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Electromagnetic Modeling in theSpectral Domain for Polarimetric

Radar ApplicationsTheoretical Framework

Ramin Sabry

Defence R&D Canada

OttawaTECHNICAL REPORT

DRDC Ottawa TR 2004-115

August 2004

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Electromagnetic Modeling in the SpectralDomain for Polarimetric RadarApplicationsTheoretical Framework

Ramin SabryDefence R&D Canada Ottawa

Defence R&D Canada OttawaTechnical ReportDRDC Ottawa TR 2004-115August 2004

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Her Majesty the Queen as represented by the Minister of National Defence, 2004

Sa majest la reine, reprsente par le ministre de la Dfense nationale, 2004

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DRDC Ottawa TR 2004-115 i

Abstract

A vectorial Fourier-based technique for electromagnetic (EM) wave reconstruction withapplication to polarimetric airborne and spaceborne radar data exploitation is presented. The

approach provides a comprehensive picture of scattering through an electromagnetic-basedformulation of equivalent sources and introduction of vector scattering functions. The methodis different from conventional modeling techniques for SAR applications as result of the fullelectromagnetic treatment of field interaction with the scatterer, the possibility of introducingnew and controllable feature classes for target classification, and the accurate decompositionof source impulse response function that avoids potential errors (e.g., loss of coherentinformation) associated with the spherical phase approximations. In addition to including thepotential for high quality image formation, the capability of extracting target or scattererinformation, such as coherent radar cross section (RCS) is explored. It is highlighted howtarget image and information processing can be optimized and tailored for a desired scenario(e.g., the required information type, the nature of the targets) by applying the developedmodel and devising appropriate inversion techniques to aid data processing in support of

exploitation.

Rsum

Le document prsente une technique vectorielle transforme de Fourier permettant dereconstituer des ondes lectromagntiques (EM) par l'application l'exploitation des donnesde radars polarimtriques aroports et spatiaux. Cette technique procure une image globalede la diffusion par la formulation de sources quivalentes, selon des principeslectromagntiques, et le recours des fonctions vectorielles de diffusion. Elle diffre destechniques classiques de modlisation des applications SAR en raison du traitementlectromagntique complet de l'interaction des champs avec le diffuseur, de la possibilitd'tablir des ensembles de caractristiques nouveaux et contrlables pour la classification descibles et de la dcomposition prcise de la fonction de rponse impulsive des sources, quilimine les erreurs possibles (p. ex. la perte d'information cohrente) lies aux approximationsde phase sphrique. Outre la formation d'images de grande qualit, la possibilit d'extraire del'information sur les cibles ou les diffuseurs, comme la surface quivalente radar (SER)cohrente, est galement examine. Le document fait ressortir comment le traitement desimages des cibles et de l'information sur les cibles peut tre optimis et adapt en fonctiond'un scnario dsir (p. ex., le type de l'information requise et la nature des cibles) parl'application du modle labor et la conception de techniques d'inversion appropriesfacilitant le traitement des donnes l'appui de l'exploitation.

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DRDC Ottawa TR 2004-115 iii

Executive summary

Polarimetric radar or SAR (Synthetic Aperture Radar) detection is a powerful remote sensingtechnique for potential military applications concerning target recognition, identification and

classification. The polarimetric approach is potent as it effectively explores radar targetfeatures (e.g., structure, change) by making use of the full vector properties of theelectromagnetic (EM) field while interacting with the target scatterers. Accordingly,exploitation of polarimetric radar data, using appropriate algorithms, results in an enhancedtarget detection and classification capability compared to single channel, single polarizationdata exploitation. Such improved functionality is desired for consistent target detection,reduction of misclassification probability, reduction of false alarm rate (FAR), and reliabletarget recognition.

It is evident from the characteristic description of radar polarimetry that the analysis of targetscattering ought to be based on electromagnetic theory. A full electromagnetic approach tomodeling the scattered vector fields and characterize the targets introduces new challenges

and requirements for more rigorous algorithm development. On the other hand, the theoreticalframework cannot be very complicated from a practical application point of view. Theoptimal trade-off between these two aspects, i.e., accuracy and feasibility, is an importantissue that has not been systematically addressed in the literature and is the underlyingmotivation for this work.

Considering the diversity of features characterizing a radar or SAR scenario for remotesensing (e.g., application types, required analysis types, target and clutter types), developmentof a unique optimized approach to model the scattering for polarimetric applications is a hugeif not insurmountable task. The intention is not to accomplish such here as this is not realistic.Instead, the objective is to construct a theoretical framework capable of accurately modelingthe process of vector EM field interaction with the target scatterer. This general formulation

will be tailored based on the requirements for the application classes of interest and will beapplied for simulations and experimental purposes in later works. The contributions of thiswork can be viewed in two folds:

The first contribution is a thorough survey of the literature in polarimetric remote sensing formodeling of EM scattering and characterizing the target scatterer. The outcome signifiesnature, objectives, required deliverables, major classes and capabilities/limitations of theexisting analytic techniques for radar target analysis. In parallel, a study is performed toidentify robust and efficient methods to model scattering and diffraction (as an EMphenomenon) that belong to different areas within the electromagnetic realm (e.g., microwavestructures) but have similar theoretical foundations. The overall objective from the aboveinvestigations is to derive a set of key requirements for the modeling formulation, and at the

same time, propose a feasible implementation of the associated algorithms.The second contribution of this work is the development of a theoretical approach to modelthe target scattering (based on the key requirements identified) that is different fromconventional methods in two ways. One aspectis the rigorous analysis (usually lacking intypical analyses) and application of source Greens function through a four-dimensional (i.e.,

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space-time) Fourier decomposition. When applying a coherent approach for targetidentification, this accurate spectral treatment of Greens function with respect to both phaseand amplitude provides the means for more accurate modeling of EM field scattering by thetarget scatterer. The other aspectis the representation and/or classification of target scatterersby vector current functions. This approach, which is based on the concept of equivalentsources in electromagnetic theory, is close to the roots of EM scattering governed by

Maxwells equations. Technically, the latter is on the same basis as the conventionalscattering matrix analysis due to formulation of scattering-induced sources. However, thepresent approach is novel (to the best of authors knowledge in the field of polarimetric SARscattering modeling for target classification) and appealing from the following points of view.

To classify the target scatterer, the associated vector current function in the space or spectraldomain is decomposed into a set of basis functions that can be viewed as features. It ispossible to perform this decomposition using different basis sets where each set represents aclass of features or sub-features. Considering the generic nature of problem formulation,efficient techniques may be adopted from other areas (e.g., RF, microwaves) for such targetcharacteristic function decompositions. As can be seen, there exist degrees of freedom and thepossibility of defining various basis sets and number of members or basis functions for each

set. The described potential of controlling the nature and number of features for targetdecomposition based on the application requirements, alleviates the misclassificationproblems associated with coarse classification/segmentation encountered through currentcoherent target decomposition techniques. It is also conceivable that study of the inducedequivalent current sources behavior on the scatterer, yield a more consistent and tangibleunderstanding of the scatterers physics. Finally, the source functions derived through thepresent approach may be readily applied to calculate the associated EM field quantities (suchas coherent radar cross section) in both near and far field zones.

The present methodology provides a core theoretical framework for polarimetric SARmodeling. The approach is general and should be tailored by investigating the SAR scenarioof interest and designing a suitable structure for required feature classes. A significant

advantage is the possibility of introducing various target classification or basis sets to the coreformulation. Moreover, is the opportunity of adopting robust and well-proven techniques(e.g., method of moments, image theory) from other EM areas to obtain the optimum (i.e.,best target representative) basis sets and associated decomposition mechanisms. The futurework will be directed towards model or sub-model development for different types ofapplications. Various EM-based numerical techniques should be studied and properly tailoredfor desired applications. Convenient experimental schemes must still be designed to validateand tune the model.

Sabry,R. 2004. Electromagnetic modeling in the spectral domain for polarimetric radarapplications. DRDC Ottawa TR 2004-115. Defence R&D Canada Ottawa.

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Sommaire

La dtection par radar polarimtrique ou SAR (radar synthse d'ouverture) constitue unepuissante technique de tldtection pour les applications militaires possibles lies la

reconnaissance, l'identification et la classification des cibles. La technique polarimtriques'avre particulirement efficace du fait qu'elle examine avec soin les caractristiques descibles radar (comme les structures et les changements) en tirant parti de toutes les propritsvectorielles des champs lectromagntiques (EM) durant leur interaction avec les diffuseursde cibles. L'exploitation des donnes de radar polarimtrique au moyen d'algorithmesappropris permet donc d'amliorer la dtection et la classification des ciblescomparativement l'exploitation des donnes avec canal unique et polarisation unique. Cetteamlioration est souhaitable pour la dtection uniforme des cibles, la rduction de laprobabilit d'erreurs de classification, la rduction du taux de fausses alarmes (TFA) et lareconnaissance fiable des cibles.

D'aprs la description caractristique de la polarimtrie radar, il est vident que l'analyse de la

diffusion des cibles devrait se fonder sur la thorie lectromagntique. Une techniquetotalement lectromagntique de modlisation des champs vectoriels diffuss et decaractrisation des cibles suscite de nouveaux dfis et impose de nouvelles exigences en vuede l'tablissement d'algorithmes plus rigoureux. Par ailleurs, le cadre thorique ne peut pastre trs compliqu du point de vue de l'application pratique. Le compromis optimal entre cesdeux aspects, soit la prcision et la faisabilit, reprsente un facteur important qui n'a pasencore t trait de faon systmatique dans la documentation publie et qui constitue lamotivation sous-jacente ces travaux.

tant donn la diversit des caractristiques des radars ou des scnarios SAR de tldtection(p. ex., types d'applications, types d'analyses requises et types de cibles et de clutter),l'tablissement d'une mthode optimise unique permettant de modliser la diffusion des

applications polarimtriques constitue une tche immense, voire un obstacle insurmontable. Ilserait irraliste de tenter l'adoption d'une telle mthode. L'objectif consiste plutt tablir uncadre thorique permettant de modliser avec exactitude le processus d'interaction des champsEM vectoriels avec le diffuseur de cible. Cette formulation gnrale sera adapte auxexigences des classes d'applications d'intrt et sera applique dans des travaux ultrieurs des fins exprimentales et de simulation. Les contributions de ces travaux sont doubles.

La premire contribution prend la forme d'un examen approfondi de la documentation entldtection polarimtrique pour la modlisation de la diffusion EM et la caractrisation dudiffuseur de cible. Le rsultat fait ressortir la nature, les objectifs, les produits livrables requisainsi que les principales classes et capacits/limites des techniques analytiques existantes pourl'analyse des cibles radar. Paralllement, une tude vise dterminer des mthodes robustes et

efficaces de modlisation de la diffusion et de la diffraction (en tant que phnomnes EM)ayant trait diffrents aspects de l'lectromagntisme (p. ex. les structures hyperfrquences)mais partageant des fondements thoriques similaires. Ces recherches ont pour objectifgnral d'tablir un ensemble d'exigences cls pour la formulation de la modlisation et,simultanment, proposer un moyen de mise en uvre des algorithmes connexes.

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titre de seconde contribution, ces travaux permettent d'laborer une mthode thoriqueservant modliser la diffusion des cibles (d'aprs les exigences cls tablies), qui diffre desmthodes habituelles deux gards.En premier lieu, cette diffrence tient l'excution d'uneanalyse rigoureuse (d'une rigueur suprieure celle des analyses typiques) et l'application dela fonction de Green pour la source, par une dcomposition de Fourier quadridimensionnelle(espace-temps). Si l'on recourt une mthode cohrente d'identification des cibles, ce

traitement spectral prcis de la fonction de Green par rapport la phase et l'amplitudepermet une modlisation plus prcise de la diffusion du champ EM par le diffuseur de cible.En second lieu, cette mthode se diffrencie par la reprsentation et/ou la classification desdiffuseurs de cibles au moyen de fonctions vectorielles de courant. Une telle faon deprocder, fonde sur le principe des sources quivalentes dans la thorie del'lectromagntisme, se rapproche des lments de la diffusion lectromagntique rgis par lesquations de Maxwell. Sur le plan technique, elle se trouve sur le mme pied que l'analysematricielle classique de la diffusion, cause de la formulation de sources induites par ladiffusion. La mthode actuelle est toutefois novatrice (autant qu'en sache l'auteur dans ledomaine de la modlisation de la diffusion par SAR polarimtrique pour la classification descibles) et intressante pour les raisons dcrites ci-dessous.

Pour la classification des diffuseurs de cibles, la fonction vectorielle de courant connexe dansle domaine spatial ou spectral est dcompose en un ensemble de fonctions de base pouvanttre considres comme des caractristiques. Il est possible d'effectuer cette dcomposition l'aide de diffrents ensembles de base, chaque ensemble reprsentant une classe decaractristiques ou de sous-caractristiques. Compte tenu de la nature gnrique de laformulation du problme, il est possible d'adopter des techniques efficaces issues d'autressecteurs (RF, micro-ondes, etc.) pour effectuer de telles dcompositions de fonctionscaractristiques des cibles. Comme on peut le constater, il existe des degrs de libert et lapossibilit de dfinir divers ensembles de base ainsi qu'un certain nombre de membres oufonctions de base pour chaque ensemble. La possibilit dcrite de dterminer la nature et lenombre de caractristiques de dcomposition des cibles en fonction des exigences del'application attnue les problmes de classification lis la classification/segmentation

approximative, qui sont typiques des techniques actuelles de dcomposition cohrente descibles. On peut aussi concevoir que l'tude de l'incidence d'une source de courant quivalentinduit sur le diffuseur procure une comprhension plus uniforme et tangible de la physique dudiffuseur. Enfin, les fonctions de source lies la mthode actuelle peuvent s'appliquerdirectement au calcul des valeurs des champs EM connexes (comme la surface quivalente deradar cohrent), tant en champ proche qu'en champ lointain.

La prsente mthode fournit un cadre thorique central pour la modlisation par SARpolarimtrique. Cette mthode est gnrale et devrait tre adapte par l'examen du scnarioSAR d'intrt ainsi que par la conception d'une structure approprie applicable aux classes decaractristiques requises. Un avantage significatif tient la possibilit de recourir diversesclassifications de cibles ou diffrents ensembles de base pour la formulation centrale. Il est

galement possible d'adopter des techniques robustes ayant fait leurs preuves (mthode desmoments, thorie des images, etc.) dans d'autres secteurs de l'EM afin d'obtenir les ensemblesde base optimaux (le plus reprsentatifs des cibles) et les mcanismes de dcompositioncorrespondants. Les travaux futurs viseront l'tablissement d'un modle ou d'un sous-modlepour les diffrents types d'applications. Diverses techniques numriques fondes surl'lectromagntisme devraient tre tudies et adaptes correctement aux applications

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dsires. Il reste encore concevoir des structures exprimentales appropries pour valider etraffiner le modle.

Sabry,R. 2004. Electromagnetic modeling in the spectral domain for polarimetric radarapplications. DRDC Ottawa TR 2004-115. R & D pour la dfense Canada - Ottawa.

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Table of contents

Abstract........................................................................................................................................ i

Executive summary ................................................................................................................... iii

Sommaire.................................................................................................................................... v

Table of contents ....................................................................................................................... ix

List of figures ............................................................................................................................ xi

1. Introduction ................................................................................................................... 1

2. General electromagnetic formulation............................................................................ 43. Polarimetric radar EM model ........................................................................................ 6

3.1 Field solution.................................................................................................... 6

3.2 Equivalent sources............................................................................................ 8

3.3 Radar field model........................................................................................... 10

4. Spectral domain modeling........................................................................................... 14

4.1 Fourier analysis .............................................................................................. 14

4.2 2-Dimensional slant plane Fourier analysis.................................................... 20

5. Fourier properties and inversion considerations.......................................................... 24

5.1 Bandwidth properties...................................................................................... 24

5.2 Inversion......................................................................................................... 32

6. Coherent radar cross section (RCS) analysis............................................................... 35

7. Conclusion................................................................................................................... 37

8. References ................................................................................................................... 40

9. Annexes ....................................................................................................................... 42

9.1 Annex A: Reference frame unit vector transformation .................................. 42

9.2 Annex B: Principle of stationary phase .......................................................... 43

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List of acronyms and initialisms............................................................................................... 45

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DRDC Ottawa TR 2004-115 xi

List of figures

Figure 1. (a) Radar-Target coordinate system. (b) Observation projection in ground plane withrespect to ground target distribution.................................................................................. 39

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DRDC Ottawa TR 2004-115 1

1. Introduction

Radar polarimetry is a fast growing and powerful remote sensing technique. It is anticipatedthat the majority of new radars will have polarimetric or scattering matrix capability [1]. The

strength of the polarimetric approach is due to many important and key features of radartargets being vectorial in nature [2]. Hence, making use of the inherent vector property ofelectromagnetic field (EM) interactions with the scatterer, results in a more comprehensivepicture of the target. There exists a considerable volume of fundamental work in classicalphysics that explores different aspects of electromagnetic wave scattering and diffraction byvarious objects that support the polarimetric approach. There is also significant researchliterature on polarimetric applications and their advantages for active microwave remotesensing [3]-[6].

Polarimetric detection in radar technology is used to create high-resolution images [7]-[9] andprovide better target detection capabilities in comparison with non-polarimetric or singlechannel operation [10]-[13]. The latter, which amounts to the analysis of a detected signal to

determine the EM characteristic (i.e., the EM signature) of the target for target identification,is an important radar application. To properly correlate the received EM signal(s) with aparticular object or scatterer (or combination of scatterers), an EM-based formulation ofscattering and propagation should be applied.

An important and early insight to radar polarimetry for target detection [14] was to represent atarget as a polarization transformer and to express its coherent properties with a 22 scattering matrix. This concept was developed by many researchers [15] and entered a newphase for systematic coherent radar detection with Huynens work on radar targetphenomenology [16]-[17], which led to target decomposition techniques. These techniquesrepresent a key theoretical approach to target recognition in current radar imagery. Theessence of target decomposition theory is to express the average scattering mechanism in

terms of independent scattering elements and to associate each of those with a known physicalmechanism or scatterer [18]. The described association can be on a coherent target detection(CTD) [17], [19]-[20], or on a partially coherent/incoherent target detection (PCTD) basis[17]-[18]. In a CTD methodology, the scattering matrix is fully polarimetric. In a PCTDapproach, however, the covariance or coherency matrices contain the polarimetric informationof partially polarized scattered waves. One widely used CTD technique is Cameronselemental decomposition of maximum symmetric scattering into scatterer classes of trihedral,dihedral, dipole, cylinder, narrow diplane and quarter wave for each resolution cell [20].There are certain inherent problems associated with Camerons classification for polarimetricSAR (Synthetic Aperture Radar) applications, including coarse segmentation and erroneouseffects due to scattering segmentation of non-coherent scattering (under a coherentassumption). To alleviate these problematic effects, new methods have been proposed to

improve the segmentation process and to make use of coherence assessment to validate thecoherent nature of scattering for the target under investigation [21]. The incoherent techniques(i.e., PCTD) provide physical insight to the nature of a partially coherent scattered field andthe degree of polarization through Kennaugh, Mueller, covariance or coherency matrixdefinitions [17]. These coherency indicators provide valuable information on the nature andclass of the scatterer [22].

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In the present work, the main objective is to establish a relatively general and accurateframework for modeling of EM propagation and scattering for airborne or spacebornepolarimetric applications. The theoretical approach is different from conventional techniquesas it combines a multidimensional Fourier analysis (i.e., decomposition) of space Greensfunction and inverse analysis of scattering using the EM theory and theprinciple of equivalentsources. Due to Fourier transformation of the exact source impulse response, coherent

information is included in the formulation. The spectral nature of the analysis also providesinsights into the Doppler effects through the space-frequency mapping mechanism betweenscatterer and observation. In addition, the forward nature of the model is advantageous for thepurpose of target identification.

The other novel aspect of the current method is the classification of target scatterers throughthe introduction of vector current functions. The latter, which are based on the concept ofscattering-induced equivalent sources in electromagnetic theory, is close to the roots of EMscattering governed by Maxwells equations. To classify the target scatterer, the associatedvector current function in the space or spectral domain is decomposed into a set of basisfunctions that can be viewed as features. Various basis sets may be used for the requireddecomposition with each set representing a class of features or sub-features. Due to the

generic nature of this formulation, well-known analytic and numerical techniques can beadopted from other areas (e.g., RF, microwaves) for such characteristic function projectionsinto the desired feature space. The possibility of defining various basis sets and number ofmembers or basis functions for each set, indicates the potential of controlling and/or designingthe nature and number of features for target decomposition. Such capability can be used tolower the misclassification probability associated with coarse classification or segmentationencountered through current coherent target decomposition techniques. Furthermore, study ofthe induced equivalent current sources behavior on a scatterer, can yield a more tangibleunderstanding of the scatterers physics. Finally, the source vector functions in the presentformulation may be readily applied to compute the associated EM field quantities (such ascoherent radar cross section) in both near and far field zones.

An important issue considered throughout this work and future related works is to identify andexamine various aspects of target recognition and imaging via their contributions in theanalytic model. Upon identification and classification of these analytic entities or blocks,comparisons can be made with the experimental results and other techniques (e.g.,simulations) for the respective observables to validate the model. Furthermore, potentialexisting analytical tools suitable for characterization of a specific block may be applied ormodified. One important category is the target scattering information with key observablessuch as the scatterers EM signature and its radar cross section (RCS).

According to the described goals and structure of present methodology, one may outline theoverall objectives as follows:

I. To develop a general Fourier-based formulation of electromagnetic propagation andscattering for polarimetric SAR applications;

II. To extract target scattering characteristics from the model and connect them toconventional features, e.g., the scattering matrix and the scatterers RCS;

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III. To investigate inversion techniques in the spectral domain and potentials oftailoring/optimizing them for certain types of applications;

IV. To study the feasibility of a systematic target classification and/or decomposition inthe spectral domain;

V. To explore frequency domain selective techniques for adaptive and targeted scatteringanalysis;

VI. To study the effects, significance and feasibility of an optimum collection path designby using the EM model; and

VII. To explore the possibility of a multi-incident-angle analysis for target recognitionenhancement.

Achievement of these objectives is a substantial task. However, these goals signify theresearch units or blocks to be investigated throughout the present and future work.Formulation of EM scattering in the Fourier domain (I), which is at the core of theinvestigation, is presented, as well as certain considerations on scattering characteristics (II)and potential inversion techniques (III). A more in-depth EM analysis of coherent scattering(e.g., RCS) and optimization of inversion techniques will be included in a later report. Targetclassification and decomposition in the spectral domain (IV), potential adaptive techniques inthe spectral domain based on the state matrix and an analytic interpolation formulation [23]-[24] to enhance target recognition (V), collection path design considerations (VI) and, multi-look analysis of scatterer (VII) using the spectral scattering model are under study and will bereported in future communications.

Governing electromagnetic field equations which are at the core of any EM field analysis aredescribed in the next section. The polarimetric model is formulated in Section 3, upon theintroduction of equivalent sources and derivation of the scattered EM field due to the inducedsources. The general relation between the described scattering-induced sources and scatteringmatrix parameters are also established in Section 3. Section 4 is devoted to the spectralanalysis of scattering through Fourier decomposition of exact point source Greens function.In particular, the important system kernels are derived that relate the polarimetric SARobservations to the spectral distribution of scatterer. Bandwidth properties are investigated inSection 5. In this section, a procedure to map the windowing effect from space to frequencydomain is analytically detailed that provides an invaluable tool to investigate effectivebandwidth and resolution. It is also highlighted how the target distribution in Fourier spacemay be explored to examine various radar incident and aspect angles by virtue of thedescribed mapping. The RCS derivations based on the knowledge of target vector distributionare presented in Section 6. Section 7 includes concluding remarks and direction for futureresearch.

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2. General electromagnetic formulation

Electromagnetic fields in a linear, isotropic and inhomogeneous medium are given by theMaxwells equations:

)1(02

srcr

r

jk JEE

=

)2(2

=

r

srcr

r

k

JH

H

In (1)-(2), HE , are the complex electric and magnetic fields with time dependence tje ,

is the temporal frequency, r and r ( 0 and 0 ) are the relative (free space) permittivity

and permeability,c

k = is the free space wavenumber, c denotes the speed of light, and

srcJ represents all the current sources within the medium of interest. The differential operator

denoted by is the Nabla or Hamilton operator.

For scattering analysis, it is convenient to convert the above equations into relations for an

equivalent scattering problem in the homogeneous medium through introduction of equivalentinduced polarization sources [25]-[26]. Using theprinciple of volumetric equivalence, (1)-(2)can be written as:

)3()( 02

KJEE = jk

)4()( 02

KJHH jk =

with the equivalent electric and magnetic current sources given by:

)5()1(0 EJJ += rsrc j

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)6()1(0 HK = rj

Equations (3)-(6) are general and provide a solid theoretical point of reference for EMpropagation and scattering modeling for radar (polarimetric in particular) applications. Acomprehensive description of scattering can be obtained by satisfying the boundary conditionsand/or identifying the sources for each scenario.

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3. Polarimetric radar EM model

3.1 Field solution

To develop a scattering model for polarimetric radar applications, an electromagnetic theoryis employed. Maxwells equations (3)-(4) are governing equations for the total electric and

magnetic fields which include both the incident ),( ii HE and scattered

),( ss HE fields. For polarimetric radars, a convenient scattering equation can be written forthe electric field as:

)7()( 02

ss

ss

jk KJEE =

In derivation of (7), we note that the incident fields satisfy the vector Helmholtz equations:

)8(022 =+ ii k EE

)9(022 =+ ii k HH

Note that vectors sJ and sK in (7) are electric and magnetic currents that represent

equivalent induced sources due to scattering.

Although modeling of the above equivalent sources introduces a challenge for certainapplications, the approach provides a robust theoretical tool for the analysis of scattering.Another attractive feature is the possibility of incorporating medium effects in definition ofthe equivalent sources (i.e., in equations (5)-(6)).

A general solution to Maxwells equations (e.g., (7)) for the scattered field at the observationpoint r , can be found by taking advantage of the well-known three-dimensional Greensfunction,

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)10(4

)(r

r

r

jke

G

=

as:

[ ] [ ] )11()(1

),( 2

0ss

s

rec GGkj

KJrE +=

where the asterisk (*) denotes three-dimensional space convolution and, as before, the time

dependence is tje .

Making use of the properties of the free-space Greens function, the far-field expression (i.e.,the radiation gauge) becomes:

)12())(()(),(0

2

TsTs

sr

rec GGj

kKJrE =

where the subscript T denotes the transversal component in a plane orthogonal to the

direction of radiation or line of sight (LOS). The radiation field in (12) can also be written as:

[ ] )13(})()({),( rKrrJrE += sssr

rec GGjk

where0

0

0

== k is the intrinsic impedance of the medium (free space) and r is

the unit vector in the direction of radiation (i.e., observation).

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3.2 Equivalent sources

There is no constraint on the type of induced current sources in (13); in general they can bevolumetric. For scattering analysis in radar applications, however, it is convenient toformulate the problem in terms of surface distributed sources. Valid electromagneticformulations of such boundary-value problems provide a comprehensive picture of scattering[25]-[28]. The main objective of these formalisms is to derive the overall field (initial andinduced or scattered) by solving respective integro-differential equations for fields and sourcedistributions when the required boundary conditions are enforced. It is crucial to use properboundary conditions for integro-differential formulations in order to produce unique fieldsolutions. In particular, care should be taken in defining equivalent sources when fields arenon-vanishing infinitesimally close to the boundary surface. The latter may cause erroneousresults even for infinitesimally thin layers of p.e.c. (perfect electric conductor) or p.m.c.(perfect magnetic conductor).

As was addressed, the fundamental theorem of electromagnetics generally known as theequivalence principle is a powerful concept that is rich in physical insight and providestheoretical ground for relating the fields to scattering through definition of equivalent induced

sources. The concept of equivalent or substitute sources is a well known approach that can betraced back to Huyghens [29]. This concept was further developed by Schelkunoff [30],whose equivalence principle asserts that use of surface equivalent sources derived from theoverall field values produces true fields at all points outside the surface and null fields inside.The StrattonChu formulation [31] provides the same result when it is used to express theelectric and magnetic fields in terms of surface integrals [32]. One can apply the describedsurface equivalence principle for target scattering modeling when the necessary boundaryconditions are satisfied.

Assume S is the scattering surface of the target or scatterer with electromagnetic fields totE

and totH uniquely defined at this surface. Employing Schelkunoffs equivalence principle, we

may write the electric and magnetic currents as:

)14()()()( stotssss rHrnrJ =

)15()()()( stotssss rErnrK =

where )( ss rn denotes the unit (outward) normal vector to the surface S at scattering point

sr . The total fields totE and totH at any point of observation space can be considered as the

sum of the incident fields, i.e., the electromagnetic fields when there is no scattering, plussecondary or scattering fields:

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)16()()()( rErErE sitot +=

)17()()()( rHrHrH sitot +=

In general, one can set:

)18()()]([)( rErSIrE iEtot +=tt

)19()()]([)( rHrSIrH iHtot +=tt

where ESt

and HSt

are dyadic quantities representing general electric and magnetic field

scattering, and It

is the unitary dyadic.

In theory, one may define the equivalent sources at any arbitrary surface S in space as:

)20(})()]([{)()( si

sHssss rHrSIrnrJ +=tt

)21(})()]({[)()( si

sEssss rErSIrnrK +=tt

The sources in (20)-(21) produce true fields outside the surface S and a null field inside.

Defining the general scattering from electric incident to magnetic scattering HEYt

with:

)22(iHEs EYH =

t

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one may recast (20) as:

)23()()()()()(

+= s

i

sHEs

iissss rErYrE

rrnrJ

t

In (23), ir is a unit vector specifying the propagation direction of the incident EM wave

),( ii HE . Application of (21) and (23) in (13) yields an expression for the scattered or the

echoed field at the space-time observation point ),( tPrec r .

At this stage, no particular form is assumed for ESt

, HSt

or HEYt

which are simply reflecting

the general scattering mechanism, i.e., the difference between the overall field and the initialor incident field. As will be discussed, however, these quantities can be directly related to thepolarimetric scattering matrix for a backscattering scenario.

3.3 Radar field model

By defining the right-handed radar system unit vectors ),,( irvh , the known incident fieldpolarization at any point in space is represented by:

)24())(sin())(cos()( vrhrru +=hv

where )(r denotes the polarization angle. That is:

)25()()()( rurrE hvii E=

Use of (25) in (21) and (23) yields:

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)26(])[( ihvEss EuSInK +=tt

)27(

ihvHEhv

iss E

+= uYu

rnJ

t

All quantities in (26)-(27) are computed at the scattering point )( sr . The vector terms in (26)-(27):

)28(])[( hvEs uSInW +=tt

)29(

+= hvHEhv

is uYu

rnF

t

can be viewed as (and hereafter referred to as) a polarimetric radar target function.

Some considerations must be made before completing the present discussion. The target

functions in (28)-(29) represent the normalized equivalent sources due to scattering. As aresult, these vectors contain the coherent scattering information and can be analyzed toestimate the scattering matrix. In fact, for a backscattering scenario where targets are modeledas reflectors, the overall scattering mechanisms in (28)-(29) reduce to conventionalpolarimetric scattering quantities. Since all components of these equations are known, exceptthe scattering quantities, target reflection functions (or equivalent source functions) can beconnected to the scattering parameters. The latter provides a means for studying scatteringcharacteristics and other target properties, such as the coherent RCS, within the same analysis.

Although possible to handle analytically, throughout the rest of this analysis magnetic currentsources (28) will be neglected (i.e., we take the electric reflector assumption). It will also beuseful to consider the following. If the scattering matrix for the electric field in a transversal

plane (or slant plane) is defined by:

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)30(ii

v

i

h

vvvh

hvhh

s

v

s

h

E

E

SS

SS

E

EES =

=

t

and if,

)31(ii

v

i

h

vvvh

hvhh

s

v

s

h

E

E

YY

YY

H

HEY =

=

t

one gets (see (8)-(9)),

)32(1

2 SYtt

=

where

)33(01

102

=

is a Pauli spin matrix.

Making use of (13), (27) and (29) gives the expression for the scattered field:

[ ] )34()(),( rrFrE = GEjk isrrec

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The convolution in (34) is performed over the coordinates of the scatterer. Hence, the integralform of (34) using (10) can be written as:

)35(||4)(),(

||

=

s

s

jkisr

rec d

e

Ejk

s

rrrrrFrE

rr

The integration in (35) is a three-dimensional volume integral whereas the equivalent sourcesare twodimensional surface currents. This, however, does not introduce inconsistency or lossof generality if the currents are properly defined. For instance, if the current surface is definedby a normal vector )( srn and coordinates ),,( 21 ntt xxx , the 3-dimensional source

distribution becomes )(),,( 21 nsnntt xxxxx J . In the latter, )(x represents the Diracdelta function.

For convenience, henceforth, the target or scatterer coordinates are denoted by r , theobservation coordinates by u , and the unit radial vector by u . Moreover, the transversalnormalized current rrF (or uuF ) is represented by TF . Accordingly:

)36(||4

),(||

rru

FuEru

de

Ejk jk

T

isr

rec =

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4. Spectral domain modeling

Multidimensional Fourier analysis of the backscattered field (36), provides valuable insight tothe nature of the scatterer. Prior to such a transformation, certain considerations are made.

In the present formulation, the centre of spatial coordinates is assumed to be the centre oftarget or scatterer coordinate system, i.e., the target is centred at the origin of the coordinate

system )0,0,0(),,( == zyxr . The incident field )( iE effects are evaluated at the targetposition r whereas the signal or wave originates from the radar position u . In order toaccount for this delay, one may consider the incident field (at the target) as:

)37()( || ur= jkii eEE

where )(iE is the original transmitted waveforms spectrum. Thus,

)38(||

),()(4

),(||2

rru

rFuEru

de

Ejk kj

T

isr

rec =

Furthermore, assuming the amplitude variations associated with Greens function (far field)for a target area are negligible, one may use:

)39(|)|(),()(4

),( ||2 urrFuE ru == udeEujk kj

T

ifar

rec

4.1 Fourier analysis

The general three-dimensional (observation) space Fourier transform of (36) is given by:

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)40(),(~

),(~

),(~

uuuTusr

rec G kkFkE =

where ),,( uzuyuxu kkk=k is the spatial frequency or wave vector, and

)41()(4

),(~

222

2 222

zyx

ujkujkujk

zyx

uuukj

i

uu dududueeeuuu

eE

jkG zuzyuyxux

zyx

++

++

=

k

)42(),(~

),(~

dzdydxeee zjkyjkxjk

TTzyx = rFkF

The integration in (41) is performed over the observation space whereas the integrations in(38) and (42) are performed over the target or scatterer coordinates. As can be seen from (40)-

(42), the three-dimensional space frequency mapping is represented through kk =u .

In order to relate the observations (in either the spatial or the spectral domain) to the Fouriercharacteristics of the scattering source, a proper Fourier decomposition of the Greensfunction in (38) is required. It is possible to handle the required derivations by using theprinciple of stationary phase (PSP), as described in Appendix B. To explore more physicalinsights and a potential slant plain analysis for SAR applications, however, the exact Fourieranalysis is carried out on the point source Greens function in separate steps for thetransversal plane and the longitudinal (or vertical) dimension.

The source response function in (38):

)43(4

)(2

rr

r

kj

u

eG

=

satisfies the scalar Helmholtz equation:

)44()()()4( 22 rr =+ uGk

Hence, the three-dimensional Fourier transform satisfies the following equation:

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)45(1),,(~

)4( 2222 =++ zyxuzyx kkkGkkkk

The polar spectrum of the Greens function can be obtained by the inverse Fourier integrationover the complex zk plane. By applying the residue theorem, one gets:

)46(42

||4exp),,(

~22

22

=

k

zkjjzkkG yxu

where ),( are the spectral polar coordinates of ),( yx kk , i.e.,

)47()cos( akx =

)47()sin( bky =

)47(222 ckk yx =+

Next, we apply (46) in (38) when making use of the generalized Parsevals theorem to obtain:

(

)48(

4

||4exp),,,(

~)(),(

22

22

dzdkdkeek

zukjzkkE yx

ujkujkz

yxT

i

recyyxx

=

FuS

where the normalized received signal is defined by:

)49(),(8

),(2

uEuS

sr

recreck

=

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Hence for 0>zuz , which introduces no loss of generality,

(

)50(

4

4exp),,,(

~)(),(

22

22

yx

ujkujkz

zyxT

i

rec dkdkeek

ukjkkkE yyxx

=

FuS

with

)51(4 22 = kkz

Consider a general observation point, or in a SAR scenario, the synthetic aperture coordinates:

)52()),sin(),cos((),,( zuuzyx urruuu ==u

where ur and are polar coordinates of observation or synthetic aperture. Making use of

(52) results in:

(

)53(

))cos(exp(4

4exp),,,(

~)(),(

22

22

ddrj

k

ukjkkkE u

z

zyxT

i

rec

= FuS

where zyx kkk ,, are defined by (47) and (51).

One can recast (53) as:

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)54(

]))cos(exp(),,,(~

[4

4exp)(),( )(

22

22

uzyxT

zi

rec rjkkkk

ukjdE

= FuS

In (54) the asterisk symbol (*) denotes the one-dimensional angular convolution (

dimension) and zyx kkk ,, are defined by (47) where is replaced with .

Equation (54) relates the spectral distribution of the polarimetric target function TF~

to anyarbitrary observation point (or synthetic aperture) u in space.

For moving platforms, as was addressed earlier, space frequency mapping to the observationspectral domain takes place in (54) (also see (40)). More specifically, taking the Fouriertransform of both sides in (54) with respect to zu :

)55(

]))cos(exp(),,,(~

)[4(4

)(),,,(~ )(22

22

uzyxTuz

i

yxuzrec rjkkkkkk

dEuuk

= FS

Taking advantage of the analytic properties of the delta function:

)56()0)(()())(()( 00 == = afwhereddfg

dafg

we find:

)57(]))cos(exp(),,,(~

[)()),sin(),cos(,(~ )(

uzyxTi

uuuzrec rjkkkkErrk = FS

where now,

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)58(uzz kk =

)59()cos(4 22 akkk uzx =

)59()sin(4 22 bkkk uzy =

with

)60(4 22 uzkkk =

Equations (58)-(60) indicate that the mapping from the observation spectral domain to targetspectral domain is, in general, non-linear. The space frequencies in (58)-(59) are located onthe surface of a sphere of radius k2 in the spectral domain, known as the Ewald sphere.

Considering the Fourier expansion of the nonlinear phase term:

)61()())cos(exp( )2/()2(

= jesHjs

where )()2( sH is the Hankel function of the second kind and order , one may transform

(57) to the spectral domain ( ) of angular dimension (which can also be considered as anaspect angle) as:

)62()(),,(~)(),,,(~ 2)2()(

juuzTi

uuzrec erkHkErk= FS

The target function Fourier transform is defined via:

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)63(),,,(~

),,(~

0

)(

dekkkk jzyxTuzT = FF

)64(),,(~

),,,(~ )(

juzTzyxT ekkkk = FF

with ),,( zyx kkk defined by (58)-(59) and, 0 representing the angular synthetic aperture

domain. It is within this domain that orthogonality of Fourier components je holds.Accordingly, for ),( , takes on integer values.

Equation (62) (or (57)) provides an analytic means to recover the target information containedin TF from the observations made through recS . A variety of methods may be used to

perform the required inverse processing which are optimal and suited for certain applications.A relatively simple and general matched filtering technique to implement the deconvolutionneeded for inverse processing will be addressed.

4.2 2-Dimensional slant plane Fourier analysis

As a typical SAR scenario, consider the imaging of the ground plane by a radar-carryingvehicle flying at a constant height hZz= . Although altitude variations from the ground plane

0=z are always present in practical operations (as included in the model developed), thoseeffects may not be of interest for certain applications. An appropriate model for suchapplications can be derived from the three-dimensional formulation by imposing theassociated conditions. We perform the required derivations as follows.

The target distribution at the ground level 0=z (i.e., ignoring the terrain variations) can bewritten as:

)65()(),,(),,,( zyxzyx TT FF =

Applying the three-dimensional Fourier transform of (65) in (57), one gets:

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)66(]))cos(exp(),,(~

[)()),sin(),cos(,(~ )(

uyxTi

uuuzrec rjkkkErrk = FS

The inverse Fourier transform of (66) with respect to )( uzz ku yields:

)67(]))cos(exp(),,(~

[)()),sin(),cos(,(~ )(

uyxTujk

uz

i

uuzrec rjkkkedkErruzuz = FS

where k is given by (60). Hence,

)68(

dkk

kdk

uz

uz =

Making use of (68) for the observation set at hz Zu = , results in:

))cos(exp(),,(

~

[4)()),sin(),cos(,(

~ 422

22

uyxT

Zkkji

uuhrec rjkkkekk

dkk

ErrZ

h

=

FS

with,

)70()cos(kkx =

)71()sin(kky =

Furthermore, the angular Fourier transform described in (63)-(64) may be used to transform(69) into:

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)72()(),,(~

4)(),,,(

~ 2)2()(22

4 22

j

uT

Zkkj

i

uhrec erkHkkk

edkkErZ

h

= FS

As can be seen from equation (72), extraction of the ground target information TF from theobservation

hZzrec =|S is more complicated than in (62) and, in principle, involves solving the

integral equation. One should note, however, that the simplicity offered in (62) is due to a fullthree-dimensional spectral domain treatment of scattering, i.e., an extra layer of Fouriertransformation in the z direction. In theory, the system defined by equation (72) can beviewed as a shift-varying system [33] with a known analytical kernel:

)73(

4

)(),(22

4 22

kk

eE

hZkkj

i

g

=

where the ground plane equivalent temporal frequency g is defined via:

)74(2c

kg

=

Setting:

)75(2)(),(~2

),,(~ 2)2()(

=

=

ckerkHkk

cr

gj

uTgug

FS

the system equation can be written as:

)76(),,(~

),,(),,,(~

ggugghuhrec drZrZ

g

SS =

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Technically, the ground plane signal (75) can be recovered by using the inverse of the systemkernel function (73):

)77(),,(),,,(~

),,(~ 1

dZrZr ghuhrecgug

= SS

Application of (77) with (75) yields the desired target function. One should note that the

required deconvolution to recover the ground plane signal gS~

could also be achieved through

matched-filtering of the system kernel scheme.

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5. Fourier properties and inversion considerations

The model developed here can be used to examine various aspects connecting the targetcharacteristics to observations from both electromagnetics and system points of view. One

important application, however, is to extract the target information from observations.According to equations (62) (or (57)) and (72) (or (62)), the latter involves inverse processingutilizing techniques ranging from matched filtering methods (as will be discussed) to moresophisticated integral equation based methods such as the method of moments (MoM).

To perform the inverse processing and conduct any spectral domain considerations (e.g.,spatial resolution), bandwidth limitations associated with factors such as finite aperture andtarget distribution must be investigated.

5.1 Bandwidth properties

Bandwidth characteristics of the received SAR signal can be studied by making use of thewindowing properties of phase-modulated (PM) signals [34]-[35]. These properties areessentially associated with the space-limited nature of the Greens function connecting thetarget (scatterer) and observation spaces. Since the Greens function and its spherical phasefront are space-limited or windowed, the Fourier transform of the Greens function is alsowindowed in the space-frequency domain. To formulate the described windowing behaviourfor bandwidth estimation, one can impose the space windowing conditions when deriving theinstantaneous frequencies.

We will derive a general relation to map the respective spatial and spectral windowing. Priorto that, however, the following should be noted. The Fourier decomposition of the Greensfunction in the present formulation represents all possible components including evanescentmodes. For SAR applications, the observation is normally within the far field region of thetarget where the evanescent or transient modes are negligible. Hence, considering thecharacteristic equation for the poles of the Greens function (Ewald sphere):

)78(04 2222 =++ kkkk zyx

the overall support band in the frequency domain (infinite aperture and target distribution) isk2 , or equally, ]2,2[),,( kkkkk zyx .

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The generalized instantaneous space frequency vector can be obtained by computing thegradient )( of the phase function )(r with respect to space coordinates:

)79()()( rrk =inst

Equation (79) provides an effective tool to evaluate the spectral domain properties of domainlimited field signals. One can show that (79) satisfies the generalized stationary phaseequation:

)80(0])([ )( = rkr rinst

that is (in scalar format) used to determine the phase center of the phase function [35].

Application of (79) for the spherical phase front associated with the Greens function:

)81(2),( ruru = k

yields the instantaneous frequency for spectral domain considerations. Spectral domainwindowing properties are examined throughout the following analysis.

Consider the SAR operation for ground plane target analysis detailed in Section 4.2. In thistypical case of operation, the observation undertakes a general motion at a certain altitude

hZz= . Furthermore, we assume the target has a limited distribution within a radius of 0r ,

i.e., 022 ryxr += (Figure 1).

Using (79) and (81), one finds the ),( yx plane instantaneous space frequencies:

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)82()()(

2)()(2222

222

hyx

xhyx

x

x

Zyuxu

xukZyuxuk

uk

++

=

++

=

)83()()(

2)()(2 222222

hyx

yhyx

y

y

Zyuxu

yukZyuxuk

uk

++

= ++=

with associated polar coordinates ),( given by:

)84()tan(xu

yu

k

k

x

y

x

y

==

)85()()(

)()(2

222

2222

hyx

yx

yx

Zyuxu

yuxukkk

++

+=+=

Define the centre slant angle associated with the centre of target distribution (centre incidentangle), as given in Figure 1(a):

)86(tan1

=

h

uz

Z

r

Inspection of Figure 1(b) reveals that the spectral polar angle defined in (84) is windoweddue to the finite target distribution as:

)87(0

where,

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)88(2

)(sin 001

0

otherwise

rrr

ru

u

=

=

Making use of (85), we find the polar phase center for any given observation point )( ur :

)89()sin(22220 z

hu

u kZr

rk =

+=

with a spread that can be approximated by:

)90()sin()(cos2 02 zk

Thus the windowing function in the spectral domain for a given observation point )( ur is:

=

)91(,0||)sin()(cos2|)sin(2|,1),,( 00

2

otherwise

andkkforW zzru

For a more particular and/or typical case of operation, one may consider a scenario when theobservation is performed along a straight line. In this case, the observation vector isrepresented by:

)92(),,( hy ZuX=u

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where Xdenotes the constant ground range associated with the center of target distributionand, LuL y represents the synthetic aperture dimension. Accordingly, L is the

synthetic aperture length. The instantaneous radial frequency for the present scenariobecomes:

)93()()(

)()(2

222

22

hy

y

ZyuxX

yuxXk

++

+=

Define the instantaneous centre incident angle associated with any observation point )( ur as:

)94()(sin22

1yz

hu

uz u

Zr

r

+=

where,

)95(22yu

uXr +=

The spectral windowing function for any observation point )( ur , is given by (89)-(91) when

definitions (94)-(95) are applied. Hence, moving along the yu direction from L to L+ ,

i.e., ],[ LLuy :

)96(

)sin())((cos2))(sin(2)sin())((cos2))(sin(2 02

02 yzyzyzyz ukukukuk +

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)97()]sin()sin(21)[cos()]sin()(cos)[sin()( 002

+==

d

d

d

d

)98()]sin()sin(21)[cos()]sin()(cos)[sin()( 002

=+=+

d

d

d

d

It can be seen that

d

d )( is positive for 20

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)102(

,)sin(4

)(sin41)()sin()(cos)sin(

,)sin()(cos)sin()()sin()(cos)sin(

max0

02

002

0

max02

002

0

>+

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One last note in the context of windowing characteristics of phase-modulated signals is thewindowing properties when the angular Fourier transform ((63)-(64)) is used. In this case, thesupport of the window for the angular direction in the respective spectral domain mustbe determined. The latter can be done by applying the instantaneous frequency relation (79) inthe angular direction and the phase definition:

)107()exp())cos(exp( ajrj u =

as,

( )bruinst 107)sin(

=

=

Thus,

)108()sin(|| 0 ur

As was asserted earlier, derivations in this subsection can be effectively used to performspectral analysis of the polarimetric radar waveform for different purposes. One typicalapplication (as detailed) is the effective bandwidth estimation. Another potentially interestingapplication is the exploration of acquired data along the paths corresponding to differentincident angles.

Consider the description of instantaneous frequencies given by (82)-(83), (85) and a similardescription for zk that can be easily derived in the same fashion. Making use of the above and(86), one can derive a similar relation for the incident angle in spectral domain, i.e.,

)109()tan(22

z

yx

z

zk

kk

k

+==

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Various paths (corresponding to certain incident and aspect angles) within the 3-dimensionalspectral distribution of the target scatterer may be explored in the spectral space

),,( zyx kkk by using (109) and (84).

5.2 Inversion

As was addressed earlier, recovery of the target characteristic vector function requiresinverse mapping according to the derived scattering model (e.g., (57), (69)). Such inverseprocessing normally involves deconvolution of the known system kernel or reference functionand may be performed through a variety of analytic and/or numerical methods. Onepotentially strong candidate is the method of moments (MoM) or its variational form,Galerkins Method[36]-[39]. Moment methods are suitable and numerically efficient forcomputations involving known analytical kernels [36]. We do not intend to discuss theanalytical details of such formulations. In general, the procedure is to solve the

inhomogeneous system:

)110() gf =(L

where L is the system operator which can be viewed as the system kernel, is the known

quantity (e.g., observation), and f denotes the unknown function, e.g., scattering

characteristics to be determined. Expanding f into a set of appropriate basis functions:

)111(mm

mfaf =

and defining a proper projection space }{ ip and moments,

)112(, = immi vu

gives:

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)113(,, == im

fp

mimim

m

m pgapfa L

The resultant linear system is solved for the unknown weightings ma . In (113), the system

operatorL is assumed to be self-adjoint. Proper selection of the basis set }{ mf , projection

space }{ ip and mechanism vu, (generalized inner product) is important to yield

convergence and accurate results. Choosing the projection space }{ ip to be the same as the

basis functions space }{ mf , results in the variational Galerkins method.

A formalism based on the method of moments can be developed and tailored for the presentSAR inverse processing to recover the polarimetric target characteristics through knowledgeof the analytic system kernel. One can explore various basis vector space design (under

investigation and to be communicated) representing scattering decomposition classes for thepurpose of target identification.

Relations (62) and (75) suggest that the target function in the spectral domain may berecovered by simply dividing the observation spectrum by the system kernel or referencefunction:

)114(),,,(~

),,,(~

),,(~ 1)(

uuzrecurefuzT rkrkSk SF=

)115(),,(~

),,(~

),(~ 1)(

gugugrefT rrkSk

SF=

where,

)116()()(),,,(~ 2)2(

j

u

i

uref erkHErkS=

)117()(2

),,(~ 2)2(

j

uugref erkHkc

rkS

=

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Technically, the operations in (114)-(115) represent the discussed deconvolution in thespectral domain with the system functions defined in (116)-(117). This recovery, however, isnot practical due to the discussed (Section 5.1) finite support band of reference

)~

,~

(grefref

SS and received )~

,~

(grec

SS signals. A practical reconstruction can be achieved by

using a matched filtering scheme such as:

)118(),,,(~

),,,(~

),,(~ )(

uuzrecurefuzTm rkrkSk SF=

)119(),,(~

),,(~

),(~ )(

gugugrefTm rrkSk

SF=

where )~

,~

( grefref SS are the complex conjugates of reference functions.

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6. Coherent radar cross section (RCS) analysis

An attractive feature of the present approach is the possibility of direct coherent RCSderivations from the knowledge of target function )(rF (or )(rFT ). The informationcontained in )(rF represents the electromagnetic identity of the target and, therefore, iscapable of producing accurate estimate of scattering-related observables such as the coherentRCS. Furthermore, derivations of the described observables are not limited to the far field ofthe scatterer (or radiator) and may be carried out in the near field region by using (11) (or aproper approximation of (11)) instead of (12)-(13). Prior to such near field considerations,however, a qualified description and/or definition of RCS in the near field has to beestablished.

The bistatic scattering cross section for an incident plane-wave iE can be expressed as [28]:

)120(|)0,0(|

|),(|4lim),,,(

2

22

i

sr

r

ii rE

E

=

According to the far field formulation (36), we have:

)121()(4

)(44lim),,,(2

||22

2||2

rrF

rrF

ru

ru

dek

deu

jk

u

jk

T

jk

Tu

ii

=

=

Applying the far field approximation:

)122(||

= uu

u

ururu

and coordinate transformation from Cartesian to Spherical systems, we find:

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)123(),(),(),,,( +=ii

where,

)124())sin()sin()cos()cos()cos((4

),( 222

dxdydzeFFFk jk

TzTyTx

ru+=

)125())cos()sin((4

),(2

22

dxdydzeFFk jk

TyTx

ru +=

In equations (124)-(125):

)126(),,(),,(),,(),,()( zyxFrF zyxFzyxFzyxFzyx TzTyTxTT ++==

and

)127()))cos()sin()sin()cos()sin((exp( zyxjkejk ++=ru

The integrations in (124)-(125) can be analytically or numerically carried out. For a MoMformulation, the integrals are carried out for an assumed set of basis functions (limitednumber of analytical integrations). The RCS is then computed by summing over the abovekernel moments using the weighting coefficients determined through MoM.

It is theoretically possible to transform the Greens function formulation (e.g., (13)) to the

spectral domain and derive the RCS from the Fourier knowledge of the target function. Thiswould require an equivalent RCS definition in the spectral domain.

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7. Conclusion

Utilization of radar or SAR polarimetry (PolSAR) for target recognition can significantlyenhance target detection, classification and recognition capabilities. To make the efficient use

of such techniques for remote sensing, appropriate data exploitation algorithms should beemployed. Considering the nature of a polarimetric approach that essentially is thecharacterization of a target or scatterer through interaction with the polarized or vectorialelectromagnetic (EM) field/wave, the required modeling analysis has to be based on anelectromagnetic theory. On the other hand, an EM-based approach to formulate the vectorfield scattering is by nature involved and complicated. Noticing the need for an optimalapproach (i.e., rigorous and feasible), the formulation here is structured to satisfy the accuracyrequirements by including the key EM scattering components and developing a precise systemkernel. The feasibility of associated algorithm implementation is addressed by formulating ageneric EM scattering problem with known (within other EM areas) analytical or numericalsolutions.

In the present theoretical framework, the target scattering and classification problem isapproached by a method which is new within the realm of conventional PolSAR targetcharacterization and classification techniques. The technique which is based on thecharacterization of target scatterers by current vector functions (i.e., the principle ofequivalent sources) and formulation of scattering using accurate Greens or system kernelfunctions in the spectral domain, is well founded in the field of classical electromagnetics.Introducing appropriate basis sets representing various scattering features for target functiondecomposition, characteristics of a target scatterer may be explored from different aspects bymaking use of the derived accurate system kernel. Theoretically, there is no restriction on thenature and number of features that can be exploited as long as they are modelled as basisfunctions. Strong and insightful techniques from other domains of computationalelectromagnetics, such as microwaves and millimetre-wave structures, may be adopted and

tailored for appropriate feature determination, analytic expression and effective targetfunction projection into the feature space. Examination of the current distribution behaviouron the target scatterers surface, which is one of the outputs of current analysis, can providegood understanding of the scatterer physical characteristics (e.g., shape). The currentdistribution on the scatterer may be readily applied to derive near and far field characteristicsand observable quantities such as RCS. The latter can also be used to extract targetdistribution information from RCS knowledge through inverse analysis and incorporation of apriori information. Accordingly, one may effectively utilize conventional radar measurementsto validate and tune the present scattering model.

Although fundamental, the analysis of scattering developed here for polarimetric radarapplications is general and should be tailored for the SAR scenario of interest. The latter is

essentially done by defining the appropriate analytic basis functions that represent targetfeatures based on the nature of scattering and the scatterer. Decomposition and/or projectionof the target scatterers current function into the features (i.e., basis functions) space shouldalso be optimized by accounting for the analytic nature of target and basis functions. Thefuture work will be focused on such target scattering feature classifications and requiredanalytic techniques for associated target function projections. A variety of EM computational

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techniques will be considered, tailored or further developed for desired applications. Resultsof the available trials and experiments will be used and (if required) new experimentalschemes will be designed to validate and tune the model by taking advantage of theestablished relations that connect measured quantities (e.g., RCS) to scattering functions.Experimental data and EM simulations for known target scatterers will be employed toinvestigate, derive and designate analytic characteristics to scattering features and/or classes

of features.

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Figure 1. (a) Radar-Target coordinate system. (b) Observation projection in ground plane with respect toground target distribution.

)(a

z

yu

ur0

x

y

xu 0r

)(b

radar

z

y

hz Zu =

ur

x

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8. References

1. F.M. Henderson, A.J. Lewis, Eds.,Principles & Applications of Imaging Radars, ThirdEdition, Volume 2, John Wiley & Sons, NY, 1998, Chapter 5.

2. Knnen, G.P.,Polarized Light in Nature, Cambridge University Press, Cambridge, UK,1985.3. Boerner, W-M., Ed.,Inverse Methods in Electromagnetic Imaging, Proceedings of NATO

Advanced Research Workshop on Inverse Methods in Electromagnetic Imaging, BadWindsheim, Federal Republic of Germany, Sept.,NATO ASI Series C, Math. & Phys. Sci.143. D. Reidel Publishing Co., Dordrecht, The Netherlands, 1985.

4. Boerner, W-M., Ed.,Direct and Inverse Methods in Radar Polarimetry, Proceedings ofNATO Advanced Research Workshop on Direct and Inverse Methods in RadarPolarimetry, Bad Windsheim, Federal Republic of Germany,NATO ASI Series C, Math.& Phys. Sci., D. Reidel Publishing Co., Dordrecht/Borton, 1992.

5. Van Zyl, J.J., Calibration of Polarimetric Radar Images using only Image Parameters andTrihedral Corner Reflectors,IEEE Trans. Geosci. Remote Sensing, Vol. 28, pp. 337-348,

1990.6. Van Zyl, J.J., Unsupervised Classification of Scattering Behaviour Using RadarPolarimetry Data,IEEE Trans. Geosci. Remote Sensing, Vol. 27, pp. 37-45, 1989.

7. Kramer, H.J., Observation of the Earth and its Environment Survey of (air/space-borne)Missions and Sensors, Third Edition, Springer Verlag, Heidelberg, NY, 1996.

8. Zebker, H.A., and Van Zyl, J.J., Imaging Radar Polarimetry: A Review,Proceedings ofthe IEEE, Vol. 79, pp. 1583-1606, 1991.

9. Zebker, H.A., Van Zyl, J.J., Durden, S.L. and Norikane, L. Calibrated Imaging RadarPolarimetry: Technique, Examples, and Applications,IEEE Trans. Geosci. RemoteSensing, Vol. 29, pp. 942-961, 1991.

10. Krogager, E., Decomposition of the Sinclair Matrix into Fundamental Components withApplication to High Resolution Radar Target Imaging,Direct and Inverse Methods inRadar Polarimetry, Part 2, Boerner, W-M., Ed. Kluwer Academic Publishers, Dordrecht,The Netherlands, pp. 1459-1478, 1992.

11. Krogager, E., and Czy, Z.H., Properties of the Sphere, Diplane, Helix (TargetScattering Matrix) Decomposition, in J. Saillard et al., Eds.,Proc. JIPR-3,Nantes,France, March 21-23, 1995, pp. 106-114.

12. Krogager, E., and Madsen, S.N., Comparison of Various Decompositions for Analysis,Interpretation, and Classification of Polarimetric SAR Images,European Conference onSynthetic Aperture Radar, EUSAR96, Knigswinter, Germany, March 26-28, 1996, pp.105-108.

13. Krogager, E., Boerner, W-M., On the Importance of Utilizing Complete PolarimetricInformation in Radar Imaging and Classifications,AGARD Symposium: Remote Sensing A valuable source of information, AGARD Proc. , 528, Toulouse, France, April 22-25,1996, pp. 17-1 17-12.

14. Sinclair, G., The Transmission and Reception of Elliptically Polarized Waves,Proceedings of the IRE, 38, pp. 148-151.

15. Kennaugh, E.M.,Effects of the Type of Polarization on Echo Characteristics, AntennaLaboratory, Ohio State University, Reports 389-4: 35 p, and 389-9: 39 p.

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16. Huynen, J.R.,Phenomenological Theory of Radar Targets, Ph.D. Thesis, Delft Universityof Technology, The Netherlands, 1970.

17. Huynen, J.R., Measurement of the Target Scattering Matrix,Proc. IEEE, Vol. 53, pp.936946, Aug. 1965.

18. Cloude, S.R., and Pottier, E., A Review of Target Decomposition Theorems in RadarPolarimetry,IEEE Trans. Geosci. Remote Sensing, Vol. 34, pp. 498518, Mar. 1996.

19. Krogager, E., New Decomposition of the Radar Target Scattering Matrix,Electron.Lett., Vol. 26, No. 18, pp. 15251527, 1990.

20. Cameron, W.L., Youssef, N., and Leung, L. K., Simulated Polarimetric Signatures ofPrimitive Geometrical Shapes,IEEE Trans. Geosci. Remote Sensing, Vol. 34, pp. 793-803, May 1996.

21. Touzi, R., and Charbonneau, F., Characterization of Target Symmetric Scattering UsingPolarimetric SARs,IEEE Trans. Geosci. Remote Sensing, Vol. 40, No. 11, pp. 2507-2516, Nov. 2002.

22. Cloude, S.R., and Pottier, E., An Entropy Based Classification Scheme for LandApplications of Polarimetric SAR,IEEE Trans. Geosci. Remote Sensing, Vol. 35, pp.6878, Jan. 1997.

23. Georgiou, T.T., Spectral Estimation Via Selective Harmonic Amplification,IEEE

Trans. Automat. Contr., Vol. 46, pp. 29-42, Jan. 2001.24. Georgiou, T.T., Spectral Analysis Based on the State Covariance: The Maximum

Entropy Spectrum and Linear Fractional Parametrization,IEEE Trans. Automat. Contr.,Vol. 47, pp. 1811-1823, Nov. 2002.

25. Harrington, R.F., Time-Harmonic Electromagnetic Fields,NY: McGraw-Hill, 1961.26. Balanis, C.A.,Advanced Engineering Electromagnetics,NY: Wiley, 1989.27. Chen, K.M., A Mathematical Formulation of the Equivalence Principle,IEEE Trans.

Microwave Theory Tech., Vol. 37, pp. 1576-1581, 1989.28. Peterson, A.F., Ray, S.L., and Mittra, R., Computational Methods for Electromagnetics,

IEEE Press, NY, 1998.29. Huyghens, C., Trait de la Lumire, 1690 (English Translation, Chicago: The University

of Chicago Press, 1945).

30. Schelkunoff, S.A., Some Equivalence Theorems of Electromagnetics and theirApplication to Radiation Problems,Bell System Tech., Jour., 15 (1936), 92-112.31. Stratton, J.A.,Electromagnetic Theory, NY: McGraw-Hill, 1941.32. Elliott, R.S.,Antenna Theory and Design, Englewood Cliffs, NJ: PrenticeHall, 1981.33. Soumekh, M., Synthetic Aperture Radar Signal Processing,NY: Wiley, 1999.34. Soumekh, M.,Fourier Array Imaging, Englewood Cliffs, NJ: PrenticeHall, 1994.35. Soumekh, M., Reconnaissance with Slant Plane Circular SAR Imaging,IEEE Trans.

Image Proc., Vol. 5, pp. 1252-1265, Aug. 1996.36. Harrington, R.F.,Field Computation by Moment Methods, Krieger Publishing Company,

Inc., Malabar, Florida, 1982.37. Harrington, R.F., Matrix Methods for Field Problems,Proc. IEEE, Vol. 55, pp. 136-

149, Feb. 1967.

38. Kantorovich, L.V., Krylov, V.I.,Approximate Methods of Higher Analysis, 4th

ed.,translated by C.D. Benster, John Wiley & Sons, Inc., NY, 1959.39. Jones, D.S., A Critique of the Variational Method in Scattering Problems, IRE Trans.,

Vol. AP-4, pp. 297-301, 1956.

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9. Annexes

9.1 Annex A: Reference frame unit vector transformation

It is often useful to express the scattered field vectors or matrices defined in the radarreference frame in terms of vectors in a universal reference frame, e.g., multi-incident-angleanalysis.

Assume ),,( lvh as the basis set for radar reference vectors. With respect to a universal

reference frame ),,( zyx , we have:

)1(

Azyxlvh UTU =t

where,

)2(

,

A

=

=

z

y

x

U

l

v

h

U zyxlvh

)3(

)cos()sin()sin()cos()sin(

0)cos()sin(

)sin()sin()cos()cos()cos(

A

iii

iii

=

Tt

In (A3), i denotes the incident angle and represents the polar angle in the transversalplane (to LOS).

Similarly, one finds for the scattering matrix:

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)4( AT TSTS zyxlvh

tttt

=

9.2 Annex B: Principle of stationary phase

The principle of stationary phase (PSP) states that [1],[34], for problems involvingintegrations of complicated waveforms with widely-varying phase functions, the phase isvariational around a certain stationary point. Considering such integrand as:

)1())(exp()( BjfIntegrand rr

=

where )(r

f is assumed to be a slowly-varying envelope, the PSP is formulated as:

)2(0)( B= r

Relation (B2) provides a powerful tool that can be effectively applied for integrationsassociated with spectral analysis of complex functions. Such complexity can be gauged by the

time-bandwidth product (TBP) of the waveform.In (B2), the stationary point

r

is considered to be a vector in general. For single-dimensionalproblems (normally the case), the stationary point satisfies:

)3(0|)(0

B=

=

Expanding )( in the vicinity of 0 :

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)4()|)()(()(!2

)()()(

02

2

00

20

0 B

=

+

and using the known result,

)5(2/1

)( 20 Bj

dej

=

one finds:

)6())(exp()()(

2))(exp()( 00

2/1

0

Bjfj

djf

The phase function is assumed to be analytic (near 0 ) in the above derivations. The accuracy

of (B6) depends on the significance of the residual terms in (B4). Effects of the residuals areinversely proportional to TBP of the waveform.

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List of acronyms and initialisms

CTD

DND

Coherent Target Detection

Department of National Defence

EM Electromagnetics

LOS Line Of Sight

MoM Method of Moments

PCTD Partially Coherent Target Detection

p.e.c perfect electric conductor p.m.c perfect magnetic conductor

PM Phase Modulated

PSP Principle of Stationary Phase

RCS Radar Cross Section

SAR Synthetic Aperture Radar

TBP Time Bandwidth Product

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