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1954): 
itokes vector 
L 
h 
U 
V 
where r-, rx, t=, and tx depend on the incidence angle as in Takashima (1984). 
The Stokes vector of all numerical “photons” reaching the top of the atmosphere within a given 
angular bin are summed. The four components of the vector yield the radiance, the polarization 
rate, and the direction of linear polarization. 
We verified the model by comparing its output with that of another radiative transfer model 
that includes polarization, the doubling-adding model of Takashima (1984). Some of the 
comparison results are presented below. In the cases selected, the solar zenith angle is 30°, the 
surface wind speed is 5 ms-1, and the wavelength is 450 nm (molecular optical thickness of 0.22). 
The aerosol layer is located below the 800 mb level and its total optical thickness is 0.25. The 
aerosols have the size distribution and refractive index of the “Water Soluble” model of the 
World Meteorological Organization (WMO) (WCP, 1986). The effect of the water body is 
simplified by assuming that 2% of the irradiance penetrating the water is reflected isotropically as 
unpolarized radiance. 
The reflectance (radiance normalized by the incident solar irradiance) and polarization rate in 
the principal plane, as estimated by both models, is presented in Fig. 1. The polarization rate is 
defined as the ratio of polarized reflectance and total reflectance. Solid circles denote the MC 
results, and open circles denote the doubling-adding results. The specular direction is on the right 
side of the figure, as indicated by the local maximum in reflectance when the viewing zenith angle 
is close to the solar zenith angle (Fig. 1, top). Since the MC values are representative of an angular 
bin, they are indicated at the bin center. On the other hand, the doubling-adding method computes 
the radiative transfer equations at precise angles. Note that the angular precision close to the limb 
is much better for the latter. 
The two radiative transfer models validate each other in these comparisons. In Fig. 1, top, both 
models show a large reflectance increase near the limb as viewing zenith angle increases, which 
results from more important atmospheric scattering. They also show a similar glitter effect, whose 
amplitude and extension depends on wind speed (through the wave slope distribution). In Fig. 1, 
bottom, the maximum polarization rate, located around 50° viewing zenith angle, results from the 
combined effect of specular reflection (maximum polarization at the Brewster angle) and 
molecular scattering (maximum polarization at 90° scattering angle). As expected, the minimum 
polarization rate is found close to backscattering. 
3- RESULTS 
For the purpose of our study, the MC simulations are performed at the wavelength of 450 nm, 
where phytoplankton absorb a substantial amount of light. The solar zenith angle is fixed at 30° 
and the wind speed at 5 ms' 1 . The aerosols are represented by the “oceanic” and “water soluble” 
models of the WMO (WCP, 1986). They are located below 800 mb and their optical thickness is 
0.2, a typical value for clear atmospheres. The hydrosols are phytoplankton particles. Their 
characteristics (size distribution, refractive index) are specified according to Zaneveld et al. (1974) 
and Aas (1981). Their volume number concentration, 1.19 10 14 nr 3 , is constant throughout the 
water column (homogeneous ocean) and corresponds to a pigment concentration (i.e., chlorophyll- 
a) of about 0.1 mgnr 3 . In Table 1, the characteristics of the aerosols and hydrosols are 
summarized, and in Fig. 2 the respective scattering phase functions are displayed. The polarization 
rates for single scattering are not shown, but they do not exceed 0.5 (compared to the maximum 
value of almost 1 for molecules). 
In Fig. 3, the total reflectance and the polarization rate of the ocean-atmosphere system is 
presented as a function of viewing geometry. The atmosphere contains “oceanic” aerosols. A basic 
feature is the large bidirectionality of the total reflectance (Fig. 3, top). Specular reflection results 
in higher reflectance values (0.25) around 30° viewing zenith angle and 180° relative azimuth 
^gle. At high viewing zenith angles, the total reflectance reaches values of about 30%. Minimum 
values (0.1-0.12) are found outside the glitter region at viewing zenith angles below 60°. The 
polarization rate also exhibits large angular variability (Fig. 3, bottom). The maximum polarization 
rate (about 70%) is obtained at 50° viewing zenith angle in the principal plane (“forward” 
scattering), not at the Brewster angle nor at 60° viewing zenith angle, which would be expected in 
fhe case of only specular reflection or molecular scattering, respectively. The mimimum 
polarization rates are obtained at large scattering angles, around the direction of the sun in 
^scattering. 
Fig. 4 displays the top-of-atmosphere reflectance and the polarization rate of photons that have 
interacted with the water body. Reflectance values are small, not exceeding 0.03-0.04, and the