Abstract

The influence of illumination by direct sunlight and the diffuse light of the sky on the apparent optical properties of seawater are studied. This study is based on the earlier self-consistent approach for solution of the radiative transfer equation. The resulting set of equations couples diffuse reflectance and diffuse attenuation coefficients and other apparent optical properties of the sea with inherent optical properties of seawater and parameters of illumination by the Sun and the sky. The resulting equations in their general form are valid for any possible values of absorption and backscattering coefficients.

© 1998 Optical Society of America

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References

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  1. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  2. C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).
  3. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the Sun zenith angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  4. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
    [CrossRef] [PubMed]
  5. V. I. Haltrin, “Propagation of light in sea depth,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Berlin, 1985), pp. 20–62 (in Russian).
  6. 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.
  7. V. I. Haltrin, V. A. Urdenko, “Propagation of light in the atmosphere and remote sensing of optical properties of seawater,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Berlin, 1985), pp. 63–90 (in Russian).
  8. V. I. Haltrin, G. W. Kattawar, “Effects of Raman scattering and fluorescence on apparent optical properties of seawater,” Rep. (Department of Physics, Texas A&M University, College Station, Texas, 1991).
  9. 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]
  10. V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
    [CrossRef]
  11. The approach of Ref. 10 and of this paper differs from that of Aas14 in that the values E1 and E2 in Eqs. (9) correspond to downward and upward irradiances by renormalized scattered components of light. In Ref. 14 irradiances Ed and Eu in two-flow equations (12) and (13) are total irradiances. In the current approach total irradiances are coupled with the renormalized irradiances E1 and E2 by the relationships (7) and (8). This means that the coefficients in Eqs. (9) should not be compared with the coefficients of Eqs. (12) and (13) by Aas.14 The comparable optical properties are only those that can be calculated with the total upward and downward irradiances Ed and Eu.
  12. V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 509–518.
  13. V. I. Haltrin, “An analytic Fournier–Forand scattering phase function as an alternative to the Henyey–Greenstein phase function in hydrologic optics,” in Proceedings of the 1998 IEEE International Geoscience and Remote Sensing Symposium, T. I. Stein, ed., IEEE Cat. No. 98CH36174 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. II, pp. 910–912.
  14. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  15. Depending on atmospheric turbidity and transmission of the seawater interface, the value q varies, approximately, between 0 and 10 (Ref. 1). The formula for q can be found, for example, in Refs. 7 and 16.
  16. V. I. Haltrin (V. I. Khalturin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 21, 452–457 (1985).
  17. V. I. Haltrin, “One-parameter model of seawater optical properties,” in Ocean Optics XIV, Kailua-Kona, Hawaii, 10–13 November 1998 (to be published).
  18. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]

1998 (1)

1993 (1)

1991 (1)

1989 (1)

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

1987 (1)

1985 (1)

V. I. Haltrin (V. I. Khalturin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 21, 452–457 (1985).

1977 (1)

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

Aas, E.

Gentili, B.

Gordon, H. R.

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

Haltrin, V. I.

V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
[CrossRef]

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 (V. I. Khalturin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 21, 452–457 (1985).

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

V. I. Haltrin, “One-parameter model of seawater optical properties,” in Ocean Optics XIV, Kailua-Kona, Hawaii, 10–13 November 1998 (to be published).

V. I. Haltrin, “An analytic Fournier–Forand scattering phase function as an alternative to the Henyey–Greenstein phase function in hydrologic optics,” in Proceedings of the 1998 IEEE International Geoscience and Remote Sensing Symposium, T. I. Stein, ed., IEEE Cat. No. 98CH36174 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. II, pp. 910–912.

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, V. A. Urdenko, “Propagation of light in the atmosphere and remote sensing of optical properties of seawater,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Berlin, 1985), pp. 63–90 (in Russian).

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

V. I. Haltrin, G. W. Kattawar, “Effects of Raman scattering and fluorescence on apparent optical properties of seawater,” Rep. (Department of Physics, Texas A&M University, College Station, Texas, 1991).

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

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, “Effects of Raman scattering and fluorescence on apparent optical properties of seawater,” Rep. (Department of Physics, Texas A&M University, College Station, Texas, 1991).

Mobley, C. D.

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

Morel, A.

Prieur, L.

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

Urdenko, V. A.

V. I. Haltrin, V. A. Urdenko, “Propagation of light in the atmosphere and remote sensing of optical properties of seawater,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Berlin, 1985), pp. 63–90 (in Russian).

Appl. Opt. (4)

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

V. I. Haltrin (V. I. Khalturin), “Self-consistent two-flux approximation to the theory of radiation transfer,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 21, 452–457 (1985).

Limnol. Oceanogr. (2)

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

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

Other (11)

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

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, V. A. Urdenko, “Propagation of light in the atmosphere and remote sensing of optical properties of seawater,” in Optical Remote Sensing of the Sea and the Influence of the Atmosphere, V. A. Urdenko, G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Berlin, 1985), pp. 63–90 (in Russian).

V. I. Haltrin, G. W. Kattawar, “Effects of Raman scattering and fluorescence on apparent optical properties of seawater,” Rep. (Department of Physics, Texas A&M University, College Station, Texas, 1991).

The approach of Ref. 10 and of this paper differs from that of Aas14 in that the values E1 and E2 in Eqs. (9) correspond to downward and upward irradiances by renormalized scattered components of light. In Ref. 14 irradiances Ed and Eu in two-flow equations (12) and (13) are total irradiances. In the current approach total irradiances are coupled with the renormalized irradiances E1 and E2 by the relationships (7) and (8). This means that the coefficients in Eqs. (9) should not be compared with the coefficients of Eqs. (12) and (13) by Aas.14 The comparable optical properties are only those that can be calculated with the total upward and downward irradiances Ed and Eu.

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

V. I. Haltrin, “An analytic Fournier–Forand scattering phase function as an alternative to the Henyey–Greenstein phase function in hydrologic optics,” in Proceedings of the 1998 IEEE International Geoscience and Remote Sensing Symposium, T. I. Stein, ed., IEEE Cat. No. 98CH36174 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), Vol. II, pp. 910–912.

Depending on atmospheric turbidity and transmission of the seawater interface, the value q varies, approximately, between 0 and 10 (Ref. 1). The formula for q can be found, for example, in Refs. 7 and 16.

V. I. Haltrin, “One-parameter model of seawater optical properties,” in Ocean Optics XIV, Kailua-Kona, Hawaii, 10–13 November 1998 (to be published).

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

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

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Equations (44)

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μ s = 1 - cos   h s n w 2 1 / 2 1 - 1 / n w 2 1 / 2 0.6656
L ˜ z ,   μ ,   ϕ = L z ,   μ ,   ϕ + L s z ,   μ ,   ϕ ,
L s z ,   μ ,   ϕ = L q μ ,   ϕ exp - α z / μ , 0 < μ 1 , 0 , - 1 μ 0 ,
L q μ ,   ϕ = E s δ μ - μ s δ ϕ ,
α = a + 2 b B < c ,
E 1 z = 0 2 π d ϕ   0 1   L z ,   μ ,   ϕ μ d μ , E 2 z = - 0 2 π d ϕ   - 1 0   L z ,   μ ,   ϕ μ d μ .
E d z = 0 2 π d ϕ   0 1   L ˜ z ,   μ ,   ϕ μ d μ = E 1 z + μ s E s exp - α z μ s ,
E u z = - 0 2 π d ϕ   - 1 0   L ˜ z ,   μ ,   ϕ μ d μ = E 2 z .
d d z + 2 - μ ¯ a + b B E 1 z - 2 + μ ¯ b B E 2 z = b B E s exp - α z / μ s , - 2 - μ ¯ b B E 1 z + - d d z + 2 + μ ¯ a + b B E 2 z = b B E s exp - α z / μ s ,
B = 0.5   - 1 0   p μ d μ ,
μ ¯ = 0 2 π d ϕ   - 1 1   L μ μ d μ 0 2 π d ϕ   - 1 1   L μ d μ = a a + 3 b B + b B 4 a + 9 b B 1 / 2 1 / 2 1 - g 1 + 2 g + g 4 + 5 g 1 / 2 1 / 2 ,
g = b B a + b B ω 0 B 1 - ω 0 + ω 0 B ,
E 1 0 = E d 0 .
E s = qE d 0 .
E i z = A i   exp - α z + C i   exp - α z / μ s ,   i = 1 ,   2 ,
- α = - a μ ¯ - 4 a a + 2 b B + μ ¯ 2 b B 2 1 / 2 - μ ¯ a + b B < 0
α 0 = μ ¯ a + b B + 4 a a + 2 b B + μ ¯ 2 b B 2 1 / 2 > 0 .
E 1 z = E d 0 exp - α z + C 1 exp - α z / μ s - exp - α z ,
E 2 z = E d 0 R   exp - α z + C 2 - R C 1 exp - α z + C 2 exp - α z / μ s - exp - α z ,
C 1 = α b B E s μ s Δ s 2 + μ ¯ μ s + 1 , C 2 = α b B E s μ s Δ s 2 - μ ¯ μ s - 1 ,
Δ s = α 0 + α μ s α - α μ s α 2 μ s 1 + μ ¯ μ s 4 - μ ¯ 2 × 1 μ 0 - 1 μ s ,
μ 0 = α α 1 + μ ¯ 2 μ ¯ 3 - μ ¯ 2 ,
C 2 - R C 1 = μ s E s R s ,
R = 1 - μ ¯ 1 + μ ¯ 2
R s = 1 - μ ¯ 2 1 + μ ¯ μ s 4 - μ ¯ 2
E d z = E d 0 exp - α z + μ s E s exp - α z / μ s + E s hR s 2 + μ ¯ μ s + 1 F s z exp - α z ,
E u z = E d 0 R   exp - α z + μ s E s   R s   exp - α z / μ s + E s hR s 2 - μ ¯ μ s - 1 F s z exp - α z ,
F s z = 1 - exp - α z 1 μ s - 1 μ 0 1 μ s - 1 μ 0 , μ s μ 0 , α z , μ s = μ 0 ;
h = 1 + μ ¯ 2 2 1 + μ ¯ 2
T z = 1 + q μ s z + hR s 2 + μ ¯ μ s + 1 F s z 1 + q μ s × exp - α z ,
z = exp - α z 1 μ s - 1 μ 0 ,     q = E s E d 0 .
R z = R + qR s μ s z + h 2 - μ ¯ μ s - 1 F s z 1 + q μ s z + hR s 2 + μ ¯ μ s + 1 F s z .
R = R + μ s qR s 1 + μ s q
R = E d 0 R + E s μ s R s E d 0 + E s μ s .
R | E s = 0 = R 1 - μ ¯ 1 + μ ¯ 2 ,
R | E d 0 = 0 = R s 1 - μ ¯ 2 1 + μ ¯ μ s 4 - μ ¯ 2 .
R = b B 4 a ,     b B a     1 .
R s = b B a + μ s 3 a - ab B , μ s = 1 - cos 2 h s / n w 2 1 / 2 ,     b B a     1 ,
R = k c b B a ,     b B a     1 ,
k c = 1 1 + q s 1 4 + q s 1 + μ s 3 - b B / a , q s = q μ s μ s E s E d 0 = E s 0 E d 0 ,
k d z = - 1 E d z d E d z d z ,     k u z = - 1 E u z d E u z d z ,
k d z = α 1 + q μ 0 z + hR s 2 + μ ¯ μ s + 1 Y s z 1 + q μ s z + hR s 2 + μ ¯ μ s + 1 F s z ,
k u z = α R + qR s μ 0 z + h 2 - μ ¯ μ s - 1 Y s z R + qR s μ s z + h 2 - μ ¯ μ s - 1 F s z ,
Y s z = μ 0   μ s 1 - ε z μ 0 - μ s - μ 0 ε z , μ s μ 0 , α z - μ 0 , μ s = μ o .

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