Abstract

The distribution function of the ocean nadir radiance, defined by the upward radiance-to-irradiance ratio, is investigated as functions of the absorption coefficient and the volume scattering function to understand their relationship rather than to develop a numerical algorithm. It is shown for oceanic waters that the distribution function is directly proportional to the volume scattering function normalized by the backscattering coefficient. The relatively small spectral variation of the distribution function is explained by the small spectral variation of the normalized volume scattering function, as well as by a function that describes the contribution of the backscattering-to-absorption ratio to the distribution function. The interpretation described was verified against in situ data, highlighting factors controlling the distribution function of oceanic waters.

© 2009 Optical Society of America

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  1. J. R. V. Zaneveld, A. Barnard, and Z.-P. Lee, “Why are inherent optical properties needed in ocean-color remote sensing?” In Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Application, Z. -P. Lee, ed. (IOCCG, 2006), pp. 3-12.
  2. J. Werdell, “Global bio-optical algorithms for ocean color satellite applications,” EOS Trans. AGU 90
    [CrossRef]
  3. T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.
  4. T. S. Kostadinov, D. A. Siegel, and S. Maritorena, “Determination of the particle size distribution using satellite ocean color imagery: applications and assessment of uncertainty,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.
  5. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864-6879 (1993).
    [CrossRef] [PubMed]
  6. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850-4862 (1996).
    [CrossRef] [PubMed]
  7. H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Rem. Sens. 22, 275-295 (2001).
    [CrossRef]
  8. A. Morel, D. Antoine, and 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]
  9. T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658-4665 (1990).
    [CrossRef] [PubMed]
  10. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503-5507 (2001).
    [CrossRef]
  11. R. A. Maffione and D. R. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36, 6057-6067 (1997).
    [CrossRef] [PubMed]
  12. M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
    [CrossRef]
  13. M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563-571 (2003).
    [CrossRef]
  14. J. F. Berthon, E. Shybanov, M. E. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189-5203 (2007).
    [CrossRef] [PubMed]
  15. M. V. Kozlyaninov and V. N. Pelevin, “On the application of a one-dimensional approximation in the investigation of the propagation of optical radiation in the sea,” J. Publ. Res. Ser. Rep. 36, 54 (1966).
  16. E. Aas, “Two-stream irradiance model for deep water,” Appl. Opt. 26, 2095-2101 (1987).
    [CrossRef] [PubMed]
  17. H. R. Gordon, O. B. Brown, and 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]
  18. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709-722 (1977).
    [CrossRef]
  19. T. Hirata and N. Højerslev, “Relationship between the irradiance reflectance and inherent optical properties of sea water,” J. Geophys. Res. 113, C03030 (2008).
    [CrossRef]
  20. J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13135-13142 (1995).
    [CrossRef]
  21. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Rep. 510, Ref. 72-78 (Scripps Institution of Oceanography, 1972).
  22. A. Morel, “Optical properties of pure water and pure seawater,” in Optical Aspects of Oceanography, N. G. Jerlov and E. Steemann Nielsen, eds. (Academic, 1974) pp. 1-24.
  23. A. Morel, “Diffusion de la lumière par les eaux de mer. Résultats experimentaux et approche théorique,” In Optics of the Sea, AGARD Lecture Series 61 (NATO, 1973) pp. 3.11-3.17.6.
  24. C. H. Whitlock, L. R. Poole, J. W. Usry, W. M. Houghton, W. G. Witte, W. D. Morris, and E. A. Gurganus, “Comparison of reflectance with backscatter and absorption parameters for turbid waters,” Appl. Opt. 20, 517-522 (1981).
    [CrossRef] [PubMed]
  25. J. E. Tyler, “Radiance distribution as a function of depth in an underwater environment,” Bull. Scripps Inst. Oceanogr. 7, 363-412 (1960).
  26. H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water,” Appl. Opt. 34, 1389-409 (1989).
  27. N. K. Højerslev and J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (Københavns Universitet, 1977).
  28. R. W. Preisendorfer, Hydrologic Optics (National Oceanic and Atmospheric Administration Environmental Research Laboratories, 1976).
  29. A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163-7180(2001).
    [CrossRef]
  30. K. J. Voss and A. L. Chapin, “Upwelling radiance distribution camera system, NURADS,” Opt. Express 13, 4250-4262(2005).
    [CrossRef] [PubMed]
  31. W. W. Gregg and N. W. Casey, “Modeling coccolithophores in the global oceans,” Deep Sea Research Part II 54, 447-477 (2007).
    [CrossRef]
  32. E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015-8024 (1999).
    [CrossRef]

2008 (1)

T. Hirata and N. Højerslev, “Relationship between the irradiance reflectance and inherent optical properties of sea water,” J. Geophys. Res. 113, C03030 (2008).
[CrossRef]

2007 (2)

2006 (1)

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

2005 (1)

2003 (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

2002 (1)

2001 (3)

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503-5507 (2001).
[CrossRef]

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Rem. Sens. 22, 275-295 (2001).
[CrossRef]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163-7180(2001).
[CrossRef]

1999 (1)

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015-8024 (1999).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13135-13142 (1995).
[CrossRef]

1993 (1)

1990 (1)

1989 (1)

1987 (1)

1981 (1)

1977 (1)

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

1975 (1)

1966 (1)

M. V. Kozlyaninov and V. N. Pelevin, “On the application of a one-dimensional approximation in the investigation of the propagation of optical radiation in the sea,” J. Publ. Res. Ser. Rep. 36, 54 (1966).

1960 (1)

J. E. Tyler, “Radiance distribution as a function of depth in an underwater environment,” Bull. Scripps Inst. Oceanogr. 7, 363-412 (1960).

Aas, E.

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015-8024 (1999).
[CrossRef]

E. Aas, “Two-stream irradiance model for deep water,” Appl. Opt. 26, 2095-2101 (1987).
[CrossRef] [PubMed]

Aiken, J.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Antoine, D.

Barlow, R.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Barnard, A.

J. R. V. Zaneveld, A. Barnard, and Z.-P. Lee, “Why are inherent optical properties needed in ocean-color remote sensing?” In Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Application, Z. -P. Lee, ed. (IOCCG, 2006), pp. 3-12.

Bernard, S.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Berthon, J. F.

Boss, E.

Brown, O. B.

Casey, N. W.

W. W. Gregg and N. W. Casey, “Modeling coccolithophores in the global oceans,” Deep Sea Research Part II 54, 447-477 (2007).
[CrossRef]

Chami, M.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

Chapin, A. L.

Dana, D. R.

Fishwick, J.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Gentili, B.

Gordon, H. R.

Gregg, W. W.

W. W. Gregg and N. W. Casey, “Modeling coccolithophores in the global oceans,” Deep Sea Research Part II 54, 447-477 (2007).
[CrossRef]

Gurganus, E. A.

Hardman-Mountford, N.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Hirata, T.

T. Hirata and N. Højerslev, “Relationship between the irradiance reflectance and inherent optical properties of sea water,” J. Geophys. Res. 113, C03030 (2008).
[CrossRef]

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Højerslev, N.

T. Hirata and N. Højerslev, “Relationship between the irradiance reflectance and inherent optical properties of sea water,” J. Geophys. Res. 113, C03030 (2008).
[CrossRef]

Højerslev, N. K.

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015-8024 (1999).
[CrossRef]

N. K. Højerslev and J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (Københavns Universitet, 1977).

Houghton, W. M.

Jacobs, M. M.

Khomenko, G.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

Korotaev, G.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

Kostadinov, T. S.

T. S. Kostadinov, D. A. Siegel, and S. Maritorena, “Determination of the particle size distribution using satellite ocean color imagery: applications and assessment of uncertainty,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Kozlyaninov, M. V.

M. V. Kozlyaninov and V. N. Pelevin, “On the application of a one-dimensional approximation in the investigation of the propagation of optical radiation in the sea,” J. Publ. Res. Ser. Rep. 36, 54 (1966).

Lee, M. E.

J. F. Berthon, E. Shybanov, M. E. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189-5203 (2007).
[CrossRef] [PubMed]

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

Lee, Z.-P.

J. R. V. Zaneveld, A. Barnard, and Z.-P. Lee, “Why are inherent optical properties needed in ocean-color remote sensing?” In Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Application, Z. -P. Lee, ed. (IOCCG, 2006), pp. 3-12.

Lewis, M. R.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

Loisel, H.

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Rem. Sens. 22, 275-295 (2001).
[CrossRef]

Maffione, R. A.

Maritorena, S.

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163-7180(2001).
[CrossRef]

T. S. Kostadinov, D. A. Siegel, and S. Maritorena, “Determination of the particle size distribution using satellite ocean color imagery: applications and assessment of uncertainty,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Marken, E.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

Martines-Vicente, V.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Morel, A.

A. Morel, D. Antoine, and 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]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163-7180(2001).
[CrossRef]

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Rem. Sens. 22, 275-295 (2001).
[CrossRef]

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

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864-6879 (1993).
[CrossRef] [PubMed]

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

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

A. Morel, “Diffusion de la lumière par les eaux de mer. Résultats experimentaux et approche théorique,” In Optics of the Sea, AGARD Lecture Series 61 (NATO, 1973) pp. 3.11-3.17.6.

Morris, W. D.

Nielsen, E. Steemann

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

Oishi, T.

Pegau, W. S.

Pelevin, V. N.

M. V. Kozlyaninov and V. N. Pelevin, “On the application of a one-dimensional approximation in the investigation of the propagation of optical radiation in the sea,” J. Publ. Res. Ser. Rep. 36, 54 (1966).

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Rep. 510, Ref. 72-78 (Scripps Institution of Oceanography, 1972).

Poole, L. R.

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics (National Oceanic and Atmospheric Administration Environmental Research Laboratories, 1976).

Prieur, L.

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

Shybanov, E.

Siegel, D. A.

T. S. Kostadinov, D. A. Siegel, and S. Maritorena, “Determination of the particle size distribution using satellite ocean color imagery: applications and assessment of uncertainty,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Smyth, T.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

Stamnes, J. J.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

Tyler, J. E.

J. E. Tyler, “Radiance distribution as a function of depth in an underwater environment,” Bull. Scripps Inst. Oceanogr. 7, 363-412 (1960).

Usry, J. W.

Voss, K. J.

Werdell, J.

J. Werdell, “Global bio-optical algorithms for ocean color satellite applications,” EOS Trans. AGU 90
[CrossRef]

Whitlock, C. H.

Witte, W. G.

Zaneveld, J. R. V.

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13135-13142 (1995).
[CrossRef]

N. K. Højerslev and J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (Københavns Universitet, 1977).

J. R. V. Zaneveld, A. Barnard, and Z.-P. Lee, “Why are inherent optical properties needed in ocean-color remote sensing?” In Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Application, Z. -P. Lee, ed. (IOCCG, 2006), pp. 3-12.

Zibordi, G.

Appl. Opt. (11)

H. R. Gordon, O. B. Brown, and 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]

C. H. Whitlock, L. R. Poole, J. W. Usry, W. M. Houghton, W. G. Witte, W. D. Morris, and E. A. Gurganus, “Comparison of reflectance with backscatter and absorption parameters for turbid waters,” Appl. Opt. 20, 517-522 (1981).
[CrossRef] [PubMed]

E. Aas, “Two-stream irradiance model for deep water,” Appl. Opt. 26, 2095-2101 (1987).
[CrossRef] [PubMed]

R. A. Maffione and D. R. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36, 6057-6067 (1997).
[CrossRef] [PubMed]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water,” Appl. Opt. 34, 1389-409 (1989).

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

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864-6879 (1993).
[CrossRef] [PubMed]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503-5507 (2001).
[CrossRef]

A. Morel, D. Antoine, and 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]

T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658-4665 (1990).
[CrossRef] [PubMed]

J. F. Berthon, E. Shybanov, M. E. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189-5203 (2007).
[CrossRef] [PubMed]

Bull. Scripps Inst. Oceanogr. (1)

J. E. Tyler, “Radiance distribution as a function of depth in an underwater environment,” Bull. Scripps Inst. Oceanogr. 7, 363-412 (1960).

Deep Sea Research Part II (1)

W. W. Gregg and N. W. Casey, “Modeling coccolithophores in the global oceans,” Deep Sea Research Part II 54, 447-477 (2007).
[CrossRef]

EOS Trans. AGU (1)

J. Werdell, “Global bio-optical algorithms for ocean color satellite applications,” EOS Trans. AGU 90
[CrossRef]

Int. J. Rem. Sens. (1)

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Rem. Sens. 22, 275-295 (2001).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

J. Geophys. Res. (5)

E. Aas and N. K. Højerslev, “Analysis of underwater radiance observations: Apparent optical properties and analytic functions describing the angular radiance distribution,” J. Geophys. Res. 104, 8015-8024 (1999).
[CrossRef]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163-7180(2001).
[CrossRef]

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, C05013 (2006).
[CrossRef]

T. Hirata and N. Højerslev, “Relationship between the irradiance reflectance and inherent optical properties of sea water,” J. Geophys. Res. 113, C03030 (2008).
[CrossRef]

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13135-13142 (1995).
[CrossRef]

J. Publ. Res. Ser. Rep. (1)

M. V. Kozlyaninov and V. N. Pelevin, “On the application of a one-dimensional approximation in the investigation of the propagation of optical radiation in the sea,” J. Publ. Res. Ser. Rep. 36, 54 (1966).

Limnol. Oceanogr. (1)

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

Opt. Express (1)

Other (8)

N. K. Højerslev and J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34 (Københavns Universitet, 1977).

R. W. Preisendorfer, Hydrologic Optics (National Oceanic and Atmospheric Administration Environmental Research Laboratories, 1976).

J. R. V. Zaneveld, A. Barnard, and Z.-P. Lee, “Why are inherent optical properties needed in ocean-color remote sensing?” In Remote Sensing of Inherent Optical Properties: Fundamentals, Tests of Algorithms, and Application, Z. -P. Lee, ed. (IOCCG, 2006), pp. 3-12.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Rep. 510, Ref. 72-78 (Scripps Institution of Oceanography, 1972).

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

A. Morel, “Diffusion de la lumière par les eaux de mer. Résultats experimentaux et approche théorique,” In Optics of the Sea, AGARD Lecture Series 61 (NATO, 1973) pp. 3.11-3.17.6.

T. Hirata, N. Hardman-Mountford, J. Aiken, T. Smyth, R. Barlow, V. Martines-Vicente, J. Fishwick, and S. Bernard, “Optical approach to derive phytoplankton size classes using ocean color remote sensing,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

T. S. Kostadinov, D. A. Siegel, and S. Maritorena, “Determination of the particle size distribution using satellite ocean color imagery: applications and assessment of uncertainty,” presented at the Ocean Optics XIX, Castelvecchio Pascoli, Italy, 6-10 Oct. 2008.

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Figures (9)

Fig. 1
Fig. 1

(a) Geometry of the radiance in a near-surface water column around an observer O; (b) geometry of the single scattering at S in a laboratory setting. In this study, the VSF is assumed azimuthally independent. The single scattering angle ψ for a radiance measured in (a) is derived from cos ( ψ ) = cos ( θ ) cos ( θ ) + sin ( θ ) sin ( θ ) cos ( φ φ ) , where ( θ , φ ) and ( θ , φ ) are the nadir and azimuth angles of the incident and scattered radiance, respectively.

Fig. 2
Fig. 2

Spectral b b ( λ ) / a ( λ ) in the Pacific Ocean and Japan Sea. Data were collected by M. Kahru and B. G. Mitchell, Scripps Institution of Oceanography, and obtained through the NASA SeaBASS program. Absorption measurements were obtained spectrophotometrically, and the backscattering using a Hydroscat-6 cast. The bold curve indicates the mean of the spectra.

Fig. 3
Fig. 3

(a)  χ g b p ( Θ ) obtained by Lorenz–Mie scattering theory. Angles Θ at which χ g b p = 1 are equal to Θ * and the value of Θ * depends on the Junge exponent ξ. At 123 Θ 127 ° and Θ = 167 ° , however, χ g b p has little dependency on ξ; the corresponding value of χ g b p ( 125 ° ) is 1.03. (b)  g b ( λ ) obtained from Eq. (15), using spectral measurements of the VSF taken at three stations A1, A2, and B2 by Whitlock et al. [24]. Values of g b ( λ ) fall in 0.489 g b ( λ ) 0.505 , as predicted by Lorenz–Mie theory. The coefficient of variation (CV) for each spectrum is <2%, indicating little spectral variation in g b ( λ ) .

Fig. 4
Fig. 4

Contour plots of spectral variability of β ˜ b b ( 180 ° , λ ) in terms of CV(%), which Petzold VSF would show over 400 to 555 nm when γ p ( 120 ° ) and γ p ( 180 ° ) are numerically varied: (a) oceanic VSF; (b) coastal VSF; (c) harbour VSF. Recent measurements of γ p ( 120 ° ) and γ p ( 179 ° ) by Berthon et al. [14] are also superimposed (*). Note that the measured γ p ( 120 ° ) and γ p ( 180 ° ) show | γ p ( 120 ° ) | < | γ p ( 180 ° ) | in general and are highly correlated with each other, implying that large values of CV found in this numerical test are unlikely in reality.

Fig. 5
Fig. 5

Surface values of χ k u (4 and 10 m ) derived from the in situ radiance data taken by Tyler [25] under clear sky conditions. Sun angle θ s is 56.6 ° and wavelength is λ = 480 nm .

Fig. 6
Fig. 6

Observed values of χ k u ( 180 ° , θ s , λ ) from the Pacific Ocean and Japan Sea data. In situ data were collected by M. Kahru and B. G. Mitchell, Scripps Institution of Oceanography, using Biospherical PRR-800 radiometer (cast), and obtained through the NASA SeaBASS program. (a)  K u ( θ s , λ ) versus k u ( θ , φ , θ s , λ ) for 23.8 ° θ s 70.4 ° and 395 λ 665     nm . The overall value of χ k u ( 180 ° , 23.8 ° , θ s 70.4 ° , 395 λ 665 nm ) for this data set is seen as the regression slope ( = 1.033 ). Data from all depths ( 200 m ) are also shown, demonstrating that depth dependency of χ k u ( 180 ° , θ s , λ ) is small; (b) Spectral variations of of χ k u ( 180 ° , 23.8 ° , θ s 70.4 ° , 395 λ 665 nm ) from the same data but for surface waters ( 5 m ). The bold curve indicates the mean of the spectra.

Fig. 7
Fig. 7

(a) Comparison χ k u ( 180 ° , θ s , λ ) derived from Eq. (19) and observations in the Pacific Ocean and Japan Sea. The reduced number of data results from less match-up data between χ k u ( 180 ° , θ s , λ ) and IOPs ( a ( λ ) and b b ( λ ) ) required for the model calculation; (b)  χ k u ( 180 ° , θ s , λ ) as a function of θ s observed at Station L4 ( 50 ° 15 N , 04 ° 13 W ) in the English Channel. No obvious solar dependency is observed. The coefficient of determination ( r 2 ) is less than 0.015 between λ = 412 and 555 nm .

Fig. 8
Fig. 8

(a) Comparison between spectral Q n 1 ( θ s = 40 ° , λ ) derived from Eq. (21) and observations off Hawaii by Voss and Chapin [30]. (b) Data from the Pacific Ocean and Japan Sea confirm the small spectral variation between 412 λ 555 nm except at λ = 555 nm , where an anomalous increase of Q n 1 ( θ s λ = 555 nm ) is observed. The bold curve indicates the mean of the spectra. For a better comparison with (a), only the wavelength range of 412 555 nm is shown although CVs were calculated for 395 λ 555 nm .

Fig. 9
Fig. 9

Solar dependency of Q n 1 ( θ s , λ = 470 nm ) derived from Eq. (21). To emphasize the dependency, Q n 1 ( θ s , λ = 470 nm ) was normalized by Q n 1 ( 0 ° , λ = 470 nm ) ( = [ Q n 1 ( θ s , λ = 470 nm ) ] N ) . Q n 1 ( Q s , 465 λ 474 nm ) regressed against in situ data collected in the Mediterranean Sea by Aas and Højerslev [32] is also superimposed for comparison.

Tables (2)

Tables Icon

Table 1 Values of χ b b ( Θ ) and χ g b ( Θ ) for Pure Seawater, Suspended Particles, and Total Seawater as well as χ k u ( θ , φ ) a

Tables Icon

Table 2 Percent Difference Between Eq. (21) and Eq. (B2) of Morel et al. [8] at θ s = 40 °

Equations (21)

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R r s ( θ , ϕ , θ s , λ ) = L u ( θ , ϕ , θ s , λ ) E d ( θ s , λ ) ,
R r s ( θ , ϕ , θ s , λ ) = L u ( θ , ϕ , θ s , λ ) E d ( θ s , λ ) R ( θ s , λ ) .
R ( θ s , λ ) = f ( θ s , λ ) b b ( λ ) a ( λ ) + b b ( λ ) ,
R r s ( θ , ϕ , θ s , λ ) = f ( θ s , λ ) Q ( θ , ϕ , θ s , λ ) b b ( λ ) a ( λ ) + b b ( λ ) ,
Q 1 ( θ , ϕ , θ s , λ ) = ( 1 + b b ( λ ) a ( λ ) ) R r s ( θ , ϕ , θ s , λ ) f ' ( θ s , λ ) .
f ( θ s , λ ) = 1 μ d ¯ ( θ s , λ ) g b ( λ ) 0.085 g b ( λ ) + 0.859 ,
μ d ¯ ( θ s , λ ) = 0 2 π L ( θ , ϕ , θ s , λ ) cos ( θ ) d ω 0 2 π L ( θ , ϕ , θ s , λ ) d ω ,
g b ( λ ) = 2 π 4 π β ( Θ , λ ) cos ( Θ ) d Ω 2 π 4 π β ( Θ , λ ) d Ω ,
R r s ( θ = 180 ° , θ s , λ ) 1 μ d ¯ ( θ s , λ ) [ 1 + k u ( θ = 180 ° , θ s , λ ) / a ( λ ) ] β ( 180 ° θ m , θ s , λ ) a ( λ ) ,
Q n 1 ( θ s , λ ) 1 + b b ( λ ) / a ( λ ) 1 + k u ( 180 ° , θ s , λ ) / a ( λ ) g b ( λ ) + 0.859 g b ( λ ) 0.085 β ˜ b b ( 180 ° θ s w , λ ) ,
g b ( λ ) = 1 2 β ( Θ # , λ ) β ( Θ * , λ ) ,
β ( Θ * , λ ) = 2 π 4 π β ( Θ , λ ) d Ω 2 π 4 π 1 d Ω = b b ( λ ) 2 π ,
β ( Θ # , λ ) = 2 π 4 π β ( Θ , λ ) cos ( Θ ) d Ω 2 π 4 π cos ( Θ d Ω ) = b b ( λ ) π g b ( λ ) .
g b ( λ ) 1 2 β w ( 135 ° , λ ) + β p ( 125 ° , λ ) β w ( 125 ° , λ ) + β p ( 120 ° , λ ) = 0.493 ± 0.135 .
β ( Θ , λ ) = β ( Θ , λ 0 ) ( λ λ 0 ) γ ( Θ ) ,
β ˜ b b ( Θ , λ ) = 1 2 π β ( Θ # , λ ) β ( Θ * , λ ) = 1 2 π β w ( Θ , λ 0 ) β w ( 125 ° , λ 0 ) × 1 + β p ( Θ , λ 0 ) β w ( Θ , λ 0 ) ( λ λ 0 ) 4.32 + γ p ( Θ ) 1 + χ b b p ( 120 ° ) β p ( 120 ° , λ 0 ) β w ( 125 ° , λ 0 ) ( λ λ 0 ) 4.32 + γ p ( 120 ° ) .
K u ( θ s , λ ) = 2 π 4 π k u ( θ , ϕ , θ s , λ ) L u ( θ , ϕ , θ s , λ ) cos ( θ ) d ω 2 π 4 π L u ( θ , ϕ , θ s , λ ) cos ( θ ) d ω = k u ( θ , ϕ , θ s , λ ) ¯ = k u ( θ * , ϕ * , θ s , λ ) ,
k u ( θ * , ϕ * , θ s , λ ) = K u ( θ s , λ ) χ k u ( θ , ϕ , θ s , λ ) .
k u ( 180 ° , θ s , λ ) = a ( λ ) + 3.312 b b ( λ ) χ k u ( 180 ° ) ,
k u ( 180 ° , θ s , λ ) a ( λ ) 1 + 3.312 [ b b ( λ ) / a ( λ ) ] χ k u ( 180 ° ) = 0.971 + 3.206 b b ( λ ) a ( λ ) ,
Q n 1 ( θ s , λ ) 1 + b b ( λ ) / a ( λ ) 1.971 + 3.206 [ b b ( λ ) / a ( λ ) ] g b ( λ ) + 0.859 g b ( λ ) 0.085 β ˜ b b ( 180 ° θ s w , λ ) = F b b a ( λ ) F g b β ˜ b b ( 180 ° θ s w , λ ) ,

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