A general analytical expression for the radiation source function of emitting and scattering media within the matrix operator method

An analytical expression has been derived for the radiation source function for a thermally emitting and scattering medium within the Matrix-Operator-Method (MOM). The final formulation is equivalent to the one found by Aronson and Yarmush (1966), who applied the transfer matrix to gamma-ray and neutron penetration and to transport problems in slab geometry. For the thermal infrared case, the general analytical expression reduces to a simple formula, which depends only on the zenith angle. The formula is incorporated in the MOM together with analytical expressions of the transmission and reflection operators following Liu (1990). With the aid of these formulations, expressions are derived as parameterizations of the scattering effects of clouds in nonscattering radiative transfer models by a modification of the emissivity and transmittance of clouds. The accuracy is better than 0.5 percent in the 11.5 micron window region for clouds of arbitrary optical depths.