Analytical Solution for a Polarimetric Surface Scattering Model

Transcription

1 Analytcal Soluton for a Polarmetrc Surface Scatterng Model Axel Breuer (1,), Irena Hajnek () (1) Alcatel Space Indutry, 6 avenue Jean-Franço Champollon, 1000 Touloue, France, Emal : axel.breuer@pace.alcatel.fr () IETR,Unverté de Renne 1, 6 avenue Général Leclerc, 504 Renne Cedex, France () German Aeropace Center, Mcrowave and Radar Inttute, PO 1116, 80 Welng, Germany, Emal : rena.hajnek@dlr.de ABSTRACT Surface parameter are mportant for applcaton uch a hydrologcal procee forecatng, agrcultural actvte plannng and clmatc model valdaton. Polarmetrc SAR data analy a promng approach for etmatng quanttatvely urface parameter over extended area. For that purpoe, fully polarmetrc nveron algorthm are under development. The am of the poter to brefly preent a drect fully polarmetrc model, the recent Facet Bragg Model (F- Bragg) [1], to compare two method to obtan the oluton and to dcu ther nveron ablte. 1 INTRODUCTION In order to retreve ol properte (manly ol moture and urface roughne) wth polarmetrc SAR data, an nvere model of the electromagnetc catterng properte of the ol needed. Wherea a drect polarmetrc model tranform the ol parameter nto a coherency matrx [T], the aocated nvere model tranform [T] nto ol parameter. To guarantee the extence of the nvere model, the amount of ol parameter mut be le or equal to the degree of freedom n [T]. [T] ha 6 degree of freedom becaue t a real by ymetrc matrx. For that reaon, a drect polarmetrc model ued for quanttatve remote enng of bare ol hould have no more than 6 parameter to decrbe the ol. The F-Bragg model [1], whch baed on a two-cale decrpton of the urface roughne, an nvertble model wth 5 ol parameter : L c : correlaton length of the large cale roughne H rm : root mean quare (rm) heght of the large cale roughne l c : correlaton length of the mall cale roughne h rm : the mall cale rm heght ol : the relatve delectrc contant of the ol Eventhought the formulaton of F-Bragg relatve mple, the computaton of t oluton not traghtforward. The frt ecton brefly ummarze the phycal and mathematcal apect of F-Bragg. It hown that the Monte-Carlo method an neffectve method to olve the F-Bragg equaton. The econd ecton expoe an alternatve oluton method by dervng the analytcal expreon of F-Bragg. The lat part compare the two method. F-BRAGG MODEL F-Bragg rele on a two-cale decrpton of the roughne derved by two charactertc of the SAR ytem: the wavelength of the emtted electromagnetc feld and the ze of a reoluton cell RR. The electromagnetc feld backcattered by a reoluton cell can be een a the um of the backcattered feld of t elementary catterer. In the F-Bragg model the elementary catterer are aumed to be facet of ze. The mall cale roughne the corrugated roughne on each facet. The large cale roughne defned by the poton of the facet center Z := x,y ) (Fg.1). The two next ubecton wll expoe the mathematcal formulaton of the contrbuton of both cale to the backcatterng of a reoluton cell. The thrd ubecton explan how the expoed formula can be olved by Monte-Carlo mulaton.

2 x k z z n y y x R Fg. 1: Schematc repreentaton of the geometrcal parameter of F-Bragg.( (x,y,z ) : poton vector of facet. vector to facet. : wave length of ncdent wave.1 Small Scale Scatterng E. k : wave vector of E. R : SAR reoluton) n : normal It well known that the backcattered feld E ha a lnear relatonhp to the determntc ncdence feld E expreed by the catterng matrx [S] (Eq 1). S S E E [ S ] E E H E HH HV H (1) V S S E VH VV V where H and V tand repectvely for horzontal and vertcal component of the polarzed wave. When E a determntc vector then [S] a determntc matrx wherea a random backcattered feld, noted E, mple that the catterng matrx [S] alo random, o that Eq.1 become E =[S] E (In the followng, all random varable are wrtten n bold). In the later cae, the relevant parameter decrbng [S] are t moment and not t realzaton. The econd order moment can be ordered n a coherency matrx [T] : [T]: 1 SVV ( SVV )( SVV )* ( SVV )* ( SVV )( SVV )* SVV SHV ( SVV )* where < > degnate the expected value and * the complex conjugate. ( SVV ) * ( SVV ) SHV * 4 SHV () The backcattered feld random becaue the mall-cale roughne a tochatc urface. In F-Bragg, the backcatterng of a facet uppoed to be a conequence of a Bragg mechanm. Conequently, the backcattered feld by the mall cale roughne calculated by a frt order perturbaton technque (alo called Small Perturbaton Model, SPM) leadng to Eq. and Eq. 4. <S HH > = <S HV > = <S VV > = 0 () HH (, ol) VV (, ol) ( HH (, ol) VV (, ol))( HH (, ol) VV (, ol)*) [ T ( )] : h (, ) Bragg rm m l c ( (, ) (, ))( (, ) (, )*) (, ) (, ) HH ol VV ol HH ol VV ol HH ol VV ol 0 0 where the ncdence angle, HH and VV the Bragg coeffcent and m an ampltude coeffcent [1]. In the next ubecton t explaned how to modfy Eq.4 n order to take nto account the fact that the facet may have varyng orentaton angle regardng the radar (4)

3 . Large Scale Scatterng The orentaton of a facet, n Cartean coordnate, gven by t normal vector n computed by a fnte dfference cheme: S x n : 1 S y (5) 1 S x S y 1 x, y ) x, y ) x where S, y ) x, y ) x : and S : y are repectvely the lope of the facet n x drecton (Fg. 1: blue pont) and the lope of the facet n y drecton (Fg. 1: green pont). The orentaton of the th facet can alo be expreed n the radar coordnate by t ncdence angle (Eq. 6) and t orentaton angle (Eq. 7) [] : S xn( ) co( ) co( ) : l l (6) S x S y 1 S y tan( ) : (7) S xco( l ) n( l ) the look angle expreed n Cartean coordnate.e. the wave vector k (Fg. 1: purple arrow) of where l the ncdence feld equal to / (n( l ), 0, co( l )). Thee angle are neceary to expre the electromagnetc behavor of a facet, or n other word to quantfy how the radar ee the facet, a hown below. The rotaton of a facet along the lne of ght of the radar can equvalently be een a change of polarzaton of the radar n tranmon and n recepton mode. For that reaon, the nfluence of on [T ] of algebrac nature. Fnally, the coherency matrx aocated to an orented facet gven by Eq. 8. [T ] := [O( )] [T Bragg ( )] [O( )] t (8) where [ O( )] : 0 co( ) n( ) a rotaton matrx and t the matrx tranpoe. 0 n( ) co( ) Furthermore, aumng that the backcattered feld of all facet are ndependent from each other then the feld add up ncoherently. A a drect conequence of that, the econd order moment of the backcattered feld by a reoluton cell equal to the average of the econd order of all facet. For ntance the Coherency matrx of the reoluton cell [T re ] defned by (Eq. 9) : N [ Tre] : 1 [ T N ] (9) 1 The tochatc nature of the mall cale roughne contaned n the cloed expreon of SPM. To take nto account the tochatc nature of the large cale roughne, the heght of the facet Z have to be condered a a realzaton of a tochatc proce Z. Th proce aumed to be a zero-mean normal proce wth autocorrelaton functon C(,j):=<x,y )x j,y j )>, where < > degnate the mean value. Indeed the dffculte n the computaton of F-Bragg are n the way to account for C(,j). The next ubecton preent the mot traghtforward computaton method and t lmtaton.

4 . Monte-Carlo Calculaton In a prevou tudy [1] the facet nde a reoluton cell were generated accordng to C(,j) and a realzaton of [T re ] wa then calculated accordng to Eq. 9. In order to etmate the mean value of [T re ], a Monte-Carlo (MC) algorthm wa ued.e. everal reoluton cell were generated and ther coherency matrx were averaged nto [ T ˆre]. The entvty of [ T ˆre] to varaton of ol parameter wa done by ung the derved tattcal polarmetrc parameter [] {Entropy, Mean Alpha Angle, Anotropy}, { H,, A }, defned a follow : { H : P logp ; 1 A : P P P P ; : P } (10) where 1 > > >0 are the three egenvalue of [ T ˆre] and P : / are the three weghted egenvalue. Each 1 1 a parameter of the angle repreentaton of the egenvector j j j e ={ co e,n co e,n n e }. e aocated to namely The entvty analy n [1] howed that the hape of C(,j), may t be gauan or fractal, doen t have any nfluence on the tattcal polarmetrc parameter derved from [ T ˆ re]. On the contrary, varaton of L c and H rm have an mportant mpact on H and. The nfluence of thee varaton on A were dffcult to confrm nce t etmated value had very mportant fluctuaton. Th noy behavor of A due to t ntrnc numercal ntablty whch requre a very hgh accuracy of the drect model. Snce the F-Bragg model wa olved by Monte-Carlo mulaton whch, by defnton, have very low convergence rate, the coherency matrx couldn t be calculated precely enough n order to oberve a realtc behavor of the expected Anotropy. We hall expoe n the next ecton how to remove the chaotc ocllaton of A by dervng the analytcal oluton of the F-Bragg model.. ANALYTICAL SOLUTION Snce the heght Z are condered a occurrence of a tochatc proce Z, the angle (, ) can mlarly be modeled by a tochatc proce (, ). A mentoned earler, the central queton to know whch properte of of (, ) nfluence F-Bragg. In fact, F-Bragg entve to the global properte of (, ) but nentve to t local properte becaue F-Bragg doen t take n account poble multple reflecton or hadowng between facet. Th tatement lead to the analytcal expreon of the expected value of the coherency matrx (Eq. 11) nce the global behavor of a random proce gven by t denty functon (wherea the local behavor gven by the correlaton functon). [ T re ] [ T re (, )] (, ) d d D, Where (,) the denty functon of (, ), whch ndependent of the facet ndex a explaned later, and D, the doman of ntegraton {0<</,-/<</}, whch the range of the angle of the llumnated facet. (11) Before applyng Eq. 11, the expreon of (,) mut be derved from the dtrbuton of the facet lope. If Z a normal tochatc proce then Z a normal random varable (NRV) for any fxed value of. The dfference of two NRV telf a NRV o that the lope S, ) defned by Eq.1 are NRV. ( x S y S S x y : : x, y ) x, x, y ) x, y ) y ) (1) If the varogram of Z, (x,y,x j,y j ) := < x,y )-x j,y j ) >, otropc uch that (x,y,x j,y j )= ((x -x j ) +(y -y j ) ) then S x and S y are ndependent and have ame dmenonle varance S = ()/(4²). For two cae, alo examned n [1], the varogram otropc and ha a mple expreon. If Z otropc tatonary wth

5 correlaton functon C( ) then ()=(C(0)-C()). If Z an otropc Fractonal Brownan Moton wth parameter H and [1] then () = H. Furthermore, the varance of S x and S y beng ndependent of, the ( x S y notaton S, ) hall from now on be replaced by S, ). ( x Sy Once the varance of S x and S y known, the remanng tak to calculate the bvarate denty functon of ( := (S x,s y ), := (S x,s y ) ) accordng to Eq Th done by applyng the fundamental theorem of functon of two random varable [4]. The reultng denty functon gven by Eq. 1. where : (, ) 1 exp( 1 ( S x,n(, ) S y, n(, ) ) Jn(, ) S n1 S S x, n(, ) S y, n(, ) J n (, ) In(, ) n( )(1 co( ) tan( ) ) ( 1) n l n()ec( ) (n( ) co( ) (1 co( ) tan( ) l l )) n( )tan( )(n( )n( ) ( 1) n l co( )co( l )ec( )) n( ) co( ) (1 co( ) tan( ) l l ) n( ) (co( )co( ) ( 1) n 1 l co( )n( )n( l )) 1 f Sx, ncot( l ) and / In(, ) 1 f Sx, ncot( l ) and / (17) 0 otherwe Eq. 1 can be mplfed, n order to overcome the ngularte at = / n Eq.14-15, by ung Eq.18. / Jn(, ) Sx, n(, ) Sy, n(, ) 1 (18) n( ) (1) (14) (15) (16) Fg. : Stattcal polarmetrc parameter for varyng large cale roughne parameter calculated by Monte- Carlo mulaton (black curve) and by the analytcal oluton (red curve).

6 4. RESULTS 4.1 Comparon of both method The Monte-Carlo mulaton ued a a reference are thoe analyzed n [1]. The fxed nput parameter for thee mulaton were ol =10, l c =5cm, h rm =0.cm. L c and H rm vared repectvely n a range of 10cm-80cm and 0cm-10cm. In order to compare thoe mulaton to the analytcal oluton, the Monte-Carlo Parameter have to be converted to analytcal parameter. Wherea ol, l c and h rm are explctly ued by the analytcal oluton, both large cale roughne parameter L c and H rm have frt to be converted nto a unque large cale parameter S accordng to Eq.1. A expected the analytcal oluton are mooth functon whch approxmate very well the ocllatng Monte-Carlo mulaton (Fg. ). The analytcal oluton alo a drectly nvertble model becaue t monotone wherea Mont-Carlo mulaton hould be artfcally regularzed before nveron. Furthermore, the analytcal oluton able to model the entvty of Anotropy to large cale roughne parameter. Th phenomenon, whch confrmed by meaured SAR data, wa not obervable n the Monte-Carlo mulaton. For the above mentoned mathematcal and phycal argument, the analytcal model oluton preferred to Monte- Carlo mulaton. 4. Properte of the analytcal oluton It ha been recaptulated n the ntroducton, that F-Bragg a model wth 5 nput parameter { ol, h rm, l c, H rm, L c }. In part, t demontrated that F-Bragg entve to only 4 parameter { ol, h rm, l c, }, where the root mean quare of the large cale lope. A remanng tak to valdate the extence of a mlar entve parameter for the mall cale urface. In the followng, t hown that the tattcal polarmetrc parameter { H,, A } derved from F-Bragg are nentve to h rm. In fact h rm a multplcatve factor n the coherency matrx [T Bragg ] of SPM (Eq.4), hence a multplcatve factor of the matrx [<T re >] of the analytcal oluton of F- Bragg (Eq.11). The averaged egenvalue P of [<T re >] are unchanged when [<T re >] multpled by a contant. The ame nvarance property hold for the egenvector e of [<T re >]. Fnally, { H,, A } are nentve to the multplcatve factor h rm nce thee parameter are derved from e and P. In other word, the tattcal polarmetrc parameter { H,, A } calculated by F-Bragg are entve to parameter { ol, l c, }. 5. CONCLUSION Th tudy ha proven that the analytcal expreon of the oluton of F-Bragg gve more nght to t behavour than t approxmaton by Monte-Carlo mulaton. For example, t ha been hown that the Anotropy of F-Bragg entve to roughne. The analytcal oluton prove that the polarmatrc parameter of F-Bragg are defned by 5 parameter but are entve to, namely : : root mean quare of the large cale lope l c : correlaton length of the mall cale roughne ol : the relatve delectrc contant of the ol Thee are the unque ol parameter whch are effectvely retrevable by the tattcal polarmetrc parameter { H,, A } calculated by F-Bragg. Th reult an eental gudelne n the elaboraton of the nveron algorthm : before nvertng, one hall know what nvertble. 6. REFERENCES 1. Breuer A., Hajnek I., Allan S., Ferro-Faml L., Potter E., Brunquel J., Polarmetrc Surface Scatterng Model for Surface Parameter Inveron, Proc. Of SPIE 9 th Internatonal Sympoum, on Remote Senng 00.. Potter E., Unuperved Clafcaton Scheme and Topography Dervaton of POLSAR Data on the H/A/ Polarmetrc Decompoton Theorem, Proc. Of 4 th Internatonal Workhop on Radar Polarmetry, France, 1998, pp Cloude S.R. and Potter E., A revew of Target Decompoton Theorem n Radar Polarmetry, IEEE Tranacton on Geocence and Remote Senng, vol. 4, no., pp , Papoul A., Probablty, Random Varable and Stochatc Procee, rd Edton, pp , Mc Graw- Hll, 1991