S
ECTION
1
INTRODUCTION
Imaging radars are active electromagnetic sensors operating from airborne or spaceborne platforms. Through the transmission and reception of electromagnetic energy, they allow to obtain a reflectivity map of the illuminated surface. Among imaging radar, since early 70’s, synthetic aperture radar (SAR) technology has been establishing itself as an essential and powerful tool for mapping surfaces with high resolution capability [Wil54, Sko90, Bam98]. The high range resolution is achieved by frequency modulation of the transmitted pulse, the high azimuth resolution by appropriate coherent combination of several pulses generated during platform motion [Bam98]. SAR sensors operate in the microwaves region of the electromagnetic spectrum, with wavelengths ranging from 1 cm to several meters. The active nature of the sensors allows for night and day operation capability, independently of the weather conditions because of the transparency of the Earth atmosphere in the microwave region. Moreover, wavelengths in L- and P- band can penetrate vegetation and also ground, collecting the backscattering contribute not only from the surface, but also from its different layers. As a consequence, SAR imaging allows for sensing either morphological characteristics or changes in the biomass features, or in the soil conductivity related for example to the moisture, and is finding many applications in topographic mapping, geology, forestry.
provides a two-dimensional (range-azimuth) image of the actual three-dimensional scenario and it is not able to recover the dynamical features of the illuminated scene. Interferometric SAR (InSAR) techniques overcome these limitations by combining interferometry with SAR imaging. In the last step of the whole processing, whereas the conventional SAR technique reduces a magnitude and phase image to a purely magnitude image, InSAR exploits also the phase information contained in the complex images in order to measure elevation (the third dimension) and changes of the backscattering elements. The InSAR technique used to derive three-dimensional elevation models is called cross-track interferometry (XTI-SAR), those used to measure the dynamical features of the scene are denoted as along-track interferometry (ATI-SAR) or differential interferometry (D-In(ATI-SAR), for short- and long-term motions, respectively.
First demonstration of InSAR capabilities for topographic mapping where made by L. C. Graham in 1974 [Gra74], followed in the 80’s by H. A. Zebker, R. M. Goldstein, and A. Gabriel experiments [Zeb86, Gab88]. Since those seminal works, XTI-SAR has established itself as a basic tool to derive digital elevation models, with high spatial resolution and elevation accuracy, finding many applications in radar, remote sensing and sonar, for topographic mapping, geology, forestry [Rod92, Hen98, Ros00]. A conventional InSAR system acquires two complex SAR images from two antennas slightly separated by a (single) cross-track
baseline. Using the phase difference between the echoes collected by the antennas at the
interferometer, the so-called interferometric phase, the elevation angle can be accurately determined for each pixel in the SAR images corresponding to the same area of ground. A basic limitation of conventional XTI-SAR is that it does not provide resolving capability along the elevation dimension, i.e. it is not able to distinguish different sources present in the same range-azimuth cell. This fact means that, when the imaged scene contains highly sloping areas or discontinuous surfaces the technique suffers from the layover phenomenon [Hen98, Wil99]. In these conditions the received signal is the superposition of echoes backscattered from the various patches of terrain that are mapped in the same range-azimuth resolution cell but have different elevation angles (see FIGURE 1.1). The result is that the height map produced by the InSAR system is affected by strong distortions.
FIGURE 1.1 –Geometry of the interferometric system in presence of layover: example with three layover sources. : baseline, : height of the system, θ : elevation angle,
: tilt angle, R: range, : range resolution. x-axes (along-track/azimuth) orthogonal to ( , plane; distance and angles not in scale.
B H
ξ ∆R
) z y
Conventional techniques to solve layover are mosaicing of height maps obtained from opposite look directions, and range Fourier analysis of the complex interferogram to separate layover areas with different slopes [Gat94]. However, multibaseline InSAR, i.e., an interferometric technique using more than two SAR images (see FIGURE 1.2), has the ability to resolve the multiple sources along the elevation angle. Therefore, several approaches based either on beamforming or superresolution techniques have been recently suggested in the literature as an alternative method of layover solution; some examples can be found in [Rös00, Rei00]. Multibaseline layover solution constitutes a way to better exploit existing interferometric data and new experimented or planned interferometric systems [Rös00].
An additional problem is the fact that the backscattering sources cannot be always represented as point-like targets, because of their possible extended nature [Rod92, Hen98]. The backscattered signal is affected by the so-called speckle phenomenon, which under some assumptions can be well modelled as a complex-valued multiplicative stochastic process [Hen98]. To counteract the deleterious effects of the speckle, i.e., the signal decorrelation along
FIGURE 1.2 –Geometry of a multichannel interferometric system in presence of layover: example with three layover sources. K complex SAR images are obtained
from K different phase centres. See FIGURE 1.1 for the legend of symbols.
the baseline and the resulting phase noise, it is often advantageous to process more than one look, combining several observations of the same terrain area [Rod92].
In some past works [Gin02, Lom03, Jak03, Bor04], and also in the present, the layover problem is cast into a spatial signal processing estimation problem. In other words, the estimation of terrain heights or, equivalently, of the interferometric phases, is formulated in terms of estimation of the spatial frequency of multiple complex exponentials corrupted by complex multiplicative noise (the speckle) with unknown spectral shape, embedded in additive white Gaussian noise. The required processing can be divided into two steps:
1. detecting the number of sources, the detection problem or model order selection (MOS) problem;
2. retrieving the parameters of each single signal component like the interferometric phase, the estimation problem.
The presence of the multiplicative noise makes the detection problem very atypical; in fact all the approaches proposed in the literature have been applied to constant amplitude sinusoidal signals. In particular, the information theoretic criteria (ITC) have been conceived to
estimate the number of signal components embedded in white noise. Some problems related to their implementation are addressed in [Lom05], including a system trade-off analysis and a baseline optimisation for detection; the authors also propose the use of diagonally loaded ITC methods as a tool for robust operation in the presence of speckle decorrelation. An original ad
hoc algorithm for MOS, termed Capon-LS, has been described in [Bor04]; it is based on the
intuitive criterion of deciding that a signal source is present if it is more powerful than the additive noise. The method find its mathematical background in the Capon spectral estimator and the least square (LS) method. The numerical analysis demonstrates the effectiveness of Capon-LS for an ideal uniform multibaseline distribution, in terms of estimation accuracy and computational efficiency; the superiority of its performance is noticeable especially with highly decorrelated speckle, even with a small number of looks.
Regarding the estimation problem, in [Gin02, Jak03, Lom03] Capon, APES (amplitude and phase estimation filter), MUSIC (multiple signal classification), RELAX, and WSF (weighted subspace fitting) algorithms were applied or extended to handle multilook data. Among these, the multilook versions of MUSIC, of RELAX RELAX), and of WSF (M-WSF) were found to be preferable, reaching comparable estimation accuracy. Overall, M-WSF generally provides the best performance, at a computational complexity comparable with that of root-MUSIC, which is much lower than that of M-RELAX [Jak03]. All the mentioned techniques are briefly recalled and their results compared also in [Gin05].
The above mentioned approaches show satisfactory performance when an ideal uniform multibaseline configuration, i.e. a uniform linear array (ULA) structure (after deramping), is available [Gin05]. The acquisition geometry affect both the spectral properties and the computational complexity of the estimation methods. In fact, nonuniform spatial sampling produces anomalous strong sidelobes in the beampattern and in the functional of spectral estimation methods [Bor04b], and in presence of multiple sources, a constructive and destructive interaction can produce spurious peaks, masking effects (threshold effect) and peak mislocation. As a consequence, to reach high elevation resolution by applying spectral estimation to nonuniform multibaseline InSAR data could be challenging [Rei00, Lom03b,
For04]. In addiction, a uniform geometry allows for computationally efficient rooting algorithms, also generally yielding superior estimates as compared with their conventional counterparts [Jak03]. More precisely, MUSIC can be implemented by the computationally simple root-MUSIC algorithm, and WSF, whose application for the general array geometry requires a cumbersome multidimensional non-linear search, can be easily realized by the efficient MODE procedure [Sto97]. Unfortunately, in practical multibaseline situations, the ULA structure is rarely encountered. In fact, the acquisition geometry is mostly determined by mechanical, structural or flight/orbital considerations which are not directly related to the estimation requirements; in fact, often such considerations will rather result in a nonuniform sampling.
The practical relevance of the nonuniform acquisition system, and the dramatic effect of nonuniform sampling on most spectral estimation methods, highlight the importance of analyzing different options enabling high elevation estimation performance of multiple components using efficient spectral estimation techniques for nonuniform multibaseline InSAR data. Until now, this problem has mostly been tackled by simple interpolation techniques matched to a single point target, which bring the data back to a uniform grid (see, for example, [Rei00, For03] and the references therein), or alternatively by superresolution or adaptive imaging approaches directly applied to the nonuniform data [Rös00, Lom03b, For04], performing better than beamforming yet still with the computation and accuracy limits previously stressed.
In [Lom03c, Bor05] the use of an interpolated array (IA) approach for multibaseline InSAR has been proposed instead. Using an IA approach, the estimation methods designed for a ULA are applied to a virtual ULA output, obtained from the actual nonuniform one by a linear interpolation transformation which minimizes the error over a sector of interest [Fri93, Bro88]. There are two main reasons for this proposal:
1. to convey in the interpolation mechanism a rough but correctly matched information about the overall backscattering sources location (the sector of interest, SOI, i.e. the spatial bandwidth), which should bring a performance improvement in layover solution.
With well interpolated data, the performance limitations intrinsic in a non ULA geometry should vanish. In a general array processing framework, this fact was originally investigated in [Ger97], where the performance of the interpolated array root-MUSIC algorithm (IA-root-root-MUSIC) is compared with that of the ordinary, ULA-based, root-MUSIC algorithm. It was also found that, asymptotically, IA-root-MUSIC is preferable to root-MUSIC;
2. the IA approach is desirable to enable the use of the spectral estimation algorithms that for ULA structures have demonstrated the best performance in term of accuracy and computational efficiency, i.e. the root-MUSIC and M-WSF algorithms [Gin02, Jak03]. It is worth recalling that WSF is not practically applicable to nonuniform linear array (NLA) data.
In [Bor05] the authors tackle the problem of interferometric phase estimation of multiple sources focusing on the case of advanced or planned single-pass acquisition systems with a low number of phase centers and NLA structure [Rös00]. The array structures under examination were obtained by thinning a full ULA, so that both geometries have the same unambiguous range (UR) for interferometric phase estimation. The analysis of interpolated methods shows that IA methods can be more effective in estimating the multiple interferometric phases than the conventional methods. They also show higher robustness to nonuniform sampling and threshold effects than conventional estimation methods, allowing for performance close to that obtainable by a ULA structure. However, an open issue was the critical trade-off related to the assumed SOI size: good performance are provided if all the sources are contained inside the same SOI, but the SOI size cannot be arbitrarily large, since its width is proportional to the interpolation error. Moreover, the proposed IA approach does not fully exploit all the potentialities offered by the NLA geometry. In fact, the maximum number of virtual ULA elements which can be estimated is equal to the number of the actual NLA elements, because numerical ill-conditioning arises in noise whitening after array interpolation when the virtual array has more elements than those of the real one. This fact causes a reduction of the UR of the virtual ULA with respect to the UR of the existing NLA. More generally, this problem occurs
for every linear interpolator.
The reconstruction of bandlimited signals from their nonuniform samples is the object of several works in literature, because this subject is of great interest in various fields of communication theory. Roughly speaking, the proposed algorithms can be divided into two big categories: iterative methods and non-iterative methods, both exploiting the a priori knowledge of the signal bandwidth.
A deep and unifying mathematical analysis of iterative methods has been carried out at the University of Wien [Cen91, Cen92, Str93]; the algorithms proposed by F. Marvasti [Mar91] and K. D. Sauer and J. P. Allebach [Sau87] are considered as particular cases of the mentioned theoretical analysis [Str93]. At the moment, despite their efficiency in interpolation, iterative methods seem not to be really suitable for the InSAR case. First of all, iterative methods are conceived to work with a relatively large amount of samples; on the contrary, the investigations in the InSAR framework are focused on system with a low number of phase centres, i.e. low number of spatial samples. Secondly, the convergence of iterative methods is demonstrated under the proper selection of some scalar parameters which appear in the algorithms; this selection is not always simple to do. Finally, the noise correlation after interpolation is almost unpredictable. As a consequence, the interpolated noise can not be whitened and this fact could have a bad effect on interferometric phase estimation, because superresolution algorithms such as MUSIC or WSF are derived in presence of spatially white noise [Sto97].
Among non-iterative reconstruction algorithms, the most known is the one derived by Yen in 1956 [Cho00] for continuous-time signals. This method is designed to minimise the distance between the interpolated signal and the original one, under a constraint on the bandwidth, considered as an a priori information about the signal. Yen conceived his algorithm in a deterministic framework; however, several authors reached the same conclusion also considering the interpolation of random processes and using completely different criteria of derivation (see for example [Win92]). More recently, Choi and Munson restated the formulation of the problem of bandlimited signal interpolation within a statistical framework; in particular, in [Cho00] they consider the interpolation as a linear estimation problem and demonstrate that
the usual Yen interpolator is a particular case of an algorithm optimal in a mean square error sense.
In this work, the use of interpolated array for interferometric phase estimation is further investigated, paying again attention to InSAR NLA structure with a low number of phase centres, with the aim of: (i) further improving the performance compared to the algorithm in [Bor05], (ii) investigating SOI selection problems, and (iii) tackling robustness issues in presence of miscalibration of the multibaseline array.
SECTION 2 presents the received data model and states the goal of the work. All the component in the model are characterised in a physical sense and in a statistical sense.
SECTION 3 opens with a brief recall of the classical non-parametric and parametric interferometric phase (IP) estimation methods such as Beamforming, Capon and MUSIC, in order to provide a theoretical background useful for the whole section. After that, the statistical interpolation algorithm proposed by Choi and Munson is applied to the multibaseline InSAR array processing case; then, the performance obtained in IP estimation by root-MUSIC applied to the virtual ULA data reconstructed with both statistical and classical interpolation methods are compared and analysed. To counteract the previously mentioned numerical ill-conditioning showed by all linear interpolation algorithms, the use in the whitening step of a diagonal loading technique is proposed. The role of the SOI (i.e., the spatial signal bandwidth for Choi-Munson) and of loading parameter are deeply nvestigated by means of Monte Carlo simulations. A final performance ranking of the interpolation methods under different grades of optimisation of the SOI is also provided.
In SECTION 4, the InSAR data model is updated to take into account the presence of either the limited accuracy of positioning instruments or estimation procedures measuring the actual flight path or orbits, or calibration errors of the multibaseline array. In the array processing literature, see e.g. [Van02], the most recent [Fer06] and references therein, it is well known that perturbations in the array result in a loss of performance in the estimation of source parameters. In the InSAR framework, an analysis of the impact of this kind of disturbance on radar texture estimation has been pointed out in [Jak05], highlighting the need of robust
estimation algorithms. In that works, it has also been showed that the estimation performance obtained by the robust Capon beamformer is often preferable to both the standard Capon and the beamforming approaches. SECTION 4 aims to investigate how calibration errors can affect the interferometric phase estimation in the NLA case, focusing on interpolated methods. Also, some total least squares approaches are proposed to make the interpolation robust and to increase the efficiency in phase estimation. Moreover, since the calibration errors are considered as random nuisance parameters on phase estimation, the Cramér-Rao bound for Gaussian vectors derived for performance analysis in SECTION 3 is no longer valid. Consequently, in SECTION 4 the hybrid Cramèr-Rao bound as formulated in [Van02] is calculated in the case of interest, accounting for the presence of calibration errors. The effectiveness of the methods proposed for robust interpolation are evaluated by means of Monte Carlo simulations.
