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Deriving the vertical diffuse attenuation coefficient (Kd) from the absorption coefficient (a).

Derivation of Kd from a
In what follows we have left off the dependence on wavelength. In addition we have left off effects due to inelastic scattering.
Modeling Kd(z) requires the inclusion of the depth dependence of the shape of the radiance
distribution. This can be accomplished by using Gershun's equation:
K(z) = a(z) / μ − (z)
where a(z) is the absorption coefficient and μ(z) is the average cosine of the light field.
Kd(z) differs from K(z) by only a few percent, so that we may set:
Kd(z) ≈ a(z) / μ − (z) .
Berwald et al. (1995) have derived a parametric model for the dependence of ⎯μ(z) on ωo for a vertical sun in a black sky. We will assume the same depth dependence for an ordinary sky. This is not precise, but the average cosine varies slowly and has a typical range of only about 10%. The model is:
ωo= b / c, where b and c are the total scattering and attenuation coefficients, including water.
τ = cz, is the optical depth.
μ∞(τ) = μ∞ + (μ(0) - μ∞ ) exp(-Ptt ).
μ∞ = - 1.59 ωo4 + 1.71 ωo3 - 0.467 ωo2 - 0.347 ωo + 1
μ(0)= cosine of refracted solar zenith angle
Pτ(ωo) = - 0.166 ωo2 + 0.341 ωo + 0.0305
Berwald et. al. Limnology and Oceanography 1995, vol. 40, no8, pp. 1347-1357 .