

Equivalence of Different Format Radar Polarimetric Data for Coherency Matrix Estimation
Riccardo Paladini^{(1)}, FABRIZIO BERIZZI^{(2)} and AMERIGO CAPRIA^{(2)}^{(1)} University of Pisa, VIA CARUSO 16, 56122 PISA, Italy
^{(2)} UNIVERSITY OF PISA, VIA CARUSO 16, 56122 PISA, Italy
Abstract
Fully Polarimetric radars allow a well characterization of target scattering properties. In particular, Polarimetric Synthetic Aperture Radar (PolSAR) system is able to produce polarimetric images where a scattering matrix is associated to each image pixel. Two main approaches have been developed to analyze the signal received by polarization diversity radar. The first approach was defined in 1949 by Kennaugh and applied later by Huynen, Cameron, Krogager and recently by Touzi, making use of a measure of the RCS among different polarization states, collected in the scattering matrix format S. By this method, a “partial” target is difficult to deal with in a unique way because of its spacetime variations. The other approach was instead developed by Huynen making use the Muller matrix analysis and later evolved to the Hermetian coherency matrix formalism by Cloude in his PhD thesis.
The target coherency matrix T has been constructed upon a kind of average of the scattering coefficient taken over an opportune domain. By averaging the contribution of different samples, the signal backscattered from every target can be described by its second order statistics. This approach is usefull for the study of coherent and not coherent targets, the properties of every reciprocal target observed by polarization diversity radar can be represented by a 3x3 Hermetian semidefined positive matrix T. Let us try to better explain what it is in mind. A target is called distributed or “partial” when its behavior is not the same along time or space. In other words, a partial target depolarizes the incident elliptical states. These partial “targets” can be considered the outcome of some dynamic process, therefore we may consider the target, to be described by a scattering matrix S at any time instant or point in the space (in case of a temporal or spatial variation).
The central key question that arises in the estimation of target coherency matrix from Ultra Wide Band (UWB) full polarimetric radar data is the following: what kind of averaging is the best to estimate the elements of T, time averaging, space averaging, or combination of them?
The contribution of this paper is to provide a scientific answer to this question. Starting from a historical background a fundamental equivalence between several methods in the estimation of the target coherency matrix T in different domains will be demostrated. Three different techniques have been defined making use of three different UWB radar data 2D domains, namely frequency/slow time (f/tau) signal, High Range Resolution Profiles/slow time (HRR/tau) and High Range Resolution/High Cross Range Resolution images (HRR/HCRR). The final goal is to demonstrate the equivalence of these 3 different methods. In other words, T estimated from radar data, is independent of the radar data used, or better of the kind of averaging operation is carried out on data.
Starting from the signal(f/tau) received by multifrequency UWB full polarimetric radar a first algorithm for the estimation of the T elements has been formulated. When the target motion is slow or locally stationary; the polarimetric range profiles HRRP are obtained as Fourier transformed coefficients of frequency/slowtime data + radial motion compensation and the coherency matrix can be estimated in the transformed HRR/tau domain with a second algorithm. The equivalence between the two approaches has been demonstrated using the BesselParseval Theorem. Finally, if the target is observed by different aspect angles, the High Cross Range Resolution processing can be done with an other Fourier Transform together with motion compensation. This new transformation defines a 2D space domain (HRR, HCRR) where the T elements can be evaluated.
Many authors consider the SAR image(HRR,HCRR) the natural domain for the estimation of target’s features, but this result is not straightforward with the original coherency matrix definition given in the time domain to describe time fluctuations of light. Considering the SAR or ISAR processing the result of an energy invariant transform, is possible to apply the Parseval equivalence to the T coefficients also in this case and extend out equivalence to three different estimation methods.
At this point another question arises: if three kind of processing give the same results, form a theoretical point of view, what is the reason that in the future will justify the use of SAR processing for the target coherency matrix analysis?
With the use of real data, we will demonstrate that radar imaging allows to limit the target analysis in a suitable spatial windows and extract the contribution of a limited target area. This reduction gives a processing gain in the SNR that can be meanly valuated as A/W, where A is the area of the scene and W the area of the averaging window.
There is also another important conclusion, if the SAR image of the target isn't available, unfocused, or the scatters positions are unknown, this equivalence shows that algorithms of target analysis, classification and identification can be developed in equivalent transformed domains.
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