Abstract

Qualitative and quantitative properties of the in-water distribution of solar radiance, as predicted by the radiative transfer equation, are examined. Two solutions of the radiative transfer equation in the small-angle limit are presented. One of the solutions is the well-known traditional small-angle solution (designated SA). The other (designated SAA) is a solution obtained recently by an approximate evaluation of an exact path-integral expression. In the limit of shallow depth the two solutions are identical, but at depths greater than a certain diffuse path length they differ substantially. Two sets of experimental data are used in comparisons of the apparent solar peak, distribution width, and magnitude as functions of depth. The SAA solution exhibits better qualitative and quantitative agreement with the experimental data than the SA solution. The depth dependence of the diffuse attenuation coefficient obtained from the SAA solution follows that predicted by a finite-difference radiative transfer calculation by Helliwell. At depths greater than approximately six total attenuation lengths, they differ by no more than 5%. The asymptotic diffuse attenuation coefficients predicted by the two approximate solutions are compared with the numerical solution of the corresponding eigenvalue problem. By discretizing the radiative transfer equation, it is shown that the asymptotic diffuse attenuation coefficient is the minimum eigenvalue of a particular matrix, which is constructed explicitly in a simplified two-stream limit. This limiting expression is complementary to the expression obtained from the SAA solution.

© 1988 Optical Society of America

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