Abstract
In order to facilitate the computations of the intensities of radiation reflected and/or transmitted by plane-parallel, vertically inhomogenous, scattering-absorbing media, we carry out the optical thickness integrations of the Cauchy systems (normally referred to as Invariant Imbedding Equations) for reflection and transmission functions originating from the first three orders of scattering: the medium in question is represented by a stack of a certain number of homogeneous slabs, each of which is characterized by a constant single scattering albedo and a constant phase function together with the optical thickness.
The results are a set of recurrence formulae involving only the angular intergrations, a convenient feature for numerical computations, and should prove useful particularly for finding approximate values of the high frequency Fourier coefficients of reflection and transmission functions of inhomogeneous media or efficiently initializing the solution for a thin layer to perform rigorous multiple scattering computations by means of other techniques such as the Doubling-Adding method.
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Kawabata, K., Ueno, S. The first three orders of scattering in vertically inhomogeneous scattering-absorbing media. Astrophys Space Sci 150, 327–344 (1988). https://doi.org/10.1007/BF00641728
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DOI: https://doi.org/10.1007/BF00641728