A Multidimensional Extension of the Concept of Coherence in Polarimetric SAR Interferometry

Interferometric synthetic aperture radar (InSAR) is a phase-based radar signal processing technique that has been addressed from a polarimetric point of view since the late 1990s, starting with Cloude and Papathanassiou´s foundational work. Polarimeric InSAR (PolInSAR) has consolidated as an active field of research in parallel to non-PolInSAR. Regarding the latter, there have been a number of issues that were discussed in an earlier paper from which some other questions related to Cloude´s PolInSAR come out naturally. In particular, they affect the usual understanding of coherence and statistical independence. Coherence involves the behavior of electromagnetic waves in at least a pair of points, and it is crucially related to the statistical independence of scatterers in a complex scene. Although this would seem to allow PolInSAR to overcome the difficulties involving the controversial confusion between statistical independence and polarization as present in PolSAR, Cloude´s PolInSAR originally inherited the idea of separating physical contributors to the scattering phenomenon through the use of singular values and vectors. This was an assumption consistent with Cloude´s PolSAR postulates that was later set aside. We propose the introduction of a multidimensional coherence tensor that includes PolInSAR´s polarimetric interferometry matrix Ω₁₂ as its 2-D case. We show that some important properties of the polarimetric interferometry matrix are incidental to its bidimensionality. Notably, this exceptional behavior in 2-D seems to suggest that the singular value decomposition (SVD) of Ω₁₂ does not provide a physical insight into the scattering problem in the sense of splitting different scattering contributors. It might be argued that Cloude´s PolInSAR in its current form does not rely on the SVD of Ω₁₂ but on other underlying optimization schemes. The drawbacks of such ulterior developments and the failu- e of the maximum coherence separation procedure to be a consistent scheme for surface topography estimation in a two-layer model are discussed in depth in this paper. Nevertheless, turning back to the SVD of Ω₁₂, the use of the singular values of a prewhitened version of Ω₁₂ is consistent with a leading method of characterizing coherence in modern Optics. For this reason, the utility of the SVD of Ω₁₂ as a means of characterizing coherence is analyzed here and extended to higher dimensionalities. Finally, these extensions of the concept of coherence to the multidimensional case are tested and compared with the 2-D case by numerically simulating the scattered electromagnetic field from a rough surface.