A linear-system model has been developed to predict irradiance distributions of visible light below an idealized optically thick atmospheric cloud, which is illuminated from above in an arbitrary manner. The model offers elegant mathematical simplicity at the expense of some precision. As such, it is applicable to a broad class of problems in which correct functional forms are required, but levels of accuracy better than a factor of 2 are not necessary. Optical thicknesses can range from about 5 to 32. One example of a problem in this class, the design of a laser communication system to operate through clouds, provided the original motivation for development of the light-transmission model. The optical effects of the cloud are calculated by means of a four-dimensional linear superposition integral, which takes account of multiple scattering. Two illustrations of the method are given in detail, with incident illumination represented by a tightly collimated beam and by a sum of infinite plane waves, respectively.
HAROLD M. HEGGESTAD, "Multiple-Scattering Model for Light Transmission through Optically Thick Clouds," J. Opt. Soc. Am. 61, 1293-1300 (1971)