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.
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