# Many of the equalities that are discussed in 42 were

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derived for the Stokes matrix in optical polarimetry. Many of the equalities that are discussed in [42] were first explicitly given by [43], then revised and extended in [42]. At the same time, this equality is a necessary but not sufficient condition to warrant that K is derived from a S matrix [41]. Thus, we have obtained the formulation for GD for all the data representations in PolSAR. Although we have a distance in the form of GD , it would be better to construct a measure of similarity from it. This formulation can be achieved by complementing it with the unit, i.e., f ref = 1 - GD ( K , K ref ) , (22) where K is an observed Kennaugh matrix and K ref is the refer- ence elementary scatterers. In this sense, f ref is a similarity and the corresponding GD is a dissimilarity. In PolSAR literature, Yang et al. [44], Touzi and Charboneau [45], and Chen et al. [46] discuss similarity-based approaches for describing scattering phenomenon from PolSAR images. Thus, the GD is advantageous in terms of its physical significance with parallel definitions across all data represen- tations in PolSAR. Its simple form makes it ideal for compu- tational implementation in several PolSAR applications [37]– [40]. II. D ATA S ETS We 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. 5 angle is specified as 28 . 02 to 29 . 82 . The original image is multi-looked by a factor of 2 in range and 4 in the azimuth resulting in a 20 m ground resolution. The other image is an L-Band ALOS-2 acquisition on 29th January 2019. The off-nadir angle is specified as 30 . 8 . The original image is multi-looked by a factor of 3 in range and 5 in the azimuth resulting in a 15 . 7 m ground resolution. Fig. 2 shows the two Pauli RGBs for these data sets. (a) Pauli RGB 1 (b) Pauli RGB 2 Fig. 2: Pauli RGB images of RS-2 C-band (on left) and ALOS- 2 L-band (on right) acquisition over San Francisco. III. N EW R OLL I NVARIANT P ARAMETERS In the phenomenon of a roll, the antenna coordinate system is rotated by an angle θ about the radar line of sight (LoS) [7]. In such a case, the observed Kennaugh matrix K transforms as follows, K ( θ ) = R ( θ ) KR ( θ ) T (23) where the (orthogonal) rotation matrix R ( θ ) is given by R ( θ ) = 1 0 0 0 0 cos 2 θ - sin 2 θ 0 0 sin 2 θ cos 2 θ 0 0 0 0 1 . (24) Let K 0 be the Kennaugh matrix for a roll-invariant target. A roll-invariant target has the property of preserving its scattering signature despite a roll i.e., R ( θ ) K 0 R ( θ ) T = K 0 (25) for any value of the θ angle. Thus, the geodesic distance between K ( θ ) and the roll invariant target K 0 can be further simplified in the following way, GD ( K ( θ ) , K 0 ) = GD ( K ( θ ) , K 0 ( θ )) = GD ( K , K 0 ) . (26) The first step was obtained by applying (25) followed by the property (P3) of GD , as discussed in Sec. I in the next step. Thus, the GD between the observation and a roll-invariant target is a roll-invariant quantity.