Theory of backscattering enhancement of random discrete isotropic scatterers based on the summation of all ladder and cyclical terms

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Abstract
Backscattering enhancement of random discrete scatterers was previously investigated by a second-order theory. In this paper it is shown that the second-order theory is inadequate for problems with appreciable albedo and optical thickness. Multiple-scattering effects are included by summing all the ladder terms and the cyclical terms for isotropic point particles. The summation of the ladder terms leads to the classical Schwarzchild–Milne integral equation. The summation of the cyclical terms leads to a two-variable cyclical-transfer integral equation. The cyclical-transfer integral equation is then solved numerically, and the results include all the multiple-scattering effects associated with cyclical terms. Numerical results are illustrated as a function of scattering angle, albedo, and optical thickness. Results demonstrate that higher-order multiple-scattering effects are important for appreciable albedo and optical thickness. The multiple-scattering solution also gives a sharper backscattering peak than the second-order solution.