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This paper summarizes the scientific legacy of Peter C. Waterman (1928-2012) who introduced concepts and theoretical techniques that have had a major impact on the fields of scattering by particles and particle groups, optical particletcharacterization, radiative transfer, and remote sensing. A biographical sketch is also included.
The minimal residual method is studied combined with polynomial preconditioning for solving large linear systems (Ax = b) with indefinite Hermitian coefficient matrices (A). The standard approach for choosing the polynomial preconditioners leads to preconditioned systems which are positive definite. Here, a different strategy is studied which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive, resp. negative eigenvalues of A around 1, resp. around some negative constant. In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of Chebyshev approximation problems. Some basic results are established for these approximation problems and a Remez type algorithm is sketched for their numerical solution. The problem of selecting the parameters such that the resulting indefinite polynomial preconditioners speeds up the convergence of minimal residual method optimally is also addressed. An approach is proposed based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.
The development of Radar Polarimetry and Radar Interferometry is advancing rapidly, and these novel radar technologies are revamping Synthetic Aperture Radar Imaging decisively. In this exposition the successive advancements are sketched; beginning with the fundamental formulations and high-lighting the salient points of these diverse remote sensing techniques. Whereas with radar polarimetry the textural fine-structure, target-orientation and shape, symmetries and material constituents can be recovered with considerable improvements above that of standard amplitude-only Polarization Radar ; with radar interferometry the spatial (in depth) structure can be explored. In Polarimetric-Interferometric Synthetic Aperture Radar (POL-IN-SAR) Imaging it is possible to recover such co-registered textural plus spatial properties simultaneously. This includes the extraction of Digital Elevation Maps (DEM) from either fully Polarimetric (scattering matrix) or Interferometric (dual antenna) SAR image data takes with the additional benefit of obtaining co-registered three-dimensional POL-IN-DEM information. Extra-Wide-Band POL-IN-SAR Imaging - when applied to Repeat-Pass Image Overlay Interferometry - provides differential background validation and measurement, stress assessment, and environmental stress-change monitoring capabilities with hitherto unattained accuracy, which are essential tools for improved global biomass estimation. More recently, by applying multiple parallel repeat-pass EWB-POL-D(RP)-IN-SAR imaging along stacked (altitudinal) or displaced (horizontal) flight-lines will result in Tomographic (Multi- Interferometric) Polarimetric SAR Stereo-Imaging , including foliage and ground penetrating capabilities. It is shown that the accelerated advancement of these modern EWB-POL-D(RP)-IN-SAR imaging techniques is of direct relevance and of paramount priority to wide-area dynamic homeland security surveillance and local-to-global environmental ground-truth measurement and validation, stress assessment, and stress-change monitoring of the terrestrial and planetary covers. In addition, various closely related topics of (i) acquiring additional and protecting existing spectral windows of the Natural Electromagnetic Spectrum (NES) pertinent to Remote Sensing; (ii) mitigating against common "Radio Frequency Interference (RFI)" and intentional Directive Jamming of Airborne & Space borne POL-IN-SAR Imaging Platforms are appraised.