derived for the Stokes matrix in optical polarimetry. Manyof the equalities that are discussed in  were first explicitlygiven by , then revised and extended in . At the sametime, this equality is a necessary but not sufficient conditionto warrant thatKis derived from aSmatrix . Thus,we have obtained the formulation forGDfor all the datarepresentations in PolSAR.Although we have a distance in the form ofGD, it wouldbe better to construct a measure of similarity from it. Thisformulation can be achieved by complementing it with theunit, i.e.,fref= 1-GD(K,Kref),(22)whereKis an observed Kennaugh matrix andKrefis the refer-ence elementary scatterers. In this sense,frefis a similarity andthe correspondingGDis a dissimilarity. In PolSAR literature,Yang et al. , Touzi and Charboneau , and Chen etal.  discuss similarity-based approaches for describingscattering phenomenon from PolSAR images.Thus, theGDis advantageous in terms of its physicalsignificance with parallel definitions across all data represen-tations in PolSAR. Its simple form makes it ideal for compu-tational implementation in several PolSAR applications –.II. DATASETSWe have utilized two PolSAR images of the San Francisco(SF) area. The first one is a C-Band RADARSAT-2 (RS-2)acquired on 9th April 2008. The near to far range incidence
TO APPEAR IN IEEE TRANS. GEOSCI. REMOTE SENS.5angle is specified as28.02◦to29.82◦. The original image ismulti-looked by a factor of2in range and4in the azimuthresulting in a20 mground resolution.The other image is an L-Band ALOS-2 acquisition on 29thJanuary 2019. The off-nadir angle is specified as30.8◦. Theoriginal image is multi-looked by a factor of3in range and5in the azimuth resulting in a15.7 mground resolution. Fig. 2shows the two Pauli RGBs for these data sets.(a) Pauli RGB 1(b) Pauli RGB 2Fig. 2: Pauli RGB images of RS-2 C-band (on left) and ALOS-2 L-band (on right) acquisition over San Francisco.III. NEWROLLINVARIANTPARAMETERSIn the phenomenon of a roll, the antenna coordinate systemis rotated by an angleθabout the radar line of sight (LoS) .In such a case, the observed Kennaugh matrixKtransformsas follows,K(θ) =R(θ)KR(θ)T(23)where the (orthogonal) rotation matrixR(θ)is given byR(θ) =10000cos 2θ-sin 2θ00sin 2θcos 2θ00001.(24)LetK0be the Kennaugh matrix for a roll-invariant target. Aroll-invariant target has the property of preserving its scatteringsignature despite a roll i.e.,R(θ)K0R(θ)T=K0(25)for any value of theθangle. Thus, the geodesic distancebetweenK(θ)and the roll invariant targetK0can be furthersimplified in the following way,GD(K(θ),K0) =GD(K(θ),K0(θ)) =GD(K,K0).(26)The first step was obtained by applying (25) followed by theproperty (P3) ofGD, as discussed in Sec. I in the next step.Thus, theGDbetween the observation and a roll-invarianttarget is a roll-invariant quantity.