Multiple scattering of a photon flux: implications for the integral average cosine of the underwater light field

It is shown that where <(mu)over bar>(s) is the average cosine of scatterin g, then for any set of photons that undergoes exactly n scatterings per pho ton, the average cosine after scattering is <(mu)over bar>(0)<(mu)over bar> (n)(s), where <(mu)over bar>(0) is the average cosine of the photon flux be fore scattering. For a set of photons that has traversed distance d through a medium with scattering coefficient b, the average cosine is <(mu)over ba r>(0) exp[-bd(1 - <(mu)over bar>(g))]. For water bodies in which loss of up ward-scattered photons through the surface is small enough to he disregarde d, the value of <(mu)over bar>(c) (the average cosine of all the photons in stantaneously present in the water column) for any given incoming flux of p hotons with average cosine <(mu)over bar>(0) is determined entirely by the inherent optical properties of the water in accordance with <(mu)over bar>( c) = <(mu)over bar>(0)/[1 + (b/a)(1 - <(mu)over bar>(s))], where a and b ar e the absorption and scattering coefficients. (C) 1999 Optical Society of A merica.