## Abstract

We have developed a Monte Carlo program that is capable of calculating both the scalar and the Stokes vector radiances in an atmosphere–ocean system in a single computer run. The correlated sampling technique is used to compute radiance distributions for *both* the scalar and the Stokes vector formulations simultaneously, thus permitting a direct comparison of the errors induced. We show the effect of the volume-scattering phase function on the errors in radiance calculations when one neglects polarization effects. The model used in this study assumes a conservative Rayleigh-scattering atmosphere above a flat ocean. Within the ocean, the volume-scattering function (the first element in the Mueller matrix) is varied according to both a Henyey–Greenstein phase function, with asymmetry factors *G* = 0.0, 0.5, and 0.9, and also to a Rayleigh-scattering phase function. The remainder of the reduced Mueller matrix for the ocean is taken to be that for Rayleigh scattering, which is consistent with ocean water measurements.

© 1993 Optical Society of America

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### Equations (10)

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(1)
$$\mathbf{\text{P}}\phantom{\rule{0.1em}{0ex}}(\phantom{\rule{0em}{0ex}}\mu \phantom{\rule{0em}{0ex}})=\frac{3}{16\phantom{\rule{0em}{0ex}}\pi}\left[\begin{array}{cccc}{\mu}^{2}+1& {\mu}^{2}-1& 0& 0\\ {\mu}^{2}-1& {\mu}^{2}+1& 0& 0\\ 0& 0& \mu & 0\\ 0& 0& 0& \mu \end{array}\right]\phantom{\rule{0.2em}{0ex}},$$
(2)
$${\mathit{\int}}_{\Omega}\mathit{\int}\phantom{\rule{0.2em}{0ex}}{P}_{11}\phantom{\rule{0.1em}{0ex}}(\phantom{\rule{0em}{0ex}}\Omega \phantom{\rule{0em}{0ex}})\phantom{\rule{0.1em}{0ex}}\text{d}\phantom{\rule{0em}{0ex}}\Omega =1\phantom{\rule{0.2em}{0ex}}.$$
(3)
$$\mathbf{\text{R}}\phantom{\rule{0.1em}{0ex}}(\phantom{\rule{0em}{0ex}}\mu \phantom{\rule{0em}{0ex}})=\left[\begin{array}{cccc}1& \frac{{\mu}^{2}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}1}{{\mu}^{2}\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}& 0& 0\\ \frac{{\mu}^{2}}{{\mu}^{2}\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}& 1& 0& 0\\ 0& 0& \frac{\mu}{{\mu}^{2}\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}& 0\\ 0& 0& 0& \frac{\mu}{{\mu}^{2}\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}\end{array}\right]\phantom{\rule{0.2em}{0ex}}.$$
(4)
$${P}_{11}(G,\mu \phantom{\rule{0em}{0ex}})=\frac{1-{G}^{2}}{4\phantom{\rule{0em}{0ex}}\pi \phantom{\rule{0em}{0ex}}{(1-2\phantom{\rule{0.1em}{0ex}}G\phantom{\rule{0em}{0ex}}\mu +{G}^{2})}^{3/2}}$$
(5)
$$G=\u3008\phantom{\rule{0em}{0ex}}\mu \phantom{\rule{0em}{0ex}}\u3009={\mathit{\int}}_{\Omega}\mathit{\int}\phantom{\rule{0.2em}{0ex}}\mu \phantom{\rule{0em}{0ex}}{P}_{11}(G,\Omega \phantom{\rule{0em}{0ex}})\phantom{\rule{0.1em}{0ex}}\text{d}\phantom{\rule{0em}{0ex}}\Omega .$$
(6)
$$\mu =\frac{1\phantom{\rule{0.2em}{0ex}}+{G}^{2}-{\left[\frac{1\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{G}^{2}}{G\phantom{\rule{0.1em}{0ex}}(2R\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}1)\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}1}\right]}^{2}}{2\phantom{\rule{0.1em}{0ex}}G}\phantom{\rule{0.2em}{0ex}},$$
(7)
$$\u3008\phantom{\rule{0.1em}{0ex}}f\phantom{\rule{0.1em}{0ex}}\u3009=\mathit{\int}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}p\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.1em}{0ex}}\text{d}x\phantom{\rule{0.2em}{0ex}}.$$
(8)
$$\u3008\phantom{\rule{0.1em}{0ex}}f\phantom{\rule{0.1em}{0ex}}\u3009=\mathit{\int}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}\frac{p\phantom{\rule{0.1em}{0ex}}(x)}{\stackrel{\sim}{p}\phantom{\rule{0.1em}{0ex}}(x)}\phantom{\rule{0.2em}{0ex}}\stackrel{\sim}{p}\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.1em}{0ex}}\text{d}x=\mathit{\int}\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}w\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}\stackrel{\sim}{p}\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.1em}{0ex}}\text{d}x\phantom{\rule{0.2em}{0ex}},$$
(9)
$${\sigma}^{2}\phantom{\rule{0.1em}{0ex}}[f\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}w\phantom{\rule{0.1em}{0ex}}(x)]=\mathit{\int}\phantom{\rule{0.2em}{0ex}}{[f\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.2em}{0ex}}w\phantom{\rule{0.1em}{0ex}}(x)-\u3008\phantom{\rule{0.1em}{0ex}}f\phantom{\rule{0.1em}{0ex}}\u3009]}^{2}\phantom{\rule{0.1em}{0ex}}\stackrel{\sim}{p}\phantom{\rule{0.1em}{0ex}}(x)\phantom{\rule{0.1em}{0ex}}\text{d}x\phantom{\rule{0.2em}{0ex}}.$$
(10)
$$B\phantom{\rule{0.2em}{0ex}}={\mathit{\int}}_{0}^{2\pi}\phantom{\rule{0.2em}{0ex}}{\mathit{\int}}_{-1}^{0}\phantom{\rule{0.2em}{0ex}}{P}_{11}(\phantom{\rule{0em}{0ex}}\mu \phantom{\rule{0em}{0ex}})\phantom{\rule{0.1em}{0ex}}\text{d}\phantom{\rule{0em}{0ex}}\mu \phantom{\rule{0em}{0ex}}\text{d}\phantom{\rule{0em}{0ex}}\varphi $$