a Method for Computing the Radiative Transfer Extinction and Phase Matrices for Liquid Water Clouds.

A detailed method is presented for computing the radiative transfer extinction and phase matrices for liquid water clouds. The approach is general and can be applied to any cloud droplet distribution and cloud liquid water content for electromagnetic radiation at any frequency and polarization. An exact expression for scattering from a finite number of droplets is derived in terms of an aggregate transition matrix (T-matrix) which is recursively computed from the isolated T-matrices for individual droplets. The isolated T-matrices were found to be diagonal and composed of Mie coefficients. Thus using relatively simple droplet spatial and size distributions, one can compute the average scattering matrix and Stokes matrix which when averaged over cloud subsections becomes the phase matrix. The optical theorem then allows one to compute the extinction matrix from the elements of the scattering matrix. Existing propagation models and a variety of approaches are also discussed. Details of some droplet size and spatial distributions are included and use of the stochastic coalescence equation is proposed as a way of introducing time variations. In fact, any additional information from measurements of cloud liquid water content, observations of cloud droplet distributions, or modeling of the refractive index of the medium between the droplets can be incorporated into this method via the associated parameters in the equations or by a Monte Carlo simulation to generate the isolated and aggregate T-matrices and ultimately the extinction and phase matrices. Violations of the addition theorem used in the recursive algorithm are expected in Monte Carlo generated droplet distributions. However, some recently developed methods for mitigating this problem are referenced.