The scattering matrix of spheroids in the Rayleigh–Gans limit is the principal topic of this chapter. While the assumption of spheroidal shape is somewhat simplistic considering the wide distribution of shapes of natural hydrometeors, it is quite remarkable that important polarimetric radar observations of precipitation can, to a large degree, be explained using the spheroidal model and Rayleigh–Gans scattering. It will also become evident that important concepts gained from Rayleigh–Gans scattering (e.g. the dependence of elevation angle, particle orientation effects), can also be extended to Mie scattering. The scattering matrix is formulated in forward scatter alignment (FSA) and back scatter alignment (BSA) conventions, following van Zyl and Ulaby (1990). Sufficient detail is provided for those readers without prior exposure to this subject matter. Simple hydrometeor orientation models are used to illustrate the behavior of linear polarization radar observables such as differential reflectivity and linear depolarization ratio.

For completeness, the Mie solution for scattering by spherical particles is formulated as a boundary value problem, with vector spherical harmonics and the multipole expansion of the electric field being covered in Appendix 2. Expressions for radar, scattering, and extinction cross sections are given in terms of Mie coefficients. This chapter concludes with a very brief discussion of numerical methods for scattering by non-spherical particles at higher frequencies, and the T-matrix method is briefly described in Appendix 3.