Computation of Domain-Averaged Irradiance Using Satellite-Derived Cloud Properties

The respective errors caused by the gamma-weighted two-stream approximation and the effective thickness approximation for computing the domain-averaged broadband shortwave irradiance are evaluated using cloud optical thicknesses derived from 1 h of radiance measurements by the Moderate Resolution Imaging Spectrometer (MODIS) over footprints of Clouds and the Earth’s Radiant Energy System (CERES) instruments. Domains are CERES footprints of which dimension varies approximately from 20 to 70 km, depending on the viewing zenith angle of the instruments. The average error in the top-of-atmosphere irradiance at a 30° solar zenith angle caused by the gamma-weighted two-stream approximation is 6.1 W m−2 (0.005 albedo bias) with a one-layer overcast cloud where a positive value indicates an overestimate by the approximation compared with the irradiance computed using the independent column approximation. Approximately one-half of the error is due to deviations of optical thickness distributions from a gamma distribution and the other half of the error is due to other approximations in the model. The error increases to 14.7 W m−2 (0.012 albedo bias) when the computational layer dividing the cloud layer is increased to four. The increase is because of difficulties in treating the correlation of cloud properties in the vertical direction. Because the optical thickness under partly cloudy conditions, which contribute two-thirds of cloudy footprints, is smaller, the error is smaller than under overcast conditions; the average error for partly cloudy condition is −2.4 W m−2 (−0.002 albedo bias) at a 30° solar zenith angle. The corresponding average error caused by the effective thickness approximation is 0.5 W m−2 for overcast conditions and −21.5 W m−2 (−0.018 albedo bias) for partly cloudy conditions. Although the error caused by the effective thickness approximation depends strongly on the optical thickness, its average error under overcast conditions is smaller than the error caused by the gamma-weighted two-stream approximation because the errors at small and large optical thicknesses cancel each other. Based on these error analyses, the daily average error caused by the gamma-weighted two-stream and effective thickness approximations is less than 2 W m−2.