Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field

The average cosine of the underwater light field ((mu)over bar) is a simple quantity that describes the angular distribution of radiance at a given point. A model of the rate of vertical change of (mu)over bar in the ocean was developed in order to examine the influences of light absorption and scattering. We made calculations of radiative transfer based on invariant imbedding theory assuming an optically homogeneous ocean with a typical scattering phase function and the simple boundary conditions of the sun overhead in a black sky and a flat ocean surface. Under such conditions, the decrease of (mu)over bar throughout the water column is well approximated by a single exponential function. The dependence of the parameter P-r, which describes the rate of change of (mu)over bar with optical depth, on the single-scattering albedo omega (o), is well approximated by a quadratic function. By applying a linearization technique to the P-tau vs. omega (o) relationship, we identified the contributions of absorption and scattering to P-tau. Our results indicate that scattering is the more important process, contributing >50% to P-tau for typical situations when omega (o) > 0.1. Absorption dominates P-tau when omega (o) <0.1 which occurs only in very clear oceanic water at long wavelengths (>650 nm). Our analysis of the effect of scattering phase function shows that the scattering into the middle angles, approximately between 20 degrees and 45 degrees. largely determines the magnitude of P-tau. Using spectral bio-optical models with several Chl concentrations, we also examined the rate of change in (mu)over bar with geometric depth P-z for various water types with realistic values of the absorption and scattering coefficients. This analysis shows large variations in both the magnitude and the spectral behavior of P-z with varying Chl concentration.