A Mathematical Extension to the General Four-Component Scattering Power Decomposition With Unitary Transformation of Coherency Matrix

Abstract

As an improvement of the four-component scattering power decomposition with rotation of coherency matrix (Y4R) and extension of volume model (S4R), the general four-component decomposition with unitary transformation (G4U) was devised to make the full use of the polarimetric information in coherency matrix. This article enables an extension to G4U by deriving the scattering balance equation system in G4U to investigate the role of unitary transformation first. Despite self-contained, the scattering balance equation system in Y4R and S4R is independent of the ${T}_{13}$ entry of coherency matrix. To include ${T}_{13}$ in decomposition, the unitary transformation in G4U adds a ${T}_{13}$ -related but redundant balance equation into the original system. As a result, ${T}_{13}$ is accounted for by G4U, and we attain no exact solution to the equation system but some approximate ones. By deducing the general expression of the approximate solutions, a generalized G4U (GG4U) is then created and denoted as ${G}({\psi })$ . The decomposition constant $\psi $ determines a GG4U by producing a $\psi $ -rotated double-bounce scattering matrix. We treat this as the scattering preference of $\mathcal {G}({\psi })$ to characterize the physical mechanism. By assigning appropriate values to $\psi $ , we attain GG4U of different preferences, while ${G}({0})$ and ${G}({+} {\pi }/{8})$ just correspond to S4R and G4U. A dual G4U ${G}(-{\pi }/{8})$ is also achieved. The duality ${G}({\pm }{\pi }/{8})$ provides us an adaptive improvement to both G4U and S4R by strengthening the double-bounce scattering over urban and building area while enhancing the surface scattering over water and land area. Both theoretical derivation and experiments on ten polarimetric synthetic aperture radar data sets validate the outperformance. Nonetheless, for whatever unitary transformation employed, there is, forever, a ${T}_{13}$ -related residual component in GG4U. Thus, the incorporation of unitary transformation into Y4R and S4R for the full modeling of polarimetric information is impossible in theory only when the canonical scattering model with nonzero $(\mathrm {1, 3})$ entry of coherency matrix is used to add the balance equation system an independent ${T}_{13}$ -related equation rather than a redundant one.