Keywords: 45 3, 6 SAR tomographic representation [56], 46 4, 1 Tomographic SAR satellite imaging geometry, 52 4, 2 MBInSAR geometrical configuration, 73

image) . . . . .

Glossary of acronyms

Résumé en français

La Terre est un système dynamique et sa surface est en constante évolution à différentes échelles de temps. L'imagerie satellitaire et les produits secondaires qui en dérivent jouent un rôle important en fournissant les informations nécessaires à l'analyse, au suivi et à la caractérisation de la dynamique de la surface de la Terre. Par conséquent, il est nécessaire d'obtenir des mesures rapides et précises de l'écosystème de la Terre. L'urbanisation et la gestion de l'environnement urbain et sa périphérie deviennent l'un des problèmes les plus cruciaux dans les pays développés et en voie de développement. Dans ces circonstances, les données de télédétection sont une source importante d'information qui reflète les interactions entre les êtres humains et leur environnement. Compte tenu de leur indépendance totale des contraintes logistiques sur le terrain, d'éclairage (lumière du jour) et de conditions météorologiques (nuages), les radars à synthèse d'ouverture (SAR) satellitaires peuvent fournir des contributions importantes d'environnements complexes à travers, par example, leur reconstruction 3-D. Les images SAR 2-D sont sujettes à d'importantes distorsions géométriques qui limitent fortement la portée de leur interprétation pour la caractérisation d'environnements complexes, comme les zones urbaines.

En particulier, à partir de 2007, la nouvelle génération de capteurs SAR à haute résolution comme l'italien COSMO-SkyMed, l'allemand TerraSAR-X et le canadien RADARSAT-2 a permis d'acquérir des images SAR jusqu'à 1 m de résolution au sol en mode SpotLight et sous divers angle d'incidence. À l'heure actuelle l'attention se pote sur la récente famille de missions appelée Sentinel (-1 pour la technologie SAR) et développée par l'Agence Spatiale Européenne (ESA) spécifiquement pour les besoins opérationnels de Copernicus, le programme d'observation de la Terre dirigé par la Commission Européenne en partenariat avec l'ESA et visant à fournir des informations précises, pertinentes et facilement accessibles pour améliorer la gestion de l'environnement, comprendre et atténuer les effets du changement climatique et assurer une mission de sécurité civile. En effet, les capteurs et les technologies SAR modernes peuvent répondre à la demande publique concernant le suivi des changements rapides sur le terrain et dans les activités anthropiques. La possibilité de réaliser en temps quasi réel la cartographie 3-D de la surface de la Terre par satellite de télédétection représente un outil précieux pour plusieurs objectifs d'Horizon 2020, le nouveau programme-cadre pour la recherche et l'innovation en Europe.

Une des applications les plus importantes de la télédétection SAR est la génération de Modèles Numériques de Surface (DSM), qui permet de modéliser en trois dimensions la surface de la Terre. Dans les zones suburbaines et urbaines, en DSM se révèle très utile pour un grand nombre d'applications, qui incluent les calculs d'optimisation de l'emplacement des antennes de télécommunication, la cartographie des risques, la génération orthoimage, la planification urbaine et la surveillance. Trois techniques différentes imagerie SAR 3-D peuvent être discernées : le radargrammétrie, l'interférométrie et la tomographie. La radargrammétrie ne requiert en prin-cipe que quelques images pour reconstruire la forme 3-D de la Terre, comme dans la vision stéréo humaine ou dans la photogrammétrie classique appliquée à l'imagerie optique, en exploitant uniquement l'information de signal d'amplitude.

L'interférométrie utilise l'information de différence de phase entre (au moins) deux images SAR pour estimer l'élévation du terrain. Ici, l'attention est mise sur la tomographie SAR : à partir d'une pile d'images recueillies en configuration interferometrique multibaselines, une telle technique permet d'extraire les informations de hauteur en formant une ouverture synthétique dans la direction d'élévation afin d'obtenir une résolution sensiblement améliorée. Les premières expériences importantes ont concerné l'analyse des structures de volume, comme les forêts, en utilisant des données aéroportés acquises à basse fréquence (par exemple les bandes L-ou P-). La première application SAR est présentée dans [START_REF] Rabus | The shuttle radar topography mission -A new class of digital elevation models acquired by spaceborne radar[END_REF], a été obtenue en utilisant des données MultiBaseline en bande L acquises par E-SAR, le système aéroportée du DLR. Au fil des ans, différents articles ont été publiés sur la tomographie SAR appliquée à l'estimation de a la structure de la forêt [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF][START_REF] Hu | Ship discrimination using polarimetric SAR data and coherent time-frequency analysis[END_REF][START_REF] Tebaldini | Single and multipolarimetric SAR tomography of forested areas: A parametric approach[END_REF], de l'épaisseur de la glace [START_REF] Stoica | Spectral Analysis of Signals[END_REF] ou la hauteur de bâtiments [START_REF] Zhu | Demonstration of super-resolution for tomographic SAR imaging in urban environment[END_REF][START_REF] Fornaro | Multilook SAR Tomography for 3-D Reconstruction and Monitoring of Single Structures Applied to COSMO-SKYMED Data[END_REF][START_REF] Homer | High resolution 3-D imaging via multi-pass SAR[END_REF], pour résoudre les problèmes liés à la géométrie SAR , en particulier dans les zones urbaines complexes [START_REF] Auer | Ray-tracing simulation techniques for understanding high-resolution sar images[END_REF][START_REF] Huang | 3-D characterization of buildings in a dense urban environment using L-band pol-insar data with irregular baselines[END_REF][START_REF] Porfiri | Building profile reconstruction using terrasar-x data time-series and tomographic techniques[END_REF][START_REF] Pisarenko | The Retrieval of Harmonics from a Covariance Function[END_REF]. Cette thèse de doctorat se concentre sur les potentialités élevées de techniques tomographiques pour la surveillance 3-D des changements et la caractérisation des zones complexes et denses bâties, en utilisant des estimateurs monodimensionnels de base comme Beamforming, Capon et MU-SIC sur la zone urbaine de Paris en utilisant des données TerraSAR-X à haute résolution et à polarisation unique. Le travail présenté ci-après limite le domaine des méthodes utilisées pour la reconstruction 3-D à des techniques d'estimation très rapides et n'aborde pas les estimateurs de signaux parcimonieux. Dans un premier temps, l'analyse de la qualité interférométrique de l'ensemble de données transformées a été réalisée. Les résultats ont montré de bonnes valeurs de cohérence moyenne au sein de l'ensemble de la pile et ont permis de détecter des images considérées comme des valeurs aberrantes en termes de niveaux de conditions temporelles et spatiales, pour finalement sélectionner l'ensemble adéquat de données multitemporelles à traiter. L'extraction des tomogrammes a montré la capacité de distinguer plus d'un diffuseur à l'intérieur de la même cellule de résolution et de reconstituer la topographie de bâtiments. Successivement, une caractérisation tridimensionnelle globale a été réalisée dans le but de développer un outil de suivi des changements 3-D des structures simples. En outre, la possibilité de corriger les distorsions géométriques en raison de la projection en distance (qui affecte fortement ce genre de scénarios) et de déterminer les informations sur le nombre de diffuseurs (jusqu'à trois) et la réflectivité correspondant à l'intérieur d'une cellule de résolution ont été évalués. Plusieurs reconstructions 3D de structures simples ont été réalisées à la fois en terme de hauteurs des bâtiments et réflectivité verticale. Par ailleurs une analyse novatrice de la stabilité temporelle de la scène observée a été réalisée afin de détecter les diffuseurs stables et instables. L'analyse effectuée en termes d'estimation de la réflectivité verticale a permis d'identifier différents comportements caractéristiques : des valeurs élevées sur l'interaction mur-sol a révélé la présence de réflexion de type double rebond ; le sol étant caractérisé par de faibles valeurs de réflectivité. L'analyse de la stabilité dans le temps 3-D a démontré la possibilité de suivre l'évolution en 3-D en fonction du temps.

Abstract

The Earth is a dynamic system and its surface is continuously changing at different time scales. Satellite imagery and derived secondary products play a relevant role in supplying the information necessary for analysing, monitoring and characterizing the Earth's surface dynamics. Accordingly, it is necessary to obtain timely and precise measurements of Earth's ecosystem. The urbanization and the management of urban environment and its periphery become one of the most crucial issues in developed and developing countries. In these circumstances, remote sensing data are an important source of information that reflects interactions between human beings and their environment. Given their complete independence from logistic constraints on the ground, illumination (daylight), and weather (clouds) conditions, Synthetic Aperture Radar (SAR) satellite systems may provide important contributions in complex environments 3-D reconstruction. SAR 2-D images are characterized by important geometric distortions, that severely limits their interpretation for the characterization of complex environments, such as urban areas.

In particular, starting from 2007 the new generation of high resolution Synthetic Aperture Radar (SAR) sensors as the Italian COSMO-SkyMed, the German TerraSAR-X and the Canadian RADARSAT-2 allowed to acquire SAR imagery up to 1 m Ground Sample Distance (GSD) in SpotLight mode, and with different incidence angle. At the present the attention is on the recent new family of missions called Sentinels (-1 for SAR technology) and developed by the ESA specifically for the operational needs of Copernicus, the Earth observation programme headed by the European Commission in partnership with ESA and aimed to provide accurate, timely and easily accessible information to improve the management of the environment, understand and mitigate the effects of climate change and ensure civil security. Indeed, modern SAR sensors and technologies can satisfy the demand for the monitoring of rapid changes on the ground and in anthropic activities. The possibility of near-real-time 3D Earth surface mapping satellite remote sensing data enable the possibility of continuous observation of the Earth's surface, with short times of acquisitions and revisit, representing a valuable tool for several objectives of Horizon 2020, the new framework programme for research and innovation in Europe starting on 2014. One of the most important applications of SAR remote sensing is the generation of Digital Surface Models (DSMs), that is, three-dimensional models of the Earth's surface. In suburban and urban areas it is very useful for a great number of applications, that include the inter-visibility calculations for the optimization of the location of telecommunication antennas, risk mapping, orthoimage generation, mission and urban planning and monitoring. Three different 3-D imaging techniques may be discerned: the radargrammetry, the interferometry and the tomography. Radargrammetry in principle requires just a couple of images to reconstruct the 3-D shape of the Earth, like in the human stereo vision or in the classical photogrammetry applied to optical imagery, exploiting just the amplitude signal information. Interferometry uses the phase differences information between (at least) two SAR images to lead the terrain elevation.

Here the attention is put on the SAR Tomography: starting from a stack of images collected using multibaseline data performed in interferometric configuration, such a technique allows to retrieve height information forming a synthetic aperture in the elevation direction in order to achieve a substantially improved resolution. First important experiments has concerned the analysis of volume structures, such as forests, using low frequencies (e.g. L-or P-band) airborne sensors. The first application is represented by [START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF]. The experimental results have been achieved using MultiBaseline L-band data sets acquired by the DLR's E-SAR (airborne) system. Succesevely, over the years, different papers have been published about the tomographic SAR techniques application in order to retrieve the forest structure [START_REF] Tebaldini | Single and multipolarimetric SAR tomography of forested areas: A parametric approach[END_REF][START_REF] Huang | Polarimetric SAR tomography of tropical forests at P-Band[END_REF][START_REF] Tebaldini | Multibaseline Polarimetric SAR Tomography of a Boreal Forest at P-and L-Bands[END_REF], to estime ice thickness [START_REF] Tebaldini | High resolution three-dimensional imaging of a snowpack from ground-based sar data acquired at X and Ku Band[END_REF] or building height [START_REF] Zhu | Demonstration of super-resolution for tomographic SAR imaging in urban environment[END_REF][START_REF] Fornaro | Imaging of single and double scatterers in Urban areas via SAR tomography[END_REF][START_REF] Huang | 3-D characterization of buildings in a dense urban environment using L-band pol-insar data with irregular baselines[END_REF], to solve the layover problem induced by strong terrain slopes or discontinuities in the imaged scene, especially in complex urban areas [START_REF] Budillon | Three-dimensional sar focusing from multipass signals using compressive sampling[END_REF][START_REF] Huang | Under-foliage object imaging using sar tomography and polarimetric spectral estimators[END_REF][START_REF] Reale | Tomographic imaging and monitoring of buildings with very high resolution sar data[END_REF][START_REF] Porfiri | Building profile reconstruction using terrasar-x data time-series and tomographic techniques[END_REF]. The present PhD thesis is focused on the high potentialities of tomographic techniques in 3-D change monitoring and characterization for complex and dense built-up areas using basic mono-dimensional estimators as Beamforming, Capon and MUSIC combined to very high satellite SAR resolution imagery. 2-D and 3-D analysis have been presented over the urban area of Paris using TerraSAR-X data at high resolution and single polarisation. The present work is mainly focused on the 3-D reconstruction by very fast estimation techniques and does not take into account parsimonious signal estima-tors.

At first, the analysis of the interferometric quality of the processed data set has been performed. The results showed good mean coherence values within the entire stack and allowed to detect images considered as outlier in terms of temporal and spatial baselines and to select the adequate set of multitemporal data to be processed. The extraction of 2-D tomograms over different azimuth-profile has showed the capabilities to distinguish more than one scatterer within the same resolution cell and to reconstruct the vertical building profiles. Successively, a global three-dimensional characterization has been performed in order to develop a monitoring tool for the 3-D changes of single structures. Moreover, the possibility to correct the geometric distortions due to the layover (that strongly affects such kind of scenarios) and to determine the information about the number of scatterers (up to three) and the corresponding reflectivity within one resolution cell have been evaluated. Several 3-D reconstructions of single structures have been performed both in term of buildings heights and vertical reflectivity. Moreover an innovative time stability analysis of the observed scene have been carried out in order to detect the stable and unstable scatterers.

Chapter 1 Introduction

Nowadays our life is strongly dependent on satellites in orbit around the Earth which make us safer, provide modern conveniences, broadcast entertainment, information about our planet gathered from space and sent around the globe in the blink of an eye. First satellite space programs have been run with military purposes and the first images of the Earth were captured in the 1960s, even though its capabilities have not been immediately understood. Indeed with respect to that date, the impact of this information on our everyday life is only a recent phenomenon. It was during the past decade (since 1999, with the launch of IKONOS) that a kind of competition started among different companies and agencies to launch optical and radar sensors with higher and higher resolution for civilian applications. Accordingly, many research groups around the world started to develop methodological and practical investigations about the large number of applications of this new satellite imagery, such as land monitoring, archaeological and environmental studies, cartography and mapping. It was clearly understood that high resolution sensors permit an easier acquisition of data of the same area at regular intervals, which is useful to monitor natural and anthropic phenomena evolving in time (e.g., urban growth, natural disasters); moreover, high resolution remote sensors allow us to obtain images of areas where it may be difficult to carry out traditional surveys for logistic reasons (e.g., developing countries, wild and remote areas). After many traditional optic sensors, radar ones acquired importance in recent years due to their improving acquisition resolution and their capacity to acquire imagery in situation where optical sensors are less useful: thanks to their active sensor nature and spectral domain they are able to acquire imagery through clouds and even during night. In particular, starting from 2007 the new generation of high resolution Synthetic Aperture Radar (SAR) sensors as the Italian COSMO-SkyMed, the German TerraSAR-X and the Canadian RADARSAT-2 allowed to acquire SAR images up to 1 m Ground Sample Distance (GSD) in SpotLight mode, and with different incidence angles. At the present the attention is on the recent new family of missions called Sentinels (-1 for SAR technology) and developed by the European Space Agency (ESA) specifically for the operational needs of Copernicus, the Earth observation programme headed by the European Commission in partnership with ESA and aimed to provide accurate, timely and easily accessible information to improve the management of the environment, understand and mitigate the effects of climate change and ensure civil security. Indeed, modern SAR sensors and technologies can satisfy the demand for the monitoring of rapid changes on the ground and in anthropic activities. The possibility of near-real-time three-dimensional (3D) Earth surface mapping satellite remote sensing data enable the possibility of continuous observation of the Earth's surface, with short times of acquisitions and revisit, representing a valuable tool for several objectives of Horizon 2020, the new framework programme for research and innovation in Europe starting on 2014.

One of the most important applications of SAR remote sensing is the generation of Digital Surface Model (DSM), that is, three-dimensional models of the Earth's surface. In modern map production, the 3-D imaging techniques have become an information source for scene analysis and understanding, for change detection, for GIS database updating, for cartographic 3-D feature extraction and reconstruction. In suburban and urban areas it is very useful for a great number of applications, that include the inter-visibility calculations for the optimization of the location of telecommunication antennas, risk mapping, orthoimage generation, mission and urban planning. Starting from the SAR data, with respect to the requirements of data accuracy and completeness and of realization efficiency and versatility, three different 3-D imaging techniques may be discerned: radargrammetry, interferometry and tomography.

Radargrammetry in principle requires just a couple of images to reconstruct the 3-D shape of the Earth, like in the human stereo vision or in the classical photogrammetry applied to optical imagery, exploiting just the amplitude signal information. Interferometry uses the phase differences information between (at least) two SAR images to lead the terrain elevation. Tomography represents an evolution of the last one, performing a 3-D imaging by the formation of an additional synthetic aperture in elevation using more than two SAR acquisitions performed in interferometric configuration.

Research objectives

The present PhD thesis is focused on the analysis of dense and complex urban areas by means of tomographic techniques applied to high resolution satellite SAR imagery. Starting from the classical second product of this approach, that is the reflectivity two-dimensional tomograms, a global three-dimensional characterization has been performed in order to develop a monitoring tool for the 3-D changes of single structures.

Moreover, the possibility to correct the geometric distortions due to the layover (that strongly affects such kind of scenarios) and to determine the information about the number of scatterers (up to three) and the SAR possesses some unique characteristics, such as all-weather functionality, capability of penetrating dense cover, and signal coherence, that gives it a considerable advantage over other infrared or optical sensors. Remote sensing is the science of obtaining and interpreting information from a distant point of view, using sensors that are not in physical contact with the object being observed [START_REF] Microimages | Introduction to remote sensing with TNTmips ®[END_REF]. The science of remote sensing in its broadest sense includes aerial, satellite, and spacecraft observations of the surfaces and atmospheres of the planets in our solar system, though the Earth is obviously the most frequent target of study. The term is customarily restricted to methods that detect and measure electromagnetic energy (see Fig. 2.1).

Imagery acquired by airborne or satellite sensors provides an important source of information for mapping and monitoring natural and manmade features on the land surface. The spatial, spectral, and temporal components of an image or set of images all provide information that we can use to form interpretations about surface materials and conditions. For each of these properties we can define the resolution of the images produced by the sensor system. These image resolution factors place limits on which information we can derive from remotely sensed images.

The spatial resolution is the smallest distance separating two objects 13 that can be discerned. It is a measure of the spatial detail in an image, which is a function of the design of the sensor and its operating altitude above the Earth's surface. Each of the detectors in a remote sensor measures energy received from a finite patch of the ground surface. The smaller these individual patches are, the more detailed will be the spatial information that we can interpret from the image. For digital images, spatial resolution is most commonly expressed as the ground dimensions of an image cell.

The spectral resolution of a remote sensing system can be described as its ability to distinguish different parts of the range of measured wavelengths. In essence, this amounts to the number of wavelength intervals (bands) that are measured, and how narrow each interval is. An image produced by a sensor system can consist of one very broad wavelength band, a few broad bands, or many narrow wavelength bands. The names usually used for these three image categories are panchromatic, multispectral, and hyperspectral, respectively.

The surface environment of the Earth is dynamic, with change oc-curring on time scales ranging from seconds to decades or longer. The seasonal cycle of plant growth that affects both natural ecosystems and crops is an important example. Repeat imagery of the same area through the growing season adds to our ability to recognize and distinguish plant or crop types. A time-series of images can also be used to monitor changes in surface features due to other natural processes or human activity. The time-interval separating successive images in such a series can be considered to define the temporal resolution of the image sequence.

Satellites detect the signal on several wavelengths depending on their spectral and spatial characteristics. Then, also the data resolution depends on these characteristics, how it is shown in table 2 In comparison with the optical remote sensing, the radar remote sensing offers some advantages, such as:

• all weather capability (small sensitivity of clouds, light rain)

• day and night operation (independence of sun illumination, i.e. active system)

• sensitivity to dielectric properties (water content , biomass, ice)

• sensitivity to surface roughness (ocean wind speed) and to dielectric properties (water content, biomass, ice)

• accurate distance measurements • sensitivity to man made objects • subsurface penetration (the longer the wavelength, the higher the transmission through a medium) and some inconveniences, such as:

• complex interactions (difficulty in understanding, complex processing)

• speckle noise effects • distortion of a scene geometry

SAR principles

In this section a basic introduction of SAR principles is given, from the fundamentals of the system to the main sensor features. Since during the last decades this technology has been deeply described and investigated, here only the concepts useful for understanding the present thesis are presented. For detailed discussions one can refer to, e.g., [START_REF] Curlander | Synthetic Aperture Radar: Systems and Signal Processing[END_REF][START_REF] Franceschetti | Synthetic Aperture Radar Processing[END_REF]. Imaging radar is an active illumination system. An antenna, mounted on a platform, transmits a radar signal in a side-looking direction towards the Earth's surface. The reflected signal, known as the echo, is backscattered from the surface and received a fraction of a second later at the same antenna (monostatic radar) (see Fig. 2.2). For coherent radar systems such as SAR, the amplitude and the phase of the received echo (which are used during the focusing process to construct the image) are recorded. Aperture means the opening used to collect the reflected energy that is used to form an image. In the case of radar imaging this is the antenna. For Real Aperture Radar (RAR) systems, only the amplitude of each echo return is measured and processed. The spatial resolution of RAR is primarily determined by the size of the antenna used: the larger the antenna, the better the spatial resolution. The spatial resolution can be defined through to directions: slant range direction, perpendicular to flight path of the spacecraft, and azimuth direction, parallel to the flight path of the spacecraft (see Fig. 2.3). Note that, defining the range as the distance from the radar to the scatterer, the slant range is the dimension in the image, different to the ground range that is the dimension along the ground. Another important element is the look angle ✓ (or off-nadir angle), defined as the vertical angle between the inclination of the antenna and the nadir. The look angle is steepest for the portion of the image nearest the flight path (near range), and decreases toward the far range. The range resolution r (see Fig. The azimuth resolution a (see Fig. 2.4b) may be approximated as:

a = R L (2.3)
where:

R is the slant range is the wavelength of the microwave L is the length of the aperture of radar antenna Azimuth resolution is dependent also on the height of the spacecraft H, and it improves ( a reduces) as the antenna length L is increased. The radar image is arranged by a pixel matrix, in which each pixel is To improve the resolution of radar images, SAR was developed in the 1960s. The SAR systems take advantage of the Doppler history of the radar echoes generated by the forward motion of the spacecraft to synthesise a large antenna (see Fig. 2.5). This allows high azimuth resolution in the resulting image despite a physically small antenna (L is drastically reduced). As the radar moves, a pulse is transmitted at each position. The return echoes pass through the receiver and are recorded in an echo store. SAR requires a complex integrated array of onboard navigational and control systems, with location accuracy provided by both Doppler and inertial navigation equipment. For sensors such as ERS-1/2 SAR and ENVISAT ASAR, orbiting about 900km from the Earth, the area on the ground covered by a single transmitted pulse (footprint) is about 5 km long in the along-track (azimuth) direction.

SAR processing is a complicated data processing of received signals Fig. 2.5 -Sbaceborne SAR system [START_REF]Earth Snapshot[END_REF] and phases from moving targets with a small antenna, the effect of which is to should be theoretically convert to the effect of a large antenna, that is a synthetic aperture length, i.e. the beam width by range which a RAR of the same length, can project in the azimuth direction. Compared to the RAR azimuth resolution described earlier, the azimuth resolution a of the SAR is much improved, that is:

a = L 2 (2.4)
Indeed, it means that the azimuth resolution a of a SAR is only determined by the length of the real aperture of an antenna L, independently of the slant range R. Each pixel of the SAR image holds the sum of the backscattered echoes toward the radar from all the objects in the resolution cell. The signal intensity received by the radar depends on the object in the resolution cell, e.g. urban or rocky areas have high intensity (white pixel), vegetated areas have medium intensity (grey pixel) and smooth areas (as basins of calm water) have low intensity (black pixel)(see Fig. 2.6). Diverse scattering mechanisms can be associated to these different media. For example double bounce reflection can occur between the ground and building façades as well as between the ground and under-foliage objects and between the object and the trunk, the volumic scattering is often related to the presence of vegetated (canopy) areas, the single bounce is derived from the building roof or the bare soil. SAR images contain a coherent, amplitude and phase, information. Depending on the information exploited for 3-D mapping, two techniques may be discerned: interferometry based on phase and radargrammetry based on amplitude. The main difference between these two techniques is that radargrammetry calculates the image range offset using the position matching of the same ground targets in two images whereas InSAR calculates the phase difference between two images. SAR imaging operating modes are:

• Spotlight Mode, in which the antenna is steered (both in the azimuth and the elevation plane) during the overall acquisition time in order to illuminate the required scene for a time period longer than the one of the standard strip side view, increasing the length of the synthetic antenna and therefore the azimuth resolution (at expense of the azimuth coverage). In such configuration the acquisition is performed in frame mode, hence it is limited in the azimuth direction due to the technical constraints deriving from the azimuth antenna pointing. The implementation allowed for this acquisition mode is the Enhanced Spotlight. In the Enhanced Spotlight mode, the spot extension is achieved by a antenna electronic steering scheme requiring the centre of the beam steering to be located beyond the centre of the imaged spot, thus increasing the observed Doppler bandwidth for each target • Stripmap Mode, which is the most common imaging mode (e.g. similar to ERS mission one), obtained by pointing the antenna along a fixed direction with respect to the flight platform path. The antenna footprint covers a strip on the illuminated surfaces as the platform moves and the system operates. The acquisition is virtually unlimited in the azimuth direction, except for the limitations deriving from the SAR instrument duty cycle

• ScanSAR Mode, which allows larger swath in range with respect to the Stripmap one, but with a less spatial resolution, obtained by periodically stepping the antenna beam to neighbouring subswaths. Since only a part of the synthetic antenna length is available in azimuth, the azimuth resolution is hence reduced. In such configuration the acquisition is performed in adjacent strip mode, hence it is virtually unlimited in the azimuth direction, but for the limitations deriving from the SAR instrument duty cycle

SAR effects

Radar geometric distortions

The SAR system measures the time delay between transmission and reception of a pulse in order to determine where targets are relative to one another in the range direction. Consequently, the SAR systems are range-based. This leads to geometric distortions due to relief displace-ment, such as foreshortening, layover, shadow, that decrease the reliability of radar imageries (see Fig. Points A, B and C are equally spaced when vertically projected on the ground (as it is done in conventional cartography). However, the distance A ′ B ′ is considerably shortened compared to B ′ C ′ , because the top of the mountain is relatively close to the SAR sensor. Also note that BC, the backslope becoming B ′ C ′ is consequently stretched. Foreshortening is a quite common effect in SAR images of mountainous areas.

Especially in the case of steep-looking spaceborne sensors, the acrosstrack slant-range differences between two points located on foreslopes of mountains are smaller than they would be in flat areas. This effect results in an across-track compression of the radiometric information backscattered from foreslope areas which may be compensated during the geocoding process if a terrain model is available. Foreshortening is obvious in mountainous areas (top left corner), where the mountains seem to 'lean' towards the sensor. Foreshortening effects are greatest in the near range (closer to the transmitter and receiver) and less pronounced in the far range. The lighter edges are the slopes facing the radar system. The narrow bright slopes (AB) illustrate increased compression in the near range.

If, in the case of a very steep slope, targets in the valley points of mountains have a larger slant range than the mountain top, then the foreslope is reversed in the slant range image (see Fig. 2.9). This phe-Fig. 2.9 -Layover in case of very steep slope [START_REF]Radar Imaging[END_REF] nomenon is called layover : the ordering of surface elements on the radar image is the reverse of the ordering on the ground. Generally, these layover zones, facing radar illumination, appear as bright features on the image due to the low incidence angle. Ambiguity occurs between targets hit in the valley and in the foreslope of the mountain, in case they have the same slant-range distance. For steep incidence angles this might also include targets on the backslope. Geocoding cannot resolve the ambiguities due to the representation of several points on the ground by one single point on the image; these zones also appear bright on the geocoded image. In built-up areas this phenomenon occurs at locations with steep elevation gradient facing towards the sensor, e.g. tall buildings. In Fig. 2.10a the layover effect is represented: different backscatters, e.g. the roof, the wall and the ground, are imaged in the same resolution cell. This causes a scattering ambiguity problem along the vertical direction that, generally, can be solved by SAR tomographic techniques (see chapter 4). An example of the layover effect is depicted in Fig. 2.10b where the well-known Tour Eiffel (captured by the satellite TerraSAR-X) is imaged 'laying down' onto the ground along the range direction (one may also note the Tower's shadow over the bottom of the figure). A slope away from the radar illumination with an angle that is steeper than the sensor depression angle causes radar shadows (see Fig. Shadow regions appear as dark (zero signal) with any changes due solely to system noise, sidelobes, and other effects normally of small importance. It should be also noted that the radar shadows of two objects of the same height are longer in the far range than in the near range.

Speckle noise

Speckle is a granular noise (see Fig. 2.12a) that inherently exists in all types of coherent imaging systems such as SAR imagery. This extremely complex pattern bears no obvious relationship to the macroscopic properties of the object illuminated. Rather it appears chaotic and unordered, and is best described quantitatively by the methods of probability and statistics [START_REF] Touzi | A review of speckle filtering in the context of the estimation theory[END_REF]. Unlike system noise (additive noise), speckle is an electromagnetic effect. Considering its statistical fluctuations, speckle may be modeled by a multiplicative gaussian noise in SAR images. It is coherent interference of waves scattered from terrain elements observed in each resolution cell. The vast majority of surfaces, synthetic or natural, are extremely rough on the scale of an optical wavelength [START_REF] Goodman | Some fundamental properties of speckle[END_REF]. Under illumination by coherent light, the wave reflected from such a surface consists of contributions from many independent scattering areas. Propagation of this reflected light to a distant observation point results in the addition of these various scattered components with relative delays which may vary from several to many wavelengths, depending on the microscopic surface and the geometry. Consequently, when illuminated by the SAR, each target contributes backscatter energy which, along with phase and power changes, is then coherently summed for all scatterers, so called random-walk (see Fig. and the filtering is one of the common methods which is used to reduce the speckle noises. The process used to apply filters to an image is known as convolution, and may be applied in either the spatial [START_REF] Mansourpour | Effects and performance of speckle noise reduction filters on active radar and SAR images[END_REF][START_REF] Lee | Digital image enhancement and noise filtering by use of local statistics[END_REF][START_REF] Kuan | Adaptive Restoration of Images with Speckle[END_REF] or frequency domain [START_REF] Xie | SAR speckle reduction using wavelet denoising and Markov random field modeling[END_REF][START_REF] Keller | Wavelets in geodesy and geodynamics[END_REF].

The influence of the preprocessing speckle filtering algorithms on the DSM generation has been taken into account, e.g., in [START_REF] Lee | A new technique for noise filtering of sar interferometric phase images[END_REF][START_REF] Porfiri | SAR speckle filtering for radargrammetric Digital Surface Model generation: implementation of several algorithms and analysis of their potentialities[END_REF][START_REF] Toutin | State-of-the-art of elevation extraction from satellite SAR data[END_REF]. More thoroughly, in [START_REF] Porfiri | SAR speckle filtering for radargrammetric Digital Surface Model generation: implementation of several algorithms and analysis of their potentialities[END_REF] its impact within the radargrammetric SAR processing chain (see section 3.1) has been studied considering the possibility to identify a greater number of homologous points characterised by a better Root Mean Square Error (RMSE) value. The work has led to infer that the speckle filtering algorithms allow to match a greater number of points, unfortunately characterizing by a lower accuracy value. This is caused by the presence of low accuracy points that alter the data and the final results. Furthermore, the relationship between the Signal to Noise Ratio (SNR) signal value and the elevation data statistics has been underlined, introducing the concept that the low accuracy points could be discarded by dynamically changing the SNR threshold value during the image matching procedure (see 3.1).

SAR platforms: TerraSAR-X

Sensors based in space are used for Earth observation, be it for research into the alterations taking place in our atmosphere, for climate research, for the monitoring of processes in our oceans, for geological investigations, and much more besides. Remote sensing data obtained from space thus complement and complete Earth-based measurements. They are always essential if a global view of our Earth is required, as is the case, for example, with the weather, or with the monitoring of the polar ice caps.

With the TerraSAR-X radar satellite, the land masses of the Earth are particularly closely inspected. This includes the mapping of our forests, the generation and current updating of land utilization maps, the recording of derelict land areas and the estimation of the maturity level of areas in agricultural use, as well as the study and monitoring of geologically active areas such as volcanic and earthquake regions. With the increasing technical capability of these sensors this kind of data can be extracted from satellites with ever more precision.

TerraSAR-X is Germany's first national remote sensing satellite that has been implemented in a public-private partnership between the German Aerospace Center (DLR) and EADS Astrium GmbH, with a significant financial participation from the industrial partner. It is designed to satisfy the steadily growing demand of the private sector for remote sensing data in the commercial market. As mentioned, TerraSAR-X products can be used for a variety of applications. Moreover, the first experimental data acquisitions have been successfully completed, among them data takes in the so-called Dual-Receive Antenna Mode, which allows motion measurements as well as fully polarimetric data acquisitions. In addition, the so-called TOPS-Mode could be demonstrated for the first time, a new method for imaging particularly wide swathes with high quality.

Today it is difficult to imagine our life without satellites. Whether the daily weather report using meteorological satellites, the live transmission of an ongoing event from remote areas using modern communication satellites or the navigation of ships on the world's oceans by means of global positioning satellites: these services that in recent times have become part of our daily life are only possible thanks to technologically advanced missions operating in space.

With TerraSAR-X it is used the Synthetic Aperture Radar sensor (see p. 15). This contributes significantly to the reliability of the system, a property that is increasingly requested by many users, since data are often required at a certain point in time. Today, these kinds of capabilities are no longer just of interest for scientific applications; such data are also being asked for increasingly on the commercial market. This is the reason why TerraSAR-X, Germany's first national remote sensing satellite, has been implemented in a so-called public-private partnership between the German Aerospace Center and EADS Astrium GmbH. In Table 2.2 are showed the main TerraSAR-X technical data. The TerraSAR-X satellite pictorial representation is given in Fig. 2.13.

The data recorded by the SAR instrument are transferred via a downlink antenna to the ground receiving station. The antenna is secured to a 3.3 meters long mast in order to avoid interferences caused by the radar antenna. The mast is folded up during the launch and is only extended after positioning of the satellite into its orbit. It allows simultaneous acquisition of new data by the radar and transmission of previously stored data to the ground. The high accuracy position control of the satellite is implemented by means of star sensors that are installed near the radar antenna, so that the required high accuracy of the antenna alignment of 65 arcsecs can be achieved. A Global Positioning System (GPS) receiver located on board enables a precise determination of the orbit. The primary payload is a Synthetic Aperture Radar with an active antenna that allows for the utilization of different imaging modes and can be used in a very flexible manner. With radar instruments various frequency ranges of the radar spectrum can be observed, the so-called bands. TerraSAR-X is operated in the X-band, which is lying at a frequency around 9.65 GHz, corresponding to a wavelength of about 3 centimetres. The technology of the active, phase-controlled antenna enables a high flexibility and mission efficiency. While in case of passive systems the whole radar antenna or even the satellite must be rotated in order to align the antenna onto the target area, the active antenna of TerraSAR-X can steer its radar pulses in a certain direction. The antenna is 4.80 meters long and 80 centimetres wide. The satellite is designed in a way that it can be installed, together with its antenna at its full size, on the launch vehicle.

In this way a complex unfolding mechanism can be avoided.

Alongside the SAR instrument two secondary payloads are flying on TerraSAR-X: Chapter 3

3-D SAR imaging

Due to the specific SAR geometry, SAR imaging represents a projection of the 3D space (x, y, z) into the bi-dimensional (2D) cylindrical zero-Doppler coordinates (x, r). For each pixel of a SAR image, the scattering response consists of a sum of contributions from all scatterers at the same range, encompassing all heights. Hence information about the vertical structure is lost. To recover such information, 3-D imaging techniques are required, whose derived secondary products consists in Digital Elevation Model (DEM), i.e. data files that contain information on the spatial distribution of the elevation data over a specified area. In particular, one may distinguish between DSM, representation of the first reflective or visible surface (including trees, buildings, and any other features which is elevated above the ground), and Digital Terrain Model (DTM), representation of the bare terrain (without objects).

In the following sections we present the basis of three 3-D imaging techniques: (i)Radargrammetric SAR (StereoSAR) [START_REF] Mèric | Radargrammetric SAR Image Processing[END_REF] and (ii)Interferometric SAR (InSAR) [START_REF] Graham | Synthetic interferometer radar for topographic mapping[END_REF], both based on the use of two complex-valued radar images acquired from, respectively, different and slightly different point of view, and (iii)Tomographic SAR (TomoSAR) [START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF], evolution of (ii) and based on the use of more than two SAR acquisitions performed in interferometric configuration.

Radargrammetric SAR

Radargrammetry was first employed in the 1950s with ground and airborne radar, then less and less, due to the quite low amplitude resolution of radar satellite imagery and, consequently, the high loss in term of accuracy [START_REF] Leberl | Radargrammetric image processing[END_REF]. Only within the last decade, thanks to the very high resolution imagery acquired by new satellite sensors, as COSMOSkyMed (Italian),

TerraSAR-X (German) and RADARSAT-2 (Canadian), which are able to supply imagery with 1 m GSD, the radargrammetric approach has been revitalized and investigations restarted [START_REF] Capaldo | DSMs generation from COSMO-SkyMed, Radarsat-2 and TerraSAR-X imagery on Beauport (Canada) test site: Evaluation and comparison of different radargrammetric approaches[END_REF][START_REF] Nascetti | chapter Radargrammetric Digital Surface Models Generation from High Resolution Satellite SAR Imagery: Methodology and Case Studies[END_REF][START_REF] Perko | Forest Assessment Using High Resolution SAR Data in X-Band[END_REF][START_REF] Toutin | 3-D Radargrammetric modeling of RADARSAT-2 ultrafine mode: Preliminary results of the geometric calibration[END_REF]. A comprehensive text on radargrammetry is [START_REF] Leberl | Radargrammetric image processing[END_REF].

Analogous to optical photogrammetry, radargrammetry is based (at least) on a stereopair that must be acquired under a suited geometric configuration. This technique exploits only the amplitude information from SAR imagery and coherence between images is not required. Respect to InSAR, the independence from the coherence is one of the main advantages of StereoSAR, especially from the point of view of the strong influence of the land cover and the atmospheric effects. Basically, atmospheric effects on the SAR imagery are same in the StereoSAR or in the InSAR processing. However, as mentioned, StereoSAR uses the magnitude (intensity) value whereas InSAR uses the phase difference in SAR imagery. Considerably, magnitude is less affected than phase by atmospheric heterogeneity. Hence, the atmospheric disturbance is undesirable for the interferometric processing but not much of a concern for the radargrammetric one.

The radargrammetric approach performs a 3-D reconstruction based on the determination of the sensor-object stereo model for which the position of each point on the object is computed as the intersection of two radar rays coming from different positions and, therefore, with two different look angles (Fig. 3.1). Radargrammetry is based on stereogrammetry, which is a classical method for relief reconstruction using optical remote sensing images. Stereo viewing reproduces the natural process of stereovision. As for photogrammetry, two steps are required for DSM generation: stereopair orientation and image matching for point cloud generation. In [START_REF] Capaldo | High Resolution Radargrammetry -3D Terrain Modeling[END_REF] a complete procedure for generating radargrammetric DSMs is proposed.

In the context of the orientation models for the stereo pair, two main kinds of models are adopted: the physical sensor models (also called rigorous models) and the generalized sensor models. In the first one, in case of SAR imagery, the image and the ground coordinates are linked through the radargrammetric equations (similarly to collinearity equations in photogrammetric approach) and the involved parameters have a physical meaning. Besides, they require knowledge on the specific satellite and orbit characteristics. On the contrary, the generalized models are usually based on the Rational Polynomial Functions (RPFs), which link image and terrain coordinates by the Rational Polynomial Coefficients (RPCs) and do not need the knowledge about the sensor and acquisition features. The RPCs can be calculated by the final users via a Least-Squares (LS) estimation directly from Ground Control Points (GCPs), or proprietarily generated by the sensor managing companies based on their own physical sensor models and distributed to users through imagery metadata.

Nevertheless, the first strategy (also called terrain-dependent) is not recommended if a reliable and accurate orientation is required. In the second strategy, they can be generated according to a terrain-independent scenario, using known physical sensor.

In [START_REF] Capaldo | Evaluation and comparison of different radargrammetric approaches for Digital Surface Models generation from COSMO-SkyMed, TerraSAR-X, RADARSAT-2 imagery: Analysis of Beauport (Canada) test site[END_REF], following from [START_REF] Leberl | Radargrammetric image processing[END_REF], is proposed a model based on two fundamental equations: the first equation of (3.1) represents the general case of zero-Doppler projection: in zero-Doppler geometry the target is acquired on a heading that is perpendicular to the flying direction of satellite; the second equation of (3.1) is the slant range constrain.

V XS ⋅ (X S -X P ) + V Y S ⋅ (Y S -Y P ) + V ZS ⋅ (Z S -Z P ) = 0 (X S -X P ) 2 + (Y S -Y P ) 2 + (Z S -Z P ) 2 -(D S + CS ⋅ I) = 0 (3.1)
X P , Y P , Z P are the coordinates of the generic Ground Point (GP) in a terrestrial coordinate system (time independent); X S , Y S , Z S are the coordinates of the satellite sensor in the same coordinate system (time dependent); V XS , V Y S , V ZS are the cartesian components of the satellite sensor velocity again in the same coordinate system (time dependent); D S is the so-called near range, a parameter related to the range measurements whose initial value is available in the metadata; CS is the slant range resolution or column spacing; I is the column position of point P on the image. The orbital arc related to an image acquisition in Spot-Light mode, since quite short (about 10 km), is conveniently interpolated with Lagrange polynomial using the orbital state vectors available in the metadata. In this way, it is possible to model the images acquisition and the stereo orientation through metadata parameters.

The image matching process heavily impacts the DSM accuracy, and its efficiency depends on the configuration for the radargrammetric appli-cation. Observing the target in opposite-site view allows to obtain a good stereo geometry, but causes large geometric and radiometric disparities that hinder image matching. Thus, in order to increase the efficiency, a good compromise is to use a same-side configuration stereo-pair with a base to height ratio ranging from 0.25 to 2 [START_REF] Mèric | Radargrammetric SAR Image Processing[END_REF]; ; anyway, it is better to limit within 0.35 and 0.70. Two are the main steps of image matching process: the definition of a matching entity, that is a primitive in the master image to be compared with a portion of other slave images, in order to identify correspondences among them; the selection of a search criteria in order to limit the possible solutions and to avoid mismatching.

In [START_REF] Nascetti | Fast terrain modelling for hydrogeological risk mapping and emergency management: the contribution of high-resolution satellite SAR imagery[END_REF] an original matching procedure is proposed. The matching strategy is based on a coarse-to-fine hierarchical solution with an effective combination of geometrical constrains and an Area Based Matching (ABM) algorithm, using a pyramidal approach and analysing the SNR.

In this way the surface model is successively refined step by step, until the last step (corresponding to the original image resolution) where the final dense and accurarate DSM is reconstructed (see Fig. 3.2).

Interferometric SAR

In the last two decades InSAR has been developed from theoretical concept to a technique that is being utilized at an increasing rate for a wide range of Earth science fields. Despite the fact that none of the currently deployed imaging radar satellite platforms was designed with interferometric applications, the high quality and large quantity of exciting results from this technique have demonstrated their potential as powerful tools for ground deformation evaluation [START_REF] Massonnet | The displacement field of the Landers earthquake mapped by radar interferometry[END_REF][START_REF] Zebker | On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake[END_REF] and DEM generation [START_REF] Ferretti | Multibaseline InSAR DEM reconstruction: the wavelet approach[END_REF][START_REF] Gamba | Detection and extraction of buildings from interferometric SAR data[END_REF].

In the last few years InSAR capability has been considerably improved by using large stacks of SAR images acquired over the same area, instead of the classical two images used in the standard configurations [START_REF] Ferretti | Permanent scatterers in SAR interferometry[END_REF][START_REF] Lanari | A small-baseline approach for investigating deformations on full-resolution differential SAR interferograms[END_REF]. Special missions, such as the Shuttle Radar Topography Mission (SRTM) [START_REF]Shuttle Radar Topography Mission (SRTM)[END_REF][START_REF] Rabus | The shuttle radar topography mission -A new class of digital elevation models acquired by spaceborne radar[END_REF], or more recently TanDEM-X (ongoing) [START_REF] Krieger | TanDEM-X: A satellite formation for highresolution SAR interferometry[END_REF], have been designed

and built to yield elevation data on a world scale. At least a pair of images of the same area acquired from slightly different look angles are needed to perform the 3-D reconstruction. Such can be obtained either simultaneously using two radar antennas mounted on the same platform (single pass interferometry) or at different times by exploiting repeated orbits of the same satellite (repeat pass interferometry). Fig. 3.3 shows the interferometric acquisition geometry and relative notations related to two radar measurements of the same point (P ) observed from slightly different looking angles (✓ 1 ,✓ 2 ). The distance between the two satellites S 1 ,S 2 in the plane perpendicular to the orbit is called the interferometer baseline B. Respect to its inclination angle ↵ B in the incidence plane, B can be decomposed into its perpendicular B ⊥ and parallel B ∥ components:

B ⊥ = Bcos(✓ 1 -↵ B ) and B ∥ = Bsin(✓ 1 -↵ B ) (3.2)
The two SAR images are composed of a regular grid with complex values, s 1 and s 2 , which can be decomposed in an amplitude and phase Fig. 3.3 -Geometry of interferometric SAR system component using:

s 1 = s 1 e -j( r1 + obj1 ) s 2 = s 2 e -j( r2 + obj2 ) (3.3)
where ri = kr i , with i = 1, 2 and k = 4 ⋅ ⇡ , where is the wavelength. Before the interferometric processing, the image pair must be at first coregistered, i.e. the slave (s 2 ) grid will be aligned and resampled to corresponding locations in the master (s 1 ) grid. After that, the SAR interferogram is generated by cross-multiplying, pixel by pixel, the first SAR image with the complex conjugate of the second [START_REF] Ferretti | InSAR Principles: Guidelines for SAR Interferometry Processing and Interpretation[END_REF]. Assuming that the proper scattering phase of the scatterer is the same in both SAR images due to the similar look angles (as true as the smaller the signal decorrelation), i.e. obj1 = obj2 , one can obtain

s 1 s * 2 = s 1 s 2 e -jk(r 1 -r 2 ) = s 1 s 2 e -jk r (3.4)
Thus, the interferogram amplitude is the amplitude of the first image multiplied by that of the second one, whereas its phase (the interfero- terrain elevation or on terrain displacement [START_REF] Toutin | State-of-the-art of elevation extraction from satellite SAR data[END_REF].

The plane wave approximation in the far-field (i.e. parallel ray approximation), can be applied to the interferometric acquisition geometry in Fig. 3.3 and we obtain

B ∥ = Bsin(✓ 1 -↵ B ) = r - r 2 2r 1 + B 2 2r 1 r (3.6)
since the baseline B and the range difference r are largely inferior to the distance between the master track and the scatter, r 1 . The interferometric phase is then proportional to the parallel part of the baseline.

SAR images have a spatial resolution defined by the pulse length, altitude of the satellite and look-angle. The slant range resolution is half the pulse length. The ground range resolution is defined as the slant range resolution divided by the sine of the look-angle. The set of phase differences inf produced for all pixels of the two SAR images is used to generate an interferogram. We can linearise this variation into two contributions respectively related to the slant range displacement, r 1 and the height variation, z,

inf = @ inf @r 1 hr r 1 + @ inf @z r 1 z = fe + topo (3.7)
The resulting two components correspond to the flat earth phase and the topographic phase respectively.

The flat-earth effect is related to the ellipsoid reference and consists of numerous fringes parallel to the flight path (see Fig. 3.5a). A shown in Fig. 3.4a, slightly moving along the ground direction y with constant height h r , the interferometric phase variation due to the slant range variation r 1 , can be written as: The topographic phase is directly linked to the surface height information. As shown in Fig. 3.4b, slightly increasing the height of the scatterer by z, while maintaining the sensor-scatterer distance at r 1 , the interferometric phase due to the height variation z, can be expressed as

fe = kB ⊥ r 1 r 1 tan✓ 1 (3.8) (
topo = kB ⊥ z r 1 sin✓ 1 (3.9)
The height sensitivity can be respresented by the vertical wavenumber k z and the height ambiguity is defined by h amb when topo = 2⇡

k z = kB ⊥ r 1 sin✓ 1 (3.10) h amb = 2⇡ k z = 2 r 1 sin✓ 1 B ⊥ (3.11)
When the scatterer is located at h r > h amb , e.g. mountains or skyscrapers, the measured topographic phase is wrapped and can be represented as

topo = k z h r 2⇡ (3.12)
with 2⇡ expressing 'modulo 2⇡'. To calculate the real height of the scatterer, a reference point P ref should be firstly selected with known height h ref .

In the neighbourhood of P ref , we compute the h r from topo by equation (3.9).

The interferometric coherence is defined by [START_REF] Bamler | Synthetic aperture radar interferometry[END_REF] as a normalized complex cross-correlation between two received signals s 1 and s

2 int = E{s 1 s * 2 } E{s 2 1 }E{s 2 2 } (3.13)
Its phase is the expected interferometric phase 0 of the selected pixel and its magnitude is related to phase noise. Thus, the coherence measures the interferogram quality and influences the estimation of interferometric phase, since it is related to phase stability or predictability. Receiver noise, for example, may render the two images to be not fully correlated, i.e. < 1. It is known from [START_REF] Bamler | Synthetic aperture radar interferometry[END_REF] that the variance of estimation errors of inf increases when int descreases. In Fig. 3.5c it is depicted the coherence image generated over a cropped area of the Berlin city: brighter areas (good coherence) correspond mainly to man-made targets, while darker areas (bad coherence) mainly to vegetated zones.

Tomographic SAR

Currently, tomographic SAR (deeply investigated in chapter 4) represents the most effective SAR technology for the monitoring of complex areas, such as, for example, urban scenarios. It is an evolution of the interferometric approach by extending one scattering phase center [START_REF] Bamler | Synthetic aperture radar interferometry[END_REF] to several scatterers by using a coherent combination of images acquired in multibaseline interferometric configuration and performing a 3-D imaging by the formation of an additional synthetic aperture in elevation [START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF]. This technique directly retrieves the distribution of the backscattered power in the vertical direction, as shown in Fig. 3.6.

TomoSAR allows also to solve the geometric distortion problems, which are intrinsic with the SAR system, in areas characterized by a high density of vertical structures, such as dense urban areas, where it happens that radar returns signal responses belonging to different parts of the single structure under observation that interfere in the same SAR resolution cell. This technique, being able to analyze the distribution at the different heights of the backscattering energy, is able to solve this interference and separate different targets inside the same pixel (i.e. to solve the layover effect). Furthermore, by analysing dataset of images distributed in time, is even possible to estimate the deformation affecting each single interfering target, through the extraction of its deformation time series.

TomoSAR is also a valuable technique to achieve detailed and veritable DSM of the observed scene, allowing the possibility to extract detailed and spatial dense 3-D models of the whole observed scene or of the single structures on ground (such as buildings). For this reason, tomographic reconstructions have started to be considered competitive with results

obtained through an analysis based on the usage of optical sensors, such as LiDAR, which are typically exploited for the 3-D reconstructions of buildings. With respect to the latter, SAR tomography offers the further advantages of a temporal monitoring as well as synoptical view. Chapter 4

SAR Tomography

Tomographic SAR focusing is a 3-D imaging technique based on the formation of an additional synthetic aperture in the elevation direction in order to achieve a substantially improved resolution. As extension of the interferometric SAR, it consists in a coherent combination of a set of images (stack) acquired from several orbit (satellite systems) or flight (airborne systems) tracks in a MB-InSAR configuration that allows a direct localization and separation of multiple scattering contributions within a range-azimuth resolution cell. TomoSAR is also a valuable technique to achieve detailed and veritable DSM of the observed scene, allowing the possibility to extract detailed 3D models of single structures on ground (such as buildings) and to monitor them in terms of temporal deformation (four-dimensional (4D)) at high resolution. These two aspects represent an important development in SAR remote sensing especially for the analysis of dense urban scenarios. Thanks to the first one, this technology allows to resolve the signal scatterings superposition due to the layover that frequently affects these areas as scenes rich of man-made structures.

The second aspect represents an important advantage in the urban and environmental monitoring and in the management of risk situations.

In the following sections an overview of this technique is given, including a short survey about the state-of-the-art, the basic principles and the spectral estimation techniques used for the present study.

State-of-the-art in urban areas

The word tomography is derived from two ancient Greek words: tomos, i.e. 'slice, section', and grapho, i.e. 'to write'. Therefore, this technique allows to generate detailed images of a specific plane section of a solid object by using changing information coming from any kind of penetrating wave. In this sense, the tomography represents a valuable tool for several scientific applications as the geophysical, the archaelogical or the medical ones. In the SAR remote sensing field, the highly investigation of the tomography is due to the desire to achieve a fully 3-D imaging starting from a 2-D SAR image. Moreover a 4-D SAR tomographic imaging allows to describe the space-time or space-velocity information and then to monitor the target deformation.

First important experiments have concerned the analysis of volume structures, such as forests, using low frequencies (e.g. L-or P-band) airborne sensors. The first application is represented by [START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF], where the concept of the formation of a second synthetic aperture in the direction of the translational movement is presented. The experimental results have been achieved using MultiBaseline (MB) L-band data sets acquired by the DLR's E-SAR (airborne) system. A method for reduction of the height ambiguities associated to the irregular and undersampled spatial distribution of the imaging positions is proposed and, as mentioned by the authors, this work represents the first step into SAR tomographic technique that showed its great potential in respect to several applications. Accordingly, over the years different papers have been published about the tomographic SAR techniques application in order to retrieve the forest structures [START_REF] Tebaldini | Single and multipolarimetric SAR tomography of forested areas: A parametric approach[END_REF][START_REF] Huang | Polarimetric SAR tomography of tropical forests at P-Band[END_REF][START_REF] Tebaldini | Multibaseline Polarimetric SAR Tomography of a Boreal Forest at P-and L-Bands[END_REF], to estimate ice thickness [START_REF] Tebaldini | High resolution three-dimensional imaging of a snowpack from ground-based sar data acquired at X and Ku Band[END_REF] or building heights [START_REF] Zhu | Demonstration of super-resolution for tomographic SAR imaging in urban environment[END_REF][START_REF] Fornaro | Imaging of single and double scatterers in Urban areas via SAR tomography[END_REF][START_REF] Huang | 3-D characterization of buildings in a dense urban environment using L-band pol-insar data with irregular baselines[END_REF], to solve the layover problem induced by strong terrain slopes or discontinuities in the imaged scene, especially in complex urban ar-eas [START_REF] Budillon | Three-dimensional sar focusing from multipass signals using compressive sampling[END_REF][START_REF] Huang | Under-foliage object imaging using sar tomography and polarimetric spectral estimators[END_REF][START_REF] Reale | Tomographic imaging and monitoring of buildings with very high resolution sar data[END_REF][START_REF] Porfiri | Building profile reconstruction using terrasar-x data time-series and tomographic techniques[END_REF]]. An overview is given in [START_REF] Ferro-Famil | Principles and Applications of Polarimetric SAR Tomography for the Characterization of Complex Environments[END_REF] and the potentialities of TomoSAR combined with polarimetric diversity are demonstrated for a wide variety of applications, such as forests remote sensing, detection of under-foliage object and urban areas characterization.

Built-up area analysis has been always a great challenge in particular starting from the availability of the first sensors able to provide sufficient resolution. Speaking about space-borne platforms, in the last decade the new satellite generation, well represented by the German TerraSAR-X (TSX) and the Italian COSMO-SkyMed (CSM) ones, provided high resolution data both in terms of space and time. Complex urban scenes, being rich of man-made structures, are considered as complex scenarios needed of a refined analysis as the observations are frequently affected by layover effects. In fact, a mixture of diverse scattering patterns (such as double bounce reflection due to wall-ground reflections, single bounce from roofs and from the ground, volumetric scattering due to potential vegetation) may interfere in the same SAR resolution cell (see subsection 2.2.1). In literature, the tomographic study of dense urban environment characterization has been mainly performed from two different point of view: using full polarimetric MB data acquired by airborne sensors (e.g. E-SAR) or using Single Polarization (SP) MB data acquired by satellite sensor (e.g. TerraSAR-X).

Regarding the former, in [START_REF] Ferro-Famil | Urban area remote sensing from Lband PolSAR data using time-frequency techniques[END_REF], [START_REF] Huang | 3-D characterization of buildings in a dense urban environment using L-band pol-insar data with irregular baselines[END_REF] and [START_REF] Sauer | Three-Dimensional Imaging and Scattering Mechanism Estimation Over Urban Scenes Using Dual-Baseline Polarimetric InSAR Observations at L-Band[END_REF] several demonstrations of the effectiveness of different spectral estimation methods for building height estimation and scatterers physical properties retrieval are given using fully polarimetric MB-InSAR data over the city of Dresden acquired by the E-SAR system. Regarding the latter, in literature different studies about the 3-D and the 4-D tomography and the layover correction have been published. In particular passing through [START_REF] Zhu | Demonstration of super-resolution for tomographic SAR imaging in urban environment[END_REF][START_REF] Zhu | Space-borne high resolution SAR tomography: experiments in urban environment using TerraSAR-X Data[END_REF][START_REF] Zhu | Very High Resolution Spaceborne SAR Tomography in Urban Environment[END_REF][73][74] one may appreciate the description of several decomposition methods and the relative comparisons presented by the authors. The papers concern the detection of multiple targets within the same azimuth-range cell and a 3-D mapping of the scattering properties of the illuminated scene [START_REF] Zhu | Space-borne high resolution SAR tomography: experiments in urban environment using TerraSAR-X Data[END_REF], the extension of the tomographic concept to the 4-D differential case [START_REF] Zhu | Very High Resolution Spaceborne SAR Tomography in Urban Environment[END_REF] and to the Compressive Sensing (CS)1 technique [START_REF] Zhu | Demonstration of super-resolution for tomographic SAR imaging in urban environment[END_REF][START_REF] Zhu | Space-borne high resolution SAR tomography: experiments in urban environment using TerraSAR-X Data[END_REF][START_REF] Zhu | Very High Resolution Spaceborne SAR Tomography in Urban Environment[END_REF] applied to the tomographic imaging of the deformations of urban infrastructures. The works represent an important demonstration of the tomographic potentialities of the high resolution TSX sensor, using data acquired over the city of Las Vegas. On the other side, several investigations about the tomographic capability to resolve layover cases, e.g. separating single and double-scattering mechanisms within imaged pixels, are performed in [START_REF] Fornaro | Multilook SAR Tomography for 3-D Reconstruction and Monitoring of Single Structures Applied to COSMO-SKYMED Data[END_REF][START_REF] Fornaro | Four-Dimensional SAR Imaging for Height Estimation and Monitoring of Single and Double Scatterers[END_REF][START_REF] Fornaro | Imaging of single and double scatterers in Urban areas via SAR tomography[END_REF][START_REF] Fornaro | CAESAR: An Approach Based on Covariance Matrix Decomposition to Improve References Multibaseline-Multitemporal Interferometric SAR Processing[END_REF][START_REF] Reale | Tomographic imaging and monitoring of buildings with very high resolution sar data[END_REF] .The results have been achieved processing datasets acquired at first from ERS-1 and ERS-2 and then from TSX and CSM satellites over different cities (such as Naples and Rome). The works aim to describe several algorithms for the reconstruction and monitoring of the whole 3-D structure of buildings, exploiting both the 4-D imaging [START_REF] Fornaro | Four-Dimensional SAR Imaging for Height Estimation and Monitoring of Single and Double Scatterers[END_REF][START_REF] Reale | Tomographic imaging and monitoring of buildings with very high resolution sar data[END_REF] and the combination of TomoSAR techniques and CAESAR, a method proposed in [START_REF] Fornaro | Multilook SAR Tomography for 3-D Reconstruction and Monitoring of Single Structures Applied to COSMO-SKYMED Data[END_REF][START_REF] Fornaro | CAESAR: An Approach Based on Covariance Matrix Decomposition to Improve References Multibaseline-Multitemporal Interferometric SAR Processing[END_REF] that allows the identification of multiple scattering mechanisms from the analysis of the covariance matrix.

In this context it is collocated the present research aimed to perform a global three-dimensional characterization of dense and complex urban areas by means of the classical spectral estimators. In particular, the possibility to correct the geometric distortions due to the layover and to determine the information about the number of scatterers (up to three) and the corresponding reflectivity within one resolution cell have been evaluated. Several 3-D reconstructions of single structures have been performed both in term of buildings heights and vertical reflectivity. Moreover an innovative time stability analysis of the observed scene have been carried out in order to detect the stable and unstable scatterers.

The processing have been performed using an adequate high resolution MB-InSAR data acquired over Paris urban area by the satellite TSX. The final goal is to demonstrate the high potentialities of tomographic techniques using basic mono-dimensional estimators as Beamforming, Capon and MUSIC combined to very high resolution data.

Basic principles

The objective of tomographic focusing is to estimate the height profile of a medium and the corresponding set of reflectivity coefficients ( (z)).

The basic principle of satellite TomoSAR focusing is illustrated in Fig.

4.1.

In order to achieve focusing in height direction and to generate a 3-D SAR image, the additional aperture is synthesized in the normal direction n, perpendicular to the range (line-of-sight) r and the azimuth

x directions. This normal direction can be projected onto the elevation direction ẑ or the ground range direction ŷ in case of, respectively, airborne or satellite systems. The former consider parallel flight tracks at different elevations; while the latter consider different, as equally spaced as possible, orbit tracks at the same elevation (Fig. 4.1). Following formulations are referred to airborne geometry.

A scatterer can be assumed as a source that emits a signal s towards the SAR. In order to characterize a scatterer (or a source), L samples (or looks or realizations) of its response are acquired and processed. The corresponding signals can be characterized from the statistical properties relating the L realizations of their response, i.e. s(l) L l=1 . For a scatterer located at the height n 0 and at a range position r(z, n 0 ), the received signal can be expressed as:

s(z, n 0 ) = a(r 0 , n 0 ) * exp{-jkr(z, n 0 )} (4.1)
r 2 0 + (z -n 0 ) 2 r 0 + (z-n 0 ) 2 2r 0 .
In equation (4.1) r 0 is the minimum range distance between the sensor and the scatter, and a(r 0 , n 0 ) is the complex reflectivity of the scatterer at the position (r 0 , n 0 ) for a given azimuth position x 0 and ′ * ′ indicates the convolution between the scattering response and the impulse response of the TomoSAR system. After the demodulation, the received signal becomes:

s d (z, n 0 ) = a(r 0 , n 0 ) * exp -jk (z -n 0 ) 2 2r 0 (4.2)
which includes the quadratic phase term as a chirp signal in the ẑ direction. Thus, the vertical spatial frequency (or vertical wavenumber) k z is given by:

k z (z, n 0 ) = @arg(s d (z, n 0 )) @z = k r 0 (n 0 -z) (4.3)
and is a function of the elevation z. After deramping which consist in multiplying the received signal with a deramping function

u(z) = exp jk z 2 2r 0 (4.4)
and the deramped signal can be obtained as follows

s dr (z, n 0 ) = a(r 0 , n 0 ) * exp -jk (n 2 0 -2zn 0 ) 2r 0 (4.5)
The vertical wavenumber k z of the deramped signal s dr is hece given by

k z (n 0 ) = @arg(s dr (z, n 0 )) @z = kn 0 r 0 (4.6)
which is independent of the z position but only depends on the scattering height n 0 inside a volume with a total height equal to H V (see Fig. 4.1b). Then, applying FFT to the deramped signal s dr (z, n 0 ) in the ẑ direction, the focused image for this scatterer may be represented by S dm (k z , n 0 )

S dm (k z , n 0 ) = a(r 0 , n 0 )L tomo exp -jk n 2 0 2r 0 sinc kL tomo 2r 0 (n 0 -n) (4.7)
The tomographic resolution depends on the width of this sinc-function and is given by

z = r 0 2L tomo (4.8)
where r 0 is the slant range distance of the master acquisition, located in the center of the tomographic constellation. L tomo is the tomographic illumination distance and equivalent to the largest baseline length in case of an MB-InSAR configuration. Thus, a better geometric tomographic resolution requires a larger tomographic aperture L tomo . In order to avoid height ambiguities, L tomo should be properly sampled and fulfil the Nyquist criterion for the vertical bandwidth. The vertical bandwidth B sd can be described by the variation of the vertical wavenumber (4.6) with respect to the observed volume, as follows:

B sd = k z (n 0 ) (4.9) with n 0 ∈ [H min , H max ], H V = H max -H min = kH V r 0 .
Therefore, the sampling distance d s of L tomo should verify the inequality

d s ≤ 2⇡ B sd = r 0 2H V (4.10)
which depends on the total height of the examined volume H V . Assuming the tracks are evenly distributed with the baseline distance d s , the tomographic resolution z is directly related to the number of acquisitions N with z ∝ 1 N and N = Ltomo ds + 1. This fact limits the geometric tomographic resolution since numerous acquisions require a high experimental effort. Another problem is that in practice tracks are unevenly distributed, implying ambiguities in height. These limitations may be fixed applying spectral analysis techniques that permit to obtain a high tomographic resolution with a moderate number of acquisitions and for a nonuniform baseline distribution.

Focusing techniques

The tomographic inversion is essentially a spectral analysis problem solved applying methods ranging from classical Fourier-based to high resolution approaches. In order to perform a refined analysis of volumetric structures of the observed scenes, these techniques are applied to MB-InSAR data sets in order to localize multiple scattering contributions within a range-azimuth resolution cell and their height z i . The joint processing of fully polarimetric data sets acquired in a Polarimetric MB-InSAR (MB-PolInSAR) configuration, using Polarimetric SAR Tomography (PolTom-SAR), permits to analyse scattering mechanisms within volumetric structures and extract some physical features [START_REF] Ferro-Famil | Principles and Applications of Polarimetric SAR Tomography for the Characterization of Complex Environments[END_REF]. After an introduction about the MB-InSAR signal model, this section introduces nonparametric and parametric spectral analysis techniques used in the present work in order to estimate the height and the reflectivity of the detected scatterers. Generally, nonparametric spectral analysis techniques are considered as robust, since they do not require any assumptions about the statistical characteristics of the received signals, but possess a limited resolution. In cases where an appropriate data model exists, model-based or parametric approaches yield generally superior estimation accuracy but require to determine the number of sources D located in each pixel under study.

MB-InSAR signal model

A SAR image resolution cell is the results of the combination of all the complex signal scattering contributions derived by real environments whose radial distances fall within the domain affected by the SAR impulse response, but with different elevation angles ✓. Therefore, in practice, the total source signal can be modelled as a sum of D scattering contributions within a range-azimuth resolution cell s(l) =

y(l) = D d=1 s d a(z d ) + n(l) = A(z)s + n(l) (4.12)
where l = 1, ..., L indicates one of the L independent realizations of the signal acquisition, also called looks. The source signal vector, s = [s 1 , ..., s D ] T , contains the unknown complex reflection coefficient of the D scatterers, and n ∈ C Mx1 represents the complex additive noise, assumed to be Gaussianly distributed with zero mean and variance 2 n , and to be white in time and space. The steering vector a(z) contains the interferometric phase information associated with a source located at the elevation position z above the reference focusing plane and is given by:

a(z) = [1, exp(jk z 2 z), ..., exp(jk z M z)] T (4.13)
where

k z j = 4⇡ B ⊥ j r 1 sin✓
is the two-way vertical wavenumber between the master and the j th acquisition tracks. The carrier wavelength is represented by , whereas ✓ stands for the incidence angle and r 1 is the slant range distance between the master track and the scatterer. The steering matrix A(z) consists of D steering vectors corresponding each to a backscattering source:

A(z) = [a(z 1 ), ..., a(z D )] (4.14)
with z = [z 1 , ..., z D ] T , the vector of unknown source heights. The data covariance matrix R can be represented as:

R = AR s A † + R n (4.15)
where R s = E(ss † ) is the source covariance matrix describing the correlations between the responses of the D scatterers and

R n = E(nn † ) = 2 n I (MxM)
is the white noise covariance matrix. Considering now interferometric decorrelation between different acquisitions, the initial model in equation (4.12) may be reformulated as a sum of contributions from random sources [START_REF] Gini | Multibaseline cross-track SAR interferometry: A signal processing perspective[END_REF]: This objective function is given by the continuous estimate of the reflectivity, i.e. P (z) = ˆ (z).

y(l) = D d=1 x d (l) ⊙ a(z d ) + n(l) =

Beamforming

The Beamforming method has been derived for estimating DOA as mentioned in [START_REF] Van Veen | Beamforming: A Versatile Approach to Spatial Filtering[END_REF][START_REF] Johnson | Array Signal Processing-Concepts and Techniques[END_REF]. In [START_REF] Homer | High resolution 3-D imaging via multi-pass SAR[END_REF], Beamforming was firstly suggested for solving the layover problem using MB-InSAR data sets. The Beamforming method for spatial frequency estimation can be derived as a Finite Impulse Response (FIR) filter [START_REF] Stoica | Spectral Analysis of Signals[END_REF], assuming a spatially white scenario, i.e. such a signal in the array output can be considered as impinging on the array with equal power from all direction ✓. Minimizing the output power of the undesired interference, the filter is designed so as to maintain undistorted the signal at a given frequency, while attenuating signals at all other frequencies even though from many of them there might be no incoming signal.

The Beamforming spectrum is associated to the PSD of the received MB-InSAR data y(l), and its objective function is given by the continuous L-look estimate of the reflectivity:

P B (z) = ˆ B (z) = 1 L L l=1 a † (z)y(l) 2 M 2 = a † (z) Ra(z) M 2 (4.20)
The beamforming DOA estimates are given by the locations of the n highest peaks of the function (4.20). Under the hypothesis of an uniform true reflectivity and for a sufficient number of looks L, the theoretical covariance R determined from the available finite sample {y(l)} L l=1 may be replaced by some estimate R such as:

R = 1 L L l=1 y(l)y † (l) (4.21)
By means of equations (4. [START_REF] Ferro-Famil | Principles and Applications of Polarimetric SAR Tomography for the Characterization of Complex Environments[END_REF]) and (4.20) it is possible to estimate, respectively, the set of scatterer elevations ẑ and the corresponding reflectivities. Nevertheless, as shown in [START_REF] Stoica | Spectral Analysis of Signals[END_REF], if this method is consistent in the single-source case (n = 1), in the general case of multiple sources it leads to biased estimates for the height and reflectivity. This is mainly due to the poor resolution of the Beamforming estimator that limits the selection of discrete sources from peaks of the reflectivity spectrum. For the n single-source case, Beamforming can be shown to provide an approximation to the nonlinear LS DOA estimation method discussed in section 4.3.3.1 [START_REF] Stoica | Spectral Analysis of Signals[END_REF].

In order to get better resolution and reduce the leakage problems, the Capon method has been proposed.

Capon

The Capon method has been derived under the same condition of minimum interfering power as the Beamforming method, but considering the general case of nonwhite interference scenario. Contrarly to the Beamforming, the Capon filter is designed in a 'data-dependent' way aiming to steer to a certain DOA (or spatial frequency) ✓ and to attenuate any other signals that actually impinge on the array from a DOA ≠ ✓. This method was applied for time series data in [START_REF] Capon | High-Resolution Frequency-Wavenumber Spectrum Analysis[END_REF][START_REF] Lacoss | Data adaptive spectral analysis methods[END_REF], and array signal processing [START_REF] Stoica | Spectral Analysis of Signals[END_REF] as well as for MB-InSAR data processing [START_REF] Gini | Multibaseline cross-track SAR interferometry: A signal processing perspective[END_REF].

In the MB-InSAR configuration, the Capon spectrum of the received data is represented by the continuous estimate of the reflectivity:

P C (z) = ˆ C (z) = 1 a † (z) R-1 a(z) (4.22)
The elevations of scatterers ẑ correspond to the locations of the D local maxima of ˆ C (z) using equation (4.19) and their reflectivities can be obtained directly from ˆ C (ẑ). It has been shown empirically in [START_REF] Stoica | Spectral Analysis of Signals[END_REF] that the Capon method exhibits better performance compared to the classical Beamforming with regard to spatial resolution and sidelobe suppression.

Whereas, the amplitude (reflectivity) estimate obtained by the Capon method is biased [START_REF] Stoica | Spectral Analysis of Signals[END_REF]. One can hence use the Least-Squares method introduced later in this chapter to estimate the reflectivity at ẑ, under the hypothesis that the scatterers are well separated by the Capon spectral estimator.

Parametric tomographic estimators

Parametric estimators exhibit an excellent capability to discriminate scatterers located at different elevations, assuming the number of scatterers D is a priori known. Hence, after determining the number of effective sources, i.e. model order, parametric methods can be applied to estimate the spatial frequencies of the detected signals with an improved resolution, compared to nonparametric methods. Here the noise is assumed to be spatially white with components having identical variance and to be uncorrelated respect to the signals. Hereafter the Least-Squares and the MUSIC high resolution parametric methods are presented.

Least-Squares method

The LS minimization criterion for amplitude estimation presented in [START_REF] Stoica | Spectral Analysis of Signals[END_REF] is based on a model of multiple complex sinusoids of known frequency embedded in additive noise. This indicates that the frequencies of signals are estimated and then put into the LS solution to obtain the corresponding amplitude estimates. Thus, the accuracy of the amplitude estimate by the LS solution relies on the accuracy of the estimated frequency. Using LS solution, one can obtain the amplitude estimate at each estimated height provided by high-resolution parametric methods. Once the height estimates are obtained by a parametric technique, for MB-InSAR cases the LS fitting can be formulated as

ŝ(l) = arg min s y(l) -A(ẑ)s 2 = (A † (ẑ)A(ẑ)) -1 A † (ẑ)y(l) (4.23)
where s(l) = [s 1 (l), ..., s D (l)] T are the unknown complex amplitudes of D scatterers at the l th observation (look) and the steering matrix A(ẑ) is calculated with ẑ derived from one tomographic estimator. From equation (4.23), one may compute a L-look estimate of the reflectivity of each source using

ˆ d = 1 L L l=1 ŝ d (l) 2 (4.24)
Using accurate height estimates given by parametric estimators, the LS method can provide a better amplitude estimate than nonparametric methods.

MUSIC method

The MUltiple SIgnal Classification (MUSIC) method [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] is a generalization of Pisarenko's harmonic retrieval method [START_REF] Pisarenko | The Retrieval of Harmonics from a Covariance Function[END_REF], and a parametric spatial frequency estimator for signals buried in white noise. MUSIC may be considered as a subspace-based technique as it uses a partition of the eigenstructure of the covariance matrix R modelled as in (4.15). Indeed the eigenstructure of R contains complete information on the frequencies related to ẑ [START_REF] Stoica | Spectral Analysis of Signals[END_REF]. MUSIC is a subspace-based single-dimension technique, whose objective function is a measure of the orthogonality between a steering vector and the noise subspace and is given by:

P M (z) = 1 a † (z) Ên Ê † n a(z) (4.25)
where Ên represents the matrix of covariance eigenvectors which span the noise subspace. Once ẑ, is determined by inserting equation (4.25) in the (4.19), the estimate of the complex reflectivity vector ŝ(l) can be obtained by solving a LS problem.

In [START_REF] Gini | Multibaseline cross-track SAR interferometry: A signal processing perspective[END_REF][START_REF] Stoica | Spectral Analysis of Signals[END_REF] a numerical analysis has been performed in order to compare the MUSIC methods with the Beamforming and Capon ones, demonstrating its effective better estimation results, even in presence of multiplicative noise. MUSIC performance quality depends mainly on the correlation between the scatterers. In fact, if they are fully correlated the source covariance matrix R s is rank deficient and the eigenvectors belonging to the signal subspace may leak into the noise subspace.

Proposed approach and innovative aspects

As mentioned in the previous section 4.1, the main goal of the present thesis is to perform a global three-dimensional characterization and change monitoring of dense and complex urban areas. In order to achieve the aim, it has been chosen to apply tomographic SAR techniques to high resolution satellite SAR data.

In particular, the strong potentialities of the classical non-and parametric spectral estimators combined to high resolution data have been evaluated, thus the Compressive Sensing method has not been taken into account. Moreover this technique, being a sparse signal analysis, is characterize by important disadvantages as:

• difficult regularization parameter solution for the Model Order Selection (MOS) techniques

• spurious atmospheric calibration (due to the gridding problem)

• necessity to process a high number of images

The proposed approach allows to correct the geometric distortions due to the layover and to determine the information about the number of scatterers and the corresponding reflectivity within one resolution cell. The analysed AOI represents the urban area of Paris, characterized by mixed features, as vegetation, infrastructures, buildings, skyscrapers, etc. In this context, in the same resolution cell one may find the superposition of the information related to a dominant scatterer and one or maximum two others minor scatterers. For that, the maximum number of the effective sources N 0 within one resolution cell has been set to three. In order to detect, for each resolution cell, the number of the effective sources, the iterative MOS technique summarized in Fig. 4.3 has been applied to the selected estimation methods. Given a "quasi" ML solution for well separated sources, it is based on the following steps: for each resolution cell and for three iterative steps, starting from the first elevation value z loc coming from the maxima local research made by means of the estimation method, a test over its intensity P (z loc ) has been evaluated as followed:

P (z loc ) P (z tot ) > I threshold (4.26)
where P (z tot ) is the total intensity value and I threshold is the customized threshold set for the analysed AOI. If the test was positive, z loc represented a real source and the process restarted in order to find the potential second or third source; if the test was negative, the process stopped.

Two different processing chains have been proposed depending on the selected estimation method. For non-parametric methods, i.e. Beamforming and Capon, the local maxima research has been performed following the proposed iterative MOS technique. For parametric methods, i.e. MUSIC, at first it has been determined a priori the number of the sources N S as a little bit bigger than N 0 within one resolution cell; successively, it has been applied the proposed iterative MOS technique. An important advantage of this solution is that the artefacts are not considered, due to their low intensity values.

Moreover, an innovative idea has been proposed for the 3-D change monitoring and detection of single structures. For this aim it has been necessary to perform a 3-D tomographic focalization using of Maximum-Likelihood Ratio Test (MLRT) detectors. This part will be better explained in the section 6.3. Chapter 5

2-D analysis and results

The principal aim of the present work is to provide a global 2-D and 3-D characterization for the analysis of dense urban areas. This chapter is focused on the analysis of the quality of the processed data set, as well as for interferometric purpose and scatterers time stability, and on the extraction of tomographic 2-D second products, i.e. tomograms. Hence, an Area Of Interest (AOI) has been selected and two azimuth-profiles with different characteristics have been chosen in order to demonstrate the high potentialities for tomographic purpose of the classical estimation techniques described in section 4.3.2 and section 4.3.3. According to the well adaption of the high-resolution imagery to the remote sensing urban applications, a MB-InSAR data set acquired by the satellite TerraSAR-X in High Resolution Spotlight (HS) and SP mode over the urban area of Paris (France) has been processed. As introduced in the previous sections, SAR imagery possesses particularly interesting features, such as foliage penetration, sensibility to scatterer shape and dielectric properties. However, SAR information over urban environments is particularly complex due to as well as the geometric distortions caused by the layover and shadowing phenomena, the complex scattering patterns within the same azimuth/range resolution cell (single/double bounce scattering, volume diffusion), the random as-67 pect due to speckle effects (see section 2.2). Here a demonstration of the potentialities of TomoSAR techniques in distortion correction and in information retrieval about the number of scatterers and the corresponding reflectivity within one resolution cell is proposed in 2-D domain using basic mono-dimensional estimators as Beamforming, Capon and MUSIC. The 3-D analysis will be discussed in chapter 6.

In this context it is important to underline that the present work is mainly focused on 3-D reconstruction by very fast estimation techniques and does not take into account parsimonious signal estimators. have not been taken into account. This chapter is structured as follow: section 5.1 reports the description and the main characteristics of the data set selected for this study; in section 5.2 the interferometric quality and time stability analysis are described; section 5.3 shows the generated 2-D tomograms.

Presentation of the data set

The analysed stack consists of 21 SP TerraSAR-X1 images acquired in HS mode over the urban area of Paris (see Fig. From the bottom of Fig. 5.3 one may appreciate the geometric distortion effect: taller buildings within the subset are 'lying down' onto the ground along the range direction, showing both their roofs, façades and bases interfering with scatterers situated below them. This phenomenon may be better figured out looking Fig. 5.4 where a simplified map of the selected subset taken from [START_REF]F4 3D Map Viewer[END_REF] is reported. Here the Tour Keller's shadow covering part of the Tour Espace 2000, Tour Perspective 2 and the ground could be seen as the layover induced by the main skyscraper over the other twos. Hence, structure parts and ground covered by the shadow may be imaged in the SAR resolution cell together to the Tour Keller ones because of the layover. In the following, we will refer to the

Interferometric quality and temporal stability analysis

A first step for the tomographic processing is the analysis of the interferometric quality of the data to be used. First of all the attention goes to the accuracy of the co-registration phase. The images, being acquired from different (even though slightly) point of view at different distances, need to be rotated, shifted and scaled relative to the master in order to align the samples for phase differencing. A sub-pixel registration of all the focused SAR images is strictly required for interferometric processing whose quality is highly dependent on co-registration accuracy. Consequently, a good coherence between images as well as a good noise phase reduction, mainly due to the atmospheric and flat earth components (see section 3.2), highly impact the quality of the secondary products as the tomograms and the DSMs. Here, starting from a co-registered and flattened data set, a calibration of the atmospheric phase component has been applied to the signals, treating the problem as a low-pass filtering one. The analysis has been performed considering all the combinations samples (with N the number of processed images). pattern and phase of the features whose height is higher than h amb , as ones affected by the layover, are wrapped every 2⇡ phase cycle.

Above results highlight the high coherence level between the majority of the images within the stack, confirming the effective capabilities of TSX high resolution data. On the basis of these preliminary analysis the two (temporally) last images have been discarded and the experimental results have been achieved processing the leftover 19 images.

In order to deeply describe the urban AOI, a time stability analysis of the in [START_REF] Ferro-Famil | Urban area remote sensing from Lband PolSAR data using time-frequency techniques[END_REF][START_REF] Hu | Ship discrimination using polarimetric SAR data and coherent time-frequency analysis[END_REF] for Polarimetric SAR (PolSAR) data based on Maximum Likelihood statistic test. There a PolSAR multidimensional information is analysed using a linear time-frequency decomposition which permits to detect man-made structures and ships and to describe their polarimetric behaviour for different azimuth angles of observation and frequencies of illumination.

The 2-D temporal coherent stability between the {s j } 19 j=1,j≠i signals has been computed accordingly to the indicator ⇢ T given by

⇢ T = 1 -R 1N (5.1)
where R = PRP, with

[P ] ii = diag( R -1 ii )
. Its values range from 1 (maximum correlated case) and 0 (decorrelated case). In Fig. 5.8a its trend shows (and confirms) that natural environments such as the river in the upper right part of the images have low ⇢ T values. On the contrary, coherent reflectors like man-made targets and buildings show the highest values of ⇢ T especially over the squared lower-left roof. The relative image over all the test site is represented in Fig. 5.9: not moving features like buildings and bridges show high ⇤ values, whereas boats and cars over main streets are affected by a very low stationary behaviour. These results show the capability of these indicators to provide complementary information, adapted to the case of man made environments.

Tomograms generation

SAR tomography can be derived from the data covariance matrix R in the frame of a classical problem of spectral estimation using mono-and multi-dimensional methods. These approaches determine ẑ, an estimate of the elevation of the scatterers under observation, as the coordinates of the ns largest local maxima of a continuous objective function P (z), ẑ = argmaxP (z). In this section SAR tomographic results have been achieved implementing and applying different mono-dimensional estimators over the data set already described in section 5.1. 2-D tomograms have been generated over several azimuth profile in order to estimate building heights and to distinguish different scatterers within the same resolution cell. In particular the following mono-dimensional methods have been used: Beamforming and Capon, non-parametric approaches generally used to globally appreciate the structure of a volumetric medium; MUSIC, parametric approach generally characterized by better resolution and performance for the analysis of discrete sources and affected by data modelling errors. Regarding the estimation of the MUSIC pseudo-spectrum, the signal subset dimensions N S has been set equal to 4, accordingly to the hypothesis on the basis of the proposed approach (see section 4.4).

Tomograms illustrated in Fig. 5.10b and 5.10c are referred to the relative image profile highlighted in Fig. 5.10a by the red line and they reflect the high quality resolution of the processed data. Indeed, it is evident how also with basic and classical spectral estimation models it is pos-sible to reconstruct building shapes and profiles. Focusing attention on slant range tomograms shown in Fig. 5.10b, moving from the top to the bottom one may note the effective vertical resolution improvement between the different approaches: sidelobes, probably due to atmospheric phase noise residuals, are particularly strong for Beamforming estimations, strongly reduced by Capon and completely resolved by MUSIC approach. From ground range normalised results depicted in Fig. 5.10c, one may appreciate the effective reconstruction of buildings shapes and the strong correction of the distortions due by the layover. Main vertical profiles may be clearly perceived thanks to the strong double bounce scattering responses related to the wall-ground interaction, as well as on the whole top of the relevant last building located at about 20 m over the referencing zero ground height. Tomograms illustrated in Fig. 5.11 are referred to the image profile highlighted in Fig. 5.11a by the red line and strongly characterized by the geometric distortions related to the layover induced by Tour Keller. Thanks to the high quality resolution of the processed data, it is possible to distinguish, for some resolution cells, more than one intensity local maxima. These peaks are related to different scatterers due mainly to the skyscraper features and the ground as depicted in the relative scheme on the right of the tomograms. In particular from Fig. 5.11b it is possible to detect two scatterers, corresponding for example to the ground and the building façade. In Fig. 5.11b one may appreciate the presence of up to three scatterers within the same resolution cell due, for example, to the addition of the building roof. This represents a valuable capability of tomographic techniques that allows to obtain complete point clouds and, hence, complete DSMs. This aspect will be better analysed in chapter 6.

Especially for MUSIC results (bottom tomograms in Fig. 5.11b and 5.11c), one may appreciate how thanks to the proposed approach, the artefacts are not taking into account as real sources, thanks to their low intensity values. Furthermore, the presence of artefacts has been

2-D analysis and results

strongly reduced filtering the signal by means of two masks obtained by a threshold computed on the parameter ⇢ and ⇤ described in the previous section 5. [START_REF]F4 3D Map Viewer[END_REF]. From the tomograms obtained by the MUSIC approach one may note how the hypothesis to set the number of the sources N S equal to 4, i.e. as a little bit bigger than N 0 = 3, allows to achieve very good results in terms of vertical resolution and accuracy and source detection.

Results reported in the present chapter have demonstrated the possibility to perform dense urban area 2-D analysis using tomographic techniques and high resolution satellite SAR data. Exploiting monodimensional spectral estimation models, the buildings heights estimation and the scatterers number extraction have been carried out. In this way it has been possible to resolve pixels affected by the layover distinguishing up to three scatterers within the same resolution cell. On the basis of these conclusions a 3-D characterization has been performed over the same subset, whose results are presented in the following chapter.

3-D analysis and results

In the present chapter a global 3-D characterization of dense and complex urban areas exploiting SAR tomographic techniques is presented. In this sense, several reconstructions in terms of building height, vertical reflectivity and time stability have been carried out processing an adequate set of multitemporal high resolution TSX images in order to deeply analyse build-up areas. Starting from experimental results obtained in the previous chapter, here the study has been widened to the urban azimuth/range tiles depicted in Fig. 6.1. In particular, Fig. 6.1a includes the azimuth-profile analysed in Fig. 5.10a and contains a mixed scenario characterized by residential buildings, streets, vegetation; 6.1b shows the tile including the Tour Keller (previously analysed in 5.11a) and it is evident its strong layover along the range direction; 6.1c represents the Tour Mirabeau, a skyscraper composed of three branches forming a triangular island. In the following sections different investigations are presented in term of built-up area and single building 3-D reconstruction and characterization. Moreover, the 3-D views have been reconstructed in ground range processing the results obtained considering up to three scatterers within one resolution cell and Beamforming,

Capon and MUSIC approaches. The investigation have been carried out for three different intensity levels, in each of which the maximum abso-lute reflectivity value has been identified from the spectrum evaluated for each resolution cell within the tile. The obtained products have led to focus the attention over the ones extracted by means of the MUSIC method. Furthermore, results shown hereafter have been preprocessed

applying several filters in terms of temporal coherent and stable scatterer analysis, in order to not consider strongly unstable pixels, e.g. cars over the main streets, that might add noisy information. Hence, tiles analysed in the following have been chosen in order to demonstrate different capabilities of tomographic techniques. In particular, single building height reconstruction and layover correction are demonstrated in section 6.1 for different man-made features, showing the extracted point clouds from different point of views; in section 6.2 the estimation of vertical building reflectivity is presented; section 6.3 shows the results coming from an innovative scatterers time stability analysis.

Height map generation

As mentioned in chapter 3, TomoSAR is a valuable technique for the extraction of the height information of a selected area/feature. In order to operate in the 3-D field, the 2-D spectral analysis described in section 5.3

for an azimuth profile is widened to the azimuth/range resolution cells within the selected tile. This process leads to the generation of sparse point clouds that, from the computational point of view, may be considered like an ASCII file in which the height values are listed in addition to the horizontal coordinates of the m detected points (sometimes up to hundreds of thousands). In this context it is important to introduce the concept of completeness and accuracy. The completeness indicates a homogeneous data distribution on the entire studied area, to minimize the zones with poor or any information. The accuracy is the standard deviation of the elevation data differences between the model surface and a referring surface.

In this section several point clouds are reported for tiles (see Fig. 6.1) characterized both by mixed features, such as man-made structures and vegetation, and by layover distortions. The 3-D views have been reconstructed in ground range coordinate system starting from results obtained considering up to three scatterers within one resolution cell and Beamforming, Capon and MUSIC approaches.

Starting from Fig. The main interesting capability of TomoSAR is the layover distortion correction, that is detecting and distinguishing different scatterers with different elevation imaged together in the same range/azimuth resolution cell of the SAR image. In order to demonstrate it, the Tour Keller layover imaging over the Tour Espace 2000 and the Tour Perspective 2 has been taken into account (the effect is well visible in Fig. 6.1b). Fig. 6.4a

reports the front and top (simplified) map views that if compared with, respectively, Fig. 6.4b and Fig. 6.4c gives an additional confirmation of the potential of this technique combined with high resolution satellite SAR data. Focusing on the biggest skyscraper, one may perceive its shape and the location of its two façades. Moreover, it is clear that resolution cells affected by layover have been georeferenced and the information related to features below them has been partially reconstructed in the others two main buildings and the ground. It is also interesting make a comparison between Fig. 6.4d and Fig. 6.4e where relative detected scatterers are highlighted respect to the point cloud populated by all the estimated sources (imaged in gray). One may note how the majority of the second source (that consists in about the 13.4% of the total detected points) are related to pixels affected by the layover, that is scatterers related to a relative peak in the reflectivity spectrum.

Eventually, in order to perform the investigation of a single building reconstruction the Tour Mirabeau has been selected, thanks to its particular shape (as one may note in Fig. 6.6a) and strong layover distortions. In this case the layover induced by the structure does not involve other important features but the ground. Hence several views of merely the point cloud extracted considering all the detected sources are represented. Fig. 6.5 shows all the scatterers detected using the three considered spectral estimation techniques. Even here Capon products reveal a noisy behaviour: the building shape is not clear and there are several sparse points within the scenario. On the contrary, the point cloud extracted by means of the Beamforming and showed in Fig. 6.5a reveals clearly the particular shape and intersection of the two façades and their elevation. MUSIC results depicted in Fig. 6.5c seem to lose information about the wall, especially at the top, adding however information about the roof of the structure (better highlighted in Fig. 6.6). Moreover, from the table 6.1 it is clear that this method allows to detected more points lying under the layover than the other two. For these reasons we focused on the MUSIC products. Comparing the top view in Fig. 6.6b with the right one in Fig. 6.6a one may note the good reconstruction of the particular shape of the two façades facing the radar signal. At the same time the frontal view depicted in Fig. 6.6c show the point detection at the different height, reconstructing partially the roof of the building. Moreover pixels imaged in layover are quite geometrically corrected and georeferenced in relation to the others demonstrating the extraction of the form of the global building. Nevertheless, ground involved by the layover has been reconstructed for a very small part, showing anyway the limits of this technique.

Globally, the proposed 3-D views show good results in terms of reconstruction of the shape and the location of man-made target and in elevation retrieval. The point clouds appear slightly noisy, i.e. some scatterers appear to have wrong elevation and/or location. It is probably due to residual errors related to the coregistration and calibration signal steps and to residual unstable scatterers coming form the pre-processing filtering applied to the point clouds. In particular, the Capon method has led to point clouds noisier and sparser respect to the others; products obtained by means of Beamforming technique ; the MUSIC approach was found to be the better in terms of layover correction and point cloud completeness. Indeed, with respect to the Capon and Beamfoming, a higher number of points related both to the total detected sources and to the second and third ones (see the statics in table 6.

Vertical reflectivity estimation

Another important characterization of the built-up areas consists in the estimation of the vertical reflectivity information. For ground imaging, the reflectivity may be modelled as the superposition of the echoes from a large number of uncorrelated scatterers randomly dispersed through the resolution cell. Hence it is expected to be characterized by low reflectivity values. For building imaging, the reflectivity is supposed to be higher at the wall-ground interaction where typically occurs the double bounce reflection. It is important to restate that the reflectivity profile is used to estimate scattering parameters, such as the number of scatterers in a pixel, their elevations, and their reflectivity strength.

Here it has been decided to focus on the analysis of the Tour Keller and Tour Mirabeau areas, due to their high elevation spread. e. discarding always one more image, is showed in Fig. 6.10b from where one may note how the majority of the unstable pixels are detected at the first step. Fig. 6.10c is a 3-D view of the image indices that more influence scatterers time stability at the first step (i.e. discarding one image) and as one may observe from Fig. 6.10d the resulting one is image with ID 6. The trend of the estimated CV m is showed in Fig. 6.11a and the 3-D reconstruction in Fig. 6.11b displays the stable and unstable scatterers in term of the estimated CV m: unstable (orange) pixels, i.e. points with CV m values higher than the estimated threshold, lie mostly on the main skyscraper façades. Focusing on the unstable images, in Fig. 6.10 is represented an example of the time analysis aiming to extract the number and the indices of images that more perturb the 3-D temporally unstable scatterers.

Also the Tour Mirabeau has been taken into account in this analysis. As

1 The Coefficient of Variation is expressed as the ratio of the standard deviation to the mean µ. It is a statistical measure of the dispersion of data points in a data series around the mean, that is the higher the CV the greater the dispersion in the variable. It is an useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other.

shown in Fig. 6.12 the analysis performed in terms of the CVm reveals that the greater number of the pixel is stable in the time domain. For that the IDs of the unstable images have not been computed. Conclusions and outlook

Conclusions

The main goal of this work was to develop a global and three-dimensional characterization of complex and dense urban areas using tomographic techniques and high resolution satellite SAR imagery. In particular it was taken into account the classical non-parametric and parametric spectral estimation techniques such as the Beamforming, Capon and MUSIC. 2-D and 3-D analysis have been presented over the urban area of Paris acquired by TerraSAR-X sensor in High Resolution SpotLight and Single Polarization mode. The present work is mainly focused on the 3-D reconstruction by very fast estimation techniques and does not take into account parsimonious signal estimators. In chapter 5 it was underlined the good quality of high resolution SAR data, such as ones acquired by TerraSAR-X, both from the interferometric and time stability and coherence points of view. The analysis of the results and the coherence maps generated over the entire stack allowed to choose the adequate set of multitemporal data to be processed, discarding images considered as outlier in terms of temporal and spatial baselines.

Further, the extraction of 2-D tomograms over different azimuth-profile has showed the capabilities to distinguish more than one scatterer within 101 the same resolution cell and to reconstruct the vertical building profiles.

The analysis have regarded the expectation, that is the parametric MU-SIC method, respect to the other non-parametric twos, revealed a better sidelobes reduction and accuracy of the estimated products.

Starting from these results the 3-D analysis has been performed in chapter 6 considering different tiles characterized both by mixed features (residential buildings, streets and vegetation) and single skyscrapers inducing a strong layover. The generated point clouds reveal the good reconstruction of building shape and location and a good estimation of their elevation. Indeed, façades intersection, structure features and main streets are well visible from the 3-D reconstruction. Moreover, being able to detect up to three scatterers within the same resolution cell, it has been possible not only to generate more complete point clouds, but also to solve the distortions due to the layover. This has been highlighted for the tile named Tour Keller area, where the second detected sources are mostly related to information lying down the strong layover induced by the tallest skyscraper. The MUSIC approach allowed to detect the highest number of points, especially in the layover cases, and the Capon's results appear to be sparser and noisier respect also to the Beamforming ones.

The analysis performed in terms of vertical reflectivity estimation has led to the identification of strong intensity of the signals at the interaction wall-ground, revealing the double bounce strong reflection occurring in that area. At the same time, the ground is characterized by low reflectivity values. The most innovative aspect of the present work is represented by the 3-D analysis of the time stability of the detected scatterers. Considering step by step all the images but the one corresponding to the relative step, a set of 3-D intensity have been generated and the stable and unstable scatterers have been distinguished and the IDs of the most perturbing contributions have been found. The study has been developed consider-ing a modified Coefficient of Variation, respect which the more unstable pixels appeared to belong to the façades of the Tour Keller. Eventually, it is possible to assert that processing high resolution SAR data allows to achieve a strong improvement in 3-D imaging capabilities.

Further investigations and improvements

Even though the inferred conclusions lead to good and encouraging results, there are still interesting issues which may enhance and deepen the developed processing chain. Future investigations should be directed towards improving tomographic technique robustness, refining 3-D change detection and monitoring, studying the integration with other techniques. Summarizing, as main topics one can list:

• Considering thermal building variations. In order to enhance the second products quality, it may be interesting to take into account the analysis of the building expansion and contraction due to the thermal loads from exposure to changes in the ambient temperature. Building shape and elevation thermal variations are related to the local weather patterns along with the heating: warmer materials expand and colder materials contract. This kind of changes, strongly depend on the different temperatures to which the structure is subjected, may occur slowly and may impact the time stability of the detected sources • Improving tomographic techniques. Spectral estimators robustness may be improved applying more sophisticated estimation techniques and a comparison with the Compressive Sensing approach may be performed in order to evaluate the main pros and cons of the two approaches • Processing data sets acquired at different spatial resolutions. Certainly, the processing chain might be applied to other stack(s) representing a complex and dense urban scenarios and acquired both at high and low resolution. An interesting and challenging investigation might be handling Sentinel-1 data, characterized by 5x20m spatial resolution (in Interferometric Wide Swath) and provided by ESA for free and on-line thought the scientific data hub tool. This may lead to study the limits of tomographic techniques respect to the spatial resolution of the precessed data • Integration with the Polarimetric SAR. Polarimetric SAR Tomography (PolTomSAR) is a well-known technique based on the combination of tomograms computed over different polarimetric channels.

In this way two different scattering mechanisms could be identified: the single-bounce reflection over rough surface (e.g. ground or building roofs) and the double-bounce reflections characteristic of ground-tree trunk or dihedral-like objects. These different scattering mechanisms, discerned in height direction by tomographic technique, will be used to characterize in a refined way volumetric environments over wide areas and within a very short amount of time. In literature the application of PolTomSAR technique to urban scenarios by means of SAR satellite data represents an innovative topic and such a technique may be used to retrieve a land cover classification, largely useful tool in different engineering civil application fields. As last goal, all final products could become a valuable tools to supply topographic information in complex environments, in particular urban areas. In this sense a TerraSAR-X proposal project has been successfully submitted and approved and a huge amount of dual polarimetric data stacks, acquired over different AOIs (as Spain, Germany and Japan), is already available to perform the tomographic and polarimetric processing chain

• DSMs validation. A DSMs assessment might be performed comparing the extracted DSMs with a more accurate ground truth one, e.g.

obtained by means of the LiDAR technology, in order to evaluate their elevation accuracy

Finalement, il est

  possible d'affirmer que le traitement des données SAR haute résolution permet d'obtenir une forte amélioration en 3-D des capacités d'imagerie. Les reconstructions 3-D présentés dans cette thèse ont permis d'effectuer une caractérisation globale des zones urbaines à travers le traitement d'un ensemble adéquat d'images multitemporelles haute résolution SAR, démontrant ainsi les potentialités de la technique TomoSAR dans la correction des distorsions, dans la détermination des informations sur le nombre de diffuseurs et de la réflectivité correspondant à l'intérieur d'une cellule de résolution et leur évolution temporelle en utilisant des estimateurs mono-dimensionnels de base comme Beamforming, Capon et MUSIC.

  The analysis performed in terms of vertical reflectivity estimation allowed to identify different features behaviour: high values at the interaction wall-ground revealed the double bounce strong reflection occurring in that area; the ground is characterized by low reflectivity values. The 3-D time stability analysis demonstrated the possibility to monitor the 3-D change depending on the time. Eventually, it is possible to assert that processing high resolution SAR data allows to achieve a strong improvement in 3-D imaging capabilities. The 3-D reconstructions presented in the present thesis enabled to perform a global characterization of build-up areas, processing an adequate set of multitemporal high resolution SAR images. It has been demonstrated the potentialities of TomoSAR technique in distortions correction, in determining information about the number of scatterers and the corresponding reflectivity within one resolution cell and in 3-D change monitoring using basic mono-dimensional estimators as Beamforming, Capon and MUSIC.

Introduction• CHAPTER 7 :

 7 corresponding reflectivity within one resolution cell have been evaluated. Several 3-D reconstructions of single structures have been performed both in terms of buildings heights and vertical reflectivity. Moreover an innovative time stability analysis of the observed scene have been carried out in order to detect the stable and unstable scatterers. The processing have been performed using an adequate high resolution Multibaseline SAR Interferometric (MB-InSAR) data acquired over Paris urban area by the satellite TerraSAR-X. The final goal is to demonstrate the high potentialities of tomographic techniques in 3-D change monitoring and characterization for built-up areas using basic mono-dimensional estimators as Beamforming, Capon and MUSIC combined to very high satellite resolution data. The present work is mainly focused on the 3-D reconstruction by very fast estimation techniques and does not take into account parsimonious signal estimators. Starting from these purposes, the thesis is structured in the following chapters: • CHAPTER 2: introduction and description of SAR imaging principles (acquisition mode, geometric distortion, speckle noise) and satellite platforms, with particular attention for TerraSAR-X • CHAPTER 3: overview of the three SAR 3-D imaging techniques: radargrammetry, interferometry and tomography • CHAPTER 4: examination of the tomographic focusing technique, including the basic principles and the state of the art for the urban analysis • CHAPTER 5: description of the data set, analysis of the data interferometric quality and of the 2-D experimental results obtained using the Beamforming, Capon and MUSIC spectral estimation techniques • CHAPTER 6: analysis of the 3-D experimental results in terms of building height retrieval, vertical reflectivity estimation and time stability analysis conclusions and perspectives Chapter 2 SAR: Synthetic Aperture Radar

Fig. 2 . 1 -

 21 Fig. 2.1 -Electromagnetic spectrum

Fig. 2 . 2 -

 22 Fig. 2.2 -Signal trasmitted from the satellite (left image) and signal backscattered from the surface (right image)

= c⌧ 2 ( 2 cFig. 2 . 3 -

 2223 Fig. 2.3 -RAR system from a satellite

Fig. 2 . 4 -

 24 Fig. 2.4 -Image resolution

Fig. 2 . 6 -

 26 Fig. 2.6 -Different backscatters from (a) an urban area, (b) a vegetated area, (c) a smooth areas

  2.7).

Fig. 2 . 7 -

 27 Fig. 2.7 -Radar geometric distortions: foreshortening, layover and shadow

Fig. 2 . 8 -

 28 Fig. 2.8 -Foreshortening [5]

  Fig. 2.10 -(a) Layover and shadow phenomena in built-up areas; (b) TerraSAR-X image captured in Spotlight mode (image taken from [7])

2

 2

Fig. 2 .

 2 Fig.2.11 -Shadow[START_REF]Radar Imaging[END_REF]

  Fig. 2.12 -Speckle effects

  The different kinds of imagery which can be acquired by TERRASAR-X are the following one[START_REF] Defence | TerraSAR-X Image Product Guide. Basic and Enhanced Radar Satellite Imagery[END_REF]:• Staring SpotLight (ST): 0.6x0.24 m rg/az resolution, 4x3.7 km rg/az scene size • HighRes SpotLight 300 MHz (HS300): 0.6x1.1 m rg/az resolution, 10x5 km rg/az scene size • HighRes SpotLight (HS): up to 1.2x1.1 m rg/az resolution, 10x5 km rg/az scene size • SpotLight (SL): up to 1.2x1.7 m rg/az resolution, 10x10 km rg/az scene size • StripMap (SM): up to 1.2x3.3 m rg/az resolution, 30x50 km rg/az scene size • ScanSAR (SC): 1.2x18.5 m rg/az resolution, 100x150 km rg/az scene size • Wide ScanSAR (WS): up to 1.7x40 m rg/az resolution, 270x200 km rg/az scene size

Fig. 2 .

 2 Fig. 2.13 -TerraSAR-X satellite

Fig. 3 . 1 -

 31 Fig. 3.1 -StereoSAR acquisition system in zero Doppler geometry

Fig. 3 . 2 -

 32 Fig. 3.2 -Geometrical constrain and voxel generation

3 -

 3 D SAR imaging metric phase) is the phase difference between the images inf = arg{s 1 s * 2 } = -k r + 2⇡n with n = 0, ±1, ±2, ... (3.5) which is 2⇡ ambiguous, i.e. inf ∈ [-⇡, ⇡]. The phase differences of radar wavelengths recorded by SAR images are able to provide precise measurement, to the sub-wavelength level, of the range or distance between the location of the observing antennas and the points of reference (pixel centres). With two observing antennas, or the same antennas at different satellites overpass times, differential phases can provide information on

  Fig. 3.4 -Interferometric phase variations with range r and height z, related respectively to the flat earth effect and the topography

Fig. 3 .

 3 Fig. 3.5 -Interferometric phase, topographic phase and interferometric coherence (Berlin urban area)

Fig. 3 . 6 -

 36 Fig.3.6 -SAR tomographic representation[START_REF] Reigber | First demonstration of airborne SAR tomography using multibaseline L-band data[END_REF]

Fig. 4 . 1 -

 41 Fig. 4.1 -Tomographic SAR satellite imaging geometry

11 )

 11 On the basis of the nature of the observed scattering, the backscattered signal can be modelled in two different ways: as deterministic and as distributed. The former regards signals affected by speckle noise and the huge number of scatters contained within one resolution cell cannot be separated; the latter is related to a dominant and stable scatterer lying within the selected resolution cell. Assuming that D scatterers are located in the same resolution cell and partially correlated, the received signal may be considered as a mixture of distributed and deterministic scattering responses, called hybrid scatterers. In Fig.4.2 the MB-InSAR acquisition geometry is represented. It consists of M SAR acquisitions, each pair of which are separated by a base-line. The baseline distance between the master position and the j th

Fig. 4 . 2 -

 42 Fig. 4.2 -MBInSAR geometrical configuration

yd 2

 2 d (l) + n(l) (4.[START_REF] Ferretti | InSAR Principles: Guidelines for SAR Interferometry Processing and Interpretation[END_REF] where x d ∈ C Mx1 is unknown and accounts for both the reflection coefficient of the d th source, s d , and its potential variations between the interferometric acquisitions over the L realizations. Depending on the type of scatterer under observation, the source signal x d possesses varying statistical properties. Considering the set of D backscattering contributions, the source covari-ance matrix R s is expressed as: and ⇢ dd ′ the correlation coefficient between two sources. The covariance matrix described in equation (4.18) is generally considered as constant over the different acquisitions and can be computed asR s (d, d ′ ) = E{y d k y * d ′ k }, ∀k, where E() is the expectation operator.The correlation of the d th source signal over the M acquisitions may be represented under the form of a normalized covariance matrix R x d containing interferometric coherences, determined by the observed source and the geometrical configuration of the acquisitions: Nonparametric tomographic estimators Generally, nonparametric spectral analysis techniques are considered as robust, since they do not require any assumptions about the statistical properties of the received signals. Accordingly, they can be used in situations with lack of information about these properties. These methods aim to estimate the Directions Of Arrival (DOA), that is, given a spatially propagating signal, to determine its angle of arrival at the array. They respond to the following assumptions: (i) the signals with a given DOA ✓ are passed undistorted and (ii) all the other DOAs different from ✓ are attenuated as much as possible. These estimators rely entirely on the characterization of the Power Spectral Density (PSD) defined by the periodogram or the correlogram, to provide the spectral estimates [59]. Their principal drawback lies in poor spectral resolution due to the high variance of the periodogram and correlogram estimates. In this section classical estimators like the Beamforming and the Capon method are presented for a MB-InSAR configuration. For tomographic purpose, the goal is to determine an estimation ẑ of the elevation of the scatterers under observation as the coordinates of the local maxima of an objective function with a scalar elevation argument P (z): ẑ = arg max z,loc P (z) (4.19)

Fig. 4 . 3 -

 43 Fig. 4.3 -Diagram of the proposed approach applied to each resolution cell and for three iterative steps (three as the maximum number of the effective sources N 0 )

  5.1) with a mean incidence angle of about 34.7 ○ with a slant-range resolution of 1.2 m and an azimuth resolution of 1.1 m. In Fig. 5.2a and 5.2b the spatial baseline and the spatial-temporal baseline distributions are respectively shown; the maximum elevation aperture size is about 583.5 m and the temporal baseline ranges from 24th January 2009 and 26th November 2010. It is possible to do the first considerations about the stack quality. From Fig. 5.2a it is clear that the image with ID 20 represents an outlier respect to the relative spatial baselines between the other images, whose value is about 334.4 m.

Fig. 5. 1 -

 1 Fig. 5.1 -Multitemporal averaged amplitude image over Paris urban area derived from TerraSAR-X data stack and the selected AOI

( a )

 a Fig. 5.2 -Distributions of the 21 acquisitions in the spatial and temporal baseline domains

Fig. 5. 4 -

 4 Fig. 5.4 -Semplified map taken from [2] of the selected subset of the AOI

Fig. 5

 5 Fig. 5.5 -Spatial-temporal trend of coherence mean values (left) evaluated over different image subsets (right) considering the first 19 (good) images within the stack

  Fig. 5.7 -Coherence image and interferogram computed between images with IDs 1 and 2

Fig. 5. 8 -

 8 Fig. 5.8 -Coherence indicator trend over the selected subset ranges from 1 (high coherence) and 0.3 (low coherence) and relative amplitude image (right)

Fig. 5. 9 -

 9 Fig. 5.9 -Image of the parameter ⇤ on Paris test site, whose values go from 0 (very low stationarity) and 1 (very high stationarity)

( a )

 a Fig. 5.10 -Tomograms generated over one profile applying Beamforming, Capon and MUSIC spectral estimators

( a )

 a Amplitude and Google Earth image of the analysed profile highlighted in red over the AOI (b) Slant range normalised reflectivity tomograms: 2 detected scatterers (c) Slant range normalised reflectivity tomograms: 3 detected scatterers

Fig. 5. 11 -

 11 Fig. 5.11 -Tomograms generated over one layover-affected profile applying Beamforming, Capon and MUSIC spectral estimators

  (a) Tile over a mixed urban area (b) Tile including Tour Keller (c) Tile including Tour Mirabeau

Fig. 6 . 1 -

 61 Fig. 6.1 -Tiles considered for 3-D reconstructions and investigations

products, in Fig. 6 .

 6 3 results obtained considering separately the dominant, the second and all the detected sources are reported. The analysed area, underlined by the red lines in Fig.6.3a, is characterized by several residential buildings and a big almost squared structure defined by different features. Fig.6.3b, 6.3c and 6.3d show the 3-D height maps generated considering, respectively, all the estimated scatterers, only the dominant (first) sources and the second ones. Comparing the 3-D views with the simplified 3-D map in the top of the figure, one may perceive the reconstruction of the main building shapes and location and the streets orientation. Especially for the big squared structure, considering all the detected sources one may recognize the principal features, such as the external wall, the internal edifice and the external patio. It is also evident that the main information is given by the dominant scatterers, even though second sources give a contribution mainly for scatterers at lower height. It is important to underline that elevation values represented in the height map and pointed out in figure legends appear to be coherent with the one estimated from Google Earth.

  Fig. 6.2 -3-D height maps represented in ground range over a mixed urban area using Beamforming, Capon and MUSIC approaches and considering all the detected sources

(a) 3 -

 3 D views of the Tour Keller's area (b) All detected scatterers: front view (c) All detected scatterers: top view (d) Dominant detected scatterers (e) Second detected scatterers

Fig. 6 .

 6 Fig. 6.4 -3-D height maps of Tour Keller area represented in ground range and extracted considering several detected scatterers using MUSIC method: layover distortions have been mostly corrected

1 )

 1 have been detected, showing the higher capabilities in scatterers detection at different reflectivity level within the same resolution cell. Further, this is congruent with the 2-D investigations performed in section 5.3, where it has been shown the MUSIC superior capability in sidelobes reduction and accuracy estimation.

( a )

 a Fig. 6.5 -3-D height maps of Tour Mirabeau represented in ground range and considering results obtained using Beamforming, Capon and MUSIC approaches

( a )

 a All detected scatterers (b) All scatterers: filtering lowest values (c) Dominant detected scatterers (d) Second detected scatterers

Fig. 6 .

 6 Fig. 6.7 -3-D reflectivity maps of the Tour Keller area represented in ground range and extracted considering several detected scatterers. Highest values are related to the double bounce reflection occurring at the wall-ground interaction

Fig. 6 . 8 -Fig. 6 .

 686 Fig. 6.8 -Tour Keller area: reflectivity histrograms for different detected sources

( a )Fig. 6 . 9 -( 6 . 3 )Fig. 6 .

 a69636 Fig. 6.9 -Reflectivity maps of the Tour Mirabeau represented in ground range and extracted considering all the detected scatterers. Highest values are related to the double bounce reflection occurring at the wall-ground interaction

  (a) Extraction of the most perturbing contributions (b) Number of the points at the different steps (c) 3-D view: indices at the first step (d) Histogram of the indices at the first step

Fig. 6 .

 6 Fig. 6.10 -Tour Keller area: analysis about the unstable image IDs

  (a) Modified Coefficient of Variation trend (b) Stable and unstable pixels

Fig. 6 .

 6 Fig. 6.11 -3-D views of the Tour Keller's area in terms of the CVm

The 3 -

 3 D reconstructions presented in the present thesis enabled to perform a global characterization of build-up areas, processing an adequate set of multitemporal high resolution SAR images. It has been demonstrated the potentialities of TomoSAR technique in distortions correction, in determining information about the number of scatterers and the corresponding reflectivity within one resolution cell and in 3-D change monitoring using basic mono-dimensional estimators as Beamforming, Capon and MUSIC. I would like to acknowledge the German Aerospace Centre (DLR) for providing TerraSAR-X data used in this work in the frame of the project ID LAN1746.

  .1

	Type OPTICAL RADAR	Satellite WorldView-3 IKONOS Sentinel-2 TerraSAR-X Sentinel-1 ALOS-2	Spectral Resolution 29 bands (400-2245 nm) 5 bands (450-900 nm) 13 bands (443-2190 nm) Microwave (X band) Microwave (C band) Microwave (L band)	Temporal Resolution < 1 days 3 days at equator 10 days (1 sat) -5 days (all sat) 2.5-11 days (depending on imaging mode) 12 days (1 sat) -6 days (all sat) 14 days	Best Spatial Resolution (rg x az) 0.31x0.31 m (panchromatic at nadir) 0.82x0.82 m (panchromatic at nadir) 10x10 m (4 visible, near-infrared) 0.6x0.24 m (Staring SpotLight mode) 5x5 m (StripMap mode) 3x1 m (Spotlight mode)

Table 2 .

 2 1 -Platform analysis at different scale of interest

Table 2 .

 2

	Operational life	5+5 years (DLR expectation: beyond 2018)
	Launch date	15 June 2007, 4:14 CEST
	Launch site	Baikonur, Kazakhstan
	Antenna size	4.788 m x 0.7 m
	Launch mass Payload mass	1230 kg ∼400 kg
	Radar frequency	9.65 GHz (X band)
	Power consumption	800 W (on average)
	Repeat Period	2.5-11 days (depending on the imaging mode)
	Orbital altitude	514 km
	Inclination	97.44°N
	ominal acquisition direction	Right side
	Polarisation	Single, dual (depending on imaging mode)

2 -TerraSAR-X technical data

Table 6 .

 6

1 -Amount of the scatterers detected at three levels using different estimation techniques over several tiles

3-D SAR imaging

The Compressive Sensive[START_REF] Candès | Compressive sampling[END_REF] is a signal processing technique for the reconstruction of super-resolved signals and images starting from far fewer data/measurements than what is usually considered necessary. As a favorable sparse reconstruction technique, it represents a valuable method to perform 4-D SAR tomography.

I would like to acknowledge the DLR for providing TerraSAR-X data used for the present research in the frame of the project ID LAN1746

2-D analysis and results(a) Coherence indicator (b) Multitemporal averaged amplitude image

2-D analysis and results

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