It is shown that where ?¯s is the average cosine of scattering, then for any set of photons that undergoes exactly n scatterings per photon, the average cosine after scattering is ?¯0?¯sn, where ?¯0 is the average cosine of the photon flux before scattering. For a set of photons that has traversed distance d through a medium with scattering coefficient b, the average cosine is ?¯0 exp[?bd(1 ? ?¯s)]. For water bodies in which loss of upward-scattered photons through the surface is small enough to be disregarded, the value of ?¯c (the average cosine of all the photons instantaneously present in the water column) for any given incoming flux of photons with average cosine ?¯0 is determined entirely by the inherent optical properties of the water in accordance with ?¯c= ?¯0/[1 + (b/a)(1 ? ?¯s)], where a and b are the absorption and scattering coefficients.
© 1999 Optical Society of America
(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics
(010.1310) Atmospheric and oceanic optics : Atmospheric scattering
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.7340) Atmospheric and oceanic optics : Water
John T. O. Kirk, "Multiple Scattering of a Photon Flux: Implications for the Integral Average Cosine of the Underwater Light Field," Appl. Opt. 38, 3134-3140 (1999)