Abstract

A new self-consistent two-stream method has been developed that allows for both elastic and inelastic processes including fluorescence. What makes this method very useful is that it contains adjustable parameters that can be selected to fit experimental data. It also has the robustness to cover a complete range of inherent oceanic parameters ranging from the very clear to the most turbid. The method also uses real solar spectral input so that one can also perform chromaticity coordinate calculations for ocean color. Apparent optical properties such as irradiance and scalar irradiance can be computed at any depth in the ocean.

© 1993 Optical Society of America

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  1. V. I. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).
  2. V. I. Haltrin, “Propagation of light in the sea,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (GDR Academy of Science, Berlin, 1985), Chap. 2, pp. 20–62.
  3. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (Nauka i Tekhnika, Minsk, 1985), Chap. 2, pp. 25–55.
  4. R. W. Preisendorfer, C. D. Mobley, “Theory of fluorescent irradiance fields in natural waters,” J. Geophys. Res. 93, 10831–10855 (1988).
    [CrossRef]
  5. V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from Irradiance Measurements,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 15, 774–776 (1979).
  6. I. D. Yefimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian ocean,” in Geophysical and Optical Studies in the Indian Ocean (Nauka, Moscow, 1975), pp. 124–132.
  7. H. W. Schrötter, H. W. Klöckner, “Raman scattering cross-sections in gases and liquids,” in Raman Spectroscopy of Gases and Liquids, A. Weber, ed. (Springer-Verlag, Berlin, 1979), pp. 123–166.
    [CrossRef]
  8. J. Aiken, Investigation of Various Fluorescence Phenomena, (Plymouth Marine Laboratory, Plymouth, UK, 1989), pp. 1–38.
  9. H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166 (1979).
    [CrossRef] [PubMed]
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, pp. 1–53.
  11. B. Davison, Neutron Transport Theory (Clarendon, Oxford, England, 1957), Chap. 19, pp. 255–284.
  12. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976), Chap. 14, pp. 175–192.
  13. C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. 106, 435–443 (1977).
    [CrossRef]
  14. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, England, 1983), Chap. 8, pp. 290–333.
  15. A. Gershun, “About photometry of turbid media,” Tr. Gos. Okeanogr. Inst. 11, 99–104 (1936).
  16. R. H. Stavn, A. D. Weidemann, “Raman scattering in ocean optics: quantative assessment of inherent radiant emission,” Appl. Opt. 31, 1295–1303 (1992).
    [CrossRef]
  17. G. W. Kattawar, Xin Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 6491–6500 (1992).
    [CrossRef] [PubMed]
  18. J. F. Potter, “The delta-function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
    [CrossRef]
  19. J. Lenoble, ed., Standard Procedures to Compute Radiative Transfer in a Scattering Atmosphere (Radiative Commission International Association Meteorology and Atmospheric Physics, National Center for Atmospheric Research, Boulder, Colorado, 1977), pp. 1–129.
  20. In this paper, “unscattered” means a sum of unscattered and forward scattered; “scattered” means scattered to all excluding forward directions.
  21. A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Marine Geod. 5, 335–349 (1982).
    [CrossRef]
  22. There is the possibility that one might infer that this method is circular in structure; i.e., the results of calculations (irradiances) depend on mean cosines, which themselves depend together with other parameters on the above-mentioned solutions (i.e., irradiances). Breaking of this circle, i.e., adoption of some hypothesis about the behavior of the mean cosines, is a sine qua non of every two-flow approximation. There are a number of reasons why our approach is in fact very effective: we seek approximate solutions only to scattered (i.e., according to the definition given in Ref. 20, scattered in all directions excluding forward) light. That scattered light behaves to great degree as the asymptotic light deep inside the medium. For that reason we can reasonably try to apply the experimentally derived asymptotic dependencies of mean cosines on inherent optical properties to our mean cosines instead of using their real dependence on unsolved irradiances by scattered light.
  23. V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), Chap. 11, pp. 171–185.
  24. All occurrences of the indeterminate form of zero divided by zero, which is unsuitable for direct numerical calculations, were rewritten with the introduction of the functions fτ(x) and fxy(x, y), which contain these indeterminate forms. These forms, however, can be precisely calculated numerically with specialized subroutines.
  25. K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), Chap. 5, pp. 155–210.
  26. P. Kubelka, F. Munk, “Ein Beitrag zur Optic der Farbanstriche,” Z. Tech. Phys. 12, 593–607 (1930).
  27. 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]
  28. V. M. Loskutov, “Lighting conditions in deep layers of a turbid medium with a strongly elongated scattering function,” Vestn. Leningr. Univ. Geol. Geogr. 3, 143–149 (1969).
  29. V. A. Timofeyeva, “Brightness in the depth regime in a turbid medium illuminated by normally incident rays,” Izv. Akad. Nauk SSSR Atm. Ocean. Phys. 11, 259–260 (1975).
  30. Unfortunately it is extremely difficult to measure the ratio of diffuse reflectance R to Gordon’s parameter x, especially when they are both very small (≤0.05). The experimental data by Timofeyeva presented in Fig. 2 are the only data known to us. In the original paper only results for R are given. It is clear that these results are less precise than Monte Carlo calculations.
  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. V. I. Haltrin, G. W. Kattawar, Light Fields With Raman Scattering and Fluorescence in Sea Water, (Department of Physics, Texas A&M University, College Station, Texas, 1991), pp. 1–74.
  33. The fortran code that we have commented on is available on request.
  34. G. E. Walrafen, “Raman spectral studies of the effects of temperature on water structure,” J. Chem. Phys. 47, 114–126 (1967).
    [CrossRef]
  35. E. D. Traganza, “Fluorescence excitation and emission spectra of dissolved organic matter in sea water,” Bull. Mar. Sci. 19, 897–904 (1969).
  36. S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
    [CrossRef]
  37. G. E. Walrafen, “Continuum model of water—an erroneous interpretation,” J. Chem. Phys. 50, 567–569 (1969).
    [CrossRef]
  38. D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
    [CrossRef]

1992 (2)

R. H. Stavn, A. D. Weidemann, “Raman scattering in ocean optics: quantative assessment of inherent radiant emission,” Appl. Opt. 31, 1295–1303 (1992).
[CrossRef]

G. W. Kattawar, Xin Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 6491–6500 (1992).
[CrossRef] [PubMed]

1989 (1)

D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
[CrossRef]

1988 (2)

R. W. Preisendorfer, C. D. Mobley, “Theory of fluorescent irradiance fields in natural waters,” J. Geophys. Res. 93, 10831–10855 (1988).
[CrossRef]

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]

1985 (1)

V. I. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).

V. I. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).

1984 (1)

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
[CrossRef]

1982 (1)

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

1979 (2)

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from Irradiance Measurements,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 15, 774–776 (1979).

H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166 (1979).
[CrossRef] [PubMed]

1977 (1)

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. 106, 435–443 (1977).
[CrossRef]

1975 (2)

V. A. Timofeyeva, “Brightness in the depth regime in a turbid medium illuminated by normally incident rays,” Izv. Akad. Nauk SSSR Atm. Ocean. Phys. 11, 259–260 (1975).

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]

1970 (1)

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

1969 (3)

V. M. Loskutov, “Lighting conditions in deep layers of a turbid medium with a strongly elongated scattering function,” Vestn. Leningr. Univ. Geol. Geogr. 3, 143–149 (1969).

G. E. Walrafen, “Continuum model of water—an erroneous interpretation,” J. Chem. Phys. 50, 567–569 (1969).
[CrossRef]

E. D. Traganza, “Fluorescence excitation and emission spectra of dissolved organic matter in sea water,” Bull. Mar. Sci. 19, 897–904 (1969).

1967 (1)

G. E. Walrafen, “Raman spectral studies of the effects of temperature on water structure,” J. Chem. Phys. 47, 114–126 (1967).
[CrossRef]

1936 (1)

A. Gershun, “About photometry of turbid media,” Tr. Gos. Okeanogr. Inst. 11, 99–104 (1936).

1930 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optic der Farbanstriche,” Z. Tech. Phys. 12, 593–607 (1930).

Acquista, C.

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. 106, 435–443 (1977).
[CrossRef]

Aiken, J.

J. Aiken, Investigation of Various Fluorescence Phenomena, (Plymouth Marine Laboratory, Plymouth, UK, 1989), pp. 1–38.

Anderson, J. L.

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. 106, 435–443 (1977).
[CrossRef]

Booth, C. R.

D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
[CrossRef]

Brown, O. B.

Chamberlin, W. S.

D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, pp. 1–53.

Davison, B.

B. Davison, Neutron Transport Theory (Clarendon, Oxford, England, 1957), Chap. 19, pp. 255–284.

Gershun, A.

A. Gershun, “About photometry of turbid media,” Tr. Gos. Okeanogr. Inst. 11, 99–104 (1936).

Gordon, H. R.

Haltrin, V. I.

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. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).

V. I. Haltrin, “Propagation of light in the sea,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (GDR Academy of Science, Berlin, 1985), Chap. 2, pp. 20–62.

V. I. Haltrin, G. W. Kattawar, Light Fields With Raman Scattering and Fluorescence in Sea Water, (Department of Physics, Texas A&M University, College Station, Texas, 1991), pp. 1–74.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (Nauka i Tekhnika, Minsk, 1985), Chap. 2, pp. 25–55.

Jacobs, M. M

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976), Chap. 14, pp. 175–192.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (Nauka i Tekhnika, Minsk, 1985), Chap. 2, pp. 25–55.

Kattawar, G. W.

G. W. Kattawar, Xin Xu, “Filling-in of Fraunhofer lines in the ocean by Raman scattering,” Appl. Opt. 31, 6491–6500 (1992).
[CrossRef] [PubMed]

V. I. Haltrin, G. W. Kattawar, Light Fields With Raman Scattering and Fluorescence in Sea Water, (Department of Physics, Texas A&M University, College Station, Texas, 1991), pp. 1–74.

Khalturin, V. I.

V. I. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).

Kiefer, D. A.

D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
[CrossRef]

Kishino, M.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
[CrossRef]

Klöckner, H. W.

H. W. Schrötter, H. W. Klöckner, “Raman scattering cross-sections in gases and liquids,” in Raman Spectroscopy of Gases and Liquids, A. Weber, ed. (Springer-Verlag, Berlin, 1979), pp. 123–166.
[CrossRef]

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optic der Farbanstriche,” Z. Tech. Phys. 12, 593–607 (1930).

Loskutov, V. M.

V. M. Loskutov, “Lighting conditions in deep layers of a turbid medium with a strongly elongated scattering function,” Vestn. Leningr. Univ. Geol. Geogr. 3, 143–149 (1969).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, England, 1983), Chap. 8, pp. 290–333.

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Theory of fluorescent irradiance fields in natural waters,” J. Geophys. Res. 93, 10831–10855 (1988).
[CrossRef]

Morel, A.

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

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optic der Farbanstriche,” Z. Tech. Phys. 12, 593–607 (1930).

Okami, N.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
[CrossRef]

Pelevin, V. N.

I. D. Yefimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian ocean,” in Geophysical and Optical Studies in the Indian Ocean (Nauka, Moscow, 1975), pp. 124–132.

Potter, J. F.

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

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Theory of fluorescent irradiance fields in natural waters,” J. Geophys. Res. 93, 10831–10855 (1988).
[CrossRef]

Schrötter, H. W.

H. W. Schrötter, H. W. Klöckner, “Raman scattering cross-sections in gases and liquids,” in Raman Spectroscopy of Gases and Liquids, A. Weber, ed. (Springer-Verlag, Berlin, 1979), pp. 123–166.
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), Chap. 5, pp. 155–210.

Smith, R. C.

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

Stavn, R. H.

R. H. Stavn, A. D. Weidemann, “Raman scattering in ocean optics: quantative assessment of inherent radiant emission,” Appl. Opt. 31, 1295–1303 (1992).
[CrossRef]

Sugihara, S.

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
[CrossRef]

Timofeyeva, V. A.

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from Irradiance Measurements,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 15, 774–776 (1979).

V. A. Timofeyeva, “Brightness in the depth regime in a turbid medium illuminated by normally incident rays,” Izv. Akad. Nauk SSSR Atm. Ocean. Phys. 11, 259–260 (1975).

Traganza, E. D.

E. D. Traganza, “Fluorescence excitation and emission spectra of dissolved organic matter in sea water,” Bull. Mar. Sci. 19, 897–904 (1969).

Vladimirov, V. S.

V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), Chap. 11, pp. 171–185.

Walrafen, G. E.

G. E. Walrafen, “Continuum model of water—an erroneous interpretation,” J. Chem. Phys. 50, 567–569 (1969).
[CrossRef]

G. E. Walrafen, “Raman spectral studies of the effects of temperature on water structure,” J. Chem. Phys. 47, 114–126 (1967).
[CrossRef]

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Raman scattering in ocean optics: quantative assessment of inherent radiant emission,” Appl. Opt. 31, 1295–1303 (1992).
[CrossRef]

Xu, Xin

Yefimenko, I. D.

I. D. Yefimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian ocean,” in Geophysical and Optical Studies in the Indian Ocean (Nauka, Moscow, 1975), pp. 124–132.

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (Nauka i Tekhnika, Minsk, 1985), Chap. 2, pp. 25–55.

Ann. Phys. (1)

C. Acquista, J. L. Anderson, “A derivation of the radiative transfer equation for partially polarized light from quantum electrodynamics,” Ann. Phys. 106, 435–443 (1977).
[CrossRef]

Appl. Opt. (5)

Bull. Mar. Sci. (1)

E. D. Traganza, “Fluorescence excitation and emission spectra of dissolved organic matter in sea water,” Bull. Mar. Sci. 19, 897–904 (1969).

Izv. Akad. Nauk SSSR Atm. Ocean. Phys. (1)

V. A. Timofeyeva, “Brightness in the depth regime in a turbid medium illuminated by normally incident rays,” Izv. Akad. Nauk SSSR Atm. Ocean. Phys. 11, 259–260 (1975).

Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. (2)

V. A. Timofeyeva, “Determination of light-field parameters in the depth regime from Irradiance Measurements,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 15, 774–776 (1979).

V. I. Khalturin (a.k.a. V. I. Haltrin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 21, 452–457 (1985).

J. Atmos. Sci. (1)

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

J. Chem. Phys. (2)

G. E. Walrafen, “Raman spectral studies of the effects of temperature on water structure,” J. Chem. Phys. 47, 114–126 (1967).
[CrossRef]

G. E. Walrafen, “Continuum model of water—an erroneous interpretation,” J. Chem. Phys. 50, 567–569 (1969).
[CrossRef]

J. Geophys. Res. (1)

R. W. Preisendorfer, C. D. Mobley, “Theory of fluorescent irradiance fields in natural waters,” J. Geophys. Res. 93, 10831–10855 (1988).
[CrossRef]

J. Oceanog. Soc. Japan (1)

S. Sugihara, M. Kishino, N. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanog. Soc. Japan, 40, 397–403 (1984).
[CrossRef]

Limnol. Oceanogr. (1)

D. A. Kiefer, W. S. Chamberlin, C. R. Booth, “Natural fluorescence of chlorophyll a: relationship to photosynthesis and chlorophyll concentration in the Western South Pacific gyre,” Limnol. Oceanogr. 34, 868–881 (1989).
[CrossRef]

Marine Geod. (1)

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

Tr. Gos. Okeanogr. Inst. (1)

A. Gershun, “About photometry of turbid media,” Tr. Gos. Okeanogr. Inst. 11, 99–104 (1936).

Vestn. Leningr. Univ. Geol. Geogr. (1)

V. M. Loskutov, “Lighting conditions in deep layers of a turbid medium with a strongly elongated scattering function,” Vestn. Leningr. Univ. Geol. Geogr. 3, 143–149 (1969).

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optic der Farbanstriche,” Z. Tech. Phys. 12, 593–607 (1930).

Other (18)

Unfortunately it is extremely difficult to measure the ratio of diffuse reflectance R to Gordon’s parameter x, especially when they are both very small (≤0.05). The experimental data by Timofeyeva presented in Fig. 2 are the only data known to us. In the original paper only results for R are given. It is clear that these results are less precise than Monte Carlo calculations.

V. I. Haltrin, G. W. Kattawar, Light Fields With Raman Scattering and Fluorescence in Sea Water, (Department of Physics, Texas A&M University, College Station, Texas, 1991), pp. 1–74.

The fortran code that we have commented on is available on request.

There is the possibility that one might infer that this method is circular in structure; i.e., the results of calculations (irradiances) depend on mean cosines, which themselves depend together with other parameters on the above-mentioned solutions (i.e., irradiances). Breaking of this circle, i.e., adoption of some hypothesis about the behavior of the mean cosines, is a sine qua non of every two-flow approximation. There are a number of reasons why our approach is in fact very effective: we seek approximate solutions only to scattered (i.e., according to the definition given in Ref. 20, scattered in all directions excluding forward) light. That scattered light behaves to great degree as the asymptotic light deep inside the medium. For that reason we can reasonably try to apply the experimentally derived asymptotic dependencies of mean cosines on inherent optical properties to our mean cosines instead of using their real dependence on unsolved irradiances by scattered light.

V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), Chap. 11, pp. 171–185.

All occurrences of the indeterminate form of zero divided by zero, which is unsuitable for direct numerical calculations, were rewritten with the introduction of the functions fτ(x) and fxy(x, y), which contain these indeterminate forms. These forms, however, can be precisely calculated numerically with specialized subroutines.

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), Chap. 5, pp. 155–210.

J. Lenoble, ed., Standard Procedures to Compute Radiative Transfer in a Scattering Atmosphere (Radiative Commission International Association Meteorology and Atmospheric Physics, National Center for Atmospheric Research, Boulder, Colorado, 1977), pp. 1–129.

In this paper, “unscattered” means a sum of unscattered and forward scattered; “scattered” means scattered to all excluding forward directions.

V. I. Haltrin, “Propagation of light in the sea,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (GDR Academy of Science, Berlin, 1985), Chap. 2, pp. 20–62.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (Nauka i Tekhnika, Minsk, 1985), Chap. 2, pp. 25–55.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, England, 1983), Chap. 8, pp. 290–333.

I. D. Yefimenko, V. N. Pelevin, “Angular distribution of solar radiation in the Indian ocean,” in Geophysical and Optical Studies in the Indian Ocean (Nauka, Moscow, 1975), pp. 124–132.

H. W. Schrötter, H. W. Klöckner, “Raman scattering cross-sections in gases and liquids,” in Raman Spectroscopy of Gases and Liquids, A. Weber, ed. (Springer-Verlag, Berlin, 1979), pp. 123–166.
[CrossRef]

J. Aiken, Investigation of Various Fluorescence Phenomena, (Plymouth Marine Laboratory, Plymouth, UK, 1989), pp. 1–38.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, pp. 1–53.

B. Davison, Neutron Transport Theory (Clarendon, Oxford, England, 1957), Chap. 19, pp. 255–284.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976), Chap. 14, pp. 175–192.

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Figures (3)

Fig. 1
Fig. 1

Relative error of the calculation of the eigenvalue [Eqs. (78)] as a function of single-scattering albedo ω0 for different values of backscattering probability B. Exact values are taken from Loskutov.28

Fig. 2
Fig. 2

Ratio of diffuse reflectance R to Gordon’s parameter x as a function of x according to different sources: (1) Monte-Carlo calculations for direct solar illumination from the nadir by Gordon et al.,27 (2) Monte-Carlo calculations for diffuse illumination by Gordon et al.,27 (3) Kubelka-Munk,26 (4) exact values for a special type of scattering phase function,31 (5) our method [see Eq. (46)], (6) experimental data by Timofeyeva.29

Fig. 3
Fig. 3

(a) Comparison of depth dependence of the elastic and Raman upward and downward irradiances calculated by Eqs. (51), (52), and (71) and the Monte-Carlo simulation for an excitation wavelength λ = 417 nm and an emission wavelength λ = 486 nm. The inherent optical properties are given in Table, 1, BHyd = 0.01798. (b) Same as (a) except that an excitation wavelength λ = 440 nm and an emission wavelength λ = 518 nm.

Tables (2)

Tables Icon

Table A1 Raman Frequency Distribution Parameters

Tables Icon

Table 1 Optical Properties Used for the Monte-Carlo Simulation and Analytical Calculations Displayed in Figs. 3(a) and 3(b)a

Equations (99)

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[ n + c ( λ ) ] L ( λ , r , Ω ) = b E ( λ ) 4 π Ω p ( cos Θ ) L ( λ , r , Ω ) d Ω + 1 4 π I = R , F λ < λ d λ σ I ( λ , λ ) × Ω p I ( cos Θ ) L ( λ , r , Ω ) d Ω ,
Ω p ( cos Θ ) d Ω 0 2 π d φ 0 π p ( cos Θ ) sin θ d θ = 4 π ,
I ( λ ) = 1 4 π λ < λ σ I ( λ , λ ) d λ Ω p I ( cos Θ ) d Ω , I = R , F .
b R ( λ ) = 1 4 π λ > λ σ R ( λ , λ ) d λ Ω p R ( cos Θ ) d Ω ,
a F l ( λ ) = 1 4 π F = C , Y λ > λ d λ σ F ( λ , λ ) Ω p F ( cos Θ ) d Ω ,
E 0 = Ω L ( Ω ) d Ω ,
E = Ω L ( Ω ) n d Ω ,
div E = - [ a t h + Δ a ] E 0 ,
[ cos θ d d z + c ( λ ) ] L ( λ , z , θ , φ ) = Q ( λ , z , θ , φ ) ,
Q ( λ , z , θ , φ ) = Q E ( λ , z , θ , φ ) + I = R , C , Y Q I ( λ , z , θ , φ ) + Q H O ( λ , z , θ , φ ) ,
L ( λ , z , θ , φ ) = L E ( λ , z , θ , φ ) + I = R , C , Y L I ( λ , z , θ , φ ) ;
Q E ( λ , z , θ , φ ) = b E ( λ ) 4 π p ( cos Θ ) L E ( λ , z , θ , φ ) d Ω ,
Q R ( λ , z , θ , φ ) = b E ( λ ) 4 π p ( cos Θ ) L R ( λ , z , θ , φ ) d Ω + 1 4 π λ < λ d λ σ R ( λ , λ ) × p R ( cos Θ ) L E ( λ , z , θ , φ ) d Ω ,
Q F ( λ , z , θ , φ ) = b E ( λ ) 4 π p ( cos Θ ) L F ( λ , z , θ , φ ) d Ω + 1 4 π λ < λ d λ σ F ( λ , λ ) × p F ( cos Θ ) L E ( λ , z , θ , φ ) d Ω , F = C , Y ,
σ R ( λ , λ ) = d ν ˜ d λ σ ν R ( ν ˜ , ν ˜ ) = ν ˜ 2 k ν β 0 R ( ν ˜ ν ˜ 0 ) 4 f R ( ν ˜ , ν ˜ ) ;
σ F ( λ , λ ) = β 0 F a 0 F ( λ ) C F f F ( λ , λ ) , F = C , Y ,
- + f R ( ν ˜ , ν ˜ ) d ν ˜ = - + f R ( ν ˜ , ν ˜ ) d ν ˜ = 1 ;
f F ( λ , λ ) = f ex F ( λ ) f em F ( λ ) ,             - + f ex F ( λ ) d λ = 1 , - + f e m F ( λ ) d λ = 1 ,             F = C , Y ,
p R ( cos Θ ) = 1 + 3 k ρ cos 2 Θ 1 + k ρ ,             k ρ 0.2 ,
T ^ φ [ cos θ d d z + c - b E 4 π d Ω p ( cos Θ ) ] φ ,
T ^ L E ( λ , z , θ , φ ) = 0 ,
T ^ L I ( λ , z , θ , φ ) = 1 4 π λ < λ d λ d Ω σ I ( λ , λ ) p I ( cos Θ ) × L E ( λ , z , θ , φ ) , I = R , F , F = C , Y .
[ cos θ d d z + c ] L E ( z , θ , φ ) = b E 4 π p ( cos Θ ) L E ( z , θ , φ ) d Ω ,
( μ d d τ + 1 ) L E ( τ , μ , φ ) = 1 2 π [ x ˜ 1 + x ˜ 0 2 π d φ - 1 1 d μ L E ( τ , μ , φ ) + Δ ( τ , μ , φ ) ] ,
Δ ( τ , μ , φ ) = ω ˜ 0 ( 1 - x ˜ ) 2 ( 1 - ω ˜ 0 ) ( 1 + x ˜ ) 0 2 π d φ - 1 1 d μ × [ p ( μ ) - p T ( μ ) ] L E ( τ , μ , φ ) ,
L E = L s + L q .
( μ d d τ + 1 ) L q ( τ , μ , φ ) = 0 ,
L q ( τ , μ , φ ) = L 0 q ( μ , φ ) exp ( - τ / μ ) .
T ^ Δ L s = g ( τ ) 2 π ,
T ^ Δ L s ( μ d d τ + 1 ) L s ( τ , μ , φ ) - 1 2 π [ x ˜ 1 + x ˜ 0 2 π d φ × - 1 1 d μ L s ( τ , μ , φ ) + Δ ( τ , μ , φ ) ] ,
L s ( 0 , μ , φ ) μ > 0 = 0 ,             lim τ L s ( τ , μ , φ ) = 0 ,
g ( τ ) = x ˜ 1 + x ˜ 0 2 π d φ - 1 1 d μ L 0 q ( μ , φ ) exp ( - τ / μ ) .
E d s = 0 2 π d φ 0 1 L s ( μ , φ ) μ d μ , E u s = - 0 2 π d φ - 1 0 L s ( μ , φ ) μ d μ ,
E 0 d s = 0 2 π d φ 0 1 L s ( μ , φ ) d μ , E 0 u s = 0 2 π d φ - 1 0 L s ( μ , φ ) d μ ,
E 0 s = 0 2 π d φ - 1 1 L s ( μ , φ ) d μ = E 0 d s + E 0 u s .
μ ¯ d = E d s E 0 d s ,             μ ¯ u = E u s E 0 u s , μ ¯ = E d s - E u s E 0 s = E d s - E u s E 0 d s + E 0 u s .
β = 1 , 2 L ^ α β ( τ ) E β s = e α g ( τ ) , L ^ ( τ ) = [ d d τ + q - - x ˜ q + - x ˜ q - - d d τ + q + ] ,
e = [ 1 1 ] , q ± = 2 ± μ ¯ 1 + x ˜ .
k = a ˜ μ ¯ 1 - x ˜ μ ¯ ( 1 + x ˜ ) .
k = μ ¯ - ( 4 - ( 4 - μ ¯ 2 ) x ˜ 2 ] 1 / 2 ( 1 + x ˜ ) .
μ ¯ = { 1 + 2 x ˜ - [ x ˜ ( 4 + 5 x ˜ ) ] 1 / 2 1 + x ˜ } 1 / 2 { 1 - x ˜ 1 + 2 x ˜ + [ x ˜ ( 4 + 5 x ˜ ) ] 1 / 2 } 1 / 2 , x ˜ = ( 1 - μ ¯ 2 ) 2 1 + 4 μ ¯ 2 - μ ¯ 4 ,
k = μ ¯ ( 3 - μ ¯ 2 ) 1 + μ ¯ 2 ,             k 0 = μ ¯ ( 4 - μ ¯ 2 ) .
L 0 q ( μ , φ ) = L 0 δ ( φ ) δ ( μ - μ s ) ,             μ s = ( 1 - sin 2 z n r 2 ) 1 / 2 ,
g ( τ ) = x ˜ L 0 1 + x ˜     exp ( - τ / μ s ) ,
E d s ( z = 0 ) = 0 ,             lim z E i s ( z ) = 0
{ E d s ( τ ) = A [ exp ( - k τ ) - exp ( - τ / μ s ) ] E u s ( τ ) = R A exp ( - k τ ) - B exp ( - τ / μ s ) ,
R = ( 1 - μ ¯ 1 + μ ¯ ) 2
A = x ˜ L 0 ( 2 + μ ¯ + 1 / μ s ) ( 1 + x ˜ ) ( 1 / μ s - k ) ( 1 / μ s + k 0 ) , B = x ˜ L 0 ( 2 - μ ¯ - 1 / μ s ) ( 1 + x ˜ ) ( 1 / μ s - k ) ( 1 / μ s + k 0 ) ,
{ E d s ( τ ) = L 0 μ s R s Q s m 1 1 - μ s k × [ exp ( - k τ ) - exp ( - τ / μ s ) ] E u s ( τ ) = L 0 μ s R s { exp ( - k τ ) + Q s m 2 1 - μ s k × [ exp ( - k τ ) - exp ( - τ / μ s ) ] } ,
m 1 = 1 + μ s ( 2 + μ ¯ ) ,             m 2 = μ s ( 2 - μ ¯ ) - 1 , Q s = 1 1 + R ( 1 + μ ¯ ) 2 2 ( 1 + μ ¯ 2 ) , R s = ( 1 - μ ¯ ) 2 1 + μ s k 0 ( 1 - μ ¯ ) 2 1 + μ s μ ¯ ( 4 - μ 2 ) ,
E d q ( τ ) = L 0 μ s exp ( - τ / μ s ) ,             E u q ( τ ) 0.
E d E ( z ) = L 0 μ s exp ( - ν z ) [ exp ( 2 z s ) + R s η Q s m 1 D 0 ( z ) ] ,
E u E ( z ) = L 0 μ s R s exp ( - ν z ) [ 1 + η Q s m 2 D 0 ( z ) ] ,
- d E d E ( z ) d z = L 0 μ s exp ( - ν z ) η [ exp ( 2 z s ) + R s Q s m 1 D z ( z ) ] ,
- d E u E ( z ) d z = L 0 μ s R s exp ( - ν z ) [ ν + η Q s m 2 D z ( z ) ] ,
k d E ( z ) = - d ln E d E ( z ) d z = η exp ( 2 z s ) + R s Q s m 1 D z ( z ) exp ( 2 z s ) + R s η Q s m 1 D 0 ( z ) ,
k u E ( z ) = - d ln E u E ( z ) d z = ν + η Q s m 2 D z ( z ) 1 + η Q s m 2 D 0 ( z ) .
( μ d d τ + 1 ) L I ( τ , μ , φ ) = 1 2 π [ x ˜ 1 + x ˜ 0 2 π d φ - 1 1 d μ L I ( τ , μ , φ ) + Δ ( τ , μ , φ ) + g s I ( τ , μ , ϕ ) + g q I ( τ , μ , φ ) ] ,
T ^ Δ L I = Q I ,             Q I = 1 2 π ( g s I + g q I ) , I = R , F ,             F = C , Y ,
g s R ( λ , τ , μ , φ ) = 3 8 ( λ ) λ < λ d λ σ R ( λ , λ ) 0 2 π d φ - 1 1 d μ × 1 + 3 k ρ cos 2 Θ 1 + k ρ L s ( λ , τ , μ , φ ) ,
g s F ( λ , τ , μ , φ ) = 1 2 ( λ ) λ < λ d λ σ F ( λ , λ ) 0 2 π d φ × - 1 1 d μ L 0 s ( λ , μ , φ ) ,
g q R ( λ , τ , μ , φ ) = 3 8 ( λ ) λ < λ d λ σ R ( λ , λ ) 0 2 π d φ 0 1 d μ × 1 + 3 k ρ cos 2 Θ 1 + k ρ L 0 q ( λ , μ , φ ) × exp [ - τ ( λ ) μ ] ,
g q F ( λ , τ , μ , φ ) = 1 2 ( λ ) λ < λ d λ σ F ( λ , λ ) 0 2 π d φ 0 1 d μ × L 0 q ( λ , μ , φ ) exp [ - τ ( λ ) μ ] ,
cos Θ = μ μ + ( 1 - μ 2 ) 1 / 2 ( 1 - μ 2 ) 1 / 2 cos ( φ - φ ) .
β = 1 , 2 L ^ α β ( τ ) E β I ( τ ) = e α g I ( τ ) , e = [ 1 1 ] ,             α , β = d , u or 1 , 2 ,
E d I ( z = 0 ) = 0 ,             lim z E α I ( z ) = 0 ,             α = d , u .
g I ( λ , z ) = 1 2 ( λ ) λ < λ d λ σ I ( λ , λ ) { L 0 ( λ ) exp [ - ( λ ) z μ s ] + [ 2 - μ ¯ ( λ ) ] E d s ( λ , z ) + [ 2 + μ ¯ ( λ ) ] E u s ( λ , z ) } ,
E I ( τ ) = A I a exp ( - k τ ) + 0 G ˜ ( τ - τ ) g I ( τ ) d τ ,
E I = [ E 1 I E 2 I ] ,             a = [ 1 R ] ,             G ˜ ( τ ) = [ G 11 ( τ ) + G 12 ( τ ) G 21 ( τ ) + G 22 ( τ ) ] ,
β = 1 , 2 L ^ α β ( τ ) G β γ ( τ ) = δ α γ δ ( τ ) ,
G ( τ ) = [ 1 R 0 R R 0 R ] θ ( τ ) exp ( - k τ ) 1 - R 0 R + [ R 0 R R 0 R 1 ] θ ( - τ ) exp ( k 0 τ ) 1 - R 0 R ,
G ˜ ( τ ) = k 1 [ a k 2 θ ( τ ) exp ( - k τ ) + b θ ( - τ ) exp ( k 0 τ ) ] ,
R 0 = R 2 + μ ¯ 2 - μ ¯ , b = [ R 0 1 ] , k 1 = 1 + R 1 - R 0 R , k 2 = 1 + R 0 1 + R ,
E I ( τ ) = k 1 ( a { k 2 0 τ g I ( τ ) exp [ - k ( τ - τ ) ] d τ - R 0 exp ( - k τ ) 0 g I ( τ ) exp ( - k 0 τ ) d τ } + b τ g I ( τ ) exp [ k 0 ( τ - τ ) ] d τ ) .
E I ( λ , z ) = k 1 2 exp ( - ν z ) a ( k 2 D R - R 0 L R ) + b L N I ,
- d E I ( λ , z ) d z = k 1 2 exp ( - ν z ) { ν [ a ( k 2 D R - R 0 L R ) + b L N ] I - ( ν + ζ ) b L N I + ( b - k 2 a ) [ ( 1 + k D 3 ) × exp ( 2 r 1 ) + n exp ( 2 r 2 ) ] I } ,
l 1 = 1 / ( ζ + η ) ,             ζ = k 0 ,             l 2 = 1 / ( ζ + ν ) , r 1 = ( ν - η ) z / 2 ,             r 2 = ( ν - ν ) z / 2 ,             r 3 = r 2 - r 1 , D i = ( z / 2 ) [ 1 + exp ( 2 r i ) ] f τ ( r i ) ,             i = 1 , 2 , 3 , D R = D 1 + n D 2 + k S ,             n = μ s R s ( 2 + μ ¯ ) , S = ( z 2 / 4 ) { [ exp ( 2 r 1 ) + exp ( 2 r 2 ) ] f τ ( r 1 ) f τ ( r 3 ) - [ 1 + exp ( 2 r 2 ) ] f x y ( r 1 , r 2 ) } , k = 2 μ s R s Q s η [ μ s ( 4 - μ ¯ 2 ) - μ ¯ ] , L R = l 1 + l 2 ( n + k l 1 ) ,             L 1 = l 1 exp ( 2 r 1 ) , L N = L 1 + l 2 N K + L K ,             N K = ( n + k l 1 ) exp ( 2 r 2 ) , L K = k D 3 L 1 ,             f x y ( x , y ) = 1 y - x [ tanh ( x ) x - tanh ( y ) y ] .
k α = - d d z ln [ E α E ( z ) + I = R , C , Y E α I ( z ) ] = k α E + I = R , C , Y ( k α I - k α E ) ρ α I 1 + I = R , C , Y ρ α I ,             α = d , u ,
ρ d I = k 1 k 2 D R - R 0 L R + R 0 L N I 2 L 0 μ s [ exp ( 2 z s ) + R s η Q s m 1 D 0 ] , ρ u I = k 1 R ( k 2 D R - R 0 L R ) + L N I 2 L 0 μ s R s [ 1 + η Q s m 2 D 0 ] ,
{ k d I = - d d z ln E d I ( z ) = ν + ( R 0 - k 2 ) [ ( 1 + k D 3 ) exp ( 2 r 1 ) + n exp ( 2 r 2 ) ] - ( ν + ζ ) R 0 L N I k 2 D R - R 0 L R + R 0 L N I k u I = - d d z ln E u I ( z ) = ν + ( 1 - k 2 R ) [ ( 1 + k D 3 ) exp ( 2 r 1 ) + n exp ( 2 r 2 ) ] - ( ν + ζ ) L N I R ( k 2 D R - R 0 L R ) + L N I .
R E u ( z = 0 ) E d ( z = 0 ) = R s + I = R , C , Y δ R I ,
δ R I = 1 + R 2 L 0 μ s l 1 + l 2 ( n + k l 1 ) I .
γ = 1 - ω 0 μ ¯ ,             where μ ¯ = { 1 + 2 x + [ x ( 4 + 5 x ) ] 1 / 2 1 + x } 1 / 2 ,
x = b B a + b B ω 0 B 1 - ω 0 ( 1 - B ) .
R G d i r = 0.0001 + 0.3244 x + 0.1425 x 2 + 0.1308 x 3 ,
R G d i f f = 0.0003 + 0.3687 x + 0.1802 x 2 + 0.0740 x 3 .
R K M = [ 1 - ( 1 - x 2 ) 1 / 2 ] x .
R H = 1 - χ 1 + χ [ ( 1 + χ 2 ) 1 / 2 - χ ] 2 , where χ = [ 1 - x 1 + ( 3 + 2 2 ) x ] 1 / 2 .
σ R ( λ , λ ) = d ν ˜ d λ σ ν R ( v ˜ , ν ˜ ) = ν ˜ 2 k ν β 0 R ( ν ˜ ν ˜ 0 ) 4 f R ( ν ˜ , ν ˜ ) ,
f R ( ν ˜ , ν ˜ ) = k R i = 1 4 α i exp [ - ( ν ˜ - ν ˜ - Δ ν ˜ i ) 2 2 σ i 2 ] ,
b R ( λ ) = β 0 R ( 400 λ ) 4 ,             R ( λ ) = β 0 R - + f R ( ν ˜ , k ν λ ) ( ν ˜ ν ˜ 0 ) 4 d ν ˜ = β 0 R n = 0 4 γ n ( 400 λ ) n ,
σ C ( λ , λ ) = β 0 C a 0 C ( λ ) C C f C ( λ , λ ) ,
f C ( λ , λ ) = f e x C ( λ ) f em C ( λ ) , f em C ( λ ) = k C exp [ - ( λ - λ C 0 ) 2 2 σ C 2 ] ,
σ ν Y ( λ , λ ) = β 0 Y a 0 Y ( λ ) C Y f Y ( λ , λ ) ,
f Y ( λ , λ ) = f ex Y ( λ ) f em Y ( λ ) , f em Y ( λ ) = k Y exp [ - ( λ - λ Y 0 ) 2 2 σ Y 2 ] ,
f e x C ( λ ) = { h ex C ( λ / λ C 0 ) , λ 1 C λ λ 2 C 0 , elsewhere ,
a Fl ( λ ) = β 0 C a 0 C em C C f ex C ( λ ) + β 0 Y a 0 Y em C Y f ex Y ( λ ) ,
Fl ( λ ) = β 0 C a 0 C ex C C f em C ( λ ) + β 0 Y a 0 Y ex C Y f em Y ( λ ) ,
a 0 F ex = - a 0 F ( λ ) f ex F ( λ ) d λ , a 0 F em = - + a 0 F ( λ ) f em F ( λ ) d λ ,             F = C , Y .

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