Basic SAR Polarimetry
Transcript of Basic SAR Polarimetry
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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
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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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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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http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf
F E
F E
F E
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http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf
F E
F E
F E
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http://earth.eo.esa.int/dragon/pottier1_SAR_polarimetry_basics.pdf
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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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)
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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.
Jong-Sen Lee and Eric Pottier , Polarimetric Radar Imaging- From Basics to ApplicationsCRC Press 2009
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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
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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
Jong-Sen Lee and Eric Pottier , Polarimetric Radar Imaging- From Basics to ApplicationsCRC Press 2009
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
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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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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 .
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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Poincare Sphere Projection on Complex Plane
From -An-Qing Xi and Wolfgang-Martin Boerner, http://dx.doi.org/10.1364/JOSAA.9.000437
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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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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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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Positive Semi-Definite Matrices
Unitary MatrixConjugate transpose
http://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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Hermitian Positive- Semi Definite Matrix
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
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Positive definite matrices
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
H i i P i i S i D fi i M i `
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Divyesh M. VaradeBasic SAR Polarimetry 27-06-2013
Positive definite matrices
Cholesky factorization
Congruence and diagonalization
Hermitian Positive- Semi Definite Matrix`
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E l
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Examples
det(A)= 6
det=12