On the equivalence of the dual-wavelength and polarimetric equations for estimation of the raindrop size distribution

In writing the integral equations for the median mass diameter and particle concentration, or comparable parameters of the raindrop size distribution, it is apparent that when attenuation effects are included, the forms of the equations for polarimetric and dual-wavelength radars are identical. In both sets of equations, differences in the backscattering and extinction cross sections appear: in the polarimetric equations, the differences are taken with respect polarization at a fixed frequency while for the dual-wavelength equations, the differences are taken with respect to wavelength at a fixed polarization. Because the forms of the equations are the same, the ways in which they can be solved are similar as well. To avoid instabilities in the forward recursion procedure, the equations can be expressed in the form of a final-value. Solving the equations in this way traditionally has required estimates of the path attenuations to the final gate: either the attenuations at horizontal and vertical polarizations at the same frequency or attenuations at two frequencies with the same polarization. This has been done for dual-frequency (air/spaceborne case) and polarimetric radars by the respective use of the surface reference technique and the differential phase shift. An alternative to solving the constrained version of the equations is an iterative procedure recently proposed in which independent estimates of path attenuation are not required. Although the procedure has limitations, it appears to be quite useful. Simulations of the retrievals help clarify the relationship between the constrained and unconstrained approaches and their application to the polarimetric and dual-wavelength equations.