2. SINGLE VS. MULTI POLARIZATION SAR DATA

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1 . SINGLE VS. MULTI POLARIZATION SAR DATA.1 Scatterng Coeffcent v. Scatterng Matrx In the prevou chapter of th document, we dealt wth the decrpton and the characterzaton of electromagnetc wave. A t wa hown, one of the man properte of a tranvere electromagnetc wave the vectoral nature of the electromagnetc feld, whch called polarmetry. An electromagnetc wave travel n tme and pace. In th voyagng through the pace, t may happen that the wave can reach a partcular target, then nteractng wth t, ee Fgure 1. A a conequence of th nteracton, part of the energy carred by the ncdent wave aborbed by the target telf, wherea the ret reradated a a new electromagnetc wave. Due to the nteracton wth the target, the properte of the reradated wave can be dfferent from thoe of the ncdent one. Then, the queton whch re at th pont f thee change could be employed to characterze or dentfy the target. In partcular, we are ntereted n the change concernng the polarzaton of the wave. In the followng, we preent the way n whch the nteracton of an electromagnetc wave and a gven target can be repreented. Incdent Wave r r r E ( ) = E e 0 ( jk r ) Scattered Wave r r r E ( ) = E e 0 ( jk r ) Target Fgure 1 Interacton of an electromagnetc wave and a target..1.1 Sngle Polarzaton Image: Scatterng Coeffcent Before to defne the nteracton of electromagnetc wave wth the nature, t neceary to ntroduce two mportant concept concernng the dea of target, nce they wll determne the way n whch they hall be characterzed. Gven a radar confguraton a depcted by Fgure 1, t may happen the target of nteret to be maller than the coverage of the radar ytem. In th tuaton, we conder the target a an olated catterer and from a pont of vew of power exchange, th target characterzed by the o-called radar cro ecton. Neverthele, we can fnd tuaton n whch the target of nteret gnfcantly larger that the coverage provded by the radar ytem. In thee occaon, t more convenent to characterze the 1

2 target ndependently of h extend. Hence, n thee tuaton, the target decrbed by the ocalled catterng coeffcent. The mot fundamental form to decrbe the nteracton of an electromagnetc wave wth a gven target the o-called radar equaton. Th equaton etablhe the relaton between the power whch the target ntercept from the ncdent electromagnetc wave E r and the power reradated by the ame target n the form of the cattered wave E r. The radar equaton preent the followng form PG A P = r σ 4π 4 (1) t t r Rt π Rr where P r repreent the power detected at the recevng ytem. The term PG t 4π R t t determned by the ncdent feld E r and t cont of t power denty expreed n term of the properte of the tranmttng ytem. The dfferent term n () are: the tranmtted power P t, the antenna gan G t and the dtance between the ytem and the target R t. On the contrary, the term A 4π R r r contan the parameter concernng the recevng ytem: the effectve aperture of the recevng antenna A r and the dtance between the target and the recevng ytem R r. The lat term n (1),.e, σ, determne the effect of the target of nteret on the balance of power etablhed by the radar equaton. Snce () a power denty,.e., power par unt area and (3) dmenonle, the parameter σ ha unt of area. Conequently, σ cont of an effectve area whch characterze the target. Th parameter determne whch amount of power ntercepted from the denty () by the target and reradated. Th reradated power fnally ntercepted by the recevng ytem (3), accordng to the dtance R t. An mportant fact whch are at th pont the way the target reradate the ntercepted power n a gven drecton of the pace. In order to be ndependent of th property, the radar cro ecton hall be referenced to and dealzed otropc catterer. Thu, the radar cro ecton of an object the cro ecton of an equvalent otropc catterer that generate the ame cattered power denty a the object n the oberved drecton r E σ = 4πR r = 4π S (4) E where E r repreent the ntenty of the electromagnetc feld and S the complex catterng ampltude of the object. The fnal value of σ a functon of a large number of parameter whch are dffcult to conder ndvdually. A frt et of thee parameter are concerned wth the magng ytem: Wave frequency f. () (3)

3 Wave polarzaton. Th dependence pecally condered later. Imagng confguraton, that, ncdent (θ,φ ) and catterng (θ,φ ) drecton. A econd et of parameter are related wth the target telf Object geometrcal tructure. Object delectrcal properte. Then, the radar cro ecton σ able to characterze the target beng maged for a partcular frequency, and magng ytem confguraton. The radar equaton, a gven by (1), vald for thoe cae n whch the target of nteret maller than the radar coverage, that, a pont target. For thoe target preentng an extend larger than the radar coverage, we need a dfferent model to repreent the target. In thee tuaton, a target repreented a an nfnte collecton of tattcally dentcal pont target. Fgure preent an cheme of th type of target. Fgure Interacton of an electromagnetc wave and an extended target. A t can be oberved n Fgure, the reultng cattered feld E r reult from the coherent addton of the cattered wave from every one of the ndependent target whch model the extended catterer. In order to expre the catterng properte of the extended target ndependently of t area extend, we conderer every elementary target a beng decrbed by a dfferental radar cro ecton dσ. In order to eparate the effect of the target extend, we conder dσ a the product of the averaged radar cro ecton per unt area σ 0 and the dfferental area occuped by the target d. Then, the dfferental power receved by the ytem due to an elementary catterer can be wrtten a PG A dp = r σ d 4π 4 (5) t t 0 r Rt πrr Hence, to fnd the total power receved from the extended target we need to ntegrate over the llumnated area A 0 3

4 P r = PG A σ d 4π 4 (6) A 0 t t 0 r Rt πrr It mut be noted that the radar equaton at (1) repreent a determntc problem, wherea (6) conder a tattcal problem. Eq. (6) repreent the average power returned from the extended target. Hence, the radar cro ecton per unt area σ 0, or mply catterng coeffcent, the rato of the tattcally averaged cattered power denty to the average ncdent power denty over the urface of the phere of radu R r r 0 σ E 4π Rr σ = = r (7) A 0 A0 E The catterng coeffcent σ 0 a dmenonle parameter. A n the cae of the radar cro ecton, the catterng coeffcent employed to characterze the cattered beng maged by the radar. Th characterzaton for a partcular frequency f, polarzaton of the ncdent and cattered wave and ncdent (θ,φ ) and catterng (θ,φ ) drecton..1. Dfferent Emon-Recepton Polarzaton State A t ha been hown n the prevou ecton, the characterzaton of a gven catterer by mean of the radar cro ecton σ or the catterng coeffcent σ 0 depend alo on the polarzaton of the ncdent feld E r. A one can oberve n (4) and (7), thee two coeffcent are expreed a a functon of the ntenty of the ncdent and cattered feld. Conequently, σ and σ 0 hall be only enble to the polarzaton of the ncdent feld through the effect the polarzaton ha over the power of the related electromagnetc wave. Hence, f we denote by p the polarzaton of the ncdent feld and by q the polarzaton of the cattered feld, we can defne the followng polarzaton dependent radar cro ecton and catterng coeffcent repectvely r E qp σqp = 4πR r = 4π S qp (8) Eqp r σ E 0 4 qp qp π Rr σ qp = = r (9) A 0 A0 E qp.1.3 General Cae: Scatterng Matrx A t ha been hown n the prevou two ecton, a gven target of nteret can be characterzed by mean of the radar cro ecton or the catterng coeffcent dependng on the nature of the catterer telf, ee (4) and (7). Addtonally, n (8) and (9) t ha been hown that thee two coeffcent depend alo on the polarzaton of the ncdent and the cattered electromagnetc feld. A cloer look to thee expreon reveal that thee two coeffcent 4

5 depend on the polarzaton of the electromagnetc feld only through the power aocated wth them. Thu, they do not explot, explctly, the vectoral nature of polarzed electromagnetc wave. Conequently, n order to take advantage of the polarzaton of the electromagnetc feld, that, ther vectoral nature, the catterng proce at the target of nteret mut be condered a a functon of the electromagnetc feld themelve. In the prevou chapter, t wa hown that the polarzaton of a plane, monochromatc, electrc feld could be repreented by the o-called Jone vector. Addtonally, a et of two orthogonal Jone vector form a polarzaton ba, n whch, any polarzaton tate of a gven electromagnetc wave can be expreed. Therefore, gven the Jone vector of the ncdent and the cattered wave, E and E repectvely, the catterng proce occurrng at the target of nteret expreed a follow S S S E = S (,//) E r = E r S// S //// jkr jkr e e // where the matrx [S (,//) ] receve the name of catterng matrx and the entre of th matrx S pq, for p, q (,//), are the o-called complex catterng coeffcent or complex catterng ampltude. The dagonal element of the catterng matrx receve the name of co-polar term, nce they relate the ame polarzaton for the ncdent and the cattered feld. Neverthele, the off-dagonal element are known a cro-polar term a they relate jkr orthogonal polarzaton tate. Fnally, the term e r take nto account the propagaton effect both, n ampltude and phae. It mut be taken nto account that the relaton expreed by (10) only vald for the far feld zone, where the planar wave aumpton condered for the ncdent and the cattered feld. Conderng (8), the element of the catterng matrx can be related wth the radar cro ecton of a gven target a follow (10) σ qp Sqp = p, q (,//) (11) 4π A t can be deduced from the prevou equaton, the characterzaton of a gven target by mean of the catterng matrx allow the poblty to explore the phae nformaton provded by the phae of complex catterng coeffcent, and no only the ntenty or ampltude. A one can oberve, the polarmetrc catterng equaton preented at (10) nvolve the Jone vector of the ncdent and the cattered feld, whch characterze ther polarzaton properte n a gven coordnate ytem. In order to be correct, thee two Jone vector mut be expreed wthn the ame coordnate reference. A a reult, the catterng matrx hall be aocated to a partcular coordnate ytem. In (10), we conder the coordnate ytem centered at the target. Hence, the ba (,//) refer to a plane repect the coordnate ytem centered n the target to whch the feld and the catterng matrx are referred. The followng example demontrate the mportance of the phae parameter. Let conder a trhedral and a dhedral. Thee two target preent the radar cro ecton coeffcent and catterng matrce gven n Fgure 3 n the polarzaton ba formed wth the horzontal and vertcal polarzaton tate, whch are parallel to the ˆx and ŷ ax, repectvely. A t can be concluded from Fgure 3, the trhedral and the dhedral cannot be dfferentated n term of the radar cro ecton coeffcent, wherea they are een a dfferent object f they are analyzed by mean of the correpondng catterng matrce. The concluon whch can be 5

6 extracted at th pont that polarmetry open the door to conder phae meaurement to characterze the target. ŷ ŷ ˆx ˆx Trhedral σxx σxy 1 0 4π σ yx σ = yy S ( xy, ) = 0 1 Fgure 3 Trhedral and dhedral polarmetrc characterzaton. Dhedral σxx σxy 1 0 4π σ yx σ = yy S ( xy, ) = 0 1 Snce the catterng matrx [S (,//) ] employed to characterze a gven target, t can be parameterzed a follow ( ϕ ϕ ) j jkr jϕ jϕ // jkr j // e S e S S S // e ϕ e e // e S = = S e S e 1443 S e S e (,// ) jϕ jϕ // //// j( // ) j r r ϕ ϕ ( ϕ ϕ //// ) // //// Abolute Phae Term // //// RelatveScatterng Matrx (1) The abolute phae term n (1) not condered a an ndependent parameter nce t preent an arbtrary value due to t dependence on the dtance between the radar and the target. Conequently, t aumed that the catterng matrx can be parameterzed by 7 ndependent parameter: the ampltude { S, S //, S //, S //// } and the relatve phae {(ϕ // - ϕ ), (ϕ // -ϕ ),(ϕ //// -ϕ )}. A a concluon, a gven target of nteret determned by 7 ndependent parameter n the mot general cae and an abolute value. It mportant, at th pont, to analyze ome partcular apect about the defnton of the matrx [S (,//) ] and the relaton about the dfferent coordnate ytem whch can be defned to decrbe the catterng proce characterzed n (10). A t wa already hghlghted n the prevou two ecton of th chapter, the radar cro ecton and the catterng coeffcent depend on the drecton of the ncdent and the cattered wave. When conderng the matrx [S (,//) ], the analy of th dependence of extreme mportance nce t alo nvolve the defnton of the polarzaton of the ncdent and the cattered feld. Snce (10) conder the polarzed electromagnetc wave themelve, t mandatory to aume a frame n whch the polarzaton defned. There ext two prncpal conventon concernng the framework where the polarmetrc catterng proce condered: Forward Scatter Algnment (FSA) and Backcatter Algnment (BSA), ee Fgure 4. In both cae, the electrc feld of the ncdent and the cattered wave are expreed n local coordnate ytem centered on the tranmttng and recevng antenna, repectvely. 6

7 All coordnate ytem are defned n term f a global coordnate ytem centered nde the target of nteret. (a) (b) Fgure 4 Reference framework: (a) FSA, (b) Btatc BSA and (c) Monotatc BSA. The FSA conventon, ee Fgure 4a, alo called wave-orented nce t defned relatve to the propagatng wave, normally condered n btatc problem, that t, n thoe confguraton n whch the tranmtter and the recever are not located at the ame patal poton. The btatc BSA conventon framework, ee Fgure 4b, defned, on the contrary, repect to the radar antenna n accordance wth the IEEE tandard. The advantage of the BSA conventon that for a monotatc confguraton, alo called backcatterng confguraton, that, when the tranmttng and recevng antenna are collocated, the coordnated ytem of the two antenna concde, ee Fgure 4c. Th confguraton preferred n the radar polarmetry communty. In the monotatc cae, the catterng matrx n the FSA conventon, [S (,//) ] FSA, can be related wth the ame matrx referenced to the monotatc BSA conventon [S (,//) ] BSA a follow 1 0 S (,//) S = (,// BSA ) 0 1 FSA (c) (13) A t ha been mentoned prevouly, n the radar polarmetry communty, the monotatc BSA conventon (backcatterng) condered a the framework to characterze the catterng 7

8 proce. The reaon to elect th confguraton due to fact that the majorty of the extng polarmetrc radar ytem operate wth the ame antenna for tranmon and recepton. One mportant property of th confguraton, for recprocal target, recprocty, whch tate that [ S // ] [ S// ] BSA [ S ] [ S ] = (14) BSA = (15) // FSA // FSA Then, the formalzaton of the catterng proce gven by (10), n the monotatc cae under the BSA conventon, reduce to S S S E = S (,//) E r = E r S // S //// jkr jkr e e // In the ame ene, equaton (1) take the form ( ϕ ϕ ) j jkr jϕ jϕ // jkr j // e S e S S S // e ϕ e e // e S = = S e S e 1443 S e S e (,// ) jϕ jϕ // //// j( // ) j r r ϕ ϕ ( ϕ ϕ //// ) // //// Abolute Phae Term // //// RelatveScatterng Matrx The man conequence of (17) that n the backcatterng drecton, a gven target no longer characterzed by 7 ndependent parameter, but by 5. Thee are: the ampltude { S, S //, S // } and the relatve phae {(ϕ // -ϕ ),(ϕ //// -ϕ )} and one addtonal abolute phae. (16) (17). Addtonal Informaton..1 Span: Jont Intenty Informaton A t wa ndcated at the begnnng of Secton.1.1, a central parameter when conderng the catterng proce occurrng at a gven target cont of the cattered power. For ngle polarzaton ytem, the cattered power determned by mean of the radar cro ecton or the catterng coeffcent. Neverthele, a polarmetrc radar ha to be condered a a mult channel ytem. Conequently, n order to determne the cattered power, t neceary to conder all the data channel, that, all the element of the catterng matrx. The total cattered power, n the cae of a polarmetrc radar ytem know a pan, beng defned n the mot general cae a ( ( ) ) trace,// (,// ) (,// ) T * SPAN S S S = = S + S // + S// + S//// (18) where trace(.) repreent the trace of a matrx, contng of the addton of the element of the prncpal dagonal. In the backcatterng cae, due to the recprocty theorem, the pan reduce to ( ( ) ) SPAN S = S + S + S,// // //// (19) 8

9 The man property of the pan that t polarmetrcally nvarable, that, t doe not depend on the polarzaton ba employed to decrbe the polarzaton of the electromagnetc wave. Now, f we conder the defnton (11) nto (19) we get ( ( ) ) = 4,// ( + + // //// ) SPAN S π σ σ σ (0) (db) S hh (db) S hv (db) S vv -30dB -15dB 0dB Span Fgure 5 Intente of the element of the catterng matrx meaured n the ba (h,v) and the reultng pan. Therefore, the pan preent the ame lmtaton a the radar cro ecton n order to repreent the polarmetrc nformaton contaned n the catterng matrx, that, the mportant 9

10 nformaton provded by the meaurement of the relatve phae completely lot. Addtonally, nce t polarmetrcally nvarable, t doe not contan polarmetrc nformaton at all. One of the man applcaton of the pan to preent the multdmenonal nformaton provded by the catterng matrx n a ngle mage. But, a hown n the prevou paragraph, th mage doe not contan any polarmetrc nformaton at all. Fgure 5 preent an example of the pan mage... Lexcographc Color-Coded Repreentaton In the prevou ecton, we howed that t wa poble to preent part of the nformaton contaned n the catterng matrx [S (,//) ] n a ngle mage by mean of the pan of the catterng matrx, but at the expene to looe all the polarmetrc nformaton. In what t follow, we ntroduce two alternatve to repreent part of the multdmenonal nformaton provded by the catterng matrx n a ngle mage. Thee two alternatve are baed on the decompoton RBG of color mage....1 RBG color codng The RGB color model an addtve color model n whch red, green, and blue lght are combned n varou way to create other color. The very dea for the model telf and the abbrevaton "RGB" come from the three prmary color n addtve lght model. The concept to retan that we can create any color of the pectra by combnng the three prmary color: red, green and blue lght, that, color can be condered a a three dmenonal pace. Fgure 6 preent the color pace, coded n the RGB bae. Blue Whte Red Green Fgure 6 RBG color cube. Conequently, three dfferent parameter can be repreented n a ngle color mage by codng each ntal parameter a one color of the RGB pace. Therefore, the color n the reultng mage can be nterpreted n term of the three electedparameter.... Polarmetrc nformaton coded a a color The color repreentaton preented above now condered to repreent the polarmetrc nformaton provded by the catterng matrx [S (,//) ]. The lexcographc color-coded 10

11 repreentaton conder the ntenty of the element of [S (,//) ], n the backcatterng drecton, a the dmenon of the color pace. A poble codfcaton could be S // //// Red S Green (1) S Blue depte any other codfcaton alo valuable. Agan, t mut be menton that, nce we are conderng the ntenty of the dfferent element of the catterng matrx, th codfcaton not able to reflect the nformaton that can be provded by the relatve phae of element...3 Importance of Relatve Phae Term: Paul Color-Coded Repreentaton In the followng we preent a color-coded repreentaton of the nformaton provded by the catterng matrx whch take nto account the nformaton provded by the relatve phae of the dfferent entre of the matrx. Any ymmetrc matrx can be parameterzed a a functon of three parameter {a, b, c} a follow S11 S1 a+ b c S1 S = c a b If we now conder the repreentaton of the catterng matrx n the backcatterng cae preented n (16), we can obtan the followng equvalence S + S a = S S b = c = S // The Paul color-coded repreentaton conder the codfcaton n color of the ntenty of the parameter n (3) a follow //// //// () (3) b c a Red Green Blue (4) Fnally, Fgure 8a preent an example or the Paul Color coded repreentaton of the mage channel preented n Fgure 5..3 Polarzaton Ba and Polarmetrc Sgnature 11

12 .3.1 Acquton Accordng to Orthogonal Jone Vector: Polarzaton Ba A t wa demontrated n the precedng chapter, the Jone vector correpondng to a gven electromagnetc feld can be expreed n an orthonormal ba (,//) a E = E + ˆ E // ˆ (5) where the untary vector ˆ and // ˆ form the o-called polarzaton ba. If now, we conder the polarmetrc catterng equaton, ee (10), the Jone vector correpondng to the ncdent and the cattered wave, E and E repectvely, are expreed n the orthonormal ba (,//). Conequently, the catterng matrx [S (,//) ] pecfe how the target tranform the polarzaton of the ncdent feld to the cattered feld when both are expreed n the ba (,//). A a reult, t ad that the catterng matrx [S (,//) ] ha been acqured wth repect to the polarzaton ba (,//). //.3. Acquton n ONE Polarzaton Ba: Repreentaton n ANY Polarzaton Ba The catterng properte of a gven target, a demontrated, are contaned wthn the catterng matrx [S (,//) ], whch, a hown prevouly, meaured n the partcular polarzaton ba (,//). Snce, there ext an nfnte number of orthonormal polarzaton bae, the queton rang at th pont that whether t poble or not to nfer the polarmetrc properte of the gven target n any polarzaton ba from the repone meaured at a partcular ba, for ntance (,//). Th queton preent an affrmatve anwer. The poblty to yntheze any polarmetrc repone of a gven target from t meaurement n a partcular orthonormal ba repreent the mot mportant property of polarmetrc ytem n comparon wth ngle-polarzaton ytem. The mot mportant conequence of th proce that the amount of nformaton about a gven catterer can be ncreaed, allowng a better characterzaton and tudy. Th polarzaton ynthe proce baed on the concept of change of polarzaton ba preented n the prevou chapter. In what t follow, we hall conder the polarzaton ynthe proce n the backcatterng drecton or monotatc confguraton of a radar ytem. 1

13 Incdent Wave r r ( jk r ) r E ( r ) = E e 0 k ˆ k ˆ = k ˆ [ S ] Target Scattered Wave r r r r ( jk r ) ( jkr ) ( ) E r = Ee = Ee 0 0 TRANSMITTER RECEIVER Fgure 7 Polarzaton ynthe proce n the backcatterng drecton. Before to decrbe the polarzaton ynthe proce n the backcatterng drecton, t neceary to analyze the catterng proce gven by (10) wth repect to the drecton of propagaton of the ncdent and the cattered feld. A oberved n Fgure 7, the ncdent feld propagate n the drecton gven by the untary vector k, wherea the cattered one propagate n the oppote drecton, gven by k ˆ. Conequently, th dfference n the propagaton drecton mut be taken nto account when defnng the polarzaton tate of the wave. Gven a Jone vector propagatng n the drecton ˆk, the Jone vector of the ame wave, but whch propagate n the drecton ˆk obtaned a * ( ) ( ) kˆ kˆ E kˆ = E ˆ k (6) ˆ From now, we hall not conder the catterng matrx referred to the coordnate ytem centered at the target, but n the ax coordnate centered n the tranmttng/recevng ytem. Conder a polarmetrc radar ytem a depcted by Fgure 7, whch tranmt the electromagnetc wave n the followng orthonormal ba (A, A ). In th partcular ba, the ncdent and cattered feld are related by the catterng matrx a follow S S AA AA E = E S SAA S = A A ( AA, ) ( AA, ) ( AA, ) ( AA, ) E (7) A hown n the prevou chapter, gven the Jone vector meaured n a partcular ba, for ntance (A, A ), t poble to derve t n any other polarzaton ba (B, B ) by mean of the followng mple mathematcal tranformaton E( ) = U BB, ( AA, ) ( BB, ) E a ( AA, ) (8) Then, f we conder (8) for the ncdent Jone vector n (7), we get = a, (9) E U E ( BB, ) ( AA, ) ( BB, ) ( AA ) 13

14 = a, (30) E U E ( AA, ) ( BB, ) ( AA, ) ( BB ) In order to apply the tranformaton ba procedure to the cattered feld E( AA, ) we need to conder that t propagate n the oppote drecton a the ncdent feld E ( AA, ). Hence, n order to conder both feld n the ame frame of reference, that, ther polarzaton are referred to the ame coordnate ytem we need to conder (6). A a reult, the tranformaton ba procedure apple to the cattered feld a follow * E U E = ( BB, ) ( AA, ) a ( BB, ) ( AA ) * E U E, ( A, A ) ( B, B ) ( A, A ) ( B, ) Now, ntroducng the reult of (30) and (3) nto (7) we get (31) = a B (3) * U E = S U E a a ( BB, ) ( AA, ) ( BB, ) ( AA, ) ( BB, ) ( AA, ) ( BB, ) * 1 E = U S U E a a, ( BB, ) ( BB, ) ( AA, ) ( AA, ) ( BB, ) ( AA, ) ( BB ) * Snce the tranformaton matrx U ( AA ) ( ) untary,.e.,, a BB, [ U] = [ U] 1 T, we get (33) (34) (a) S hh + S vv S hh - S vv S hv 14

15 (b) S aa + S bb S aa - S bb S ab (c) S ll + S rr S ll - S rr S lr Fgure 8 Polarzaton ynthe. (a) Meaured repone at the lnear polarzaton ba (h,v). (b) Synthe of the repone n the lnear ba rotated 45 degree (a,b). (c) Synthe n the crcular polarzaton ba (l,r). T E U S U E a a, = ( BB, ) ( BB, ) ( AA, ) ( AA, ) ( BB, ) ( AA, ) ( BB ) (35) In the prevou equaton, we can clearly dentfy, then, T S ( ) U ( ) ( ) S BB, BB, AA, ( AA, ) U = a ( BB, ) a ( AA, ) (36) The tranformaton expreed n (36) receve the name of con-mlarty tranformaton. Th tranformaton allow to yntheze the catterng matrx n an arbtrary ba (B, B ), from t meaure n the ba (A, A ). If ntead of an arbtrary ba, we conder the lnear orthogonal polarzaton ba (ˆ x, y ˆ, the tranformaton matrx from ( ˆ, ˆ) x y to an arbtrary ellptcal ba can be parameterzed a U U U U = // ( ˆ ˆ) ( ) ( φ) ( τ xy, A, A ) ( α a ) jα coφ nφ coτ jnτ e 0 = jα nφ coφ jnτ coτ 0 e Fgure 8 preent an example of the applcaton of the con-mlarty tranformaton gven at (36) to yntheze the polarmetrc repone at dfferent polarzaton ba. The polarmetrc nformaton repreented by mean of the Paul color-coded repreentaton. The orgnal polarmetrc et, preented n Fgure 8a obtaned n the lnear polarzaton ba (h,v), where h tand for the horzontal polarzaton and v for the vertcal polarzaton. Ung (37) the repone to two dfferent polarzaton ba ynthezed. Fgure 8b preent the repone to the orthonormal ba (a,b) where a ndcate the lnear polarzaton at 45 degree ) (37) 15

16 and b the lnear polarzaton at -45 degree. Fnally, Fgure 8c how the repone to the crcular polarzaton ba (l,r), where l refer to the left crcular polarzaton and r to the rght crcular polarzaton..3.3 Complete Polarmetrc Characterzaton of Each Pxel: Polarzaton Sgnature In the prevou ecton, we have employed the powerful con-mlarty tranformaton to yntheze the polarmetrc repone jut n two alternatve polarzaton bae: the 45 degree rotated lnear ba (a, b) and the crcular polarzaton ba (l, r). Neverthele, th ecton am to explot all the nformaton whch can be extracted from th tranformaton through the exploraton of the pace of all poble polarzaton bae. The objectve of th proce to preent a new way to characterze a gven catterer. Fgure 9 preent the cheme of a general polarmetrc radar employed to meaure a gven target, characterzed by a partcular catterng matrx. In th cheme, the Jone vector A refer to the polarzaton emtted by the tranmttng ytem and B ndcate the Jone vector contanng the polarzaton wth repect to whch the recevng antenna receve the cattered feld E. In th context, the power receved at the recevng ytem obtaned by mean of T B E P = α (38) where α contan all thoe term whch do not depend on the polarzaton of the antenna A and B. Ung the characterzaton of the target by mean of the catterng matrx, t poble to rewrte (38) a follow [ S ] Target A E B TRANSMITTER RECEIVER Fgure 9 General polarmetrc ytem confguraton. [ ] B T PA α B S = A (39) In the frame of (39), two dfferent power meaurement are defned: 16

17 The co-polarzed power. In th confguraton, we aume that the tranmttng and recevng antenna are characterzed by the ame polarzaton tate. Hence, (39) turn out to be [ ] T PCO α A S = A (40) The cro-polarzed power. In th cae, t aumed that the recever antenna receve wth the orthogonal polarzaton of the tranmttng ytem. Thu, we wrte (39) a [ ] T PX α A S = A (41) The co- and cro-polarzed power can be generate ynthetcally, nce, gven the lnear polarzaton ba (ˆ x, y ˆ, we can wrte an arbtrary polarzaton tate and ) coφ nφ coτ jnτ A = xˆ nφ coφ jnτ coτ π π co φ + n φ + co( τ) jn( τ) A= xˆ π π jn( τ) co( τ) n φ co φ + + Conequently, we can obtan the co- and cro-polarzed power, P CO and P X, a a functon of the angle φ and τ, that, the ellpe orentaton and the ellpe aperture whch defne the polarzaton ellpe of an electromagnetc wave. We have not condered here the phae term α. Secton.5 preent the polarzaton gnature for the canoncal catterng mechanm. (4) (43).4 Optmal Polarzaton State and The Poncaré Sphere.4.1 Optmal Polarzaton State: Deducton Orented Parameterzaton A t ha been hown n the prevou ecton of th chapter, the receved power at the recever ytem vare accordng to the polarzaton of the tranmtted and ncdent wave, and on the charactertc of the target under tudy through t catterng matrx [S], (39). In the cae of the polarzaton gnature, we partcularzed the tudy of the receved power to two pecal cae. On the one hand, we have condered the tuaton n whch the tranmttng and recevng ytem employ the ame polarzaton. On the other hand, the cae n whch both ytem ue orthogonal polarzaton ha been analyzed. The polarzaton gnature for the co-polar and cro-polar confguraton, gven by (40) and (41) repectvely, explore all the 17

18 pace of poble polarzaton. Neverthele, the characterzaton of a partcular catterer can be alo performed by tudyng the o culled charactertc polarzaton tate. Thee tate are defned a uch wave polarzaton tate gvng a a reult maxmum or mnmum receved power. Gven a partcular polarzaton tate for an electromagnetc wave, the correpondng Jone vector can be repreented n term of the polarzaton rato ρ a preented n the prevou chapter. Then for the Jone vector A jα e 1 A = 1 ρ ρ + The Jone vector of the orthogonal tate A (44) A + jα * e ρ = 1+ ρ 1 (45) If now, we conder the catterng matrx a [ S] S S XX XY = SXY S YY the co-polar P CO and cro-polar P X can be repectvely wrtten a follow (46) CO α XX ρ XY ρ YY P = S + S + S (47) ( 1 ) P = αρs ρ S + ρsyy (48) * X XX XY Then, n order to derve the charactertc polarzaton tate, we need to calculate the followng dervatve P CO = 0 ρ P X = 0 ρ A one can oberve, (47) and (48) cont of blnear form. Conequently, the procee preented at (49) and (50) to derve the charactertc polarzaton tate hall preent two oluton. The next two ecton analyze the charactertc tate reultng from (49) and (50) Charactertc Polarzaton State n the Co-polar Confguraton The co-polar power P CO preent two polarzaton tate reultng n maxmum receved power. Thee two tate are called COPOL MAX and are repreented by the par of polarzaton tate (K,L). Then, K PK L PL The COPOL MAX polarzaton tate are orthogonal. (49) (50) Global Maxmum (51) Local Maxmum (5) 18

19 Addtonally, the co-polar power P CO ha two charactertc polarzaton tate for whch the receved power zero. Th par of polarzaton tate are named COPOL NULLS and are repreented by (O 1,O ). Thee polarzaton tate gve a a reult P O = (53) O1 0 1 O P O = (54) 0 The man charactertc of thee tate that they are not mutually orthogonal Charactertc Polarzaton State n the Cro-polar Confguraton In the cae of the cro-polar power P X, t poble to fnd three par of orthogonal polarzaton tate whch reult nto charactertc polarzaton tate. The frt par of polarzaton tate reult nto maxmum receved power at the recever ytem. Th par of polarzaton tate receve de name of XPOL MAX and are repreented by the par (C 1,C ). Thee tate reult nto P C1 C P 1 C C The tate (C 1,C ) are mutually orthogonal. Global Maxmum (55) Local Maxmum (56) The econd par of polarzaton tate gve null revved power. Th et of polarzaton tate known a XPOL NULL and are repreented by (X 1,X ), reultng nto PX = (57) X1 0 1 X PX = (58) 0 A frt etablhed by Kennaugh, the XPOL NULL and the COPOL MAX repreent the ame par of polarzaton tate. Conequently, (X 1,X ) cont alo n orthogonal polarzaton tate. Fnally, n the cae of the cro-polar power, t poble to defne a thrd par of polarzaton tate whch can be taken a charactertc. Thee polarzaton tate reult nto a mnmum receved power and are know a XPOL SADDLE. They are repreented by (D 1,D ) and P D1 D P 1 D D Global Mnmum (59) Local Mnmum (60) Agan, (D 1,D ) cont of mutually orthogonal polarzaton tate..4. Repreentaton on The Poncare Sphere In the former chapter, we preented the repreentaton of a gven polarzaton tate nto the ocalled Poncare Sphere. In the followng, we are gong to make ue of th repreentaton to how that the charactertc polarzaton tate par gven prevouly preent a partcular hape nto the Poncare phere. Fgure 10 preent the repreentaton of the charactertc polarzaton tate wthn the Poncare phere. 19

20 V C 1 O 1 L X D 1 D U Q K X 1 O Fgure 10 Repreentaton n the Poncare Sphere of the fve par of charactertc polarzaton tate. C In the prevou fgure, t can be clearly oberved that all the par except (O 1,O ) repreent par of orthogonal polarzaton tate nce they cont of antpodal pont of the Poncare Sphere. Addtonally, t poble to recognze that the orthogonal par of polarzaton tate XPOL NULL (equal to the COPOL MAX) and XPOL MAX defne a plane wthn the Poncare phere whch alo contan the non-orthogonal par of COPOL NULLS..4.3 Polarzaton Fork In the prevou ecton, we have condered the repreentaton of the fve par of charactertc polarzaton tate wthn the Poncare Sphere. Huynen ntroduced the concept of Polarzaton Fork by conderng only the repreentaton of the par (K,L) or (X 1,X ) and the non-orthogonal et (O 1,O ). Fgure 11 gve the repreentaton of the polarzaton fork. In order to better undertand th concept t helpful to conder the parameterzaton of the catterng matrx propoed by Hyunen. Gven an arbtrary catterng matrx, we want to fnd the rotaton matrx [U], n the ene of (37), whch gve a a reult a dagonal catterng matrx. In general, the dagonalzaton of a matrx performed by the tandard egen-decompoton of matrce. Neverthele, n the backcatterng cae under the BSA conventon, we have to be aware that we are dealng wth electromagnetc wave travelng n oppote drecton. Conequently, the dagonalzaton of the catterng matrx mut be done accordng to the con-mlarty tranformaton, a preented n Secton.3.. In th partcular tuaton, the dagonalzaton of the catterng matrx done wth the peudo egen-decompoton, that, 0

21 [ ] S X * = λ X (61) where λ refer to the peudo-egenvalue of [S] and X to the correpondng peudoegenvector. Snce the catterng matrx cont of a complex ymmetrc matrx n the backcatterng cae under the BSA conventon, (61) preent two oluton. A demontrated by Huynen, the peudo-egenvector of the catterng matrx correpond to the XPOL NULL,.e., (X 1,X ). Snce thee two polarzaton tate are orthogonal, t only neceary to pecfy one of them. Therefore, the con-mlarty tranformaton to dagonalze the catterng matrx S U S = U ( X, X ) ( X, X ) a( X, Y) ( X, Y) ( X, X ) a ( X, Y) U X X X Y where, the untary matrx ( 1, ) a(, ) take the form T ( ) ( ) [ X X, X X, Y ] (6) U 1 = 1 X a (63) and t can be parameterzed a gven by (37),.e., t can be expreed n term of the three angle {φ, τ, α}. Huynen parameterzed the reultng dagonal matrx a follow 1 0 j S ( X1, X) = m e ξ 0 tan γ where m repreent the maxmum polarzaton and the angle γ the target hp angle. Fnally ξ cont of an abolute phae. A a reult, Huynen parameterzed the catterng matrx n term of the 5 Euler parameter {m, γ, φ, τ, α} and the abolute phae ξ. A can be oberved n Fgure 11, the angle γ determne the angle between the lne formed by the tate (X 1,X ) and (O 1,O ), both condered to be over the ame ecton or plane of the Poncare phere. Addtonally, the three Euler parameter {φ, τ, α} cont of the rotaton to be done over every one of the three ax defnng the pace n order to brng the polarzaton tate gven by x, correpondng to a lnear polarzed wave, to the polarzaton tate gven by X 1. (64) 1

22 φ $z$z X 1 0 γ O $y$y τ $x$x ν X O 1 Fgure 11 Polarzaton fork..5 Canoncal Scatterng Mechanm A real target preent alway a complex catterng repone a a conequence of t complex geometrcal tructure and t reflectvty properte. Conequently, the nterpretaton of th repone obcure. A t hall be preented n a future chapter, a poble oluton to nterpretate th repone to decompoe t nto the repone of canoncal mechanm. Thee catterng mechanm are characterzed by preentng a mple catterng repone. Th ecton preent the lt of th canoncal catterng mechanm and t characterzaton by the catterng matrx and the polarmetrc gnature. The catterng matrx hall be preented n three dfferent orthogonal polarmetrc bae: Lnear polarzaton ba (h,v) where h and v tand for the horzontal and vertcal polarzaton repectvely. Lnear rotated ba (a,b) where a and b tand for the 45 degree lnear and the -45 degree lnear polarzaton repectvely. Crcular polarzaton ba (l,r) where l and r tand for the left crcular and the rght crcular polarzaton repectvely.

23 .5.1 Sphere, Plane, Trhedral ˆ v t ˆ h t Fgure 1 Trhedral orentated horzontally. Scatterng matrce of a phere, a plane or a trhedral n the three polarzaton ba: Lnear polarzaton ba (h,v) [ S] 1 0 = 0 1 Lnear rotated polarzaton ba (a,b) [ S] 1 0 = 0 1 Co-polar and cro-polar gnature of a phere, a plane or a trhedral: Crcular polarzaton ba (l,r) [ S] 0 j = j 0 P CO P X Fgure 13 Co-polar and Cro-polar gnature correpondng to a phere, plane or trhedral. 3

24 .5. Dpole.5..1 Dpole orentated n the vertcal ax ˆ v t ˆ h t Fgure 14 Dpole orentated n the vertcal ax. Scatterng matrce of a dpole orentated n the vertcal ax n the three polarzaton ba: Lnear polarzaton ba (h,v) [ S] 1 0 = 0 0 Lnear rotated polarzaton ba (a,b) [ S] = 1 1 Crcular polarzaton ba (l,r) [ S] Co-polar and cro-polar gnature of a dpole orentated n the vertcal ax: 1 1 j = j 1 P CO P X Fgure 15 Co-polar and Cro-polar gnature correpondng to a dpole orented vertcally. 4

25 .5.. Orented dpole In th cae, we conder the dpole orented wth an angle φ: ˆ v t l φ ˆ h t Fgure 16 Dpole orented wth a φ angle. Scatterng matrce of a dpole orented wth an angle φ n the three polarzaton ba: Lnear polarzaton ba (h,v) [ S ] 1 co φ n φ = 1 n n φ φ [ S] Lnear rotated polarzaton ba (a,b) coφ nφ co φ = 1 1 co φ coφnφ Co-polar and cro-polar gnature of a dpole orented wth an angle φ: Crcular polarzaton ba (l,r) [ S] jφ 1 e j = jφ j e P CO P X Fgure 17 Co-polar and Cro-polar gnature correpondng to a dpole orented wth a φ angle. 5

26 .5.3 Dhedral ˆ v t ˆ h t Fgure 18 Dhedral orented n the horzontal ax. Scatterng matrce of a dhedral orented n the horzontal ax n the three polarzaton ba: Lnear polarzaton ba (h,v) [ S] 1 0 = 0 1 Lnear rotated polarzaton ba (a,b) [ S] 0 1 = 1 0 Crcular polarzaton ba (l,r) [ S] 1 0 = 0 1 Scatterng matrce of a dhedral orented wth an angle φ n the three polarzaton ba: Lnear polarzaton ba (h,v) [ S] co φ n φ = n φ coφ Lnear rotated polarzaton ba (a,b) [ S] n φ coφ = co φ n φ Crcular polarzaton ba (l,r) [ S ] j φ e 0 = j φ 0 e Co-polar and cro-polar gnature of a dhedral orented n the horzontal ax: P CO P X Fgure 19 Co-polar and Cro-polar gnature correpondng to a dhedral orented n the horzontal ax. 6

27 .5.4 Rght Helx ˆ v t ˆ h t Fgure 0 Rght helx. Scatterng matrce of a rght helx orented wth an angle φ n the three polarzaton ba: Lnear polarzaton ba (h,v) j = j 1 1 j φ 1 [ S] e Lnear rotated polarzaton ba (a,b) 1 j φ j 1 [ S] e = 1 j Co-polar and cro-polar gnature of a rght helx orented at 0 degree: Crcular polarzaton ba (l,r) 0 0 = φ 0 e [ S] j P CO P X Fgure 1 Co-polar and Cro-polar gnature correpondng to a rght helx. 7

28 .5.5 Left Helx ˆ v t ˆ h t Fgure Co-polar and Cro-polar gnature correpondng to a left helx. Scatterng matrce of a left helx orented wth an angle φ n the three polarzaton ba: Lnear polarzaton ba (h,v) j = j 1 1 j φ 1 [ S] e Lnear rotated polarzaton ba (a,b) 1 j φ j 1 [ S] e = 1 j Co-polar and cro-polar gnature of a left helx orented at 0 degree: Crcular polarzaton ba (l,r) [ S ] j φ e 0 = 0 0 P CO P X Fgure 3 Co-polar and Cro-polar gnature correpondng to a left helx. 8