Abstract

Remote sensing of ocean color, applied to the estimation of chlorophyll biomass, is discussed for the case where the vertical distribution of phytoplankton pigments is nonuniform. Using a spectral model of reflectance, the consequences of vertical structure are evaluated by sensitivity analysis on a generalized pigment profile. It is shown that the assumption of a vertically homogeneous chlorophyll distribution can lead to significant errors (relative error exceeding 100%) in the estimation from satellite data of photic depth and total pigment content in the photic zone. The errors are shown to be functions of the parameters of the pigment profile. It is further shown that, if the shape of the pigment profile is known from independent data, the entire pigment profile may be recovered from the satellite data by making slight changes in the existing algorithms for chlorophyll retrieval.

© 1989 Optical Society of America

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References

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  1. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery (Springer-Verlag, New York, 1983), p. 114.
  2. S. Sathyendranath, A. Morel, “Light Emerging from the Sea—Interpretation and Uses in Remote Sensing,” in Remote Sensing Application in Marine Science and Technology, A. P. Cracknell, Ed. (Reidel, Dordrecht, 1983), pp. 323–357.
    [CrossRef]
  3. M. V. Kozlyaninov, V. N. Pelevin, “Sur l’Utilisation de l’Approximation Unidirectionelle pour l’Étude de la propagation du Flux Lumineux en Mer,” Acad. Nauka SSR Tr. Oceanogr. Inst. 77, 73 (1965).
  4. H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed Relationship Between the Inherent and Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  5. L. Prieur, A. Morel, “Relations Théoriques Entre le Facteur de Réflexion Diffuse de l’Eau de Mer à Diverses Profondeurs et les Caractéristiques Optiques (Absorption, Diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).
  6. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  7. J. T. O. Kirk, “Monte Carlo Study of the Nature of the Underwater Light Field in, and Relationship Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
    [CrossRef]
  8. L. Prieur, “Transfert Radiatitif dans les Eaux de Mer. Application à la Détermination de Paramètres Optiques Caractérisant leur Teneur en Substances Dissoutes et leur Contenu en Particules,” D. Sci. Thesis, U.P. & M. Curie2 (1976).
  9. S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
    [CrossRef]
  10. S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur l’Absorption et la Réflectance. Modélisation et Applications à la Télédétection,” Thesis 3è Cycle, U.P. & M. Curie1 (1981).
  11. S. Sathyendranath, L. Prieur, A. Morel, “Interpretation of Ocean Colour Data with Special Reference to OCM,” Mid Term Report, contract ESA 4726-81-F-DD-SC (1982).
  12. S. Sathyendranath, L. Prieur, A. Morel, “A Three-Component Model of Ocean Colour and its Application to Remote Sensing in ‘Case 2’ Waters,” submitted to Int. J. Remote Sensing (1988).
  13. L. Prieur, 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,” Limmol. Oceanogr. 26, 671 (1981).
    [CrossRef]
  14. S. Sathyendranath, T. Platt, “The Spectral Irradiance Field at the Surface and in the Interior of the Ocean: A Model for Applications in Oceanography and Remote Sensing,” J. Geophys. Res. 93, 9270 (1988).
    [CrossRef]
  15. A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary Layer Meteorol. 18, 177 (1980).
    [CrossRef]
  16. D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 227–238.
    [CrossRef]
  17. H. R. Gordon, W. R. McCluney, “Estimation of the Depth of Sunlight Penetration in the Sea for Remote Sensing,” Appl. Opt. 14, 413 (1975).
    [CrossRef] [PubMed]
  18. H. R. Gordon, D. K. Clark, “Remote Sensing of Optical Properties of a Stratified Ocean: An Improved Interpretation,” Appl. Opt. 19, 3428 (1980).
    [CrossRef] [PubMed]
  19. S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).
  20. T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
    [CrossRef]
  21. T. Platt, S. Sathyendranath, “Oceanic Primary Production: Estimation by Remote Sensing at Regional and Larger Scales,” Science 241, 1613 (1988).
    [CrossRef] [PubMed]

1988 (3)

S. Sathyendranath, T. Platt, “The Spectral Irradiance Field at the Surface and in the Interior of the Ocean: A Model for Applications in Oceanography and Remote Sensing,” J. Geophys. Res. 93, 9270 (1988).
[CrossRef]

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

T. Platt, S. Sathyendranath, “Oceanic Primary Production: Estimation by Remote Sensing at Regional and Larger Scales,” Science 241, 1613 (1988).
[CrossRef] [PubMed]

1987 (1)

S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
[CrossRef]

1983 (1)

S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).

1981 (2)

J. T. O. Kirk, “Monte Carlo Study of the Nature of the Underwater Light Field in, and Relationship Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

L. Prieur, 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,” Limmol. Oceanogr. 26, 671 (1981).
[CrossRef]

1980 (2)

1977 (1)

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1975 (3)

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed Relationship Between the Inherent and Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
[CrossRef] [PubMed]

L. Prieur, A. Morel, “Relations Théoriques Entre le Facteur de Réflexion Diffuse de l’Eau de Mer à Diverses Profondeurs et les Caractéristiques Optiques (Absorption, Diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

H. R. Gordon, W. R. McCluney, “Estimation of the Depth of Sunlight Penetration in the Sea for Remote Sensing,” Appl. Opt. 14, 413 (1975).
[CrossRef] [PubMed]

1965 (1)

M. V. Kozlyaninov, V. N. Pelevin, “Sur l’Utilisation de l’Approximation Unidirectionelle pour l’Étude de la propagation du Flux Lumineux en Mer,” Acad. Nauka SSR Tr. Oceanogr. Inst. 77, 73 (1965).

Brown, O. B.

Caverhill, C.

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

Clark, D. K.

H. R. Gordon, D. K. Clark, “Remote Sensing of Optical Properties of a Stratified Ocean: An Improved Interpretation,” Appl. Opt. 19, 3428 (1980).
[CrossRef] [PubMed]

D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 227–238.
[CrossRef]

Gordon, H. R.

Jacobs, M. M.

Kirk, J. T. O.

J. T. O. Kirk, “Monte Carlo Study of the Nature of the Underwater Light Field in, and Relationship Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

Kozlyaninov, M. V.

M. V. Kozlyaninov, V. N. Pelevin, “Sur l’Utilisation de l’Approximation Unidirectionelle pour l’Étude de la propagation du Flux Lumineux en Mer,” Acad. Nauka SSR Tr. Oceanogr. Inst. 77, 73 (1965).

Lewis, M. R.

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

McCluney, W. R.

Morel, A.

S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).

A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary Layer Meteorol. 18, 177 (1980).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, A. Morel, “Relations Théoriques Entre le Facteur de Réflexion Diffuse de l’Eau de Mer à Diverses Profondeurs et les Caractéristiques Optiques (Absorption, Diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

S. Sathyendranath, L. Prieur, A. Morel, “Interpretation of Ocean Colour Data with Special Reference to OCM,” Mid Term Report, contract ESA 4726-81-F-DD-SC (1982).

S. Sathyendranath, A. Morel, “Light Emerging from the Sea—Interpretation and Uses in Remote Sensing,” in Remote Sensing Application in Marine Science and Technology, A. P. Cracknell, Ed. (Reidel, Dordrecht, 1983), pp. 323–357.
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, “A Three-Component Model of Ocean Colour and its Application to Remote Sensing in ‘Case 2’ Waters,” submitted to Int. J. Remote Sensing (1988).

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery (Springer-Verlag, New York, 1983), p. 114.

Pelevin, V. N.

M. V. Kozlyaninov, V. N. Pelevin, “Sur l’Utilisation de l’Approximation Unidirectionelle pour l’Étude de la propagation du Flux Lumineux en Mer,” Acad. Nauka SSR Tr. Oceanogr. Inst. 77, 73 (1965).

Platt, T.

T. Platt, S. Sathyendranath, “Oceanic Primary Production: Estimation by Remote Sensing at Regional and Larger Scales,” Science 241, 1613 (1988).
[CrossRef] [PubMed]

S. Sathyendranath, T. Platt, “The Spectral Irradiance Field at the Surface and in the Interior of the Ocean: A Model for Applications in Oceanography and Remote Sensing,” J. Geophys. Res. 93, 9270 (1988).
[CrossRef]

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

Prieur, L.

S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).

L. Prieur, 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,” Limmol. Oceanogr. 26, 671 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, A. Morel, “Relations Théoriques Entre le Facteur de Réflexion Diffuse de l’Eau de Mer à Diverses Profondeurs et les Caractéristiques Optiques (Absorption, Diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

S. Sathyendranath, L. Prieur, A. Morel, “Interpretation of Ocean Colour Data with Special Reference to OCM,” Mid Term Report, contract ESA 4726-81-F-DD-SC (1982).

L. Prieur, “Transfert Radiatitif dans les Eaux de Mer. Application à la Détermination de Paramètres Optiques Caractérisant leur Teneur en Substances Dissoutes et leur Contenu en Particules,” D. Sci. Thesis, U.P. & M. Curie2 (1976).

S. Sathyendranath, L. Prieur, A. Morel, “A Three-Component Model of Ocean Colour and its Application to Remote Sensing in ‘Case 2’ Waters,” submitted to Int. J. Remote Sensing (1988).

Sathyendranath, S.

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

S. Sathyendranath, T. Platt, “The Spectral Irradiance Field at the Surface and in the Interior of the Ocean: A Model for Applications in Oceanography and Remote Sensing,” J. Geophys. Res. 93, 9270 (1988).
[CrossRef]

T. Platt, S. Sathyendranath, “Oceanic Primary Production: Estimation by Remote Sensing at Regional and Larger Scales,” Science 241, 1613 (1988).
[CrossRef] [PubMed]

S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).

L. Prieur, 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,” Limmol. Oceanogr. 26, 671 (1981).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, “Interpretation of Ocean Colour Data with Special Reference to OCM,” Mid Term Report, contract ESA 4726-81-F-DD-SC (1982).

S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur l’Absorption et la Réflectance. Modélisation et Applications à la Télédétection,” Thesis 3è Cycle, U.P. & M. Curie1 (1981).

S. Sathyendranath, L. Prieur, A. Morel, “A Three-Component Model of Ocean Colour and its Application to Remote Sensing in ‘Case 2’ Waters,” submitted to Int. J. Remote Sensing (1988).

S. Sathyendranath, A. Morel, “Light Emerging from the Sea—Interpretation and Uses in Remote Sensing,” in Remote Sensing Application in Marine Science and Technology, A. P. Cracknell, Ed. (Reidel, Dordrecht, 1983), pp. 323–357.
[CrossRef]

Acad. Nauka SSR Tr. Oceanogr. Inst. (1)

M. V. Kozlyaninov, V. N. Pelevin, “Sur l’Utilisation de l’Approximation Unidirectionelle pour l’Étude de la propagation du Flux Lumineux en Mer,” Acad. Nauka SSR Tr. Oceanogr. Inst. 77, 73 (1965).

Adv. Space Res. (1)

S. Sathyendranath, L. Prieur, A. Morel, “An Evaluation of the Problems of Chlorophyll Retrieval from Ocean Colour, for Case 2 Waters,” Adv. Space Res. 7(2), 27 (1987).
[CrossRef]

Appl. Opt. (3)

Aust. J. Mar. Freshwater Res. (1)

J. T. O. Kirk, “Monte Carlo Study of the Nature of the Underwater Light Field in, and Relationship Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

Boundary Layer Meteorol. (1)

A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary Layer Meteorol. 18, 177 (1980).
[CrossRef]

Deep Sea Res. (1)

T. Platt, S. Sathyendranath, C. Caverhill, M. R. Lewis, “Ocean Primary Production and Available Light: Further Algorithms for Remote Sensing,” Deep Sea Res., 35, 855 (1988).
[CrossRef]

IAPSO-IGGU XVI General Assembly (Grenoble) (1)

L. Prieur, A. Morel, “Relations Théoriques Entre le Facteur de Réflexion Diffuse de l’Eau de Mer à Diverses Profondeurs et les Caractéristiques Optiques (Absorption, Diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

J. Geophys. Res. (1)

S. Sathyendranath, T. Platt, “The Spectral Irradiance Field at the Surface and in the Interior of the Ocean: A Model for Applications in Oceanography and Remote Sensing,” J. Geophys. Res. 93, 9270 (1988).
[CrossRef]

Limmol. Oceanogr. (1)

L. Prieur, 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,” Limmol. Oceanogr. 26, 671 (1981).
[CrossRef]

Limnol. Oceanogr. (1)

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1)

S. Sathyendranath, L. Prieur, A. Morel, Rapport Complémentaire, contract ESA 4726-81-F-DD-SC (1983).

Science (1)

T. Platt, S. Sathyendranath, “Oceanic Primary Production: Estimation by Remote Sensing at Regional and Larger Scales,” Science 241, 1613 (1988).
[CrossRef] [PubMed]

Other (7)

D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981), pp. 227–238.
[CrossRef]

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery (Springer-Verlag, New York, 1983), p. 114.

S. Sathyendranath, A. Morel, “Light Emerging from the Sea—Interpretation and Uses in Remote Sensing,” in Remote Sensing Application in Marine Science and Technology, A. P. Cracknell, Ed. (Reidel, Dordrecht, 1983), pp. 323–357.
[CrossRef]

L. Prieur, “Transfert Radiatitif dans les Eaux de Mer. Application à la Détermination de Paramètres Optiques Caractérisant leur Teneur en Substances Dissoutes et leur Contenu en Particules,” D. Sci. Thesis, U.P. & M. Curie2 (1976).

S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur l’Absorption et la Réflectance. Modélisation et Applications à la Télédétection,” Thesis 3è Cycle, U.P. & M. Curie1 (1981).

S. Sathyendranath, L. Prieur, A. Morel, “Interpretation of Ocean Colour Data with Special Reference to OCM,” Mid Term Report, contract ESA 4726-81-F-DD-SC (1982).

S. Sathyendranath, L. Prieur, A. Morel, “A Three-Component Model of Ocean Colour and its Application to Remote Sensing in ‘Case 2’ Waters,” submitted to Int. J. Remote Sensing (1988).

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Figures (6)

Fig. 1
Fig. 1

(a) Relationship between the reflectance ratio for wavelength pair 440/550 and pigment concentration in water: 1, Eq. (1) with parameters A = 1.13 and B = −1.71 after Gordon and Morel,1 the NASA algorithm for retrieval at low chlorophyll concentrations; 2, Eq. (2) with parameters A = 1.92 and B = −1.8 from Morel15; 3, analytical relation proposed by Gordon and Morel1; 4, results of reflectance model used in this study. (b) Relationship between reflectance ratio for wavelength pair 520/550 and pigment concentration in the water: 1, Eq. (1) with parameters A = 3.326 and B = −2.439, the NASA algorithm for retrieval at high chlorophyll concentrations; 2, Eq. (2) with parameters A = 1.69 and B = −4.45 after Clark16; 3, result of the reflectance model used in this study.

Fig. 2
Fig. 2

Reflectance ratio (440/550) plotted against the satellite-weighted surface concentration for 520 nm, Cs (mg m−3) for various nonuniform pigment profiles. In each of the subplots, one of the parameters of the pigment profile is varied while the other three are held constant at their assigned typical values. In each subplot, the relationship is plotted for eleven values of the parameter at equal intervals within the range over which the parameter is varied: (a) h is varied from 0.1 to 250 (mg m−2); (b) σ is varied from 2 to 22 m; (c) zm is varied from 0 to 85 m; (d) C0 is varied from 0.05 to 1.5 (mg m−3).

Fig. 3
Fig. 3

Reflectance ratio (440/550) plotted against the parameter values of the pigment profiles. When one of the parameters is varied, the others retain their constant typical values. Dotted line—nonuniform pigment profile; continuous line—uniform pigment profile with concentration the same as the mean photic-zone concentration in the nonuniform case.

Fig. 4
Fig. 4

Photic depth zp (m) plotted against the four profile parameters. Continuous line—photic depth estimated from the satellite-weighted surface concentration, assuming uniform pigment profile; dotted line—real photic depth computed with the given nonuniform pigment profile; dashed line—relative error (%) in estimated photic depth computed as (estimated depth–real depth)/real photic depth. As in the previous figure, when one parameter is changed, the others are held constant.

Fig. 5
Fig. 5

Total pigment in the photic zone (mg m−2) plotted against the four parameters of the pigment profile. Continuous line—estimated pigment content, calculated as product of Cs and the estimated photic depth; dotted line—the actual pigment content [integral of Cs(z) from surface to bottom of photic zone]; dashed line—relative error in estimated photic-zone pigment content computed as (estimated value–real value)/real value.

Fig. 6
Fig. 6

Blue-green reflectance ratio plotted against {C0 + h/[σ√(2π)]} (mg m−3) for a given profile shape defined by zm 20 m, σ = 5 m and h/[C0σ√(2π)] = 10.

Equations (8)

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C = A [ L ( λ 1 ) / L ( λ 2 ) ] B ,
R ( λ ) = 0.33 b b ( λ ) / a ( λ ) ,
a ( λ ) = a w ( λ ) + C a c * ( λ ) + Y a y * ( λ ) ,
b b ( λ ) = b ˜ b w b w ( λ ) + b ˜ b c b c ( λ ) ,
z 90 ( λ ) = 1 / K ( λ ) .
C s ( λ ) = 0 z 90 ( λ ) C ( z ) f ( z ) d z / 0 z 90 ( λ ) f ( z ) d z ,
f ( z ) = exp [ - 0 z 2 K ( λ , z ) d z ] .
C ( z ) = C 0 + h σ 2 π exp [ - ( z - z m ) 2 2 σ 2 ] .

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