Abstract

A full multiple-scattering algorithm for inverting profiles of the upwelling and downwelling irradiances to yield profiles of the absorption and backscattering coefficients in a vertically stratified water body is described and tested with simulated data. The algorithm does not require knowledge of the scattering phase function of the medium. The results are better the closer the phase function assumed in the retrievals is to the true phase function, although excellent retrievals of the absorption coefficient can still be obtained with an inaccurate phase function. Simulations show that the algorithm is capable of determining the vertical structure of a stratified water body and usually provides the absorption coefficient profile with an error ≲2% and the backscattering coefficient profile with an error ≲10%, as long as the spacing between pseudodata samples is sufficiently small that the necessary derivatives of the irradiances can be accurately computed. The performance is only slightly degraded when the upwelling radiance (nadir viewing) is substituted for the upwelling irradiance.

© 1998 Optical Society of America

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References

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  1. H. R. Gordon, G. C. Boynton, “Radiance–irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]
  2. C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, New York, 1994).
  3. 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]
  4. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  5. H. R. Gordon, W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
    [CrossRef] [PubMed]
  6. H. R. Gordon, “Remote sensing of optical properties in continuously stratified waters,” Appl. Opt. 17, 1893–1897 (1978).
    [CrossRef] [PubMed]
  7. H. R. Gordon, D. K. Clark, “Remote sensing optical properties of a stratified ocean: an improved interpretation,” Appl. Opt. 19, 3428–3430 (1980).
    [CrossRef] [PubMed]
  8. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).
  9. 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 (1993).
    [CrossRef] [PubMed]
  10. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  11. J. Dera, H. R. Gordon, “Light field fluctuations in the photic zone,” Limnol. Oceanogr. 13, 697–699 (1968).
    [CrossRef]
  12. D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).
  13. H. R. Gordon, “The sensitivity of radiative transfer to small-angle scattering in the ocean: a quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993).
    [CrossRef] [PubMed]
  14. G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
    [CrossRef]
  15. H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [CrossRef]
  16. Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
    [CrossRef]

1997 (1)

1995 (1)

Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
[CrossRef]

1993 (2)

1989 (2)

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

1980 (1)

1978 (1)

1977 (1)

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

1975 (3)

1968 (1)

J. Dera, H. R. Gordon, “Light field fluctuations in the photic zone,” Limnol. Oceanogr. 13, 697–699 (1968).
[CrossRef]

Boynton, G. C.

Brody, E. A.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Brown, O. B.

Clark, D. K.

Davis, C. O.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Dera, J.

J. Dera, H. R. Gordon, “Light field fluctuations in the photic zone,” Limnol. Oceanogr. 13, 697–699 (1968).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).

Ge, Y.

Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
[CrossRef]

Gentili, B.

Gordon, H. R.

H. R. Gordon, G. C. Boynton, “Radiance–irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
[CrossRef] [PubMed]

Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
[CrossRef]

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 (1993).
[CrossRef] [PubMed]

H. R. Gordon, “The sensitivity of radiative transfer to small-angle scattering in the ocean: a quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993).
[CrossRef] [PubMed]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

H. R. Gordon, D. K. Clark, “Remote sensing optical properties of a stratified ocean: an improved interpretation,” Appl. Opt. 19, 3428–3430 (1980).
[CrossRef] [PubMed]

H. R. Gordon, “Remote sensing of optical properties in continuously stratified waters,” Appl. Opt. 17, 1893–1897 (1978).
[CrossRef] [PubMed]

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]

H. R. Gordon, W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
[CrossRef] [PubMed]

J. Dera, H. R. Gordon, “Light field fluctuations in the photic zone,” Limnol. Oceanogr. 13, 697–699 (1968).
[CrossRef]

Hooker, S. B.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Jacobs, M. M.

Jin, Z.

Kattawar, G. W.

Konnoff, D. A.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

McCluney, W. R.

Mobley, C. D.

Morel, A.

Mueller, J. L.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

O’Brien, M. C.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).

Prieur, L.

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

Reinersman, P.

Rhea, W. G.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Siegal, D. A.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Sorensen, J. C.

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

Stamnes, K.

Stavn, R. H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).

Voss, K. J.

Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
[CrossRef]

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Zibordi, G.

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Appl. Opt. (7)

J. Geophys. Res. (1)

Y. Ge, K. J. Voss, H. R. Gordon, “In situ measurements of inelastic scattering in Monterey Bay using solar Fraunhofer lines,” J. Geophys. Res. 100, 13,227–13,236 (1995).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Limnol. Oceanogr. (3)

J. Dera, H. R. Gordon, “Light field fluctuations in the photic zone,” Limnol. Oceanogr. 13, 697–699 (1968).
[CrossRef]

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

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?,” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

Remote Sensing Environ. (1)

G. Zibordi, K. J. Voss, “Geometrical and spectral distribution of sky radiance: comparison between simulations and field measurements,” Remote Sensing Environ. 27, 343–358 (1989).
[CrossRef]

Other (3)

C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, New York, 1994).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in FORTRAN (Cambridge University, Cambridge, England, 1992).

D. A. Siegal, M. C. O’Brien, J. C. Sorensen, D. A. Konnoff, E. A. Brody, J. L. Mueller, C. O. Davis, W. G. Rhea, S. B. Hooker, “Results of the SeaWiFS Data Analysis Round-Robin (DARR-94),” Vol. 26 of SeaWiFS Technical Report Series NASA Tech. Memor. 104566 (July1994).

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Figures (6)

Fig. 1
Fig. 1

R(z) for a medium in which the a(z) = 0.1 m-1 and b(z) = 0.9 m-1. The filled circles are the values of R(z) introduced into the algorithm as data (the pseudodata). The solid curve joins the points of the value of R(z) computed with the values of a(z) and b b (z) that were retrieved by the algorithm: (a) θ0 = 0, (b) θ0 = 60°.

Fig. 2
Fig. 2

Error in the recovered values of (a) a(z) and (b) b b (z). The solid circles represent the true values, whereas the solid curves and broken curves join the errors in the recovered values for θ0 = 0 and 60°, respectively.

Fig. 3
Fig. 3

Retrieval results for the case aub2 in Table 2. (a) R(z) pseudodata (solid circles) and reconstructed values by use of a(z) and b b (z) retrieved with the correct (solid curve) and an incorrect (dashed curve) phase function. (b) Percent error in a(z) retrieved with the correct (solid curve) and an incorrect (dashed curve) phase function. (c) Percent error in b b (z) retrieved with the correct (solid curve) and an incorrect (dashed curve) phase function.

Fig. 4
Fig. 4

Profiles of quantities retrieved in the presence of a thin absorbing layer at z = 2 m. Filled circles are the true values, the dashed curves refer to retrievals when pseudodata at z = 2 m are missing, and the solid curves are for retrievals made when pseudodata at z = 2 m are present: (a) absorption coefficient, (b) backscattering coefficient, (c) downwelling irradiance, (d) irradiance reflectance.

Fig. 5
Fig. 5

Variation of the true phase function for the medium in Section 5 from the smallest (dashed curve) to the largest (solid curve) value of C. The dotted curves correspond to the HG phase functions for g = 0.80, 0.85, 0.90, and 0.95 that were assumed in the retrievals. P(Θ) at large Θ is smaller for larger values of g.

Fig. 6
Fig. 6

Retrievals of a(z) and b b (z) for a situation in which the scattering phase function of the medium varies strongly with depth. Filled circles are the exact values, solid curves are retrievals by use of g = 0.80, and dashed curves are retrievals by use of g = 0.90: (a) absorption coefficient, (b) backscattering coefficient, (c) percent error in the absorption coefficient, (d) percent error in the backscattering coefficient.

Tables (5)

Tables Icon

Table 1 Summary of the Average and Maximum Errors in Percent Obtained for a(z) and bb(z) for a Homogeneous Mediuma

Tables Icon

Table 2 Gaussian Profile Parameters and Identification Code

Tables Icon

Table 3 Depth Averages of the Absolute Error in a(z) and bb(z) (percent) When the Correct Phase Function is Used in the Retrievalsa

Tables Icon

Table 4 Depth Averages of the Absolute Error in a(z) and bb(z) (percent) When an Incorrect Phase Function is Used in the Retrievalsa

Tables Icon

Table 5 Depth-Averaged Absolute Error [Eq. (6)] in Percent Obtained for a(z) and bb(z)a

Equations (18)

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a z = μ ¯ z K v z ,
b b z = 3 R z a z
b b ˜ = 2 π   π / 2 π   P Θ sin   Θ d Θ .
δ n = 1 2 N i = 1 N   | ln E d n z i - ln E d z i | + 1 2 N × i = 1 N   | ln E u n z i - ln E u z i | ,
R 0 - X 0 - 3 ,
X 0 - 0 z 90   X z g z d z 0 z 90   g z d z , g z = E d z / E d 0 2 , X z = b b z / a z .
R z X z 3 ,
X z = z z 90   X z g z ,   z d z z z 90   g z ,   z d z , g z ,   z = E d z / E d z 2 ,
X z = 3 R z - d R z d z z z max d z E d z E d z 2 .
a 0 z = μ 0 K v m z ,   b b 0 z = a 0 z X m z ,
b 0 z = b b 0 z b b ˜ z .
a i z = μ ¯ i - 1 z K v m z ,
Δ X i z = X m z - X i z .
Δ b b i z = Δ X i z a i z ;
b b i z = b b i - 1 z + f Δ b b i z ,
E u m z = Q i - 1 z L u m z ,
Δ a 1 N i = 1 N a retrieved z i - a true z i a true z i ,
a z = a 0 + a 1 exp - z - z a 2 2 σ a 2 , b b z = b 0 + b 1 exp - z - z b 2 2 σ b 2 ,

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