DEVELOPMENTS OF MULTIDIMENSIONAL SAR IMAGING FOR THE ANALYSIS OF LAYOVER, VOLUMETRIC AND DYNAMIC SCENARIOS

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Sommario

In seguito alla maturazione delle tecniche di interferometria SAR, basate esclusivamente su dati di sola fase, un sensibile interesse sta crescendo su tecniche di combinazione coerente di dati SAR complessi (modulo e fase) per l’estrazione di una maggiore informazione sulla scena osservata e per sfruttare appieno gli archivi di immagini SAR (sempre in espansione). Tra queste, la Tomografia SAR 3-D (Tomo SAR) è una tecnica interferometrica sperimentale basata su acquisizioni multiple (multibaseline) in grado di ottenere un imaging 3D completo nel dominio range-azimuth-quota attraverso tecniche di stima spettrale spaziale, cioè beam-forming lungo la quota. La Tomografia SAR permette di risolvere retrodiffusori multipli all’interno della stessa cella SAR, consentendo il superamento di una limitazione della Inter-ferometria SAR convenzionale e permettendo l’analisi di scenari complessi. Recentemente, è stata introdotta sotto il nome di Tomografia SAR Differenziale (Diff-Tomo) una nuova tecnica di combinazione dati la quale integra sinergicamente i concetti di Tomografia e in-terferometria differenziale per permette l”’apertura” del pixel SAR in scenari complessi non stazionari, chiarificandone il contenuto informativo. La Tomografia Differenziale è basata su di una analisi spettrale spazio-tempo, sfruttando pienamente le diversità spaziali e temporali dei dati. Permette la risoluzione congiunta di quote e velocità di deformazione di scatteratori multipli presenti in un pixel SAR.

In questo lavoro sono presentati avanzamenti di queste tecniche per scenari volumetrici complessi, includendo inoltre variazioni temporali del segnale, originate sia da decorre-lazione temporale che da moti di deformazione dei retrodiffusori. In particolare, sono ri-portati nuovi risultati riguardanti le potenzialità della Tomografia-Differenziale per l’analisi di scenari forestali, sfruttando il concetto originale delle firme spettrali spazio-tempo della decorrelazione temporale. Più specificatamente, tre nuove funzionalità tomografiche sono state sviluppate per migliorare o per estrarre un nuovo tipo di informazione da scenari fore-stali. Per primo è introdotto un metodo orientato alla riduzione dei fenomeni di defocaliz-zazione, dovuta a decorrelazione temporali, nel processing di profili tomografici. In seguito, è introdotta una nuova tecnica basata, su Tomografia Differenziale, per estendere le fun-zionalità dell’Interferometria Differenziale anche a scenari di tipo forestale. Infine, l’ultima funzionalità tomografica proposta riguarda la possibilità di identificare il grado di decorre-lazione temporale che agisce ai vari livelli di un volume forestale. Queste funzionalità, esten-sioni del imaging SAR multidimensionale, sono discusse attraverso estese analisi simulate e confermate con primi risultati sperimentali.

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Abstract

After the maturation of the interferometric SAR (InSAR, D-InSAR) techniques based on phase-only data, much interest is growing in techniques based on the coherent combination of complex (i.e. amplitude and phase) SAR data for the extraction of even more rich in-formation on the observed scene, to fully exploit existing SAR data archives, experimented multi-antenna airborne and unmanned air vehicle systems and new multistatic satellite clus-ters. Among these techniques, 3-D SAR Tomography (Tomo-SAR) is an experimental multi-baseline (MB) interferometric mode achieving full 3-D imaging in the range-azimuth-height space through elevation beamforming, i.e. spatial (baseline) spectral estimation. Tomo-SAR resolves multiple scatterers in height in the same cell, overcoming a limitation of con-ventional InSAR processing and allowing the analysis of complex scenarios. Recently, a novel coherent data combination mode termed Differential SAR Tomography (Diff-Tomo) has been recently originated, synergically integrating the D-InSAR and the Tomo-SAR con-cepts to allow “opening” the SAR pixel in complex non-stationary scenes, de-garbling its information content. Diff-Tomo is based on two-dimensional space-time spectral analysis, fully exploiting the MB-multipass data. It allows the joint resolution of multiple heights and deformation velocities of the scatterers mapped in a SAR pixel.

In this work, advances of this new framework are investigated for complex volume scat-tering scenarios including temporal signal variations, both from scatterer temporal decor-relation and deformation motions. In particular, new results are reported concerning the potentials of Diff-Tomo for the analysis of forest scenarios, based on the original concept of the space-time signatures of temporal decorrelation. More specifically, we propose three new challenging tomographic functionalities aimed to improve or to extract new information from forest scenarios. First, we introduce a new processing method aimed to reduce the blurring effects from temporal decorrelation in tomographic profiles. Second, we present a method aimed to extend the differential interferometric functionality to monitor subsidence in forest area. Finally, the last challenging functionality regards the possibility of discriminating the amount of temporal decorrelation that acts on the various layers of the forest volume. These developments of multidimensional SAR imaging are discussed trough extensive simulated analyses and first E-SAR P-band data results.

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List of Figures

2.1 SAR Tomography, acquisition geometry . . . 14 2.2 Baseline-time acquisition pattern. (circle: monostatic; cross: multistatic) . . . 20 2.3 Tomo adaptive BF spectra, no temporal decorrelation, (a) monostatic

acqui-sition, (b) multistatic acquisition . . . 22 2.4 Tomo MUSIC pseudo-spectra, no temporal decorrelation, (a) monostatic

ac-quisition, (b) multistatic acquisition . . . 22 2.5 Temporal coherence, (a) τc= 44 r.t.u., (b) τc= 2.8 r.t.u., (+ ground, x canopy,

o overall signal) . . . 24 2.6 Tomographic profiles, τ = 44, multistatic acquisition,(a) Tomo adaptive BF

spectra, (b) Tomo MUSIC pseudo-spectra, . . . 25 2.7 Tomo adaptive BF spectra, τ = 2.8, (a) monostatic acquisition, (b) multistatic

acquisition . . . 25 2.8 Tomo MUSIC pseudo-spectra, τ = 2.8, (a) monostatic acquisition, (b)

mul-tistatic acquisition . . . 25 2.9 Resolution probability vs. Coherence time, (a) monostatic acquisition, (b)

multistatic acquisition . . . 27 2.10 Forest height RMSE vs. Coherence time, (a) monostatic acquisition, (b)

multistatic acquisition . . . 27 3.1 Temporal decorrelation signature, Diff-Tomo adaptive beam, (a): monostatic

acquisition pattern, (b): multistatic acquisition pattern . . . 32 3.2 Temporal coherence, τ = 11.5 r.t.u. (ground only +, canopy only x, overall

signal o) . . . 36 3.3 Diff-Tomo images, multistatic, long term decorrelation, τ = 11.5 revisit time

units, Diff-Tomo image, (a) MUSIC, (b) gen. MUSIC . . . 37 3.4 Robust tomographic profiles from generalized MUSIC Diff-Tomo, τ = 11.5

revisit time units, (a) multistatic , (b) monostatic . . . 38 3.5 Temporal coherence, long term temporal decorrelation,τ = 2.8 revisit times

(ground only +, canopy only x, overall signal o) . . . 39 3.6 Generalized MUSIC, multistatic, long term decorrelation, τ = 2.8 revisit

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3.8 Few pass multistatic acquisition pattern, g/v=3, (a) baseline-time acquisition

pattern, (b) temporal coherence . . . 43

3.9 Diff-Tomo image, gen. MUSIC, few pass multistatic acquisition pattern, g/v=3. 43 3.10 Temporal coherence, few pass multistatic acquisition pattern, g/v=3, long and short term temporal decorrelation (ground only +, canopy only x, overall signal o) . . . 45

3.11 BIOSAR test-site, (a) Google image,(b) saturated radar image . . . 46

3.12 Temporal coherence from a forested cell, BIOSAR-I campaign . . . 47

3.13 BIOSAR, Diff-Tomo images, (a) MUSIC, (b) gen. MUSIC . . . 49

3.14 BIOSAR, Tomographic profile, (a) MUSIC, (b) Robust profile . . . 49

3.15 BIOSAR, Robust tomographic profiles, results from other cells . . . 50

3.16 Range line #1, Tomogram, (a) MUSIC (b) gen. MUSIC . . . 51

3.17 Range line #1, Forest height estimates, (a) MUSIC (b) gen. MUSIC . . . 52

3.18 Robust tomography, range line #2, (a) Tomogram,(b) Forest height estimates . 52 3.19 Robust tomography, range line #3, (a) Tomogram,(b) Forest height estimates . 53 3.20 BIOSAR, LIDAR forest height map . . . 54

3.21 Robust Tomography, map of forest height estimates . . . 55

3.22 Maps of forest height RMSE, (a) Robust tomography, (b) CLEAN Tomo-graphic method . . . 56

3.23 Radar-LIDAR correlation maps (a) Robust tomography, (b) CLEAN Tomo-graphic method . . . 56

3.24 Robust Tomography, improved gen. MUSIC, map of forest height estimates . 57 4.1 Sample sketch of a space-time spectrum relative to a forest scenario with subsiding ground . . . 64

4.2 Temporal coherence, τ = 5.6r.u., (+ ground, x canopy, o overall signal) . . . . 67

4.3 Simulated analysis, Diff-Tomo image . . . 68

4.4 Sub-canopy subsidence, RMSE vs. temporal bandwidth . . . 70

4.5 Sub-canopy subsidence, RMSE vs ground-canopy power ratio . . . 72

4.6 Diff-Tomo image, forested cells with injected motion, P-band data . . . 75

4.7 Scatterplot of estimated ground-canopy temporal frequency . . . 76

4.8 BIOSAR-I dataset, analyzed azimuth cuts location . . . 76

4.9 Estimated ground subsidence over different azimuth cuts . . . 77

4.10 Ground/volume power ratio analysis, (a) g/v histogram, (b) scatterplot of LIDAR measurements-Ground/Canopy power ratio values . . . 79

4.11 Diff-Tomo sub-canopy subsidence estimation, accuracy gain over classical motion monitoring technique, g/v masking . . . 80

4.12 2D histograms of the canopy elevation, extracted from LIDAR measure-ments, and the estimated backscattered power. . . 81

4.13 Diff-Tomo sub-canopy subsidence estimation, accuracy gain over classical motion monitoring technique, radar intensity masking . . . 81

4.14 Pattern of synthetic injected motion, (a) planar view, (b) 3D view . . . 82 4.15 Diff-Tomo sub-canopy subsidence estimation, estimated ground velocity map

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5.1 Space-time signature of a temporal decorrelating volume over ground, adap-tive Diff-Tomo . . . 87 5.2 Diff-Tomo dynamic functionals, elevation-bandwidth domain, multistatic

ac-quisition (a) ESC method, (b) semi-parametric method . . . 91 5.3 Diff-Tomo dynamic functionals, elevation-bandwidth domain, monostatic

acquisition pattern, (a) ESC method, (b) semi-parametric method . . . 92 5.4 Mean estimated temporal bandwidth vs. true bandwidth, multistatic

acquisi-tion, (a) ESC method, (b) semi-parametric method . . . 94 5.5 Accuracy of temporal bandwidth estimates vs. true bandwidth, multistatic

acquisition, (a) ESC method, (b) semi-parametric method . . . 95 5.6 Mean estimated temporal bandwidth vs. true bandwidth, multistatic

acquisi-tion, (a) ESC method, (b) semi-parametric method . . . 96 5.7 Accuracy of temporal bandwidth estimates vs. look number N, multistatic

acquisition, (a) ESC method, (b) semi-parametric method . . . 96 5.8 Volume scatterer, height varying temporal decorrelation, theoretical

spec-trum . . . .100 5.9 Height varying temporal decorrelation, Diff-Tomo elevation-bandwidth

func-tional . . . .100 5.10 Height varying temporal coherence, estimated temporal bandwidths (a),

es-timated temporal coherence . . . .101 5.11 Forested cell, (a): Diff-Tomo output, (b): Height-velocity-temporal

band-width functional . . . .102 5.12 Diff-Tomo - Height-velocity-temporal bandwidth “5D” functional, three

dif-ferent forested cells . . . .103 5.13 BIOSAR-I dataset, Analyzed azimuth cuts positions . . . .104 5.14 Range line #1, (a) temporal bandwidth estimates, (b) temporal coherence

estimates . . . .104 5.15 Range line #2, (a) temporal bandwidth estimates, (b) temporal coherence

estimates . . . .105 5.16 Range line #1, (a) temporal bandwidth estimates, (b) temporal coherence

estimates . . . .105 5.17 Histogram of overall temporal coherence vs. (a) canopy coherence, (b) ground

coherence (b) . . . .106 5.18 Temporal decorrelating forested cell, Diff-Tomo adaptive image, (a) linear

scale, (b) normalized linear scale . . . .108 5.19 Temporal decorrelating forested cell, normalized profiles extracted at

canopy-height centroid canopy-heights . . . .108 5.20 Diff-Tomo Temporal decorrelation signature. 2nd forested cell, (a)

normal-ized adaptive beam image, (b) normalnormal-ized profiles extracted at canopy-height centroid heights . . . .109 5.21 Diff-Tomo Temporal decorrelation signature. 3rd forested cell, (a)

normal-ized adaptive beam image, (b) normalnormal-ized profiles extracted at canopy-height centroid heights . . . .109

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List of Tables

2.1 Forest height RMSE, ideal case, multistatic acquisition . . . 24

2.2 Forest height RMSE, temporal decorrelation, multistatic acquisition . . . 24

3.1 Robust tomography, forest height RMSE (Rayleigh r.u.), multistatic, long term decorrelation, τ = 11.5 r.t.u. . . 39

3.2 Robust tomography, forest height RMSE (Rayleigh r.u.), multistatic, long term decorrelation, τ = 2.8 r.t.u. . . 41

3.3 Robust tomography, forest height RMSE (Rayleigh r.u.), few pass acquisition pattern, long term decorrelation, τ = 2.8 r.t.u. . . 44

3.4 Robust tomography, forest height RMSE (Rayleigh r.u.), few pass multi-static, long term decorrelation, τ = 2.8 r.t.u., short term decorrelation, ρo= 0.7 45 3.5 Orthogonal baselines and acquisition times of the BIOSAR-I dataset . . . 47

3.6 BIOSAR-I, parameters characterizing the acquisition; . . . 47

3.7 Forest height statistics, Robust tomography . . . 53

3.8 Forest height statistics, MUSIC . . . 53

3.9 Forest height statistics, adaptive beam . . . 54

3.10 Robust tomography ,statistics from extensive analysis . . . 58

3.11 Robust tomography, reduced baseline analysis, statistics . . . 59

4.1 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, multistatic acqui-sition pattern . . . 69

4.2 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, monostatic acqui-sition pattern . . . 69

4.3 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, N=32, multistatic acquisition pattern . . . 72

4.4 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, N=32, monostatic acquisition pattern . . . 73

4.5 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, canopy-ground relative motion, multistatic acquisition pattern . . . 74

4.6 RMSE (Fourier r.u.) of estimated sub-canopy subsidence, BIOSAR dataset . 78 5.1 Parameters of the simulated forest . . . 90

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5.2 Temporal bandwidth estimates statistics . . . 93 5.3 Temporal bandwidth estimates, multistatic acquisition, looks number N=8 . . 97 5.4 Temporal bandwidth estimates, multistatic acquisition, looks number N=32 . 97 5.5 g/v = 3 . . . 98 5.6 BIOSAR simulation . . . 99 5.7 Mean values of estimated temporal bandwidth (phase cycles/month) and

cor-relation factor (month) . . . .106 5.8 Forest cells classification based on the analysis of adaptive Diff-Tomo

spec-trums, Nh number of cells with higher canopy temporal bandwidth, Nequ number of cells with similar temporal bandwidth from canopy and ground scatterers . . . .110 5.9 Forest cells classification based on semi-parametric adaptive Diff-Tomo

tem-poral bandwidth estimates, Nh number of cells with higher canopy tempo-ral bandwidth, Nequnumber of cells with similar temporal bandwidth from canopy and ground scatterers . . . .110

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Chapter

1

Introduction

1.1 Remote sensing and Synthetic Aperture Radar

Imag-ing

Remote sensing is the science of obtaining and interpreting information from a distance, using sensors that are not in physical contact with the object being observed. This science has assumed in the years, thank to its wide range of applications, a prominent role for the understanding of the natural and human processes that affects the Earth. In fact, examples of these applications can be found in many different fields of study, such as geology, mineral and petroleum exploration, monitoring of urban subsidence, monitoring of forest, oceonagraphy, metheorology. From the physical point of view, remote sensing exploits electromagnetic radiation to retrieves parameters and geo-physical information on the Earth surface an its dynamic. Depending on the source of the wave, Remote Sensing can be divided in two major classes: active and passive remote sensing. Passive RS makes use of sensors which exploit natural radiations, either reflected or emitted from the earth. On the other hand, active RS exploits sensors which produce their own electromagnetic radiation (e.g. LIDAR, RADAR). Among active remote sensing sensors, imaging radars, such as real aperture radar (RAR) and Synthetic Aperture Radar (SAR), have a prominent role thank to the fact that offer a day-night global coverage and in all weather condition. Moreover they offer a sensitivity to dieletric properties and to surface roughness, allowing applications such as monitoring of biomass, ice, ocean wind speed, soil moisture mapping. 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). The brightness, or radar reflectivity (Intensity, Power, Amplitude) of the received echo is measured and recorded. For coherent radar systems such as Synthetic Aperture Radar (SAR), the phase of the received echo is also measured. Radar echoes and related phase are used during the focusing process to construct the image.

However, expecially for spaceborne campaign, the use of a real aperture radar for a re-mote sensing campaign is deprecated, due to its intrinsic limitation for what concern az-imuth resolution, which is inversely proportional to the physical dimension of the antenna

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and directly proportional to the flight height of the platform on which the sensor is mounted. Consequently, the antenna length requirements of a RAR show problems when detailed res-olution are sought. For longer wavelength radar, often needed in practice, the situation will be even worse.

To overcome this constraint, sinthetic aperture radar, or SAR, has been developed and now is widely used. The basic concept of a SAR system is to synthesize an apparently long antenna aperture in the along track dimension, through the motion of a physical an-tenna (fixed on the platform). A complex (amplitude and phase) coherent combination of the echo, recorded as the platform moves in the along track direction, allows to obtain a narrow beam-width pattern of the synthetic antenna and consequently to achieve high resolution in the azimuth direction. The azimuth resolution obtainable with SAR is independent of slant range, and thus platform altitude, and independent of operating wavelength. Since ground range resolution is also height independent a SAR can, in principle, operate at any altitude with no variations in resolution. Consequently, spaceborne operation is acceptable. These characteristics have made SAR the most important and used sensor in high resolution active remote sensing campaign. SAR data are increasingly applied to geophysical problems, either by themselves or in conjunction with data from other remote sensing instruments [1].

1.2 Interferometric SAR techniques

SAR interferometry is a particular remote sensing technique aimed to overcome some limita-tion of SAR techniques. SAR systems have been used extensively in the past two decades for fine resolution mapping and other remote sensing applications. In particular it was demon-strated by the early missions that synthetic aperture radar is able to reliably map the Earth’s surface and acquire information about it’s physical properties. However, for each imaged point, what we retrieve with a SAR sensor is the projection of the three dimensional coor-dinate vector (range azimuth and height) on a two-dimensional plane (range-azimuth). The target location in a SAR image are then distorted relative to a planimetric view. For many applications this height-dependent distorsion adversely affects the interpretation of the im-agery. SAR Interferometry (or InSAR) enables the measurement of the third dimension, in other words it enables topographic mapping of the scene. The basic concept of SAR inter-ferometry (or InSAR) is to employ at least two complex-valued SAR images to derive more information about the sensed scene by exploiting the phase of the SAR signals. The two SAR images should differ for at least one imaging parameter. Depending on this imaging param-eter (e.g. flight path, acquisition time wavelength), different type of interferomparam-eters can be implemented [2]. The SAR Interferometer used to retrieve information on surface’s topogra-phy is across-track interferometer (XTI-InSAR), exploiting two different images acquired in spatial diversity (e.g. from two different flight path).

The first report of an InSAR system applied to Earth observation was by Graham [3]. However, for terrestrial applications it was only in the 1980s that the first results were pub-lished, see . The use of spaceborne SARs as interferometers has ben estenxively investigated as an enourmous amount of SAR data set were made available after the launch of the ESA satellite ERS-1 in 1991 [2].

In XTI-InSAR the whole information is contained in the interferometric phase φ , the phase difference between the two SAR images and that can be expressed as:

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1.2 Interferometric SAR techniques

∆φ =4π

λ Bsin (θ ) (1.1)

where λ is the radar wavelength, B is the baseline of the two SAR sensors and θ is the tilt angle of the sensor. From the acquisition geometry it can be demonstrated that the topog-raphy height h is directly related to the interferometric phase. However, it is worth noting that the formation of a InSAR image requires a certain amount of post-processing step, each one representing a wide field of study. Before the estimation of the interferometric phase, a coregistration step is needed to ensure that the two images are aligned, reducing the relative distorsion due the slight different acquisition geometries. Succesively, the interferometric phase, corrupted by phase noise due to the speckle in the SAR resolution cell (modeled as a stochastic process), has to be estimated. The maximum likelihood estimator of the interfero-metric phase is the phase of an averaged interferogram:

ˆ ∆φ = arg ( _N

∑

n=1 g(n) g∗(n) ) (1.2)

The N interferogram samples averaged in this equation can be obtained by local spatial averaging of a single interferogram. This technique is usually adressed as spatial multilook-ing. After that, the phase has to be unwrapped to suppress the residual 2π ambiguties. The latter process can be seen as the most difficult step in the formation of an interferogram. Finally, the topography height can be extimated.

Another important class of SAR interferometer is the one that enables the monitoring of the motion of the surface. The idea is to exploit the interferogram between two image acquired at different time. If a scatterer on the ground slightly changes its relative position in the time interval between two SAR acquisitions (e.g. subsidence, landslide, earthquake . . . ), an additive phase term, independent of the baseline, appears. This new contribution to the interferometric phase can be expressed as:

∆φdisplacement= 4π

λ d (1.3)

where d is the relative scatterer displacement projected on the slant-range direction. Air-borne along-track interferometric (ATI) SAR has been widely investigated for its potential in measuring ocean surface currents and waves. The ATI–SAR employs two antennas that are separated physically along the platform flight path (along-track) direction. Goldstein and Zebker [4] first demonstrated the ability of the ATI–SAR to measure surface ocean current velocity in San Francisco Bay, Mission Bay and San Diego Bay. Further studies by Marom et al. have developed the method of applying ATI–SAR to measure directional and power spectra of wave fields. Generally, in repeat-pass interferometry, the temporal separation of days, months, or even years can be used to advantage for long-term monitoring of geody-namic phenomena [5]. In this case, the overall interferometric phase is the combination of the phase due to topography and the surface deformation motion. Several approaches have been developed to compensate the nuisance phase due topography, such as using a pre-computed topography or exploiting three SAR images, in which one pair is used to retrieve the topography. Since the wavelength is in the order of centimetres, D-InSAR can measure displacements down to millimetre accuracy and have thus demonstrated to posses unique ca-pabilities in the global monitoring of glacier and ice sheet dynamics, seismic deformations,

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volcanic activities, and land subsidence in mining areas. Nowadays, D-InSAR is a mature technique based on possibly more than three SAR acquisitions. In this context, two opera-tive techniques are persistent scatterer interferometry (PSI) [6] and D-InSAR stacking. The former operates at a high horizontal resolution scale, and focuses on point-like scatterers, while the latter operates at a low resolution scale, assuming distributed scattering. Another technique is Small Baseline Subset (SBAS) [7] in which is exploited an appropriate combi-nation of differential interferograms created by using SAR image pairs characterized by a small orbital separation (baseline).

1.3 Multidimensional SAR imaging: new possibilities for

complex non-stationary scenarios

In the past three decades, the interferometric techniques based on synthetic aperture radar (SAR) images have demonstrated to posses unique capabilities for geoscience applications. Cross-track SAR interferometry allows to overcome the imagery misinterpretations due to the 3-D scattering projection onto the 2-D range-azimuth plane. By exploiting the pixel-by-pixel phase variation between two or more SAR images acquired in viewing angle diver-sity (producing one or more interferometric baselines), InSAR has become a well-assessed technique for the generation of accurate digital elevation models (DEM) [1]. However, In-SAR techniques can not sense scatterer movements. For this reason, Differential InIn-SAR (D-InSAR) has been proposed based on the analysis of temporal phase changes. Nowadays, D-InSAR is a mature technique based on multiple pass satellite SAR acquisition to accu-rately detecting and mapping centimeter to millimeter scale deformations of the ground, and monitoring buildings, glacier flows and slope instabilities [4]- [2]. Nonetheless, when the surface topography is steep enough to generate critical projection of the scatterers in the slant imaging geometry, or the imaged area is characterized by a high spatial density of strong scat-terers, the signal collected in a SAR pixel may contain the superposition of responses from multiple scatterers (layover) [8]- [9]. This condition is common also for volumetric scenar-ios such as forested areas, glaciers, and arid zones, especially for low frequency SARs, and it is frequent when data are acquired over complex scenarios such as urban areas or large structures and infrastructures. Existing InSAR and D-InSAR algorithms can not separate the multiple scattering phenomena in the same pixel, thus degrades or can not operate in these conditions.

Consequently, much interest recently concerned on the concept of SAR 3-D tomography (Tomo-SAR) [8,10]. This is an advanced multibaseline (MB) interferometric mode which overcomes the limitation of standard InSAR techniques by achieving focused fully 3-D im-ages through elevation beamforming, i.e. spatial (baseline) spectral estimation. In detail, SAR Tomography employs many (on the order of ten) passes over the same area, producing an aperture synthesis also along the vertical plane (cross-track array) to get full 3D imag-ing. In particular, Tomo-SAR can add more features for applications involving estimation of forest biomass and height, sub-canopy topography, soil humidity, and ice thickness, and for solving InSAR height and reflectivity misinterpretation caused by layover geometries in natural or urban areas.

Applications of ice thickness monitoring, whose importance is significant given the prob-lem of global warming, might also benefit from SAR Tomography. It can be potentially adopted jointly with or alternatively to Polarimetric Interferometry, to obtain polarimetric

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1.3 Multidimensional SAR imaging: new possibilities for complex non-stationary scenarios

forest model validation [11], classification, and improvements, or measurements in a direct imaging framework, respectively. The tomographic concept has been firstly demonstrated trough a laboratory experiment carried out by Politecnico di Milano in cooperation with JRC-Ispra . Other experiments have been conducted exploiting repeat passes of an airborne monostatic L-band SAR sensor with Fourier based tomography [10]. Research efforts have concentrated so far on improving the typical unsatisfactory 3-D imaging quality of conven-tional beamforming (BF)-based Tomo-SAR due to the typically limited and sparse baseline distribution.

However, the elevation focusing process relies on the sensed scene staying coherent, while physical changes typically occur during the repeat pass MB acquisition, especially for spaceborne sensors. As it is well known for conventional azimuth SAR focusing, partially coherent scatterers can lead to blurring effects. In the Tomo-SAR case, elevation defocus-ing or misplacement effects can arise in the 3-D imagdefocus-ing process [12,13]. Such temporal decorrelation effects are recognized by ESA and NASA-JPL to be a possible development and application barrier of MB 3-D tomographic imaging, especially for spaceborne sensors. Moreover, Tomo-SAR can separate multiple scatterers, but it has no measuring sensitivity to their deformation motions.

To overcome these limitations, recently, the Differential InSAR and Tomo-SAR con-cepts have been deeply integrated to furnish a new coherent data combination framework, termed Differential Tomography (Diff-Tomo) [14].This is a coherent complex data fusion technique based on two-dimensional space (baseline)-time spectral analysis, fully exploit-ing the information content of multipass multibaseline data, allowexploit-ing “openexploit-ing” the SAR pixel in complex non-stationary scenes. It results in joint elevation-deformation velocity resolution of multiple scattering components in a same range-azimuth cell, and this new con-cept has been demonstrated by processing ERS-1/2 data over an urban area with layover elevation-concentrated scatterers , also handling the possible relative motion case. More generally, Diff-Tomo has the potential to identify scattering components intensity distribu-tions in the joint domain of elevation and temporal frequency (phase rate of change) of 2D harmonics in which a signal from a scattering component with temporal variations can be decomposed [12,14].

A first direct implication of the joint and continuous scattering profiling in the elevation-deformation velocity domain allowed by Diff-Tomo is the potential for the development of a “volumetric differential interferometry” technique. In other word it can be exploited for continuous velocity profiling of non-rigid volume sliding (like a cold glacier flow sliding), or for investigations on possible buried scatterers and subsurface dynamical processes (e.g. sub-canopy ground subsidence estimation), coupling the velocity dimension to 3D reflectivity maps.

The techniques of SAR Tomography and of Differential SAR Tomography can be seen as a part of a new framework, termed multidimensional SAR imaging. Exploiting the capabil-ity of continuous 3D imaging along the height dimension and the newly added 4D imaging capability (3D + time ), the framework of multidimentional SAR imaging allows new in-teresting functionalities for remote sensing of multiple complex non-stationary scenes (e.g. volumetric or urban scenarios). They allow the extraction of a more rich and/or accurate information on the observed scene w.r.t. the phase-only InSAR techniques.

In this framework, both existing SAR data archives, experimented multi-antenna airborne systems, and future multistatic satellite clusters [15] may be efficiently exploited by use of

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Diff-Tomo for 3D imaging and monitoring of complex surface scattering scenarios (3D+time or 4D imaging).

1.4 Contribution and outline of the thesis

This thesis focuses on the development and the experiment of new functionalities for to-mographic SAR techniques aimed to extend the analysis of complex scenarios, in which multiple scattering contribution are interfering from different heights in the same SAR cell. In particular, the attention is dedicated to the simulation and experiment with real data of new functionalities for the monitoring of structrue and dynamic process of volumetric scenarios, in particular of forest. The main theme of this paper regards the application of Diff-Tomo based techniques to forest scenarios, characterized by the presence of a temporal decorrelat-ing volumetric canopy scatterer, in layover with a ground scatterer.

In this context the contributions of this thesis are the following:

• the estensive analysis of temporal decorrelation effects on 3D SAR Tomographic fo-cusing. The analysys, conducted over a tipical simulated forest scenario, has the sco-pus of quantifying the blurring effects originated by the temporal decorrelation of the canopy volume in a forest scenario.

• indication of the criticalities of an adaptive and model-based tomographic methods with respect to temporal decorrelation condition. The indication of the these critical-ities, expressed in term of the revisit time of the sensor employed to synthesize the tomographic array, may be useful in the planning of future spaceborne campaigns. • the proposal of algorithms exploiting the Differential SAR Tomography framework for

the 3-D analysis of non-stationary volumetric scatterers, as forested areas subjected to temporal decorrelation. In particular, solutions are proposed for the tomographic pro-filing robust to the blurring effects due to the temporal decorrelation, and first results are shown of the accuracy obtained in the estimation of the forest canopy height with real data.

• the proposal of a new functionality based on Differential SAR Tomography to moni-tor the motion of buried scatterers under a volume scatterer. In particular, it has been allowed the monitoring of ground subsidence under the forest volume. Extensive sim-ulated results and first real data experiment are conducted, proving the capability of this newly developed functionality.

• the development of another new functionality aimed to detect different temporal scat-tering mechanisms inside the same SAR resolution cell. The resulting functionality, developed trough a special model-based Diff-Tomo processor, furnishes with the inter-esting possibility of discriminating the amount of temporal decorrelation of the canopy and ground scatterers.

• phenomenological analysis of temporal decorrelation, through an adptive Diff-Tomo processor, in forested cells from real P-band data. In particular it will be demon-strated for the first time, at the best of the author knowledge, the concept of space-time signature of a temporal decorrelating volume over ground, which has been already demonstrated in simulations.

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1.4 Contribution and outline of the thesis

The remainder of this thesis is structured as follows.

In Chapter 2 the concept of Tomo SAR is briefly reported, giving in particular the theoret-ical basis of the tomographic focusing along the height dimension. After that, the theory of a model-based (MUSIC) and an adaptive (Capon) tomographic method is briefly recalled. Be-fore analyzing the temporal decorrelation effect, the temporal decorrelation model adopted for the simulated scenario is introduced. In the second part of the chapter, the performance model-based and the adaptive tomographic methods are evaluated over a simulated forest scenario, characterized by a temporal decorrelating canopy volume in layover with a steady ground scatterer. The corresponding tomographic profiles and the accuracy on estimated scatterer position are compared under two different temporal decorraltion conditions: one light and one heavy (total loss of coherence at the last pass of the SAR sensor.). Finally, the results from the two methods are compared and indications are hinted on their criticalities with respect to the revisit time of the sensor.

In Chapter 3 we introduce the new functionality of tomography robust to temporal decor-relation effects. First, the concept of Differential SAR Tomography is briefly recalled, to-gether with some hints on its principal applications. In particular, we focus to the concept of the spatio-temporal signature from temporal decorrelation, upon which the rationale of the new functionality is based. After that, we introduce a special model-based Diff-Tomo processor matched to a model with continuos spectral components, aimed to fit the tem-poral spectral signature from decorrelating scatterers and reduce their misinterpretation in Tomo processing. Then, this new functionality is used for the analysis of forested scenar-ios affected by temporal decorrelation. In this case, two issues are addressed. Firstly, we show how Diff-Tomo processing can furnish a non-blurred Tomo-SAR profile in presence of temporal decorrelation, restoring the possibility of resolution of the ground and the canopy scatterers. Secondly, we show that a Diff-Tomo processor can lead to an enhanced parame-ter estimation accuracy w.r.t. a Tomo-SAR processor with temporal decorrelated data. The new functionality is extensively tested on a simulated forest scenario with varying temporal decorrelation conditions and acquisition parameters. Finally, the first experimental results obtained with a real P-band dataset are presented. The results, obtained in mild temporal decorrelating scenario, are encouraging and shows how the robust tomography lead to en-hanced performance w.r.t. classical tomographic processors.

In Chapter 4 we show the how the functionality of “volumetric differential interferom-etry”, already hinted in the previous section, can be implemented trough proper Diff-Tomo processing. More specifically, we exploit the Diff-Tomo technique to derive a tomographic functionality for monitoring under-volume deformations, in particular of sub-canopy ground subsidence. After introducing the problem of subsidence monitoring in forest area, we intro-duce another special Diff-Tomo processor aimed to decouple the interfering volume effect from the ground signal and thus allowing a correct monitoring of the ground subsidence. The new functionality has been firstly demonstrated, under the assumption of perfectly cal-ibrated data, on a simulated forest scenario. Finally, first real data experiments on a real P-band forest scenario has been conducted, considering also an injected pattern of motion. The preliminary results shows that the sucanopy ground subsidence may be monitored, with sensible improvements over classical motion monitoring techniques. Finally, the challenging possibility of retrieving canopy-ground relative motion, possibly simulating the tree growth phenomenon, is hinted

In Chapter 5 is introduced the last new functionality, proposed in this thesis work, for monitoring dynamic processes inside a SAR resolution cell. This functionality concerns the

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possibility of discriminating the amount of temporal decorrelation of canopy and ground scatterer. We will show how this kind of information may be retrieved from nuisance param-eters of the special Diff-Tomo processors used for the first two functionalities. In particular, we exploits parameter estimates of the assumed continuos spectral model, related to the ban-width of the spectral signature from temporal decorrelation and from which we can retrieve the desidered temporal coherence. The functionality has been demonstrated trough simulated analyses on a forest scenario and trough first preliminary experimental results on real P-band forest. The interesting results, altough biased, are consistent with the canopy scatterer being the expected decorrelating one. In the second part of the chapter we again refer to the concept of temporal decorrelation signature, already discussed in Chapter 3. In particular we report a phenomenological analysis on real data forested cells aimed to detect the temporal decorre-lation signature of the canopy scatterer through an adaptive Diff-Tomo processor. The results show a wider spectral component at the height of the canopy scatterer. Moreover, we show how that these results are in accord with that from the first part of the chapter.

Finally, Chapter 6 draws some conclusions of the entire work, discussing open issues and further perspectives for the tomographic imaging.

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Chapter

2

Temporal Decorrelation in 3D SAR

Tomography

Motivation

Much interest is continuing to grow in advanced SAR methods of full 3D imaging [10,16] of volumetric scatterers, in particular forests. Multibaseline SAR tomographic elevation beam forming is a promising technique in this framework. This technique, exploiting different ac-quisition (baselines) of the same scene with slightly different look angles, is able to retrieve a profile of the backscattered power along the elevation dimension. This capability allow to resolve multiple scatterer in the same SAR pixel, and can thus be usefully exploited to im-prove standard interferometric products [17,18], that are obtained assuming the presence of only one scatterer in the pixel. However, the tomographic process relies on the sensed scene staying coherent during the repeat pass multibaseline acquisition, which can be long hours to month or even years (for airborne and spaceborne sensors respectively). As it is well known for conventional azimuth SAR focusing, partially coherent or moving scatterers can lead to blurring effects. In this chapter the important effects of temporal decorrelation of volumetric scenarios on SAR tomographic processing is investigated. In a first section we give a brief recall of the concept of SAR Tomography and some hints on its wide application field. Then the tomographic processing is tested with an adaptive and a model-based method on a simu-lated typical forest scenario (canopy+ground scatterers), affected by temporal decorrelation from the canopy scatterer. The test are conducted exploiting different temporal decorrelation conditions and for two different acquisition pattern, one monostatic (one phase center for each pass) and one multistatic (more than one phase centers for each pass). The results will clearly show the important effect of the temporal decorrelation, heavily affecting the tomo-graphic scene reconstruction. In the last part of this chapter some hints on criticalities of tomographic methods are given, in particular we give an indication on the possible choice of mission parameters to mitigate the temporal decorrelation effects.

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2.1 SAR Tomographic concept

The basic principle of tomographic SAR focusing is to perform an aperture synthesis in the elevation direction out of different flight tracks [10]. Let us refer to the system geometry depicted in2.1in which we have K passes over the interested area. The sensor is supposed to move in the azimuth direction (along-track), labeled as x , and, for the sake of simplicity, all the sensor are aligned along z. The orbits are not necessarily co-planar and uniformly-spaced The 2D scene reflectivity function is obtained trough the classical SAR focusing of raw data. The integration of the echoes along the elevation direction is intrinsically performed by the system. Following this assumptions, the generic signal (image) at the k-th sensor can be modeled in the following form [8]:

gk(x, r) = ˆ ˆ f x0− x, r0− r dxdr ˆ γh(x, r, s) exp − j4π λ Rk(r, s) ds (2.1)

where x0and r0are the discrete variables associated to the azimuth and range position of the focused data, λ is the operating wavelength and γh(; ) is the three-dimensional reflectivity function of the imaged scene. The term Rkrepresents the path length (two-way) between the k− th sensor and a scattering element at height s and can be expressed as

R_k(r, s) ≈ r − b_qk+(s − b⊥k) 2

2 (r − b_qk) (2.2)

where b_qk is the baseline, between the k + 1 and k sensor, component parallel to the slant range direction, while b⊥kis the normal to slant range component. In 2.2the Fresnel or paraxial approximation has been exploited.

In the signal model of2.1is also present the post-focusing 2-D point spread function (PSF), determined by the range-azimuth resolution of the SAR system

f(x0, r0) = sinc r 0 ∆r sinc x 0 ∆x (2.3) where ∆r and ∆x are the range and azimuth resolution,respectively. For the sake of simplicity, we can assume the bandwidth to be ideally infinite, thus allowing the PSF to be approximated with a 2-D Dirac functions. Under these assumptions the problem can be shift from a 3-D (range-azimuth-elevation) to a one-dimensional (elevation) estimation problem and2.1can be rewritten as [8]: gk= ˆ a −a γh(s) exp − j4π λ Rk(s) ds (2.4)

where 2a is the extent, in the elevation direction, of the imaged scene and the dependence from the range r has been omitted. The phase term in2.4comprises a quadratic distortion due to the aforementioned paraxial approximation. This first step is to compensate this quadratic phase variation by multiplying the received signal by a complex conjugate quadratic phase function. This operation is denoted as deramping in the radar jargon and is implemented as:

hk= gkexp + j4π λ Rk(0) (2.5)

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2.1 SAR Tomographic concept

where Rk(0)is the distance from the center of elevation target. Substituting2.5in2.4, after some trivial manipulation we get

hk≈ a ˆ −a γh(s) exp − j4π λ s2 2 (r − b_qk) exp j4π λ b⊥ks r− b_qk ds (2.6)

Interestingly, 2.6contains a phase term proportional to s2. This term corrupts the im-age phase, which should ideally be independent from the position of the target. For simple imaging, where only the amplitudes are of interest, this is irrelevant. But for polarimetric and other applications that require a phase-preserving image reconstruction, it is necessary to remove this term by multiplying an inverse phase term after the final image has been gen-erated [10]. Hereafter, we incorporate this factor in the elevation reflectivity function, thus obtaining the final relationship

hk= ˆ γh(s) exp j4π λ b⊥ks r− b_qk ds (2.7)

Equation 2.7states that the complex 3D reflectivity function and the K SAR acquisitions are related trough a 1D Fourier relation. The reflectivity function γh(s) can thus be esti-mated through a simple Fourier transformation of the sequence hk, k = 1 · · · K in the height-dependent spatial frequency domain. These spatial frequency can be expressed as:

ωs= 2s

λ r (2.8)

. To efficiently compute the inversion, FFT algorithms can be used, but this requires that the involved signal is sampled with uniform spacing along the elevation co-ordinate, at the proper uniform sampling rate. This assumption is not usually satisfied, since the orbits are usually not uniformly spaced along the s’ direction. However, problems of cost and possible temporal decorrelation of the scattering prevent from using a large number of passes, while navigation/orbital considerations may not allow obtaining ideal planned uniformly spaced flight tracks. As a consequence, the point spread function (PSF) along the height dimension is distorted. In particular, the non-uniform baseline (spatial) sampling causes bad imaging quality results with the classical Fourier-based focusing in terms of contrast and ambiguities, in addition to the intrinsic limited resolution, as anomalous sidelobes affect the PSF. The Sin-gular Values Decomposition (SVD) [8] is a reliable linear technique for the reconstruction of the unknown. Besides, the SVD procedure allows accounting for a priori information about the support of the object in order to slightly improve the resolution of the reconstruction and achieve more robustness in the reconstruction. Alternatively to this processing, an adaptive (Capon) and a model-based (MUSIC) beamforming techniques were proposed [9,19], pro-viding height superresolution and sidelobe suppression capabilities. More recently, other so-lutions have been proposed and tested for the 3-D focusing and parameter extraction [20,21], comprising the computationally burdensome compressive sensing (CS) [21]. In a differ-ent class of tomographic processors, the MB data are pre-processed by filling the gaps in the available baseline distribution. Under the simplifying assumption that a single height backscattering contribution is dominant in the range-azimuth cell, a simple gap filling algo-rithm has been proposed in [10]. However, unsatisfactory results are obtained in the scenarios of interest comprising more than one point-like spaced scatterers , because of the assumption mismatch.

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Figure 2.1 SAR Tomography, acquisition geometry

However, the elevation focusing process relies on the sensed scene staying coherent, while physical changes typically occur during the repeat pass MB acquisition, especially for spaceborne sensors. As it is well known for conventional azimuth SAR focusing, partially coherent or moving scatterers can lead to blurring effects. In the Tomo-SAR case, elevation defocusing or misplacement effects can arise in the 3D imaging process, the distortions being worsened due to the typically space-irregular (non monotonic) temporal synthesis of the cross-track array [22], [12,13]. Such effects are recognized by ESA and NASA-JPL to be a possible development and application barrier of MB 3D tomographic imaging, especially for spaceborne sensors and forest areas.

2.2 Superresolution Tomographic methods

According to what discussed in the previous section, the reflectivity function in elevation (tomographic profile) can thus be estimated trough spatial spectral estimation techniques. In particular, in this section we adresses two different methods for spectral analysis both charac-terized by good superresolution and sidelobe reduction capabilities: adaptive beamforming and subspace-based MUSIC (Multiple Signal Classification).

2.2.1 Adaptive beamforming (Capon)

The Capon spectral estimator is non-parametric and belongs to the class of filter bank ap-proaches for spectral estimation. The basic idea of Capon method is to design a finite im-pulse response filter that passes the spatial frequency ωs in without distortion and, at the same time, attenuates all the other frequencies as much as possible [23]. The optimization problem can be expressed in formula as

min ωs

hHRhˆ Hsubject to hHa (ωs) = 1 (2.9) In the previous formula, a (ωs) is the so-called steering vector. It collects the phase his-tory (linear with the orthogonal baseline length after deramping) generated at the interfer-ometric MB array by the elevation position s. Its generic element is given by [a (ωs)]_k= exp { j2πb⊥kωs}. With h we denote the vector containing the weights of the filter and with

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2.2 Superresolution Tomographic methods

ˆ

R the sample covariance matrix, defined as ˆ R = 1 N N

∑

n=1 y(n) yH(n) (2.10)

, where with y we denote the signal sampled at the multibaseline array. Once the optimal weight have been found (e.g. trough Lagrange multipliers), the final formula of the Capon spectral estimator results in:

PC(γs) =

1 aH_(ω

s) ˆR−1a (ωs)

(2.11) . Interestingly, the resulting spatial filter changes both with s and, notably, with the inten-sity spectrum along elevation, thus being data-adaptive. This capability of obtaining more focused nulls in the direction where other source are present allows to reduce the spatial leak-age from closely spaced sources thus augmenting the resolution . Moreover, it is known in the array processing literature that the mismatch between the nominal array response (i.e. the one assumed by the processor) and the real one can cause a sensible performance degrada-tion. In the interferometric applications, this mismatch can be due, for example, to spurious phase contributions produced by (residual) atmospheric effects, and deramping errors. ABF could see the useful signal component at the height of interest as an interference, and par-tially suppress it instead of leave unaltered. This phenomenon is generally named signal self-nulling or self-cancellation. To mitigate the ABF non-linear radiometric effects due to the selfcancellation phenomenon, and to ensure stability in the inversion of the multilook covariance matrix estimate in2.10, a diagonal loading of can be included in the filter calcu-lation, that is should be used ˆRL= ˆR + δ σv2I instead.

2.2.2 Model-based method (MUSIC)

The MUSIC algorithm is a parametric method for frequency estimation of sinusoidal signals embedded in additive white noise and it relies on the eigen-decomposition of the covari-ance matrix [23]. The set of eigenvalues of R can be split into two subsets. Denote with {λ1, . . . , λK} the eigenvalues of R arranged in non-increasing order. In the absence of multi-plicative noise, the following property holds:

( λk> σv2 λk= σv2 for k = 1 . . . Ns for k = Ns+ 1 . . . K (2.12)

where we assumed Ns is the number of sources in a scene and σv2 is the noise variance. Denote with {s1, . . . , sNs} the eigenvectors corresponding to the first Nseigenvalues, and with

{g1. . . gNs} the eigenvectors corresponding to the remaining K − Nseigenvalues. The matrix

of eigen vectors can be partitioned as E = [S G], where S = [s1· · · sNs] and G = [g1· · · gNs]

are the matrices which collect the “signal” and “noise” eigenvectors, respectively. The range space of S equals the space spanned by the signal, so we refer to S as the signal subspace. Similarly, G is the noise subspace. The MUSIC algorithm is based on the fact that the signal and noise subspace are orthogonal, thus for a given source it follows that:

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From this condition it can be derived the expression for the spectral MUSIC algorithm PM(ωS) = 1 aH_(ω S) GGHa (ωS) (2.14)

Theoretically2.14will have peaks (infinitely high in the noise-free case) at the signal direc-tions. In practical case the covariance, the signal and noise subspace can be derived trough the eigenvalue-analysis of the estimate covariance matrix2.10. The estimation accuracy of the MUSIC algorithm , as well as Capon algorithm, strongly depends on a reliable estimate of this covariance matrix. To form a reliable estimate of ˆR, it is necessary to collect a suffi-ciently large number of snapshots N—a simple rule of thumb is, at the very least, N > 3K. However, to separate closely spaced signals a much larger number of snapshots may be re-quired—typically tens or hundreds.

In the rest of the work , Capon and Music methods are used as reference results, so they are sometimes adressed as classical technique.

2.3 Sources of decorrelation in SAR Tomography

Radar interferometric techniques have proved to posses unique capabilities for remote sens-ing application. Moreover, the implementation approach, that utilize a ssens-ingle SAR sensor in repeating orbits, is attractive not only for cost and complexity reasons but also in that it permits inference of changes in the surface over the orbit repeat cycle. However, both amplitudes and phase of radar backscatter can be perturbed by changes of geometry and di-electric constant, arising from different phenomena acting on various time scale. This, in techniques such as SAR Tomography, that exploits a coherent combination of radar echoes (amplitude and phase) to retrieve the elevation power distribution of scatterers, can badly affect the estimation process. Generally, the quality of various interferometric products is highly determined by the degree of correlation between the interferometric pairs. From this point of view the study of decorrelation mechanisms is an important task in repeat pass radar remote sensing. In this section we adress and characterize the major sources of decorrelation that can affect radar echoes statistics, then we focuses on temporal decorrelation.

Considering two radar signals g1and g2acquired by two antenna observing the same target at different times and different acquisition geometry. The degree of correlation, or co-herence, between these single look complex (SLC) images of an interferometer is measured as the magnitude of the complex cross correlation between the images

ρtot= hg1, g∗2i q g1, g∗1 g2, g∗2 (2.15)

where withh; iwe denotes ensemble averaging and with * the conjugate-transpose operator. It will take the value 1 when the images are fully correlated over the region chosen to compute the average, and zero if there is no statistical relationship between the images, in which case they are said to be fully decorrelated. Equ. 2.15represents the total observed correlation factor, however, it can be demonstrated that is the product of correlation arising from three different factors

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2.3 Sources of decorrelation in SAR Tomography

where the components are the decorrelation factor due to thermal noise, to different acquisi-tion geometries and to physical changes in the acquisiacquisi-tion time, respectively. It is possible to give expression for the different contributions.

The noise coherence is intrinsically in the radar system and depends on the signal to noise ratio. It can be expressed as

ρthermal= 1

1 + SNR−1 (2.17)

. For a very high receiver signal to noise ratio, which is to be expected, this term should be close to unity. Equivalently the term related to the spatial baseline decorrelation can be expressed in therms of geometric parameters

ρspatial= 1 −

2B⊥∆r cos (θ ) λ Ro

(2.18) where B is the interferometer baseline, ∆x is the range resolution, λ is the radar wavelength and r is the slant range distance. Interestingly,2.18suggests that there is a limit in the choice of the interferometric baseline, also leading to a trade-off in the performances of interfero-metric techniques. For example, in SAR Tomography there is a direct relationship between the length of the baseline and the resolution in the elevation dimension. Equ.2.18states that the resolution can not be raised trough system parameters. A similar trade off can be found in SAR interferometry, where the length of the baseline affects the height accuracy. These consideration suggest the concept of critical baseline, defined as the baseline between two acquisitions sufficient to make the relative radar echoes completely uncorrelated. Specifi-cally,

Bc= λ Ro

2∆r cos (θ ) (2.19)

. The final decorrelation source of interest is the temporal effect, which follows from phys-ical change in the surface over the time period between observations. Since the temporal decorrelation effect on SAR Tomography is the main theme of this chapter, a detailed model will be discussed in the next session.

2.3.1 Spatio-Temporal decorrelation model

Modeling of the partial scene coherence, which follows from physical changes in the surface over the time period between the observation, is a difficult task. Calculation of this effect is application-specific, depending on detailed changes of a given surface type [24]. Both ampli-tude and phase of radar backscatter can be perturbed by changes of geometric and dielectric constant, arising from different phenomena acting on various time scales. In this work we ex-ploit the analytical model in [22] for the decorrelation due to change in the scatterer position. We consider time-homogeneous surface changes with both short-term random movements and long-term correlated random and deterministic movements. The motion component that affects the beamforming process is the change in slant range scatter distance δ rk,l between acquisitions k and l, assumed to be taken at time tk and tl ,respectively. White zero-mean Gaussian displacement of power g/2 takes account for movements completely decorrelated in the short term (seconds to minute). These arise, as an example, from wind action on vegetation . To model slow-correlated random movements Brownian motion with squared jump-size to time-step limit ratio d. Each scatterer within the cell undergo a random walk

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along the line of sight. A possible uniform motion component with velocity e, that may be useful to represent scatterer movement, such as that from vegetation growth, is also taken into account. From this motion model results thatδ rk,l∈ N

ηk,l, σ_k,l2

with σ_k,l2 = g + d |t_k− t_l|and ηk,l= e (tk− tl) for tk6= tland 0 for tk= tl. From the expression for backscatter decorrelation in [24] , following this motion model the temporal decorrelation model can be derived

R(k, l) =      γ exp (−τ |tk− tl|) exp ( jε (tk− tl)) ,tk6= tl 1 ,t_k= t_l (2.20)

where γ = exp − 4π/λ2 g/2 is a steady short-term correlation coefficient, τ = (4π/λ )2d/2 is the inverse of the long-term decorrelation time constant and ε = (4π/λ ) e is a Doppler shift.

An alternative model in [25] makes the assumption that the elemental scatterers in the resolution cell undergo sudden changes in the reflectivity, passing from perfect coherence to zero coherence (Birth and Death process). However, the derived temporal decorrelation model still holds an exponential behavior.

In forest volumetric scenario the effects of geometric decorrelation (i.e., the one due to the perspective effects arising from the projection of the slant range cell along the elevation direction) has to be take into account when modeling the covariance matrix. The model assumed for the covariance matrix of the data vector can be quite complicated. Several models have been proposed in the literature; one of the most famous is the so called random volume over ground (RVoG) [26]. In this model, the volumetric scattering of the canopy has an exponential power distribution along height, and the decadence of this exponential is governed by the extinction coefficient, which is measured in dB/m. When the extinction coefficient equals 0, the power distribution of the canopy becomes uniform between the height of the ground and the height of the canopy.

Here we treat this decorrelation effect from a signal point of view. Assume a large number of wavefronts impinging on the MB array, all originating from independent reflections in the vicinity of the source, i.e. coming from different layers of the volume. We can write the received signal at the k acquisition as:

y(k) = s

_∑

n

pne− j2πb⊥,k(ωs+ωn)+ v(k) (2.21) where s is the transmitted signal, pnis the complex ray gain, which is assumed to be inde-pendent from snapshot to snapshot as well as from ray to ray, ωs is the spatial frequency relative to volume centroid, ωnis a small deviation from the centroid and v is thermal noise. It can be argued that y is a zero-mean complex Gaussian random-vector. Assuming indigent rays and including in the model the temporal decorrelation, the (k, l) − th element of array covariance matrix is given by

Ey(k)yH(l) = |s|2E

∑

n |pn|2e− j2π(b⊥,k−b⊥,l)(ωs+ωn)ej 4πδ rk,l λ . (2.22)

For the sake of simplicity, assume that the distribution of rays, i.e the power distribution of the canopy (volume), is uniform, that is equivalent to assuming an extinction coefficient equals 0 in the RVoG model. However, without loss of generality, canopy power distribution

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2.4 Tomography of volumetric forest scenarios: simulated analysis

with different shapes can be exploited. Further assume that change in the scatterer distance due to temporal decorrelation and the spatial the deviation ωnare independent, then2.22can be rewritten as Rk,l = Ey(k)yH(l) ≈ |s|2e− j2π(b⊥,k−b⊥,l)ωs (2.23) 1 ∆ ˆ ∆/2 −∆/2 e− j2π(b⊥,k−b⊥,l)ωs_dω n ˆ ej4πδ rk,lλ pd f(δ r_k,l)dδ r_k,l

where ∆is the extension in height of the volume and p(.) is the probability density function (pdf) of the change in slant-range scatterer distance. The integrals can be evaluated resulting in

R = a(ωs)aH(ωs) Rs RT (2.24)

where a(ωs) = 1 e− j2πb⊥,1ωs · · · e− j2πb⊥,Kωs T

is the array steering vector , the ele-ments of RTare defined in2.20, the k,l-th element of RSis sinc(4π(b⊥,k− b⊥,l)∆) and is the Schur-Hadamard or elementwise product.

2.4 Tomography of volumetric forest scenarios: simulated

analysis

In this section, the effects are quantified of temporal decorrelation on the formation of To-moSAR profiles from repeat pass MB data, considering in particular model-based and also adaptive BF methods, typically employed in critical resolution conditions. The analysis are carried out for a statistically simulated forest scenario, in terms of resolution probability and estimated elevation accuracies for varying decorrelation conditions and acquisition configu-rations. Indications of temporal criticalities will be derived.

2.4.1 Simulated forest scenario

The analyses presented in this and in the following sessions concern a typical forest sce-nario with canopy and ground contributions. Here, we report a description of the scesce-nario including a brief description of the physical parameters that enters in the model of the forest. The multibaseline-multitemporal signal from the ground and canopy layover scatterer is simulated by a statistical simulator. This simulator assumes the knowledge of the statistical properties of the scene, i.e. the spatio-temporal model of the previous section, and from this information reconstruct the multibaseline-multitemporal signal received from a generic array. It is based only on the statistical properties of the scene and does not make any assumption on the physical ones.

In the simulated scenario, two baseline-time acquisition patterns are considered, one with a single phase center track per pass, tipically monostatic, one with three phase center tracks per pass, e.g. multistatic. In particular, this latter type of acquisition pattern has been sim-ulated in view of the incoming and future satellite cluster configurations, such as those of COSMO-Skymed, TanDEM-X, and TanDEM-L [15,27,28], that will provide more than one array phase center per pass. The reported (ortho) baselines, normalized to the minimum one, are reported in Fig. 2.2, where with circles we denoted the monostatic acquisition pattern, whereas the crosses identify the multistatic one; the acquisition times are normalized to the

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Figure 2.2 Baseline-time acquisition pattern. (circle: monostatic; cross: multistatic)

revisit time. It is worth noting the sparse and irregular sampling, especially in the monostatic case.

Canopy contribution is modelled in the simulator as originated from a homogeneous non-lossy volumetric scatterer. Also, exploiting the model of the previous section, a long-term temporal decorrelation factor is considered, leading to exponential decaying temporal coherence. In our analyses we consider three different temporal decorrelation conditions : an ideal condition where the scene maintains the full coherence during the acquisition time, a mild temporal decorrelation condition (τ=44 revisit times) and a strong temporal decorrelation condition (τ=2.8 revisit times). The short term decorrelation is not considered here. The canopy volumetric scatterer width is 0.2 Rayleigh resolution units (r.u.), with a centroid elevation over the reference height of 0.4 r.u. For the sake of simplicity it is assumed a uniform vertical volumetric distribution; however it is worth noting that the shape of the volumetric distribution is not a constrain of spatio-temporal model. Ground contribution is considered from an elevation compact speckled scatterer. This scatterer is electrically stable and steady, and its power is 1/5 of that from canopy, which is representative of a L-band acquisition.

The temporal correlation coefficients of the overall signal as a function of the acquisition index are reported in Fig. 2.5aand Fig. 2.5bfor the two mentioned temporal coherences. The elevation of the compact ground scatterer with respect to the reference is -0.3 r.u., thus separation from the canopy centroid is below the Rayleigh limit, amounting to 0.7 r.u. The number of independent looks is 16, and total SNR is 15dB.

The simulated scenario is thus a two-component random volume over ground scattering with both short and long term temporal decorrelation for the volume (canopy).

2.4.1.1 Performance criteria

Here we introduce the two performance criteria used in this work with which the mentioned tomographic methods will be judged and compared. The first criteria is related to the res-olution probability of the method under test. In general, we say that the two scatterers are resolved when both the heights of canopy and ground centroid can be retrieved in the to-mographic profile. In this work, resolution probability is the most relevant as performance criteria since one the main objective is to evaluate the superresolution capabilities of the adaptive and model based method in critical decorrelation condition condition. Moreover,