Abstract

The approach of ocean optical radiance to an approximate asymptotic dependence with increasing depth in spatially uniform waters is numerically examined for a variety of sea surface illumination conditions.

© 2004 Optical Society of America

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References

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  1. C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), Secs. 4.2, 5.7, and 9.6.
  2. B. D. Piening, N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003).
    [CrossRef] [PubMed]
  3. R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
    [CrossRef]
  4. L. K. Sundman, R. Sanchez, N. J. McCormick, “Ocean optical source estimation with widely spaced irradiance measurements,” Appl. Opt. 37, 3793–3803 (1998).
    [CrossRef]
  5. 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]
  6. 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]
  7. N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
    [CrossRef]
  8. A. H. Hakim, N. J. McCormick, “Ocean optics estimation for absorption, backscattering, and phase function parameters,” Appl. Opt. 42, 931–938 (2003).
    [CrossRef] [PubMed]
  9. C. D. Mobley, L. K. Sundman, Hydrolight 4.1 (Sequoia Scientific, Redmond, Wash., 2000).
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.
  11. I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
    [CrossRef]
  12. G. W. Kattawar, ed., Selected Papers on Multiple Scattering in Plane Parallel Atmospheres and Oceans: Methods, Vol. MS42 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1991).
  13. C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
    [CrossRef]
  14. M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
    [CrossRef]
  15. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  16. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).
  17. V. I. Haltrin, V. I. Mankovsky, “Analytical representation of experimental light scattering phase functions measure in seas, oceans and Lake Baykal,” proceedings of the 2002 IEEE International Geoscience and Remote Sensing Symposium, 24–28 June 2002, Toronto, Canada, available on CD ROM from the Geoscience and Remote Sensing Society, http://www.ewh.ieee.org/soc/grss/igarss.html .
  18. R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
    [CrossRef]
  19. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  20. 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, 7484–7504 (1994).
    [CrossRef]
  21. E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
    [CrossRef]
  22. J. R. V. Zaneveld, E. Boss, A. Barnard, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40, 1442–1449 (2001).
    [CrossRef]
  23. G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), App. E.

2003 (3)

B. D. Piening, N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003).
[CrossRef] [PubMed]

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

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

2001 (1)

2000 (1)

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[CrossRef]

1999 (2)

1998 (1)

1997 (1)

1994 (2)

1991 (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

1985 (1)

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

1979 (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

1977 (1)

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

1941 (1)

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

Barnard, A.

Boss, E.

Campos Velho, H. F.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Chalhoub, E. S.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.

Garcia, R. D. M.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

Gentili, B.

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.

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.

Kušcer, I.

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

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, 7484–7504 (1994).
[CrossRef]

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

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), Secs. 4.2, 5.7, and 9.6.

Morel, A.

Nakajima, T.

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

Piening, B. D.

Reinersman, P.

Roesler, C. S.

Sanchez, R.

Siewert, C. E.

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

Stamnes, K.

Stavn, R. H.

Sundman, L. K.

Tanaka, M.

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), App. E.

Vilhena, M. T.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

Zaneveld, J. R. V.

Appl. Opt. (9)

B. D. Piening, N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003).
[CrossRef] [PubMed]

R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8698 (1997).
[CrossRef]

L. K. Sundman, R. Sanchez, N. J. McCormick, “Ocean optical source estimation with widely spaced irradiance measurements,” Appl. Opt. 37, 3793–3803 (1998).
[CrossRef]

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]

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]

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

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

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, 7484–7504 (1994).
[CrossRef]

J. R. V. Zaneveld, E. Boss, A. Barnard, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40, 1442–1449 (2001).
[CrossRef]

Astrophys. J. (1)

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

J. Math. Phys. (1)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[CrossRef]

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Transp. Theory Stat. Phys. (3)

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, M. T. Vilhena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

Other (7)

G. W. Kattawar, ed., Selected Papers on Multiple Scattering in Plane Parallel Atmospheres and Oceans: Methods, Vol. MS42 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1991).

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

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.

C. D. Mobley, Light and Water Radiative Transfer in Natural Waters (Academic, New York, 1994), Secs. 4.2, 5.7, and 9.6.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 71–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972).

V. I. Haltrin, V. I. Mankovsky, “Analytical representation of experimental light scattering phase functions measure in seas, oceans and Lake Baykal,” proceedings of the 2002 IEEE International Geoscience and Remote Sensing Symposium, 24–28 June 2002, Toronto, Canada, available on CD ROM from the Geoscience and Remote Sensing Society, http://www.ewh.ieee.org/soc/grss/igarss.html .

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), App. E.

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Figures (5)

Fig. 1
Fig. 1

Largest three eigenvalues for the m = 0 mode of the transfer equation as a function of the albedo of single scattering for the Petzold phase function (continuous curves). Selected results for the HG phase function are also shown: crosses, ν 1 0; open circles, ν 2 0; pluses, ν 3 0.

Fig. 2
Fig. 2

Largest eigenvalues for the m = 0, 1, 2 modes of the transfer equation as a function of the albedo of single scattering for the Petzold phase function.

Fig. 3
Fig. 3

Azimuthal average percent difference metric of Eq. (8) as a function of optical depth for the HG phase function. Results with incident beam angle θ = 60°, solid curves; θ = 45°, dashed curves; θ = 30°, dashed–dotted curves.

Fig. 4
Fig. 4

Same as Fig. 3 but for the azimuthal maximum percent difference metric of Eq. (11).

Fig. 5
Fig. 5

Polar average percent difference metric of Eq. (12) as a function of optical depth for the HG phase function. Results with incident beam angle θ = 60°, solid curves; θ = 45°, dashed curves; θ = 30°, dashed–dotted curves.

Tables (3)

Tables Icon

Table 1 First 260 Expansion Coefficients for the Petzold Offshore-California Phase Function for Use in Computing pn from Eqs. (21) and (22)

Tables Icon

Table 2 Azimuthal Average Percent Difference Metric of Eq. (8) for Selected Optical Depths for the Petzold Phase Function with Incident Beam Directions of 30°, 45°, and 60°

Tables Icon

Table 3 Polar Average Percent Difference Metric of Eq. (12) for Selected Optical Depths for the Petzold Phase Function with Incident Beam Directions of 30°, 45°, and 60°

Equations (30)

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μ Lτ, μ, φτ+Lτ, μ, φ=ϖ 02πdφ -11 ×dμβ˜μ, φμ, φ×Lτ, μ, φ, τ0,
L0-, μ, φ=Fiμ, φ, 0μ1, i=1, 2,
F1μ, φ=Ed0-δμ-μ0,aδφ-φ0/μ0,a, 0μ0,a1,
F2μ, φ=π-1Ed0-, 0μ1.
Lτ, μ, φ=Lcτ, μ, φ+Luτ, μ, φ.
Lcτ, μ, φ=m=0M2-δm,0Lcmτ, μcosmφ-φ0,
Lcmτ, μ=1/2π02π Lcτ, μ, φcosmφ-φ0dφ,
100 maxφ0,2π1-LdτLdτ, φφ,aτ,
Ldτ=01 Lτ, μdμ,
Ldτ, φ=01 Lτ, μ, φdμ.
1001-minφ0,2πLτ, μ0, φmaxφ0,2πLτ, μ0, φφ,mτ,
100 maxμ-1,11-ϕ0ν10, μ/g1+ν10Lτ, μ/Edτμ,aτ,
g1+ν10=01 μϕ0ν10, μdμ,
Edτ=01 μLτ, μdμ.
β˜cos Θ=n=0N β˜nPncos Θ,
β˜n=4π-12n+1fn,
β˜cos Θ=expn=06 cnΘn/2
β˜cos Θ=2αδ1-cos Θ+1-αpcos Θ,
ϖ=ϖ1-α1-ϖα,
τ=τ1-ϖα.
pn=fn-α1-ϖα.
α=fM.
L0+, μ, φ=02πdφ 01dμL0+, -μ, φ×Rwwμ, μ, φ, φ+02πdφ×01dμL0-, μ, φ×Tawμ, μ, φ, φ, 0μ1,
Rwwμ, μ, φ, φ=rwwμδμ-μδφ-φ,μ1-1/n21/2,
=δμ-μδφ-φ,μ<1-1/n21/2,
Tawμ, μ, φ, φ=n2tawgμδgμ-μδφ-φ,
rwwμ=12μ-ngμμ+ngμ2+nμ-gμnμ+gμ2,
tawμ=2nμfμ1μ+nfμ2+1nμ+fμ2,
fμ=1-1-μ2/n21/2,
gμ=1-n21-μ21/2.

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