BASICS OF SAR POLARIMETRY I. Basics of Radar Polarimetry

Transcription

1 BAIC OF A OLAIMY I Wolfgang-Martn Boerner UIC-C Communcaton, enng & Navgaton Laboratory 9 W. aylor t., L (67) W-4, M/C 54, CHICAGO IL/UA mal: boerner@ece.uc.edu Bac of adar olarmetry Abtract A comprehenve overvew of the bac prncple of radar polarmetry preented. he relevant fundamental feld euaton are frt provded. he mportance of the propagaton and catterng behavor n varou freuency band, the electrodynamc foundaton uch a Mawell euaton, the Helmholtz vector wave euaton and epecally the fundamental law of polarzaton wll frt be ntroduced: he fundamental term whch repreent the polarzaton tate wll be ntroduced, defned and eplaned. Man pont of vew are the polarzaton llpe, the polarzaton rato, the toke arameter and the toke and Jone vector formalm a well a t preentaton on the oncaré phere and on relevant map projecton. he olarzaton Fork decrptor and the aocated van Zyl polarmetrc power denty and Agrawal polarmetrc phae correlaton gnature wll be ntroduced alo n order to make undertandable the polarzaton tate formulaton of electromagnetc wave n the freuency doman. he polarzaton tate of electromagnetc wave under catterng condton.e. n the radar cae wll be decrbed by matr formalm. ach catterer a polarzaton tranformer; under normal condton the tranformaton from the tranmtted wave vector to the receved wave vector lnear and th behavor, prncpally, wll be decrbed by a matr called catterng matr. h matr contan all the nformaton about the catterng proce and the catterer telf. he dfferent relevant matrce, the repectve term lke Jone Matr, -matr, Müller M- matr, Kennaugh K-matr, etc. and t nterconnecton wll be defned and decrbed together wth change of polarzaton bae tranformaton operator, where upon the optmal (Charactertc) polarzaton tate are determned for the coherent and partally coherent cae, repectvely. he lecture concluded wth a et of mple eample.. Introducton: A evew of olarmetry adar olarmetry (olar: polarzaton, Metry: meaure) the cence of acurng, proceng and analyzng the polarzaton tate of an electromagnetc feld. adar polarmetry concerned wth the utlzaton of polarmetry n radar applcaton a revewed mot recently n Boerner [] where a hot of pertnent reference are provded. Although polarmetry ha a long htory whch reache back to the 8 th century, the earlet work that related to radar date back to the 94. In 945 G.W. nclar ntroduced the concept of the catterng matr a a decrptor of the radar cro ecton of a coherent catterer [], []. In the late 94 and the early 95 major poneerng work wa carred out by.m. Kennaugh [4, 5]. He formulated a backcatter theory baed on the egenpolarzaton of the catterng matr ntroducng the concept of optmal polarzaton by mplementng the concurrent work of G.A. Dechamp, H. Mueller, and C. Jone. Work contnued after Kennaugh, but only a few notable contrbuton, a thoe of G.A. Dechamp 95 [6], C.D. Grave 956 [7], and J.. Copeland 96 [8], were made untl Huynen tude n 97. he begnnng of a new age wa the treatment preented by J.. Huynen n h doctoral the of 97 [9], where he eploted Kennaugh optmal polarzaton concept [5] and formulated h approach to target radar phenomenology. Wth th the, a renewed nteret for radar polarmetry wa raed. However, the full potental of radar polarmetry wa never fully realzed untl the early 98, due n no mall part to the advanced radar devce technology [, ]. echnologcal problem led to a ere of negatve concluon n the 96 and 97 about the practcal ue of radar ytem wth polarmetrc capablty []. Among the major contrbuton of the 97 and 98 are thoe of W-M Boerner [, 4, 5] who ponted out the mportance of polarzaton frt n addreng vector electromagnetc nvere catterng []. He ntated a crtcal analy of Kennaugh and Huynen work and etended Kennaugh optmal polarzaton theory [6]. He ha been nfluental n caung the radar communty to recognze the need of polarmetry n remote enng applcaton. A detaled overvew on the htory of polarmetry can be found n [, 4, 5], whle a htorcal revew of polarmetrc radar technology alo gven n [, 7, 8]. Boerner, W.-M. (7). In adar olarmetry and Interferometry (pp. - -4). ducatonal Note O-N--8b, aper. Neully-ur-ene, France: O. Avalable from: O-N--8b -

2 olarmetry deal wth the full vector nature of polarzed (vector) electromagnetc wave throughout the freuency pectrum from Ultra-Low-Freuence (ULF) to above the Far-Ultra-Volet (FUV) [9, ]. Whenever there are abrupt or gradual change n the nde of refracton (or permttvty, magnetc permeablty, and conductvty), the polarzaton tate of a narrow band (ngle-freuency) wave tranformed, and the electromagnetc vector wave re-polarzed. When the wave pae through a medum of changng nde of refracton, or when t trke an object uch a a radar target and/or a catterng urface and t reflected; then, charactertc nformaton about the reflectvty, hape and orentaton of the reflectng body can be obtaned by mplementng polarzaton control [, ]. he comple drecton of the electrc feld vector, n general decrbng an ellpe, n a plane tranvere to propagaton, play an eental role n the nteracton of electromagnetc vector wave wth materal bode, and the propagaton medum [,,, 4, 6]. Wherea, th polarzaton tranformaton behavor, epreed n term of the polarzaton ellpe named llpometry n Optcal enng and Imagng [, ], t denoted a olarmetry n adar, Ldar/Ladar and A enng and Imagng [, 4, 5, 9] - ung the ancent Greek meanng of meaurng orentaton and object hape. hu, ellpometry and polarmetry are concerned wth the control of the coherent polarzaton properte of the optcal and rado wave, repectvely [, 9]. Wth the advent of optcal and radar polarzaton phae control devce, ellpometry advanced rapdly durng the Forte (Mueller and Land [4, ]) wth the aocated development of mathematcal ellpometry,.e., the ntroducton of the coherent Jone forward catterng (propagaton) and the aocated 4 4 average power denty Mueller (toke) propagaton matrce []; and polarmetry developed ndependently n the late Forte wth the ntroducton of dual polarzed antenna technology (nclar, Kennaugh, et al. [,, 4, 5]), and the ubeuent formulaton of the coherent nclar radar back-catterng matr and the aocated 4 4 Kennaugh radar back-catterng power denty matr, a ummarzed n detal n Boerner et al. [9, 5]. nce then, ellpometry and polarmetry have enjoyed teep advance; and, a mathematcally coherent polarzaton matr formalm n the proce of beng ntroduced for whch the lecographc covarance matr preentaton [6, 7] of gnal etmaton theory play an eually mportant role n ellpometry a well a polarmetry [9]. Baed on Kennaugh orgnal poneerng work on dcoverng the properte of the pnoral olarzaton Fork concept [4, 5], Huynen [9] developed a henomenologcal Approach to adar olarmetry, whch had a ubtle mpact on the teady advancement of polarmetry [, 4, 5] a well a ellpometry by developng the orthogonal (group theoretc) target catterng matr decompoton [8, 9, ] and by etendng the charactertc optmal polarzaton tate concept of Kennaugh [, 4, 5], whch lead to the renamng of the pnoral polarzaton fork concept to the o called Huynen olarzaton Fork n adar olarmetry []. Here, we emphaze that for treatng the general btatc (aymmetrc) catterng matr cae, a more general formulaton of fundamental llpometry and olarmetry n term of a pnoral group-theoretc approach trctly reured, whch wa frt eplored by Kennaugh but not further purued by hm due to the lack of pertnent mathematcal formulaton [, ]. In ellpometry, the Jone and Mueller matr decompoton rely on a product decompoton of relevant optcal meaurement/tranformaton uantte uch a dattenuaton, retardence, depolarzaton, brefrngence, etc., [4, 5,, 8, 9] meaured n a chan matr arrangement,.e., multplcatvely placng one optcal decompoton devce after the other. In polarmetry, the nclar, the Kennaugh, a well a the covarance matr decompoton [9] are baed on a group-theoretc ere epanon n term of the prncpal orthogonal radar calbraton target uch a the phere or flat plate, the lnear dpole and/or crcular helcal catterer, the dhedral and trhedral corner reflector, and o on - - oberved n a lnearly upermpoed aggregate meaurement arrangement [6, 7]; leadng to varou canoncal target feature mappng [8] and ortng a well a catter-charactertc decompoton theore [9, 7, 4]. In addton, polarzaton-dependent peckle and noe reducton play an mportant role n both ellpometry and polarmetry, whch n radar polarmetry were frt purued wth rgor by J-. Lee [4, 4, 4, 44]. he mplementaton of all of thee novel method wll fal unle one gven fully calbrated catterng matr nformaton, whch apple to each element of the Jone and nclar matrce. It here noted that t ha become common uage to replace ellpometry by optcal polarmetry and epand polarmetry to radar polarmetry n order to avod confuon [45, 8], a nomenclature adopted n the remander of th paper. - O-N--8b

3 Very remarkable mprovement beyond clacal non-polarmetrc radar target detecton, recognton and dcrmnaton, and dentfcaton were made epecally wth the ntroducton of the covarance matr optmzaton procedure of ragl [46], Novak et al. [47-5], Lüneburg [5-55], Cloude [56], and of Cloude and otter [7]. pecal attenton mut be placed on the Cloude-otter olarmetrc ntropy H, Anotropy A, Feature-Angle (α ) parametrc decompoton [57] becaue t allow for unuperved target feature nterpretaton [57, 58]. Ung the varou fully polarmetrc (catterng matr) target feature ynthee [59], polarzaton contrat optmzaton, [6, 6] and polarmetrc entropy/anotropy clafer, very conderable progre wa made n nterpretng and analyzng OL-A mage feature [6, 57, 6, 64, 65, 66]. h nclude the recontructon of Dgtal levaton Map (DM) drectly from OL-A Covarance-Matr Image Data ake [67-69] net to the famlar method of DM recontructon from IN-A Image data take [7, 7, 7]. In all of thee technue well calbrated catterng matr data take are becomng an eental pre-reute wthout whch lttle can be acheved [8, 9, 45, 7]. In mot cae the mult-look-compreed A Image data take MLC- formattng uffce alo for completely polarzed A mage algorthm mplementaton [74]. However, n the ub-aperture polarmetrc tude, n olarmetrc A Image Data ake Calbraton, and n OL-IN-A Imagng, the LC (ngle Look Comple) A Image Data ake Formattng become an abolute mut [9, ]. Of coure, for LCformatted Image data, n partcular, varou peckle-flterng method mut be appled alway. Implementaton of the Lee Flter eplored frt by Jong-en Lee - for peckle reducton n polarmetrc A mage recontructon, and of the olarmetrc Lee-Whart dtrbuton for mprovng mage feature characterzaton have further contrbuted toward enhancng the nterpretaton and dplay of hgh ualty A Imagery [4 44, 75].. he lectromagnetc Vector Wave and olarzaton Decrptor he fundamental relaton of radar polarmetry are obtaned drectly from Mawell euaton [86, 4], where for the ource-free otropc, homogeneou, free pace propagaton pace, and aumng I tandard [] tme-dependence ep( + jωt), the electrc and magnetc H feld atfy wth µ beng the free pace permeablty and ε the free pace permttvty whch for the tme-nvarant cae, reult n r ( ) jωµ Hr ( ), Hr ( ) jωε r ( ) (.) ep( jkr) ep( jkr) r H r (.) r r ( + k ), ( ), ( ) H for an outgong phercal wave wth propagaton contant k ω ( ε µ ) / and c ( εµ ) / pace velocty of electromagnetc wave beng the free No further detal are preented here, and we refer to tratton [86], Born and Wolf [4] and Mott [76] for full preentaton.. olarzaton Vector and Comple olarzaton ato Wth the ue of the tandard phercal coordnate ytem ( r, θφ, ; u ˆ,u ˆ, uˆ r θ φ ) wth r,θ, φ denotng the radal, polar, azmuthal coordnate, and u,u, ˆ ˆ uˆ r θ φ the correpondng unt vector, repectvely; the outward travellng wave epreed a / uˆ ˆ r u r u ˆ u ˆ uˆ H u ˆ H, µ θ θ φ φ θ θ φ φ, Z π [ ] Z ε + H + H Ω (.) O-N--8b -

4 wth denotng the oyntng power denty vector, and Z beng the ntrnc mpedance of the medum (here vacuum). Far from the antenna n the far feld regon [86, 76], the radal wave of (.) take on plane wave charactertc, and aumng the wave to travel n potve z-drecton of a rght-handed Cartean coordnate ytem (, yz, ), the electrc feld, denotng the polarzaton vector, may be rewrtten a y uˆ u ˆ ep( ){uˆ uˆ + y y jφ + y ep( jφ )} (.4) wth, y beng the ampltude, φ, φ y the phae, φ φ y φ the relatve phae; / y tanα wth φ, φy, α and φ defnng the Dechamp parameter [6, ]. Ung thee defnton, the normalzed comple polarzaton vector p and the comple polarzaton rato can be defned a u ˆ ˆ + u y y p + ( u ˆ ˆ uy) (.5) wth + and defne the wave ampltude, and gven by y y y ( jφ ), φ φ y φ ep (.6). he olarzaton llpe and t arameter he tp of the real tme-varyng vector, orp, trace an ellpe for general phae dfferenceφ, where we dtnguh between rght-handed (clockwe) and left-handed (counter-clockwe) when vewed by the oberver n drecton of the travellng wave [76, 9], a hown n Fg.. for the commonly ued horzontal H (by replacng ) and vertcal V (by replacng y) polarzaton tate. here et unue relaton between the alternate repreentaton, a defned n Fg.. and Fg.. wth the defnton of the orentaton ψ and ellptcty χ angle epreed, repectvely, a y α, α π / and tan ψ tan( α)co φ π / ψ + π / (.7) tan χ ± mnor a/major a, n χ n αn φ, π / 4 χ π / 4 (.8) where the + and gn are for left- and rght-handed polarzaton repectvely. For a par of orthogonal polarzaton p and p p p * p, ψ ψ + χ χ (.9) In addton, the followng ueful tranformaton relaton et: co χ n ψ + j n χ tanα ep( jφ) (.) + co χ co ψ π - 4 O-N--8b

5 where ( α, φ) and ( ψ, χ) are related by the followng euaton: co α co ψ coχ, tanφ tan χ / n ψ (.) and nverely ψ e{ } arctan + π Im{ }... mod( π ) χ arcn (.) (a) otaton ene (Courtey of rof.. otter) v e H H e V V jφh jφv V (b) Orentaton ψ and llptcty χ Angle. α H (c) lectrc Feld Vector. h Fg.. olarzaton llpe. O-N--8b - 5

6 Fg.. olarzaton llpe elaton (Courtey of rof.. otter) Another ueful formulaton of the polarzaton vector p wa ntroduced by Huynen n term of the parametrc formulaton [9, 4], derved from group-theoretc conderaton baed on the aul U() matr et ψ { [ σ ],,,,} a further purued by otter [5], where accordng to (.) and (.), for ψ, and then rotatng th ellpe by ψ. p co n co ( ψ ψ χ, φψ,, χ) ep( jφ) n ψ co ψ j n χ (.) whch wll be utlzed later on; and { [ ],,,,} matrce [ σ ] a ψ σ defned n term of the clacal untary aul j j [ σ ], [ σ ], [ σ ], [ σ ] where the [ σ ] matrce atfy the untarty condton a well a commutaton properte gven by (.4) [ σ ] [ σ], Det {[ σ] }, [ σ] σ j σ j [ σ], [ σ][ σ] [ σ] atfyng the ordnary matr product relaton. (.5). he Jone Vector and Change of olarzaton Bae If ntead of the ba { y} or {H V}, we ntroduce an alternatve preentaton {m n} a a lnear combnaton of two arbtrary orthonormal polarzaton tate and for whch m n uˆ + u ˆ (.6) m m n n - 6 O-N--8b

7 and the tandard ba vector are n general, orthonormal,.e. uˆ uˆ, uˆ uˆ uˆ uˆ (.7) m n m m n n wth denotng the hermtan adjont operator [, 5, 5]; and the Jone vector mn may be defned a m m ep jφm ep( jφ ) co m α mn m ep( jφ ) ep * m n n jφ n nαep( jφ) (.8) + wth tan α n / m and φ φ n φm. h tate that the Jone vector poee, n general, four degree of freedom. he Jone vector decrpton for charactertc polarzaton tate are provded n Fg... m A mn ( mn, ) j (, j) n and AB ( AB, ) j (.) B he unue tranformaton from the { uˆ u ˆ } to the arbtrary{ uˆ uˆ } or { uˆ u ˆ } bae ought whch a lnear tranformaton n the two-dmenonal comple pace o that m n j A B [ U ] or (, j) [ U ] ( m, n) wth j mn [ U ][ U ] [ I ] (.) atfyng wave energy conervaton wth [ I ] beng the dentty matr, and we may chooe, a hown n [8], wth φ j φj + φ + π o that ep( jφ ) ep( jφ ) ep( jφ ) u ˆ and uˆ ˆ j u (.) ep( jφ) ep( jφj) [ U ] + ep( jφ) ep( jφj) (.) yeldng Det{ [ U ] } ep{ j( φ + φ j)} wth φ + φ j nce any monochromatc plane wave can be epreed a a lnear combnaton of two orthonormal lnear polarzaton tate, defnng the reference polarzaton ba, there et an nfnte number of uch bae { j} or {A B} for whch uˆ + uˆ uˆ + uˆ uˆ + uˆ (.9) m m n n j j A A B B wth correpondng Jone vector preented n two alternate, mot commonly ued notaton O-N--8b - 7

8 Fg.. Jone Vector Decrpton for Charactertc olarzaton tate wth drecton of propagaton out of the page (Courtey of rof.. otter) nce [ U ] a pecal untary comple matr wth unt determnant, mplyng that () the ampltude of the wave reman ndependent of the change of the polarzaton ba, and that () the phae of the (abolute) wave may be contently defned a the polarzaton ba changed, we fnally obtan, ep( jφ ) [ U] ep( jφ ) + (.4) poeng three degree of freedom mlar to the normalzed Jone vector formulaton, but n mot cae the phae reference taken a φ whch may not be o n polarmetrc nterferometry [96]. For further - 8 O-N--8b

9 detal on the group-theoretc repreentaton of the proper tranformaton relaton ee the formulaton derved by otter n [6]..4 Comple olarzaton ato n Dfferent olarzaton Bae Any wave can be reolved nto two orthogonal component (lnearly, crcularly, or ellptcally polarzed) n the plane tranvere to the drecton of propagaton. For an arbtrary polarzaton ba {A B} wth unt vector â and ˆb, one may defne the polarzaton tate ( AB) aˆ + bˆ (.5) where the two component A and B are comple number. he polarzaton rato AB n an arbtrary ba {A B} alo a comple number, and t may be defned a A B B B AB ep{ j( B A)} AB ep{ j AB} φ φ φ (.6) A A where AB the rato of magntude of two orthogonal component of the feld A and B and φab the phae dfference between A and B. he comple polarzaton rato AB depend on the polarzaton ba {A B} and can be ued to pecfy the polarzaton of an electromagnetc wave B B + A A A ( AB) A ep{ jφa} A ep{ jφa} A AB B B AB + ep{ jφ A} + AB AB AB A A (.7) where A A + B B the ampltude of the wave ( AB). If we chooe and dregard the abolute phae φ A, the above repreentaton become ( AB) + (.8) AB h repreentaton of the polarzaton tate ung the polarzaton rato AB very ueful. For eample, f we want to repreent a left-handed crcular (LHC) polarzaton tate and a rght-handed crcular (HC) polarzaton tate n a lnear ba {H V} ung the polarzaton rato. For a left-handed crcular (LHC) polarzaton, H V, φ φ π V φh, and accordng to (.6), the polarzaton rato j. Ung (.8) wth j, we obtan for the left-handed crcular (LHC) polarzaton AB AB ( ) j (.9) O-N--8b - 9

10 mlarly, the polarzaton rato of a rght-handed crcular (HC) polarzaton tate n a lnear ba {H V} j becaue the relatve phaeφ π, and t repreentaton ( ) j (.) he comple polarzaton rato mportant n radar polarmetry. However, the value of the polarzaton rato defned n a certan polarzaton ba dfferent from that defned n the other polarzaton ba even f the phycal polarzaton tate the ame..4. Comple olarzaton ato n the Lnear Ba {H V} In the lnear {H V} ba wth unt vector ĥ and ˆv, a polarzaton tate may be epreed a: he polarzaton rato, accordng to (.6), can be decrbed a: ( ) hˆ + vˆ (.) V V ep( j ) tan ep ( j ), V H φ α φ φ φ φ (.) H H where the angle α defned n Fg..c, only n the {H V} ba and H V + coα H H V + nα V H V (.) Alo, for a ngle monochromatc, unform M (tranvere electromagnetc) travelng plane wave n the potve z drecton, the real ntantaneou electrc feld wrtten a ε (,) co( ) zt ωt kz+ φ ε (,) zt ε y(,) zt y co( ωt kz+ φy) (.4) ε z (,) zt In a cartean coordnate ytem, the + -a commonly choen a the horzontal ba (H) and the + y - a a the vertcal ba (V) ubttutng (.) nto (.4), we fnd H + V coα co( ωt kz+ φ ) H ε(,) zt H V nα co( ωt kz φv) + + coα H + V ep ep{ j( ωt kz+ φh)} nα ep( jφ) (.5) where φ φv φh the relatve phae. he epreon n the uare bracket a pnor [] whch ndependent of the tme-pace dependence of the travelng wave. he pnor parameter ( α, φ ) are eay to - O-N--8b

11 be located on the oncaré phere and can be ued to repreent the polarzaton tate of a plane wave. In Fg..4c, the polarzaton tate, decrbed by the pont on the oncaré phere, can be epreed n term of thee two angle, where α the angle ubtended by the great crcle drawn from the pont on the euator meaured from H toward V; and φ the angle between the great crcle and the euator. From euaton, (.7) and (.8) for the {H V} ba we have n χ n α nφ tan ψ tan( α )coφ (.6) whch decrbe the ellptcty angle χ and the tlt or orentaton angle ψ n term of the varable α and φ. Alo, from (.) for the {H V} ba an nvere par that decrbe the α and φ n term of χ and ψ gven n (.7) co α co ψ co χ tanφ tan χ n ψ (.7) It convenent to decrbe the polarzaton tate by ether of the two et of angle ( α, φ) or ( χ, ψ ) whch decrbe a pont on the oncaré phere. he comple polarzaton rato can be ued to pecfy the polarzaton of an electromagnetc wave epreed n the {H V} ba. ome common polarzaton tate epreed n term of ( χ, ψ ),, and the normalzed Jone vector are lted n able. at the end of th ecton..4. Comple olarzaton ato n the Crcular Ba {L } In the crcular ba {L }, we have two unt vector ˆL (left-handed crcular) and ˆ (rght- handed crcular). Any polarzaton of a plane wave can be epreed by ( L) Lˆ+ ˆ (.8) A unt ampltude left-handed crcular polarzaton ha only the L component n the crcular ba {L }. It can be epreed by L ˆ ˆ ( L) L+ (.9) he above repreentaton of a unt (LHC) polarzaton n the crcular ba {L } dfferent from that n the lnear ba {H V} of (.9). mlarly, a unt ampltude rght-handed crcular polarzaton ha only the component n the crcular ba {L } whch dfferent from that n the lnear {H V} ba. ˆ ˆ ( L) L+ (.4) O-N--8b -

12 he polarzaton rato L, accordng to (.6) where L ep{ j( L)} L ep{ j L} tan L ep{ j L} φ φ φ α φ (.4) L L L the rato of magntude of the two orthogonal component L and dfference. he angle angle, and φ L the phae α L and φ L are alo eay to be found on the oncaré phere (ee Fg..6) lke the, are lted n able.. α and φ. ome common polarzaton tate n term of L.4. Comple olarzaton ato n the Lnear Ba {45 5 } In the lnear {45 5 } ba wth unt vector 45ˆ o and 5ˆ o, a polarzaton tate may be epreed a where 45 o and accordng to (.6) (45 o 5 o ) 45 o o + o5 o 45 5 ˆ ˆ (.4) 5 o are the 45 component and the 5 component, repectvely. he polarzaton rato o o 5 5 j j j φ φ φ α φ o o o o o o o o o o o o o 45 o 45 ep{ ( )} ep{ } tan ep{ }(.4) where 45 o 5 o the rato of magntude of the two orthogonal component the phae dfference. he angle Fg..6) α45 o 5 o and 5 o and 45 o, and φ45 o 5 o φ45 o 5 o are alo eay to be found on the oncaré phere (ee ABL. OLAIZAION A IN M OF ( χ, ψ ), OLAIZAION AIO AND NOMALIZD JON VCO OLAIZAION χ ψ Lnear Horzontal Lnear Vertcal 45 Lnear 5 Lnear Left-handed Crcular ght-handed Crcular π 4 π π 4 π 4 π j 4 {H V} ba {45 5 } ba {L } ba 45 o 5 o L j j j j j j j j j j j j j j j j j - O-N--8b

13 .5 he toke arameter o far, we have een completely polarzed wave for whch, and φ AB are contant or at leat A, B lowly varyng functon of tme. If we need to deal wth partal polarzaton, t convenent to ue the toke parameter,, and ntroduced by toke n 85 [7] for decrbng partally polarzed wave by obervable power term and not by ampltude (and phae)..5. he toke vector for the completely polarzed wave For a monochromatc wave, n the lnear {H V} ba, the four toke parameter are H H + coφ nφ V V H V H V (.44) For a completely polarzed wave, there are only three ndependent parameter, whch are related a follow + + (.45) he toke parameter are uffcent to characterze the magntude and the relatve phae, and hence the polarzaton of a wave. he toke parameter alway eual to the total power (denty) of the wave; eual to the power n the lnear horzontal or vertcal polarzed component; eual to the power n the lnearly polarzed component at tlt angle ψ 45 o or 5 o ; and eual to the power n the lefthanded and rght-handed crcular polarzed component. If any of the parameter,, or ha a non-zero value, t ndcate the preence of a polarzed component n the plane wave. he toke parameter are alo related to the geometrc parameter A, χ, and ψ of the polarzaton ellpe H + V A H V A coψ coχ (.46) H V coφ A n ψ co χ H V nφ A n χ whch for the normalzed cae e e + e and H V H + V eh + e V e H V eh ev e co ψ co χ (.47) e{ } ehev co e n co H + φ ψ χ V Im{ } ehev nφ e n χ.5. he toke vector for the partally polarzed wave he toke parameter preentaton [4] poee two man advantage n that all of the four parameter are meaured a ntente, a crucal fact n optcal polarmetry, and the ablty to preent partally polarzed wave n term of the comple hermtan potve em-defnte wave coherency matr [ J ] alo called the Wolf coherence matr [4], defned a: O-N--8b -

14 J J + + j [ J ] H H H V HH J V H V V VH J VV j (.48) where... lm... dt ndcatng temporal or enemble averagng aumng tatonarty of the wave. We can aocate the toke vector wth the coherency matr [ J ] coφ + J + J nφ j j jj jj H V H H V V HH VV + + J + J H V H H V V HH VV J J H V H V V H VH H V H V V H VH (.49) and nce [ J ] potve emdefnte matr Det {[ J ]} or + + (.5) the dagonal element preentng the ntente, the off-dagonal element the comple cro-correlaton between H and V, and the race{[ J ]}, repreentng the total energy of the wave. For J no correlaton between H and V et, [ J ] dagonal wth JHH JVV, (.e. the wave unpolarzed or completely depolarzed, and poee one degree of freedom only : ampltude). Wherea, for Det{[ J ]} we fnd that JVH J JHH JVV, and the correlaton between H and V mamum, and the wave completely polarzed n whch cae the wave poee three degree of freedom: ampltude, orentaton, and ellptcty of the polarzaton ellpe. Between thee two etreme cae le the general cae of partal polarzaton, where Det{[ J ]} > ndcatng a certan degree of tattcal dependence between H and polarzaton V whch can be epreed n term of the degree of coherency µ and the degree of D p a J µ µ ep( jβ ) (.5) J J HH VV D p ( ) / / 4 Det{[ J ]} + + ( race{[ J ]}) (.5) where µ D p for totally depolarzed and µ D p for fully polarzed wave, repectvely. However, under a change of polarzaton ba the element of the wave coherency matr [ J ] depend on the choce of the polarzaton ba, where accordng to [5, 5], [ J ] tranform through a untary mlarty tranformaton a [ J ] [ U ] [ J ] [ ] j mn U (.5) - 4 O-N--8b

15 he fact that the trace and the determnant of a hermtan matr are nvarant under untary mlarty tranformaton mean that both, the degree of polarzaton a well a the total wave ntenty are not affected by polarmetrc ba tranformaton. Alo, note that the degree of coherence µ mn doe depend on the polarzaton ba. able. gve the Jone vector, Coherency Matr [ J ], and toke Vector for pecal cae of purely monochromatc wave feld n pecfc tate of polarzaton. ABL. JON VCO, COHNCY MAIX [ J ], AND OK VCO FO OM A OF OLAIZAION OLAIZAION Lnear Horzontal Lnear Vertcal {H V} BAI [ J ] 45 Lnear 5 Lnear Left-handed Crcular ght-handed Crcular j j j j j j.6 he oncaré olarzaton phere he oncaré phere, hown n Fg..4 for the repreentaton of wave polarzaton ung the toke vector and the Dechamp parameter ( α, φ ) a ueful graphcal ad for the vualzaton of polarzaton effect. here one-to-one correpondence between all poble polarzaton tate and pont on the oncaré phere, wth the lnear polarzaton mapped onto the euatoral plane ( ) wth the zenth preentng O-N--8b - 5

16 left-handed crcular and the nadr rght-handed crcular polarzaton tate accordng to the I tandard notaton ep( + jωt) [], and any et of orthogonally fully polarzed polarzaton tate beng mapped nto antpodal pont on the oncaré phere [8]. (a) (b) (c) Fg..4 oncaré phere epreentaton (Courtey of rof.. otter).6. he polarzaton tate on the oncaré phere for the {H V} ba In the oncaré phere repreentaton, the polarzaton tate decrbed by a pont on the phere, where the three Cartean coordnate component are,, and accordng to (.46). o, for any tate of a completely polarzed wave, there correpond one pont (,, ) on the phere of radu, and vce vera. In Fg..5, we can ee that the longtude and lattude of the pont are related to the geometrc parameter of the polarzaton ellpe and they are ψ and χ repectvely. - 6 O-N--8b

17 Fg..5 he oncaré phere and the parameter α and φ In addton, the pont on the oncaré phere can alo be repreented by the angle α and φ. From (.7) and (.46) we fnd that coψ coχ coα (.54) where co α the drecton cone of the toke vector wth repect to the X-a,.e., the angle α the angle between and the X-a. he angle φ the angle between the euator and the great crcle wth ba dameter through the pont, and t eual to the angle between the XOY plane and the XO plane. Drawng a projectng lne from pont to the YOZ plane, the nterectng pont on the XO plane, o φ YO ( φ φ n Fg..5). On the YOZ plane we fnd that whch atfe euaton (.46) and (.7). tanφ tan YO tan χ (.55) n ψ.6. he polarzaton rato on the oncaré phere for dfferent polarzaton bae Alo, t can be hown that a polarzaton tate can be repreented n dfferent polarzaton bae. Any polarzaton ba cont of two unt vector whch are located at two correpondng antpodal pont on the oncaré phere. Fg.6 how how the polarzaton tate on the oncaré phere can be repreented n three polarzaton bae, {H V}, {45 5 }, and {L }. he comple polarzaton rato are gven by O-N--8b - 7

18 π tanα ep( ) jφ < α < ep ( jφ ) tanα ep( jφ ) π < α < π (.56) π tanα ep( jφ ) < α o o < o o o o 45 5 ep o o o o ( jφ o o 45 5 ) tan ep ( j ) π < < α o o φ o o α o o π 45 5 (.57) π tanα ep( ) L L jφ < α < L ep L L ( jφl ) tanαl ep( jφl ) π < α L < π (.58) where tanα, tanα45 o 5 o, and tanα L are the rato of the magntude of the correpondng orthogonal component, and φ, φ45 o 5 o, and φ L are the phae dfference between the correpondng orthogonal component Fg.6 he olarzaton tate n Dfferent olarzaton Bae - 8 O-N--8b

19 .6. he relatonhp between the toke vector and the polarzaton rato for dfferent polarzaton bae Frt, conder the polarzaton rato defned n the {H V} ba. Becaue co α the drecton cone of the toke vector wth repect to the X-a, we fnd tan α co α + + tan α (.59) the traght forward oluton for + (.6) from (.54), we fnd φ YO tan (.6) Combnng above two euaton yeld + ep( j ) ep jtan φ (.6) For a completely polarzed wave, we may obtan the toke vector n term of the polarzaton rato applyng by + + co α + coφ tanα coφ n( α + + tanα nφ n( α + )nφ )coφ (.6) he gn of the three component of the toke vector ummarzed n able.. econdly, conder the polarzaton rato 45 o 5 o defned n the {45 5 } ba. he co α 45 o 5 o the drecton cone of the toke vector wth repect to the Y-a. o mlarly, wth O-N--8b - 9

20 φ o o 45 5 o o tan (.64) ABL. H IGN OF H,, AND AAM IN H {H V} BAI φ α π < α < π + + π < α < π < φ < π < α < π + < α < π π + + < α < π + π < α < π < φ < π + < α < π + + < α < π hen the polarzaton rato 45 o 5 o can be determned by the toke vector ep tan o o j 45 5 (.65) + Alo, the toke vector can be determned by the polarzaton rato 45 o 5 o a follow: coφ o o o o n α coφ o o o o o o 45 5 o o 45 5 co α o o o o 45 5 nφ o o o o n α nφ o o o o o o 45 5 (.66) - O-N--8b

21 Fnally, conder the polarzaton rato L defned n the {L } ba. mlarly, becaue the co α L the drecton cone of the toke vector wth repect to the Z-a, the polarzaton rato L can be determned by the toke vector a: L j + ep tan (.67) Inverely, L coφl n α + L nφ n α + L L L co α + L L L L L coφ nφ L L (.68) ABL.4 ALNA XION FO NOMALIZD OK VCO NAION ON H OLAIZAION H χ, ψ α, φ α o o, φ o o, L α φ co χ co ψ co α co χ n ψ n α coφ n χ n α nφ n α o o coφ o o n α coφ co α o o n α nφ 45 5 n α o o nφ o o co α L L L L L L.6.4 he oncaré polarzaton phere and comple polarzaton rato plane Ung the emann tranformaton, oncaré ntroduced the polarzaton phere repreentaton of Fg..5 whch gve a relatonhp between the polarzaton rato and t correpondng phercal coordnate on the oncaré phere. Frt we need to ntroduce an aulary comple parameter u( ), whch defned by the emann tranformaton [4] of the urface of the phere onto the polar grd a follow, j u( ) (.69) + j n the {H V} ba, tanα ep{ jφ } tan α (coφ + jn φ ), then ( + tanα n φ ) j tanα coφ u ( tanα n φ ) + j tanα coφ u + tanα n φ ) + tan α tanα n φ ) + tan α O-N--8b -

22 u tanα nφ n α nφ u + + tan α accordng to (.6) and Fg..4b, the polar angle Θ π χ can be obtaned from o that u n χ n( π Θ ) co Θ u + u Θ co (.7) u + alo, accordng to (.6) and Fg..4b, the phercal azmuthal angle Φ ψ can be obtaned Im{u} tanα coφ from tan ψ tan Φ, o that the phercal azmuthal angle Φ become e{u} tan α Im{u} Φ tan e{u} (.7) Fg..7 oncaré phere and the Comple lane.7 Wave Decompoton heorem he dagonalzaton of J j, under the untary mlarty tranformaton euvalent to fndng an orthonormal polarzaton ba n whch the coherency matr dagonal or - O-N--8b

23 J J mm nm J J mn nn e e e e λ e λ e e e (.7) where λ and λ are the real non-negatve egenvalue of [ J ] wth λ λ ê e e and ê [ e e] are the comple orthogonal egenvector whch defne [ U ] and a polarzaton ba { eˆ, e ˆ} n whch [ J ] dagonal. [ J ] Hermtan and hence normal and every normal matr can be untarly dagonalzed. Beng potve emdefnte the egenvalue are nonnegatve., and [ ].8 he Wave Dchotomy of artally olarzed Wave he oluton of (.7) provde two euvalent nterpretaton of partally polarzed wave [8]: ) eparaton nto fully polarzed[ J ], and nto completely depolarzed [ J ] component [ J ] ( λ λ [ J ] + λ [ I ] (.7) ) where [ I ] the dentty matr ; ) non-coherency of two orthogonal completely polarzed wave tate repreented by the egenvector and weghed by ther correpondng egenvalue a [ ] λ )[ J ] + λ [ J ] λ (ˆ e eˆ ) + λ (ˆ e ˆ ) ( e J (.74) where Det{[ J]} Det{[ J]} ; and f λ λ the wave totally depolarzed (degenerate cae) wherea for λ, the wave completely polarzed. Both model are unue n the ene that no other decompoton n form of a eparaton of two completely polarzed wave or of a completely polarzed wth noe poble for a gven coherency matr, whch may be reformulated n term of the degree of polarzaton D a p λ λ D p, ( λ λ) and ( λ ) (.75) λ + λ for a partally unpolarzed and completely polarzed wave. he fact that the egenvalue λ and λ are nvarant under polarzaton ba tranformaton make D p an mportant ba-ndependent parameter..9 olarmetrc Wave ntropy Alternately to the degree of wave coherency µ and polarzaton D p, the polarmetrc wave entropy H ω [8] provde another meaure of the correlated wave tructure of the coherency matr[ J ], where by ung the logarthmc um of egenvalue λ Hω { log } wth (.76) λ+ λ o that + and the normalzed wave entropy range from H ω where for a completely polarzed wave wth λ and H ω, whle a completely randomly polarzed wave wth λ λ poee mamum entropy H ω. O-N--8b -

24 . Alternate Formulaton of the olarzaton roperte of lectromagnetc Vector Wave here et everal alternate formulaton of the polarzaton properte of electromagnetc vector wave ncludng; () the Four-vector Hamltonan formulaton freuently utlzed by Zhvotovky [9] and by Czyz [], whch may be ueful n a more conce decrpton of partally polarzed wave ; () the pnoral formulaton ued by Bebbngton [], and n general gravtaton theory [] ; and () a peudopnoral formulaton by Czyz [] n development whch are mot eental tool for decrbng the general b-tatc (non-ymmetrc) catterng matr cae for both the coherent (-D oncaré phere and the -D polarzaton pherod) and the partally polarzed (4-D Zhvotovky phere and 4-D pherod) cae [9]. Becaue of the eorbtant eceve addtonal mathematcal tool reured, and not commonly acceble to engneer and appled centt, thee formulaton are not preented here but deerve our fullet attenton n future analye.. he lectromagnetc Vector catterng Operator and the olarmetrc catterng Matrce he electromagnetc vector wave nterrogaton wth materal meda decrbed by the catterng Operator [ ( k / k )] wth k, k repreentng the wave vector of the cattered and ncdent, (), r () r repectvely, where related to he catterng operator [ / k )] ( r) ep( jk r) e ep( jk r) (.) ( ( r) ep( jk r) e ep( jk r) (.) ep( jk r) () r [ ( k / k) ] () r (.) r k obtaned from rgorou applcaton of vector catterng and dffracton theory, to the pecfc catterng cenaro under nvetgaton whch not further dcued here, but we refer to [97] for a thought-provokng formulaton of thee tll open problem.. he catterng cenaro and the catterng Coordnate Framework k / k ) appear a the output of the catterng proce for an arbtrary nput ( he catterng operator [ ] whch mut carefully be defned n term of the catterng cenaro; and, t proper unue formulaton of ntrnc mportance to both optcal and radar polarmetry. Wherea n optcal remote enng manly the forward catterng through tranlucent meda condered, n radar remote enng the back catterng from dtant, opaue open and cloed urface of nteret, where n radar target backcatterng we uually deal wth cloed urface wherea n A magng one deal wth open urface. hee two dtnct cae of optcal veru radar catterng are treated eparately ung two dfferent reference frame; the Forward (ant-monotatc) catterng Algnment (FA) veru the Back Btatc catterng Algnment (BA) from whch the Monotatc Arrangement derved a hown n Fg... In the followng, eparately detaled for both the FA and BA ytem are hown n Fg.. and.., - 4 O-N--8b

25 Fg.. Comparon of the FA, BA, and MA Coordnate ytem ẑ ĥ f kˆ f ĥ vˆ kˆ vˆ f θ θ φ ŷ ˆ φ kˆ vˆ ĥ f f f n θ co θ n φ co φ co φ ˆ + n θ ˆ + co φ ˆ + co θ ŷ n φ n φ ŷ + co θ ẑ ŷ n θ ẑ Fg.. Detaled Forward catterng Algnment (FA) O-N--8b - 5

26 ẑ ĥ ĥ b vˆ kˆ kˆ b vˆ b θ θ φ ŷ ˆ φ kˆ vˆ ĥ b b b n θ co θ n φ co φ co φ ˆ co φ ˆ n θ ˆ + co θ ŷ n φ n φ ŷ co θ ŷ n θ ẑ ẑ Fg.. Detaled Back catterng Algnment (BA). he Jone Forward [J] veru nclar [] Back-catterng Matrce nce we are dealng here wth radar polarmetry, nterferometry and polarmetrc nterferometry, the btatc BA reference frame more relevant and here ntroduced only for reaon of brevty but dealng both wth the btatc and the monotatc cae ; where we refer to [5, 5], [76] and [9] for a full treatment of the ant-monotatc FA reference frame. Here, we refer to the dertaton of apathanaou [97], the tetbook of Mott [76], and metculou dervaton of Lüneburg [5] for more detaled treatment of the ubject matter, but we follow here the dervaton preented n [9]. Ung the coordnate of Fg.. wth rght-handed coordnate ytem; y z, y z, y z ; denotng the tranmtter, catterer and recever coordnate, repectvely, a wave ncdent on the catterer from the tranmtter gven by the tranvere component and y n the rght-handed coordnate ytem y z wth the z a ponted at the target. he catterer coordnate ytem y z rght-handed wth z pontng away from the catterer toward a recever. BA Coordnate ytem y z rght-handed wth z pontng toward the catterer. It would concde wth the tranmtter ytem y z f the tranmtter and recever were co-located. he wave reflected by the target to the recever may be decrbed n ether the tranvere component and y or by the revered component and y. Both conventon are ued, leadng to dfferent matr formulaton. he ncdent and tranmtted or reflected (cattered) feld are gven by two-component vector; therefore the relatonhp between them mut be a matr. If the cattered feld epreed n y z coordnate (BA), the feld are related by the nclar matr [ ], thu y e y 4π r y yy y - j kr (.4) and f the cattered feld n y z coordnate (FA), t gven by the product of the Jone matr [ J ] wth the ncdent feld, thu - 6 O-N--8b

27 y 4π r y y y y y e - j kr (.5) In both euaton the ncdent feld are thoe at the target, the receved feld are meaured at the recever, and r the dtance from target to recever. he nclar matr[ ] motly ued for back-catterng, but readly etended to the btatc catterng cae. If the name catterng matr ued wthout ualfcaton, t. In the general btatc catterng cae, the element of the nclar normally refer to the nclar matr [ ] matr are not related to each other, ecept through the phyc of the catterer. However, f the recever and tranmtter are co-located, a n the mono-tatc or back-catterng tuaton, and f the medum between target and tranmtter recprocal, manly the nclar matr [ ( AB )] ymmetrc,.e. AB BA. he Jone matr ued for the forward tranmon cae; and f the medum between target and tranmtter, wthout Faraday rotaton, the Jone matr uually normal. However, t hould be noted that the Jone matr not n general normal,.e., n general the Jone matr doe not have orthogonal egenvector. ven the cae of only one egenvector (and a generalzed egenvector) ha been condered n optc (homogeneou and nhomogeneou Jone matrce). If the coordnate ytem beng ued are kept n mnd, the numercal ubcrpt can be dropped. It clear that n the btatc cae, the matr element for a target depend on the orentaton of the target wth repect to the lne of ght from tranmtter to target and on t orentaton wth repect to the target-recever lne of ght. In the form (.4) and (.5) the matrce are abolute matrce, and wth ther ue the phae of the cattered wave can be related to the phae of the tranmtted wave, whch trctly reured n the cae of polarmetrc nterferometry. If th phae relatonhp of no nteret, a n the cae of mono-tatc polarmetry, the dtnct phae term can be neglected, and one of the matr element can be taken a real. he reultng form of the nclar matr called the relatve catterng matr. In general the element of the catterng matr are dependent on the freuency of the llumnatng wave [9, 4, 5]. Another target matr parameter that hould be famlar to all who are ntereted n mcrowave remote enng the radar cro ecton (C). It proportonal to the power receved by a radar and the area of an euvalent target that ntercept a power eual to t area multpled by the power denty of an ncdent wave and re-radate t eually n all drecton to yeld a recever power eual to that produced by the real target. he radar cro ecton depend on the polarzaton of both tranmttng and recevng antenna. hu the radar cro ecton may be pecfed a HH (horzontal recevng and tranmttng antenna), (horzontal recevng and vertcal tranmttng antenna), etc. When conderng ground reflecton, the cro ecton normalzed by the ze of the ground patch llumnated by the wave from the radar. he cro ecton not uffcent to decrbe the polarmetrc behavor of a target. In term of the nclar matr [ ], and the normalzed effectve length of tranmttng and recevng antenna, h ˆ t and h ˆ r, repectvely, the radar cro ecton [ ] σ rt hˆ hˆ (.6) r t A polarmetrcally correct form of the radar euaton that pecfe receved power n term of antenna and target parameter W W G( θ, φ ) A ( θ, φ ) r r t t er rt (4 π ) hˆ [ ] hˆ r t (.7) O-N--8b - 7

28 where W t the tranmtter power and ubcrpt t and r dentfy tranmtter and recever, and t properte are h ˆ θ,φ, defned va the electrc defned n more detal n Mott [76] and n [9]. he effectve antenna heght ( ) t feld ( r,θ,φ ), radated by an antenna n t far feld, a t jz I r, (.8) λ r (, θ, φ ) ep( jkr) hˆ ( θ, φ ) wth Z the charactertc mpedance, λ the wavelength, and I the antenna current.. Ba ranformaton of the nclar catterng Matr [] edefnng the ncdent and catterng cae n term of the tandard {H V} notaton wth H, V y and wth proper re-normalzaton, we redefne (.) a * [ ] or ( ) [ ( )] ( ) (.9) where the comple conjugaton reult from nveron of the coordnate ytem n the BA arrangement whch nvte a more rgorou formulaton n term of drectonal nclar vector ncludng the concept of tme reveral a treated by Lüneburg [5]. Ung thee nclar vector defnton one can how that the tranformaton from one orthogonal polaraton ba {H V} nto another { j} or {A B} a untary congruence (untary conmlarty) tranformaton of the orgnal nclar catterng matr [ ] nto [ ] j, where [ ] [ U ][ ] [ U ] j or [ j] [ U] [ ] [ U] ( ) ( ) (.) wth [ U ] gven by (.), o that the component of the general non-ymmetrc catterng matr for the btatc cae n the new polarzaton ba, characterzed by a comple polarzaton rato, can be wrtten a [8, 5] j j [ ] HH VH + + VV [ + ] HH VH + VV [ + ] HH VH + VV (.) [ + ] jj HH VH + + here et three nvarant for the general btatc cae (BA) under the change-of-ba tranformaton a gven by (.5): () pan ] { } { } 4 [ HH VH VV j j jj κ (.) confrm that the total power conerved, and t known a Kennaugh pan-nvarant κ 4 ; VV - 8 O-N--8b

29 (), for monotatc cae (.) VH j j warrantng ymmetry of the catterng matr n any polarzaton ba a long a the BA for the trctly mono-tatc but not general btatc cae mpled; () Det ] } Det{[ ] } or Det{[ ( )]} Det{[ ( j)]} (.4) {[ j due to the fact that Det {[ U ]} mple determnantal nvarance. In addton, dagonalzaton of the catterng matr, alo for the general btatc cae, can alway be obtaned but reure med ba repreentaton by ung the ngular Value Decompoton heorem (VD) [5, 5] o that the dagonalzed catterng matr [ D ] can be obtaned by the left and rght ngular vector, where λ [ D ] [ QL ][ ][ Q ] wth [ D ] (.5) λ and Det {[ Q ]} Det{[ Q ]} L and λ and λ denote the dagonal egenvalue of [ ], and the dagonal element λ and λ can be taken a real nonnegatve and are known a the ngular value of the matr [ ]. For the ymmetrc catterng matrce n the mono-tatc cae (MA), dagonalzaton acheved accordng to the untary conmlarty tranform for whch [ Q ] [ Q ] (.6) L and above euaton wll mplfy due to the retrcton of ymmetry j. j.4 he 44 Mueller (Forward catterng) [M] and the 44 Kennaugh (Back-catterng) [K] ower Denty Matrce For the partally polarzed cae, there et an alternate formulaton of epreng the cattered wave n term of the ncdent wave va the 44 Mueller[ M ] and Kennaugh [ K ] matrce for the FA and BA coordnate formulaton, repectvely, where [ ] [ M ][ ] (.7) For the purely coherent cae, [ M ] can formally be related to the coherent Jone catterng Matr [ ] a [ ] [ A] ([ ] [ ] )[ A] [ A]([ ] [ ] )[ ] [ M ] A (.8) wth ymbolzng the tandard Kronecker tenoral matr product relaton [] provded n (A.), Append A, and the 44 epanon matr [ ] A gven by [76] a [ A ] (.9) j j O-N--8b - 9

30 wth the element M j of [ M ], gven n (B.), Append B. pecfcally we fnd that f [ ] normal,.e. [ M ][ M ] [ M ] [ M ]., then [ M ] alo normal,.e. * * [ ][ ] [ ] [ ] mlarly, for the purely coherent cae [76], [ K ] can formally be related to the coherent nclar matr [ ] a [ K] [ A] ([ ] [ ] )[ A ] (.) where [ A] [ A] (.) and for a ymmetrc nclar matr [ ], then [ K ] ymmetrc, keepng n mnd the mathematcal formalm [ M ] dag[ ] [ K], but great care mut be taken n trctly dtnguhng the phycal meanng of [ K ] veru [ M ] n term of [ ] veru [ ] repectvely. hu, f [ ] ymmetrc, VH, then [ K ] ymmetrc, K j K j ; and the correct element for [ M ], [ K ] and the ymmetrc cae are preented n (B. B.5), Append B..5 he Grave olarzaton ower catterng Matr [G] Kennaugh ntroduce, net to the Kennaugh matr[ K ], another formulaton[ G ], for epreng the power n the cattered wave to the ncdent wave for the coherent cae n term of the o-called Grave polarzaton coherent power catterng matr [ G ], where (.) [ G] 8 π Zr o that n term of the Kennaugh element K j, defned n the append, for the mono-tatc cae [ G] [ ] K [ ] K + K + jk 4 K K jk K 4 (.) By ung a ngle coordnate ytem for (, y, z ) and (, y, z ) for the monotatc cae, a n Fg.., and alo decrbed n detal n [9], t can be hown that for a catterer enemble (e.g. precptaton) for whch ndvdual catterer move lowly compared to a perod of the llumnatng wave, and uckly compared to the tme-averagng of the recever, tme-averagng can be adjuted to fnd the decompoed power catterng matr [ G ], a HH HH VH VH VV [ G ] [ ( t)] [ ( t)] [ GH ] + [ GV ] + (.4) HH VH VV VV - O-N--8b

31 h how that the tme averaged Grave ower catterng Matr [ G ], frt ntroduced by Kennaugh [4, 5], can be ued to dvde the power that are receved by lnear horzontally and vertcally by polarzed antenna, a dcued n more detal n [9] and n []. It hould be noted that a mlar decompoton alo et for the Muller/Jone matrce, commonly denoted a FA power catterng matr HH HH VH VHVV F H V + (.5) HH VHVV VV + [ ] [ ( t) ] [ ( t) ] [ F ] + [ F ] whch not further analyzed here []..6 Co/Cro-olar Backcatterng ower Decompoton for the One-Antenna (rancever) and the Matched wo-antenna (Qua-Monotatc) Cae Aumng that the catterer placed n free unbounded pace and that no polarzaton tate tranformaton occur along the propagaton path from the tranmtter () to the catterer ncdence (), and along that from the catterer() to the recever (), then the value of the termnal voltage of the recever, V, nduced by an arbtrarly cattered wave at the recever, defned by the radar brghtne functon V, and the correpondng receved power epreon V * h ˆ V V (.6) wth the defnton of the Kennaugh matr [ K ] n term of the nclar matr [ ], the receved power or radar brghtne functon may be re-epreed where ˆ [] [ K] and the correpondng normalzed toke vector. h (.7) For the one-antenna (trancever) cae the co-polar channel (c) and the cro-polar channel () power become: wth c h ˆ [ ] [ Kc ] (.8) ˆ [] [ K] h (.9) ([ A] ) ([ ] [ ] )[ A] [ C][ K ] [ K ] (.) c and ([ A] ) ([ Y ][ ] [ ] )[ A] [ C ][ K ] [ K ] (.) O-N--8b -

32 [C] [Y ] [X ] (.) For the wo-antenna Dual olarzaton cae, n whch one antenna erve a a tranmtter and the other a the recever, the optmal receved power m for the matched cae become by ung the matchng condton ˆ / (.) * h m o that m [ K ], where [ K ] [ K ] + [ K ] [ K ][ K ], and m m c [ K ] (.4) whch repreent an eental relatonhp for determnng the optmal polarzaton tate from the optmzaton of the Kennaugh matr..7 he catterng Feature Vector : he Lecographc and the aul Feature Vector Up to now we have ntroduced three decrpton of the catterng procee n term of the Jone veru nclar, [ ] veru [ ], the power catterng matrce, [ F] veru [ G ], and the 44 power denty Muller veru Kennaugh matrce, [ M ] veru [ K ]. Alternatvely, the polarmetrc catterng problem can be addreed n term of a vectoral feature decrptve formulaton [4] borrowed from vector gnal etmaton theory. h approach replace the catterng matrce [ ] veru[ ], the power catterng matrce [ F] veru [ G ], and the 44 Muller [ M ] veru Kennaugh [ K ] matrce by an euvalent four-dmenonal comple catterng feature vector f 4, formally defned for the general b-tatc cae a HH [ ] 4 F{[ ]} race{[ ] ψ} [ f f f f] VH f (.5) VV where F{[ ]} the matr vectorzaton operator race{[ ]} the um of the dagonal element of [ ], and ψ a complete et of comple ba matrce under a hermtan nner product. For the vectorzaton of any complete orthonormal ba et [97] of four matrce that leave the (ucldean) norm of the catterng feature vector nvarant, can be ued, and there are two uch bae favored n the Ψ, and the other aul pn matr et polarmetrc radar lterature; one beng the lecographc ba [ ] L [ Ψ ]. We note here that the dtncton between the lecographc and aul-baed feature vector repreentaton related to rncpal and Independent Component Analy (CA/ICA) whch an nteretng topc for future reearch. () he Lecographc Feature vector : ung [ Ψ L ], wth f 4L, obtaned from the mple lecographc epanon of [ ] - O-N--8b