Abstract

The relationships between the apparent optical properties (AOP’s) and the inherent optical properties (IOP’s) of oceanic water bodies have been reinvestigated by solution of the radiative transfer equation. This reexamination deals specifically with oceanic case 1 waters (those for which phytoplankton and their associated particles or substances control their inherent optical properties). In such waters, when the chlorophyll content is low enough (in most of the entire ocean), the influence of molecular scattering by water molecules is not negligible, leading to a gradual change in the shape of the phase function. The effect of this change on the AOP’s is analyzed. The effect of the existence of diffuse sky radiation in addition to the direct solar radiation on AOP–IOP relationships is also examined. Practical parameterizations are proposed to predict in case 1 waters, and at various depths, the vertical attenuation coefficient for downward irradiance (K d) as a function of the IOP’s and solar angle. These parameterizations are valid for the spectral domain where inelastic scattering does not significantly occur (wavelengths below 590 nm).

© 1998 Optical Society of America

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References

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  1. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif.1994).
  2. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Mongr. Intl. Union Geod. Geophys. Paris 10, 11–30 (1961).
  3. H. R. Gordon, W. R. McCluney, “Estimation of the depth of sun light penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
    [CrossRef] [PubMed]
  4. H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water,” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [CrossRef]
  5. H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  6. J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
    [CrossRef]
  7. J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
    [CrossRef]
  8. J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
    [CrossRef]
  9. T. J. Petzold, “Volume scattering functions for selected natural waters,” Scripps Inst. Oceanogr. Contrib.72–78 (Scripps Institution of Oceanography, San Diego, Calif, 1972).
  10. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  11. G. H. R. Morel, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York1983), p. 114.
  12. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angles as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
    [CrossRef] [PubMed]
  13. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
    [CrossRef] [PubMed]
  14. S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
    [CrossRef]
  15. V. I. Haltrin, G. W. Kattawar, A. D. Weidemann, “Modeling of elastic and inelastic scattering effects in oceanic optics,” in Ocean Optics XIII, S. G. Ackleson, ed. Proc. SPIE2963, 597–602 (1996).
    [CrossRef]
  16. C. D. Mobley, B. Gentili, H. R. Gordon, J. Zhonghai, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  17. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  18. A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
    [CrossRef]
  19. L. Elterman, “UV, visible, and IR attenuation for altitude to 50 km,” Rep. AFCRL-68-0153 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).
  20. E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” environmental res. paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).
  21. C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).
  22. L. Prieur, A. Morel, “Etude théorique du régime asymptotique: relation entre caractéristiques optiques et coefficient d’extinction relatif à la pénétration de la lumière du jour,” Cah. Oceanogr. 23, 35–47 (1971).
  23. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
    [CrossRef] [PubMed]
  24. The optical thicknesses for the aerosol assemblages considered for the present computations and in Figs. 8 and 9 are 0.230, 0.222, 0.211, 0.198, 0.182 for λ = 410, 443, 490, 560, 665 nm, respectively, when τa(550) = 0.2. When τa(550) = 0.4 or τa (550) = 0.8 the corresponding τa(λ) values are 0.444, 0.433, 0.417, 0.397, and 0.374 or 0.871, 0.853, 0.828, 0.796, and 0.758, respectively.
  25. J. H. Ryther, “Photosynthesis and fish production in the sea,” Science 166, 72–76 (1969).
    [CrossRef] [PubMed]
  26. D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
    [CrossRef]
  27. R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
    [CrossRef]
  28. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
    [CrossRef]
  29. A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10,749–10,768 (1988).
    [CrossRef]

1996

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[CrossRef] [PubMed]

D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
[CrossRef]

1993

1991

1989

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

1988

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10,749–10,768 (1988).
[CrossRef]

1984

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
[CrossRef]

1981

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

1978

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

1977

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

1975

1971

L. Prieur, A. Morel, “Etude théorique du régime asymptotique: relation entre caractéristiques optiques et coefficient d’extinction relatif à la pénétration de la lumière du jour,” Cah. Oceanogr. 23, 35–47 (1971).

1969

J. H. Ryther, “Photosynthesis and fish production in the sea,” Science 166, 72–76 (1969).
[CrossRef] [PubMed]

1961

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Mongr. Intl. Union Geod. Geophys. Paris 10, 11–30 (1961).

1955

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

André, J. M.

D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
[CrossRef]

Antoine, D.

D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
[CrossRef]

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

Bricaud, A.

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Cox, C.

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

Elterman, L.

L. Elterman, “UV, visible, and IR attenuation for altitude to 50 km,” Rep. AFCRL-68-0153 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” environmental res. paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

Gentili, B.

Gordon, H. R.

C. D. Mobley, B. Gentili, H. R. Gordon, J. Zhonghai, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

H. R. Gordon, W. R. McCluney, “Estimation of the depth of sun light penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
[CrossRef] [PubMed]

Haltrin, V. I.

V. I. Haltrin, G. W. Kattawar, A. D. Weidemann, “Modeling of elastic and inelastic scattering effects in oceanic optics,” in Ocean Optics XIII, S. G. Ackleson, ed. Proc. SPIE2963, 597–602 (1996).
[CrossRef]

Kattawar, G. W.

C. D. Mobley, B. Gentili, H. R. Gordon, J. Zhonghai, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

V. I. Haltrin, G. W. Kattawar, A. D. Weidemann, “Modeling of elastic and inelastic scattering effects in oceanic optics,” in Ocean Optics XIII, S. G. Ackleson, ed. Proc. SPIE2963, 597–602 (1996).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

Kishino, M.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
[CrossRef]

McCluney, W. R.

Mobley, C. D.

Morel, A.

D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[CrossRef] [PubMed]

C. D. Mobley, B. Gentili, H. R. Gordon, J. Zhonghai, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angles as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10,749–10,768 (1988).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

L. Prieur, A. Morel, “Etude théorique du régime asymptotique: relation entre caractéristiques optiques et coefficient d’extinction relatif à la pénétration de la lumière du jour,” Cah. Oceanogr. 23, 35–47 (1971).

Morel, A. Y.

G. H. R. Morel, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York1983), p. 114.

Morel, G. H. R.

G. H. R. Morel, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York1983), p. 114.

Munk, W.

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

Okami, N.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected natural waters,” Scripps Inst. Oceanogr. Contrib.72–78 (Scripps Institution of Oceanography, San Diego, Calif, 1972).

Preisendorfer, R. W.

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Mongr. Intl. Union Geod. Geophys. Paris 10, 11–30 (1961).

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

L. Prieur, A. Morel, “Etude théorique du régime asymptotique: relation entre caractéristiques optiques et coefficient d’extinction relatif à la pénétration de la lumière du jour,” Cah. Oceanogr. 23, 35–47 (1971).

Reinersman, P.

Ryther, J. H.

J. H. Ryther, “Photosynthesis and fish production in the sea,” Science 166, 72–76 (1969).
[CrossRef] [PubMed]

Sathyendranath, S.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” environmental res. paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

Stamnes, K.

Stavn, R. H.

Sugihara, S.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
[CrossRef]

Weidemann, A. D.

V. I. Haltrin, G. W. Kattawar, A. D. Weidemann, “Modeling of elastic and inelastic scattering effects in oceanic optics,” in Ocean Optics XIII, S. G. Ackleson, ed. Proc. SPIE2963, 597–602 (1996).
[CrossRef]

Zhonghai, J.

Appl. Opt.

Aust. J. Mar. Freshwater Res.

J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in, and the relationships between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

Cah. Oceanogr.

L. Prieur, A. Morel, “Etude théorique du régime asymptotique: relation entre caractéristiques optiques et coefficient d’extinction relatif à la pénétration de la lumière du jour,” Cah. Oceanogr. 23, 35–47 (1971).

Global Biogeochem. Cycles

D. Antoine, J. M. André, A. Morel, “Ocean primary production. 2. Estimation at global scale from satellite (coastal zone color scanner) chlorophyll,” Global Biogeochem. Cycles 10, 57–69 (1996).
[CrossRef]

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10,749–10,768 (1988).
[CrossRef]

J. Mar. Res.

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

J. Oceanogr. Soc. Jpn.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Jpn. 40, 397–404 (1984).
[CrossRef]

Limnol. Oceanogr.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

Mongr. Intl. Union Geod. Geophys. Paris

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Mongr. Intl. Union Geod. Geophys. Paris 10, 11–30 (1961).

Science

J. H. Ryther, “Photosynthesis and fish production in the sea,” Science 166, 72–76 (1969).
[CrossRef] [PubMed]

Other

The optical thicknesses for the aerosol assemblages considered for the present computations and in Figs. 8 and 9 are 0.230, 0.222, 0.211, 0.198, 0.182 for λ = 410, 443, 490, 560, 665 nm, respectively, when τa(550) = 0.2. When τa(550) = 0.4 or τa (550) = 0.8 the corresponding τa(λ) values are 0.444, 0.433, 0.417, 0.397, and 0.374 or 0.871, 0.853, 0.828, 0.796, and 0.758, respectively.

V. I. Haltrin, G. W. Kattawar, A. D. Weidemann, “Modeling of elastic and inelastic scattering effects in oceanic optics,” in Ocean Optics XIII, S. G. Ackleson, ed. Proc. SPIE2963, 597–602 (1996).
[CrossRef]

L. Elterman, “UV, visible, and IR attenuation for altitude to 50 km,” Rep. AFCRL-68-0153 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” environmental res. paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

G. H. R. Morel, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: a Review (Springer-Verlag, New York1983), p. 114.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif.1994).

T. J. Petzold, “Volume scattering functions for selected natural waters,” Scripps Inst. Oceanogr. Contrib.72–78 (Scripps Institution of Oceanography, San Diego, Calif, 1972).

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Figures (10)

Fig. 1
Fig. 1

Phase functions for molecular scattering, for particles, and for mixtures [Eq. (2)] with variable η values in percent (1, 3, 5, 7, 10, 15 and 20 from bottom to top).

Fig. 2
Fig. 2

Variations of b/ a (and ω̅, right-hand scale) with η within the 400–700-nm spectral domain and for various [Chl] values from 0.02 to 2 mg m-3, as indicated.

Fig. 3
Fig. 3

In the b/ a - η plane, isopleths of the AOP beneath the air–water interface (ζ = 0) for the Sun at zenith (and in a black sky).

Fig. 4
Fig. 4

K d /a beneath the air–water interface for a Sun at zenith as a function of b/ a (and ω̅) for various η values.

Fig. 5
Fig. 5

As in Fig. 3 but at the first attenuation depth (ζ = 1).

Fig. 6
Fig. 6

(a) Relative change of the AOP over the interval ζ = 0 and ζ = 2.3 for a Sun at zenith and η = 5% as a function of b/ a. Data are computed through Eq. (15). (b) As in (a) but for η varying and b/ a = 5.

Fig. 7
Fig. 7

(a) Variations of G with μ w , the cosine of the Sun’s zenith angle after refraction, for several η values: open symbols, results of computations; curves, calculated from Eq. (17). (b) K d /a as function of b/ a at ζ = 2.3 with a light incident at 30°. Solid curves, computed according to Eq. (17); dashed curves (Kirk’s result), according to Eq. (16).

Fig. 8
Fig. 8

(a) Evolution of K d /a, μ̅ d , and μ̅ u beneath the sea surface (ζ = 0) at the wavelength 445 nm and with a [Chl] of 0.1 mg m-3 as functions of the Sun’s zenith angle θ and for several aerosol optical thicknesses τ a (550). For comparison, the value of K d /a at 665 nm is also shown (note that K d is not influenced by Raman scattering at the depth considered). (b) Evolution of the reflectance beneath the sea surface for a [Chl] of 0.1 mg m-3 at 445 nm as a function of θ and for various τ a (550) values.

Fig. 9
Fig. 9

K d /a4.6 averaged over Δζ = 4.6 and for selected wavelengths (in nanometers) as indicated, as a function of [Chl] when the Sun is at zenith. Solid curves, K d /a values computed through Eq. (19) (Kirk’s expression); dashed and dotted curves, K d /a derived from Eqs. (20) and (21), and by use of the coefficients for a real sky (Δζ = 4.6, Table 4).

Fig. 10
Fig. 10

Relative deviations (expressed as percent) between the present results [Eqs. (20) and (21)] and Kirk’s result [Eq. (19)] for selected wavelengths and various sky conditions, plotted as a function of [Chl].

Tables (4)

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Table 2 Changes (in %) in μ u , μ d , Kd/a, and R When η Decreases from 20 to 0% [Eq. (9)] and for Selected b/a Values at Various ζ Values or for Δζ = 1 as Indicateda

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Table 3 G Values for the Layer of Optical Thickness Δζ = 1 for μ w = 1

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Table 4 Values for the Four Coefficients of Eq. (21)a

Equations (24)

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ω ¯ = b / a 1 + b / a     or     b / a = ω ¯ 1 - ω ¯ .
b = 4 π   β ϕ d Ω ,
β ϕ = b β ¯ ϕ .
β ¯ ϕ = η β ¯ w ϕ + 1 - η β ¯ p ϕ .
g = 4 π   β ¯ ϕ cos ϕ d Ω .
g = η g w + 1 - η g p = 1 - η 0.924 .
ζ = K d Z ,
b λ = b w λ + 550 / λ 0.3 Chl 0.62 ,
a λ = [ a w λ + 0.06 A λ Chl 0.65 ] 1 + 0.2 Y λ ,
Y λ = exp - 0.014 λ - 440 .
a = K d 1 / μ ¯ d + R / μ ¯ u 1 - R + 1 K d d R d z ,
δ AOP = AOP η = 20 % - AOP η = 0 % AOP η = 0 % .
K d / a = μ w - 1 1 + G μ w b a 1 / 2 ,
G μ w = 1 ,   ζ = 0 = 0.0527 + 1.371 η .
G μ w ,   ζ = 0 = 0.131 + 1.039 η + - 0.077 + 0.344 η μ w .
K d = μ w - 1   1.0395 a + b b .
K d / a = μ w - 1   1.0395 1 + b a 0.481 - 0.019 η .
= AOP ζ = 2.3 - AOP ζ = 0 AOP ζ = 0 ,
G μ w ,   ζ = 2.3 = 0.473 μ w - 0.218 .
G μ w ,   ζ = 2.3 = 0.451 + 2.584 η μ w - 0.2046 + 0.521 η .
G μ w ,   Δ ζ = 1 = 0.1304 + 0.272 η + - 0.01414 + 1.343 η μ w .
K d / a 4.6 = μ w - 1 1 + 0.425 μ w - 0.190 b a 1 / 2 .
K d / a Δ ζ = μ w - 1 1 + Γ   b a 1 / 2 ,
Γ Δ ζ ,   τ a = γ 1 + γ 2 η + γ 3 + γ 4 η μ w .

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