Intensity of radiation reflected by a vertically inhomogeneous medium through multiple scattering |
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Authors: | Kiyoshi Kawabata Ryuko Hirata |
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Institution: | (1) Department of Physics, College of Science, Science University of Tokyo, Shinjuku-ku, Tokyo, Japan;(2) Department of Astronomy, College of Science, University of Kyoto, Japan |
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Abstract: | To facilitate the computation of the radiative intensity reflected, upon multiple scattering, by a vertically inhomogeneous medium, an implicit formula for integrating the invariant imbedding equation for Fourier-decomposed reflection function is derived starting with the formal solution: the height variations of the single-scattering albedo and the phase function characterizing the degree of inhomogeneity are thereby approximated by piece-wise, but continuous, linear functions of optical height , while the reflection function is approximated by a piece-wise quadratic polynomial in over each integration step. Using these approximations, the integration involved in the formal solution is then carried out analytically, yielding a correctortype formula for finding the reflection function at each step of . It is expected that this formula is capable of handling general cases of inhomogeneous media where both single-scattering albedo and phase function are allowed to vary continuously with height.Similar, but explicit expressions are also derived for the single and the second-order scattering solutions, with which the higher-order Fourier terms of reflection function are to be approximated, thereby enabling us to avoid the iterative process. |
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