Basic SAR Polarimetry

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Divyesh M. VaradeBasic SAR Polarimetry 27-06-2013

Basic SAR Polarimetry

Basic SAR Polarimetry

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Doctorand M.Sc.-Ing Divyesh M. Varade Dr. Avik BhattacharyaEditorial Board NERD, IIT Kanpur

Geoinformatics Division,Department of Civil EngineeringIndian Institute of Technology,

Kanpur-208016

Centre for Studies in ResourceEngineering CSRE, IIT Bombay,

Powai, MumbaiMaharashtra400 076


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ContentsPolarization Ellipse

Stokes ParametersEffect of Axis Rotation on Stokes ParametersEntropy vs DOPJones Vector Formulation of EM WaveComplex Polarization RatioDeschamps ParametersComplex Planimetric Projection of Polarization States on PoincareSphere

Appendix

Spherical TrigonometryTime Averaging and Ensemble AveragingPSD Hermitian Matrices

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Polarization EllipseDefines an ellipsoidal space in which the Horizontal and Verticalcomponents of the EM wave oscillate following an elliptical locus.

22( , ) ( , ) ( , )( , )

2 cos sin x o y o y o x oox ox oy oy

E z t E z t E z t E z t E E E E

Equation of an EllipseAx2 +By2+Cxy+Dx+Ey+F = 0

http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf

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http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf


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Polarization EllipseAmplitude (A)

Ellipticity ( )

Orientation Angle ( )

22oyox E E

)2,2(

),(2

)4/,0(

90

0)2/,0(2

Circle

Linear

cos22tan 22

oyox

oyox

E E

E E

E E

E E

oyox

oyox sin22sin 22

2cos2sintan2sin

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Polarization Ellipse

Courtesy Eric Pottier


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Deschamps ParametersSimilar to the ellipticity and the orientation angle,Deschamps parameters are angles that specify thepolarization state of the Electromagnetic wave on thePoincre Sphere.

cos22tan 22oyox

oyox

E E E E

E E

E E

oyox

oyox

sin22sin 22

2cos2sintan2sin http ://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf

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http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdfhttp://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf


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Deschamps ParametersWe have to use spherical trigonometry now

Correlated with Spherical Trigonometry(see Appendix)

cos 2 cos 2 cos 2

tan tan 2 cos 2ec

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Stokes Parameters


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Effect of Axis Rotation on Stokes Parameters

= -

No change in

Changes

Remains Same

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Changed Parameters

Let us introduce a shift in orientation angle by , then

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Conservation of Energy

We know that S3 is the invariant parameter

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Coherency MatrixCoherency Matrix

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Divyesh M. VaradeBasic SAR Polarimetry 27-06-2013 13


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F E

F E

F E

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F E

F E

F E

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F E

F E

F E

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P E on the Surfaceof the Sphere

P E inside the Sphere

P E at the centre ofthe Sphere


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Entropy vs DOPPolarized EM wave after scattering splits into two components

Polarized Component

Unpolarized Component

EM wave after scattering is partly depolarized.

An average measure of the depolarizing power of the medium isgiven by the so called depolarization index

Amount of depolarization is estimated using the Entropy

and the degree of polarization (P F)

The field quantities E F and P F are related by a single-valued function:

A. Aiello and J.P. Woerdman : Physical Bounds to the Entropy -Depolarization Relation in RandomLightScattering , Physical Review Letters (2004) http://cds.cern.ch/record/782196/files/0407234.pdf

Partially Polarized EM wave

)( F E

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http://cds.cern.ch/record/782196/files/0407234.pdfhttp://cds.cern.ch/record/782196/files/0407234.pdf


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Polarized light (P F = 1) has E F = 0 while partially polarized light(0 P

F< 1) has 1 E

F> 0.

Non-depolarizing media are characterized by D M = 1, whiledepolarizing media have 0 D M < 1.Non-depolarizing media are characterized by E M = 0, while fordepolarizing media 0 < E M 1.

Depolarizing systemswhich decrease the degree of polarization of the incident wave.Propagation through a depolarizing medium is defined by a non-deterministic Mueller matrix M.

Non-depolarizing systems

which do not decrease PFpropagation through non-depolarizing media can be described by adeterministic Mueller (or Mueller-Jones) matrix M J

Note : Subscript F single valued field relation

while M multi- valued media relation

A. Aiello and J.P. Woerdman : Physical Bounds to the Entropy -Depolarization Relation in RandomLightScattering, Physical Review Letters (2004) http://cds.cern.ch/record/782196/files/0407234.pdf

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Deterministic Mueller matrix M J

S q and S qwe denotes the Stokes parameters of the beambefore and after the scattering and relates Mueller-Jones Matrixas

J is the CoherencyMatrix derivablefrom the StokesParameters

*

**

( ) J J

J J

M J J J J

' 0,1,2,3 J q qS M S q

J M q

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Comes from Normalized Pauli Matrices


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Non-deterministic Mueller matrix MDefined by ensemble average of Jones Matrix with

definite power density i.e. P(< E>) >= 0

*

*

* *

( )

( ) ( ) J J

n J J n

M J

J J J p J n J n

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Mueller matrix(psd) defined as a parallelcomposition of pure Mueller matrices orCoherency matrix

EigenValues of H

0 1 2 3

Jos J. Gil Workshop on Light scattering from microstructures Laredo , Spain (11/09/1998)

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, , are indexes that relateselectron spin and momentum withpolarization. It is a stochastic process (due to

random nature of TEC inIONOSPHERE) A. Aiello and J.P. Woerdman : Physical Bounds to theEntropy-Depolarization Relation in RandomLightScattering, Physical Review Letters (2004)http://cds.cern.ch/record/782196/files/0407234.pdf

The simulation of curves is thus carried throughMonte Carlo Modelling with ( , ) being sampledrandomly in the model by varying and between0 and 1.

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Jones Vector General equation for electric fieldFor a travelling wave that has undergone k cycles atand with a phase difference of

cos E(z,t)=| E | ( t kz )

y joy

x jox

yoy

xox

e E

e E E(z,t)=

)kz t ( | E

)kz t ( ||E E(z,t)=

cos|

cos

Propagating in ZHence no oscillations in z-direction Ez = 0

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Defines the Polarsation State of the EM wave onPoincare Sphere

Jones Vector

)kz t ( oy

kz)t ( ox

oy

ox

e E(y,t)=E

e E(x,t)=E

)kz t ( E(y,t)=E

kz)t ( E(x,t)=E

cos

cos

e E(y,t)=E

E(x,t)=E e

e E(y,t)=E

e E(x,t)=E

e E(y,t)=E

e E(x,t)=E

e E(y,t)=E

e E(x,t)=E

oy

ox

)( oy

ox

)( oy

ox

)kz t ( oy

kz)t ( ox

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Jones Vector

cos

sinox x

j joy y

E(x,t)= E E =e e

E(y,t)= E e E = e

eeee E E E= j y xoyox .sin.cos.22

ipticalRight Ell pticalLeft Elli

.sincos

je Ae E

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cos cos sin cos( )sin . sin cos sin . x j j

E Ae R Aee e

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Special Unitary Group SU (2)Pauli Matrices

Form a set of basis matrices from which any real or complex matrixproblem can be decomposed into simple form.significant for simplifying complex matrix equations.

1 * , det( )=1T i i i i j j i

0i i

.

0 cos sin p j

p A e j

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Special Unitary Group SU (2)

Jong-Sen Lee and Eric Pottier , Polarimetric Radar Imaging- From Basics to ApplicationsCRC Press 2009

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Change of Polarimetric Basis

Orthogonal polarization states and polarization basisTwo Jones vectors E 1 and E 2 are orthogonal if their Hermitian

scalar product is equal to 0.


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Change of Polarimetric Basis

Jones Vector in Cartesian BasisJones Vector in orthonormal (, ) polarimetric basis T

Inverse Special Unitary Transformation


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Instead of using L-R or R-L basis we must use L-L or R-R basis.

Linear to Circular Basis

1, 2 j

Not Special Unitary

Jong-Sen Lee and Eric Pottier , Polarimetric Radar Imaging- From Basics to ApplicationsCRCPress 2009

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Complex Polarimetric RatioThe polarization of a wave can also be described by thecomplex polarization ratio.

from the Jones vectors we have

ox

joy

E(x,t)= E E

E(y,t)= E e

tanoy j j

ox

E E(y,t)e e

E(x,t) E

cos sin cossin cos sin E Ae j

cos sin cos cos cos sin sin

sin cos sin sin cos cos sincos cos sin sin 1 tan tan

by cos cossin cos cos sin tan tan

j

j j j j

j j

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Complex Polarimetric Ratio

It is possible to denote the SU(2) matrices in terms of

tan tan

1 tan tan

j

j

cos 2 cos 2 sin 2

tan1 cos 2 cos 2 j j

e


Absolute Phase term often taken as 0

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in an Arbitrary Polarization Basis

Any wave can be resolved into two orthogonal components

(linearly, circularly, or elliptically polarized) in the plane transverseto the direction of propagationFor an arbitrary polarization basis {A B} with unit vectors

where E A

and EB

are complex numbers. {AB} is also a complex number.

is the phase difference

cos 2 cos 2 sin 2tan

1 cos 2 cos 2 j j e

http://earth.eo.esa.int/polsarpro/Manuals/LN_Basic_Concepts.pdf

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in an Arbitrary Polarization Basis

*2 2 2 *

*2

coscos tancossin .

1cos Let 1

1 1 1tan = cos

111

1

j j AB

AB

B A

A B B A B B AB AB

A A A

E E E ee

E E E

E E

E E E E E E E E E

E

*

1

1 AB AB AB

Energy term cancels

je

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in an Various Polarization Basis


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on Poincare Sphere


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Polarization State in different basis


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The Poincar polarization sphere and

Mapping Function

Follows Reimanns Theory


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Poincare Sphere Projection on Complex Plane

From -An-Qing Xi and Wolfgang-Martin Boerner, http://dx.doi.org/10.1364/JOSAA.9.000437

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http://dx.doi.org/10.1364/JOSAA.9.000437http://dx.doi.org/10.1364/JOSAA.9.000437


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NullsCorresponds to the polarizationstates for which the energy ratio(or )

Saddle Points corresponds to polarizationstates at which the energy ratioincreases in some directionssymmetric to the point.Decreases in other orthogonaldirections in some otherorthogonal directions thereceived power will decreasedepending on both the modulus

and the phase of '. - Gives the complexpolarization ratio for the differentpolarization states in thetransformed basis .



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The family of fourpoints T1 , T2, X1, and

X2 lies on one greatcircle because the pointspossess the samephase.

The location of T1 andT2 on the Poincare

sphere can be found byrotating Si and S2 by anangle of /2 about thebase diameter X1X2.

the three pairs X 1 X2 ,S1S 2, and T1T2 are

perpendicular to oneanother

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Polarization fork constructed byusing the mapping function as

discussed previously withRiemann Transformations


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APPENDIX

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Spherical Trigonometry A spherical triangle is defined when three planes passthrough the surface of a sphere and through thesphere's center of volume.

'A', 'B', and 'C' labelthe surface angles

'a', 'b', and 'c'label thecentralangles

. The surface anglescorrespond to the angle atwhich two planes intersecteach other

www.rwgrayprojects.com/rbfnotes/trig/strig/strig.html

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http://www.rwgrayprojects.com/rbfnotes/trig/strig/strig.htmlhttp://www.rwgrayprojects.com/rbfnotes/trig/strig/strig.html


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Napier's Rules

1. The sine of an angle is equal to the productof cosines of the opposite two angles.2. The sine of an angle is equal to the product

of tangents of the two adjacent angles.

From Napier's Rule #1 and #2 respectively

sin cos(90 )cos(90 )

sin tan( ) tan(90 )

a A c

a b B

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Time vs Ensemble AveragingSimilar to noise, the unpolarized light varies stochastically in time andspace.

The same holds for the unpolarized component in partially polarized EMwave.The we cannot speculate the polarization state of the unpolarizedcomponent in time and space, an averaged approximation in is defined.

Over a certain time (or space) interval Time averaging

Over various samples taken at certain time instance (or spatialposition) - Ensemble Averaging

An ensemble average is directly related to the probability density functionderived from statistical analysis.

A time average is more directly related to real experiments.

Theoretical predictions based on ensemble averaging are equivalent toexperimental measurement results corresponding to time averaging when,and only when, the system is a so- called ergodic ensemble .Statistically Stationary systems are ergodic in nature. Thus ergodicityimplies stationarity

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Ergodicity of the mean implies stationarity of the mean. However,stationarity of the mean does not imply ergodicity of the mean

Ergodicity

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Contrasting Examples

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l


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0 500 1000 1500 2000 2500 3000-10

0

10Example of ensemble vs. time average for a noisy signal that contains a periodic compone

x ( t )

0 100 200 300 400 5000.8

1

1.2

t i m e a v g

.

0 500 1000 1500 2000 2500 3000-2

0

2

e n s .

a v g .

points

Example

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f


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Positive Semi-Definite Matrices

Unitary MatrixConjugate transpose

http://college.cengage.com/mathematics/larson/elementary_linear/4e/shared/downloads/c08s5.pdf

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http://college.cengage.com/mathematics/larson/elementary_linear/4e/shared/downloads/c08s5.pdfhttp://college.cengage.com/mathematics/larson/elementary_linear/4e/shared/downloads/c08s5.pdf


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Comparison of Hermitian and Symmetric Matrices

Positive Semi-Definite Matrices


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Hermitian Positive- Semi Definite Matrix

Hermitian Matrix


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f


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Positive definite matrices

Courtesy- Magnus Jansson/ Bhavani Shankar: http://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdf

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i i i i S i fi i i
http://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdfhttp://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdfhttp://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdfhttp://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdf


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Courtesy- Magnus Jansson/ Bhavani Shankar:http://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdf

H i i P i i S i D fi i M i `


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Cholesky factorization

Congruence and diagonalization

Hermitian Positive- Semi Definite Matrix`

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Courtesy- Magnus Jansson/ Bhavani Shankar: http://www.kth.se/polopoly_fs/1.123121!/Menu/general/column-content/attachment/lec6.pdf

E l


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Examples

det(A)= 6

det=12

Basic SAR Polarimetry

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