The subject of polarization for completely and partially polarized waves has been extensively treated throughout the literature (e.g. Mott 1992). This chapter presents a classical approach to wave and antenna polarization for readers who have not been previously exposed to this topic. The first goal is to derive the voltage form of the dual-polarized radar range equation for scattering by a single particle in both the linear and circular bases. This is followed by a generalization of the usual (single-polarized) power form of the radar range equation to include the concepts of copolar and cross-polar response surfaces and characteristic polarizations.

For scattering by a large number of distributed particles, the ensemble-averaged Mueller matrix is formally defined and conventional radar observables, such as reflectivity, differential reflectivity, the linear depolarization ratio, and the copolar correlation coefficient, are defined in terms of Mueller matrix elements. The time-averaged polarimetric covariance matrix is formally defined from the concept of an “instantaneous” back scatter matrix and the associated “feature” vector. Dual-polarized radars which are configured to measure the three power elements and three (complex) correlation elements of the covariance matrix offer a complete characterization of scattering from precipitation particles. The relation between the covariance matrices in the linear (h/v) and circular bases is examined in detail, including simplifications afforded by symmetry arguments. A number of examples of radar measurements are presented to illustrate the theory.