A model of a clear atmosphere is presented based upon two assumptions: (1) the point-function equilibrium radiance for a given path of sight does not change with altitude; (2) there is no absorption. As a result of these assumptions, the equation of transfer can be integrated. The path radiance for any slant path becomes a function of the equilibrium radiance and the beam transmittance of that path. In addition, the equilibrium radiance is a function of the scalar irradiance from the sun, sky, and earth and the proportional directional scattering coefficient for ground level. Sky radiances, and path radiances through the atmosphere for both upward and downward paths are determined by four parameters; the proportional directional scattering function for ground level, the total vertical beam transmittance of the atmosphere, the scalar albedo, and the solar zenith angle.
There is evidence that the real atmosphere does on some days conform to the above two assumptions to a useful extent for the visible portion of the spectrum.PDF Article