Analytical Solutions of Inverse Radiative Transfer Problems by Using the Structural Angular Method: Cloudy Atmospheres

The application of a structural angular methodStructural angular method developed by the author to analytical solutionsAnalytical solutions of inverse problemsInverse problems in the optics of cloudy planetary atmospheres has been considered for combined retrieval of a full set of their optical parameters: optical thicknessOptical thickness, phase functionPhase function, and single scattering albedoSingle scattering albedo in the visible region of spectrum. The input information for solving such problems includes azimuthal harmonicsAzimuthal harmonics of the diffuse reflection coefficientDiffuse reflection coefficient for a cloudy atmosphereCloudy atmosphere in dependence on arbitrary values of solar zenith distanceSolar zenith distance and viewing directions in the case of infinite optical thicknesses. The chosen mathematical models for angular distribution of diffuse reflection were based on its exact representations given by V.A. Ambartsumian for the case of a homogeneous semi-infinite cloudy atmosphereSemi-infinite cloudy atmosphere and corresponding Sobolev—van de Hulst strict asymptotic formulas for finitely large optical thicknesses in the case of conservative and almost conservative scatteringConservative scattering of photons. Exact linear singular integral equationsSingular integral equations obtained by (Sobolev in Radiant energy transport in atmospheres of stars and planets, State publishing house of tech-theor. literature, Moscow (in Russian), 1956; Sobolev in Light scattering in planetary atmospheres, Pergamon Press, Oxford, 1975) and (Mullikin in Astroph J 139:379–388, 1964) were also used for solving the above-mentioned problems. Special attention was paid to evaluation of the quality of elaborated algorithms and obtained solutions of inverse problems in the case of real models of the Earth’s cloudy atmosphere in dependence on active angular variablesAngular variables, dimensions of input and retrieved phase functions as well as perturbation parameters of input optical data.