Advanced Backscattering Simulation Methods for the Design of Spaceborne Radar Sounders

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DEPARTMENT OF INFORMATION ENGINEERING AND COMPUTER SCIENCE ICT International Doctoral School

Advanced Backscattering

Simulation Methods for the Design

of Spaceborne Radar Sounders

Christopher Gerekos

Advisor:

Prof. Lorenzo Bruzzone, Universit`a degli Studi di Trento

Committee:

Prof. Elena Pettinelli, Universit`a degli Studi Roma Tre

Prof. Dirk Plettemeier, Technische Universit¨at Dresden

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Abstract

Spaceborne radar sounders are an important class of remote sensing instruments which operate by recording backscattered electromagnetic waves in the vicinity of a solid planetary body. The incoming waves are generally transmitted by the radar itself (active sounding), although external signals of opportunity can also be used (passive sounding). There are currently two major planetary radar sounders under development, both headed to the Jo-vian icy moons (Europa, Ganymede and Callisto). Designing a radar sounder is a very challenging process involving careful leveraging of heritage and predictive tools, and in which backscattering simulators play a central role. This is especially true for coherent simulators, due to their higher accuracy and the possibility they offer to apply advanced processing techniques on the resulting simulated data, such as synthetic aperture radar fo-cusing, or any other operation which requires field amplitude, phase and polarisation. For this reason, designing computationally-efficient coherent simulators is an important and active research area. The first contribution of this thesis is a novel multilayer coherent simulator based on the Stratton-Chu equation and the linear phase approximation, which can generate realistic simulated radar data on a wide range of surface and subsurface dig-ital elevation models (DEM), using only a fraction of the computational resources that a finite-difference time-domain method would need. Thorough validation was conducted against both theoretical formulations and real data, which confirmed the accuracy of the method. The method was then generalised to noisy active and passive sounding, which is an important capability in the context of the proposed use of passive sounding on the Jovian icy moons. Provided that representative information about the surface and this external field exists, the simulator could compare the relative scientific value of active and passive sounding of a given target under given conditions. However, quality DEMs of the Jovian icy moons are scarce. For this reason we also present a comparative study of the fractal roughness of Europa and Mars (a much better studied body), where we derive frac-tal analogue maps of twelve types of Europan terrains on Mars. These maps could be used to guide the choice of Martian DEMs on which to perform representative backscattering simulations for future radar missions on Europa. Finally, we explore the possibility of entirely new radar architectures with the novel concept of the distributed radar. In a dis-tributed sounder, very large across-track antennas can be synthesised from smallsats flying on selected orbits, providing a way to obtain a highly-directive antenna without the need to deploy large and complex structures in space. We develop an analytical formulation to treat the problem of beamforming with an array affected by perturbations on the positions of its array elements, and propose a set of Keplerian parameters that enable the concept.

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Acknowledgements

Science is a human endeavour. What I learnt from a scientific point of view is amply described in this document, and I would thus like to say a few word about what I learnt on a more personal level. First, this PhD is arguably my first direct contact with the world of research, and as such, was an enriching experience in ways too numerous to be listed here. I had the occasion to travel to so many places, and meet so many extraordinary people. I would like to thank my advisor Lorenzo Bruzzone for having made all that possible. And although, it must be said, the practice of science was not entirely as I was idealising it, this was the occasion to discover new qualities to myself. There were periods of joy, excitement and triumph, and periods of delusion, doubt and adversity. I feel grown by all of them.

Another defining aspect of this doctoral experience was living in a different country. Although adapting to some aspects of life in Italy has sometimes been challenging, I feel privileged to have been able to live these three years in a country so full of art, history, natural beauty, great cuisine, and most importantly, kind and talented people. In Varietate Concordia.

I would first like to thank Boris Segret, system engineer at the Observatoire de Paris, France, for it is our conversation on the hills around the Physikzentrum Bad Honnef, Germany, hosting the RelGeo2016 conference at the time, that ultimately convinced me, then an undegraduate physics student eager to make his way into space exploration, that a PhD in planetary science instrumentation was the way forward.

Then, although our lives took different directions, I would like to thank Bianca Stelzer for having made me feel less of a stranger in Italy, for the beautiful moments we had together and for her support during critical periods of this PhD.

These three years would have also been very different without my informal scientific collaborators, hiking partners, and dear friends, Ana-Maria Ilisei and Adamo Ferro. They are the first people I would contact when I needed help, and I have never been let down. Ana-Maria was the first one to welcome me in the lab as a friend, and I always enjoyed our frank conversations about research and everything else over lunch, and later on, over din-ner. It was particularly interesting to have her unique point of view of on many subjects. Adamo’s deep understanding of radar science (and of life) helped me see several seemingly intractable problems under a different light, from which solutions would become obvious. I am indebted to his advice, and I am particularly grateful for his careful proofreading of this thesis. I will dearly miss our conversations and the stunning mountainous landscapes of Trentino-Alto-Adige in which they would usually take place.

I also express my gratitude to Yady Tatiana Solano Correa, who was there to listen when I needed someone to talk to, who would sometimes believe in me more than I believed

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life at the lab certainly would not have been as fun. I am glad I got to supervise her Master’s thesis work, and to get to know her joyful and overexcitable side when she was no longer my student. And to Milad Niroumand-Jadidi, whom I was always happy to randomly meet at the bus stop at the end of the day, and deciding on a whim to get dinner in the city centre, where we would vent about our administrative difficulties. On Elena’s impulse, the four of us eventually established a new tradition, the weekly “Movie club”, which would be held every Monday evening, and achieved the remarkable feat of making Monday a day I would look forward to.

I would also like to thank Yann Berquin, whose work constitutes the foundation of much of my thesis. Although we have never met, he provided me with vital guidance in the early stages of my PhD, when I, a theoretical physicist by training, had only tenuous notions of radar science and backscattering theory. Another person I have never physically met but who has helped me greatly is Alessandro Tamponi. He started the multilayer simulator project I have been working on, and provided very helpful information to help me get started with it. Similarly, I would also like to thank Massimo Santoni and many others in our team for their advice in the early stages of this doctorate.

My time at the University of Texas at Austin, United States, was likewise marked by unforgettable encounters. I would first like to thank Donald Blankenship for his hospitality and for welcoming me amongst his team, and Gregory Ng, for hosting me at his place and helping me around town. My colleagues of the Science Data System (but not only), Cyril Grima, Gregor Steinbr¨ugge, Kirk Scanlan, Anja Rutishauer, Duncan Young, Dillon Buhl, Natalie Wolfenbarger, Constantinos Panagopoulos, also became my friends. I would also like to thank Mark Haynes from the Jet Propulsion Laboratory, Pasadena, for his interest in my work, and the subsequent invitation to his workplace, where I got the chance to see the flight model of the Mars 2020 rover.

Austin is also the place where I met my girlfriend, Bianca Viel, who brings a lot to my life, and with whom I learn so much about myself. Wherever the next step of my career will be, I hope there will not be an ocean between us anymore.

Finally, I express my deepest gratitude to my family, and to my mother in particular, for their unwavering and unconditional support. This doctorate, along with all the previous degrees that lead to it, I would not have obtained them if it wasn’t for their love and their wisdom.

This thesis was supported by the grant ASI/INAF n.2013-056-R.O “Partecipazione italiana alla fase A/B1 della missione JUICE”, the grant ASI n. 2016-14-U.O. “SaTellite Radar sounder for eArTh sUb-surface Sensing (STRATUS)”, the grant

ASI/INAF n. 2018-25-HH.0 “Attivit`a scientifiche per JUICE fase C/D”, and by the REASON investigation within the

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List of Tables

2.1 Oceanus Procellarum: average power levels of the recorded layers . . . 51

2.2 Oceanus Procellarum: statistical properties of the recorded layers . . . 51

2.3 Mare Crisium: average power levels of the recorded layers . . . 53

2.4 Mare Crisium: statistical properties of the recorded layers . . . 54

2.5 Computational resources: 3D-FDTD vs. multilayer coherent simulator . . . 57

3.1 Main characteristics of the RIME instrument. . . 72

4.1 Analysis of the average rangelines shown in figure 4.11. . . 108

4.2 Darkspot ridged terrain fractal analogues on the HiRISE dataset. . . 120

4.3 Yelland HighRes ridged terrain fractal analogues on the HiRISE dataset. . 121

4.4 Powys chaos matrix fractal analogues on the HiRISE dataset. . . 121

5.1 Assumptions for the parameters of the distributed array performance analysis.141 5.2 Distributed radar sounder performance summary. . . 143

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List of Figures

1.1 (a) Illustration of the general geometry of radar sounder acquisition on a two-layer terrain. (b) Illustration of the general aspect of the radargram from such a terrain (lighter colours represent higher intensities). . . 10 1.2 Illustration of the relevant quantities intervening in the analysis of plane

waves interacting with a flat dielectric interface. (a) Situation with E perpendicular to the plane of incidence, (b) situation with E parallel to plane of incidence. . . 14 1.3 Illustration of the relevant quantities intervening in the definition of the

field generated by a dipole antenna. . . 16 1.4 (a) A generic volume V enclosing a domain of interest D. (b) A more

carefully chosen volume V which allows easy application of Huygens’ prin-ciple to planetary radar backscattering. The surface enclosing V is given by ∂V = S = {SA, SB, SC, SD}. . . 19

1.5 Illustration of vector quantities used in three possible configurations. (a) a bistatic configuration where reflection is analysed, (b) a monostatic config-uration where reflection is analysed, and (c) a bistatic configconfig-uration where transmission is analysed. . . 21 1.6 Graphical representation of the integral given in (1.49). A, B, and C are

the coordinates triangle vertices, while rα is the position of the triangle

incentre. r0 and the dotted lines represent several examples of paths ap-pearing in the integral in (1.49). In (a) we represent a reflection and in (b) a transmission. . . 23 1.7 (a) Illustration of the relevant quantities intervening in the definition of

the first Fresnel zone radius. (b) Illustration of the relevant quantities intervening in the definition of the pulse-limited radius. . . 25 1.8 Illustration of the relevant quantities intervening in the definition of Friis’

formula. . . 27

2.1 Block diagram showing the architecture of the proposed simulator. . . 35 2.2 Graphical representation of transmitted, reflected and backscattered

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analysis (axes are not to scale). . . 42 2.4 Simulated range lines obtained with different facet sizes. (a) Comparison

between the obtained range lines. (b) Comparison between the peak power of each layer (taken from the (a) graph). DEM shown in figure 2.3. . . 43 2.5 Simulated range lines obtained with different frequencies and bandwidth.

(a) Comparison between fc = 10 MHz and fc = 20 MHz, for a fixed

bandwidth of 5 MHz. (b) Comparison between Bw = 5 MHz and Bw = 2.5

MHz, for a fixed central frequency of 10 MHz. DEM shown in figure 2.3. . 43 2.6 Simulated range lines obtained with different interlayer dielectric constants,

r,1 = 2 + i0.004, r,2 = 3 + i0.0006, r,3 = 2.5 + i0.05, and r,4 = 4, (dotted

curve), and with different layer depths, d1 = 106 m, d2 = 212 m, d3 = 318

m, (dashed curve), along with the base case (solid curve). DEM shown in figure 2.3, with adapted depths in the case of the dashed curve. . . 44 2.7 Simulated range lines obtained with a random Gaussian surface with RMS

height (a) σ0(L = 200 m) = 0.13 m, (b) 0.64 m, (c) 1.3 m, and (d) 6.4 m. 45

2.8 Simulated range lines obtained with various layers presenting a Gaussian height distribution with RMS height σ0(L = 200 m) = 1.3 m to various

layers : (a) rough first layer, (b) rough surface and rough first layer, (c) rough second layer, (d) rough surface and rough second layer, (e) rough third layer, (f) rough surface and rough third layer. . . 46 2.9 (a) Ground track the Kaguya probe during acquisition of LRS radargram

20080604124958 (red line) superimposed on a shaded relief map of the surface, an area in Oceanus Procellarum, Moon. (b) Similar image for LRS radargram 20071227143959 of area in Mare Crisium, Moon. “start” and “end” labels refer to the beginning and end of the simulated radargram. . . 48 2.10 Comparison between (a) simulation result and (b) LRS radargram

20080604-124958 of Oceanus Procellarum. . . 49 2.11 Azimuth-direction average range-line of the simulated and actual

radar-grams for (a) LRS radargram 20080604124958 (Oceanus Procellarum), and (b) LRS radargram 20071227143959 (Mare Crisium). . . 50 2.12 Oceanus Procellarum radagram: echo brightness histograms for all three

layers. The range bin analysed in each case is the one of the corresponding peak in figure 2.11-(a). . . 52 2.13 Comparison between (a) simulation result and (b) LRS radargram

20071227-143959 of Mare Crisium. . . 53 2.14 Mare Crisium radagram: echo brightness histograms for all three layers.

The range bin analysed in each case is the one of the corresponding peak in figure 2.11-(b). . . 55

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2.15 Ground track the MRO probe during acquisition of SHARAD radargram 0589803 (red line) superimposed on a shaded relief map of the surface and the subsurface, an area in Elysium Planitia, Mars. . . 56 2.16 (a) Incoherent clutter simulation. (b) Simulation result using the proposed

method. (c) RDR SHARAD radargram 0589803. . . 57

3.1 The two types of time-dependence for the HOM/DAM radiation snoise

that will be considered in this study can be divided into two categories: synthetically-generated and experimentally-measured. . . 64 3.2 Implementation of incoming and backscattered fields in simulations with a

given sampled model of external (Jovian) noise, along with typical values for T , W0 and W1 for the RIME instrument. . . 70

3.3 Average A-scan for incoming noise flux densities of 10−19to 10−15W/m2_/Hz

for the validations scenario described in the text, using active sounding and unfocused SAR processing. . . 74 3.4 Average A-scan for incoming noise flux densities of 10−19to 10−15W/m2_/Hz

for the validations scenario described in the text, using passive sounding and the autocorrelated signals. . . 74 3.5 Surface peak power versus surface roughness (expressed with surface RMS

slope [◦]), relative to a simulation with a flat layer. The surface data points for active and passive modes are represented as blue and orange dots, respectively. As a reference, the active and passive spreading factors ga and gp are shown as blue and orange dashed lines, respectively. . . 76

3.6 Digital terrain model used in the simulations for the rough multilayer val-idation test described in section 3.5.3 (axes not to scale). . . 77 3.7 Simulated radargrams for the terrain model shown in figure 3.6 with a

RIME-like instrument, for (a) noise-free active, (b) noisy active, and (c) passive sounding, assuming a flux of 10−18W/m2/Hz. For each acquisition mode, the power collected by the antenna is shown (in dBW) on the top, and the phase of that signal (in degrees) on the bottom. . . 78 3.8 Incoherently-averaged rangelines of the simulations shown in figure 3.7 for

active sounding and for passive sounding using different incoming noise listening windows. . . 78 3.9 (a) Coloured hill-shade elevation map of the surface DEM used in the

“Ganymede” simulation, which is coming from the chaotic terrain sur-rounding Olympus Mons, Mars. (b) Coloured hill-shade elevation maps of the DEMs of the surface and subsurface used in the “Europa” simula-tion; the surface represents an area of Tempe Terra, Mars, whereas the subsurface is procedurally-generated with a fBm process. The spacecraft trajectory during the acquisition is shown as the thick red line superim-posed over all maps. . . 79

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(b) autocorrelation. . . 80 3.11 LWA1 waveforms used in the “Europa” simulation: (a) spectral density,

(b) autocorrelation. . . 81 3.12 Simulation results for the Ganymede simulations. (a) Noise-free active, (b)

active in presence of Jovian noise modelled with synthetically-generated Gaussian noise, (c) passive simulation based on the LWA1 signals shown in figure 3.10, (d) passive simulation using white noise of comparable duration. 82 3.13 Simulation results for the Europa simulations. (a) Noise-free active, (b)

active in presence of Jovian noise modelled with synthetically-generated Gaussian noise, (c) passive simulation based on the LWA1 signals shown in figure 3.11, (d) passive simulation using white noise of comparable duration. 84

4.1 Example of experimentally-derived Allan profile (blue curve) and its three-segment linear fitting function (orange curve), alongside with an illustration of the fitting parameters p. The profile itself comes from a 35 km × 35 km area centred on (-73.5◦N, 293.0◦E) coming from the MOLA dataset. . . 92 4.2 Overview of the fractal characterisation of the surface of Mars using the

proposed methodology on the MOLA dataset, presented in cylindrical pro-jection. (a) Hurts coefficient of the first segment of the obtained Allan profiles. (b) root mean square height difference at 463m baseline. (c) scale of the first inflexion point observed in the Allan profile, if any. (d) Hurst coefficient after the first inflexion point. If the profile presents no inflexion point, the corresponding pixel is black in the two bottom figures. . . 95 4.3 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan ridged terrains found through the MOLA dataset. . . 97 4.4 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan chaos matrices found through the MOLA dataset. . . 97 4.5 An area of Aeolis Planum, Mars, imaged by MRO’s CTX (6 metres per

pixel), alongside Pwyll, Europa, imaged by Galileo’s SSI (242 metres per pixel), shown at the same scale. This area of Pwyll consists mostly of ridged terrain, although band terrain can be seen on the lower right quarter.101 4.6 Echus Chaos, Mars, imaged by MRO’s CTX (6 metres per pixel), alongside

ridged terrain of Darkspot, Europa, imaged by Galileo’s SSI (20 metres per pixel), shown at the same scale. . . 101 4.7 Linear and polygonal landforms of West Tharsis, imaged by MRO’s CTX

(6 metres per pixel), alongside ridged terrain of Yelland, Europa, imaged by Galileo’s SSI (48 metres per pixel), shown at the same scale. . . 102

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4.8 An area west of Gordii Dorsum, Mars, imaged by MRO’s CTX (6 metres per pixel), alongside Conamara terrain, Europa, imaged by Galileo’s SSI (54 metres per pixel), shown at the same scale. This area of Conamara comprises chaos matrix terrain, intertwined with ridged plates and irregular ridged plates. . . 103 4.9 An area of East Tharsis and of the smooth terrain south of Olympus Mons,

Mars, imaged by MRO’s CTX (6 metres per pixel), alongside Powys, Eu-ropa, imaged by Galileo’s SSI (28 metres per pixel). All images are shown at the same scale. This area of Powys consists mostly of ridged terrain, although band terrain can be seen on the right. . . 104 4.10 (a) Portion of SHARAD track 1358201 passing over East Tharsis. (b)

Portion of SHARAD track 0438202 passing over Aeolis Planum. (c) Portion of SHARAD track 2630101 passing over Gordii Dorsum and Lycus Sulci. (d) Portion of SHARAD track 2623902 passing over Echus Chaos. Top picture shows the track superimposed on a shaded relief topographic map of Mars, whereas bottom picture shows the acquisition itself. . . 106 4.11 Topography-compensated average rangelines of the radargrams shown in

figure 4.10, along with a radargram over a quasi-flat area of Amazonis Planitia, which acts as a reference by providing an impulse-like response. . 108 4.12 Hillshade topographic map of the HiRISE DTEEC 012444 2065 014000 206

5 U01 digital terrain model, showing a terrain strip centred around 5.04◦S, 173.89◦W. The area comprises several types of geological units, which the proposed classification algorithm will attempt to separate. . . 110 4.13 Inputs fed into the k-means classifier for the DEM shown in figure 4.12: (i)

normalised, average slope-corrected topography, (ii) the x-direction gradi-ent, (iii) the y-direction gradigradi-ent, and (iv) height standard deviation on a 9 × 9 pixel sliding window. The DEM used to generate these maps has a resolution of 100 m, and thus, so do these maps. . . 110 4.14 Steps of the classification algorithm applied on the DEM shown in figure

4.12: (i) result of the raw k-means clustering, (ii) classification after remov-ing spurious categories, (iii) classification after first dilation/erosion, (iv) final classification. . . 111 4.15 Segmented DEM, obtained by selecting the areas obtained by the

classifi-cation algorithm, shown on figure 4.14-(iv) for the DEM shown in figure 4.12. . . 112 4.16 Location of the main Martian regions mentioned in this study (background

image: Mars Viking colorised global mosaic). . . 114 4.17 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan crater peaks found through the MOLA dataset. . . 115

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the different sub-geological units of Europan crater floors found through the MOLA dataset. . . 115 4.19 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan crater rims found through the MOLA dataset. . . 116 4.20 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan crater ejectae found through the MOLA dataset. . . 116 4.21 Cylindrically-projected map of Mars highlighting the fractal analogs of the

different sub-geological units of Europan smooth terrains found through the MOLA dataset. . . 117 4.22 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan undifferentiated terrains found through the MOLA dataset. . . 117 4.23 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan ridged plates found through the MOLA dataset. . . 118 4.24 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan irregular ridged plates found through the MOLA dataset. . . 118 4.25 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan band terrains found through the MOLA dataset. . . 119 4.26 Cylindrically-projected map of Mars highlighting the fractal analogues of

the different sub-geological units of Europan double ridges found through the MOLA dataset. . . 119

5.1 (a) Distributed radar sounder geometry, green arrows indicate the sensors direction of movement. (b) Top-view illustration depicting how the equiv-alent antenna (gray area) is formed along with its shape. The value of θp

is given by (5.1). . . 125 5.2 (a) Across-track section view. (b) Geometry sketch for θp,max definition. . . 127

5.3 Normalised two-way radiation intensity ˆU2_{(x, y) contour plots in dB for (a)}

θp = 0, (b) θp = 10◦, (c) θp = 33◦ and (d) θp = 74◦. The plots are relative

to the all sensors simultaneously transmitting and receiving configuration. The case displayed in (c) corresponds to θp,max. Red circles correspond to

the maximum equivalent surface radius for which subsurface returns are expected. The parameters considered in this plot are λ = 6.66 m, h = 500 km, N = 21, zp = 4 km, θg = 11.5◦, r = 3.1 and θb = 0.59◦. . . 128

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5.4 Example of behaviour of the signal to clutter improvement Cg(z) as a

func-tion of the penetrafunc-tion depth z. The figure has been computed considering λ = 6.66 m, N = 92, θg = 13.44◦, h = 500 km. . . 131

5.5 Comparisons between unperturbed and average perturbed patterns for an array characterised by f = 10 MHz, λ = 30 m, h = 200 km, Lant = 80λ,

dact = 3λ, N = 27. (a) Unperturbed pattern. (b) σz = λ/4. (c) σx =

r1F/60 (d) σy = r1F/60. (e) σx = r1F/50, σy = r1F/50. (f) σx = r1F/50,

σz = λ/10. (g) σx = r1F/100, σy = r1F/100, σz = λ/10. The variable

denoted as r1F =pλh/2 is the first Fresnel radius. . . 133

5.6 Illustration of the effect of SAR focusing on the perturbed pattern, shown on a single realisation of the perturbation of figure 5.5-(g). (a) Pattern of a physical array. (b) Pattern of the same array with unfocused SAR processing (Lat = r1F, dat = 20 m). The considered array is the same as

that in figure 5.5, with a rotation angle θp = 30◦. . . 135

5.7 Comparison between the simplified pattern for nadir h ˜U inad _{(blue curve)}

and that for off-nadir zones h ˜U ibg _{(orange curve) from (5.29) and (5.30)}

against the actual perturbed-averaged pattern h ˜U i (purple curve) from (5.25). An array with the following characteristics is considered : f = 5 MHz, λ = 60 m, h = 100 km, Lant = 40λ, dact = 0.4λ, N = 101. For

reference, the unperturbed pattern U from (5.15) is shown in yellow. . . 137 5.8 Schematic representation of the array configurations varying: (a) only the

orbital inclinations in (ξn = 0 and Ωn = 0). (b) both in and ξn. (c) in, ξn

and Ωn. . . 139

5.9 Comparison of the normalised across-track radiation intensity for (a) Dis-tributed sounder, trade-off configuration, (b) single configuration with dipole antenna and (c) single configuration with Yagi antenna. . . 142 5.10 Distributed radar sounder ground track for about one orbital period on the

first orbit (black), the 50th orbit (red) and the 100th orbit (green). The nodal precession has the effect of increasing the latitude of the crossing points and the separation between sensors in across-track (not at the poles). 145 5.11 Plot of (a) rotation angle θp (absolute value), (b) equivalent across-track

antenna length Lact, (c) across-track equivalent antenna beam-width θb

and (d) first ambiguity angle θg versus latitude for 200 orbits assuming

the trade-off configuration (N = 21). The acquisition points correspond to poles (icy regions) and the equator (arid regions). The red line indicates the performance values for the first orbit. . . 146 5.12 Two-way peak power losses P2

L,nadir as function of σz for different number

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N = 11, (b) N = 21, (c) N = 45 and (d) N = 65 as function of σx, σy and

σz evaluated at ˆθ = θg/2. . . 147

5.14 Surface and subsurface DEM of the desertic area simulations. The ground track of the sounder is shown in red. . . 148 5.15 Desert region backscattering simulations for (a) single and (b) distributed

sounder configuration. There is a clear reduction in clutter contribution in the latter case . . . 149 5.16 Surface and subsurface DEM of the icy area simulations. The ground track

of the sounder is shown in red. . . 150 5.17 Icy region backscattering simulations for (a) single and (b) distributed

sounder configuration. Also in this case the distributed configuration pro-vides a clear advantage in clutter suppression capabilities. . . 151

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List of Abbreviations

ALSE Apollo Lunar Sounder Experiment

ASI Azienda Spaziale Italiana (Italian Space Agency)

CONSERT Comet Nucleus Sounder Experiment by Radiowave Transmission CPU Central processing unit

CReSIS Center for Remote Sensing of Ice Sheets CTX Context Camera

DAM Decametric emissions DEM Digital elevation model EDR Experimental Data Product ESA European Space Agency fBm Fractional Brownian motion FDTD Finite-difference time-domain GPU Graphics processing unit HF High frequency

HiCARS High Capability Radar Sounder

HiRISE High Resolution Imaging Science Experiment HOM Hectometric emissions

JAXA Japan Aerospace Exploration Agency JUICE Jupiter Icy Moons Explorer

LEO Low Earth orbit

LOLA Lunar Orbiter Laser Altimeter LPA Linear phase approximation LRS Lunar Radar Sounder LWA1 Long-Wavelength Array 1

MARSIS Mars Advanced Radar for Subsurface and Ionospheric Sounding MEGDR Mission Experiment Gridded Data Records

MOLA Mars Orbiter Laser Altimeter MRS Multilayer radar simulator

NASA National Aeronautics and Space Administration OASIS Orbiting Arid Subsurface and Ice Sheet Sounder

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PSTD Pseudospectral time-domain RAAN Right ascension of ascending node RAM Random-access memory

RDR Reduced Data Record

REASON Radar for Europa Assessment and Sounding: Ocean to Near-Surface RIME Radar for Icy Moons Exploration

RMS Root-mean square

RPWI Radio and Plasma Wave Investigation SAR Synthetic aperture radar

SCR Signal-to-noise ratio SHARAD Shallow Radar

SI Syst`eme International (International System of Units) SNR Signal-to-noise ratio

SPICE Spacecraft Planet Instrument C-matrix Events SSI Solide-State Imager

STRATUS Satellite Radar Sounder for Earth Subsurface Sensing TE Transverse electric

TM Transverse magnetic UHF Ultra high frequency VHF Very high frequency

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List of Symbols

E Electric field

B Magnetic field

D Auxiliary electric field H Auxiliary magnetic field c = 299792485 m/s Speed of light in vacuum µ0 = 4π × 10−7 H/m Vacuum permeability

0 = µ0/c2 Vacuum permittivity

Z = cµ0 Impedance of free space

kB = 1.380649 × 10−23 J/K Boltzmann constant

ω Angular frequency

k, k Wavevector, wavenumber λ = 2π/k Wavelength

µr Relative permeability

r Relative permittivity/dielectric constant

v = c/√rµr Speed of light in medium

ζ Attenuation constant

tan δ Loss tangent

ri Emitter position

rr, xq Receiver position (general, or as position in radargram)

r0 Position on surface used as integration variable rα Incentre of facet ∆α

ˆ

n Local outgoing normal (unit vector) ki Wavevector of emitted field

ks Wavevector of scattered field

Φα(ri, rr) Phase integral on facet ∆α

A(·) Area of an object

I Identity tensor

δij Kronecker delta of indices {i, j} ∈ Z

Θ(x) _{Heaviside function of variable x ∈ R}

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fc Radar central frequency

Bw Radar signal bandwidth

Pi Radar radiated power

V0 Radar emitted field amplitude factor

Ts Chirp duration, emission window duration

W1 Reception window duration

fs Sampling frequency

Tf Sampling period

s(t) Time-dependent signal W (t) Windowing function

Q(xq) Range-compressed rangeline

PRF Pulse repetition frequency PRI Pulse repetition interval Aeff Antenna effective area

r1F Radius of the first Fresnel zone

rP ltd Radius of the pulse-limited zone

G Antenna gain

σ0 _{Normalised radar cross-section}

h Radar platform altitude vs Radar platform speed

R Distance between scatterer and radar L Length of triangle cathetus, facet size Teq Equivalent temperature

H Hurst coefficient

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Introduction

This chapter introduces the topics of the thesis. An overview of planetary radar sounders is presented, which serves as a backdrop for the motivation and scope of this work. The novel contributions are highlighted within their context, and the structure of the document is explained.

Background

A radar is a device that can detect and localise objects through the use of electromagnetic waves [1], and radar sounders are a class of scientific instruments making use of this prin-ciple to gather valuable information about the surface and subsurface of a given planetary body from a great distance, a capability that makes them particularly important in the field of remote sensing. They can be airborne, in the case of Earth, or spaceborne, in the context of planetary exploration. Radar sounders are usually active, monostatic, nadir-looking instruments working with low-frequency electromagnetic waves –typically in the MHz range– allowing for deep penetration within the subsurface. In most solid planetary bodies, past or present geophysical processes often cause inhomogeneities in the dielectric constant of the subsurface. Examples of such inhomogeneities includes geological layers (in stratified terrains), discreet reflectors such as inclusions, or extended inhomogeneities such as underground pockets of water. Such dielectric discontinuities produce a reflec-tion if illuminated by an incoming electromagnetic wave, which may be detected by the receiver of a radar instrument. Analysing and interpreting these backscattered echoes forms the basis of science with radar sounders. This field of study is at the crossroads of geology, physics, mathematics, electronics, and more recently, computer science.

Alongside radar sounders, another class of remote sensing radars are imaging radars. These instruments typically use a side-looking geometry and work at much higher frequen-cies, often in the GHz range. They are able to generate precise surface backscattering maps, but are not designed for subsurface penetration. Ground-penetrating radars are radar sounders operated in situ (and are therefore not remote sensing instruments), typ-ically mounted on wheels and placed as close as possible to the ground, and have a lot of commonalities with radar sounders.

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whom credit for “inventing the radar” should be attributed.1 Rather, in the last hundred years, technological breakthroughs have almost always been the culmination of the efforts of a large number of individuals, research groups, and governmental actors from many countries. The radar is no exception. The discovery of the operating principles of radars is usually attributed to Heinrich Hertz, who demonstrated in 1886 that electromagnetic waves could be reflected by solid surfaces, whereas the first patent on the subject was filed by Christian H¨ulsmeyer in 1904. In 1925, Gregory Breit and Merle Tuve succesfully measured the height of the ionosphere using electromagnetic pulses [1]. In parallel, as World War I had shown that aerial warfare would radically transform future conflicts [2], governmental research institutes such as the Naval Research Laboratory in the United States were deploying considerable efforts to develop electromagnetic systems able to de-tect aircrafts at great distances, and the first successful attempts were made in the early 1930’s [1]. By the beginning of the 1940’s, most great powers, such as the United States, the United Kingdom, Germany, France, Russia, Italy and Japan, had indepentendly de-veloped their own systems. The word radar itself was coined by the United States Navy in 1940 as an acronym for Radio Detection And Ranging [3].

Radars, as ranging and tracking devices, have been an integral part of planetary ex-ploration since the earliest days of the Space Age, essentially spearheaded by the Soviet programmes Sputnik (Sputnik 1, 1957, first Earth orbiter), Luna (Luna 1, 1959, first lunar flyby), Venera (Venera 1, 1961, first probe to another planet, Venus), and Mars (Mars 1, 1962, first probe to Mars), but it was not until the 1970’s that they would be used as scientific intruments. Indeed, the first spaceborne radar sounder was the Apollo Lunar Sounder Experiment (ALSE) [4], flown on the service module of Apollo 17, an American crewed mission to the Moon launched in 1972. Although not specifically carrying radar sounders, the National Aeronautics and Space Administration (NASA) Venus-bound Magellan mission, launched in 1989, and the international mission Cassini-Huygens, launched in 1998 towards the Saturnian system, should be noted for their use of imaging radars. More recently, two successful examples of Martian radar sounders are the Mars Advanced Radar for Subsurface and Ionosphere (MARSIS) [5], and the Shal-low Radar (SHARAD) [6], both developed under leadership of the Italian Space Agency (ASI). MARSIS has been flying on the European Space Agency’s (ESA) Mars Express probe, launched in 2003, whereas SHARAD is an instrument of the Mars Reconnaissance Orbiter (MRO) of NASA, launched in 2005. The moon was once again studied through the antennas of a radar sounder with the Lunar Radar Sounder (LRS) instrument on-board the Kaguya probe, operated by the Japan Aerospace Exploration Agency (JAXA), and launched in 2007. In 2014, the Comet Nucleus Sounder Experiment by Radiowave Transmission (CONSERT), a bistatic radar intrument flown on ESA’s Rosetta/Philae mission (launched in 2004), made the first radar measurements of the interior of a comet, 67P/Churyumov-Gerasimenko [7]. Currently, two more radar sounder instruments are

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Introduction

der development, an European and an American one, both headed to the Jovian system. On the European side, the Radar for Icy Moons Exploration (RIME) will fly onboard the Jupiter Icy Moons Explorer (JUICE), ESA’s future spacecraft to Jupiter’s icy moons Eu-ropa, Ganymede, and Callisto [8]. Across the Atlantic, the Radar for Europa Assessment and Sounding: Ocean to Near-surface (REASON) instrument will fly on NASA’s upcom-ing Europa Clipper mission, targetupcom-ing the Jovian moon Europa [9]. JUICE is currently slated to launch in 2022, whereas Europa Clipper’s current schedule indicates a launch in 2025.

On Earth, many radar sounder instruments have been developed for airborne platforms. Such platforms include, e.g., the multi-instrument sounding system for polar research developed by CReSIS [10], the POLARIS instrument of ESA [11], the bistatic radar flown by the British Antarctic Survey [12], the HiCARS instrument developed at the University of Texas at Austin [13], and the multimode P-band sounder funded by the Italian Space Agency [14]. No orbital radar sounder has been operated around Earth yet, although there have been several proposals, such as the Satellite Radar Sounder for Earth Subsurface Sensing (STRATUS) mission concept in Europe [15] and the Orbiting Arid Subsurface and Ice Sheet Sounder (OASIS) mission concept in the United States [16]. Most of the Earth-related radar sounding campaigns focus on the study of the cryosphere – radar sounding is particularly effective on ice, which is characterised by very low dielectric attenuation – and a few of them on the detection of subsurface water in desertic areas.

Designing a radar sounder is a complex process which requires very careful cross-consideration of all aspects that have an influence on the final scientific return of the instrument, given the highly critical environment spacecrafts have to evolve in, and the very limited possibilities there are to “fix” an unforeseen issue. Applying lessons learnt from the design and operation of previous instruments is thus an important part of the design of a new one. In the case of RIME and REASON, this heritage is provided by the Martian sounders MARSIS and SHARAD, be it for the design of the instruments themselves or for the science and operations on the Jovian moons. On the other hand, each instrument is unique, being tailored to a specific environment and subject to specific engineering constrains. This is the reason for which heritage must be used in conjunction with predictive tools, such as backscattering simulators, to ensure a successful mission. Indeed, there are many obstacles to a correct interpretation of radar signals, which are highly dependent on the targets of the mission. The major challenges are related to: (i) surface clutter, which is caused by off-nadir echoes arising from terrain topography ap-pearing as subsurface features in a given range line, (ii) optical deformations of subsurface features due to the varying indices of refraction of the terrain, (iii) subsurface clutter, and (iv) the presence of unwanted electromagnetic radiation (noise) in the considered acqui-sition environment. Additionally, complex subsurface features may result in signals that are difficult to interpret. Simulation techniques are very well-suited to address all these issues simultaneously, as they attempt to reproduce the radar response of the the terrain

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itself (or the portion of it relevant to the considered analysis). The central role that radar sounder simulators play in both the design of a mission and the subsequent analysis of the acquired data is thus easy to comprehend. This is particularly true for coherent simulators, which include amplitude, phase, and polarisation information (as opposed to incoherent simulators, which only simulate intensity), as they also allow for the testing of radar data processing algorithms. Of the aforementioned issues, clutter is a particularly relevant one for radar sounders to address, and although simulators are routinely used to detect or remove clutter echoes from a radargram, it is always preferable to have a radar architecture that is intrinsically less sensitive to clutter in the first place. One of the most direct ways to achieve this is the use of directional antennas. Unfortunately, this type of antennas are complex to build, test, and deploy, and engineering constrains in space are such that only the simplest (and least selective) antenna designs have so far been considered for radar sounders. In light of this problem, innovative radar architectures that can mimic the selectivity of a large antenna have recently become another facet of radar sounder design, which further requires precise electromagnetic simulations.

In this thesis, we propose several highly-efficient approaches to the simulation of radar backscattered fields. The cornerstone of this dissertation is a novel coherent multilayer simulator of radar sounder echoes, which can generate realistic radargrams of a given terrain as seen by a given radar sounder instrument with minimal computational require-ments. The simulator requires a digital elevation model (DEM) of the terrain of interest and represents the subsurface as a series of layers. The flexibility of the method makes it particularly well-suited to the design phase of an instrument or the post-acquisition analysis of its data. While the method was originally devised to simulate noiseless active acquisitions, it was later generalised to include noisy active and passive acquisitions for a given external field. This generalisation paves the way for comparing the relative scientific value of active and passive acquisitions on the Jovian icy moons by RIME and REASON, given that the Jovian magnetosphere is a source of intense emissions in frequency bands relevant for these instruments. We also present a comparative study of the roughness of Europa and Mars, where we identify the areas of Mars which share similar fractal prop-erties as Europan terrains, and thus, are likely to scatter radar waves in a similar way. Amongst other things, this opens up the possibility to select highly representative Martian DEMs to conduct simulations that could support the operation of RIME and REASON at Europa. Lastly, we present a proof-of-concept study for an Earth-orbiting distributed sounder, a novel radar sounder architecture comprised of a large number of smallsats carrying radar transmitters flying in formation. By coherently assembling the signals transmitted/received by these array elements, we are able to synthesise a very selective and flexible antenna that shows highly reduced sensitivity to clutter and electromagnetic noise.

Additional information about the novel contributions of this thesis is given in the following section.

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Introduction

Scope and novel contributions of the thesis

The overarching purpose of this thesis is to advance the state of the art regarding electro-magnetic and diffraction simulation techniques for radar sounders. It comprises the four novel contributions listed below:

1. an efficient coherent radar simulator which generates radargrams considering noise-less active radar sounding scenarios over multi-layered terrains;

2. an extension of this simulator to noisy active and passive sounding which takes into account the precise characteristics of the external fields, and is centred around RIME acquisitions;

3. a comparative study of the roughness of Europa and Mars in terms of fractal be-haviour, which aims at predicting the Martian areas which are likely to scatter elec-tromagnetic waves in a similar way to Europan terrains;

4. a proof-of-concept study for an Earth-orbiting radar sounder formed of an array of satellites, constituting a novel radar architecture showing reduced clutter sensitivity.

In the following, we describe these four contributions in greater detail.

A coherent multilayer simulator for noiseless active radar sounding

As stated before, reliable and computationally-efficient backscattering simulation meth-ods are highly important assets in the design process of radar sounder instruments and the subsequent analysis of their data. A popular formula to simulate coherent backscattering of radar echoes is the Stratton-Chu integral. Several coherent simulators based on the Stratton-Chu integral exist in the literature, but are limited to surface-only backscattering or, at most, to DEMs containing one subsurface layer only. However, stratified terrains are ubiquitous in nature, and there is a need to be able to coherently simulate backscat-tering from them. Method such as finite-difference time-domain (FDTD) algorithms are able to generate very realistic data on a wide range of terrains, but the huge computa-tional resources they demand in radar sounding scenarios makes them impractical to use in most cases. For this reason, we developed a novel, efficient Stratton-Chu radar sounder simulator for active acquisitions being able to consider DEMs with an arbitrary number of layers. We additionally generalised the linear phase approximation for subsurface prop-agation, which allows the use of DEMs with resolutions as low as hundreds of metres. The simulator was validated against radar data coming from multilayered terrains of the Moon and of Mars, acquired by the LRS instrument of JAXA’s Kaguya probe, and the SHARAD instrument onboard NASA’s MRO, respectively. The simulated radargrams match with the real ones to a high degree of accuracy, and were obtained with a fraction of the computational resources typically required by finite-element methods.

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A coherent multilayer simulator for active and passive radar sounding of the Jovian icy moons

The possibility of making use of Jupiter’s strong radio emissions to perform passive sound-ing on the Jovian icy moons with RIME and REASON has been a very active research topic in recent years. Indeed, the magnetosphere of Jupiter is the source of powerful ra-dio waves that could interfere with the planned active acquisitions of the aforementioned sounders. Passive sounding could thus potentially result in scientifically-useful acquisi-tions in situaacquisi-tions where regular active sounding signals would be drowned in noise. For this reason, starting from the method presented in the first contribution, we devised a multilayer coherent simulator being able to consider noiseless active, noisy active, and passive acquisitions, with a special emphasis on the case of the Jovian environment. The external emissions are modelled as plane waves. Their amplitude, polarisation, direction, and time-dependence can be freely-chosen. In the case of Jupiter, this time-dependence can be taken from experimental measurements of the Jovian noise. Both autocorrelation and cross-correlation processing is considered. Aspects related to noisy active and passive sounding are thoroughly validated against the existing literature, which mostly uses the radar equation. To illustrate the capabilities of the simulator, we present a comparison of actively- and passively-acquired radargrams using DEMs of Martian areas believed to be analogues of areas of Ganymede and Europa. In the future, the method could be used to evaluate and compare the relative scientific value of active and passive sounding under otherwise identical conditions, making it a precious planning tool.

A comparative study of the fractal roughness of Mars and Europa

Radar sounder simulation algorithms are only as accurate as the information about the targeted terrain is complete, and in the case of the upcoming Jovian sounders, there are only fractional data about the topography and roughness of the Jovian icy moons. Available DEMs are either very local or with low resolutions, due to the relative scarcity of missions that have flown past these bodies. On the other hand, Mars is a much better studied planet, and many global and local DEMs are available for running backscattering simulations. For this reason, our goal in this contribution is to identify in a systematic way the fractal analogues of Europan terrains on Mars, which are Martian areas susceptible of producing a similar electromagnetic response. DEMs from these areas could then be used to run representative backscattering simulations for RIME and REASON on Europa. To this end, we analysed the roughness of Mars in terms of piecewise monofractal behaviour, and compared the derived Allan profiles to those of twelve types of Europan terrains. The most common ones are chaos matrix and ridged terrain, and our analysis is thus more focused on them. We have found that the Martian areas of East Tharsis, Echus Chaos and Aeolis Planum have a high density of Europan ridged terrain fractal analogues, whereas Gordii Dorsum, South of Olympus Mons, is rich in chaos matrix

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Introduction

terrain analogues. Examples of SHARAD radargrams over these are presented, and their properties qualitatively discussed. This contribution is part of the effort to leverage the rich heritage of Martian radar sounders to support the operations of RIME and REASON on Europa.

A novel architecture for radar sounding using sensors in formation flight: the distributed radar

As mentioned above, radar systems that are by design less sensitive to clutter and/or noise through the use of narrow-beamwidth antennas will always make better scientific acquisitions than those with less selective antennas. Since such systems are complex to deploy in space, we examine in this contribution the possibility of forming a very large virtual antenna through the use of smallsats flying in formation carrying very simple ra-diating elements. In particular, we propose a novel architecture for an Earth-orbiting radar sounder in 3-300 MHz band comprised of a few tens of array elements aligned to one another. Amongst other things, we examine the very interesting problem of beam-forming with such an array in the presence of a statistically-defined error on the positions of each array element. We analytically derive a formula of the ensemble-averaged per-turbed pattern, and further reduce it to obtain practical engineering metrics, such as the peak-to-background power ratio degradation, as functions of the properties of the three-dimensional perturbation. We also examine the problem of orbital deployment, by proposing a set of Keplerian parameters for each array element which ensures that the array always forms a line with sufficient across-track baseline over the targets of inter-est. We additionally consider and analyse the two main orbital perturbations that such a constellation would be subject to. We finally present a numerical application of these principles based on the proposed STRATUS mission, along with coherent radar simula-tions illustrating the vast improvement in the detection of subsurface features when using a constellation of smallsats instead of a single sounder configuration.

Structure of the thesis

This thesis is divided into five chapters. Chapter 1 introduces the fundamental concepts behind radar sounders and scattering of electromagnetic waves, whereas the four remain-ing chapters present the four contributions introduced above. Chapter 2 describes the proposed multilayer Stratton-Chu radar backscatter simulation algorithm that is used or built upon in subsequent chapters. Chapter 3 illustrates the generalisation of the multi-layer Stratton-Chu simulator to passive acquisitions, with emphasis on sounding of the Jovian icy moons. Chapter 4 presents the comparative study of Mars and Europa in terms of fractal roughness. Finally, chapter 5 proposes the proof-of-concept study for an Earth-orbiting distributed sounder, which also illustrates how the simulator proposed in

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chapter 2 can be used in the design of a novel radar architecture. An overall critical view of all four contributions and remarks on possible future work conclude the thesis.

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Chapter 1 Fundamentals

In this chapter we formulate the theoretical foundations of the work presented in this thesis. We start by introducing the fundamental principles of radar sounding. Then, we present Maxwell’s equations and some of their oscillatory solutions. We describe the problem of diffraction of electromagnetic waves, and derive several important results such as the Stratton-Chu formula. Finally, we introduce some fundamental quantities used in radar remote sensing.

1.1 Introduction

A radar sounder traditionally operates by emitting electromagnetic waves towards a solid planetary object and by recording the subsequent reflections of those waves. As described in the introduction, reflections can be produced when a wave encounters an abrupt change of the index of refraction in the propagation volume. This can be, for instance, the surface of a planetary object, possible subsurface layers and/or discrete diffractors. Airborne radar sounders acquire at several hundreds or thousands of metres from the surface, whereas spaceborne radar sounders typically operate at altitudes of several hundreds of kilometres.

A radar sounder acquisition usually proceeds as follows. The antenna of the radar emits a very short pulse in the direction of the body of interest. This wavefront will interact with the surface and the subsurface of the target body, and part of the signal will be reflected towards the instruments [figure 1.1-(a)]. The radar will collect these signals in a way that ensures the delay of the echoes is recorded at all times. If the platform is moving, consecutive acquisitions are placed side-by-side, forming a two-dimensional object known as a radargram [figure 1.1-(b)]. Informations such as the intensity, the delay, and the along-track position of a given echo are then used to infer the properties of the investigated terrain. Due to the practical limitations of generating very short pulses of high intensity, most radars actually transmit a extended waveform. The received signal is then processed so that any possibly-present echoes can be detected as if they were reflections of a very

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Along-track d irection Across-tr ack directi on Depth directi on Emitted waveform Reﬂected waveforms Surface Subsurface (a) Depth [m] Along-track direction [m] Intensity [ dBW] Surface echo Suburface echo (b)

Figure 1.1: (a) Illustration of the general geometry of radar sounder acquisition on a two-layer terrain. (b) Illustration of the general aspect of the radargram from such a terrain (lighter colours represent higher intensities).

short pulse, a process known as range-compression. The most commonly-used type of signal is the linear chirp, due to the favourable properties of its autocorrelation function. In addition to range-compression, along-track processing is also usually applied to the radargram. This consists in combining a certain number of successive acquisitions (with or without additional operations) in order to greatly increase the amount of information vis-ible in a radargram. Along-track processing is often called focusing, or synthetic aperture radar (SAR) processing, for reasons that will become clearer later in this chapter.

We consider in this thesis nadir-pointing radar sounders only, and we consider the emitted fields to be spherical waves unless otherwise mentioned.

1.2 Electromagnetic fields and propagation

The theory of electromagnetism is fundamental to radar, and its understanding a central pillar of radar science. Thus, the concepts mentioned above can only be properly under-stood within electromagnetic theory. In this section we will review the equations ruling the relationship between electromagnetic fields, sources, and matter. In particular, we will examine the plane wave solution that electromagnetic theory admits, the problem of interaction with a planar interface, and the modelling of electromagnetic wave sources.

Throughout this section and the following ones, bold quantities will represent vectors or tensors unless otherwise mentioned. Hatted vectors represent vectors of unit norm. It is understood that the electromagnetic field components E and B are complex, and that the physical fields are obtained by considering the real part of those fields. We will thus

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1.2. Electromagnetic fields and propagation

omit the Re {} operator that should apply to every electromagnetic field solution.

1.2.1 Maxwell’s equations

Maxwell’s equations are a set of first-order partial differential equations that determine the three components of the electric field E(r, t) and magnetic field B(r, t) at any position r and any time t, for a given medium and a given set of boundary conditions. The four Maxwell’s equations are comprised of Gauss’ law, Gauss’ flux law, Maxwell-Faraday’s law and Maxwell-Amp`ere’s law. In “microscopic” form, that is, when interactions with matter are not incorporated in the fields, these four equations read:

∇ · E = η 0 , (1.1) ∇ · B = 0, (1.2) ∇ × E + ∂B ∂t = 0, (1.3) ∇ × B − 1 c2 ∂E ∂t = µ0j. (1.4)

In the above equations, η represents the charge density in the considered volume (in SI units of C/m3_{), and j the current density (in SI units of A/m}2_{). The quantity}

c ≡ √1 0µ0

(1.5)

is a characteristic speed (= 299792458 m/s in SI units) which, as will be shown in the next subsections, corresponds to the speed of propagation of electromagnetic waves in vacuum. Lastly, µ0 = 4π10−7 H/m is the vacuum permeability, and 0 = µ0/c2 is the

vacuum permittivity (SI units of F/m).

The “macroscopic” form of Maxwell’s equations is obtained by defining the auxiliary fields D and H, which relate to the original electric and magnetic fields as follows:

D ≡ 0E + P, H ≡

1 µ0

B − M, (1.6)

where P is the medium polarisation, and M the medium magnetisation. The relationship between P and D, and M and H is determined through constitutive equations. D and P have units of charge per area (C/m−2), while H and M have units of current per length (A/m). With the D and H fields, Maxwell’s equations read as follows:

∇ · D = ρ, (1.7) ∇ · B = 0, (1.8) ∇ × E + ∂B ∂t = 0, (1.9) ∇ × H − ∂D ∂t = J. (1.10)

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where ρ and J are the free charge density and free current density, respectively.

The Poynting vector S is a vectorial quantity describing the energy flux of a given time-varying electromagnetic field. It has units of power per surface unit (W/m2_{) and}

depends on the cross-product of the electric and the magnetic field in auxiliary form:

S(r, t) = E(r, t) × H(r, t). (1.11)

The total radiated power is recovered by integrating the energy flux given by the Poynting vector across the solid sphere:

P (t) =

Ω

S(r, t) · ˆndΩ, (1.12)

where ˆn is the outgoing normal. The radiated power defined above corresponds to the power factor used in the so-called radar equation, although, as we will see in the next sections, a time-averaged version is better suited for harmonic fields.

1.2.2 Plane wave solutions

This discussion closely follows that of [17]. We look for solutions to eqs. (1.7)-(1.10) of harmonic time-dependence:

E(r, t) B(r, t)

∝ eiωt_, _(1.13)

where ω is the angular frequency. We assume an absence of free charges and currents, ρ = 0, J = 0, an infinite, isotropic, and homogeneous medium where D = E and B = µH , where and µ, being referred to as permittivity and permeability, respectively, are generally complex functions of ω. If the considered medium is vacuum, we have = 0 and µ = µ0. It is commonplace to additionally define relative permittivity and

permeability as r = 0 , (1.14) µr= µ µ0 . (1.15)

By inserting (1.13) in (1.9) and (1.10) and combining them, the two equations reduce to the Helmholtz wave equation,

(∇2+ µω2)E(r, t) B(r, t)

= 0. (1.16)

It can be seen that the so-called plane wave solution respects the above equation. We can write: E(r, t) B(r, t) =E0 B0 eik·r−iωt, (1.17)

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1.2. Electromagnetic fields and propagation

where E0, B0 and k = kˆk are constant vectors. The norm of the latter is called the

wavenumber, and is given by

k = ω√µ. (1.18)

The complex nature of k depends on whether or µ are complex. If and µ are real, the amplitude of the waves (1.17) will be constant along the direction of propagation. A medium exhibiting such a characteristic is called a lossless medium. The subject of lossy media and attenuation will be discussed in subsequent paragraphs. Note that the unit vector ˆk is always considered real in this thesis. The solution (1.17) describes a plane wave travelling in the direction ˆk at a speed (µ)−1/2.

By reinserting the above expressions into Maxwell’s equations, it can be seen that: ˆ k · E0 = 0, (1.19) ˆ k · B0 = 0, (1.20) B0 = √ µˆk × E0. (1.21)

The first two conditions describe a wave that is transverse, i.e., which is characterised by an amplitude oscillating perpendicularly to the direction of propagation. The last condition shows that the magnetic field is perpendicular to both the electric field and the direction of propagation, and that its amplitude is not free.

Since the vector E0 evolves only in a given plane, it is possible to decompose it into a

linear sum of orthonormal vectors: E0 ≡ E11+ E22. The corresponding decomposition

for B0 in terms of the same basis vectors can be obtained from (1.21). If E1 and E2

have the same phase, the corresponding wave is called linearly-polarised, with a direction given by the relative value of E1 and E2. If their phase is different, the wave is called

elliptically-polarised. The special case of a ±π/2 difference (i.e., E1 = 0, E2 = ±i)

describes a circularly-polarised wave.

Let us commence our analysis of wave amplitude by first considering that µr = 1, as

most common media studied in with radar (ice, sand, regolith) do not display magnetisa-tion, and focus the discussion on the properties of the dielectric constant. We postulate that the wavenumber can be decomposed as k ≡ kR+ ikI, meaning the amplitude of a

wave will evolve as E(d) ∼ e−dkI_{, where d represents the distance travelled by the wave}

in the direction of propagation. Assuming kI kR, we obtain [17] from (1.18) that

kI ≈ nω tan δ/(2c), where

n ≡ pRe {r}, (1.22)

tan δ ≡ Im {r} Re {r}

. (1.23)

n is called the index of refraction of the medium, while tan δ, the ratio of the imaginary to the real part of the dielectric constant, is referred to as the loss tangent of the medium. We have n = 1 for vacuum, by definition, whereas classical media are characterised by

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k kr k_t θinc θ_tra θrﬂ = θinc n1 n2

⊗

E B

⊗

Br Er

⊗

E_t Bt (a) k kr k_t θinc θ_tra θrﬂ = θinc n1 n2

⊙

E B

⊙

B_r E_r

⊙

Et Bt (b)

Figure 1.2: Illustration of the relevant quantities intervening in the analysis of plane waves interacting with a flat dielectric interface. (a) Situation with E perpendicular to the plane of incidence, (b) situation with E parallel to plane of incidence.

n ≥ 1. Since the phase velocity of a wave is given by v = ω/k, we may write

v = c

n. (1.24)

From the discussion in this section, we can thus see that the quantity c appearing in (1.4) is nothing but the phase velocity of harmonic electromagnetic fields in vacuum, but that in presence of matter, the velocity of those waves will be reduced by a factor n. The index of refraction thus corresponds to the ratio of the speed of light in vacuum to the speed of light in the considered media.

In radar remote sensing, it is usual to define an attenuation coefficient as follows:

ζ ≡ π

cn tan δ, (1.25)

so that the amplitude of the field can be written as E(d) ∼ e−ζf d, where f = ω/(2π) is the frequency of the harmonic wave under consideration.

1.2.3 Flat dielectric interface

The interaction of plane waves with planar surfaces is a direct application of the principles derived above, and the results that can be derived in such a setting are universally used in computational electromagnetics. Let us consider an incident medium characterised by an index of refraction n1 and a second medium characterised by an index of refraction n2,

as shown in figure 1.2. We assume for simplicity that µr,1 = µr,2 = 1 and we define the

incident, transmitted, and reflected fields as,

E(r, t) = E0eik·r−iωt, Et(r, t) = Et0e

ikt·r−iωt_, _E

r(r, t) = Er0e

Advanced Backscattering Simulation Methods for the Design of Spaceborne Radar Sounders