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

[Optical Society of America ]

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