A method is described for numerical solution with spherical harmonics of the equation of transfer for plane-parallel atmospheric models with arbitrary inhomogeneities in the vertical direction and with axially-symmetric phase functions. Our method has the following features: (1) high efficiency in finding eigenvalues and eigenvectors associated with characteristics of the scattering layer; (2) stable expressions for the condition linking the radiance at the top to the bottom of the layer; (3) economy in calculations for a multi-layer problem. A smoothing method to obtain reasonable intensity patterns is also discussed. Representative results are presented for monochromatic diffuse fluxes and for the intensity of the radiation emerging from a single layer. © 1988 Pergamon Press plc.
Takeuchi, Y. (1988). Use of spherical harmonics in the solution of the radiation transfer problem. Journal of Quantitative Spectroscopy and Radiative Transfer, 39(3), 237–245. https://doi.org/10.1016/S0022-4073(88)90086-6