Abstract

We develop two algorithms for determining two inherent optical properties (IOPs) from radiometric measurements in vertically homogeneous waters. The first algorithm is for estimation of the ratio of the backscattering to absorption coefficients from measurements of only the vertically upward radiance and the downward planar irradiance at depths where the light field is in the asymptotic regime. The second algorithm enables estimation of the absorption coefficient from measurement of the diffuse attenuation coefficient in the asymptotic regime after use of the first algorithm. Multiplication of the two estimates leads to an estimate for the backscattering coefficient. The algorithms, based upon the use of a simplified phase function and the asymptotic eigenmode, are shown to potentially provide good starting conditions for iteratively determining the absorption and backscattering coefficients of a wide variety of waters. The uncertainty in the estimates defines a subspace for IOPs that may reduce ambiguity in such iterative solutions. Because of the ease of estimating the backscattering to absorption ratio from in-water measurements, this IOP deserves further investigation as a proxy for biogeochemical quantities in the open ocean.

© 2011 Optical Society of America

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References

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  27. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 194–201 (1994).
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  35. C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express, submitted (2011).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  45. L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981).
    [Crossref]
  46. A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
    [Crossref]
  47. M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res I 50, 1537–1549 (2003).

2009 (1)

G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009).
[Crossref]

2008 (1)

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

2007 (1)

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?,” J. Geophys. Res. 112, C03004 (2007).
[Crossref]

2004 (1)

2003 (2)

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

M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res I 50, 1537–1549 (2003).

2002 (4)

H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002).
[Crossref]

S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global scale applications,” Appl. Opt. 41, 2705–2714 (2002).
[Crossref] [PubMed]

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

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

2000 (3)

1999 (2)

1997 (2)

1996 (2)

Q. Min and L. C. Harrison , “An adjoint formulation of the radiative transfer method,” J. Geophys. Res. 101, 1635–1640 (1996).
[Crossref]

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

1992 (2)

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

N. J. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

1991 (1)

A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[Crossref]

1989 (1)

J. R. V. Zaneveld, “An asymptotic closure theory of irradiance in the sea and its inversion to obtain the vertical structure of inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[Crossref]

1983 (1)

B. Efron and G. Gong, “A leisurely look at the bootstrap, the jackknife, and cross-validation,” Am. Stat. 37, 36–48 (1983).
[Crossref]

1981 (1)

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981).
[Crossref]

1977 (1)

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

1976 (1)

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[Crossref]

1975 (1)

1973 (2)

Babin, M.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

Behrenfeld, M. J.

G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009).
[Crossref]

M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res I 50, 1537–1549 (2003).

Bell, G. I.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand-Reinhold, 1970).

Boss, E.

G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009).
[Crossref]

M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res I 50, 1537–1549 (2003).

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

Boynton, G. C.

Brown, O. B.

Carder, K. L.

T. G. Peacock, K. L. Carder, C. O. Davis, and R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 303–319 (1990).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chami, M.

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?,” J. Geophys. Res. 112, C03004 (2007).
[Crossref]

Claustre, H.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

DallOlmo, G.

G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009).
[Crossref]

Davis, A. B.

A. B. Davis and Y. Knyazikhin, “A Primer in Three-Dimensional Radiative Transfer,” in Three-Dimensional Radiative Transfer in the Cloudy Atmosphere, A. Davis and A. Marshak, eds. (Springer-Verlag, 2005), pp. 153–242.
[Crossref]

Davis, C. O.

T. G. Peacock, K. L. Carder, C. O. Davis, and R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 303–319 (1990).

Defoin-Platel, M.

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?,” J. Geophys. Res. 112, C03004 (2007).
[Crossref]

Deschamps, P.

H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002).
[Crossref]

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Efron, B.

B. Efron and G. Gong, “A leisurely look at the bootstrap, the jackknife, and cross-validation,” Am. Stat. 37, 36–48 (1983).
[Crossref]

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 194–201 (1994).

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 194–201 (1994).

Franz, B. A.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

Frouin, R.

H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002).
[Crossref]

Glasstone, S.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand-Reinhold, 1970).

Gong, G.

B. Efron and G. Gong, “A leisurely look at the bootstrap, the jackknife, and cross-validation,” Am. Stat. 37, 36–48 (1983).
[Crossref]

Gordon, H.

Gordon, H. R.

Gray, D.

Harrison, L. C.

Q. Min and L. C. Harrison , “An adjoint formulation of the radiative transfer method,” J. Geophys. Res. 101, 1635–1640 (1996).
[Crossref]

Jacobs, M. M.

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[Crossref]

Kaczmarek, S.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

Kitchen, J. C.

J. R. V. Zaneveld, J. C. Kitchen, and C. C. Moore, “Scattering error correction of reflecting tube absorption meter,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 44–55 (1994).

Knyazikhin, Y.

A. B. Davis and Y. Knyazikhin, “A Primer in Three-Dimensional Radiative Transfer,” in Three-Dimensional Radiative Transfer in the Cloudy Atmosphere, A. Davis and A. Marshak, eds. (Springer-Verlag, 2005), pp. 153–242.
[Crossref]

Kušcer, I.

N. J. McCormick and I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology7, E. J. Henley and J. Lewins, eds. (Academic, 1973), pp. 181–282.

Leathers, R. A.

Lewis, M. R.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

Loisel, H.

H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002).
[Crossref]

H. Loisel and D. Stramski, “Estimation of inherent optical properties of natural waters from the irradiance attenuation coefficient and reflectance in the presence of Raman scattering,” Appl. Opt. 39, 3001–3011 (2000).
[Crossref]

Maritorena, S.

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

McCormick, N. J.

N. J. McCormick, “Analytic inverse radiative transfer equations for atmospheric and hydrologic optics,” J. Opt. Soc. Am. A 21, 1009–1017 (2004).
[Crossref]

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

R. A. Leathers, C. S. Roesler, and N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[Crossref]

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

N. J. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

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

N. J. McCormick and I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology7, E. J. Henley and J. Lewins, eds. (Academic, 1973), pp. 181–282.

Min, Q.

Q. Min and L. C. Harrison , “An adjoint formulation of the radiative transfer method,” J. Geophys. Res. 101, 1635–1640 (1996).
[Crossref]

Mitchell, B. G.

M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

Mobley, C. D.

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
[Crossref] [PubMed]

M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

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

C. D. Mobley and L. K. Sundman, Hydrolight 5.0, Ecolight 5.0 Technical Documentation (Sequoia Scientific, Inc., 2008).

C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express, submitted (2011).
[Crossref] [PubMed]

Moore, C. C.

J. R. V. Zaneveld, J. C. Kitchen, and C. C. Moore, “Scattering error correction of reflecting tube absorption meter,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 44–55 (1994).

Morel, A.

A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[Crossref]

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

Nicolas, J.

H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002).
[Crossref]

Peacock, T. G.

T. G. Peacock, K. L. Carder, C. O. Davis, and R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 303–319 (1990).

Pegau, W. S.

Peterson, A. R.

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep. SIO 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972), pp. 1–78.

Piening, B. D.

Prieur, L.

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981).
[Crossref]

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

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[Crossref]

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Röttgers, R.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

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L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981).
[Crossref]

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D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

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[Crossref]

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[Crossref]

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D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

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[Crossref]

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D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
[Crossref]

M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley, “Estimation of the absorption and backscattering coefficients from in-water radiometric measurements,” Limnol. Oceanogr. 45, 628–641 (2000).
[Crossref]

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D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
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Biogeosciences (2)

G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009).
[Crossref]

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008).
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Figures (5)

Fig. 1
Fig. 1 An IOP and a radiometric profile from the NAB08 dataset. (A) Typical IOPs were nearly homogeneous in the lit portion of the water column. (B) Radiometric profiles of Lu and Ed at 488 nm show little curvature until 40 m, another indication of vertically homogeneous water.

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Fig. 2
Fig. 2 Estimates of F using rrs = Lu/Ed and IOP measurements from the NAB08 dataset. A) Values of fitted parameter F at 488 nm for all training set combinations (black x’s) and the linear fit for each value of k. Color indicates number of profiles k used in the training set. (B) fitted parameter F at selected wavelengths (color), created by linear fit for all k = 6 training set combinations.

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Fig. 3
Fig. 3 (A) Matchups of estimated and measured bb/a values show convergence towards 1:1 line (black dashed line) as k increases; each colored dot represents a single matchup. (B) Depth-averaged absolute error for each validation set ( | ψ i | ¯ , colored dots) remains variable irrespective of the number of profiles k used to determine F. The overall average absolute error | ψ | ¯ for the algorithm (black dashed line) decreases most from k = 1 to 4. (C) Similarly, relative error depth-averaged for each validation set ( ψ i ¯ , colored dots) shows variability but little average bias ψ̄ (black dashed line). (D) Absolute error averaged across all validation set combinations at each depth | ψ ( z j ) | ¯ decreases markedly in deep water.

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Fig. 4
Fig. 4 Values of absorption a (A) and average error (B–C) at 488 nm, estimated from F, bb/a, ν0, and Kd (zas), using Eq. (41), and (D–F) estimated from bb/a and measurements of bb(470) and bb(700) using Eq. (44). Colors indicate the number of profiles k used to determine the value of F; remaining profiles are used to estimate bb/a, Kd (zas) and a.

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Fig. 5
Fig. 5 Values of backscattering bb (A) and average error (B) and (C) at 488 nm, estimated from bb/a, F, and Kd (zas). Colors indicate the number of profiles k used to determine the value of F; remaining profiles are used to estimate bb/a, Kd (zas) and a.

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Tables (6)

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Table 1 Example training and validation sets of NAB08 calibration profiles for k = 1,2 used for cross validation.

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Table 2 Average relative error ψ̄ and absolute error | ψ | ¯ (in percent) for estimates of bb/a from radiative transfer simulations at 490 nm. Chl in mg m−3. Parenthetical values indicate the range of | ψ | ¯ over all depths.

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Table 3 Slope, intercept and R2 for fitted parameter F = p1(bb/a) + p2 for AC-9 wavelengths ≤ 532 nm with a linear least squares fit using all k = 6 training set combinations.

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Table 4 Relative and absolute error (in percent) for estimates of bb/a using different numbers of profiles k in the training set for fitted parameter F

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Table 5 Relative and absolute error (in percent) for estimates of a using Eqs. (41) and (44) using different numbers of profiles k in the training set for fitted parameter F

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Table 6 Relative and absolute error (in percent) for estimates of bb using different numbers of profiles k in the training set for fitted parameter F

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Equations (58)

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μ ∂ ∂ z L ( z , μ , φ ) + c L ( z , μ , φ ) = b ∫ 0 2 π ∫ − 1 1 β ˜ ( μ ′ , φ ′ , μ , φ ) L ( z , μ ′ , φ ′ ) d μ ′ d φ ′ , z ≥ 0 .
L ( z , μ ) = ( 2 π ) − 1 ∫ 0 2 π L ( z , μ , φ ) d φ
μ ∂ ∂ z L ( z , μ ) + c L ( z , μ ) = b ∫ − 1 1 β ˜ ( μ ′ , μ ) L ( z , μ ′ ) d μ ′ ,                 z ≥ 0 ,
β ˜ ( μ ′ , μ ) = ∫ 0 2 π β ˜ ( μ ′ , φ ′ , μ , φ = 0 ) d φ ′ .
E d ( z m ) = 2 π ∫ 0 1 μ L ( z m , μ ) d μ .
r r s ( z m ) = L u ( z m ) / E d ( z m ) .
β ˜ ( μ ′ , μ ) = 1 2 [ b ˜ b F + 2 ( 1 − b ˜ b F ) δ ( μ ′ − μ ) ]
∫ − 1 1 β ˜ ( μ ′ , μ ) d μ = 1 .
τ = ( c − b + b b ˜ b / F ) z = a z [ 1 + b b / a F ] ,
μ ∂ ∂ τ L ( τ , μ ) + L ( τ , μ ) = ϖ 2 ∫ − 1 1 L ( τ , μ ′ ) d μ ′ ,                 τ ≥ 0
ϖ = b b / a b b / a + F .
ϖ 1 − ϖ = b b / a F .
d d τ ∫ − 1 1 μ 2 L ( τ , μ ) L ( τ , − μ ) d μ = − ϖ ∫ − 1 1 μ L ( τ , μ ) d μ ∫ − 1 1 L ( τ , μ ′ ) d μ ′ .
∫ − 1 1 L ( τ , μ ) d μ = − ( 1 − ϖ ) − 1 d d τ ∫ − 1 1 μ L ( τ , μ ) d μ .
d G ( τ ) d τ = d d τ [ 4 ∫ 0 1 μ 2 L ( τ , μ ) L ( τ , − μ ) d μ − ϖ 1 − ϖ ( ∫ − 1 1 μ L ( τ , μ ) d μ ) 2 ] = 0 .
ϖ 1 − ϖ [ ∫ − 1 1 μ L ( z m , μ ) d μ ] 2 = 4 ∫ 0 1 μ 2 L ( z m , μ ) L ( z m , − μ ) d μ .
b b / a F [ 1 − 2 π ∫ 0 1 μ L ( z m , − μ ) E d ( z m ) d μ ] 2 = 16 π 2 ∫ 0 1 μ L ( z m , μ ) E d ( z m ) μ L ( z m , − μ ) E d ( z m ) d μ .
Λ ( z m ) = 2 π ∫ 0 1 μ L ( z m , − μ ) L u ( z m ) d μ = E u ( z m ) L u ( z m ) ,
Ω ( z m ) = 16 π 2 ∫ 0 1 μ 2 L ( z m , μ ) E d ( z m ) L ( z m , − μ ) L u ( z m ) d μ ,
E u ( z m ) = 2 π ∫ 0 1 μ L ( z m , − μ ) d μ .
b b / a F = Ω ( z m ) r r s ( z m ) [ 1 − Λ ( z m ) r r s ( z m ) ] 2 .
b b / a F = Ω ( z a s ) r r s ( z m ) [ 1 − Λ ( z a s ) r r s ( z m ) ] 2
Λ ( z a s ) = 2 π ∫ 0 1 μ L ( z a s , − μ ) L u ( z a s ) d μ = E u ( z a s ) L u ( z a s )
Ω ( z a s ) = 16 π 2 ∫ 0 1 μ 2 L ( z a s , μ ) L ( z a s , − μ ) d μ E d ( z a s ) L u ( z a s ) .
L ( τ a s , μ ) = A ( ν 0 ) ϕ ( ν 0 , μ ) exp ( − τ a s / ν 0 ) ,                       − 1 ≤ μ ≤ 1 ,
L ( z a s , μ ) = A ( ν 0 ) ϕ ( ν 0 , μ ) exp { − [ 1 + b b / ( a F ) ] a z a s / ν 0 } ,                 − 1 ≤ μ ≤ 1 .
( 1 − μ / ν 0 ) ϕ ( ν 0 , μ ) = ( ϖ / 2 ) ∫ − 1 1 ϕ ( ν 0 , μ ′ ) d μ ′ .
∫ − 1 1 ϕ ( ν 0 , μ ) d μ = 1
ϕ ( ν 0 , μ ) = ϖ ν 0 / 2 ν 0 − μ ,               − 1 ≤ μ ≤ 1 .
1 − ϖ ν 0 2 ln ( ν 0 + 1 ν 0 − 1 ) = 0 .
∫ − 1 1 μ ϕ ( ν 0 , μ ) d μ = ν 0 ( 1 − ϖ ) = ν 0 [ 1 + b b / ( a F ) ] − 1 .
Λ ( z a s ) = 2 π ( ν 0 + 1 ) ∫ 0 1 μ ( ν 0 + μ ) − 1 d μ = 2 π ( ν 0 + 1 ) [ 1 − ν 0 ln ( ν 0 + 1 ν 0 ) ] .
Ω ( z a s ) = 16 π 2 ( ν 0 + 1 ) ∫ 0 1 μ 2 ( ν 0 2 − μ 2 ) − 1 d μ ∫ 0 1 μ ( ν 0 − μ ) − 1 d μ = 8 π 2 ( ν 0 + 1 ) [ − 1 + ν 0 ln ( ν 0 ν 0 − 1 ) ] − 1 ∫ − 1 1 μ ( ν 0 − μ ) − 1 d μ = F b b / a 16 π 2 ( ν 0 + 1 ) [ − 1 + ν 0 ln ( ν 0 ν 0 − 1 ) ] − 1 ,
a = K E ( z m ) μ ¯ ( z m ) ,
K E ( z m ) = − d [ ln ∫ − 1 1 μ L ( z m , μ ) d μ ] / d z m = − d { ln [ E d ( z m ) − E u ( z m ) ] } / d z m
μ ¯ ( z m ) = [ ∫ − 1 1 μ L ( z m , μ ) d μ ] / [ ∫ − 1 1 L ( z m , μ ) d μ ] .
a = K ^ d ( z a s ) μ ¯ ( z a s ) ,
K ^ d ( z a s ) ≈ Δ [ ln E d ( z a s ) ] Δ z a s .
K ^ d ( z a s ) = K d ( z 1 ) K d ( z 3 ) − [ K d ( z 2 ) ] 2 K d ( z 1 ) + K d ( z 3 ) − 2 K d ( z 2 )
μ ¯ ( z a s ) = [ ∫ − 1 1 μ ϕ ( ν 0 , μ ) d μ ] / [ ∫ − 1 1 ϕ ( ν 0 , μ ) d μ ] = ν 0 [ 1 + ( b b / ( a F ) ] − 1 .
a = K ^ d ( z a s ) ν 0 [ 1 + ( b b / ( a F ) ] − 1 ,
L ( z a s , − 1 ) = L u ( z a s ) = A ( ν 0 ) ϕ ( ν 0 , − 1 ) exp [ − a z a s [ 1 + b b / ( a F ) ] / ν 0 ] .
a = ν 0 ( z 2 − z 1 ) [ 1 + b b / ( a F ) ] ln [ L u ( z 1 ) L u ( z 2 ) ] .
a = b b , meas b b / a .
b b = ( b b / a ) a .
X F = [ ν 0 b b / a F ] .
G F = [ 1 − ϖ ν 0 2 ln ( ν 0 + 1 ν 0 − 1 ) b b / a F − Ω ( z a s ) r r s ( z m ) [ 1 − Λ ( z a s ) r r s ( z m ) ] 2 ( b b / a ) − ( b b / a ) measured ] .
X F 0 = [ 1.0005 0.2 0.2 ] .
F ( b b / a ) = p 1 ( b b / a ) + p 2 .
X = [ ν 0 b b / a ] .
G = [ 1 − ϖ ν 0 2 ln ( ν 0 + 1 ν 0 − 1 ) b b / a F − Ω ( z a s ) r r s ( z m ) [ 1 − Λ ( z a s ) r r s ( z m ) ] 2 ] .
X 0 = [ 1.0005 0.2 ] .
ψ i ( z j ) = 100 I O P i , est ( z j ) − I O P i , meas ( z j ) I O P i , meas ( z j ) ,
| ψ ( z j ) | ¯ = 1 M k ∑ i = 1 M k | ψ i ( z j ) |
| ψ i | ¯ = 1 8 ∑ j = 1 8 | ψ i ( z j ) | ,
ψ i ¯ = 1 8 ∑ j = 1 8 ψ i ( z j )
| ψ | ¯ = 1 8 M k ∑ i = 1 M k ∑ j = 1 8 | ψ i ( z j ) | ,
ψ ¯ = 1 8 M k ∑ i = 1 M k ∑ j = 1 8 ψ i ( z j )

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