scattering polarimetry - esa · 2013-08-01 · fsa convention bsa convention. back scattering...

© E. Pottier, L. Ferro-Famil (01/2004)

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SCATTERINGPOLARIMETRYSCATTERINGSCATTERING

POLARIMETRYPOLARIMETRY

0

$x

$y( )rE z t,

$z


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WAVE POLARIMETRYWAVE POLARIMETRY

TRANSMITTER: XRECEIVERS: X & Y

X

X

AX

=

YX

XXS S

SE

YXS

XXS

YXS

XXS

YXS

XXS

YXS

XXS

X X X X

JONES VECTORS

WAVE POLARIMETRY

RX

TAY

RY

T

RX

RY


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SCATTERING POLARIMETRYSCATTERING POLARIMETRY

X

Y

AX

[ ]

=

YY

XY

YX

XX

SS

SS

S

YXS

XXS

YYS

XYS

YXS

XXS

YYS

XYS

X Y X Y

SINCLAIR MATRICES

SCATTERING POLARIMETRY

RX

TAY

RY

T

RX

RY

TRANSMITTER: X & YRECEIVERS: X & Y


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SCATTERING PROBLEMSCATTERING PROBLEM

Far Field Approximation

|'r||r| rr >>

λ>>|r| rand

( )rjki0

i ieE)r(Errr

=Incident Wave

Scattered Wave

( ) [ ] [ ]([ ]) ( ) 'dSe)'r(Hn)'r|r(G...

...)'r(EnkkkIer4

kj)r(E

rjkS

sssrjkss

s

s

′×+

××−= ∫rrrrr

rrrrr

π

Electromagnetic Theory - Integral Representation


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Far Field Approximation

|'r||r| rr >>

λ>>|r| rand

( )rjki0

i ieE)r(Errr

=Incident Wave

Scattered Wave

=

⊥

⊥

⊥⊥⊥⊥

)r(E)r(E

)r(S)r(S)r(S)r(S

re

)r(E)r(E

i

ijkr

s

s

r

r

rr

rr

r

r

////////

//

//

2x2 Complex Scattering MatrixElectromagnetic Theory - Matrix Representation


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Incidence Plane

rE k tI

I($ , )

$k I

θ I

$n

$uIθ

$uIφ

$urI

$uI⊥

$||uI

$p

$zS

$yS

$xS

$urI

$q

′θ I

′ = ′φ φI R

0S

Infinite Plane Surface

φ I

Oriented Transmission AntennaCoordinates System

Incident Wave

= i

ii

EE

Eφ

θ

= ⊥

i

ii

EE

E//// ( )[ ]

=

⊥

⊥⊥⊥⊥

//////

//// SS

SSS ,

Oriented Incidence WaveCoordinates System


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rE k tI

I($ , )

$k I

θ I

$n

$uIθ

$uIφ

$urI

$uI⊥

$||uI

$p

$zS

$yS

$xS

$urI

$q

′θ I

′ = ′φ φI R

0S

φ I

′ = ′θ θR I

$uR⊥

$urR

$||uR

$kR

rE k tR

R($ , )

Reflexion Plane

Incident Wave

= i

ii

EE

Eφ

θ

= ⊥

i

ii

EE

E////

Incidence Plane

Infinite Plane Surface

Oriented Transmission AntennaCoordinates System

= ⊥

s

ss

EE

E////

Oriented Reflected WaveCoordinates System

Oriented Incidence WaveCoordinates System

( )[ ]

=

⊥

⊥⊥⊥⊥

//////

//// SS

SSS ,


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SCATTERING COORDINATE FRAMEWORK


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BISTATIC SCATTERING CASE

INCIDENCE PLANE

si θθ

iu //

su//iu⊥ ii p,k s

u⊥

ss p,k

INCIDENCE PLANE

si θθ

iu //

su//iu⊥ ii p,k s

u⊥

sp

sk

BSA CONVENTIONFSA CONVENTION

Back Scattering Alignment Conventionor Antenna Oriented Coordinate System

Forward Scattering Alignment Conventionor Wave Oriented Coordinate System

ii pk = ss pk = ii pk = ss pˆ −=kandand

jpwhere

( )[ ]FSA,S //⊥

is the direction of wave propagation

( )[ ]BSA,S //⊥


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INCIDENCE PLANE

sp

iu //

iu⊥ ii p,k

su//s

u⊥

sk

su//s

u⊥

ss p,kForward Scattering Alignment Convention

or Wave Oriented Coordinate System

isis kkandpp −=−=

Back Scattering Alignment Conventionor Antenna Oriented Coordinate System

isis kkandpp =−= FSA

BSA

BACKSCATTERING CASE

−=

⊥

⊥⊥⊥

⊥

⊥⊥⊥FSAFSA

FSAFSA

BSABSA

BSABSA

SSSS

1001

SSSS

//////

//

//////

//


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POLARIMETRIC DESCRIPTORSPOLARIMETRIC DESCRIPTORS

X

YTHE DIFFERENT

TARGET POLARIMETRICDESCRIPTORS

[S] SINCLAIR Matrixk, Ω Target Vectors[K] KENNAUGH Matrix[T] Coherency Matrix[C] Covariance MatrixTRANSMITTER: X & Y

RECEIVERS: X & Y


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SCATTERING MATRIXSCATTERING MATRIX

BISTATIC CASE

SCATTERING MATRIX or JONES MATRIX

=

iY

iX

YYYX

XYXXjkr

sY

sX

EE

SSSS

re

EE

DEFINED IN THE LOCAL COORDINATES SYSTEM

[S] IS INDEPENDENT OF THE POLARISATION STATE OF THE INCIDENCE WAVE

[S] IS DEPENDENT ON THE FREQUENCY AND THE GEOMETRICAL ANDELECTRICAL PROPERTIES OF THE SCATTERER

TOTAL SCATTERED POWER

[ ]( ) [ ][ ]( ) 2YY

2YX

2XY

2XX

*T |S||S||S||S|SSTraceSSpan +++==


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SCATTERING MATRIXSCATTERING MATRIX

[ ]

=

=

YYYX

XYXX

jφYY

jφYX

jφXY

jφXX

jkr

YYYX

XYXXjkr

eSeSeSeS

re

SSSS

reS

[ ]( )

( ) ( )

= −−

−

XXYYXXYX

XXXYXX

φφjYY

φφjYX

φφjXYXX

jjkr

eSeSeSS

reeS

φ

ABSOLUTE SCATTERING MATRIX

RELATIVE SCATTERING MATRIXSeven Parameters: 4 Amplitudes and 3 Phases

Absolute PhaseFactor

SCATTERER POLARIMETRIC DIMENSION = 7


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BACKSCATTERING MATRIXBACKSCATTERING MATRIX

MONOSTATIC CASE

BACKSCATTERING MATRIX or SINCLAIR MATRIX

In the case of Backscattering from Reciprocal Scatterers:

FSAYX

FSAXY

BSAYX

BSAXY SSSS −=⇔=RECIPROCITY THEOREM

=

iY

iX

YYXY

XYXXjkr

sY

sX

EE

SSSS

re

EE

(BSA CONVENTION)

TOTAL SCATTERED POWER

[ ]( ) [ ][ ]( ) 2YY

2XY

2XX

*T |S||S|2|S|SSTraceSSpan ++==


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BACKSCATTERING MATRIXBACKSCATTERING MATRIX

[ ]

=

=

YYXY

XYXX

jφYY

jφXY

jφXY

jφXX

jkr

YYXY

XYXXjkr

eSeSeSeS

re

SSSS

reS

[ ]( )

( ) ( )

= −−

−

XXYYXXXY

XXXYXX

φφjYY

φφjXY

φφjXYXX

jjkr

eSeSeSS

reeS

φ

ABSOLUTE BACKSCATTERING MATRIX

RELATIVE BACKSCATTERING MATRIXFive Parameters: 3 Amplitudes and 2 Phases

Absolute PhaseFactor

SCATTERER POLARIMETRIC DIMENSION = 5


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ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION

ELLIPTICAL BASIS TRANSFORMATION EXPRESSED INTHE ORIENTED ANTENNA COORDINATES SYSTEM

( ) ( ) ( )[ ] ( )i

B,BB,BA,Ai

A,A EUE⊥⊥⊥⊥

= aEMISSION:

SU(2) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX

IN THE BSA CONVENTION PROPAGATES IN THE:( )s

A,AE is pp −=RECEPTION:⊥

Time reversal = Complex conjugation

PROPAGATES IN THE:( )( )*sA,AE

⊥is pp =

( )( ) ( ) ( )[ ] ( )( )*sB,BB,BA,A

*sA,A EUE

⊥⊥⊥⊥= a


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( )

( ) ( )[ ] ( )

( )

( )[ ] ( )

( )[ ] ( ) ( )[ ] ( )

( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ] ( )i

B,BB,BA,AA,A

1*B,BA,A

iB,BB,BA,AA,A

iA,AA,A

sB,B

sB,B

*B,BA,A

sA,A

EUSU

EUS

ES

E

EU

E

⊥⊥⊥⊥⊥⊥

⊥⊥⊥⊥

⊥⊥

⊥

⊥⊥⊥

⊥

−=

=

=

aa

aa

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]⊥⊥⊥⊥⊥⊥

= B,BA,AA,AT

B,BA,AB,B USUS aa

CON-SIMILARITY TRANSFORMATION

SU(2) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX ( ) ( )[ ]

⊥⊥ B,BA,AU a


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ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATIONSPECIAL UNITARY SU(2) GROUP

( ) ( )[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

⊥⊥ α

α

ττττ

φφφφ

j

j

B,BA,A e00e

cossinjsinjcos

cossinsincos

U a

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

( ) ( )[ ]

−

+=

−

⊥⊥ ξ

ξ

ρρ

ρ j

j

B

*B

2B

B,BA,A e00e

11

1

1U a

( )[ ]ρ2U ( )[ ]ξ2U

With: ( ) ( )( ) ατφξ −= − tantantan 1


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FROBENIUS NORM OF

( )[ ]( ) ( )[ ] ( )[ ]( ) 2AA

2AA

2AA

*TA,AA,AA,A |S||S|2|S|SSTraceSSpan

⊥⊥⊥⊥⊥⊥++==

( )[ ]⊥A,AS

FROBENIUS NORM OF

( )[ ]( ) ( )[ ] ( )[ ]( ) 2BB

2BB

2BB

*TB,BB,BB,B |S||S|2|S|SSTraceSSpan

⊥⊥⊥⊥⊥⊥++==

( )[ ]⊥B,BS

[ ][ ] [ ] [ ]( ) 1UdetIUU 2D*T +==SPECIAL UNITARY SU(2) GROUP

( )[ ]( ) ( )[ ]( )⊥⊥

= B,BA,A SSpanSSpanFROBENIUS NORM OF A SCATTERING MATRIX

IS INVARIANT UNDER BASIS ELLIPTICAL TRANSFORMATION


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CHARACTERISTIC POLAR. STATESCHARACTERISTIC POLAR. STATES

E

BA

TRANSMITTER RECEIVER

RECEIVED POWER

[ ] 2TBA ASBP α=

CO-POLARIZEDPOWER

CROSS-POLARIZEDPOWER


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+=

−

ρρ

ξ 1

1eA

2

j

−

+=

+

⊥11

eA*

2

j ρ

ρ

ξ

CROSS-POLARIZED POWER :

[ ] 2TAAX ASAPP ⊥== ⊥ α

CO-POLARIZED POWER :

[ ] 2TAACO ASAPP α==

2YY

2XYXXCO SS2SP ρρα ++= ( ) 2

YYXY2

XX*

X SS1SP ρρρα +−−=

0P0P XCO =∂

∂=

∂∂

ρρ

MAXIMISATION / MINIMISATION


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B = A

EA

RECEIVERTRANSMITTER

CO-POLARIZED POWER : [ ] 2TAACO ASAPP α==

MAXIMISATION :MaximumLocalP

MaximumGlobalPL

L

KK

=

= (K , L) COPOL MAX


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B = A

EA

RECEIVERTRANSMITTER

CO-POLARIZED POWER : [ ] 2TAACO ASAPP α==

0PP 2

2

1

1

OO

OO == (O1 , O2)

COPOL NULLSMINIMISATION :


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EA

B = A⊥

RECEIVERTRANSMITTER

CROSS-POLARIZED POWER : [ ] 2TAAX ASAPP ⊥== ⊥ α

MaximumLocalP

MaximumGlobalP2

2

1

1

CC

CC

=

=⊥

⊥

MAXIMISATION : (C1 , C2)XPOL MAX


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EA

B = A⊥

RECEIVERTRANSMITTER


(D1 , D2)XPOL SADDLEMINIMISATION :

MinimumLocalP

MinimumGlobalP2

2

1

1

DD

DD

=

=⊥

⊥


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EA

B = A⊥

RECEIVERTRANSMITTER


(X1 , X2)XPOL NULLS

NULL :(UNDER CONDITION) 0P

0P2

2

1

1

XX

XX

=

=⊥

⊥


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K

L

C 2

X 1

O 2

X 2

C 1

D 1 D 2

O 1

Q

U

V

C1 C2XPOLL MAX

O1 O2COPOLL NULLS D1 D2

XPOLL SADDLE

K LCOPOLL MAX

X1 X2XPOLL NULLS


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EXAMPLE: [S] DIAGONALISATION

CROSS POL POWER MINIMISATION

[ ]( )

( )[ ]( )S

S SS S

SS

SH VHH HV

HV VV

A B

A BAA

BB,

, ?

,=

=

⇒

00

PSEUDO EIGENVECTORS[ ]( )S X XH V,

*= λ

PSEUDO EIGENVECTORS = XPOLL NULLS (X1, X2)

[U] = [X1 X2] BASIS CHANGEMENT UNITARY MATRIX


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GRAVES MATRIX

[ ] [ ] [ ]

++

++== 2

YY2

XYXY*YYXX

*XY

YY*XYXY

*XX

2XY

2XX*

SSSSSSSSSSSS

SSG

HERMITIAN MATRIX

[ ] [ ] [ ]2

Gdet4GtraceGtrace 2

2,1−±

=λ

EIGENVALUES

−

−+

=12

112,12

12

112,1

j

2,1

GG

1

GG

1

eX λλ

αEIGENVECTORS


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[ ] [ ]21 XXU =BASIS CHANGEMENT UNITARY MATRIX

[ ] [ ] [ ][ ]UGUG 1D

−=DIAGONALISED GRAVES MATRIX

[ ] [ ] [ ]SSG *= [ ] [ ] [ ]D*

DD SSG =

[ ] [ ] [ ][ ]USUS TD =

[ ]( )

( ) ( )

=

=

−−

+

χν

χν

γ 2j2

2j

2

1D etan0

0ems00s

S

DIAGONALISED SINCLAIR MATRIX


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EULER PARAMETERSEULER PARAMETERS

( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

ν

ν

ττττ

φφφφ

j

j

1e00e

cossinjsinjcos

cossinsincos

X $ux

[ ]( )ξ

γj

2X,X etan0

01mS

21

=

DIAGONALISED SINCLAIR MATRIX[ ]( ) [ ] [ ]( )[ ]USUS Y,X

TX,X 21

=

m, φ, τ, ν, γ : 5 EULER PARAMETERS

(ξ ABSOLUTE PHASE)


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EULER PARAMETERSEULER PARAMETERS

m : MAXIMAL TARGET R.C.S

φ : TARGET ORIENTATION

τ : TARGET SYMMETRY

ν : TARGET SKIP ANGLE

γ : TARGET POLARISABILITY ANGLE


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HUYNEN POLARISATION FORKHUYNEN POLARISATION FORK

2 ν

0

$z

$x

2γ

2φ

X1

X2

O1

O2$y

2τ


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POLARIMETRIC SIGNATURESPOLARIMETRIC SIGNATURES

CONVENIENT GRAPHICAL WAY TO DISPLAY THE RECIVED POWERAS A FUNCTION OF POLARIZATION

( ) ( )( ) ( )

( ) ( )( ) ( ) xu

cossinjsinjcos

cossinsincos

A

−=

ττττ

φφφφ

CROSS-POLARIZED POWER :

[ ] 2AA

2TX SASAP

⊥== ⊥α

CO-POLARIZED POWER :

[ ] 2AA

2TCO SASAP ==

CO-POLARIZEDCORRELATION PHASE:

⊥⊥−= AAAACO SargSargΨ

CROSS-POLARIZEDCORRELATION PHASE :

⊥−= AAAAX SargSargΨ


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CO-POLARIZED POWER : CROSS-POLARIZED POWER :

CROSS-POLARIZEDCORRELATION PHASE :

CO-POLARIZEDCORRELATION PHASE:

Courtesy of Pr WM Boerner


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[ ]

=1001

cS

ˆ v t

ˆ h t

TRIHEDRAL CORNER REFLECTOR

DIHEDRAL CORNER REFLECTORˆ v t

ˆ h t

[ ]

−

=1001

cS

Courtesy of Pr WM Boerner


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[ ]

−

−=

1111

cS

ˆ v t

ˆ h tl

ˆ v t

ˆ h t

[ ]

=1000

cS

SHORT THIN CYLINDER

ORIENTED SHORT THIN CYLINDER


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[ ]

−

=1ii1

21S

ˆ v t

ˆ h t

LEFT-HANDED HELIX

RIGHT-HANDED HELIX

[ ]

−−−

=1jj1

21S

ˆ v t

ˆ h t


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X

YTHE DIFFERENT

TARGET POLARIMETRICDESCRIPTORS

[S] SINCLAIR Matrixk, Ω Target Vectors[K] KENNAUGH Matrix[T] Coherency Matrix[C] Covariance MatrixTRANSMITTER: X & Y

RECEIVERS: X & Y


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TARGET VECTORSTARGET VECTORS

VECTORIAL FORMULATION OF THE SCATTERING PROBLEM

[ ]

=

YYYX

XYXX

SSSS

SSCATTERING MATRIX

MATRIX VECTORISATION OPERATOR

SET OF ORTHOGONAL 2x2 MATRICES

[ ]( ) [ ][ ]( ) 4C

4S3S2S1S

STrace21SV:S ∈

=== Ψr

[ ]( )[ ]Ψ

SV

SCATTERING VECTOR

With:

[ ]( ) 2YY

2XY

2YX

2XX

24

23

22

21

*T2

|S||S||S||S|SSpan|S||S||S||S|SS||S||

+++==+++=⋅=

rrrFROBENIOUS NORM OF :S

r


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[ ]( ) [ ][ ]( )PSTrace21SVk ψ==PAULI SCATTERING VECTOR

SET OF 2x2 COMPLEX MATRICES FROM THE PAULI MATRICES GROUP

[ ]

−

−

=0j

j02,

0110

2,10

012,

1001

2Pψ

( )[ ]TYXXYYXXYYYXXYYXX SSjSSSSSS

21k −+−+=

Advantage: Closer related to physical properties of the scatterer

Note: Also known as k4P


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[ ]( ) [ ][ ]( )LSTrace21SV ψΩ ==LEXICOGRAPHIC SCATTERING VECTOR

SET OF 2x2 COMPLEX MATRICES FROM THE LEXICOGRAPHIC MATRICES GROUP

[ ]

=1000

2,0100

2,0010

2,0001

2Lψ

[ ]TYYYXXYXX SSSS=Ω

Advantage: Directly related to the system measurables

Note: Also known as k4L


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SCATTERING VECTOR TRANSFORMATIONS

Pauli Scattering Vector: Lexicographic Scattering Vector:

[ ]

−

−=

0jj001101001

1001

21D4

[ ] [ ] [ ] kDkDandDk *T4

144 === −ΩΩ

UNITARY TRANSFORMATION

( )

−+−+

=

YXXY

YXXY

YYXX

YYXX

SSjSSSSSS

21k

=

YY

YX

XY

XX

SSSS

Ω

WHERE [D4] IS A SU(4) MATRIXIN ORDER TO PRESERVE THE NORM

OF THE SCATTERING VECTOR


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MONOSTATIC CASE

Pauli Scattering Vector:

−+

=

XY

YYXX

YYXX

S2SSSS

21k

( )

−+−+

=

YXXY

YXXY

YYXX

YYXX

SSjSSSSSS

21k

Note: Also known as k3P

Lexicographic Scattering Vector:

=

YY

YX

XY

XX

SSSS

Ω

=

YY

XY

XX

SS2

SΩ

Note: Also known as k3L


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SCATTERING VECTOR TRANSFORMATIONS

Pauli Scattering Vector: Lexicographic Scattering Vector:

−+

=

XY

YYXX

YYXX

S2SSSS

21k

=

YY

XY

XX

SS2

SΩ

[ ] [ ] [ ] kDkDandDk T3

133 === −ΩΩ


[ ]

−=020101

101

21D3

WHERE [D3] IS A SU(3) MATRIXIN ORDER TO PRESERVE THE NORM

OF THE SCATTERING VECTOR


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THE DIFFERENT TARGET POLARIMETRIC

DESCRIPTORS

[S] SINCLAIR Matrixk, W Target Vectors[K] KENNAUGH Matrix[T] Coherency Matrix[C] Covariance Matrix

STATISTICAL DESCRIPTION

PARTIAL SCATTERING POLARIMETRY

X

Y



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PARTIAL SCATTERING POLARIMETRYPARTIAL SCATTERING POLARIMETRY

0

$x$y

$z 0

$x$y

$z

DETERMINISTIC SCATTERERS PARTIAL SCATTERERS

DETERMINISTIC SCATTERING RANDOM SCATTERING(Variation in Time / Space)

COMPLETELY POLARISED SCATTERING PARTIALLY POLARISED SCATTERING

COMPLETELY DESCRIBED BY [S] CAN NOT BE DESCRIBED BY [S]STATISTICAL DESCRIPTION


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MUELLER MATRIXMUELLER MATRIX

[ ] is ESE =BISTATIC CASE JONES MATRIX :

[ ]is EE gMg =

[ ] [ ] [ ] [ ] [ ]

⊗= V*SSV

21M T [ ]

−+−

=

0011j100j100

0011

V

MUELLER MATRIX :

[ ]

−+−−−−+−−−−++−+−+++++

=

ABAJDMGLFJDABAKENHMGKEABAICLFNHICABA

M

00

0

0

00

HUYNEN PARAMETERS


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KENNAUGH MATRIXKENNAUGH MATRIX

[ ] is ES=MONOSTATIC CASE ESINCLAIR MATRIX :

[ ]is EE gKg =

[ ] [ ] [ ] [ ] [ ]

⊗= V*SSV

21K T [ ]

−+−

=

0011j100j100

0011

V

KENNAUGH MATRIX :

[ ]

+−−

++

=

00

0

0

00

BADGFDBAEHGEBACFHCBA

K

HUYNEN PARAMETERS


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HUYNEN PARAMETERSHUYNEN PARAMETERS

PHYSICAL INTERPRETATIONMAN-MADE TARGET DECOMPOSITION

IDENTIFICATION and ANALYSIS

« PHENOMENOLOGICAL THEORY OF RADAR TARGETS » (1970)

A0 : GENERATOR OF TARGET SYMMETRY

B0+B : GENERATOR OF TARGET NON-SYMMETRY

B0-B : GENERATOR OF TARGET IRREGULARITY

C : GENERATOR OF TARGET GLOBAL SHAPE (LINEAR)

D : GENERATOR OF TARGET LOCAL SHAPE (CURVATURE)

E : GENERATOR OF TARGET LOCAL TWIST (TORSION)

F : GENERATOR OF TARGET GLOBAL TWIST (HELICITY)

G : GENERATOR OF TARGET LOCAL COUPLING (GLUE)

H : GENERATOR OF TARGET GLOBAL COUPLING (ORIENTATION)


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STOKES VECTORSTOKES VECTOR

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

α

α

ττττ

φφφφ

[U2(φ)] [U2(τ)] [U2(α)]

HOMOMORPHISM SU(2) - O(3)

(σp, σq) : Pauli Matrices

( )[ ] ( )[ ] ( )[ ]( )q2p*T

2q,p3 UUTr212O σθσθθ =

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

xu

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

2E gAg

= −

−−

αα

ααττ

ττ

φφ

φφ

STOKES VECTOR

[O4(2φ)] [O4(2τ)] [O4(2α)]


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( ) ( )[ ] ( )i

A,AA,As

A,A ESE⊥⊥⊥

=

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]⊥⊥⊥⊥⊥⊥

= B,BA,AA,AT

B,BA,AB,B USUS aa


( ) ( )[ ] ( )i

B,BB,Bs

B,B ESE⊥⊥⊥

=SINCLAIR MATRIX

( ) ( )[ ]( )i

A,As

A,A EA,AE gKg⊥⊥⊥

=

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] 1B,BA,A4A,AB,BA,A4B,B OKOK −

⊥⊥⊥⊥⊥⊥= aa

SIMILARITY TRANSFORMATION

( ) ( )[ ]( )i

B,Bs

B,B EB,BE gKg⊥⊥⊥

=KENNAUGH MATRIX

O(4) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX ( ) ( )[ ]

⊥⊥ B,BA,A4O a


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[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

α

α

ττττ

φφφφ

j

j

e00e

cossinjsinjcos

cossinsincos

U

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

HOMOMORPHISM SU(2) - O(3)

(σp, σq) : Pauli Matrices

( )[ ] ( )[ ] ( )[ ]( )q2p*T

2q,p3 UUTr212O σθσθθ =

O(4) UNITARY GROUP

[O4(2φ)] [O4(2τ)] [O4(2α)]

[ ] ( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

= −

−−

αα

ααττ

ττ

φφ

φφ

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

4O


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DESCRIPTORS

[S] SINCLAIR Matrixk, Ω Target Vectors[K] KENNAUGH Matrix[T] Coherency Matrix[C] Covariance Matrix



X

Y



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COHERENCY MATRIXCOHERENCY MATRIX

BISTATIC CASE

PAULI SCATTERING VECTOR k( )[ ]T

YXXYYXXYYYXXYYXX SSjSSSSSS2

1k −+−+=

COHERENCY MATRIX [T]

[ ]

−+++−−−−+++−+−

=⋅=

A2jIJjNMjKLjIJBBjFEjGHjNMjFEBBjDCjKLjGHjDCA2

kkT0

0

0

T*

HERMITIAN POSITIVE SEMI DEFINITE MATRIX - RANK 1


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COHERENCY MATRIXCOHERENCY MATRIX

MONOSTATIC CASE

[ ]TXYYYXXYYXX S2SSSS

21k −+=

PAULI SCATTERING VECTOR k

[ ]

−−−++++−

=⋅=BBjFEjGH

jFEBBjDCjGHjDCA2

kkT

0

0

0T*

HERMITIAN MATRIX - RANK 1


A0, B0+B, B0-B : HUYNEN TARGET GENERATORS


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TARGET VECTOR : TARGET VECTOR : kk

[ ]

−−−++++−

=⋅=BBjFEjGH

jFEBBjDCjGHjDCA2

kkT

0

0

0T*


−

+=

−+=

−+

=

+

−

HGarctanj

0

CDarctanj

0

0

j0

0

j

XY

YYXX

YYXX

eBB

eBB

A2

ejGHjDC

A2

A2e

S2SSSS

21k φ

φ

( )YYXX SSarg +=φWith:

PAULI SCATTERING VECTOR k


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TARGET GENERATORS TARGET GENERATORS

PHYSICAL INTERPRETATION

SINGLE BOUNCESCATTERING

(ROUGH SURFACE)

DOUBLE BOUNCESCATTERING

VOLUMESCATTERING

2YYXX011 SSA2T +== 2

XY033 S2BBT =−=

2YYXX022 SSBBT −=+=


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( )dB0A2 ( )dB0 BB −( )dB0 BB +

20dB 40dB0dB


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( )dB0A2 ( )dB0 BB −( )dB0 BB +

-15dB 0dB-30dB


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[ ] kSS

SS

SYY

XY

YX

XX a

=PARTIAL SCATTERERS

[ ] [ ]∑=

=⋅=N

1ii

T* TN1kkTSTATISTICAL DESCRIPTION

Describes completely the polarimetric properties of distributed scatterers

Diagonal Elements:

Off-Diagonal Elements:

2YXXY44

2YXXY33

2YYXX22

2YYXX11

SSTSST

SSTSST

−=+=

−=+=

( )( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( ) ( )*XYYXYXXY34

*XYYXYYXX24

*YXXYYYXX23

*XYYXYYXX14

*YXXYYYXX13

*YYXXYYXX12

SSjSSTSSjSST

SSSSTSSjSST

SSSSTSSSST

−+=−−=

+−=−+=

++=−+=


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( ) ( )[ ] ( )i

A,AA,As

A,A ESE⊥⊥⊥

= ( ) ( )[ ] ( )i

B,BB,Bs

B,B ESE⊥⊥⊥

=SINCLAIR MATRIX

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]⊥⊥⊥⊥⊥⊥

= B,BA,AA,AT

B,BA,AB,B USUS aa


COHERENCY MATRIX

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] 1B,BA,A3A,AB,BA,A3B,B UTUT −

⊥⊥⊥⊥⊥⊥= aa


U(3) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX ( ) ( )[ ]

⊥⊥ B,BA,A3U a


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[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

α

α

ττττ

φφφφ

j

j

e00e

cossinjsinjcos

cossinsincos

U

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

[U3(2φ)] [U3(2τ)] [U3(2α)]

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

−

−

−

10002cos2sinj02sinj2cos

2cos02sinj010

2sinj02cos

2cos2sin02sin2cos0

001αα

αα

ττ

ττ

φφφφ

SPECIAL UNITARY SU(3) GROUP


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[ ]

−

+=

−

ξ

ξ

ρρ

ρ j

j

E

*E

2E

e00e

11

1

1U

( )[ ]ρ2U ( )[ ]ξ2U


SPECIAL UNITARY SU(2) MATRIX


( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

−ℜ−ℑℜℜ−ℑ+−ℑℑ−−ℜ+

+−−−

−−−

2

j2j22j22

j2j22j22

2

12j2e2e2cosej2sinjej2ej2sinje2cos

11

ρρρρρξρξρρξρξ

ρξξξ

ξξξ


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DESCRIPTORS




X

Y



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COVARIANCE MATRIXCOVARIANCE MATRIX

BISTATIC CASE

LEXICOGRAPHIC SCATTERING VECTOR Ω[ ]T

YYYXXYXX SSSS=Ω

COVARIANCE MATRIX [C]

[ ]

=⋅=

*YYYY

*YXYY

*XYYY

*XXYY

*YYYX

*YXYX

*XYYX

*XXYX

*YYXY

*YXXY

*XYXY

*XXXY

*YYXX

*YXXX

*XYXX

*XXXX

T*

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

C ΩΩ



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COVARIANCE MATRIXCOVARIANCE MATRIX

MONOSTATIC CASE

[ ]TYYXYXX SS2S=ΩLEXICOGRAPHIC SCATTERING VECTOR Ω


[ ]

=⋅=

*YYYY

*XYYY

*XXYY

*YYXY

*XYXY

*XXXY

*YYXX

*XYXX

*XXXX

T*

SSSS2SSSS2SS2SS2

SSSS2SSC ΩΩ



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[ ] Ωa

=

YY

XY

YX

XX

SS

SS

SPARTIAL SCATTERERS

[ ] [ ]∑=

=⋅=N

1ii

T* CN1C ΩΩSTATISTICAL DESCRIPTION

Describes completely the polarimetric properties of distributed scatterers

2YY44

2YX33

2XY22

2XX11

SCSC

SCSC

==

==Diagonal Elements:

Off-Diagonal Elements:

*YYYX34

*YYXY24

*YXXY23

*YYXX14

*YXXX13

*XYXX12

SSCSSC

SSCSSC

SSCSSC

==

==

==


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( ) ( )[ ] ( )i

A,AA,As

A,A ESE⊥⊥⊥

=

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]⊥⊥⊥⊥⊥⊥

= B,BA,AA,AT

B,BA,AB,B USUS aa


( ) ( )[ ] ( )i

B,BB,Bs

B,B ESE⊥⊥⊥

=SINCLAIR MATRIX

COVARIANCE MATRIX

( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] 1B,BA,A3A,AB,BA,A3B,B UCUC −

⊥⊥⊥⊥⊥⊥= aa


U(3) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX ( ) ( )[ ]

⊥⊥ B,BA,A3U a


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[ ]

−

+=

−

ξ

ξ

ρρ

ρ j

j

E

*E

2E

e00e

11

1

1U

( )[ ]ρ2U ( )[ ]ξ2U




( )[ ]

−−−

+=

+++

−−−

ξξξ

ξξξ

ρρρρρ

ρρ

ρξρ

j2j2*j22*

2*

j22j2j2

2

ee2e212ee2e

11,U


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COVARIANCECOVARIANCE--COHERENCY MATRICESCOHERENCY MATRICESBISTACTIC CASE

k ΩPauli Scattering Vector: Lexicographic Scattering Vector:


144 === −ΩΩ


[ ] [ ] [ ] [ ][ ][ ][ ] [ ] [ ] [ ] [ ][ ]4

*T44

**T4

*T

*T44

*T4

*4

*T

DTDDkkDC

DCDDDkkT

=⋅=⋅=

=⋅=⋅=

ΩΩ

ΩΩUNITARY TRANSFORMATION


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COVARIANCECOVARIANCE--COHERENCY MATRICESCOHERENCY MATRICESMONOSTACTIC CASE

k ΩPauli Scattering Vector: Lexicographic Scattering Vector:


133 === −ΩΩ


[ ] [ ] [ ] [ ][ ][ ][ ] [ ] [ ] [ ] [ ][ ]3

*T33

**T3

*T

*T33

*T3

*3

*T

DTDDkkDC

DCDDDkkT

=⋅=⋅=

=⋅=⋅=

ΩΩ

ΩΩUNITARY TRANSFORMATION


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COVARIANCECOVARIANCE--COHERENCY MATRICESCOHERENCY MATRICESCOHERENCY MATRIX COVARIANCE MATRIX

[ ] T*kkT ⋅= [ ] T*C ΩΩ ⋅=

UNITARY TRANSFORMATION[ ] [ ][ ][ ] *T

4or34or3 DCDT =

[T] and [C] HAVE THE SAME EIGENVALUES

Both contain the same information about Polarimetric Scattering Amplitudes,Phase Angles and Correlations

[T] is closer related to Physical and Geometrical Properties of the ScatteringProcess, and thus allows a better and direct physical interpretation

[C] is directly related to the system measurables

[T] is directly related to the Kennaugh matrix and the Huynen parameters


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SINCLAIR MATRIX

[ ]S =

SXX SXY

SYX SYY

KENNAUGH MATRIX

[ ] [ ] [ ] [ ] [ ]K V S S VT= ⊗

12

*

EQUIVALENCE ?

[ ]TXYYYXXYYXX S2SSSS

21k −+=


[ ] T*kkT ⋅=

SCATTERING VECTOR k

[ ]TYYXYXX SS2S=Ω


[ ] *TC ΩΩ=

SCATTERING VECTOR Ω


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[ S’ ]=[ U2 ]T[ S ] [ U2 ]

[ S ]SINCLAIR

SU(2)

[ T ]COHERENCY

SU(3)

[ C ]COVARIANCE

SU(3)

[ K ]KENNAUGH

O(4)

[ C’ ]=[ U3 ] [ C ] [ U3 ]-1

[ T’ ]=[ U3 ] [ T ] [ U3 ]-1 [ K’ ]=[ O4 ] [ K] [ O4 ]-1


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[U3(2φ)] [U3(2τ)] [U3(2α)]

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

−

−

−

10002cos2sinj02sinj2cos

2cos02sinj010

2sinj02cos

2cos2sin02sin2cos0

001αα

αα

ττ

ττ

φφφφ

SPECIAL UNITARY SU(3) GROUP (T Matrix)

SPECIAL UNITARY SU(2) GROUP( ) ( )( ) ( )

( ) ( )( ) ( )

− −

α

α

ττττ

φφφφ

j

j

e00e

cossinjsinjcos

cossinsincos

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

O(4) UNITARY GROUP

[O4(2φ)] [O4(2τ)] [O4(2α)]

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

−

−−

αα

ααττ

ττ

φφ

φφ

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001


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POLARIMETRIC TARGET DIMENSIONPOLARIMETRIC TARGET DIMENSION

[ ]

=

YYYX

XYXX

SSSS

S

BISTATIC CASE MONOSTATIC CASE

XXYYXXYXXXXY

YYYXXYXX

,,S,S,S,S

−−− φφφ

7 DEGREES OF FREEDOM

XXYYXXXY

YYXYXX

,S,S,S

−− φφ


TARGET BISTATIC POLARIMETRIC « DIMENSION »

77=

TARGET MONOSTATIC POLARIMETRIC « DIMENSION »

55

=


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TARGET EQUATIONSTARGET EQUATIONS

PURE TARGET – MONOSTATIC CASE

SINCLAIR MATRIX [S]POLARIMETRIQUE TARGET

« DIMENSION » = 5

KENNAUGH MATRIX [K]COHERENCY MATRIX [T]

9 HUYNEN REAL PARAMETERS(A0, B0, B, C, D, E, F, G, H)

COVARIANCE MATRIX [C]9 REAL PARAMETERS

|XX|, |XY|, |YY|, Re(XXXY*), Im(XXXY*)Re(XXYY*), Im(XXYY*)Re(XYYY*), Im(XYYY*)

9 - 5 = 4 TARGET EQUATIONS


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[ ]

−−−++++−

=⋅=BBjFEjGH

jFEBBjDCjGHjDCA2

kkT

0

0

0T*

3x3 HERMITIAN MATRIX - RANK 1

9 PRINCIPAL MINORS = 0

( ) ( )

( ) ( )( )

( ) 0DFCEBBH0EDFCBBG0DHCGFA20GEFHBBD0GFEHBBC

0FEBB0DGCHEA20HGBBA20DCBBA2

0

00

00

222200

2200

2200

=−−+=−++−=−−=−+−−=−−−

=−−−=−+−=−−−=−−+


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MONOSTATIC TARGET DIAGRAMMONOSTATIC TARGET DIAGRAM

(C,D

)

(E,F)

( ) 2200 DCBBA2 +=+

( ) 2200 HGBBA2 +=−

(G,H)

2A0

B0 + B

B0 - B

[ ]

=⋅= T*kkT

2A0 C - jD H + jGC + jD B0 + B E + jFH - jG E - jF B0 - B

( )( ) 2200 FEBBBB +=−+


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MONOSTATIC TARGET DIAGRAMMONOSTATIC TARGET DIAGRAM

[ ]

=⋅= T*kkT

2A0 C - jD H + jGC + jD B0 + B E + jFH - jG E - jF B0 - B

( ) EGFHDBB0 −=−

DGCHEA2 0 −=

DHCGFA2 0 +=

( ) DECFGBB0 −=+

( ) DFCEHBB0 +=+

(C,D

)

(E,F)

(G,H)

2A0

B0 + B

B0 - B

( ) FGEHCBB0 +=−


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9 HUYNEN REAL PARAMETERS(A0, B0, B, C, D, E, F, G, H)


|XX|, |XY|, |YY|, Re(XXXY*), Im(XXXY*)Re(XXYY*), Im(XXYY*)Re(XYYY*), Im(XYYY*)

POLARIMETRIQUE TARGET« DIMENSION » = 5

SINCLAIR MATRIX [S]


( )( )

DHCGFA2DGCHEA2HGBBA2DCBBA2

0

0

2200

2200

+=−=+=−+=+


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POLARIMETRIC TARGET DIMENSIONPOLARIMETRIC TARGET DIMENSION

[ ]

=

YYYX

XYXX

SSSS

S

BISTATIC CASE MONOSTATIC CASE

XXYYXXYXXXXY

YYYXXYXX

,,S,S,S,S

−−− φφφ


XXYYXXXY

YYXYXX

,S,S,S

−− φφ


TARGET BISTATIC POLARIMETRIC « DIMENSION »

77=

TARGET MONOSTATIC POLARIMETRIC « DIMENSION »

55

=


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BISTATIC TARGET EQUATIONSBISTATIC TARGET EQUATIONS

PURE TARGET – BISTATIC CASE

SINCLAIR MATRIX [S]POLARIMETRIQUE TARGET

« DIMENSION » = 7


16 HUYNEN REAL PARAMETERS(A, A0, B0, B, C, D, E, F, G, H, I, J, K, L, M, N)


|XX|, |XY|, |YX|, |YY|, Re(XXXY*), Im(XXXY*)Re(XXYX*), Im(XXYX*)Re(XXYY*), Im(XXYY*)Re(XYYX*), Im(XYYX*)Re(XYYY*), Im(XYYY*)Re(YXYY*), Im(YXYY*)



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4x4 HERMITIAN MATRIX - RANK 1

16 PRINCIPAL MINORS = 0

[ ]

−+++−−−−+++−+−

=⋅=

A2jIJjNMjKLjIJBBjFEjGHjNMjFEBBjDCjKLjGHjDCA2

kkT0

0

0

T*


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[ ]T k k T= ⋅ =

*

2A0 C - jD H + jG L - jKC + jD B0 + B E + jF M - jNH - jG E - jF B0 - B J + jIL + jK M + jN J - jI 2A


( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

2 2

2 2

2 2

2 2

4 2

2 2

2 2

2 2

2

02 2

02 2

0

02 2

0 0

0 0

0 0

0

A B B I J HN FL EK GM

A B B N M IC FL EK DJ

AE JM IN KB JG NC HI MD

AF JN IM LB MC JH IG DN

AA L K IC HN DJ GM

I B B EN FM AC IA K N H L M G

N B B EI JF AH NA K C I L J D

M B B EJ IF AG MA K D J L C I

J B B EM FN AD

− − − + − −

+ − − + + −

− + − + + + +

+ + − − + +

− − + + +

+ + + + − − − −

− + + + − − − −

− − + − + + + +

+ − + ( ) ( ) − − + + +2 0JA K G M L N H

+

+

+

+

+

+

+

+

+

( ) ( )

( ) ( ) ( ) ( )

2

2

2

2

0

0 02 2

0 02 2

0

02 2 2 2

0

0

0

0

A B B C D

A B B G H

A E CH DG

A F CG DH

B B E F

C B B EH GF

H B B CE DF

G B B FC ED

D B B FH GE

+ − −

− − −

− + −

− −

− − −

− − −

+ − −

− + + −

− − + −

=

=

=

=

=

=

=

=

=

0

0

0

0

0

0

0

0

0

+

+

+

+

+

+

+

+

+


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BISTATIC TARGET DIAGRAMBISTATIC TARGET DIAGRAM

[ ]T k k T= ⋅ =

*


(C,D

)(G,H)

(E,F)

(L,K)

(I,J)

(M,N)

2A0

B0 + B

B0 - B

2A


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[ ]T k k T= ⋅ =

*


(C,D

)

(E,F)

(L,K)

(M,N)

( )2 0 02 2A B B C D+ = +

( )2 0 02 2A B B G H− = +

2 02 2A A L K= +

B B E F02 2 2 2− = +

( )2 02 2A B B I J− = +

( )2 02 2A B B M N+ = +

2A0

B0 + B

B0 - B

2A

(G,H)

(I,J)


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[ ]T k k T= ⋅ =

*


(L,K)

(M,N)

( )22

AE JM INAF JN IM

= −

= − +

2A0

B0 + B

B0 - B

2A

(G,H)

(C,D

)

(E,F)

(I,J)


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[ ]T k k T= ⋅ =

*


(L,K)

(M,N) ( )22

AG IL JKAH JL KI

= − +

= −

2A0

B0 + B

B0 - B

2A

(G,H)

(C,D

)

(E,F)

(I,J)


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[ ]T k k T= ⋅ =

*


(L,K)

(M,N)

22

AC LM KNAD KM LN

= += −

2A0

B0 + B

B0 - B

2A

(G,H)

(C,D

)

(E,F)

(I,J)


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[ ]T k k T= ⋅ =

*


(L,K)

(M,N)

22

AC LM KNAD KM LN

= += −

( )22

AE JM INAF JN IM

= −

= − +

( )22

AG IL JKAH JL KI

= − +

= −

2A0

B0 + B

B0 - B

2A

(G,H)

(C,D

)

(E,F)

(I,J)


SAPHIR

SAPHIR

BISTATIC POLARISATION FORKBISTATIC POLARISATION FORK

K

L

MC 2

X 1

O 2N'

X 2

E 2

E 1

M'

C 1

N

D 1

D 2

O 1

Q

U

V(D1 , D2)

XPOL SADDLE(K , L)

COPOL MAX

(O1 , O2)COPOL NULLS

(C1 , C2)XPOL MAX

Z.H. CZYZP.I.T Warsaw

(E1 , E2) (X1 , X2)XPOL NULLS

TRANSMISSION (M, N = M⊥) , RECEPTION (M ’ , N ’ = M ’⊥)


SAPHIR

SAPHIR

MONOSTATIC POLARISATION FORKMONOSTATIC POLARISATION FORK

K

L

M

C 2

X 1

O 2

X 2

E 2

E 1

C 1

N

D 1 D 2

O 1

Q

U

V


SAPHIR

SAPHIR

SCATTERING POLARIMETRYSCATTERING POLARIMETRY

X

Y

SINCLAIR MATRIX [S]


DESCRIPTORS


YY

XYYX

XX


POLARIMETRIC REMOTE SENSING


SAPHIR

SAPHIR

Questions ?Questions ?

scattering polarimetry - esa · 2013-08-01 · fsa convention bsa convention. back scattering...

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