Asymptotic optical attenuation in sea water

Abstract

A simple method is proposed for calculating the asymptotic attenuation coefficient K∞ of sea water. The method is based on the scaling analysis of the radiative transfer equation within the small-angle approximation. From the scaling, it follows that K∞ is analytically expressed in terms of the absorption coefficient, the transport scattering coefficient, and two dimensionless numeric constants (scaling exponents) depending on a specific scattering phase function. For a given phase function, the scaling exponents can be determined by numerical calculations of the downwelling irradiance. We test the method on a number of oceanlike scattering models. A direct numerical integration of the radiative transfer equation is carried out with the DISORT code for the Petzold, Morel et al., Fournier–Forand and Kopelevich phase functions. Our numerical results agree perfectly with the small-angle scaling and allow us to establish an explicit expression for K∞ for each phase function. We find that the scaling exponents depend rather weakly on the specific angular profile of the phase function and can be taken equal to their values for the Henyey–Greenstein function (this approximation leads to a relative error in the value K∞ less than 5%). So that, within a few percent accuracy, the coefficient K∞ is governed only by the absorption and transport scattering coefficients and is described by a universal formula.