Radiative transfer algorithms are developed to estimate the optical thickness of clouds using an irradiance detector located above, deep within, and beneath a cloud. Both monodirectional and diffuse illumination cases are considered. For each detector position, one algorithm is derived from asymptotic radiative transfer theory and another is derived from transport-corrected diffusion theory. The algorithms for an above-cloud or below- cloud detector with monodirectional illumination can be utilized without radiative transfer calculations and are much less sensitive to the phase function than a related algorithm for an above-cloud detector that requires measurements of downward and upward radiances.
Radiative transfer calculations with the FN method are used to numerically test the above-cloud detector algorithms for monodirectional illumination for the Haze-L and Fair Weather Cumulus cloud models. Both algorithms show good agreement with the FN results for optically thick clouds; the transport-corrected diffusion algorithm also agrees with the FN method for optically thin clouds, but is less accurate for intermediate cloud thicknesses than the asymptotic algorithm.
An error analysis for the above-cloud detector algorithms is included that confirms the difficulty of estimating the optical thickness if the surface albedo is close to an “exclusion” albedo, which is the irradiance ratio for a semi-infinite cloud for the transport-corrected diffusion algorithm or a value close to it for the asymptotic algorithm. A sensitivity analysis provides a means to estimate retrieval ranges for the algorithms and it shows that these ranges dramatically increase as the cloud absorption decreases. The useful retrieval range generally tends to be broader for non-normal incident illumination directions. The general trends of the sensitivity analysis also are applicable to the estimation of optical thickness using the bidirectional reflection function.