TUTORIAL ON SAR POLARIMETRY
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TUTORIAL ON SAR POLARIMETRY

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ENVISAT/ASAR Dual Polarisation Case 6. ENVISAT/ASAR DUAL POLARISATION CASE 6.1 ASAR dual polarization data formats 6.1.1 Supported ASAR data formats 6.1.1.1 Alternating Polarization modes The ASAR instrument may be configured to acquire data according to one of the Alternating Polarization (AP) modes. Classical fully polarimetric SAR devices perform data acquisition with two emission and reception polarization channels. ASAR, instead, emits signals on a single polarization channel (H or V) while reception is performed on one or two orthogonal polarizations (H and/or V). The different possible configurations are summarized in the following figure. Received Emitted PP configuration polarizations polarizationH PP 1 H V H PP2 V V 1 ENVISAT/ASAR Dual Polarisation Case H H PP3 V V where PP stands for Partial Polarization or Partially Polarimetric. 6.1.1.2 Supported AP data formats Partial polarization data sets are delivered under different formats adapted to specific types of application. Among the different AP data formats, three are supported by PolSARpro and can be exploited to provide an interpretation of the polarimetric properties of scattering. • Alternating Polarization mode Single-look complex (APS) data format Data type Sets of complex scattering coefficients corresponding to the selected PP configuration Geometry Slant Range Coverage 100 km along-track, 56 - 100 km across-track Pixel spacing Equal to Resolution ...

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ENVISAT/ASAR Dual Polarisation Case
6 . E N V I S AT / A S A R D U A L P O L A R I S AT I O N C A S E
6.1
6.1.1
6.1.1.1 
ASAR dual polarization data formats
Supported ASAR data formats
Alternating Polarization modes
The ASAR instrument may be configured to acquire data according to one of the Alternating Polarization (AP) modes.
Classical fully polarimetric SAR devices perform data acquisition with two emission and reception polarization channels. ASAR, instead, emits signals on a single polarization channel (H or V) while reception is performed on one or two orthogonal polarizations (H and/or V).
The different possible configurations are summarized in the following figure.
Received
Emitted
PP configuration polarizations polarization
PP 1
PP2
H
V
H
V
H
V
1
 
 
 
 
 
Geometry
Coverage
Pixel spacing
Equal to Resolution: e.g 7.8m * 4m
Images ofreal reflectivitycorresponding to the selected PP configuration
 Alternating Polarization mode Precision (APP) image format
Geometry
Ground Range
Alternating Polarization mode Single-look complex (APS) data format
Among the different AP data formats, three are supported by PolSARpro and can be exploited to provide an interpretation of the polarimetric properties of scattering.
 
 
Data type
Sets ofcomplex scattering coefficients corresponding to the selected PP configuration 
100 km along-track, 56 - 100 km across-track
Slant Range
ENVISAT/ASAR Dual Polarisation Case
H
H
V
where PP stands for Partial Polarization or Partially Polarimetric.
6.1.1.2 
Supported AP data formats
Partial polarization data sets are delivered under different formats adapted to specific types of application.
 
 
Pixel spacing
 
12.5 m * 12.5 m
100 km along-track, 56 - 100 km across-track
Coverage
Data type
2
 
 
 
PP3
V
ENVISAT/ASAR Dual Polarisation Case
 ASAR Alternating Polarization mode ellipsoid Geocoded (APG) image
Data type
Geometry
Coverage
Pixel spacing
 
6.1.2
Images ofreal reflectivitycorresponding to the selected PP configuration
Geocoded Ground Range
100 km * 100 km
12.5 m * 12.5 m
Relation with fully polarimetric representations
Fully polarimetric quantities are measured from two acquisitions performed with orthogonal polarization states (generally H and V) in order to form each pixel scattering matrix [S]
The coherent scattering matrix modifies the polarization of an incident wave according to the following expression
Es=[S]Ei 
The scattering matrix is built from two acquisitions as follows
(1)
= [S]SShhhvSSvhvv=[EshEsv] S S0S(2) E withEsh=SSvhhhSSvvhv10=SShhvhandsv=SvhhhSvvhv1=SvhvvwhereEshandEsvcorrespond to the scene response to normalized incident horizontally and vertically polarized waves respectively.
6.1.2.1 APS format
APS format images are composed of coherent scattering coefficients and my then be expressed directly from the expression given in (2) according to the selected PP configuration
PP1  
A pixel is represented by a Jones vectors formed with two of the scattering matrix coefficients
 
 PP2
= EPP1SShhhv
⎤⎡ SSvhvv01=SShhhv 
(3)
A pixel is represented by a Jones vectors formed with two of the scattering matrix coefficients
3
ENVISAT/ASAR Dual Polarisation Case
 
 PP3
EPP2=SShhhv
SSvvvh10=SSvvvh 
(4)
A pixel is represented by a two element complex vector formed with two of the scattering matrix coefficients
[Shh EPP3= [Svh
SShvvv]]1001=SShvhv 
(5)
One may note that, oppositely to the vectorEs, the Jones vectorsEshandEsvare received in a polarization basis. It is then possible to representEshandEsvinto any desired polarization basis using classical Special Unitary transformation matrices E[U(φ,τ,α)]E =
(6)
Such a transformation may also be applied toEs PP3 configuration, but without the in physical interpretation associated to the formalism of changes of polarization basis. This might be particularly useful to perform a transformation similar to the one linking [C] and [T] matrices
6.1.2.2 
vv Es211111Es12SShh+SS = ⎥ = vv hh
APP and APG formats
(7)
APP and APG format images are composed of intensities and my then be expressed from the square moduli of the Sinclair matrix elements, according to the selected PP configuration
 
 
 
 
 
PP1
PP2
PP3
IPP1=SShhvh22 
IPP2
=SSvvhv22 
4
(8)
(9)
ENVISAT/ASAR Dual Polarisation Case
IPP3=SShvhv22 
(10)
One may note that APP and APG mode data do not exactly correspond to intensities derived from equivalent APS data, due to the slant range to ground range mapping transformation.
6.2
6.2.1
Speckle filtering
APP and APG format data filtering
As it was seen in the former paragraph, APP and APG format data consist of incoherent intensity values that are fit to incoherent speckle filtering.
The formulation of the boxcar and Lee filters can be adapted to intensity variables.
6.2.1.1 APP and APG boxcar filter Filtered intensity vector estimates,IPPi, are constructed by computing the sample mean over each pixel neighborhood, defined by a sliding window of(NwNw)pixels ~ IPPi=IPiNPw (11)
6.2.1.2 
APP and APG Lee filter
The adaptation of Lee MMSE filter to APP and APG data formats leads to the following intensity vector estimate expression ~ IPPi=IiPPNw+k IPPiIPPNiw (12) where k is an adaptive filtering coefficient based on local statistics and given by vIE2Iσ k=ar2n2 varI[1n1] σn2=Lcn e()3pe sriiaar vleht 1oirp a e with ck where represents the sum of the components of the vectorIPPi. The two components of the intensity vectorIPPi be filtered independently by applying may separate scalar filters.
6.2.2
APS format data filtering
APS format data consist of coherent scattering coefficients that cannot be directly used as inputs of a speckle filter.
5
ENVISAT/ASAR Dual Polarisation Case
Similarly to the case of target vectors filtering, a incoherent covariance matrix may be built from the APP coherent vector,EPPi, as [C2]PPi=EPPiEP†Pi (14) The express of the boxcar and Lee filters are similar to those introduced for the (3×3) [C] and [T] matrices.
6.2.2.1 APP and APG boxcar filter
Filtered estimates, [C2]PPi, are constructed by computing the sample mean over each pixel neighborhood, defined by a sliding window of(Nw×Nw)pixels ~ [C2]PPi=[C2]Pi(15) PNw 
6.2.2.2 
APP and APG Lee filter
The adaptation of Lee MMSE filter to APP and APG data formats leads to the following intensity vector estimate expression ~ [C2]PPi=[C2]PNiPw+k[C2]PPi[C2]PPiNw (16)
where k is an adaptive filtering coefficient, based on local statistics, given by varspanE2spanσn2 k= varspan[1n2] withσn2=1Lthe a priori speckle variance wherespanrepresents the trace of the covariance matrix.
6.3
Incoherent decomposition
(17)
Averaged [C2 do not correspond to an PP scattering mechanism and may the be] matrices decomposed in order to extract a pure PP representation.
A [C2] mayon its eigenvector basis as follows be decomposed
† [C2]=[V2][Λ2][V2]†= λ1v1v1†+ λ2v2v2 
(18)
where [Λ2] and [V2] represent (2× eigenvalue and special unitary eigenvector2) real matrices respectively. Each unitary eigenvector,vi, may be parameterized using 3 real parameters
vi=ejξ[cosαisinαiejδi]T 
The eigenvalue set may also be parameterized using two real variables 1+2=Tr[C2]=spanandH= −P1 log2(P1)P2 log2(P2)
6
(19)
(20)
ENVISAT/ASAR Dual Polarisation Case
withPi=λ1+λiλ2 
where H represents the scattering mechanism entropy whilePi is the pseudo-probability of one of the decomposed orthogonal scattering mechanisms, by conventionP1P2. The anisotropy, A, may also be used to characterize the scattering phenomenon P1P221) A=P1+P2 (
One may note that the joint use of H and A is redundant since the characterization of a two probability set requires only a single real parameter.
The physical signification of H and A is identical to the one derived for the decomposition of (3×3) [C] and [T] matrices.
Similarly to the (3×3) case, an average scattering mechanism may be rebuilt from the pseudo-probability set as
( , )=P1(1,1)+P2(2,2)
On the opposite, the interpretation of ( , the other.
6.4
(22)
) is totally different from one PP configuration to
Supervised segmentation
It has been demonstrated, in the chapter dedicated to polarimetric SAR data classification, that statistical supervised classification of data requires the derivation of data ML statistics.
6.4.1
6.4.1.1 
APS format data segmentation
Segmentation of coherent data
APS format consist of complex two-element vector composed of coherent scattering coefficients and may be represented, in a general way, as follows E=SSrspq (23)
where the subscripts p, q, r, s represent polarization channels.
It has been verified that when the radar illuminates an area of random surface of many elementary scatterers,Ecan be modeled as having a multivariate complex circular gaussian probability density functionNC(0,[Σ])of the form
expE†[ΣE P k= ( ) (πq[Σ]]1)
7
(24)
ENVISAT/ASAR Dual Polarisation Case
where q stands for the number of elements ofE, equal to three in the monostatic case, represents the determinant, and [Σ]=E(E E†) is the global (3×3) covariance matrix ofE.
The likelihood of a target vectorEgiven a clusterΘiis then given by
†exp(E[Σi]1E) P EΘ = iπq[Σi]
(25)
 
With[Σi] the covariance matrix of clusterΘi computed during the learning phase. In practice the actual value of[Σi] unknown and the covariance matrix is replaced by remains its maximum likelihood estimate [Σˆi as] defined
[Σˆi]=1E E† ni p∈Θi
Withnithe number of pixels belonging to the training clusterΘi.
(26)
From a computational point of view, it is generally preferable to deal with log-likelihood function,L EΘiinstead of the expression given inErreur ! Source du renvoi introuvable. 
L EΘi= −ln [Σˆi]tr[Σˆi]1E E†qlnπ 
(27)
The logarithm function being strictly increasing with its argument, the optimal decision rule is transformed, in the case ofEvector segmentation, to :
Decidep∈ ΘiifΘi=Argmind E[Σˆi] with d E[Σˆi]= +ln [Σˆi]+tr[Σˆi]1E E† 
(28)
Where the variabled E[Σˆi] may be assimilated to a statistical distance, derived from the opposite of log-likelihood function without constant terms.
The supervised segmentation algorithm may be summarized as follows
 
 
 
8
ENVISAT/ASAR Dual Polarisation Case
 
Initia lize pixel distribution over M custers fro m tra ining data sets
For each cluster [Σˆi]=N1iE E†∈ Θi
For each pixel,E       Θ  m if d(E,Θi)<d(E/Θj)j=1,L,M
Figure 1Supervised ML segmentation scheme
 
6.4.1.2 
 
Segmentation of incoherent data
ji
 
It has been shown that assuming thatE vectors have aNC(0,[ ]) a sample n- distribution, look covariance matrix [C2]=1/nE E†, follows a complex Wishart distribution with n n degrees of freedom,WC(n,[Σ]), given by
1 p([C])nqn[C2]n qexp(tr(n[Σ][C2])) 2= K(n,q) [Σ]n q withK(n,q)= πq(q1) / 2Γ(ni+1) i=1
WhereΓ(.) represents the Gamma function.
(29)
A development similar to the one presented in the case of coherent vectors leads to the following decision rule
Decidep∈ ΘiifΘi=ArgmaxL[C2] [Σˆi] with L[C2][Σ]ˆ= −nln [Σ]ˆntr[Σ]ˆ1[C2]+qnlnn+(nq) ln [C2]lnK(n,q) i i i
With [Σˆi the maximum likelihood estimate of the coherency matrix defined as] ,
[Σ]ˆ=1[C2] i ni p∈Θi
(30)
(31)
Taking the opposite of the lower expression of relation (30) and removing terms that do not depend on the cluster under test, the optimal decision rule becomes : Decidep∈ ΘiifΘi=Argmind[C2] [Σˆi] with d([C2][Σˆi]= +ln [Σˆi]+tr[Σˆi]1[C2)32 ](
9
ENVISAT/ASAR Dual Polarisation Case
The supervised segmentation algorithm may be summarized as follows
 
Initia lize pixel distribution over M custers
fro m tra ining data sets
For each cluster [Σˆi]=N1[C2]∈ Θi i
For each pixel, [C  2]   Θ   m if d([C2],Θi)<d([C2]/Θj)j=1,L,M
Figure 2Supervised ML segmentation scheme
6.4.2
APP and APG data segmentation
ji
 
The segmentation presented is simplified by assuming that APP and APG intensities are independent. In this case, their joint probability is given by ([2])=nqn[C2]nqexp(tr(n[Σ]1[C2])) I I I p CIK(n,q) [Σ]n I(33) q withK(n,q)=q(q1) / 2Γ(ni1) π + i=1
Where [CI2] and [ΣI] are (2×2) real matrices with off-diagonal elements equal to zero and built as follows
I [CI2]=01
0and [ΣI]=E(0I1) I2
0E(I2) 
The optimal decision rule for APP and APG format data is then given by Decidep∈ ΘiifΘi=Argmind[CI2] [ΣˆI i] with d([CI2][ΣˆI i]= +ln [ΣˆI i]+tr[ΣˆI i]1[CI2]  
   
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