Number of Terms Required in the Fourier Expansion of the Reflection Function for Optically Thick Atmospheres

Computational results have been obtained for the separate terms in the Fourier expansion of the reflection function of an optically thick, conservatively scattering, atmosphere composed of cloud particles. The computations were obtained by successive applications of the invariant imhedding. doubling and asymptotic fitting methods to cover the range from very thin to very thick atmospheres. Result\ are presented which illustrate the magnitude of the separate terms in the Fourier expansion of the phase function and the Fourier expansion of the reflection function of a semi-infinite atmosphere as a function of the zenith angles of incidence and reflection. The azimuthally independent reflection fun&n i\ enhanced by as much as a factor of I15 over the first-order reflection function, whereas the azimuth-dependent reflection functions generally result from less multiple scattering. These results are compared with those for an atmosphere having a Henyey-Greenstein phase function with the same asymmetry factor (4 = 0.84123) as in the cloud model. The relative difference in the escape function and azimuthally independent reflection function is generally less than a few per cent, though differences up to 70% occur in the reflection function at angles where single scattering is important, Results are also presented which chow the number of terms required in the Fourier expansion of the reflection function to be assured an accuracy of tI.lci. The number of terms required depends strongly on the zenith angles of incidence and reflection a\ well as on details of the phase function.