Abstract

A self-consistent variant of the two-flow approximation that takes into account strong anisotropy of light scattering in seawater of finite depth and arbitrary turbidity is presented. To achieve an appropriate accuracy, this approach uses experimental dependencies between downward and total mean cosines. It calculates irradiances, diffuse attenuation coefficients, and diffuse reflectances in waters with arbitrary values of scattering, backscattering, and attenuation coefficients. It also takes into account arbitrary conditions of illumination and reflection from the bottom with the Lambertian albedo. This theory can be used for the calculation of apparent optical properties in both open and coastal oceanic waters, lakes, and rivers. It can also be applied to other types of absorbing and scattering medium such as paints, photographic emulsions, and biological tissues.

© 1998 Optical Society of America

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  1. W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
    [CrossRef]
  2. B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
    [CrossRef] [PubMed]
  3. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  4. V. I. Haltrin (V. I. Khalturin), “Propagation of light in sea depth,” in Remote Sensing of the Sea and the Influence of the AtmosphereV. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Moscow–Berlin–Sevastopol, 1985), pp. 20–62, in Russian.
  5. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Tech. Phys. 12, 593–607 (1931).
  6. G. A. Gamburtsev, “On the problem of the sea color,” Zh. Russ. Fiz. Khim. Ova. Chast Fiz. (J. Russ. Phys. Chem. Soc. Phys. Ser.) 56, 226–234 (1924).
  7. C. Sagan, J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res. 72, 469–477 (1967).
    [CrossRef]
  8. J. A. Coakley, P. Chylek, “The two-stream approximation in radiative transfer: including the angle of the incident radiation,” J. Atmos. Sci. 32, 409–418 (1975).
    [CrossRef]
  9. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  10. K. Schwarzschield, “Über das Gleichgewicht des Sonnenatmosphäre,” Göttingen Nachrichten 41, 3–32 (1906).
  11. E. P. Zege, Preprint On the Two-Flow Approximation in Radiative Transport (Byelarus Academy of Sciences Institute of Physics, Minsk, 1971).
  12. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  13. V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Opt. 32, 5356–5367 (1993).
    [CrossRef] [PubMed]
  14. 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, R. Frouin, eds., Proc. SPIE2963, (1997), pp. 597–602.
  15. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  16. V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).
  17. V. A. Timofeyeva, “The diffuse reflection coefficient and its relation to the optical parameters of turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 467–469 (1971).
  18. I. D. Efimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian Ocean,” in Geophysical and Optical Studies in the Indian Ocean, L. M. Brekhovskikh, K. S. Shifrin, eds. (Nauka, Moscow, 1975), pp. 124–132, in Russian.
  19. 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]
  20. F. R. Gantmakher, Lectures on Analytical Mechanics (Chelsea, New York, 1970).
  21. V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.
  22. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1.
  23. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 122–123.
  24. J. P. Potter, “The delta-function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
    [CrossRef]
  25. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Media (Springer-Verlag, Berlin, 1991).
  26. C. D. Mobley, Light and Water (Academic, San Diego, 1994).
  27. N. J. McCormick, “Asymptotical optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  28. A. A. Gershun, “Transmission of light through a flat layer of a light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) 11, 43–68 (1936).
  29. T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).
  30. The theory, that is widely known by the name of Kubelka and Munk,5 was originally proposed by Schuster,9 and Schwarzschield.10 In 1924 it was applied to marine conditions by Gamburtsev.6
  31. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
    [CrossRef] [PubMed]
  32. L. F. Gate, “Comparison of the photon diffusion model and Kubelka-Munk equation with the exact solution of the radiative transfer equation,” Appl. Opt. 13, 236–238 (1974).
    [CrossRef] [PubMed]
  33. S. E. Orchard, “Reflection and transmission of light by diffusing suspensions,” J. Opt. Soc. Am. 59, 1584–1597 (1969).
    [CrossRef]
  34. J. I. Gordon, Directional Radiance (luminance) of the Sea Surface, SIO Ref. 69-20 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., October1969).
  35. V. I. Haltrin, “A real-time algorithm for atmospheric corrections of airborne remote optical measurements above the ocean,” in Proceedings of the Second International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1996), Vol. III, pp. 63–72.
  36. V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.
  37. V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments: Technology and Applications (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 502–508.
  38. V. I. Haltrin, “Diffuse reflectance of the optically deep sea under combined illumination of its surface,” in Proceeding of the 1997 International Geoscience and Remote Sensing Symposium, IEEE Cat. No. 97CH36042 (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 296–298.
  39. J. T. O. Kirk, Monte-Carlo Procedure for Simulating the Penetration of Light into Natural Waters, Tech. paper 36 (Division of Plant Industry, Commonwealth Scientific and Industrial Research Organization, Australia, 1981).
  40. V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 15, 774–776 (1979).
  41. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  42. V. A. Timofeyeva, “Optical characteristics of turbid media of sea-water type,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 863–865 (1971).

1994 (1)

1993 (1)

1992 (1)

N. J. McCormick, “Asymptotical optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1988 (1)

1987 (1)

1986 (1)

1980 (1)

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

1979 (1)

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 15, 774–776 (1979).

1977 (1)

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

1975 (2)

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]

J. A. Coakley, P. Chylek, “The two-stream approximation in radiative transfer: including the angle of the incident radiation,” J. Atmos. Sci. 32, 409–418 (1975).
[CrossRef]

1974 (1)

1972 (1)

V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).

1971 (2)

V. A. Timofeyeva, “The diffuse reflection coefficient and its relation to the optical parameters of turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 467–469 (1971).

V. A. Timofeyeva, “Optical characteristics of turbid media of sea-water type,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 863–865 (1971).

1970 (1)

J. P. Potter, “The delta-function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

1969 (1)

1967 (1)

C. Sagan, J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res. 72, 469–477 (1967).
[CrossRef]

1936 (1)

A. A. Gershun, “Transmission of light through a flat layer of a light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) 11, 43–68 (1936).

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Tech. Phys. 12, 593–607 (1931).

1924 (1)

G. A. Gamburtsev, “On the problem of the sea color,” Zh. Russ. Fiz. Khim. Ova. Chast Fiz. (J. Russ. Phys. Chem. Soc. Phys. Ser.) 56, 226–234 (1924).

1906 (1)

K. Schwarzschield, “Über das Gleichgewicht des Sonnenatmosphäre,” Göttingen Nachrichten 41, 3–32 (1906).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Aas, E.

Brown, O. B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chylek, P.

J. A. Coakley, P. Chylek, “The two-stream approximation in radiative transfer: including the angle of the incident radiation,” J. Atmos. Sci. 32, 409–418 (1975).
[CrossRef]

Coakley, J. A.

J. A. Coakley, P. Chylek, “The two-stream approximation in radiative transfer: including the angle of the incident radiation,” J. Atmos. Sci. 32, 409–418 (1975).
[CrossRef]

Efimenko, I. D.

I. D. Efimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian Ocean,” in Geophysical and Optical Studies in the Indian Ocean, L. M. Brekhovskikh, K. S. Shifrin, eds. (Nauka, Moscow, 1975), pp. 124–132, in Russian.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 122–123.

Gamburtsev, G. A.

G. A. Gamburtsev, “On the problem of the sea color,” Zh. Russ. Fiz. Khim. Ova. Chast Fiz. (J. Russ. Phys. Chem. Soc. Phys. Ser.) 56, 226–234 (1924).

Gantmakher, F. R.

F. R. Gantmakher, Lectures on Analytical Mechanics (Chelsea, New York, 1970).

Gate, L. F.

Gershun, A. A.

A. A. Gershun, “Transmission of light through a flat layer of a light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) 11, 43–68 (1936).

Gordon, H. R.

Gordon, J. I.

J. I. Gordon, Directional Radiance (luminance) of the Sea Surface, SIO Ref. 69-20 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., October1969).

Gouesbet, G.

Haltrin, V. I.

V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Opt. 32, 5356–5367 (1993).
[CrossRef] [PubMed]

V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
[CrossRef] [PubMed]

V. I. Haltrin, “A real-time algorithm for atmospheric corrections of airborne remote optical measurements above the ocean,” in Proceedings of the Second International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1996), Vol. III, pp. 63–72.

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.

V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments: Technology and Applications (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 502–508.

V. I. Haltrin, “Diffuse reflectance of the optically deep sea under combined illumination of its surface,” in Proceeding of the 1997 International Geoscience and Remote Sensing Symposium, IEEE Cat. No. 97CH36042 (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 296–298.

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, R. Frouin, eds., Proc. SPIE2963, (1997), pp. 597–602.

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

V. I. Haltrin (V. I. Khalturin), “Propagation of light in sea depth,” in Remote Sensing of the Sea and the Influence of the AtmosphereV. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Moscow–Berlin–Sevastopol, 1985), pp. 20–62, in Russian.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Media (Springer-Verlag, Berlin, 1991).

Jacobs, M. M.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Media (Springer-Verlag, Berlin, 1991).

Kattawar, G. W.

V. I. Haltrin, G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Opt. 32, 5356–5367 (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, R. Frouin, eds., Proc. SPIE2963, (1997), pp. 597–602.

Kirk, J. T. O.

J. T. O. Kirk, Monte-Carlo Procedure for Simulating the Penetration of Light into Natural Waters, Tech. paper 36 (Division of Plant Industry, Commonwealth Scientific and Industrial Research Organization, Australia, 1981).

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Tech. Phys. 12, 593–607 (1931).

Maheu, B.

McCormick, N. J.

Meador, W. E.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

Mobley, C. D.

C. D. Mobley, Light and Water (Academic, San Diego, 1994).

Morel, A.

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

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 122–123.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Tech. Phys. 12, 593–607 (1931).

Orchard, S. E.

Pelevin, V. N.

I. D. Efimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian Ocean,” in Geophysical and Optical Studies in the Indian Ocean, L. M. Brekhovskikh, K. S. Shifrin, eds. (Nauka, Moscow, 1975), pp. 124–132, in Russian.

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

Pollack, J. B.

C. Sagan, J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res. 72, 469–477 (1967).
[CrossRef]

Potter, J. P.

J. P. Potter, “The delta-function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

Prieur, L.

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

Sagan, C.

C. Sagan, J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res. 72, 469–477 (1967).
[CrossRef]

Sanchez, R.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Schwarzschield, K.

K. Schwarzschield, “Über das Gleichgewicht des Sonnenatmosphäre,” Göttingen Nachrichten 41, 3–32 (1906).

Tao, Z.

Timofeyeva, V. A.

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 15, 774–776 (1979).

V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).

V. A. Timofeyeva, “The diffuse reflection coefficient and its relation to the optical parameters of turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 467–469 (1971).

V. A. Timofeyeva, “Optical characteristics of turbid media of sea-water type,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 863–865 (1971).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1.

Weaver, W. R.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[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, R. Frouin, eds., Proc. SPIE2963, (1997), pp. 597–602.

Zege, E. P.

E. P. Zege, Preprint On the Two-Flow Approximation in Radiative Transport (Byelarus Academy of Sciences Institute of Physics, Minsk, 1971).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Media (Springer-Verlag, Berlin, 1991).

Appl. Opt. (7)

Astrophys. J. (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Göttingen Nachrichten (1)

K. Schwarzschield, “Über das Gleichgewicht des Sonnenatmosphäre,” Göttingen Nachrichten 41, 3–32 (1906).

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (4)

V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).

V. A. Timofeyeva, “The diffuse reflection coefficient and its relation to the optical parameters of turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 467–469 (1971).

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from irradiance measurements,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 15, 774–776 (1979).

V. A. Timofeyeva, “Optical characteristics of turbid media of sea-water type,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 7, 863–865 (1971).

J. Atmos. Sci. (3)

J. A. Coakley, P. Chylek, “The two-stream approximation in radiative transfer: including the angle of the incident radiation,” J. Atmos. Sci. 32, 409–418 (1975).
[CrossRef]

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

J. P. Potter, “The delta-function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

J. Geophys. Res. (1)

C. Sagan, J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res. 72, 469–477 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (2)

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

N. J. McCormick, “Asymptotical optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) (1)

A. A. Gershun, “Transmission of light through a flat layer of a light scattering medium,” Tr. Gos. Opt. Inst. (Proc. State Opt. Inst.) 11, 43–68 (1936).

Zeit. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Tech. Phys. 12, 593–607 (1931).

Zh. Russ. Fiz. Khim. Ova. Chast Fiz. (J. Russ. Phys. Chem. Soc. Phys. Ser.) (1)

G. A. Gamburtsev, “On the problem of the sea color,” Zh. Russ. Fiz. Khim. Ova. Chast Fiz. (J. Russ. Phys. Chem. Soc. Phys. Ser.) 56, 226–234 (1924).

Other (19)

V. I. Haltrin (V. I. Khalturin), “Propagation of light in sea depth,” in Remote Sensing of the Sea and the Influence of the AtmosphereV. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Moscow–Berlin–Sevastopol, 1985), pp. 20–62, in Russian.

E. P. Zege, Preprint On the Two-Flow Approximation in Radiative Transport (Byelarus Academy of Sciences Institute of Physics, Minsk, 1971).

F. R. Gantmakher, Lectures on Analytical Mechanics (Chelsea, New York, 1970).

V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), pp. 509–518.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, pp. 122–123.

T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

The theory, that is widely known by the name of Kubelka and Munk,5 was originally proposed by Schuster,9 and Schwarzschield.10 In 1924 it was applied to marine conditions by Gamburtsev.6

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, R. Frouin, eds., Proc. SPIE2963, (1997), pp. 597–602.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer Through a Scattering Media (Springer-Verlag, Berlin, 1991).

C. D. Mobley, Light and Water (Academic, San Diego, 1994).

I. D. Efimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian Ocean,” in Geophysical and Optical Studies in the Indian Ocean, L. M. Brekhovskikh, K. S. Shifrin, eds. (Nauka, Moscow, 1975), pp. 124–132, in Russian.

J. I. Gordon, Directional Radiance (luminance) of the Sea Surface, SIO Ref. 69-20 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., October1969).

V. I. Haltrin, “A real-time algorithm for atmospheric corrections of airborne remote optical measurements above the ocean,” in Proceedings of the Second International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1996), Vol. III, pp. 63–72.

V. I. Haltrin, “Algorithm for computing apparent optical properties of shallow waters under arbitrary surface illumination,” in Proceedings of the Third International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 463–470.

V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” in Proceedings of the Fourth International Conference Remote Sensing for Marine and Coastal Environments: Technology and Applications (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 502–508.

V. I. Haltrin, “Diffuse reflectance of the optically deep sea under combined illumination of its surface,” in Proceeding of the 1997 International Geoscience and Remote Sensing Symposium, IEEE Cat. No. 97CH36042 (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 296–298.

J. T. O. Kirk, Monte-Carlo Procedure for Simulating the Penetration of Light into Natural Waters, Tech. paper 36 (Division of Plant Industry, Commonwealth Scientific and Industrial Research Organization, Australia, 1981).

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