The vertical attenuation of irradiance as a function of the optical properties of the water

Average vertical attenuation coefficients, K(av), for irradiance calculated by linear regression of In E(z) on z through the euphotic zone or from two irradiance values in a certain depth interval, are useful but somewhat arbitrary procedures for estimating these important apparent optical properties of the ocean. A more fundamental approach is to calculate an irradiance-weighted coefficient, K-w(av), integrated over the whole water column, in which for each increment of depth, the corresponding irradiance value is used to weight the estimate of the irradiance coefficient in accordance with K-w(av) = integral(0)(infinity) K(z)E(z) dz/integral(0)(infinity) E(z) dz. Attenuation coefficients calculated in this way exhibit some interesting relationships both to certain other properties of the light field and to the inherent optical properties of the water. In particular, I find that for all types of irradiance K-w(av) = E(0)/integral(0)(infinity) E(z) dz, where E(0) is the value of irradiance just below the water surface. For net downward irradiance, K-w(E)(av) = a/mu(c) and K-w(E)(av) approximate to (a/mu(o))[1 + (b/a)(1 - mu(s))], where mu(c) is the integral average cosine for the water column, muo is the average cosine of the incoming flux just below the water surface, a is the absorption coefficient, b is the scattering coefficient, and As is the average cosine (asymmetry factor) of the scattering phase function. For scalar irradiance, K-w(o)(av) = a/mu(0), where mu(0) is the average cosine of the light field just below the water surface. The extent to which these and conventionally calculated attenuation coefficients reproduce the depth variation of irradiance is explored using Monte Carlo modeling.