Abstract

This paper is devoted to the derivation of a fast and accurate model of scalar irradiance for stratified Case 2 waters. Five strategies are formulated and employed in the new model, including (1) reallocating the sky radiance, (2) approximating the influence of the air-water interface, (3) constructing a look-up table of average cosine based on the single-scattering albedo and the backscatter fraction, (4) calculating the phase function of surrogate particles in Case 2 waters, and (5) using the average cosine as an index to cope with stratified waters. A comprehensive model-to-model comparison shows that the new model runs more than 1,400 times faster than the commercially-available Hydrolight model, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.

© 2006 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 Press, San Diego, CA, 1994), p. 592.
  2. C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, "Fast and accurate model of underwater scalar irradiance," Appl. Opt. 41, 4962-4974 (2002).
    [CrossRef] [PubMed]
  3. A. Morel and L. Prieur, "Analysis of variations in ocean color," Limnol. Oceanogr. 22, 709-722 (1977).
    [CrossRef]
  4. H. R. Gordon and A. Y. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), Vol.  4, p. 114.
  5. J. C. Pernetta and J. D. Milliman, "Land-ocean interactions in the coastal zone implementation plan," IGBP Report 33 (Stockholm, 1995).
  6. S. Sathyendranath, ed., Remote sensing of ocean colour in coastal, and other optically-complex, waters, IOCCG Report (MacNab Print, Dartmouth, Canada, 2000), Vol. 3, p. 140.
  7. A. Albert and C. D. Mobley, "An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters," Opt. Express 11, 2873-2890 (2003).
    [CrossRef] [PubMed]
  8. R. W. Preisendorfer, Hydrologic optics (U. S. Department of Commerce, Seattle, WA, 1976).
  9. R. L. Fante, "Relationship between radiative-transport theory and Maxwell's equations in dielectric media," J. Opt. Soc. Am. 71, 460-468 (1981).
  10. R. M. Measures, Laser remote sensing: fundamentals and applications (Kreiger, Malabar, Florida, 1992), p. 510.
  11. K. Stamnes, "The theory of multiple scattering of radiation in plane parallel atmospheres," Rev. Geophys. 24, 299-310 (1986).
    [CrossRef]
  12. G. N. Plass and 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]
  13. C.-C. Liu and J. Woods, "Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method," Int. J. Remote Sens. 25, 921-936 (2004).
    [CrossRef]
  14. K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
    [CrossRef]
  15. R. W. Preisendorfer and C. D. Mobley, "Unpolarized irradiance reflectances and glitter patterns of random capillary waves on lakes and seas, by Monte Carlo simulation," NOAA Technical Memorandum, ERL PMEL-63 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1985).
  16. C. D. Mobley and R. W. Preisendorfer, "A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces," NOAA Technical Memorandum, ERL PMEL-75 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1988).
  17. C. D. Mobley, "A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces," Limnol. Oceanogr. 34, 1473-1483 (1989).
    [CrossRef]
  18. C. D. Mobley, "Hydrolight 3.0 Users' Guide," (SRI International, Menlo Park, CA, 1995).
  19. C.-C. Liu, J. D. Woods, and C. D. Mobley, "Optical model for use in oceanic ecosystem models," Appl. Opt. 38, 4475-4485 (1999).
    [CrossRef]
  20. G. R. Fournier and J. L. Forand, "Analytic phase function for ocean water," presented at the SPIE: Ocean Optics XII, 1994.
  21. N. J. McCormick, "Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient," Limnol. Oceanogr. 40, 1013-1018 (1995).
    [CrossRef]
  22. A. Gershun, "The light field," J. Math. Phys. 18, 51-151 (1939).
  23. V. I. Haltrin, "Chlorophyll-Based Model of Seawater Optical Properties," Appl. Opt. 38, 6826-6832 (1999).
    [CrossRef]
  24. C. D. Mobley, L. K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
    [CrossRef] [PubMed]
  25. M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
    [CrossRef]
  26. L. Prieur and S. Sathyendranath, "An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials," Limnol. Oceanogr. 26, 671-689 (1981).
    [CrossRef]
  27. A. Morel, "Light and marine photosynthesis: a spectral model with geochemical and climatological implications," Prog. Oceanogr. 26, 263-306 (1991).
    [CrossRef]
  28. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, "Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship determinations and CZCS estimates," Appl. Opt. 22, 20-36 (1983).
    [CrossRef] [PubMed]
  29. A. Morel, "Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water)," J. Geoph. Res. 93, 10749-10768 (1988).
    [CrossRef]
  30. R. M. Pope and E. S. Fry, "Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements," Appl. Opt. 36, 8710-8723 (1997).
    [CrossRef]
  31. R. C. Smith and K. Baker, "Optical properties of the clearest natural waters," Appl. Opt. 20, 177-184 (1981).
    [CrossRef] [PubMed]
  32. R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical properties and remote sensing of inland and coastal waters (CRC Press, 1995), p. 384.

2004 (1)

C.-C. Liu and J. Woods, "Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method," Int. J. Remote Sens. 25, 921-936 (2004).
[CrossRef]

2003 (2)

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

A. Albert and C. D. Mobley, "An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters," Opt. Express 11, 2873-2890 (2003).
[CrossRef] [PubMed]

2002 (2)

1999 (2)

1997 (1)

1995 (1)

N. J. McCormick, "Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient," Limnol. Oceanogr. 40, 1013-1018 (1995).
[CrossRef]

1991 (1)

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

1989 (1)

C. D. Mobley, "A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces," Limnol. Oceanogr. 34, 1473-1483 (1989).
[CrossRef]

1988 (1)

A. Morel, "Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water)," J. Geoph. Res. 93, 10749-10768 (1988).
[CrossRef]

1986 (1)

K. Stamnes, "The theory of multiple scattering of radiation in plane parallel atmospheres," Rev. Geophys. 24, 299-310 (1986).
[CrossRef]

1983 (3)

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, "Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship determinations and CZCS estimates," Appl. Opt. 22, 20-36 (1983).
[CrossRef] [PubMed]

M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
[CrossRef]

H. R. Gordon and A. Y. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), Vol.  4, p. 114.

1981 (3)

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

R. L. Fante, "Relationship between radiative-transport theory and Maxwell's equations in dielectric media," J. Opt. Soc. Am. 71, 460-468 (1981).

R. C. Smith and K. Baker, "Optical properties of the clearest natural waters," Appl. Opt. 20, 177-184 (1981).
[CrossRef] [PubMed]

1977 (1)

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

1972 (1)

G. N. Plass and 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]

1939 (1)

A. Gershun, "The light field," J. Math. Phys. 18, 51-151 (1939).

Albert, A.

Baker, K.

Boss, E.

Broenkow, W. W.

Brown, J. W.

Brown, O. B.

Carder, K. L.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, "Fast and accurate model of underwater scalar irradiance," Appl. Opt. 41, 4962-4974 (2002).
[CrossRef] [PubMed]

Chen, R. F.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Clark, D. K.

Cullen, J. J.

M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
[CrossRef]

Davis, C. O.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

English, D. C.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Evans, R. H.

Fante, R. L.

Fry, E. S.

Gershun, A.

A. Gershun, "The light field," J. Math. Phys. 18, 51-151 (1939).

Gordon, H. R.

H. R. Gordon and A. Y. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), Vol.  4, p. 114.

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, "Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship determinations and CZCS estimates," Appl. Opt. 22, 20-36 (1983).
[CrossRef] [PubMed]

Haltrin, V. I.

Ivey, J. E.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, "Fast and accurate model of underwater scalar irradiance," Appl. Opt. 41, 4962-4974 (2002).
[CrossRef] [PubMed]

Kattawar, G. W.

G. N. Plass and 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]

Lee, Z. P.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Lewis, M. R.

M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
[CrossRef]

Liu, C.-C.

C.-C. Liu and J. Woods, "Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method," Int. J. Remote Sens. 25, 921-936 (2004).
[CrossRef]

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, "Fast and accurate model of underwater scalar irradiance," Appl. Opt. 41, 4962-4974 (2002).
[CrossRef] [PubMed]

C.-C. Liu, J. D. Woods, and C. D. Mobley, "Optical model for use in oceanic ecosystem models," Appl. Opt. 38, 4475-4485 (1999).
[CrossRef]

McCormick, N. J.

N. J. McCormick, "Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient," Limnol. Oceanogr. 40, 1013-1018 (1995).
[CrossRef]

Miller, R. L.

Mobley, C. D.

Morel, A.

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

A. Morel, "Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water)," J. Geoph. Res. 93, 10749-10768 (1988).
[CrossRef]

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

Morel, A. Y.

H. R. Gordon and A. Y. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), Vol.  4, p. 114.

Patten, J.

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Plass, G. N.

G. N. Plass and 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]

Platt, T.

M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
[CrossRef]

Pope, R. M.

Prieur, L.

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

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

Sathyendranath, S.

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

Smith, R. C.

Stamnes, K.

K. Stamnes, "The theory of multiple scattering of radiation in plane parallel atmospheres," Rev. Geophys. 24, 299-310 (1986).
[CrossRef]

Sundman, L. K.

Woods, J.

C.-C. Liu and J. Woods, "Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method," Int. J. Remote Sens. 25, 921-936 (2004).
[CrossRef]

Woods, J. D.

Appl. Opt. (7)

Int. J. Remote Sens. (1)

C.-C. Liu and J. Woods, "Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method," Int. J. Remote Sens. 25, 921-936 (2004).
[CrossRef]

J. Geoph. Res. (2)

A. Morel, "Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water)," J. Geoph. Res. 93, 10749-10768 (1988).
[CrossRef]

M. R. Lewis, J. J. Cullen, and T. Platt, "Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile," J. Geoph. Res. 88, 2565-2570 (1983).
[CrossRef]

J. Math. Phys. (1)

A. Gershun, "The light field," J. Math. Phys. 18, 51-151 (1939).

J. Opt. Soc. Am. (1)

J. Phys. Oceanogr. (1)

G. N. Plass and 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. (5)

N. J. McCormick, "Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient," Limnol. Oceanogr. 40, 1013-1018 (1995).
[CrossRef]

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

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

K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

C. D. Mobley, "A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces," Limnol. Oceanogr. 34, 1473-1483 (1989).
[CrossRef]

New York (1)

H. R. Gordon and A. Y. Morel, Remote assessment of ocean color for interpretation of satellite visible imagery: a review, Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), Vol.  4, p. 114.

Opt. Express (1)

Prog. Oceanogr. (1)

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

Rev. Geophys. (1)

K. Stamnes, "The theory of multiple scattering of radiation in plane parallel atmospheres," Rev. Geophys. 24, 299-310 (1986).
[CrossRef]

Other (10)

C. D. Mobley, Light and water: radiative transfer in natural waters (Academic Press, San Diego, CA, 1994), p. 592.

C. D. Mobley, "Hydrolight 3.0 Users' Guide," (SRI International, Menlo Park, CA, 1995).

R. W. Preisendorfer and C. D. Mobley, "Unpolarized irradiance reflectances and glitter patterns of random capillary waves on lakes and seas, by Monte Carlo simulation," NOAA Technical Memorandum, ERL PMEL-63 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1985).

C. D. Mobley and R. W. Preisendorfer, "A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces," NOAA Technical Memorandum, ERL PMEL-75 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1988).

R. W. Preisendorfer, Hydrologic optics (U. S. Department of Commerce, Seattle, WA, 1976).

R. M. Measures, Laser remote sensing: fundamentals and applications (Kreiger, Malabar, Florida, 1992), p. 510.

J. C. Pernetta and J. D. Milliman, "Land-ocean interactions in the coastal zone implementation plan," IGBP Report 33 (Stockholm, 1995).

S. Sathyendranath, ed., Remote sensing of ocean colour in coastal, and other optically-complex, waters, IOCCG Report (MacNab Print, Dartmouth, Canada, 2000), Vol. 3, p. 140.

G. R. Fournier and J. L. Forand, "Analytic phase function for ocean water," presented at the SPIE: Ocean Optics XII, 1994.

R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical properties and remote sensing of inland and coastal waters (CRC Press, 1995), p. 384.

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of the validity of Eq. (18). The computational conditions are set as typical Case 1 waters with θ s=30°, Cloudiness=0.0, Vwind =5.0 ms-1, and (a) Chl=l.0 mg∙m-3, BFp =0.00915, (b) Chl=10.0 mg∙m-3, BFp =0.00915, (c) Chl=1.0 mg∙m-3, BFp =0.0183, (d) Chl=10.0 mg∙m-3, BFp =0.0183, (e) Chl=l.0 mg∙m-3, BFp =0.0366, (f) Chl=l0.0 mg∙m-3, BFp =0.0366. The values of the atmospheric parameters used are the default settings in Hydrolight. Both Hydrolight and the new model are employed to simulate μ̄(ζ).

Fig. 2.
Fig. 2.

(a) The analytical phase functions for ocean water proposed by Fournier and Forand [20]. A large range in β(ψ) can be obtained by varying the value of BF. (b) An example of applying the technique of optimization to calculate BFp for simulating the phase function of surrogate particles in Case 2 waters.

Fig. 3.
Fig. 3.

Illustration of using average cosine as an index to cope with stratified waters. The computational conditions are θS = 30°, Cloud = 0.0, λ = 440 nm and Vwind = 5m∙s-1. The atmospheric parateters are those default settings in Hydrolight without consideration of inelastic scattering. (a) A profile of chlorophyll with one meter intervals. (b) A set of μ̄(z) that are calculated by considering eight cases of homogeneous bodies of water, each with one of those chlorophyll concentrations under the same conditions of incident light. (c) Combining these “layer solutions” of μ̄(z) to approximate the full solution of μ̄(z) for the corresponding case of an inhomogeneous body of water under the same conditions of incident light. (d) Comparison of the approximated E 0 to the Hydrolight simulated E 0.

Fig. 4.
Fig. 4.

Twenty cases of stratified Case 2 waters with vertical profiles of (a) Chl, (b) a g,440 and (c) M. The corresponding profiles of IOPs at wavelength 440 nm are illustrated as (d) a, (e) b and (f) BFp .

Fig. 5.
Fig. 5.

Comparison of accuracy and speed in simulating E 0,PAR (z) (W m-2) between the new model and Hydrolight. A very high correlation (r = 0.999924) as well as a large CSR of 1402.8 was obtained for our model. The percentage error ε% is 2.03% and the maximum relative error ε max is not more than 6.81%.

Tables (5)

Tables Icon

Table 1. A section of LUT CW for quick reference to the corrective factor CW based on wind speed Vwind and albedo ω 0.

Tables Icon

Table 2. A section of the LUT μ for quick reference to a set of parameters (B 0, B 1, P, B 2, Q) used by the McCormick five-parameter model [21] of μ̄(ζ). This new LUT is based on two non-dimensional variables BF (0.0001 – 0.5) and ω 0(0.01 – 0.99).

Tables Icon

Table 3. A section of the LUT BF for quick reference to the value of backscattering fraction BFp simulating the phase function of surrogate particles in Case 2 waters. The LUT BF is based on three variables: ratiol (0.0 – 1.0), the backscattering fraction for large particles BFl (0.0001 – 0.0 the backscattering fraction for small particles BFs (0.018 – 0.3).

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Table 4. The randomly specified values of parameters for the 20 cases that are used in the model-to-model comparison to examine the accuracy, flexibility and applicability of the new model. The parameters include the solar zenith angle θS , cloudiness, surface wind speed Vwind , the backscattering fraction for large particles BFl , the backscattering fraction for mineral particles BFS , and the exponential coefficient of CDOM absorption γ.

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Table 5. The randomly specified values of parameters in Eq. (23) for describing the vertical profiles of chlorophyll concentration Chl(z) and the mineral particle concentration M(z), respectively. The CDOM absorption at a reference wavelength a g,440(z) is set to be varied linearly from the surface value ag,440Surface to the bottom value ag,440Bottom, which are specified randomly as well.

Equations (30)

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DL ( x ; ξ ̂ ; λ ) Dr = c ( x ; ξ ̂ ; λ ) L ( x ; ξ ̂ ; λ ) + L * + L * S ,
D Dr = 1 v D Dt = 1 v ( t + v ) = 1 v t + ξ ̂ = ξ ̂ ,
ξ ̂ L ( x ; ξ ̂ ; λ ) = c ( x ; λ ) L ( x ; ξ ̂ ; λ ) + ξ′ ̂ ∈Ξ L ( x ; ξ′ ̂ ; λ ) β ( x ; ξ ̂ ξ ̂ ; λ ) d Ω ( ξ ̂ )
cos θ dL ( z ; θ ; ϕ ; λ ) dz = c ( z ; λ ) L ( z ; θ ; ϕ ; λ ) + ϕ′ = 0 2 π θ′ = 0 π L ( z ; θ ; ϕ ; λ ) β ( z ; θ θ ; ϕ ϕ ; λ ) sin θ .
E 0 ( z ) = Ξ L ( z ; ξ ̂ ) d Ω = ϕ = 0 2 π θ = 0 π L ( z ; θ ; ϕ ) sin θd θdθ .
E d ( 0 + ) = ϕ = 0 2 π θ = 0 π 2 L ( 0 + ; θ ; ϕ ) cos θ sin θd θdϕ = θ = 0 π 2 L ̄ ( 0 + ; θ ; ϕ 0 ) cos θ sin θd θ ,
L ̅ ( 0 + ; θ ; ϕ 0 ) = ϕ = 0 2 π L ( 0 + ; θ ; ϕ ) .
E 0 ( z ) = θ = 0 π 2 E ̅ 0 ( z ; θ ; ϕ 0 ) L ̄ ( 0 + ; θ ; ϕ 0 ) cos θ sin θd θ .
CW ( θ ; ϕ 0 ) V wind V wind E ̄ 0 ( 0 ; θ ; ϕ 0 ) V wind E ̄ 0 ( 0 ; θ ; ϕ 0 ) V wind ,
E 0 ( z ) V ' wind = θ = 0 π 2 E ̄ 0 ( z ; θ ; ϕ 0 ) L ̄ ( 0 + ; θ ; ϕ 0 ) CW ( θ ; ϕ 0 ) V wind V ' wind cos θ sin θd θ .
1 μ ̄ ( ζ ) = B 0 + B 1 exp ( ) + B 2 exp ( ) .
μ ̄ ( z ) E ( z ) E 0 ( z ) .
K NET ( z ) = a ( z ) μ ̄ ( z ) .
K NET ( z ) d In E ( z ) dz = d In [ E d ( z ) E u ( z ) ] dz ,
BF ( λ ) b w ( λ ) b ( λ ) × BF w + b p ( λ ) b ( λ ) BF p ,
Δ ( 0 ) = ( 1 BF p BF ) [ μ ̄ p ( 0 ) μ ̄ ( 0 ) ] ,
Δ ( ζ ) = 1 2 [ μ ̄ p ( ζ ) μ ̄ p ( ζ ) ] .
μ ̄ ( ζ ) = μ ̄ p ( ζ ) + Δ ( ζ ) .
Δ ( ζ ) = [ Δ ( 0 ) Δ ( ζ ) ] [ μ ̄ p ( ζ ) μ ̄ p ( ζ ) μ ̄ p ( 0 ) μ ̄ p ( ζ ) ] .
β ˜ ( ψ ; λ ) b w ( λ ) b ( λ ) β ˜ w ( ψ ) + b l ( λ ) b ( λ ) β ˜ l ( ψ ; λ ) + b s ( λ ) b ( λ ) β ˜ s ( ψ ; λ ) .
β ˜ p ( ψ ; λ ) = b l ( λ ) b l ( λ ) + b s ( λ ) β ˜ l ( ψ ; λ ) + b s ( λ ) b l ( λ ) + b s ( λ ) β ˜ s ( ψ ; λ ) .
ratio l b l ( λ ) b l ( λ ) + b s ( λ ) .
C ( z ) = C o + h s 2 π exp [ 1 2 ( z z max s ) 2 ] .
a g z λ = a g , 440 ( z ) exp [ γ ( λ 440 ) ] ,
a c ( z ; λ ) = 0.06 a c * ( λ ) Chl ( z ) 0.65 ,
b c ( z ; λ ) = 0.30 ( 550 λ ) Chl ( z ) 0.62 ,
a m ( z ; λ ) = M ( z ) a m * ( λ ) ,
b m ( z ; λ ) = M ( z ) b m * ( λ ) .
ε % = 10 RMSE log 10 1 ,
RMSE log 10 = n = 1 N ( log 10 E 0 , PAR Model log 10 E 0 , PAR Hydrolight ) 2 N .

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