An analytical solution is discussed for the nadir radiance as measured from a satellite, based on a simplified single-scattering approximation in which the scattered radiation is not subject to extinction. In the solution, terms can be identified as due to a reflection from the vicinity of the object pixel, and, respectively, (1) upward scattering to zenith above the object pixel, and (2) downward scattering from the entire atmosphere to the object pixel. The first term is referred to as the cross radiance, the second as the cross irradiance. The cross radiance is proportional to the forward scattering optical thickness, as defined, and the cross irradiance to the backscattering optical thickness. The cross radiance usually constitutes the predominant effect. The effect, even at low atmospheric turbidity, can be large enough to constitute a significant fraction of the radiance registered at the satellite, thus hampering determination of spectral signature of the object pixel or identification of pixels with inherently the same spectral signature. Explicit expressions and computer solutions for the cross radiance from annular or from rectangular reflecting areas are presented. The effect depends on the height distribution and on the sharpness of the forward peak of the scattering particles.
J. Otterman and R. S. Fraser, "Adjacency effects on imaging by surface reflection and atmospheric scattering: cross radiance to zenith," Appl. Opt. 18, 2852-2860 (1979)