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A Muceller-matrix-based approach is described for calculating polarimetric radar parameters such as reflectivity, differential reflectivity, linear depolarization ratio, propagation differential phase, and backscatter differential phase. Particle scattering is calculated using the T matrix, which is ideally suited for spheroids in the Mie region. The T matrix, is calculated once for a given wavelength, particle size, shape, and dielectric constant, making full use of symmetry through use of a body coordinate system attached to the particle. For various orientations of the particle symmetry axis and for various radar elevation angles, a separate laboratory coordinate system is used. The scattering amplitudes and polarization basis vectors are transformed between the two coordinate systems using geometrical transformations. Two particle orientation distributions, which are two-dimensional in θ, ϕ, are considered. These are the Gaussian and simple harmonic in θ with random ϕ. Sample calculations are presented for a 5:1 prolate spheroid, which is used to model failing needles/columns preferentially oriented due to vertical electric field. An oblate ice particle (axis ratio = 0.8) is used to model the fall mode of large hailstones. Whereas the horizontal-vertical polarization basis is used here, the Mueller matrix for other orthogonal bases can be easily derived.
Vivekanandan et al. (1991) studied this question.