The consortium will implement new polarimetric decompositions including the Huynen, Krogager and Cameron decompositions. In addition, we believe it will be useful and relatively easy to implement the related Kennaugh Matrix and the Yang decomposition.Huynen Decomposition: A time-dependent target is described by the target coherency matrix <T3> consisting of nine independent real parameters, whereas a stationary target is determined by five real parameters. The basic idea of the Huynen decomposition is to decompose the nine-parameter target coherency matrix <T3> into an average stationary effective target described by T0 with five parameters and a residue part given by TN which contains the four remaining degrees of freedom: (1)Where (2)And (3)Here the subscript T and N denote the equivalent single target (T) and N-target (N). The N-target corresponds to a perfectly non-symmetric target which is defined with parameters (B0N, BN, EN, FN). The parameters (A0, C, D, H, G) are fixed and the parameters (B0T, BT, ET, FT) corresponding to the equivalent single target can be obtained from the following equations: 2A_0 (B_0T+B_T )=C^2+D^2 (4)2A_0 (B_0T-B_T )=G^2+H^2 (5)2A_0 E_T=CH-DG (6)2A_0 F_T=CG+DH (7)Then the parameters (B0N, BN, EN, FN) are determined from equation (1), i.e.B_0=B_0T+B_0N (8)B=B_T+B_N (9)E=E_T+E_N (10)F=F_T+F_N (11)The three generators of the equivalent single target T0 are T_11T=2A_0 (12)T_22T=B_0T+B_T (13)T_33T=B_0T-B_T (14)They can also be displayed in color-coded image with Red=T22T, Green=T33T, and Blue=T11T. Krogager Decomposition: Krogager decomposition is a coherence decomposition in which the symmetric scattering matrix S is decomposed into three coherent components corresponding to the sphere, diplane and helix contributions: (15)where kS, kD, and kH correspond to the sphere, diplane, and helix contribution, Î¸ is the orientation angle, and Ï† is the absolute phase. It is important to note that the three Krogager decomposition parameters (kS, kD, kH) can be expressed in function of three roll invariant Huynen parameters (A0, B0, F), following: k_S^2=2A_0 (16)k_D^2=2(B_0-|F|) (17)k_H^2=4(B_0-âˆš(B_0^2-F^2 ))=2(âˆš(B_0+F)-âˆš(B_0-F))^2 (18)The Krogager decomposition can be displayed in color-coded image as: Red=kD, Green=kH and Blue=kS. Cameron Decomposition: In the Cameron decomposition, the scattering matrix S is decomposed in terms of basis invariant target features using the Pauli matrices. It performs a factorization of the measured scattering matrix S based on two basic properties of radar targets: reciprocity and symmetry, by means of projections of the measured scattering vectors onto the relative subspace of reciprocal, symmetric, etc., scattering vectors. The first step is to decompose the scattering matrix S into reciprocal and nonreciprocal components, by projecting the scattering S matrix onto the Pauli matrices and separating the symmetric and nonsymmetric components of the matrix. The second step then considers decomposition of the reciprocal term into two further components. The Cameron decomposition takes the following form: (19)where the scalar a=span(S), the angle Î¸rec represents the degree to which the scattering matrix obeys the reciprocity principle, and the angle Ï„sym represents the degree to which the scattering matrix deviates from the set of scattering matrices corresponding to symmetric scatterers. S Ì‚_nonrec represents the normalized nonreciprocal component, S Ì‚_sym^max the normalized maximum symmetric component, and S Ì‚_sym^min the normalized minimum symmetric component.For the convenience of derivation, the scattering matrix S above is expressed in vector form: (20)Then the Pauli decomposition can be formulated in a vector form as follows: (21)The next step consists of defining different projectors PQ as the direct product of the different basis vector S Ì‚_(Qâˆˆ{A,B,C,D} ) with its transpose, following: (22)Then angle Î¸rec is given by (23)Scattering S matrix with Î¸rec=0 corresponds to a scatterer which strictly obeys the reciprocity principle, whereas scattering matrices with Î¸rec=Ï€/2 corresponds to a fully nonreciprocal scatterer. Defining the operator DX âƒ— as the follows: (25)The scattering matrix S which corresponds to a reciprocal scatterer can be further decomposed into maximum and minimum symmetric components with (26)It follows that the last three Cameron decomposition parameters are given by a=ã€–â€–S âƒ—â€–ã€—_2^2=span(S) (27)S Ì‚_nonrec=((S âƒ—,S Ì‚_D ))/|(S âƒ—,S Ì‚_D )| S Ì‚_D (28) (29)Kennaugh MatrixThe Kennaugh matrix is related to the Mueller matrix, but defined for the radar backscattering case. All elements of the Kennaugh matrix are measurable quantities in power rather than amplitudes and phases of the scattering matrix and coherency matrix. The classical representation of a target using a scattering matrix describes a single physical event. The representation in terms of power allows evaluating the same physical event in different ways, by considering mainly that this results from independent measurements. The representations of data in terms of power to describe backscattering mechanisms are generally more powerful. This is because the elimination of the absolute phase from the target means that the power-related parameters become incoherently additive parameters. The 4x4 Kennaugh K matrix is defined as follows: where is the complex conjugate operation, corresponds to the Kronecker tensori matrix product given by and where the scattering matrix S and matrix A are given respectively by and All the Kennaugh elements can be calculated with the equations below: Yang Decomposition:For some special cases, mainly when the parameter A0 is relatively small, the Huynen decomposition cannot be used to extract a desired target from an averaged Kennaugh matrix or coherency T3 matrix.To overcome this disadvantage, Yang et al. [R-14] modified the Huynen decomposition based on a simple transform of the Kennaugh matrix. From equations (4) to (7) in the Huynen decomposition above, we can see that if the parameter A0 is small or null, the parameters (B0T, BT, ET, FT) become very sensitive to the averaged Kennaugh matrix.The modified Huynen decomposition is the following: If the parameter A0 associated with the averaged Kennaugh matrix K is not small, for example, A0 â‰¥ m00/10 where m00 is the first (row, column) element of K, then use the classical Huynen target decomposition theorem. Else if A0 < m00/10, then define: and where and If A01 â‰¥ A02 then apply the Huynen target decomposition theorem to the Kennaugh matrix K1 and denote It follows the modified Huynen decomposition: Else then apply the Huynen target decomposition theorem to the Kennaugh matrix K2 and denote It follows the modified Huynen decomposition:

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