## Abstract

Using the reciprocal equation derived by Yang and Gordon [Appl. Opt. **36**, 7887–7897 (1997)] for atmospheric diffuse
transmittance of the ocean–atmosphere system, I examined the accuracy
of an analytical equation proposed by Gordon *et al*. [Appl. Opt. **22**, 20–36 (1983)] in computing the atmospheric
diffuse transmittance for wavelengths from 412 to 865 nm for both a
pure Rayleigh and a two-layer Rayleigh-aerosol atmosphere overlying a
flat Fresnel-reflecting ocean surface. It was found that for
viewing angles up to approximately 40°, the analytical formula
produces errors usually between 2% and 3% for nonabsorbing and weakly
absorbing aerosols and for aerosol optical thicknesses
τ_{a} ≤ 0.4. The error increases with an
increase in aerosol absorption, aerosol optical thickness, and viewing
angle, and with the decrease of wavelength. By a simple numerical
fit to modify the analytical formula, the atmospheric diffuse
transmittance can be accurately computed usually to within ∼1%
(∼0.5% in most cases) for a variety of aerosol models, aerosol
optical thicknesses τ_{a} ≤ 0.6, viewing angles
θ ≤ 60°, different aerosol vertical structure distribution, and
for wavelengths from 412 to 865 nm.

© 1999 Optical Society of America

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### Equations (11)

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(1)
$${L}_{t}\left(\mathrm{\lambda}\right)={L}_{r}\left(\mathrm{\lambda}\right)+{L}_{a}\left(\mathrm{\lambda}\right)+{L}_{\mathit{ra}}\left(\mathrm{\lambda}\right)+t\left(\mathrm{\lambda}\right){L}_{w}\left(\mathrm{\lambda}\right),$$
(2)
$${t}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right)=exp\left[-{\mathrm{\tau}}_{r}\left(\mathrm{\lambda}\right)/2cos\mathrm{\theta}\right],$$
(3)
$$t_{r}{}^{\left(C\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right)=exp\left[-{C}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right){\mathrm{\tau}}_{r}\left(\mathrm{\lambda}\right)/2cos\mathrm{\theta}\right],$$
(4)
$${C}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right)={a}_{1}\left(\mathrm{\theta}\right)+{a}_{2}\left(\mathrm{\theta}\right){log}_{e}\left[{\mathrm{\tau}}_{r}\left(\mathrm{\lambda}\right)\right]+{a}_{3}\left(\mathrm{\theta}\right)log_{e}{}^{2}\left[{\mathrm{\tau}}_{r}\left(\mathrm{\lambda}\right)\right],$$
(5)
$${a}_{j}\left(\mathrm{\theta}\right)={a}_{0j}+{a}_{1j}/cos\mathrm{\theta}+{a}_{2j}/{cos}^{2}\mathrm{\theta}+{a}_{3j}/{cos}^{3}\mathrm{\theta},$$
(6)
$$\mathrm{\Delta}{t}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right)={t}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right)-t_{r}{}^{\left(m\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right)$$
(7)
$$t\left(\mathrm{\lambda},\mathrm{\theta}\right)={t}_{r}\left(\mathrm{\lambda},\mathrm{\theta}\right){t}_{a}\left(\mathrm{\lambda},\mathrm{\theta}\right),{t}_{a}\left(\mathrm{\lambda},\mathrm{\theta}\right)=exp\left\{-\left[1-{\mathrm{\omega}}_{a}\left(\mathrm{\lambda}\right){F}_{a}\left(\mathrm{\lambda}\right)\right]{\mathrm{\tau}}_{a}\left(\mathrm{\lambda}\right)/cos\mathrm{\theta}\right\},$$
(8)
$${F}_{a}\left(\mathrm{\lambda}\right)=1/2{\int}_{0}^{1}{P}_{a}\left(\mathrm{\Theta},\mathrm{\lambda}\right)dcos\mathrm{\Theta},$$
(9)
$${t}^{\left(C\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right)=t_{r}{}^{\left(C\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right)t_{a}{}^{\left(C\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right),t_{a}{}^{\left(C\right)}\left(\mathrm{\lambda},\mathrm{\theta}\right)=exp\left\{-{a}_{o}\left(\mathrm{\lambda}\right)\left[1+{\mathrm{\omega}}_{a}\left(\mathrm{\lambda}\right){C}_{a}\left(\mathrm{\lambda},\mathrm{\theta}\right)\right]/cos\mathrm{\theta}\right\},$$
(10)
$${C}_{a}\left(\mathrm{\lambda},\mathrm{\theta}\right)={b}_{1}\left(\mathrm{\lambda},\mathrm{\theta}\right)+{b}_{2}\left(\mathrm{\lambda},\mathrm{\theta}\right){log}_{e}\left[{a}_{o}\left(\mathrm{\lambda}\right)\right]+{b}_{3}\left(\mathrm{\lambda},\mathrm{\theta}\right)log_{e}{}^{2}\left[{a}_{o}\left(\mathrm{\lambda}\right)\right],$$
(11)
$${b}_{j}\left(\mathrm{\lambda},\mathrm{\theta}\right)={b}_{0j}\left(\mathrm{\lambda}\right)+{b}_{1j}\left(\mathrm{\lambda}\right)/cos\mathrm{\theta}+{b}_{2j}\left(\mathrm{\lambda}\right)/{cos}^{2}\mathrm{\theta}+{b}_{3j}\left(\mathrm{\lambda}\right)/{cos}^{3}\mathrm{\theta}+{b}_{4j}\left(\mathrm{\lambda}\right)/{cos}^{4}\mathrm{\theta},$$