## Abstract

The radiance and polarization of multiple scattered light is calculated from the Stokes’ vectors by a Monte Carlo method. The exact scattering matrix for a typical haze and for a cloud whose spherical drops have an average radius of 12 *μ* is calculated from the Mie theory. The Stokes’ vector is transformed in a collision by this scattering matrix and the rotation matrix. The two angles that define the photon direction after scattering are chosen by a random process that correctly simulates the actual distribution functions for both angles. The Monte Carlo results for Rayleigh scattering compare favorably with well known tabulated results. Curves are given of the reflected and transmitted radiances and polarizations for both the haze and cloud models and for several solar angles, optical thicknesses, and surface albedos. The dependence on these various parameters is discussed.

© 1968 Optical Society of America

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### Equations (6)

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(1)
$$\begin{array}{l}\left(\begin{array}{l}I\hfill \\ Q\hfill \\ U\hfill \\ V\hfill \end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \text{cos}2{i}_{2}& -\text{sin}2{i}_{2}& 0\\ 0& \text{sin}2{i}_{2}& \text{cos}2{i}_{2}& 0\\ 0& 0& 0& 1\end{array}\right)\\ \times \left(\begin{array}{cccc}{\scriptstyle \frac{1}{2}}({M}_{2}+{M}_{1})& {\scriptstyle \frac{1}{2}}({M}_{2}-{M}_{1})& 0& 0\\ {\scriptstyle \frac{1}{2}}({M}_{2}-{M}_{1})& {\scriptstyle \frac{1}{2}}({M}_{2}+{M}_{1})& 0& 0\\ 0& 0& {S}_{21}& -{D}_{21}\\ 0& 0& {D}_{21}& {S}_{21}\end{array}\right)\\ \times \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \text{cos}2{i}_{1}& -\text{sin}2{i}_{1}& 0\\ 0& \text{sin}2{i}_{1}& \text{cos}2{i}_{1}& 0\\ 0& 0& 0& 1\end{array}\right)\hspace{0.17em}\left(\begin{array}{c}{I}^{\prime}\\ {Q}^{\prime}\\ {U}^{\prime}\\ {V}^{\prime}\end{array}\right).\end{array}$$
(2)
$${f}^{*}({\u220a}_{1},\dots ,{\u220a}_{m})=f({\u220a}_{1},\dots ,{\u220a}_{m})\times [{p}_{1}({\u220a}_{1})\dots {p}_{m}({\u220a}_{m})]/[{{p}_{1}}^{*}({\u220a}_{1})\dots {{p}_{m}}^{*}({\u220a}_{m})].$$
(3)
$$\begin{array}{l}E{f}^{*}({\u220a}_{1},\dots ,{\u220a}_{m})={\int}_{{x}_{1},\dots ,{x}_{m}}^{\hspace{0.17em}}f({x}_{1},\dots ,{x}_{m})\{[{p}_{1}({x}_{1})\dots {p}_{m}({x}_{m})]/[{{p}_{1}}^{*}({x}_{1})\dots {{p}_{m}}^{*}({x}_{m})]\}\hspace{0.17em}[{{p}_{1}}^{*}({x}_{1})\dots {{p}_{m}}^{*}({x}_{m})]d{x}_{1}\dots d{x}_{m}\\ ={\int}_{{x}_{1\dots m}x}^{\hspace{0.17em}}f({x}_{1},\dots ,{x}_{m})[{p}_{1}({x}_{1})\dots {p}_{m}({x}_{m})]d{x}_{1}\dots d{x}_{m}\\ =Ef({\eta}_{1},\dots ,{\eta}_{m}).\end{array}$$
(4)
$$I(\mathrm{\Theta},{i}_{1})={\scriptstyle \frac{1}{2}}{I}^{\prime}({M}_{1}+{M}_{2})+{\scriptstyle \frac{1}{2}}({M}_{2}-{M}_{1})\times ({Q}^{\prime}\text{cos}\hspace{0.17em}2{i}_{1}-{U}^{\prime}\text{sin}\hspace{0.17em}2{i}_{1}),$$
(5)
$$n(r)=0.00108\hspace{0.17em}{r}^{6}\hspace{0.17em}\text{exp}(-0.5\hspace{0.17em}r).$$
(6)
$$P=Q/I=({I}_{r}={I}_{l})/({I}_{r}+{I}_{l}).$$