Abstract

A spectral model of the inherent optical properties (absorption and scattering coefficients a and b, respectively) of oceanic case 1 waters with varying chlorophyll concentrations C is operated. It provides the initial conditions for Monte Carlo simulations aimed at examining the diffuse reflectance directly beneath the surface R and its variations with the solar zenith angle ζ. In most oceanic waters, molecular scattering is not negligible, and molecular backscattering may largely exceed backscattering. The variable contributions (depending on C and wavelength) of water molecules and particles in the scattering process result in considerable variations in the shape of the volume-scattering function. R(ζ) is sensitive to this shape. From the simulations, R (which increases as ζ increases) appears to be linearly related to cos ζ, with a slope that is strongly dependent on ηb, the ratio of molecular backscattering to particle backscattering. The value of the single-scattering albedo (ω¯=b/a+b) has a negligible influence on the R(ζ) function provided that ω¯<0.8, a condition that is always fulfilled when dealing with oceanic case 1 waters. Practical formulas for R(ζ) are proposed. They include the influence of the diffuse sky radiation. The history of each photon and the number of collisions it experiences before exiting have been recorded. These histories and also a probabilistic approach allow the variations of R with cos ζ, ηb, and ω¯ to be understood.

© 1991 Optical Society of America

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References

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  1. R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
    [CrossRef]
  2. R. C. Smith, K. S. Baker, “Optical classification of natural waters and remote sensing,” Limnol. Oceanogr. 23, 260–267 (1978).
    [CrossRef]
  3. H. R. Gordon et al., “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
    [CrossRef]
  4. A. Morel, “Optical modelling of upper ocean in relation to its biogenous matter content (Case 1 waters),” J. Geophys. Res. 93, 10,749–10,768 (1988).
    [CrossRef]
  5. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  6. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Monogr. Int. Union Geod. Geophys. Paris, 10, 11–30 (1961).
  7. A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geod. 5, 335–349 (1982).
    [CrossRef]
  8. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  9. 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]
  10. G. N. Plass, G. W. Kattawar, “Radiative transfer in an atmosphere–ocean system,” Appl. Opt. 8, 455–466 (1969).
    [CrossRef] [PubMed]
  11. G. N. Plass, G. W. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth's atmosphere ocean system: I, Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
    [CrossRef]
  12. H. R. Gordon, O. B. Brown, “Irradiance reflectivity of a flat ocean as a function of its optical properties,” Appl. Opt. 12, 1549–1551 (1973).
    [CrossRef] [PubMed]
  13. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 666.
  14. A. Morel, “Optical properties of pure water and pure sea-water,” N. G. Jerlov, E. Steemann Nielsen, Eds. in Optical Aspects of Oceanography (Academic, New York, 1974), pp. 1–24.
  15. A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
    [CrossRef]
  16. H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
    [CrossRef] [PubMed]
  17. H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery; a Review, (Springer-Verlag, New York, 1983).
  18. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  19. 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]
  20. T. J. Petzold, “Volume scattering functions for selected natural waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, San Diego, Calif., 1972).
  21. A. Morel, “Diffusion de la lumière par les eaux de mer; résultats théorique et approche théorique,” in Optics of the Sea, AGARD Lect. Ser. Sec. 3 63, 1–76 (1973).
  22. 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]
  23. H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
    [CrossRef] [PubMed]
  24. A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing application,” in Oceanography from Space (Plenum, New York, 1981), pp. 313–327.
    [CrossRef]
  25. A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
    [CrossRef]
  26. A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
    [CrossRef]
  27. C. Cox, W. Munk, “Some problems in optical oceanography,” J. Mar. Res. 14, 63–78 (1955).
  28. R. W. Austin, “The remote sensing of spectral radiance from below the ocean surface,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1976), pp. 317–344.
  29. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [CrossRef] [PubMed]
  30. D. Tanré, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric modeling for space measurements of ground reflectances including bidirectional properties,” Appl. Opt. 18, 3587–3594 (1979).
    [CrossRef] [PubMed]
  31. R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
    [CrossRef]
  32. R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
    [CrossRef]
  33. L. Prieur, “Transfert radiatif dans les eaux de mer: application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et particulaires,” Thèse d'Etat (Université Pierre et Marie Curie, Villefranche sur Mer, France, 1976).
  34. A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery; use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).
  35. 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]

1991 (2)

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[CrossRef]

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

1989 (2)

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

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

1988 (2)

H. R. Gordon et al., “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[CrossRef]

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

1987 (2)

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery; use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).

H. R. Gordon, “Bio-optical model describing the distribution of irradiance at the surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

1986 (1)

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

1984 (1)

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]

1983 (1)

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

1982 (1)

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geod. 5, 335–349 (1982).
[CrossRef]

1981 (4)

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]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption 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]

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]

1979 (1)

1978 (2)

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

R. C. Smith, K. S. Baker, “Optical classification of natural waters and remote sensing,” Limnol. Oceanogr. 23, 260–267 (1978).
[CrossRef]

1977 (1)

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

1975 (1)

1973 (3)

1972 (1)

G. N. Plass, G. W. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth's atmosphere ocean system: I, Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

1969 (1)

1961 (1)

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

1955 (1)

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

Ahn, A. Y.

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

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. Steemann Nielsen, eds. (Academic, New York, 1976), pp. 317–344.

Baker, K. S.

R. C. Smith, K. S. Baker, “Optical classification of natural waters and remote sensing,” Limnol. Oceanogr. 23, 260–267 (1978).
[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, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery; use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
[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, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing application,” in Oceanography from Space (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

Brown, O. B.

Clark, D. K.

Cox, C.

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

de Leffe, A.

Deschamps, P. Y.

Gordon, H. R.

Herman, M.

Jacobs, M. M.

Kattawar, G. W.

G. N. Plass, G. W. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth's atmosphere ocean system: I, Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

G. N. Plass, G. W. Kattawar, “Radiative transfer in an atmosphere–ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef] [PubMed]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 666.

Kirk, J. T. O.

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]

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]

Morel, A.

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[CrossRef]

A. Morel, “Optical modelling of 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, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery; use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geod. 5, 335–349 (1982).
[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]

A. Morel, “Diffusion de la lumière par les eaux de mer; résultats théorique et approche théorique,” in Optics of the Sea, AGARD Lect. Ser. Sec. 3 63, 1–76 (1973).

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery; a Review, (Springer-Verlag, New York, 1983).

A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing application,” in Oceanography from Space (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

A. Morel, “Optical properties of pure water and pure sea-water,” N. G. Jerlov, E. Steemann Nielsen, Eds. in Optical Aspects of Oceanography (Academic, New York, 1974), pp. 1–24.

Munk, W.

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

Petzold, T. J.

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

Plass, G. N.

G. N. Plass, G. W. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth's atmosphere ocean system: I, Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

G. N. Plass, G. W. Kattawar, “Radiative transfer in an atmosphere–ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef] [PubMed]

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]

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

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption 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]

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

L. Prieur, “Transfert radiatif dans les eaux de mer: application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et particulaires,” Thèse d'Etat (Université Pierre et Marie Curie, Villefranche sur Mer, France, 1976).

Sathyendranath, S.

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

Smith, R. C.

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geod. 5, 335–349 (1982).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical classification of natural waters and remote sensing,” Limnol. Oceanogr. 23, 260–267 (1978).
[CrossRef]

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

Stavn, R. H.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Tanré, D.

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Appl. Opt. (7)

Aust. J. Mar. Freshwater Res. (1)

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]

J. Geophys. Res. (2)

H. R. Gordon et al., “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[CrossRef]

A. Morel, “Optical modelling of 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. (2)

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

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

J. Phys. Oceanogr. (2)

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

G. N. Plass, G. W. Kattawar, “Monte-Carlo calculations of radiative transfer in the earth's atmosphere ocean system: I, Flux in the atmosphere and ocean,” J. Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

Limnol. Oceanogr. (9)

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplanktons,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

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

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

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

R. C. Smith, K. S. Baker, “Optical classification of natural waters and remote sensing,” Limnol. Oceanogr. 23, 260–267 (1978).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption 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]

Mar. Geod. (1)

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geod. 5, 335–349 (1982).
[CrossRef]

Monogr. Int. Union Geod. Geophys. Paris (1)

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

Oceanol. Acta (1)

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery; use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).

Optics of the Sea, AGARD Lect. Ser. Sec. 3 (1)

A. Morel, “Diffusion de la lumière par les eaux de mer; résultats théorique et approche théorique,” in Optics of the Sea, AGARD Lect. Ser. Sec. 3 63, 1–76 (1973).

Prog. Oceanogr. (1)

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[CrossRef]

Other (7)

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery; a Review, (Springer-Verlag, New York, 1983).

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 666.

A. Morel, “Optical properties of pure water and pure sea-water,” N. G. Jerlov, E. Steemann Nielsen, Eds. in Optical Aspects of Oceanography (Academic, New York, 1974), pp. 1–24.

L. Prieur, “Transfert radiatif dans les eaux de mer: application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et particulaires,” Thèse d'Etat (Université Pierre et Marie Curie, Villefranche sur Mer, France, 1976).

R. W. Austin, “The remote sensing of spectral radiance from below the ocean surface,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1976), pp. 317–344.

A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing application,” in Oceanography from Space (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

(a) Phase functions for molecular scattering w [Eq. (2)] for particles p and for mixtures (1,2,7) with variable η values (see Table II). (b) Probability distribution function corresponding to the phase function adopted for particles; the insets show details of the small angle (0–1°) and large angle (120–180°) domains.

Fig. 2
Fig. 2

Isopleths of ω ¯ (= b/c) in the wavelength λ pigment concentration C plane, as computed through Eqs. (7)(10).

Fig. 3
Fig. 3

As in Fig. 2 but for the isopleths of η (= bb/b), expressed in percentages.

Fig. 4
Fig. 4

(a) As in Fig. 2, but for the isopleths of bb/a when the backscattering probability for particles b ¯ bp is constant and equal to 1.90%. (b) As in (a) but b ¯ bp varies with C according to the empirical equation given by Morel.4

Fig. 5
Fig. 5

Relative contribution of water molecules to backscattering ηb as a function of the relative contribution of water molecules to scattering η [Eq. (12)].

Fig. 6
Fig. 6

As in Figs. 2 and 3 with the selected cases (numbered in reference to Table II) shown along with the closest isopleths for ω ¯(= 0.65) and η(= 0.049).

Fig. 7
Fig. 7

(a) Ratio of the reflectances for various Sun–zenith angles (ζ) to the reflectance when ζ = 0, plotted as a function of μ0 = cos ζ. The numbers in (a), (b), and (c) refer to those in Table II. (b) The f factor [Eq. (13)] as a function of μ0 (c) The f′ factor [Eq. (13′)] as a function of μ0. (d) The f factor for cases 6 and 7 only as a function of μw = cos ζ′ (in water) with ζ′ = sin−1 (sin / ζ /1.34); the dashed straight line was proposed by Kirk.9

Fig. 8
Fig. 8

(a)–(c) As in Figs. 7(a)–7(c) and for the other cases (11–17) are as in Table II. (d) As in (a) but for case 7 and a hypothetical case 7′ (not given in Table II) for which η = 0 and ω ¯ = 0.9. The dashed curves, G.8 and G.9, are redrawn from Fig. 2 of Ref. 8 for ω ¯ = 0.8 and 0.9, respectively.

Fig. 9
Fig. 9

(a) As in Fig. 7(a) except that only cases 2 and 6 are redrawn. In addition, the curves labeled 6′, 6″ and 2′, 2″ account for natural illumination conditions where sun and sky radiances are added according to their respective proportions (see text); 2′ and 6′ are for a visibility of 23 km, 2″ and 6″ for a visibility of 5 km. (b) As in Fig. 7(b); the curves are numbered as in (a), (c), (d) As in Figs. 7(a) and 7(b) but for cases 2 and 7 only and with varying wind speeds. The solid curves are redrawn from Figs. 7(a) and 7(b); the dashed and dotted curves are for 2.5 and 10 m s−1, respectively, with corresponding wave slopes of ±7° and ±12.7° (at ±σ).

Fig. 10
Fig. 10

Relative proportions pn (in percentages and with a log scale) of exiting photons having experienced a given number of collisions (n, from 1 to 30). The solid stepped curves are for the results of Monte Carlo computations; the dashed continuous curves are for predictions from Eq. (18). (a), (b) The two sets of computations are numbered as in Table II.

Fig. 11
Fig. 11

Average number n ¯ of collisions [computed according to Eq. (17)] experienced by the entire population of emerging photons as a function of the ratio ω ¯. The hyperbola corresponds to the predictions of Eq. (19), and the circles are the results of Monte Carlo experiments.

Tables (2)

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Table I Significant Symbols, Definitions, and Units

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Table II Selected Case Studies and Relevant Information

Equations (37)

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R = E u / E d
β ¯ w ( θ ) = 3 4 π ( 3 + p ) ( 1 + p cos 2 θ ) ,
D ( θ ) = 2 π 0 θ β ¯ ( θ ) sin θ d θ ,
D w ( θ ) = 1 2 [ 1 cos ( θ ) + ( p / 3 ) cos 3 θ 1 + ( p / 3 ) ] .
ω ¯ n t , then n = ln t / ln ( ω ¯ ) .
n ¯ euph = ln ( 10 2 ) / ln ( ω ¯ ) ,
n ¯ = 13.8 / ln ( ω ¯ ) ,
N . n 3.5 × 10 8 ( photons × collisions ) .
R = ( μ d μ ) ( μ u + μ ) 1 ( μ u / μ d ) ,
a = E ˚ 1 d ( E d E u ) / d z ,
b ( λ ) = b w ( λ ) + ( 550 / λ ) 0.3 ( C ) 0.62 ,
a ( λ ) = [ a w ( λ ) + 0.06 A ( λ ) ( C ) 0.65 ] [ 1 + 0.2 Y ( λ ) ] ,
Y ( λ ) = exp [ 0.014 ( λ 440 ) ] ,
c ( λ ) = a ( λ ) + b ( λ ) .
β ¯ ( θ ) = η β ¯ w ( θ ) + ( 1 η ) β ¯ p ( θ ) ,
β ( θ ) = b β ¯ ( θ ) = b w β ¯ w ( θ ) + b p β p ( θ ) .
η b = η [ η + 0.038 ( 1 η ) ] 1
R = f b b a ,
f ( μ 0 , η b ) = ( 0.6279 0.2227 η b 0.0513 η b 2 ) + ( 0.3119 + 0.2465 η b ) μ 0 .
R = f b b a + b b ,
f = f ( 1 + b b a ) ,
α = E diff / E tot , ( 1 α ) = E dir / E tot ,
R = α R diff + ( 1 α ) R dir .
f = α f diff + ( 1 α ) f dir .
μ 0 = α μ sky + ( 1 α ) μ dir ,
R = 0.33 ( 1 + Δ ) ( b b / a )
n ¯ = p 1 + 2 p 2 + + n p n .
p n = ω ¯ n / 1 ω ¯ n ,
p n = ω ¯ n 1 ( 1 ω ¯ ) .
n ̅ = 1 / ( 1 ω ¯ ) .
b ¯ b
β ¯ ( θ )
E ˚
μ d = E d / E ˚ d
μ u = E u / E ˚ u
ω ¯
ω ¯

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