Theoretical Developments on the Optical Properties of Highly Turbid Waters and Sea Ice

The photon diffusion equation is derived in a direct manner from the radiative transfer equation and is shown to be an asymptotic equation that can be directly related to asymptotic radiative transfer theory. Diffusion theory predicts that the asymptotic diffuse attenuation coefficient, K!infty, is related to the beam attenuation coefficient, c, the single scattering albedo, w0, and the asymmetry parameter, g, of the scattering phase function by K!infty = c!sqrt 3[1 - w0- g(w0- w0 2)]. Kirk has previously published a K relationship based entirely on Monte Carlo radiative transfer simulations that can be expressed in the form K!infty = c!sqrt 1 - 2w0+ W0 2+ G(w0- w0 2), where G is a regression parameter. Equating these two results gives G = 3(1-g) + 2(1/w0- 1), showing explicitly, as Kirk found numerically, how G is a function of w0and g. These results are expected to be valid for highly turbid water where$w_0 > 0.95$. Comparison of the analytical expression for G with Kirk's regression value, using w0of 0.99, differed by only 2%.