Abstract

Many spaceborne sensors have been deployed to image the ocean in the visible portion of the spectrum. Information regarding the concentration of water constituents is contained in the water-leaving radiance—the radiance that is backscattered out of the water and subsequently propagates to the top of the atmosphere. Recognizing that it depends on the viewing and Sun geometry, ways have been sought to normalize this radiance to a single Sun-viewing geometry—forming the normalized water-leaving radiance. This requires understanding both the bidirectional nature of the upwelling radiance just beneath the surface and the interaction of this radiance with the air–water interface. I believe that the latter has been incorrectly computed in the past when a water surface roughened by the wind is considered. The presented computation suggests that, for wind speeds as high as 20 m/s, the influence of surface roughness is small for a wide range of Sun-viewing geometries, i.e., the transmittance of the (whitecap-free) air–water interface is nearly identical (within 0.01) to that for a flat interface.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. R. W. Austin, “The remote sensing of spectral radiance from below the ocean surface,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, London, 1974), pp. 317–344.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  11. C. Cox, W. Munk, “Measurements of the roughness of the sea surface from photographs of the Sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
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    [CrossRef]
  13. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).
  14. V. I. Haltrin, “Algorithm and code to calculate specular reflection of light from a wavy water surface,” in Proceedings of the Seventh International Conference on Remote Sensing for Marine and Coastal Environments (Veridian, Ann Arbor, Mich., 2002), pp. 1–7.
  15. N. Ebuchi, S. Kizu, “Probability distribution of surface slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58, 477–486 (2002).
    [CrossRef]

2002

N. Ebuchi, S. Kizu, “Probability distribution of surface slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58, 477–486 (2002).
[CrossRef]

A. Morel, D. Antoine, B. Gentili, “Bidirectional reflectance of oceanic waters: accounting for Raman emission and varying particle scattering phase function,” Appl. Opt. 41, 6289–6306 (2002).
[CrossRef] [PubMed]

1997

1996

1995

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13143–13150 (1995).
[CrossRef]

1993

1991

1988

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

1986

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

1981

1957

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

1954

Antoine, D.

Austin, R. W.

R. W. Austin, “The remote sensing of spectral radiance from below the ocean surface,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, London, 1974), pp. 317–344.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

Case, K. M.

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

H. R. Gordon, D. K. Clark, “Clear water radiances for atmospheric correction of coastal zone color scanner imagery,” Appl. Opt. 20, 4175–4180 (1981).
[CrossRef] [PubMed]

Cox, C.

Ebuchi, N.

N. Ebuchi, S. Kizu, “Probability distribution of surface slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58, 477–486 (2002).
[CrossRef]

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

Gentili, B.

Gordon, H. R.

Haltrin, V. I.

V. I. Haltrin, “Algorithm and code to calculate specular reflection of light from a wavy water surface,” in Proceedings of the Seventh International Conference on Remote Sensing for Marine and Coastal Environments (Veridian, Ann Arbor, Mich., 2002), pp. 1–7.

Kizu, S.

N. Ebuchi, S. Kizu, “Probability distribution of surface slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58, 477–486 (2002).
[CrossRef]

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

Morel, A.

Munk, W.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

Voss, K. J.

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13143–13150 (1995).
[CrossRef]

Yang, H.

Appl. Opt.

J. Geophys. Res.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10909–10924 (1988).
[CrossRef]

A. Morel, K. J. Voss, B. Gentili, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100C, 13143–13150 (1995).
[CrossRef]

J. Oceanogr.

N. Ebuchi, S. Kizu, “Probability distribution of surface slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58, 477–486 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Oceanogr.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Rev. Mod. Phys.

K. M. Case, “Transfer problems and the reciprocity principle,” Rev. Mod. Phys. 29, 651–663 (1957).
[CrossRef]

Other

R. W. Austin, “The remote sensing of spectral radiance from below the ocean surface,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, London, 1974), pp. 317–344.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

V. I. Haltrin, “Algorithm and code to calculate specular reflection of light from a wavy water surface,” in Proceedings of the Seventh International Conference on Remote Sensing for Marine and Coastal Environments (Veridian, Ann Arbor, Mich., 2002), pp. 1–7.

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Figures (6)

Fig. 1
Fig. 1

Definitions of the unit vectors used in the text to describe the direction of propagation of various rays, along with the surface normal.

Fig. 2
Fig. 2

Schematic illustrating the incident radiances L1(ρ, ξ ^) and L2(ρ, ξ ^) for application of the reciprocity principle to radiative transfer across the air–water interface. (a) Reflects discussion in Subsection 3.A (reflection of uniform radiance incident from above). (b) Reflects discussion in Subsection 3.B (transmittance of radiance incident from below). The normals to the surfaces ρA and ρB are directed away from the interface. L2(ρA, ξ ^) is uniform; L2(ρB, ξ ^ is not necessarily uniform.

Fig. 3
Fig. 3

Difference between the true viewing angle θc and the viewing angle for a flat surface θν′ as a function of wind speed. The upper scale is the viewing angle θν (above the surface). Note that θc can be greater than the critical angle for a flat surface.

Fig. 4
Fig. 4

tν), the transmittance of uniform radiance incident from below the air–water interface, as a function of wind speed computed with the Cox and Munk11 probability distribution of the wave facet normal.

Fig. 5
Fig. 5

ℜ(θν, θs) as a function of wind speed computed with the Cox and Munk11 probability distribution of the wave facet normal (solid curves). The dashed curves are from Morel and Gentili7 (their Fig. 11). As in Morel and Gentili, I suppressed any dependence on θs by using mean values for t ¯ fs) and 1 − r ¯Rs).

Fig. 6
Fig. 6

ℜ(θν, θs) as a function of wind speed recomputed by Morel and Gentili (as described in Appendix A) with the Ebuchi and Kizu15 probability distribution of the wave facet normal (solid curves). The dashed curves are from Morel and Gentili7 (their Fig. 11).

Equations (29)

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L w ( λ ) = a - 2 [ L w ( λ ) ] N cos θ s exp { - [ τ R ( λ ) 2 + τ O z ( λ ) ] ( 1 cos θ s ) } ,
[ L w ( λ ) ] N = F ¯ 0 [ ( 1 - ρ f ) ( 1 - ρ ¯ f ) m 2 ] R Q ,
[ L w ( λ ) ] N = F ¯ 0 [ ( 1 - ρ f ) ( 1 - ρ ¯ f ) m 2 ( 1 - r ¯ R ) ] R Q ,
[ L w ( ξ ^ ν ) ] N = F ¯ 0 { [ 1 - ρ f ( ξ ^ ν ) ] [ 1 - ρ ¯ f ( ξ ^ s ) ] m 2 [ 1 - r ¯ R ( ξ ^ s ) ] } R ( ξ ^ s ) Q ( ξ ^ ν , ξ ^ s ) ,
R ( ξ ^ ν , ξ ^ s ) = { [ 1 - ρ f ( ξ ^ ν ) ] [ 1 - ρ ¯ f ( ξ ^ s ) ] m 2 [ 1 - r ¯ R ( ξ ^ s ) ] } .
[ L w ] N exact = [ L w ( n ^ h ) ] N = F ¯ 0 R ( n ^ h , - n ^ h ) R ( - n ^ h ) Q ( n ^ h , - n ^ h ) .
[ L w ] N field = [ L w ( n ^ h ) ] N = F ¯ 0 R ( n ^ h , ξ ^ s ) R ( ξ ^ s ) Q ( n ^ h , ξ ^ s ) .
[ L w ] N space = [ L w ( ξ ^ ν ) ] N = F ¯ 0 R ( ξ ^ ν , ξ ^ s ) R ( ξ ^ s ) Q ( ξ ^ ν , ξ ^ s ) .
[ L w ] N field = R ( n ^ h , ξ ^ s ) R ( n ^ h , - n ^ h ) Q ( n ^ h , - n ^ h ) Q ( n ^ h , ξ ^ s ) R ( ξ ^ s ) R ( - n ^ h ) [ L w ] N exact ,
[ L w ] N space = R ( ξ ^ ν , ξ ^ s ) R ( n ^ h , - n ^ h ) Q ( n ^ h , - n ^ h ) Q ( ξ ^ ν , ξ ^ s ) R ( ξ ^ s ) R ( - n ^ h ) [ L w ] N exact .
[ L w ] N field [ L w ] N space = R ( n ^ h , ξ ^ s ) R ( ξ ^ ν , ξ ^ s ) Q ( ξ ^ ν , ξ ^ s ) Q ( n ^ h , ξ ^ s ) ,
R ( ξ ^ ν , ξ ^ s ) = { t f ( ξ ^ ν ) t ¯ f ( ξ ^ s ) m 2 [ 1 - r ¯ R ( ξ ^ s ) ] }
t f ( ξ ^ ν ) = 1 - ρ f ( ξ ^ ν ) , t ¯ f ( ξ ^ s ) = 1 - ρ ¯ f ( ξ ^ s ) ;
S d S ξ ^ · n ^ < 0 ξ ^ · n ^ [ L 1 ( ρ , ξ ^ ) L 2 ( ρ , - ξ ^ ) m 2 ( ρ ) - L 1 ( ρ , - ξ ^ ) L 2 ( ρ , ξ ^ ) m 2 ( ρ ) ] d Ω ( ξ ^ ) = 0 ,
L 2 ( ρ A , - ξ ^ 0 ) L 0 = ξ ^ · n ^ < 0 L 1 ( ρ A , - ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) F 0 ξ ^ 0 · n ^ = ξ ^ · n ^ L 1 ( ρ A , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) F 0 ξ ^ 0 · n ^ .
L 2 ( ρ A , - ξ ^ 0 ) L 0 = E 1 u ( ρ A , ξ ^ 0 ) F 0 ξ ^ 0 · n ^ .
L 2 ( ρ A , - ξ ^ 0 ) = ξ ^ · n ^ < 0 L 1 ( ρ B , - ξ ^ ) L 2 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) m 2 F 0 ξ ^ 0 · n ^ = ξ ^ · n ^ > 0 L 1 ( ρ B , ξ ^ ) L 2 ( ρ B , - ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) m 2 F 0 ξ ^ 0 · n ^ .
E 1 d ( ρ B , ξ ^ 0 ) = ξ ^ · n ^ > 0 L 1 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) .
t - ( - ξ ^ 0 ) = def m 2 L 2 ( ρ A , - ξ ^ 0 ) L 0 = E 1 d ( ρ B , ξ ^ 0 ) F 0 ξ ^ 0 · n ^ ,
E 1 u ( ρ A , ξ ^ 0 ) + E 1 d ( ρ B , ξ ^ 0 ) = F 0 ξ ^ 0 · n ^ ,
r + ( - ξ ^ 0 ) + t - ( - ξ ^ 0 ) = 1 ,
r - ( - ξ ^ 0 ) + t + ( - ξ ^ 0 ) = 1 ,
L 2 ( ρ B , - ξ ^ ) = L 2 ( ρ B , - ξ ^ c ) + ( - ξ ^ + ξ ^ c ) [ L 2 ( ρ B , - ξ ^ ) ] ξ ^ = ξ ^ c + 1 2 ( - ξ ^ + ξ ^ c ) T [ H ] ξ ^ = ξ ^ c ( - ξ ^ + ξ ^ c ) + ,
ξ ^ c = ξ ^ · n ^ > 0 ξ ^ L 1 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) ξ ^ · n ^ > 0 L 1 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) ,
L 2 ( ρ A , - ξ ^ 0 ) = L 2 ( ρ B , ξ ^ c ) × ξ ^ · n ^ > 0 L 1 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) m 2 F 0 ξ ^ 0 · n ^ ,
E 1 d ( ξ ^ 0 ) = ξ ^ q · n ^ > 0 L 1 ( ρ B , ξ ^ ) ξ ^ · n ^ d Ω ( ξ ^ ) .
m 2 L 2 ( ρ A , - ξ ^ 0 ) L 2 ( ρ B , - ξ ^ c ) = E 1 d ( ξ ^ 0 ) F 0 ξ ^ 0 · n ^ = def t f ( ξ ^ 0 ) ,
[ L w ( ξ ^ ν ) ] N = F ¯ 0 R ( ξ ^ ν , ξ ^ s ) R ( ξ ^ s ) Q ( ξ ^ c , ξ ^ s ) ,
R ( ξ ^ ν , ξ ^ s ) = { T f ( ξ ^ ν ) T f ( ξ ^ s ) m 2 [ 1 - r ¯ R ( ξ ^ s ) ] } ,

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