Abstract

The depth dependence for which the downward diffuse attenuation coefficient, the upward-to-downward plane irradiance ratio, the vertically upward radiance-to-downward plane irradiance ratio, and the mean cosine of the radiance depend negligibly on the surface incident illumination have been examined. The depths at which these coefficients approach to within a specified percent of their asymptotic values depends significantly on the characteristics of the incident illumination and on the inherent optical properties of the water. This information is useful when solving inverse ocean optics problems with a method for which the radiance is assumed to be approximately in the asymptotic regime.

© 2003 Optical Society of America

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References

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  1. R. A. Matzner, ed. Dictionary of Geophysics, Astrophysics, and Astronomy (CRC Press, Boca-Raton, Fla., 2001), p. 27.
  2. N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” (University of Copenhagen, Denmark, April1977).
  3. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  4. R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
    [CrossRef]
  5. R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
    [CrossRef]
  6. A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
    [CrossRef] [PubMed]
  7. L. K. Sundman, R. Sanchez, N. J. McCormick, “Ocean optical source estimation with widely spaced irradiance measurements,” Appl. Opt. 37, 3793–3803 (1998).
    [CrossRef]
  8. R. A. Leathers, N. J. McCormick, “Algorithms for ocean-bottom albedo determination from in-water natural-light measurements,” Appl. Opt. 38, 3199–3205 (1999).
    [CrossRef]
  9. H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).
  10. P. J. Flatau, J. Piskozub, J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Exp. 5, 120–124 (1999).
    [CrossRef]
  11. C. D. Mobley, L. K. Sundman, Hydrolight 4.1 (Sequoia Scientific, Inc., Redmond, Wash., 2000).
  12. C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 457–469 and 512–521.
  13. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7505 (1993).
    [CrossRef] [PubMed]
  14. H. R. Gordon, K. Ding, W. Gong, “Radiative transfer in the ocean: computations relating to the asymptotic and near-asymptotic daylight field,” Appl. Opt. 32, 1606–1619 (1993).
    [CrossRef] [PubMed]
  15. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  16. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  17. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
    [CrossRef]
  18. A. H. Hakim, B. D. Piening, N. J. McCormick, Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, are preparing a manuscript to be called “Near-asymptotic azimuthal angular dependence of ocean optical radiance.”

2003 (1)

2002 (1)

H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

1999 (3)

1998 (1)

1997 (1)

1996 (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

1993 (2)

1992 (1)

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

1941 (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Chandrasekhar, S.

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

Ding, K.

Flatau, P. J.

P. J. Flatau, J. Piskozub, J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Exp. 5, 120–124 (1999).
[CrossRef]

Gentili, B.

Gong, W.

Gordon, H. R.

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hakim, A. H.

A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
[CrossRef] [PubMed]

A. H. Hakim, B. D. Piening, N. J. McCormick, Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, are preparing a manuscript to be called “Near-asymptotic azimuthal angular dependence of ocean optical radiance.”

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Jin, Z.

Kattawar, G. W.

Leathers, R. A.

McCormick, N. J.

Mobley, C. D.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7505 (1993).
[CrossRef] [PubMed]

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 457–469 and 512–521.

C. D. Mobley, L. K. Sundman, Hydrolight 4.1 (Sequoia Scientific, Inc., Redmond, Wash., 2000).

Morel, A.

Piening, B. D.

A. H. Hakim, B. D. Piening, N. J. McCormick, Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, are preparing a manuscript to be called “Near-asymptotic azimuthal angular dependence of ocean optical radiance.”

Piskozub, J.

P. J. Flatau, J. Piskozub, J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Exp. 5, 120–124 (1999).
[CrossRef]

Reinersman, P.

Roesler, C. S.

Sanchez, R.

Stamnes, K.

Stavn, R. H.

Sundman, L. K.

Zaneveld, J. R. V.

P. J. Flatau, J. Piskozub, J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Exp. 5, 120–124 (1999).
[CrossRef]

Ann. Nucl. Energy (1)

N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996).
[CrossRef]

Appl. Opt. (7)

Astrophys. J. (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Limnol. Oceanogr. (1)

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

Oceanologia (1)

H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

Opt. Exp. (1)

P. J. Flatau, J. Piskozub, J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt. Exp. 5, 120–124 (1999).
[CrossRef]

Other (6)

C. D. Mobley, L. K. Sundman, Hydrolight 4.1 (Sequoia Scientific, Inc., Redmond, Wash., 2000).

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), pp. 457–469 and 512–521.

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

R. A. Matzner, ed. Dictionary of Geophysics, Astrophysics, and Astronomy (CRC Press, Boca-Raton, Fla., 2001), p. 27.

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” (University of Copenhagen, Denmark, April1977).

A. H. Hakim, B. D. Piening, N. J. McCormick, Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, are preparing a manuscript to be called “Near-asymptotic azimuthal angular dependence of ocean optical radiance.”

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Figures (9)

Fig. 1
Fig. 1

Downward diffuse attenuation coefficient asymptotic optical depth τ Kd versus single-scattering albedo ω0 for a black sky with solar illumination zenith angles of 0, 30, and 60 deg and a diffuse (diff) solar illumination, each with ε Kd = 3% and 5%.

Fig. 2
Fig. 2

Downward diffuse attenuation coefficient K d (τ) versus optical depth τ for a black sky with solar illumination zenith angles of 0, 30, and 60 deg for single-scattering albedo values of 0.3, 0.6, and 0.8.

Fig. 3
Fig. 3

Depth-dependent downward diffuse attenuation coefficient K d (τ) versus optical depth τ for a diffuse (diff) illumination for single-scattering albedo values of 0.3, 0.6, and 0.8.

Fig. 4
Fig. 4

Irradiance ratio asymptotic optical depth τ R versus single-scattering albedo ω0 for a black sky and with solar illumination zenith angles of 0, 30, and 60 deg and for a diffuse (diff) solar illumination, each with ε R = 3% and 5%.

Fig. 5
Fig. 5

Irradiance ratio R(τ) versus optical depth τ for a black sky and with solar illumination zenith angles of 0, 30, and 60 deg for single-scattering albedo values of 0.3, 0.6, and 0.8.

Fig. 6
Fig. 6

Irradiance ratio R(τ) versus optical depth τ for a diffuse (diff) illumination for single-scattering albedo values of 0.3, 0.6, and 0.8.

Fig. 7
Fig. 7

Radiance-irradiance ratio asymptotic optical depth τ RL versus single-scattering albedo ω0 for a black sky and solar illumination zenith angles of 0, 30, and 60 deg and for a diffuse (diff) solar illumination with ε RL = 1% and 3%.

Fig. 8
Fig. 8

Mean cosine of the radiance asymptotic depth τ μ̅ versus single-scattering albedo ω0 for a black sky and solar illumination zenith angles of 0, 30, and 60 deg and for a diffuse (diff) solar illumination with ε μ̅ = 1% and 3%.

Fig. 9
Fig. 9

Mean cosine of the radiance μ̅(τ) versus optical depth τ for a black sky and solar illumination zenith angle of 0 deg and for a diffuse (diff) solar illumination for single-scattering albedo values of 0.3, 0.6, and 0.8.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

μLz, μ, ψ/z+czLz, μ, ψ=bz02πdψ-11dμβ¯z, μ, ψμ, ϕ×Lz, μ, ψ, z0,
μLτ, μ, ψ/τ+Lτ, μ, ψ=ω002πdψ-11 ×dμβ˜μ, ψμ, ϕ×Lτ, μ, ψ, τ0.
Lτ, μ, ψ|τ largeLasτ, μ=Lasμexp-Kτ,
Kdτ=-dln Edτ/dτ,
Rτ=Euτ/Edτ,
Euτ=02πdψ-10dμ|μ|Lτ, μ, ψ,
Edτ=02πdψ01dμμLτ, μ, ψ;
RLτ=2πLuτ/Edτ2πLτ, -1/Edτ;
μ¯τ=Eτ/E0τ,
Eτ=02πdψ-11dμμLτ, μ, ψ, E0τ=02πdψ-11dμLτ, μ, ψ.
100KdτKdK-1εKd,
100RτRRas-1εR,
100RLτRLRL, as-1εRL,
100μ¯τμ¯μ¯as-1εμ¯,
AOPτ=AOPas+AOPτr-AOPτas×exp-Pτ-τr, τ>τr,

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