A Direct Solution of the Spherical Harmonics Approximation to the Radiative Transfer Equation for an Arbitrary Solar Elevation. Part I: Theory

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Abstract
The spherical harmonics approximation to the transfer equation for an azimuth-dependent component of intensity of scattered radiation is reduced by finite differences to a block algebraic system of particularly simple structure. This algebraic system can be solved numerically for homogenous or nonhomogenous models of a plane-parallel atmosphere, using the finite-differences analogue of the simple-shooting technique based on initial-value problems or the multiple-shooting technique for the solution of two-point boundary-value problems.