Inverse solutions to radiative-transfer problems with partially transparent boundaries and di*use re+ection

Analytical techniques are used to solve a class of inverse radiative-transfer problems relevant to 1nite and semi-in1nite plane-parallel media. While the assumption of isotropic scattering is made, di*use re+ection is allowed at the surface, for the semi-in1nite case, and at both surfaces for the case of a 1nite layer. For the general case based on a semi-in1nite medium, a cubic algebraic equation is used to de1ne the basic result, but for the speci1c case of a semi-in1nite medium illuminated by a constant incident distribution of radiation, very simple exact expressions are developed for the albedo for single scattering $ and the coe7cient for di*use re+ection . Analytical results are also developed (again in terms of a cubic algebraic equation) for the case of a 1nite layer with equal re+ection coe7cients relevant to the two surfaces. For the general case of a 1nite layer with unequal re+ection coe7cients, two speci1c formulations are given. The 1rst algorithm is based on a system of three quadratic algebraic equations for the two re+ection coe7cients 1 and 2 and the single-scattering albedo $. Secondly, an elimination between these three algebraic equations is carried out to yield two coupled algebraic equations for 1 and 2 plus an explicit expression for $ in terms of 1 and 2. In addition, an exact expression for 0, the optical thickness of the 1nite layer, is developed in terms of $, 1 and 2. As is typical with the considered class of inverse problems in radiative transfer, all surface quantities are either speci1ed or considered available from experimental measurements. All basic results are tested numerically. ? 2001 Elsevier Science Ltd. All rights reserved.