Abstract

A differential equation of a Riccati type for the diffuse reflection coefficient of a stratified sea is proposed. For a homogeneous sea with arbitrary inherent optical properties this equation is solved analytically. For an inhomogeneous sea it is solved approximately for any arbitrary stratification. The resulting equation expresses the diffuse reflection coefficient of the sea through vertical profiles of absorption and backscattering coefficients, bottom albedo, and sea depth. The results of calculations with this equation are compared with Monte Carlo computations. It was found that the precision of this approach is in the range of 15%.

© 1999 Optical Society of America

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References

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  1. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Media (Springer-Verlag, Berlin, 1991).
    [CrossRef]
  2. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  3. C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).
  4. C. D. Mobley, “Hydrolight 3.0 user’s guide,” (SRI International, Menlo Park, Calif., 1995).
  5. The approach outlined in Refs. 6 and 7 and in this paper differs from the approach of Aas22 in that the values Ed and Eu in Eqs. (2) correspond to downward and upward irradiances by renormalized components of light. In Ref. 22 irradiances Ed and Eu in two-flow equations (12) and (13) are total irradiances. This means that the coefficients in Eqs. (2) should not be compared with the coefficients of Eqs. (12) and (13) of Ref. 22.
  6. 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]
  7. 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]
  8. 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.
  9. V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).
  10. The left-hand side of Eqs. (4) were derived by the author from the experimental results published by Timofeyeva.9
  11. K. Schwarzschild, “Über das Gleichgewicht der Sonnenatmosphere,” Gottingen Nachrichten 41, 1–24 (1906).
  12. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  13. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–607 (1931).
  14. G. C. Pomraning, “An extension of the Eddington approximation,” J. Quant. Spectrosc. Radiat. Transfer 9, 407–422 (1969).
    [CrossRef]
  15. 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]
  16. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  17. H. R. Gordon, O. B. Brown, “Diffuse reflection of the ocean: some effects of vertical structure,” Appl. Opt. 14, 2892–2895 (1975).
    [CrossRef] [PubMed]
  18. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vols. 1 and 2.
  19. V. I. Haltrin, “Apparent optical properties of the sea illuminated by Sun and sky: case of the optically deep sea,” Appl. Opt. 37, 8336–8340 (1998).
    [CrossRef]
  20. The values of a and bB used here correspond to the 530-nm wavelength band. We should note that in the elastic radiative transfer the wavelength is a parameter. This means that use of values of a and bB at certain wavelengths to estimate possible errors does not restrict this theory to a certain wavelength.
  21. After the submission of this paper the resulting Eqs. (27)–(29) were tested with the Monte Carlo simulations with a modified23 J. T. O. Kirk’s code.24,25 The inherent optical properties were adopted from T. J. Petzold.8,26 The results of simulations confirm the conclusion of Section 4 that the precision of calculation with Eqs. (27)–(29) is in the range of 15%.
  22. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  23. V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” 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. 502–508.
  24. J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” (Commonwealth Scientic and Industrial Research Organization, Canberra, Australia, 1981).
  25. J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte-Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
    [CrossRef]
  26. T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

1998

1994

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte-Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

1993

1988

1987

1977

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

1975

1972

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

1969

G. C. Pomraning, “An extension of the Eddington approximation,” J. Quant. Spectrosc. Radiat. Transfer 9, 407–422 (1969).
[CrossRef]

1931

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

1906

K. Schwarzschild, “Über das Gleichgewicht der Sonnenatmosphere,” Gottingen Nachrichten 41, 1–24 (1906).

1905

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

Aas, E.

Brown, O. B.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vols. 1 and 2.

Gordon, H. R.

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, “Apparent optical properties of the sea illuminated by Sun and sky: case of the optically deep sea,” Appl. Opt. 37, 8336–8340 (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, “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, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” 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. 502–508.

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.

Ivanov, A. P.

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

Jerlov, N. G.

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

Katsev, I. L.

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

Kattawar, G. W.

Kirk, J. T. O.

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte-Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” (Commonwealth Scientic and Industrial Research Organization, Canberra, Australia, 1981).

Kubelka, P.

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

Mobley, C. D.

C. D. Mobley, “Hydrolight 3.0 user’s guide,” (SRI International, Menlo Park, Calif., 1995).

C. D. Mobley, Light and Water (Academic, San Diego, Calif., 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), Vols. 1 and 2.

Munk, F.

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

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).

Pomraning, G. C.

G. C. Pomraning, “An extension of the Eddington approximation,” J. Quant. Spectrosc. Radiat. Transfer 9, 407–422 (1969).
[CrossRef]

Prieur, L.

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

Schuster, A.

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

Schwarzschild, K.

K. Schwarzschild, “Über das Gleichgewicht der Sonnenatmosphere,” Gottingen Nachrichten 41, 1–24 (1906).

Timofeyeva, V. A.

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

Zege, E. P.

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

Appl. Opt.

Astrophys. J.

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

Gottingen Nachrichten

K. Schwarzschild, “Über das Gleichgewicht der Sonnenatmosphere,” Gottingen Nachrichten 41, 1–24 (1906).

Izv. Acad. Sci. USSR Atmos. Oceanic Phys.

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

J. Quant. Spectrosc. Radiat. Transfer

G. C. Pomraning, “An extension of the Eddington approximation,” J. Quant. Spectrosc. Radiat. Transfer 9, 407–422 (1969).
[CrossRef]

Limnol. Oceanogr.

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

J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte-Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
[CrossRef]

Z. Tech. Phys. (Leipzig)

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

Other

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 values of a and bB used here correspond to the 530-nm wavelength band. We should note that in the elastic radiative transfer the wavelength is a parameter. This means that use of values of a and bB at certain wavelengths to estimate possible errors does not restrict this theory to a certain wavelength.

After the submission of this paper the resulting Eqs. (27)–(29) were tested with the Monte Carlo simulations with a modified23 J. T. O. Kirk’s code.24,25 The inherent optical properties were adopted from T. J. Petzold.8,26 The results of simulations confirm the conclusion of Section 4 that the precision of calculation with Eqs. (27)–(29) is in the range of 15%.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vols. 1 and 2.

V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” 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. 502–508.

J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” (Commonwealth Scientic and Industrial Research Organization, Canberra, Australia, 1981).

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

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

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

C. D. Mobley, “Hydrolight 3.0 user’s guide,” (SRI International, Menlo Park, Calif., 1995).

The approach outlined in Refs. 6 and 7 and in this paper differs from the approach of Aas22 in that the values Ed and Eu in Eqs. (2) correspond to downward and upward irradiances by renormalized components of light. In Ref. 22 irradiances Ed and Eu in two-flow equations (12) and (13) are total irradiances. This means that the coefficients in Eqs. (2) should not be compared with the coefficients of Eqs. (12) and (13) of Ref. 22.

The left-hand side of Eqs. (4) were derived by the author from the experimental results published by Timofeyeva.9

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.

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Figures (1)

Fig. 1
Fig. 1

Diffuse reflectance R (2) of the two-layer sea for various values of parameter b B /a (numbers to the right-hand side of the curves). The solid curves represent values calculated with Eqs. (30)–(32), and the circles correspond to the values taken from Ref. 17.

Equations (32)

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R=k bBa, or R=k bBa+bB, bBa0.3,
ddz+2-μ¯a+bBEdz-2+μ¯bBEuz=0,  -2-μ¯bBEdz+-ddz+2+μ¯a+bB×Euz=0,
B=0.5 π/2π pcos θsin θdθ
Lμ=1-μ¯221-μ¯μ3, μ¯=-11 Lμμdμ-11 Lμdμ, μ=cos θ.
μ¯d=01 Lμμdμ01 Lμdμ=12-μ¯,
μ¯u=--10 Lμμdμ-10 Lμdμ=12+μ¯.
α=4aa+2bB+μ¯2bB21/2-μ¯a+bB.
μ¯=aa+3bB+bB4a+9bB1/21/21-g1+2g+g4+5g1/21/2,  g=bBa+bBBω01-ω0+Bω0,
α=c1-ω01-ω01-3B+Bω04-4ω0+9Bω01/21/2,
Euz=RzEdz.
dRzdz-4az+bBzRz=-bBz2-μ¯z+R2z2+μ¯z.
R=1-μ¯1+μ¯2a+3bB+bB4a+9bB1/21/2-aa+3bB+bB4a+9bB1/21/2+a2.
k=1+4μ¯2-μ¯41+μ¯4,  0.25k1.
Rz=R+ξz1+R0ξz,
R0=2+μ¯2-μ¯ R,
ξz=AB-R1-R0AB exp-νzB-z,
ν=2aμ¯1-μ¯21-g=a 7+2μ¯2-μ¯4μ¯3-μ¯2.
-dRzdz+4az+bBzRz=bBz2-μ¯z.
τz=4 0zaz+bBzdz.
dRτdτ+Rτ=Rτ,
Rτ=1-μ¯τ1+μ¯τ22-μ¯τ4bBτaτ+bBτ.
Rpτ= Gτ-τRτdτ.
Gτ=Hτexp-τ
dGτdτ+Gτ=δτ,
δτ=,τ=0,0,τ0,  -+ δτdτ=1,  Hτ=1,τ0,0,τ<0,  dHτdτ=δτ.
Rτ|z=zB=AB.
Rz=4 zzB Rzexp-4 zzaz+bBzdz×az+bBzdz+AB exp-4 zzBaz+bBzdz.
R=4 0zB Rzexp-4 0zaz+bBzdzaz+bBzdz+AB exp-4 0zBaz+bBzdz.
R=4 0 Rzaz+bBzexp-4 0zaz+bBzdzdz.
R2=R1+R2-R1exp-4ha1+bB1,
R1=1-μ¯11+μ¯12,  R2=1-μ¯21+μ¯22,
μ¯1μ¯a1, bB1,  μ¯2μ¯a2, bB2.

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