1

Abstract— In this paper we develop a dual polarizedversion of the entropy/alpha decomposition method.We first develop the basic algorithms and then applythem to theoretical models of surface and volumescattering to demonstrate the potential fordiscrimination, classification and parameterestimation. We then apply the formalism to dualpolarized data from the ALOS/PALSAR systemoperating in PLR and FBD modes to illustrateapplication to forest classification, urban scatteringcharacterization and point target signature analysisfor ship classification.

I. INTRODUCTION

While the entropy/alpha approach was originally designedto simplify multi-parameter depolarization [1,2], as occursfor example in Quadpol radar backscatter, it can also beapplied to the simpler case of dual polarization. In thisscenario the radar transmits only a single polarization andreceives, either coherently or incoherently, two orthogonalcomponents of the scattered signal. This corresponds (inthe coherent case) to measurement of the full state ofpolarization of the scattered signal for fixed illumination.The new ALOS/PALSAR sensor has just such a fullycoherent-on-receive mode. This motivates us in this paperto investigate development and application of a dualpolarized entropy/alpha technique that can be used to takeadvantage of such coherent dual polarized systems.

In radar there are two important special cases when thedual formalism becomes important. It is oftenadvantageous in radar design (from a cost, data rate andcoverage point of view) to employ a single transmittedpolarization state and a coherent dual channel receiver tomeasure orthogonal components of the scattered signal.Such dual polarized radars are not capable ofreconstructing the complete scattering matrix but insteadcan be used to reconstruct a column of the [S] matrix.From this we can then construct a 2 x 2 wave coherencymatrix [J] to estimate depolarization. One key decision inthe design of such radars is the best single polarization toemploy (the reference point X on the Poincaré sphereshown in figure 1).

For example, it is widely acknowledged that circularpolarization would be a good choice, since the coherencebetween co and cross circular polarization can be used todirectly estimate orientation of the scatterer. However,circular polarized transmitters are not widely employed inradar imaging systems, where there is a preference forlinear transmitters. For example, several radar systemsemploy horizontal (H) or vertical (V) polarization transmitand dual channel reception of H and V components.

Figure 1 : The Poincaré Sphere interpretation ofthe dual polarized alpha angle

These radars can then be used, via local averaging, toestimate the following forms of the 2x2 wave coherencymatrix

€

JH[ ] =SHHSHH

* SHHSHV*

SHV SHH* SHV SHV

*

JV[ ] =SVV SVV

* SVV SVH*

SVHSVV* SVHSVH

*

-(1)

Note that some radars drop the coherent-on-receive mode(for example the European ENVISAT/ASAR system inalternating polarization (AP) mode). In this case wecannot measure the off-diagonal elements of [J] and obtainonly the two diagonal terms. It is important to realize thatfor such non-coherent polarimetric radars we cannot applythe entropy/alpha or polarized/depolarizeddecompositions.

2. THE DUAL POLARISED ENTROPY/ALPHADECOMPOSITION

Using the standard interpretation of normalisedeigenvalues of [J] as probabilities Pi, together with thefact that in 2x2 problems the second eigenvector can bederived from the principal eigenvector usingorthogonality, we obtain an entropy/alphaparameterisation of the wave coherency matrix [J] asshown in equation 2. From this we can again define anentropy/alpha plane as shown in figure 2. In this case theupper and lower bounds of alpha for a given entropy canbe found by considering the parameterised diagonal 2 x 2matrix shown in equation 3. In this case the lower andupper curves show a symmetry about α = 45o. This canbe traced to the 2 dimensional eigenvector space, one ofwhich has α = 0 and the other π/2.

THE DUAL POLARISATION ENTROPY/ALPHA DECOMPOSITION:A PALSAR CASE STUDY

Shane CLOUDE, AEL Consultants, Cupar, Scotland, UK, e-mail: aelc@mac.com

δ

2α PPP

2θ

2τ

X

2

€

J[ ] =Jxx JxyJxy* Jyy

⇒

U2[ ] =cosα −sinαe− iδ

sinαeiδ cosα

[D] = λ1 + λ2( )P1 00 P2

⇒α2 = P1α + P2(

π2−α) =α(P1 − P2) + P2

π2

H2 = Pi log2 Pii=1

2

∑

- (2)

€

1 00 m

0 ≤ m ≤1⇒αmin =

mπ2(1+ m)

m 00 1

0 ≤ m ≤1⇒αmax =

π2(1+ m)

-(3)

By contrast, in the quadpol case we encounter a 2:1 biasin favour of the π/2 eigenvector subspace that tends to liftthe alpha range with increasing entropy. To distinguishthe two cases we refer to the dual polarized diagram asH2α.

Figure 2: Dual Polarised Entropy/Alpha Diagram

In this dual polarized case we have a simple interpretationof α on the Poincaré sphere (see figure 1). The angle α isthe angular separation between the received wavepolarization state P and the reference state X used forconstruction of the wave coherency matrix. Hence theorigin of the H2α plane corresponds to a replica of thetransmitted polarization, and all other received states canbe mapped into the diagram, depending on both theirtransformation of state and degree of polarization (throughentropy). Indeed, the angles α and δ are related to theorientation and ellipticity angles θ,τ of the receivedwave’s polarization ellipse via a spherical triangleconstruction on the Poincaré sphere as shown in figure 1.One reason for employing this approach as opposed to theconventional Stokes vector/degree of polarization [1,2] isthe continuity it provides with the entropy/alpha diagramsfor quadpol backscatter [1] and bistatic polarimetry [4] asshown in figure 3. In this way we can use the samephenomenology to describe all polarized scatteringproblems. The averaging implied by the entropy/alphaapproach does not pick out the state P corresponding tothe maximum eigenvalue as in the polarized/depolarizeddecomposition but instead forms an average based on aprobabilistic interpretation of making measurements onthe wave and obtaining polarization X and Y withprobabilities P1 and P2 respectively.

Figure 3: Entropy/Alpha Diagrams for Dual (red) ,Monostatic (green)and Bistatic (blue) scattering

Hence the average polarization state would have acorresponding alpha value given by

€

α . For example,when the coherency matrix approaches the identity (noise)then

€

α =π4

, being an equal mixture of the state X (α = 0)

with its antipodal orthogonal state Y (α = π/2).

3. SURFACE AND VOLUME SCATTERING ANDTHE H2α DECOMPOSITION

Returning to the coherent dual polarised case of equation1, we can summarise the polarimetric information contentof these 2 x 2 matrices by relating them to the full 3x3scattering coherency matrix [T] [1,2] as shown in equation4

€

k H =121 1 00 0 1

k

⇒ JH[ ] = k H k H*T =

121 1 00 0 1

T[ ]

1 01 00 1

=12t11 + t22 + 2Re(t12) t13 + t23

(t13 + t23)* t33

kV =121 −1 00 0 1

k

⇒ JV[ ] = kV kV*T =

121 −1 00 0 1

T[ ]

1 0−1 00 1

=12t11 + t22 − 2Re(t12) t13 − t23

(t13 − t23)* t33

- (4)

These can still be used to characterize (albeit in a limitedsense compared to full [S] matrix systems) depolarizationby random surface and volume scatterers. For example,Bragg scattering from smooth surfaces will have 2x2coherency matrices of the form shown in equation 5 [3].Note that these imply zero scattering entropy and α2 = 0for all angles of incidence and dielectric constants. Thisstands in contrast to the Quadpol H/α decompositionwhich still maintains zero entropy but where α can berelated to dielectric constant for known angle of incidence[3].

3

€

JH[ ]Bragg =12a + c + 2Re(b) 0

0 0

JV[ ]Bragg =12a + c − 2Re(b) 0

0 0

-(5)

€

E pqs = i 2k cosθ Bpq ˆ Z 2k sinθ( )

⇒

BHH = B⊥ =cosθ − εr − sin2θ

cosθ −+ εr − sin2θ

BVV = B|| =εr −1( ) sin2θ −εr(1+ sin2θ)[ ]εr cosθ + εr − sin2θ( )

2

BHV = BVH = 0

⇒

a = BHH + BVV2

b = BHH + BVV( ) BHH − BVV( ) *c = BHH − BVV

2

We can extend this approach to include depolarizationcaused by surface roughness by considering the X-Braggmodel [3]. This takes the projected form into 2x2matrices shown in equation 6

€

JH[ ]XBragg =12

a +c2(1+

sin4Δ4Δ

) + 2Re(b) sin4Δ4Δ

0

0 c2(1− sin4Δ

4Δ)

JV[ ]XBragg =12

a +c2(1+

sin4Δ4Δ

) − 2Re(b) sin4Δ4Δ

0

0 c2(1− sin4Δ

4Δ)

-(6)

Here we see nonzero scattering entropy, but note amixture of roughness and dielectric constant dependencein the terms. Contrast this again with the full 3x3coherency matrix formalism where we are able to separateroughness and moisture effects [3]. Still, the level ofdepolarisation in equation 6 is small and to investigatehow high the entropy can become in dual polarizedsystems, we turn to the case of volume scattering from arandom cloud of anisotropic particles. The dualpolarisation coherency matrices can be expressed in termsof the particle anisotropy factor A as shown in equation 7[2]

€

JH[ ]Vol = JV[ ]Vol =1

153A2 + 4A + 8 0

0 (A −1)2

-(7)

where A is defined as the ratio of principal particlepolarisabilities and is given in terms of the particledielectric constant ε r and shape factors Li as shown inequation 8. For dipole scatterers, the strongest source ofdepolarization, A tends to infinity. For oblate particles onthe other hand A = 0 and the depolarisation is weaker.These two cases may be distinguished in dual polarisedsystems using their limiting form of 2x2 coherencymatrix as shown in equation 9, derived directly from 7and 8. We see that the maximum dual polarised entropyfrom such a cloud is 0.811 and that dual polarisedsystems do offer some potential for the separation ofdifferent types of volume scattering behaviour.

€

α i =εo εr −1( )V1+ εr −1( )Li

⇒ A =L2 +

1εr −1

L1 +1

εr −1

L1 =

1− e2

e2(−1+

12eln1+ e1− e

)

x1 > x2 = x3 e2 =1− x2

2

x12 (prolate)

1+ f 2

f 2(1− 1

farctan f )

x1 < x2 = x3 f2 =

x22

x12 −1 (oblate)

L2 = L3 =12(1− L1)

- (8)

€

JH[ ]prolate = JH[ ]prolate ∝3 00 1

⇒ H2 = 0.811

JH[ ]oblate = JV[ ]oblate ∝8 00 1

⇒ H2 = 0.503

-(9)

In all the above examples we have seen prediction of adiagonal 2 x 2 coherency matrix. This may give theimpression that the phase of the received signal containszero information and hence may be dispensed with in dualpolarised systems (as in the ENVISAT/ASAR AP modefor example). To see that this is not always the case,consider a simple extension of these ideas to includerotations about the line of sight, as occurs for example inscattering from a sloped surface. To see this we considerthe form of the dual polarisation scattering vectors forcoherent point scatterers in terms of a rotation about theline of sight. These are shown in equation 10 asprojections of the coherent Pauli k scattering vectors [2].

€

k H =121 1 00 0 1

.1 0 00 cos2θ −sin2θ0 sin2θ cos2θ

.k0k10

=12k0 + k1 cos2θk1 sin2θ

kV =121 −1 00 0 1

.1 0 00 cos2θ −sin2θ0 sin2θ cos2θ

.k0k10

=12k0 − k1 cos2θk1 sin2θ

-(10)

This applies for example to a titled Bragg surface when k0

= BHH + BVV and k1 = BHH – BVV. It follows that thecoherency matrices for reflection symmetric random mediatake the form shown in equation 11

€

JH[ ] = k H k H*T =

12

k02

+ k12 cos2 2θ + 2Re(k0k1

*)cos2θ k12 cos2θ sin2θ + k0k1

* sin2θk1

2 cos2θ sin2θ + k1k0* sin2θ k1

2 sin2 2θ

JV[ ] = kV kV*T =

12

k02

+ k12 cos2 2θ − 2Re(k0k1

*)cos2θ k12 cos2θ sin2θ − k0k1

* sin2θk1

2 cos2θ sin2θ − k1k0* sin2θ k1

2 sin2 2θ

-(11)

4

from which we note the presence of complex off-diagonalterms. Only for zero tilt (θ = 0) or uniformly random θ(azimuthal symmetry) do these off-diagonal termsdisappear.

This suggests that the phase terms will also be importantfor identifying coherent point scatterers. Using a coherentassumption, we can relate the dual polarisation alphaparameter to the scattering matrix elements for asymmetric point as shown in equation 12

€

tanα2 =SXX − SYY( )sin2θ

SXX + SYY( ) + SXX − SYY( )cos2θ

δ2 = argSXX − SYY( )sin2θ

SXX + SYY( ) + SXX − SYY( )cos2θ

- (12)

In the more general case of a non-symmetric pointscatterer (helices for example), these relations take theform shown in equation 13

€

tanα2 =2SXY cos2θ + SXX − SYY( )sin2θ

SXX + SYY( ) + SXX − SYY( )cos2θ − 2SXY sin2θ

δ2 = arg2SXY cos2θ + SXX − SYY( )sin2θ

SXX + SYY( ) + SXX − SYY( )cos2θ − 2SXY sin2θ

-(13)

We now turn to consider application of the H2α ideas tointerpretation of dual polarized data from theALOS/PALSAR sensor.

4. ALOS-PALSAR CASE STUDY

JAXA successfully launched the ALOS satellite onJanuary 24th 2006 and obtained their first SAR imagefrom the PALSAR sensor on February 15th. One keyinnovation of the PALSAR sensor is its ability to operatewith full polarization diversity. Table I summarises themain modes of operation of the system. Key for ourpurposes are two modes, Fine Beam Dual polarization(FBD) which employs a single transmit polarisation (H)and dual coherent reception of H and V and the fullypolarimetric mode (PLR) which enables full coherentscattering matrix collection by alternating H and Vpolarizations on transmit. We use the PLR mode data tosimulate FBD by taking only the HH and HV scatteringterms.

In order to view the products of the entropy/alphadecomposition we employ a colour-coding scheme asshown in figure 4. We employ an HSV coding with thehue equal to the alpha parameter (in range 0 to 90o)ranging from blue for values around zero to green for 45degrees and red for 90 degrees. For the phase term we usea similar coding but scaled to the range –180o to 180o sogreen now corresponds to zero phase. To modulate this inboth cases we employ the entropy as a saturation term, sothe colour saturation decreases with increasing entropy asshown in figure 4.

Finally we employ the scattered power to modulate theintensity. In this way areas of the image with highentropy will resort to conventional gray scale imagerywhile low entropy regions will be coloured with theappropriate value.

Figure 4 : HSV Colour Coding for Display of Entropy/Alpha (upper)and Entropy/Phase (lower) products

We illustrate three potential application areas of the H2αtechniques. In the first case we consider forestclassification, in the second urban area scattering and inthe third ship classification [5]. We now consider each ofthese in turn.

a) Forest Classification Studies

We use PALSAR data for Glen Affric in Scotland(lat/long 57.25oN/5oW) collected on 16/10/06. This is aremote mountainous region in NW Scotland where thereis a mixture of open moorland, fresh water Lochs andforest cover. Figure 5 shows an HH image of the region(taken from PLR mode at 21.5 degrees angle ofincidence). We note good water/land contrast but poordiscrimination between open moorland and forest.Contrast this with figure 6. Here we show a dualpolentropy/alpha image. We can see clear discrimination offorested and non-forested regions. The latter have highentropy while the former show a return signal closelypolarized to the incident wave (and hence appear as blueregions in figure 6). Finally in figure 7 we show theentropy/phase image. Here we see, in addition to thegood forest/nonforest classification, close correlationbetween the phase and surface slopes.

b) Urban Scattering

We turn now to consider data collected in the dualpolarized FBD mode at 41.5 degrees angle of incidence.We choose an urban scene over the city of Adelaide inSouth Australia (Lat/Long 35oS/138.5oW) collected on20/06/06. Figure 8 shows an HH image of the city.Figure 9 shows an H2α image, where we immediately

5

see interesting colour modulations, particularly from themain city area in the centre of the image.

Figure 5 L-Band HH Image of Glen Afffric Region, Scotland

Figure 6 : HH/HV Entropy/Alpha Image for Glen Affric Scene

Figure 7 : HH/HV Entropy/Phase Image for Glen Affric Scene

Figure 10 shows the corresponding entropy/phase image,which again shows interesting structure. These earlyresults point to a level of scattering complexity in theurban environment that supports further development ofclassification and interpretation algorithms based on dualpol phase analysis.

c) Ship Classification

Finally we consider application to ship classification. Inthe same Adelaide scene were located two ships andseveral small boats near the harbour region. Figure 11shows how the large ships have different signatures in theentropy/phase domain. Figure 12 shows a larger view ofthe scene with a range of small boats around the harbourmouth. Again we note significant variation of scatteringbehaviour from the various craft, which indicates thepossibility of discrimination and classification based onphase.

5. CONCLUSIONS

In this paper we have introduced the dual polarizationentropy/alpha decomposition and shown how it relates tothe conventional entropy/alpha approach. We havedemonstrated application of the technique toALOS/PALSAR data. In particular we have used the

coherent-on-receive FBD dual polarized and PLR modesto demonstrate the importance of phase information inpolarimetry by considering three important applicationareas namely forest classification, urban characterizationand ship classification.

6. REFERENCES

[1] S.R. Cloude, E. Pottier, "An Entropy Based Classification Schemefor Land Applications of Polarimetric SAR", IEEE Transactionson Geoscience and Remote Sensing, Vol. 35, No. 1, pp 68-78 ,January 1997

[2] S.R. Cloude, E. Pottier, "A Review of Target DecompositionTheorems in Radar Polarimetry", IEEE Transactions onGeoscience and Remote Sensing, Vol. 34 No. 2, pp 498-518,March 1996

[3] I.Hajnsek, E. Pottier, S.R. Cloude”, Inversion of SurfaceParameters from Polarimetric SAR”, IEEE Transactions onGeoscience and Remote Sensing, Vol 41/4, April 2003, pp 727-744

[4] S R Cloude, “Information Extraction in Bistatic Polarimetry”,Proceedings of 6th European SAR Conference, EUSAR 06,Dresden, May 2006

[5] J.S. Lee, M.R. Grunes, T.L Ainsworth, L. J. Du, D. L. Schuler,S.R Cloude, “Unsupervised Classification using PolarimetricDecomposition and the Complex Wishart Distribution”, IEEETransactions Geoscience and Remote Sensing, Vol 37/1, No. 5, p2249-2259, September 1999

Figure 8 : L-band HH Image of Adelaide, Australia

Figure 9 : HH/HV Entropy/Alpha Image for Adelaide Scene

Figure 10 : HH/HV Entropy/Phase Image for Adelaide Scene

6

Figure 11 : Entropy/Phase Images of 2 large ships in Adelaide HarbourScene

Figure 13 : Entropy/Alpha (upper) and Entropy/Phase (lower) Imagesof small boats in Adelaide Harbour Scene

Mode Polarisation Incidence Angle Data Rate SwathFBS(28 MHz) HH 21,34,41 degrees 240 Mbps 56kmFBD(14 MHz) HH,HV 34,41 degrees 240 Mbps 56kmPLR (14 MHz) HH,HV,VH,VV 21.5 degrees 240 Mbps 15kmDirect(14 MHz) HH 21,34,43 degrees 120 Mbps --

SCANSAR HH 5 scans 120 Mbps --Table I : Operating Modes of the ALOS PALSAR sensor with polarimetric modes shown in red