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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References

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  1. R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), Chap. 7, pp. 297–343.
  2. M. S. Longuet-Higgins, “On the statistical distribution of the heights of sea waves,” J. Mar. Res. 11, 245–266 (1952).
  3. S. Q. Duntley, “Measurements of the distribution of water wave slopes,” J. Opt. Soc. Am. 44, 574–575 (1954).
    [CrossRef]
  4. C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
  5. C. Cox, W. Munk, “Slopes of the sea surface deduced from photographs of Sun glitter,” Scripps Inst. Oceanogr. Bull. 6, 401–487 (1956).
  6. G. Plass, G. Kattawar, J. Guinn, “Radiative transfer in the Earth’s atmosphere and ocean: influence of ocean waves,” Appl. Opt. 14, 1924–1946 (1975).
    [CrossRef] [PubMed]
  7. J. Gordon, “Directional radiance (luminance) of the sea surface,” Scripps Inst. Oceanogr. Visibility Lab. Rep.69–20, 1–50 (Scripps Institution of Oceanography, San Diego, Calif., 1969).
  8. R. Preisendorfer, Surfaces, Vol. 6 of Hydrologic Optics (National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce, Honolulu, Hawaii, 1976), pp. 263 ff.
  9. C. R. Zeisse, “Radiance of the ocean horizon,” Naval Command, Control and Ocean Surveillance Center Research, Development, Test and Evaluation Division Tech. Rep.1660, 1–30 (1994). This reference also contains the fortran code for evaluating the equations used throughout this paper and for generating ASCII files to create many of these figures. The source code is available on disk through correspondence with the author, whose e-mail address is zeisse@nosc.mil.
  10. The tolerance ellipse contains all those slopes capable of reflecting a ray from any part of the solar disk into the receiver.
  11. One can see that this is the case by imagining a receiver on the shore looking directly into a Sun whose center is exactly on the far horizon. Only the upper half of the Sun’s disk would be visible. A facet with nearly zero slope would be required for reflection of the very top of this half-disk into the receiver, but facets with infinite slope would be required if one wished to reflect into the receiver those sides of the half-disk that touch the horizon.
  12. C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).
  13. P. Saunders, “Radiance of sea and sky in the infrared window 800–1200 cm−1,” J. Opt. Soc. Am. 58, 645–652 (1968).
    [CrossRef]

1975 (1)

1968 (1)

1956 (1)

C. Cox, W. Munk, “Slopes of the sea surface deduced from photographs of Sun glitter,” Scripps Inst. Oceanogr. Bull. 6, 401–487 (1956).

1955 (1)

C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).

1954 (2)

1952 (1)

M. S. Longuet-Higgins, “On the statistical distribution of the heights of sea waves,” J. Mar. Res. 11, 245–266 (1952).

Cox, C.

C. Cox, W. Munk, “Slopes of the sea surface deduced from photographs of Sun glitter,” Scripps Inst. Oceanogr. Bull. 6, 401–487 (1956).

C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

Duntley, S. Q.

Gordon, J.

J. Gordon, “Directional radiance (luminance) of the sea surface,” Scripps Inst. Oceanogr. Visibility Lab. Rep.69–20, 1–50 (Scripps Institution of Oceanography, San Diego, Calif., 1969).

Guinn, J.

Kattawar, G.

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “On the statistical distribution of the heights of sea waves,” J. Mar. Res. 11, 245–266 (1952).

Munk, W.

C. Cox, W. Munk, “Slopes of the sea surface deduced from photographs of Sun glitter,” Scripps Inst. Oceanogr. Bull. 6, 401–487 (1956).

C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the Sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

Plass, G.

Preisendorfer, R.

R. Preisendorfer, Surfaces, Vol. 6 of Hydrologic Optics (National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce, Honolulu, Hawaii, 1976), pp. 263 ff.

Saunders, P.

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), Chap. 7, pp. 297–343.

Zeisse, C. R.

C. R. Zeisse, “Radiance of the ocean horizon,” Naval Command, Control and Ocean Surveillance Center Research, Development, Test and Evaluation Division Tech. Rep.1660, 1–30 (1994). This reference also contains the fortran code for evaluating the equations used throughout this paper and for generating ASCII files to create many of these figures. The source code is available on disk through correspondence with the author, whose e-mail address is zeisse@nosc.mil.

Appl. Opt. (1)

J. Mar. Res. (2)

M. S. Longuet-Higgins, “On the statistical distribution of the heights of sea waves,” J. Mar. Res. 11, 245–266 (1952).

C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).

J. Opt. Soc. Am. (3)

Scripps Inst. Oceanogr. Bull. (1)

C. Cox, W. Munk, “Slopes of the sea surface deduced from photographs of Sun glitter,” Scripps Inst. Oceanogr. Bull. 6, 401–487 (1956).

Other (6)

R. E. Walker, Marine Light Field Statistics (Wiley, New York, 1994), Chap. 7, pp. 297–343.

J. Gordon, “Directional radiance (luminance) of the sea surface,” Scripps Inst. Oceanogr. Visibility Lab. Rep.69–20, 1–50 (Scripps Institution of Oceanography, San Diego, Calif., 1969).

R. Preisendorfer, Surfaces, Vol. 6 of Hydrologic Optics (National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce, Honolulu, Hawaii, 1976), pp. 263 ff.

C. R. Zeisse, “Radiance of the ocean horizon,” Naval Command, Control and Ocean Surveillance Center Research, Development, Test and Evaluation Division Tech. Rep.1660, 1–30 (1994). This reference also contains the fortran code for evaluating the equations used throughout this paper and for generating ASCII files to create many of these figures. The source code is available on disk through correspondence with the author, whose e-mail address is zeisse@nosc.mil.

The tolerance ellipse contains all those slopes capable of reflecting a ray from any part of the solar disk into the receiver.

One can see that this is the case by imagining a receiver on the shore looking directly into a Sun whose center is exactly on the far horizon. Only the upper half of the Sun’s disk would be visible. A facet with nearly zero slope would be required for reflection of the very top of this half-disk into the receiver, but facets with infinite slope would be required if one wished to reflect into the receiver those sides of the half-disk that touch the horizon.

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Figures (15)

Fig. 1
Fig. 1

Geometry of facet reflection. Unit vectors Us and Ur point toward the source and the receiver, respectively, from the origin of a right-handed coordinate system located at the reflection point. The X axis points upwind, the Y axis points crosswind, and the Z axis points toward the zenith. The facet normal Un has been left out of the figure for clarity. Azimuths are positive when measured counterclockwise, looking down on the XY plane.

Fig. 2
Fig. 2

Plot of the occurrence probability density p as a function of slope for a wind speed of 1 m s−1. The coordinate system shown in Fig. 1 has been inserted at the top of this figure to illustrate the relationship between coordinates and slopes. Note that the first quadrant contains negative slopes.

Fig. 3
Fig. 3

Plot of the occurrence probability density p throughout slope space for a wind speed of 10 m s−1.

Fig. 4
Fig. 4

(a) Representative facet of area dF from the pixel footprint. Its projection dA onto the horizon is proportional to the probability of finding this facet within the footprint. Its projection dB in the direction U gives the area of the facet seen from U and hence gives the relative importance of this facet with respect to all the other facets (not shown here) that populate the footprint. (b) Coordinate system for (a). During the integration of Eqs. (7), (10), and (11), U remains fixed, Un and ω vary, and no sort of reflection is considered to occur.

Fig. 5
Fig. 5

Area of the ergodic cap projected toward a ray (or beam) for various values of wind speed. The ray is upwind (zero azimuth), and a unit area has been assumed for the footprint. The footprint contains a rough surface that can be seen even from a glancing direction (zenith angle 90°). In contrast, the flat surface representing mean sea level inside the footprint has a projection that varies as the cosine of the beam zenith angle. It vanishes at a glancing angle.

Fig. 6
Fig. 6

Plot of q in slope space for a wind speed of 10 m s−1 and for a beam pointing in the direction (80°, 270°).

Fig. 7
Fig. 7

Plot of q in slope space for a wind speed of 10 m s−1 and for a beam pointing in the direction (89.75°, 270°).

Fig. 8
Fig. 8

Instantaneous specular reflection by a single facet. The radiance of the marine sky is redirected by the facet into a receiver looking down onto the ocean surface.

Fig. 9
Fig. 9

Pixel footprint seen by the receiver shown in Fig. 8. Conditions are the same as for Fig. 6. The dashed line labeled FWHM indicates those positions in the sky that are reflected into the receiver by those slopes given in Fig. 6 for which q has the value 3.5, half its maximum value of 7.

Fig. 10
Fig. 10

Expanded view of Fig. 6, with glint columns included. The receiver is fixed at (80°, 270°). The solar position for each column is (θs, 90°), with θs being given at the base of each column. The solar azimuth has been chosen so that the receiver is looking directly along the center of the glint pattern.

Fig. 11
Fig. 11

Same as Fig. 7, but showing the caps of a line of glint columns. The zenith angle of the solar center progresses from 78° to 90° in 2° steps at a fixed azimuth of 90°. The approximation associated with relation (23) amounts to the assumption that each column has a flat top.

Fig. 12
Fig. 12

Glint ratio versus solar zenith angle for a horizontal view of a sunset over a rough sea. Parameters: ϕs = 90°; (θr, ϕr) = (89.75°, 270°); W = 10 m s−1; ρ = 100%. The solid curve labeled Exact gives the volume of the glint columns whose caps are shown in Fig. 11. The curves labeled Flattop and Cox–Munk show the behavior of relations (23) and CM(9), respectively.

Fig. 13
Fig. 13

Ratio of Sun glint radiance leaving the footprint to solar radiance arriving at the footprint as a function of the zenith angle of the beam leaving the footprint for the receiver. Conditions: (θs, ϕs) = (88°, 90°); ϕr = 270°; W = 10 m s−1; ρ = 100%.

Fig. 14
Fig. 14

Ratio of glint radiance leaving the footprint to solar radiance arriving at the footprint as a function of wind speed. Conditions: (θs, ϕs) = (88°, 90°); (θr, ϕr) = (88°, 270°); ρ = 100%. The higher wind speeds given in this figure and in Fig. 5 have been included only to reveal trends: above 10 m s−1 the inevitable presence of foam and gravity waves reduces the accuracy of these results.

Fig. 15
Fig. 15

Glint pattern for a Sun elevated by 2° over a rough sea. The Sun is to the right of the diagram, near the positive Y axis, with an azimuth of 90°. The receiver is to the left of the diagram, near the negative Y axis, looking toward the Sun. The wind speed is 10 m s−1. The reflectivity is 100%.

Equations (33)

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P p ( ζ x , ζ y , W ) d ζ x d ζ y
p ( ζ x , ζ y , W ) 1 2 π σ u σ c exp [ - 1 2 ( ζ x 2 σ u 2 + ζ y 2 σ c 2 ) ] , σ u 2 = 0.000 + 3.16 × 10 - 3 W , σ c 2 = 0.003 + 1.92 × 10 - 3 W .
N r H = ρ ( ω ) p ( ζ x , ζ y , W ) sec 4 θ n 4 cos θ r .
U s + U r = ( 2 cos ω ) U n .
d A A p d ζ x d ζ y .
Q q ( θ , ϕ , ζ x , ζ y , W ) d ζ x d ζ y
d B B q d ζ x d ζ y .
B = ω π / 2 U = const . d B .
d B = d F cos ω , d A = d F cos θ n ,
q = A B cos ω cos θ n p .
B A = ω π / 2 U = const . cos ω cos θ n p d ζ x d ζ y .
q = cos ω cos θ n p ω π / 2 U = const . cos ω cos θ n p d ζ x d ζ y ,
d P s = N s d B R 2 d S = N s d S R 2 d B .
d P r = ρ d P s .
P r = A d P r = A ρ N s d S R 2 d B .
N r = P r B r ( d S / R 2 ) .
N r = ω π / 2 U r = const . ρ ( ω ) N s ( θ s , ϕ s ) q ( θ r , ϕ r , ζ x , ζ y , W ) d ζ x d ζ y .
N r = U r = const . ρ N s q r J d θ s d ϕ s ,
J ( ζ x , ζ y ) ( θ s , ϕ s ) = sec ω sec 3 θ n sin θ s 4 .
N r = A 4 B r U r = const . ρ ( ω ) N s ( θ s , ϕ s ) p ( ζ x , ζ y , W ) sec 4 θ n × sin θ s d θ s d ϕ s .
N r sun N = ellipse U r - const . ρ q r d ζ x d ζ y ,
N r sun N = A 4 B r = disk U r = const . ( ρ p sec 4 θ n ) sin θ s d θ s d ϕ s ,
N r sun N ρ 100 %
N r sun H A 4 B r ( ρ p sec 4 θ n )
B r A = U r = const . cos ω cos θ n p d ζ x d ζ y = cos θ r
N r sun H ( ρ p sec 4 θ n 4 cos θ r ) ,
cos ω = - sin θ r ( ζ x cos ϕ r + ζ y sin ϕ r ) + cos θ r ( 1 + ζ x 2 + ζ y 2 ) 1 / 2 ,
cos θ s = - 2 ζ x sin θ r cos ϕ r + 2 ζ y sin θ r sin ϕ r + ( - 1 + ζ x 2 + ζ y 2 ) cos θ r 1 + ζ x 2 + ζ y 2 ,
tan ϕ s = ( 1 + ζ x 2 - ζ y 2 ) sin ϕ r + 2 ζ y cot θ r - 2 ζ x ζ y cos ϕ r ( 1 - ζ x 2 + ζ y 2 ) cos ϕ r + 2 ζ x cot θ r - 2 ζ x ζ y sin ϕ r .
2 cos 2 ω = 1 + sin θ s sin θ r cos ( ϕ s - ϕ r ) + cos θ s cos θ r ,
tan 2 θ n = sin 2 θ s + sin 2 θ r + 2 sin θ s sin θ r cos ( ϕ s - ϕ r ) ( cos θ s + cos θ r ) 2 ,
ζ x = - sin θ s cos ϕ s + sin θ r cos ϕ r cos θ s + cos θ r ,
ζ y = - sin θ s sin ϕ s + sin θ r sin ϕ r cos θ s + cos θ r .

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