Page updated: April 14, 2020
Author: Curtis Mobley
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# Whitecaps

The contribution of white caps and foam to the TOA radiance depends on two factors: the reﬂectance of whitecaps per se and the fraction of the sea surface that is covered by whitecaps.

Following Gordon and Wang (1994b), the contribution of whitecaps and foam at the TOA is (recall Eq. 13 of the Normalized Reﬂectances page)

 $t\left({𝜃}_{v},\lambda \right){\rho }_{wc}\left(\lambda \right)={\left[{\rho }_{wc}\left(\lambda \right)\right]}_{N}\phantom{\rule{0.3em}{0ex}}t\left({𝜃}_{s},\lambda \right)\phantom{\rule{0.3em}{0ex}}t\left({𝜃}_{v},\lambda \right)\phantom{\rule{0.3em}{0ex}},$

where $t\left({𝜃}_{v},\lambda \right)$ is the diﬀuse atmospheric transmission in the viewing direction, $t\left({𝜃}_{s},\lambda \right)$ is the diﬀuse transmission in the Sun’s direction, and ${\left[{\rho }_{wc}\left(\lambda \right)\right]}_{N}$ is the non-dimensional normalized whitecap reﬂectance. ${\left[{\rho }_{wc}\left(\lambda \right)\right]}_{N}$ is deﬁned in the same manner as was the normalized water-leaving reﬂectance ${\left[{\rho }_{w}\left(\lambda \right)\right]}_{N}$ in Eq. (3.3) of the Normalized Reﬂectances page, namely

 ${\left[{\rho }_{wc}\right]}_{N}\equiv \frac{\pi }{{F}_{o}}{\left[{L}_{wc}\right]}_{N}=\pi \frac{{\left(\frac{R}{{R}_{o}}\right)}^{2}{L}_{wc}\left({𝜃}_{s}\right)}{{F}_{o}\phantom{\rule{0.3em}{0ex}}cos{𝜃}_{s}\phantom{\rule{0.3em}{0ex}}t\left({𝜃}_{s}\right)}\phantom{\rule{0.3em}{0ex}},$ (1)

where ${L}_{wc}$ is the whitecap radiance. It is assumed that the whitecaps are Lambertian reﬂectors, so that (unlike for ${L}_{w}$) ${L}_{wc}$ does not depend on direction ${𝜃}_{v},\varphi$. This gives the interpretation (Gordon and Wang (1994b), page 7754) that ”$\rho$ is the reﬂectance–the reﬂected irradiance divided by the incident irradiance–that a Lambertian target held horizontally at the TOA would have to have to produce the radiance $L$.” ${\left[{\rho }_{wc}\right]}_{N}$ can be interpreted as the average reﬂectance of the sea surface that results from whitecaps in the absence of atmospheric attenuation.

The eﬀective whitecap irradiance reﬂectance is taken from Koepke (1984) to be 0.22 (albeit with $±50%$ error bars). This reﬂectance is independent of wavelength. This gives ${\left[{\rho }_{wc}\right]}_{N}=0.22{F}_{wc}$, where ${F}_{wc}$ is the fraction of the sea surface that is covered by whitecaps. The fractional coverage is taken from Stramska and Petelski (2003), who give two models for for ${F}_{wc}$:

$\begin{array}{lll}\hfill {F}_{wc}=& \phantom{\rule{1em}{0ex}}5.0×1{0}^{-5}{\left({U}_{10}-4.47\right)}^{3}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}developed\phantom{\rule{1em}{0ex}}seas\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill {F}_{wc}=& \phantom{\rule{1em}{0ex}}8.75×1{0}^{-5}{\left({U}_{10}-6.33\right)}^{3}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}undeveloped\phantom{\rule{1em}{0ex}}seas\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\end{array}$

where $W$ is the wind speed in $m\phantom{\rule{1em}{0ex}}{s}^{-1}$ at 10 m. Formula (3) for undeveloped seas is used on the assumption that if the seas are well developed it is probably stormy, hence cloudy, so that remote sensing is not possible. The blue curve in Fig. (4) shows ${F}_{wc}$ for undeveloped seas.

The ﬁnal model for ${\left[{\rho }_{wc}\right]}_{N}$ is then taken to be

$\begin{array}{llll}\hfill {\left[{\rho }_{wc}\right]}_{N}\left(\lambda \right)=& \phantom{\rule{1em}{0ex}}{a}_{wc}\left(\lambda \right)×0.22×{F}_{wc}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}{a}_{wc}\left(\lambda \right)×1.925×1{0}^{-5}{\left({U}_{10}-6.33\right)}^{3}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\end{array}$

A whitecap correction is applied for wind speeds in the range $6.33\le {U}_{10}\le 12\phantom{\rule{1em}{0ex}}m\phantom{\rule{1em}{0ex}}{s}^{-1}$. The factor ${a}_{wc}\left(\lambda \right)$ is a normalized whitecap reﬂectance that describes the decrease in reﬂectance at red and NIR wavelengths. This factor is taken from Figs. 3 and 4 of Frouin et al. (1996); the values are

 $\lambda$ = 412 443 490 510 555 670 765 865 ${a}_{wc}$ = 1.0 1.0 1.0 1.0 1.0 0.889 0.760 0.645

Linear interpolation is used as needed between these values. Figure 1 shows the whitecap reﬂectance as given by Eq. (4) when ${a}_{wc}=1$.