## Abstract

Cox and Munk used aerial photographs of Sun glint to determine the statistical distribution of ocean capillary wave slopes as a function of wind velocity [
J. Opt. Soc. Am. **44**,
838 (
1954)]. When their equation connecting the slope distribution with Sun glint is used on the horizon, however, an infinite glint is predicted even though Sun glint never exceeds solar radiance. An integral equation connecting the capillary wave slope distribution with ocean radiance is derived. The integral predicts a finite Sun glint on the ocean horizon and, away from the horizon, reduces to the algebraic form used by Cox and Munk.

© 1995 Optical Society of America

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### Equations (33)

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(1)
$$P\equiv p({\zeta}_{x},{\zeta}_{y},W)\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}$$
(2)
$$\begin{array}{c}p({\zeta}_{x},{\zeta}_{y},W)\approx \frac{1}{2\pi {\sigma}_{u}{\sigma}_{c}}\text{exp}\left[-\frac{1}{2}\left(\frac{{{\zeta}_{x}}^{2}}{{{\sigma}_{u}}^{2}}+\frac{{{\zeta}_{y}}^{2}}{{{\sigma}_{c}}^{2}}\right)\right],\\ {{\sigma}_{u}}^{2}=0.000+3.16\times {10}^{-3}W,\\ {{\sigma}_{c}}^{2}=0.003+1.92\times {10}^{-3}W.\end{array}$$
(3)
$$\frac{{N}_{r}}{{H}_{\odot}}=\frac{\rho (\omega )p({\zeta}_{x},{\zeta}_{y},W){\text{sec}}^{4}\hspace{0.17em}{\theta}_{n}}{4\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{r}}.$$
(4)
$${\mathbf{U}}_{s}+{\mathbf{U}}_{r}=(2\hspace{0.17em}\text{cos}\hspace{0.17em}\omega ){\mathbf{U}}_{n}.$$
(5)
$$\frac{\text{d}A}{A}\equiv p\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}.$$
(6)
$$Q\equiv q(\theta ,\varphi ,{\zeta}_{x},{\zeta}_{y},W)\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}$$
(7)
$$\frac{\text{d}B}{B}\equiv q\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}.$$
(8)
$$B={\int}_{\begin{array}{l}\omega \le \pi /2\\ \mathbf{U}=\text{const}.\end{array}}\text{d}B.$$
(9)
$$\begin{array}{c}\text{d}B=\text{d}F\hspace{0.17em}\text{cos}\hspace{0.17em}\omega ,\\ \text{d}A=\text{d}F\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{n},\end{array}$$
(10)
$$q=\frac{A}{B}\frac{\text{cos}\hspace{0.17em}\omega}{\text{cos}\hspace{0.17em}{\theta}_{n}}p.$$
(11)
$$\frac{B}{A}={\iint}_{\begin{array}{l}\omega \le \pi /2\\ U=\text{const}.\end{array}}\frac{\text{cos}\hspace{0.17em}\omega}{\text{cos}\hspace{0.17em}{\theta}_{n}}p\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}.$$
(12)
$$q=\frac{\frac{\text{cos}\hspace{0.17em}\omega}{\text{cos}\hspace{0.17em}{\theta}_{n}}p}{{\iint}_{\begin{array}{l}\omega \le \pi /2\\ \mathbf{U}=\text{const}.\end{array}}\frac{\text{cos}\hspace{0.17em}\omega}{\text{cos}\hspace{0.17em}{\theta}_{n}}p\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}},$$
(13)
$$\text{d}{P}_{s}={N}_{s}\frac{\text{d}B}{{R}^{2}}\text{d}S={N}_{s}\frac{\text{d}S}{{R}^{2}}\text{d}B.$$
(14)
$$\text{d}{P}_{r}=\rho \text{d}{P}_{s}.$$
(15)
$${P}_{r}={\int}_{A}\text{d}{P}_{r}={\int}_{A}\rho {N}_{s}\frac{\text{d}S}{{R}^{2}}\text{d}B.$$
(16)
$${N}_{r}=\frac{{P}_{r}}{{B}_{r}(\text{d}S/{R}^{2})}.$$
(17)
$${N}_{r}={\iint}_{\begin{array}{l}\omega \le \pi /2\\ {\mathbf{U}}_{r}=\text{const}.\end{array}}\rho (\omega ){N}_{s}({\theta}_{s},{\varphi}_{s})q({\theta}_{r},{\varphi}_{r},{\zeta}_{x},{\zeta}_{y},W)\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}.$$
(18)
$${N}_{r}={\iint}_{{\mathbf{U}}_{r}=\text{const}.}\rho {N}_{s}{q}_{r}J\text{d}{\theta}_{s}\text{d}{\varphi}_{s},$$
(19)
$$J\equiv \frac{\partial ({\zeta}_{x},{\zeta}_{y})}{\partial ({\theta}_{s},{\varphi}_{s})}=\frac{\text{sec}\hspace{0.17em}\omega \hspace{0.17em}{\text{sec}}^{3}\hspace{0.17em}{\theta}_{n}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{s}}{4}.$$
(20)
$${N}_{r}=\frac{A}{4{B}_{r}}{\iint}_{{\mathbf{U}}_{r}=\text{const}.}\rho (\omega ){N}_{s}({\theta}_{s},{\varphi}_{s})p({\zeta}_{x},{\zeta}_{y},W){\text{sec}}^{4}\hspace{0.17em}{\theta}_{n}\times \text{sin}\hspace{0.17em}{\theta}_{s}\text{d}{\theta}_{s}\text{d}{\varphi}_{s}.$$
(21)
$$\frac{{N}_{r}^{\text{sun}}}{{N}_{\odot}}={\iint}_{\begin{array}{l}\text{ellipse}\\ {\mathbf{U}}_{r}-\text{const}.\end{array}}\rho {q}_{r}\text{d}{\zeta}_{x}\text{d}{\zeta}_{y},$$
(22)
$$\frac{{N}_{r}^{\text{sun}}}{{N}_{\odot}}=\frac{A}{4{B}_{r}}={\iint}_{\begin{array}{l}\text{disk}\\ {\text{U}}_{r}=\text{const}.\end{array}}(\rho p\hspace{0.17em}{\text{sec}}^{4}\hspace{0.17em}{\theta}_{n})\text{sin}\hspace{0.17em}{\theta}_{s}\text{d}{\theta}_{s}\text{d}{\varphi}_{s},$$
(23)
$$\frac{{N}_{r}^{\text{sun}}}{{N}_{\odot}}\approx \rho \le 100\%$$
(24)
$$\frac{{N}_{r}^{\text{sun}}}{{H}_{\odot}}\approx \frac{A}{4{B}_{r}}{(\rho p\hspace{0.17em}{\text{sec}}^{4}\hspace{0.17em}{\theta}_{n})}_{\odot}$$
(25)
$$\frac{{B}_{r}}{A}={\iint}_{{\mathbf{U}}_{r}=\text{const}.}\frac{\text{cos}\hspace{0.17em}\omega}{\text{cos}\hspace{0.17em}{\theta}_{n}}p\hspace{0.17em}\text{d}{\zeta}_{x}\text{d}{\zeta}_{y}=\text{cos}\hspace{0.17em}{\theta}_{r}$$
(26)
$$\frac{{N}_{r}^{\text{sun}}}{{H}_{\odot}}\approx {\left(\frac{\rho p\hspace{0.17em}{\text{sec}}^{4}\hspace{0.17em}{\theta}_{n}}{4\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{r}}\right)}_{\odot},$$
(27)
$$\text{cos}\hspace{0.17em}\omega =\frac{-\text{sin}\hspace{0.17em}{\theta}_{r}({\zeta}_{x}\hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{r}+{\zeta}_{y}\hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{r})+\text{cos}\hspace{0.17em}{\theta}_{r}}{{(1+{{\zeta}_{x}}^{2}+{{\zeta}_{y}}^{2})}^{1/2}},$$
(28)
$$\text{cos}\hspace{0.17em}{\theta}_{s}=-\frac{2{\zeta}_{x}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{r}+2{\zeta}_{y}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{r}+(-1+{{\zeta}_{x}}^{2}+{{\zeta}_{y}}^{2})\text{cos}\hspace{0.17em}{\theta}_{r}}{1+{{\zeta}_{x}}^{2}+{{\zeta}_{y}}^{2}},$$
(29)
$$\text{tan}\hspace{0.17em}{\varphi}_{s}=\frac{(1+{{\zeta}_{x}}^{2}-{{\zeta}_{y}}^{2})\text{sin}\hspace{0.17em}{\varphi}_{r}+2{\zeta}_{y}\hspace{0.17em}\text{cot}\hspace{0.17em}{\theta}_{r}-2{\zeta}_{x}{\zeta}_{y}\hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{r}}{(1-{{\zeta}_{x}}^{2}+{{\zeta}_{y}}^{2})\text{cos}\hspace{0.17em}{\varphi}_{r}+2{\zeta}_{x}\hspace{0.17em}\text{cot}\hspace{0.17em}{\theta}_{r}-2{\zeta}_{x}{\zeta}_{y}\hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{r}}.$$
(30)
$$2\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\omega =1+\text{sin}\hspace{0.17em}{\theta}_{s}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{cos}({\varphi}_{s}-{\varphi}_{r})+\text{cos}\hspace{0.17em}{\theta}_{s}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta}_{r},$$
(31)
$${\text{tan}}^{2}\hspace{0.17em}{\theta}_{n}=\frac{{\text{sin}}^{2}\hspace{0.17em}{\theta}_{s}+{\text{sin}}^{2}\hspace{0.17em}{\theta}_{r}+2\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{s}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{cos}({\varphi}_{s}-{\varphi}_{r})}{{(\text{cos}\hspace{0.17em}{\theta}_{s}+\text{cos}\hspace{0.17em}{\theta}_{r})}^{2}},$$
(32)
$${\zeta}_{x}=-\frac{\text{sin}\hspace{0.17em}{\theta}_{s}\hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{s}+\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{cos}\hspace{0.17em}{\varphi}_{r}}{\text{cos}\hspace{0.17em}{\theta}_{s}+\text{cos}\hspace{0.17em}{\theta}_{r}},$$
(33)
$${\zeta}_{y}=-\frac{\text{sin}\hspace{0.17em}{\theta}_{s}\hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{s}+\text{sin}\hspace{0.17em}{\theta}_{r}\hspace{0.17em}\text{sin}\hspace{0.17em}{\varphi}_{r}}{\text{cos}\hspace{0.17em}{\theta}_{s}+\text{cos}\hspace{0.17em}{\theta}_{r}}.$$