by a blue dashedline) between 0 and 3 m of depth in a subsurface of dielectric constant 4 (SNR = 30 dB). The values for the number of elements which are breaking points candidates depending on the chosen threshold are indicated with red arrows.

Figure 5 . 13 :

 513 Figure 5.13: Estimation of the retrieved number of scatter points by the AIC, BIC and MDL in soundings generated for number of echoes from 1 to 60, SNR of 30 or 10 dB, and 1000 cases of noise considered (Monte-Carlo approach); (a) estimation of the number of scatter points for a SNR = 30 dB, (b) estimation of the number of scatter points for a SNR = 10 dB, (c) estimation error on the number of scatter points for a SNR = 30 dB, (d) estimation error on the number of scatter points for a SNR = 10 dB.

3. 2 .

 2 Validation on synthetic WISDOM soundings generated with analytical models 3.2.1. Synthetic WISDOM soundings with two point scatters The SSBWE technique has first been applied to the same synthetic soundings as previously used to validate the BWE technique (see Section 4 of Chapter 4) featuring two echoes from two reflectors separated by an increasing distance d ranging from 0 to 15 cm with an SNR = 30 dB.

Figure 5 . 14 :

 514 Figure 5.14: Synthetic radargram generated with an analytical signal model for 2 reflectors in a white Gaussian noise (SNR=30 dB), the same radargram after the application of the BWE, and the same radargram after the application of the SSBWE.

Figure 5 .

 5 Figure 5.14 shows, for this simple example, a clear improvement between the BWE and the SSBWE:the effect of the noise is greatly reduced. Indeed, the oscillations in phase it amplifies in the radargram after BWE have disappeared in the radargram after SSBWE. This induces a gain of the SSBWE technique with respect to the BWE in terms of estimation of the distance d between reflectors, as will be shown in the following.

Figure 5 . 15 :

 515 Figure 5.15: Synthetic radargram generated with an analytical signal model for 2 reflectors in a white Gaussian noise (SNR = 30 dB), the position and amplitudes and decay of the echoes detected when generating the State-Space signal models, the AIC was used to estimate the number of echoes in each sounding of the synthetic radargram.

Figure 5 . 16 :

 516 Figure 5.16: Error on the estimate of the distance between the two reflectors from time delays of their echoes, before application of the BWE, after BWE, and after SSBWE (SNR = 30 dB).

Figure 5 .

 5 Figure 5.18 displays the error on the estimated distance between the two reflectors from the time delays of their corresponding echoes, as a function of the SNR and of the distance d between them.

Figure 5 .

 5 Figure 5.17: a. Measured amplitude coefficient ratio between the two synthetic echoes, after application of the BWE, and after SSBWE, b. Retrieved amplitude coefficient of the two synthetic echoes, with a set amplitude of 1 for each of them.

Figure 5 . 18 :

 518 Figure 5.18: Error on the estimation of the distance between the two reflectors after application of the SSBWE, as a function of the SNR and the distance d between the reflectors: the mean value of the error (on the left), and the standard deviation on the error (on the right).

Figure 5 . 19 :

 519 Figure 5.19: Amplitude coefficients of the first (a,b) and second (c,d) synthetic echoes after application of the SSBWE as a function of the distance d between the two reflectors and the SNR. 100 synthetic cases with a Gaussian noise for a SNR between 1 and 40 dB have been considered. The first echo has a random phase. Both echoes have been initially generated with an amplitude of 1. The mean value of the amplitude (a,c) and the standard deviation (b,d) are displayed.

CHAPTER 5 :

 5 ASSESSMENT OF IMPROVEMENTS TO THE BANDWIDTH EXTRAPOLATION TECHNIQUE FOR WISDOM RADAR SOUNDINGS 268 distances superior or equal to a third of the WISDOM resolution in the subsurface. The equivalent distance in the subsurface of each reflector zk which is required for the estimation of ξ is calculated from the time-delays $ found by the model, and a prior knowledge on the dielectric constant, to guarantee a realistic scenario.

Figure 5 . 20 :

 520 Figure 5.20: Synthetic radargram generated with an analytical signal model, with 5 couples of reflectors at different depths and having set amplitudes of 1, distances between them ranging from 0 to 15 cm, and considering a homogeneous permittivity of 4+0.03j. The position, amplitude coefficients and decay of the echoes estimated with the State-Space models are also displayed. The AIC was used to estimate the number of echoes.

Figure 5 . 21 :

 521 Figure 5.21: Estimation of the imaginary part of the permittivity from the amplitude of the first echo for 3 depths, at distances for which they are all well separated. The input subsurface permittivity is 4+0.03j.

Figure 5 . 3 . 3 .

 533 Figure 5.22: Estimation of the imaginary part of the permittivity from the amplitude of the first echo for 3 depths, at distances for which they are all separated. The input subsurface permittivity is 3+0.0015j.

Figure 5 . 23 :

 523 Figure 5.23: Subsurface model used to generate FDTD simulated radargrams, with a first homogeneous layer of thickness d and dielectric constant 3, lying over a second homogeneous layer of dielectric constant 6.

Figure 5 .

 5 Figure 5.24 displays the original synthetic radargram, the same radargram after application of the classical BWE processing, and after SSBWE processing. At first glance, it is clear that both the BWE and the SSBWE lead to an improvement of the spatial resolution by a factor ~3. As expected, the radargram after BWE shows errors in the phase of the echoes, oscillating with the distance between the two interfaces (an effect already shown on synthetic soundings generated with analytical models in Section 3.2). Such oscillations are absent from the radargram obtained after SSBWE, where the two separating echoes appear as nearly straight lines, as would be expected from this scenario.

Figure 5 .

 5 Figure 5.24: FDTD simulated radargram for the presented two-layers subsurface model, the radargram after application of the BWE, and after application of the SSBWE.

Figure 5 . 25 :

 525 Figure 5.25: Error on the thickness of the first layer, measured from time-delays between the two echoes, for the original FDTD simulated radargram, after application of the BWE, and after SSBWE.

Figure 5 .

 5 Figure 5.26: Estimation of the dielectric constant of the first layer from the amplitude of the surface echo using FDTD simulated radargram, after application of the BWE, and after SSBWE.

Figure 5 . 27 :

 527 Figure 5.27: Slice of the subsurface model generated by FDTD simulations, displaying two heterogeneous layers of dielectric constants 3 and 6, characteristic lengths of the heterogeneities 2.5 and 16.6 cm, and a thickness of 5 cm for the first layer.

Figure 5 .

 5 Figure 5.28: Shallow FDTD simulated radargrams with the subsurface model for two heterogeneous subsurface layers (Figure 5.27), before application of the BWE, after BWE and after SSBWE.

Figure 5 .

 5 Figure 5.28 confirms that, with the original generated radargram, as expected, it is nearly impossible to

Figure 5 . 29 :

 529 Figure 5.29: Estimation of the dielectric constant of the first layer from the amplitude of the surface echo, before application of the BWE, after BWE and after SSBWE.

Figure 5 .

 5 30 is shown the current state of the application of the SSBWE to the "2 reflectors" experiment performed at LATMOS, to which the BWE was applied in Section 5.1 of Chapter 4. The technique is efficient on down to ~4.5 cm of distance between the reflectors, where the estimation of the order of the SSBWE visibly becomes more difficult, leading to an unstable extrapolation in frequency domain and thus artefacts of very strong amplitude in time domain.

Figure 5 . 30 : 3 . 5 .

 53035 Figure 5.30: Experimental radargram from Section 5.1 of Chapter 4 (on the left), and application of the SSBWE to this radargram (on the right).

  Sounding radars are remote sensing instruments capable of unveiling the structure and dielectric properties of a planetary subsurface over a few meters to a several kilometres of depth. The interpretation of acquisitions from this type instrument provides essential clues on the geologic history of planetary objects, complementary to surface observations. It is therefore understandable that space agencies such as NASA (United-States), ESA (Europe), CNSA (China) and JAXA (Japan) all selected sounding radars for planetary missions between 1972 and today, with a variety of targets: The Moon, Mars, the 67P comet, Jupiter's icy moons and, more recently, Venus. Subsurface penetrating radars have first been operated from orbit, acquiring soundings at a global scale. Since the selection in 2003 of the WISDOM GPR for the ExoMars rover mission, several sounding radars operating at the surface, for the study of a local geological site, have been selected.

  The WISDOM Flight Model functional tests occurred at the very beginning of this thesis in late 2018, and the instrument was integrated to the Rosalind Franklin rover in 2019. Measurement campaigns with models of the instrument very close to the flight model were thus possible. In addition to a series of laboratory measurements, two main campaigns have been organized during the thesis and have provided valuable data sets: 1) on a well-documented artificial environment in Germany in collaboration with the TUD (namely the "TU Dresden 06/2019"), 2) in the Desert of Atacama during the ExoFit simulation of operations campaign ("ExoFit Atacama 02/2019") that provided the possibility to acquire 3 long parallel traverses in a Martian analogue terrain with a rover relatively similar to the Rosalind Franklin.

1 .

 1 Introduction au sondage des sous-surfaces planétaires par radar Sonder la sous-surface au radar : Un radar est un instrument actif éclairant des cibles ou une scène avec des signaux électromagnétiques micro-ondes ou radio via une antenne émettrice, et mesurant la part de ces signaux renvoyés dans la direction d'une antenne réceptrice. Le premier radar remonte au brevet de Hülsmeyer en 1904, et à peine 6 ans plus tard Leimbach et Löwy proposent, dans un nouveau brevet, l'application de cette technologie au sondage du proche sous-sol terrestre. En effet, les ondes électromagnétiques envoyées vers le soussol interagissent avec celui-ci, et l'interprétation des signaux reçus par le radar peut en révéler les discontinuités, la structure. Pour des milieux non-magnétiques, les propriétés intrinsèques des matériaux sondés ayant une influence sur la propagation des ondes électromagnétiques sont les suivantes : • La permittivité diélectrique : Elle caractérise la propension d'un milieu à modifier la position d'équilibre de ses particules chargées en réaction à l'application d'un champ électrique. En régime harmonique, elle est une fonction complexe de la fréquence notée : = + " . Elle est souvent exprimée en terme de « permittivité relative », normalisée par celle du vide . On la note alors = . • La conductivité électrique : La conductivité caractérise la propension d'un matériau à laisse circuler des charges libres à travers lui. En régime harmonique, c'est aussi une valeur complexe : = + " . En pratique, les effets des quantités et " sur l'accumulation d'énergie dans le matériau, ainsi que les effets de " et sur les pertes d'énergie dans le matériau ne pouvant être séparés par simple mesure, on considère souvent une permittivité « effective », que représentera par la suite et dont nous noterons la partie réelle et " la partie imaginaire. Les radars sondeurs pour l'exploration planétaire : Il faut attendre 1972 pour que des « radars sondeurs » ou GPR (pour Ground Penetrating Radar) soient pour la première fois appliqués au sondage d'un autre corps planétaire que la Terre : la Lune avec la mission Apollo 17. Avec le développement des technologies numériques, de nombreux autres radars sondeurs ont été sélectionnés pour des missions d'exploration planétaire ces 20 dernières années, dévoilant la sous-surface de la Lune (missions Apollo 17 et SELENE), Mars (missions Mars Express et Mars Reconnaissance Orbiter), de la comète 67/P (mission Rosetta) ou même de Titan (mission Cassini, bien que le sondage de ce dernier n'était à l'origine pas intentionnel). Les premiers radars sondeurs planétaires envoyés opéraient en orbite autour de l'objet d'étude, ce qui permet le sondage de la sous-surface à l'échelle globale. Plus récemment des GPR opérant depuis la surface ont été envoyés sur la Lune (missions Chang'E 3, 4 et 5) et sur Mars (missions Mars 2020 et Tianwen-1), pour la majorité embarquée sur des robots nommés « rovers ». Ils permettent de réaliser des sondages du proche sous-sol de l'objet d'étude à échelle locale, lors des déplacements du rover. De nouveaux radars sondeurs orbitaux devraient également partir dans les décennies à venir, vers les lunes glacées de Jupiter (missions JUICE et Europa Clipper), ainsi que Venus (mission EnVision). Cet engouement pour les GPR planétaires s'explique par l'apport de ces instruments qui ouvrent une troisième dimension, celle de la profondeur, dans l'exploration des cibles visées. A leur actif on peut, entre autres, citer : les études stratigraphiques des calottes polaires Martiennes, la découverte d'un lac souterrain hyper-salé dans la région d'Ultimi Scopuli sur Mars ou la première bathymétrie des lacs de Titan. La conception d'un radar sondeur planétaire : Tous ces radars étant voués à sonder des sous-sols de nature variée, avec des objectifs et contraintes de mission différents, il est intéressant de comparer les choix de conception retenus pour chacun d'entre eux, afin d'en comprendre les points communs et les différences. Les principaux paramètres d'un radar sondeur à définir sont les suivants : • Fréquence centrale et profondeur de pénétration : La « fréquence centrale » correspond à la médiane de la gamme de fréquences dans le spectre du signal émis par le radar. La profondeur de pénétration théorique d'un radar sondeur en dépend (ainsi que de la permittivité de la sous-surface), d'où l'intérêt de choisir une fréquence centrale adaptée. Une bonne approximation de la profondeur de pénétration est la « profondeur de peau électrique » qui est la profondeur à laquelle la puissance du signal émis est atténuée d'un facteur 1/e (soit ~0.37). Elle s'exprime, dans des milieux à faibles pertes : centrale est basse, plus le radar sondera donc profondément. Cette dernière est cependant limitée par d'autres facteurs, comme la taille maximale possible pour les antennes, qui est aussi inversement proportionnelle à la fréquence.

•

  Puissance transmise et bilan de liaison : On appelle bilan de liaison le compte de tous les gains et pertes qui s'appliquent à un signal électromagnétique de son émission à sa réception. Pour les radars sondeurs, il permet de vérifier que le signal reçu pour la profondeur de pénétration visée sera bien détecté par l'instrument et suffisamment élevé par rapport au niveau de bruit. Les gains des antennes, des amplificateurs dans les chaînes d'émission et de réception, ainsi que la puissance du signal émis entrent donc dans ce bilan. WISDOM, sujet de cette thèse : En 2003, dans le cadre du programme russo-européen ExoMars, le radar sondeur WISDOM (Water Ice and Subsurface Deposits Observation on Mars) était sélectionné pour être intégré au premier rover Martien de l'Agence Spatiale Européenne (ESA), Rosalind Franklin. L'instrument est intégré au rover depuis janvier 2020, jusqu'à son lancement qui devrait avoir lieu en septembre 2022, pour un atterrissage sur Mars prévu en juin 2023. L'instrument WISDOM est déjà intégré au rover, dans l'attente de son lancement. En attendant, les équipes instrumentales du LATMOS (France) et de la TU de Dresde (Allemagne) préparent l'interprétation des futurs observations martiennes en développant des outils logiciels, qui sont validés sur des données WISDOM synthétiques et expérimentales. Cette thèse porte sur la mise en place et la validation de chaînes de traitement et d'interprétation des données de WISDOM, en vues des futures opérations sur Mars.

  qui a motivé cette mission est l'idée que des biomarqueurs, témoins d'une période où Mars était habitable et peut-être habitée il y a 4 milliards d'années, pourraient avoir été conservés dans le sous-sol de la planète et protégés des conditions hostiles de la surface (en termes de radiations et d'oxydation notamment). C'est pourquoi le rover de la mission ExoMars sera équipé d'une foreuse capable de collecter des échantillons jusqu'à 2 m de profondeur, une première sur Mars. Afin d'optimiser les chances de trouver des traces de vie dans le sous-sol martien, le site d'atterrissage de la mission a été soigneusement choisi : l'ESA a sélectionné Oxia Planum, une plaine située entre les vallées de débâcle d'Ares Vallis et de Mawrth Vallis. Ce site garantit à la fois un atterrissage sûr, et la présence de formations géologiques d'intérêt réparties dans toute la zone d'incertitude sur le site d'atterrissage. Oxia Planum est, en effet, parsemée de matériaux argileux altérés par l'eau (70% de la surface du site), et formés il y a environ 3.9 milliard d'années. Les dépôts d'argiles sont connus sur Terre pour capturer et préserver les biomarqueurs, et l'on espérer que le même processus se soit produit Mars. Le plan d'action pour la collecte d'échantillons est le suivant : Rosalind Franklin devra trouver un affleurement rocheux composé de matériaux argileux hydratés par le passé, chercher si une portion de cette roche continue jusqu'à une profondeur suffisante dans le sous-sol, forer, collecter les échantillons et les analyser dans le corps du rover. Dans le but de réaliser ces différentes étapes, le rover embarque 9 instruments ont été séléctionnés, dont le radar à pénétration de sol WISDOM. La conception de WISDOM : WISDOM a été conçu par le LATMOS, la TU de Dresde et le LAB (Laboratoire d'Astrophysique de Bordeaux) pour révéler la structure des premiers mètres du sous-sol d'Oxia Planum avec une résolution centimétrique, et fournir des indices sur sa composition avant toute opération de forage (Ciarletti et al., 2017). Ainsi pourra-t-il aider à reconstituer le contexte géologique des échantillons collectés, et orienter le choix d'un site de forage intéressant et sûr pour la foreuse. Sa conception a aussi dû s'adapter aux contraintes en termes de masse, volume et consommation d'énergie de la mission. Tous ces éléments ont amené à un choix de fréquence centrale, bande de fréquence, modulation, antennes et puissance d'émission comme expliqué dans la Section 1. Afin de pouvoir guider le forage avec une précision centimétrique (les échantillons collectés ayant une hauteur ~3 cm), et ce jusqu'à une profondeur de l'ordre de 2 m (la taille de la foreuse), WISDOM opérera sur une large bande de fréquences de 2.5 GHz, autour d'une fréquence centrale de 1.75 GHz. WISDOM se distingue ainsi de ses homologues américain et chinois, RIMFAX (Mars 2020) et RoSPR (Tianwen-1), arrivés sur Mars en 2021, qui n'ayant pas d'opérations de forage similaires ont choisi des fréquences centrales et bandes de fréquences visant des profondeurs de pénétration plus importantes mais au détriment de la résolution. WISDOM est aussi le seul de ces radars à utiliser une modulation « à sauts de fréquences », qui consiste à envoyer une série de signaux harmoniques entre 0.5 et 3 GHz avec un pas de 2.5 MHz. Cette technique permet de réduire la puissance nécessaire pour émettre la quantité d'énergie désirée vers le sol. La réponse impulsionnelle du sous-sol doit alors être reconstituée par transformée de Fourier inverse. L'instrument WISDOM est composé de deux éléments principaux : • L'unité électronique ou WEU (WISDOM Electronic Unit) : L'électronique interne de WISDOM (voir Figure 1.b) est divisée en trois cartes électroniques reliées entre elles. La 1 ère est le convertisseur, qui permet de fournir toutes les tensions nécessaires au fonctionnement de WISDOM. La 2 nde est l'ordinateur de WISDOM, permettant le contrôle des fréquences émises, ainsi que la numérisation, le stockage et la transmission des données. La 3 ème est la carte Radio Fréquences (RF), qui génère le signal à émettre, et réceptionner le signal retour vers les antennes. • L'assemblage antennes ou WAA (WISDOM Antennas Assembly) : Le WAA est un assemblage d'antennes, relié au WEU par des câbles coaxiaux (voir Figure 1.c). Afin d'émettre sur la large gamme de fréquences de WISDOM, des antennes Vivaldi ont été retenues. Elles permettent l'émission et la réception dans 2 polarisations différentes, autorisant ainsi 2 acquisitions co-polaires, et 2 acquisitions cross-polaires à chaque sondage. Le WAA est situé à l'arrière du rover et est incliné de 7° vers l'extérieur pour réduire l'effet des réflexions du signal sur le corps et les roues du rover.

Figure 1 :

 1 Figure 1 : a. Modèle en images de synthèses du rover Rosalind Franklin, avec les antennes de WISDOM indiquées en jaune, b. Vue de la WEU intégrée au rover, c. Vue de la WAA intégrée au rover (crédits : ESA, Mlabspace, LATMOS, TUD, Airbus).

  obtenir l'écho de plus haute amplitude recevable par le radar d'un réflecteur à la même distance, et d'obtenir la forme de l'impulsion équivalente du radar. En utilisant la forme de cette impulsion, un spectre WISDOM peut être « blanchi », c'est-à-dire corrigé des différences de puissance émise à chaque fréquence.

Figure 2 : 3 .

 23 Figure 2 : a. Sondage expérimental WISDOM, b. Représentation en niveaux de couleurs de ce sondage, c. Radargramme généré par l'assemblage horizontal de différents sondages. Données expérimentales et données synthétiques :

Figure 3 :

 3 Figure 3 : a. Schéma décrivant le modèle de propagation dans le cas d'antennes sur le sol, b. Schéma décrivant le modèle de propagation dans le cas d'antennes au-dessus du sol.La détection automatique d'une hyperbole mathématique parfaite dans une image peut se faire en utilisant un algorithme appelé la « transformée de Hough »(Capineri et al., 1998), que nous avons décidé d'appliquer aux radargrammes de WISDOM.

Figure 4 :

 4 Figure 4 : a. Résultat d'application de la version modifiée de la transformée de Hough dans le cas simulé de réflecteurs enfouis dans un milieu homogène de constante diélectrique 4, b. Résultat obtenu sur des données expérimentales pour un réflecteur dans l'air (constante diélectrique attendue de 1), c. Résultat obtenu sur des données expérimentales pour un réflécteur enfoui dans une couche de glace d'eau (constante diélectrique attendue de 3.2 pour de la glace pure). Précisions de 5 cm pour X et 0.1 pour ′. Application à la campagne ExoFit 2019, dans le désert d'Atacama :

Figure 5 : 4 .

 54 Figure 5 : a. Portion d'un des 3 longs radargrammes acquis dans le désert d'Atacama. Les échos provenant d'un potentiel réflecteur en sous-sol sont indiqués par une flèche blanche, b. Courbe de diffraction provenant d'un réflecteur à la surface détecté dans l'un des 3 radargrammes, c. Images du site sur lequel ont été relevés les 3 radargrammes, avec de potentiels affleurements rocheux indiqués par des flèches blanches.

Figure 6 :

 6 Figure 6 : a. Radargramme synthétique obtenu pour 2 échos à différentes distances l'un de l'autre, b. Application de la BWE à ce radargramme synthétique, c. Radargramme expérimental obtenu avec 2 réflecteurs devant les antennes de WISDOM, d. Application de la BWE au radargramme expérimental. L'expérience de laboratoire la plus déterminante a été réalisée avec 2 plaques métalliques l'une à côté de l'autre en face des antennes de WISDOM, et différents décalages en distance d entre les plaques. Ainsi, deux échos plus ou moins proches sont visibles dans les sondages, reproduisant les données synthétiques générées précédemment pour tester la BWE (voir les résultat en Figure 6.c et d) et

Figure 7 : 5 .

 75 Figure 7 : Application de la BWE à un radargramme de la campagne du Dachstein, a. Radargramme original, b. Radargramme après application de la BWE, c. Interprétation du radargramme après BWE.

Figure 8 :

 8 Figure 8 : Radargrammes synthétiques générés pour 2 voies polarimétriques 00 et 11, avec 2 échos à différentes distances l'un de l'autre, le second ayant un déphasage différent de 180° entre les 2 voies (à droite). Application de la BWE à ces radargrammes (au milieu), et application de la PBWE à ces radargrammes (à gauche).

Figure 9 :

 9 Figure 9 : Radargramme généré par FDTD pour un fine couche de matériau homogène d'épaisseur d comprise entre 0 et 15 cm de constante diélectrique 3 au-dessus d'un milieu homogène de constante diélectrique 3 (à gauche), Application de la BWE à ce radargramme (au milieu), Application de la SSBWE à ce radargramme (à droite).

Table 1 . 1 :

 11 Summary of the different modes of the MARSIS instrument.

		Dayside modes	Nightside modes
	Subsurface mode 1	3.3 -4.3 MHz	1.3 -2.3 MHz
	Subsurface mode 2	4.3 -5.3 MHz	2.3 -3.3 MHz
	Subsurface mode 3		3.3 -4.3 MHz
			4.3 -5.3 MHz
	Ionosphere mode		0.8 -5.3 MHz

Table 1

 1

	.2 have working frequencies between 1 and 100 MHz,

Table 1 .2: Perspective

 1 of all presented radar sounders operating in-orbit, their design and mission.

		Planetary	Working	Bandwidth	Free-space	Beginning
		body	frequencies		resolution	of
						operations
	ALSE	Moon	HF1: 5 MHz	HF1: 0.53 MHz	HF1: 300 m	1972
			HF2: 15 MHz	HF2: 1.6 MHz	HF2: 100 m
			VHF: 150MHz	VHF: 16 MHz	VHF: 10 m
	LRS	Moon	5 MHz (with 1 and	2 MHz	75 m	2007
			15 MHz options)
	PRISM	Mars	12 MHz	6 MHz	25 m
	MARSIS	Mars	1.3 -5.3 MHz	1 MHz	150 m	2005
	SHARAD	Mars	15 -25 MHz	10 MHz	15 m	2006
	MOSIR	Mars	LF1: 10 -15 MHz	LF1: 5 MHz	LF1: 30 m	2021
			LF2: 15 -20 MHz	LF2: 5 MHz	LF2: 30 m
			HF: 30 -50 MHz	HF: 20 MHz	HF: 7.5 m
	CONSERT	67P nucleus 90 MHz	10 MHz	30 m	2014
	Cassini radar Titan	13.78 GHz	4.25 MHz	35 m	2005
	RIME	Jupiter's icy	9 MHz	3 MHz	50 m	2030
		moons
	REASON	Jupiter's icy	HF: 9 MHz	HF: 1 MHz	HF: 150 m	2030
		moons	VHF: 60 MHz	VHF: 10 MHz	VHF: 15 m
	SRS	Venus	9 -30 MHz (9 MHz	6 -20 MHz (6 MHz	7.5 -25 m	~2034
			as baseline)	as baseline)

Table 1 . 3 :

 13 Perspective of all presented sounding radars operating from the surface, their design and mission.

		LPR	LRPR		RoSPR	RIMFAX	WISDOM
	Radar design	Nanosecond	Nanosecond		Linear	Frequency	Stepped
		impulse	impulse		Frequency	Modulated	Frequency
					Modulation	Continuous	Continuous
					(chirp)	Wave	Wave
	Working	CH1: 40 -80	0.5 -4 GHz		CH1: 15 -95	0.15 -1.2 GHz 0.5 -3 GHz
	frequencies	MHz			MHz
		CH2: 250 -750			CH2: 0.45 -
		MHz			2.15 GHz
	Antennas design	CH1:	Arrays	of	CH1:	Bow-tie	Vivaldi
		Monopole	Vivaldi		Monopole	antenna	antennas
		antennas	antennas		antennas
		CH2: Bow-tie			CH2: Vivaldi
		antennas			antennas
	Free-space	CH1: 3.75 m	4.3 cm		CH1: 1.88 m	14.3 cm	6 cm
	resolution (worst	CH2: 30 cm			CH2: 8.8 cm
	case)
	Expected	CH1: Hundreds	≥ 2 m		CH1: 10 -100	Tens of meters,	3 -10 m
	penetration depth	of meters			m	depending on
	(material	CH2: Tens of			CH2: 3 -10 m	the mode
	dependent)	meters
	Dynamic range	≥ 90 dB	90 dB		?	~100 dB for a	~84 dB of
						single mode,	effective range
						160 dB together
	Power	9.8 W	15 W		26.5 W	9.5 W	12.5 W
	consumption
	Mass	≤ 5.5 kg	≤ 3.1 kg		6.1 kg	3.46 kg	1.36 kg

Table 2 . 1 :

 21 Characteristics of the WISDOM instrument.

	Parameter of the instrument	Typical values for WISDOM
	Frequency Bandwidth	0.5 to 3 GHz (2.5 GHz)
	Frequency steps	2.5 MHz (1001 steps)
	Repetition time	200 μs
	Transmitted power	0 dBm
	Noise factor	8 dB
	Receiving channel gain (adjustable)	-7 to 25 dB
	Antenna gain	1 to 8 dB
	Effective range of the ADC (16 bits) ~84 dB
	IFT gain	27 dB
	Gain for 10 coherent additions	10 dB (see (Hervé, 2018))
	Transmitter and receiver efficiency	0.1

Table 2 .

 2

	2 Mean scattering losses measured with echoes from a perfect reflector below 2.5 m of a
	heterogeneous layer, in WISDOM synthetic soundings.
	Homogeneous 6 = 7. 4 89 6 = :. 4 89 6 = ;. 4 89	∆1 = 2. 4 0 dB -0.5 dB -1 dB -0.8 dB	∆1 = 5 0 dB -1.6 dB -3.1 dB -4.1 dB	∆1 = 5. 4 0 dB -5.0 dB -15.3 dB -12.2 dB

Table 2 . 3 :

 23 Main characteristics of 5 windows among the most commonly used in data processing.

	Window	Main lobe width at -6	Maximum	side-lobe	Side-lobe roll-off rate
		dB (bins)	height (dB)		(dB/decade)
	Rectangular	1.21	-13		20
	Hann	2	-32		60
	Hamming	1.81	-43		20
	Blackman	2.35	-58		60
	Flat-top	3.56	-44		20

Table 2 . 4 :

 24 Summary of the different validation tests performed on the WISDOM instrument prior to its integration

	Validation test performed	Dates of the test	Test facilities
	Shock test	November 2018	Mecano I&D, Toulouse
	Mass/inertia test	November 2018	IAS, Orsay
	Vibration test	November 2018	IAS, Orsay
	Thermal/vacuum test	November 2018	LATMOS
	Dimensions measurement	December 2018	LATMOS by G2metric
	Electromagnetic compatibility test	December 2018	Gerac, Trappes
	Sterilization	December 2018	IAS, Orsay

Table 2 .

 2 5.

Table 2 . 5 :

 25 Summary of the different possible scales of heterogeneities generated with the Diamond-

	Square algorithm in Temsi-FD
	Nit	1	2	3	4	5	6	7
	L (cm)	1.5	2.5	4.5	8.5	16.6	32.5	64.7

  Equation(3.12) is a fourth degree equation with 4 solutions, but only one is physically possible: the solution has to be real and in the interval [x;X] if x<X or in the interval [X;x] if X<x. Solving a fourth degree equation analytically is a complicated process. Therefore, in the present study, this equation is always solved numerically.

	In the same article, it is shown that the limits of the equation when the time delay 6 → 0 or 6 → ∞ are lim > ? → T x = 2 c x -X + H ( 3.13 )
					E1F G ? →H		= 6 +	2 √	-	+ !	( 3.14 )
	4 -2 +	-	5 +		+	+ 4 +	4 1 -6 -6	-	( 3.12 )
			-2	+ 2	+	6 2 1 -	-	6 2 1 -	-+
			+	6 4 1 -	-	6 4 1 -	= 0
	where 6 =	7 "	and 6 =	√# 8 "

are the time delays associated with the distances H and Z, respectively.

  and (1.5 m, 21 ns) respectively have estimated dielectric constants of 3.0±0.1 and 3.8±0.1, close to the 3.2 expected for pure water ice. One of the fits obtained for the diffraction curve around (2.8, 17 ns) returns a dielectric constant of 4.1±0.1, in the higher limit of the manually estimated values. However, the second obtained fit has an estimated dielectric constant of 2.5±0.1, inconsistent with the expected values. It seems that this estimation, if fitting the right branch of this diffraction curve, fits echoes of other origins on the left side. It is thus not a strictly speaking a false positive, but a wrong fit of the detected curve. Lastly, the detection at (2.3 m, 17 ns) could possibly correspond to an existing diffraction curve, even if unclear, as the corresponding estimated dielectric constant of 3.8±0.1 is geologically plausible. Nevertheless, this result cannot be confirmed. This application of the implemented Hough transform to the Dachstein campaign radargrams validates its capacity to detect and characterize diffraction curves. With all these encouraging results on both synthetic and experimental WISDOM radargrams, even from natural environments such as the Dachstein Giant Ice caves, we are confident the technique will prove useful when interpreting Martian radargrams in 2023.

Table 3 .

 3

1: Measured dielectric properties of the main compounds identified by the RAMAN-LS

instrument in the sample collected on-site, for frequencies in the WISDOM bandwidth.

Material Frequencies (GHz) Dielectric constants Source

	Quartz	0.5-1	3.85-3.97	(Church et al., 1988)
		2.45	4.72±0.09	(Zheng et al., 2020)
	K-feldspar	2.45	5.55±0.06	(Zheng et al., 2020)
	Muscovite	0.5-1	3.97-4.46	(Church et al., 1988)
		2.45	6.50±0.47	(Zheng et al., 2020)
	Calcite	0.5-1	8.66-9.1	(Church et al., 1988)
		2.45	8.46±0.21	(Zheng et al., 2020)

  RÉSUMÉ EN FRANÇAIS 300 toutes antennes, il génèrera dans les spectres acquis pour chacune des combinaisons 00, 01, 10 et 11 une sinusoïde complexe de même fréquence, mais d'amplitudes et de phases différentes pour 00, 01, 10 et 11. On peut donc modéliser le spectre acquis pour . réflecteurs de la sorte comme : , I 8,2 , I 8 ,2 et I 88,2 les amplitudes associées au R-ième réflecteur pour chaque polarisation, et S ,2 , S 8,2 , S 8 ,2 and S 88,2 les déphasages induits par la réflexion.

	avec I ,2	D	/ / 8 / 8 / 88	E = 0 6 278	⎣ ⎢ ⎢ ⎢ ⎡	I ,2 3 BKL ,M I 8,2 3 BKL N,M I 8 ,2 3 BKL N ,M I 88,2 3 BKL NN,M ⎦ ⎥ ⎥ ⎥ ⎤	3$4 -2	5 2

Plaut, J.J.,Picardi, G., Safaeinili, A., Ivanov, A.B., Milkovich, S.M., Cicchetti, A., Kofman, W., Mouginot, J.,

Remerciements / Acknowledgements

WISDOM experimental dataset

The design and implementation of a processing and interpretation chain for the WISDOM instrument required an experimental dataset. Since 2010, a variety of measurement campaigns have been organized by the WISDOM team, to acquire soundings in both controlled environments (with an a-priori knowledge on the reflectors in sight of the antennas) and natural sites. In this section will be presented a list of the main WISDOM measurement campaigns of the last decade, with an emphasis on the campaigns performed during this thesis.

Acquisitions in controlled or semi-controlled environments

Laboratory experiments

In order to test the different WISDOM models, from prototypes to the flight and spare models, in controlled environments (with reflectors at knows positions from the antennas), measurements have been performed in the LATMOS and TU Dresden facilities on simple targets such as plates, spheres or multihedral shapes. Such experiments provide soundings on very simple set-ups, allowing the assessment of the expected instrument characteristics and performances. In addition to experiments performed on an advanced prototype of WISDOM, validation and functional tests have been performed on the flight model of the instrument by the WISDOM team at the very beginning of this thesis.

The WISDOM flight model first underwent a variety of validation tests: Metrology of the instrument (measuring its exact dimensions), its physical properties (mass, center of gravity, moments of inertia), shock tests (with shock levels in agreement with what the instrument may endure during its journey to Mars, with margins), vibration tests (to measure the resonant frequencies of the instrument, with levels of vibration in agreement with what the instrument is expected to endure), thermal tests (the instrument acquiring soundings at temperatures and air pressures representative of what it may encounter during its journey and once on Mars), electromagnetic compatibility (to asses that the WISDOM operations will not affect and will not be affected by other rover operations), and sterilization (to respect planetary protection rules). These different validation tests are summarized in Table 2.4. Prior to the integration of the WISDOM flight model to the Rosalind Franklin rover, additional functional tests have been performed with the WEU and WAA in the LATMOS clean room (see Figure 2.34). Various types of reflectors (metallic spheres, metallic plates, di-hedral reflectors, Plexiglas plates, …) were placed at various distances in front of the WISDOM antennas. The WISDOM acquisitions performed during these functional tests allowed the measurement of the instrument performances (Hervé et al., 2020). The different field tests are briefly described in Table 4.1. In the following, a radargram from the semicontrolled environment will be presented, then the radargrams from icy environments, and lastly a radargram from a lithic environment.

Semi-controlled environment: The Dresden "Mars Yard" campaign

In June 2019, a measurement campaign took place in the Technische Universität of Dresden (TUD, Germany) facilities with a copy of the instrument flight model (electronic unit and antennas). Free-space measurements and calibrations on metallic plates were performed in an anechoic chamber. WISDOM acquisitions were performed across a 75 cm deep soil trench, where different reflectors were buried. This type of campaign is important to validate the interpretation of WISDOM radargrams, as the reflectors locations are known prior to the acquisition.

where 00, 01, 10 and 11 indices correspond to the two co-polar (00 and 11) and the two cross-polar (01 and 10) channels of the radar. , , , , , and , are the real amplitudes associated with the reflector k. ' , , ' , , ' , and ' , are the phase shift due to the reflection on reflector k associated.

The vector

is often referred to as the "scattering coefficients" associated to reflector k. This is the basis for the Polarimetric Bandwidth Extrapolation, or PBWE.

If the response from a point scatter has different scattering coefficients in the different considered channels, more information on the point scatter will be contained in the samples used to model and extrapolate the signal. It is therefore expected that the effect of noise, and the oscillations due to the interference between the different echoes, will be reduced using such model instead of the classical one.

However, in the case of WISDOM, some scatters can be present in only one polarimetric channel if they are located inside the radiation pattern of an antenna, and outside the radiation pattern of the other one.

Extrapolating the frequency responses of such scatters would not lead to any improvement. It is important to keep in mind this limitation when interpreting radargrams extrapolated with this technique.

The PBWE multichannel version of the Burg algorithm, described in the following subsection, has been adapted to the WISDOM data, and tested on both synthetic and experimental data.

For the sake of compactness, the AR-model used to extrapolate the WISDOM polarimetric spectra is presented with a matrix formalism. Instead of one set of N spectrum values ( ) = ( ) ) as in Chapter 4, a WISDOM sounding is now represented as four sets of N spectrum values

and ( ) ). The data matrix is thus defined as

We also defined, for an order of the model the Y , ,-matrix, that contents all polarimetric data used to obtain an estimate of * ) :

. ) ,/ = 0* ) -

The multichannel AR-model is therefore:

/ : (7) = * ) -* 9 ) : = * ) -; / : 2 . ) ,/ ( 5.10 ) and + 1 < 7 < 8 / F (7 -) = * ) / -* 9 ) / F = * ) / -; / F 2 . ),/ ( 5.11 ) The error matrices are defined as

H

)&/I / : (7) J ( 5.12 )

( 5.13 )

)&/I / F (7 -) J

( 5.14 ) with † representing the conjugate transpose of a matrix.

The multichannel version of the Burg algorithm consists of the following steps:

1) Initialization of prediction errors as : (7) = * ) and F (7 -1) = * ) for 7 ∈ [2; 8].

2) The value of the order is incremented by one.

3) From the prediction errors for the order -1, the coefficients for an order are determined:

( 5.15 ) ( 5.16 ) 4) The matrices of coefficients are determined for the order :

; / : = ; / : -; / F < /,/ : 2 ; / F = ; / F -; / : < /,/ F 2

( 5.17 ) ( 5.18 ) 5) Prediction errors for order are determined:

/ : (7) = * ) -; / : 2 . ) ,/ / F (7 -) = * ) / -; / F 2 . ),/

( 5.19 ) ( 5.20 ) promising results obtained on synthetic WISDOM soundings, such a criterion would be worth the search.

Recapitulative table

In order to ease the comparison between the 3 Bandwidth Extrapolation techniques presented in this manuscript (the classic BWE, the PBWE and the SSBWE), a recapitulative table summarizing their characteristics and obtained results is proposed in Table 5.1.

All techniques having assets and drawbacks, one of them cannot be considered as the best choice in all situations for the moment being. When operated on Mars, we plan to automatically produce radargrams without BWE and with BWE to enhance the vertical resolution. Then manually, more resourcedemanding algorithms such as the PBWE and SSBWE will be applied to try and get the best from the dataset.

Yes No No

Error levels between -2.3 and 1.6 cm for the estimation of the distance between reflectors in free-space have been experimentally measured. According to the results obtained for synthetic soundings simulating two reflectors in free-space, amplitudes are reconstructed with an error level below 7%.

If the BWE provided impressive results on WISDOM radargrams, the technique is limited by the signal model it uses. The effects of a white noise corrupting the signal, and/or of an exponential decay in amplitudes with frequency due to the absorption of the signal in the subsurface are indeed not accounted for in the BWE model. In order to improve the performances of the BWE technique, two more sophisticated models were proposed in Chapter 5.

• The first model named "Polarimetric Bandwidth Extrapolation" (PBWE) benefitting from the polarimetric capacity of WISDOM. For echoes arising from reflectors with different scattering behaviours in the different polarimetric channels of the instrument, the model is more accurate and less sensitive to noise. • The second model accounts for the presence of an additive white noise and, most of all, of an exponential decay of the amplitude due to electric losses in the subsurface. It uses a "State-Space" representation of the model and is named as "State-Space Bandwidth Extrapolation" (SSBWE).

Both techniques were validated on synthetic WISDOM soundings, with promising results: the errors on the estimated distances between reflectors and amplitudes of echoes are significantly reduced. Even for SNR levels down to 3 dB, with the PBWE or the SSBWE technique, the maximum average error on distances between reflectors obtained for synthetic WISDOM soundings is ~0.5 cm. For the same SNR, the maximum average error on the amplitudes of echoes is ~10%.

To our knowledge, this is the first time such models are applied for BWE to a planetary radar sounder.

Chapter 5 consisted in a preliminary validation of the PBWE and SSBWE techniques on synthetic WISDOM soundings, and modified experimental soundings. Future validations on experimental WISDOM radargrams are planned. For the PBWE technique they will be done on an experimental setup including reflectors of different scattering properties for the 2 WISDOM co-polar channels. For the SSBWE a better criterion to determine the order of the model must be found first. With an improved BWE technique to enhance the vertical resolution of WISDOM radargrams, the soundings could be even more accurately interpreted in terms of distances between echoes or amplitudes. This should therefore be a major axis of research following this thesis.

New campaigns are planned before the arrival of the rover on Mars in 2023 in collaboration with other instrument teams, in order to prepare the synergy between the instruments required for the ExoMars 2022 mission. They will generate a series of data sets on which the BWE techniques will be systematically applied. To provide a geological context prior to the Rosalind Franklin operations, the WISDOM team also initiated the analysis of SHARAD acquisitions performed in the region of Oxia Planum, which remain to this day the best radar soundings in term of resolution from the future ExoMars 2022 landing site (with a free-space resolution of 15 m). No clear subsurface structures have been identified to date, but the application of the BWE technique to the selected radargrams may reveal new details. The analysis of the surface echo, in amplitude and pulse width, will also provide information on the composition and roughness of the surface.

The different interpretation tools developed for the interpretation of WISDOM radargrams during this thesis are virtually applicable to any radar sounder. We could for instance imagine applying the automated detector of diffraction curves we proposed to the radargrams of the RIMFAX and RoSPR Martian Ground Penetrating Radars, as they are also air-coupled GPRs. The BWE and potentially the SSBWE could also be used on these GPR radargrams, and to the future acquisitions of the RIME and REASON radars, as the technique already returned impressive results on icy subsurface on the MARSIS, the SHARAD and the WISDOM radargrams. Since the MOSIR Martian radar has, like WISDOM different polarimetric channels, the PBWE could enhance even more than the classic BWE the radargrams from this instrument.

The rising interest in subsurface sounding radars for planetary exploration, recently confirmed by the selection of SRS for the EnVision mission, is justified considering the past and potential scientific return of such instruments. To adapt this technology to the constraints of space missions, ingenuous designs have been proposed since the 1970s. New post-processing and interpretation techniques are also being adapted, to learn as much as possible on planetary subsurface with the acquisitions from these instruments. In the next two decades, the interpretation of the planetary radar soundings could be even more fruitful than we dreamed of. Abstract : From summer 2023, the Rosalind Franklin rover of the ExoMars 2022 mission (ESA / ROSCOSMOS) will seek out past or present traces of life down to 2 m in the Martian subsurface with a drill, where collected samples could have been shielded from the harsh conditions of the surface. In order to give an insight into the structure and composition of the shallow subsurface prior to any drilling operations, and to provide clues on the geological context of the rover landing site, the Ground Penetrating Radar WISDOM (Water Ice and Subsurface Deposits Observation on Mars) will sound the first meters of the Martian shallow subsurface with a vertical resolution of a few cm. This thesis was dedicated to the assessment and implementation of new and sophisticated tools in the processing and interpretation chain of the WISDOM data, enhancing the quality of future Martian observations. Data acquired by WISDOM consist in "radargrams", displaying the amplitude of received signals (after reflection / scattering on subsurface structures) as a function of their time delays. During this thesis was developed an automated tool of detection / characterization of diffraction curves in radargrams, which shape gives the propagation speed of the signals, thus guarantying the display of radargrams as a function of distance. This thesis also focused on the enhancement of the vertical resolution (in range / depth) of WISDOM radargrams. A super-resolution technique based on the "Bandwidth Extrapolation" (BWE) was tested and implemented to the WISDOM data processing chain, resulting in an enhancement of the range resolution by a factor of 3. Potential improvements of the BWE, based on more sophisticated signal models, were also explored.