Abstract

A simple, empirically derived algorithm for estimating oceanic chlorophyll concentrations from spectral radiances measured by a low-flying spectroradiometer has proved highly successful in field experiments in 1980–82. The sensor used was the Multichannel Ocean Color Sensor, and the originator of the algorithm was G. W. Grew, NASA CP-2188 (1981). This paper presents an explanation for the algorithm based on the optical properties of waters containing chlorophyll and other phytoplankton pigments and the radiative transfer equations governing the remotely sensed signal. The effects of varying solar zenith, atmospheric transmittance, and interfering substances in the water on the chlorophyll algorithm are characterized, and applicability of the algorithm is discussed.

© 1983 Optical Society of America

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References

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  1. G. W. Grew, “Real-Time Test of MOCS Algorithm during Superflux 1980,” in NASA CP-2188 (1981), p. 301.
  2. J. W. Campbell, J. P. Thomas, Eds., “The Chesapeake Bay Plume Study: Superflux 1980,” NASA CP-2188 (1981).
  3. W. E. Esaias, J. W. Campbell, Trans. Am. Geophys. Union 63, No. 3, 69 (1982).
  4. W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).
  5. F. E. Hoge, R. N. Swift, Appl. Opt. 19, 871 (1980).
    [CrossRef] [PubMed]
  6. R. C. Smith, K. S. Baker, Limnol. Oceanogr. 23, 260 (1978).
    [CrossRef]
  7. P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).
  8. Since the algorithm is evaluated over the 440–529-nm region, the spectral region where accessory photopigments absorb strongly along with chlorophylls a, b, and c, and their degradation products, use of the term chlorophyll may be somewhat inaccurate. However, we will refer to chlorophyll concentration measured in situ and make the assumption that all blue-absorbing photopigments covary with chlorophyll.
  9. G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), p. 296.
  10. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
    [CrossRef] [PubMed]
  11. D. K. Clark, “Phytoplankton Algorithms for Nimbus-7 CZCS,” in Oceanography from Space, J. R. F. Gower, Ed. (Plenum, New York, 1981), p. 227.
    [CrossRef]
  12. X. X. Govindjee, R. Govindjee, Sci. Am. xx, 000 (Dec.1974).
  13. A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography (Academic, New York, 1974), p. 4.
  14. R. W. Preisenderfer, Hydrologic Optics, Vol. I, Introduction, PB-259 793 (U.S. Department of Commerce, Washington, D.C., 1976), p. 57.
  15. R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
    [CrossRef]
  16. J. E. Tyler, R. C. Smith, J. Opt. Soc. Am. 57, 595 (1967).
    [CrossRef]
  17. R. C. Smith, J. E. Tyler, Photochemistry and Photobiology Reviews, Vol. 1 (Plenum, New York, 1976), p. 147.
  18. H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).
  19. R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).
  20. T. A. Talay, L. R. Poole, “Spectral Atmospheric Observations at Nantucket Island, May 7–14, 1981,” NASA Tech. Memo. 83196 (1981).
  21. A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  22. H. R. Gordon, Appl. Opt. 17, 1631 (1978).
    [CrossRef] [PubMed]
  23. H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  24. N. G. Jerlov, “Significant Relationships between Optical Properties of the Sea,” in Optical Aspects of Oceanography, N. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), p. 77.
  25. J. Kitchen, Proc. Soc. Photo-Opt. Instrum. Eng. 160, 31 (1978).
  26. A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
    [CrossRef]
  27. K. S. Baker, R. C. Smith, Limnol. Oceanogr. 27, 3 (1982).
    [CrossRef]

1982 (3)

W. E. Esaias, J. W. Campbell, Trans. Am. Geophys. Union 63, No. 3, 69 (1982).

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

K. S. Baker, R. C. Smith, Limnol. Oceanogr. 27, 3 (1982).
[CrossRef]

1981 (1)

A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
[CrossRef]

1980 (2)

F. E. Hoge, R. N. Swift, Appl. Opt. 19, 871 (1980).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

1979 (1)

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

1978 (3)

J. Kitchen, Proc. Soc. Photo-Opt. Instrum. Eng. 160, 31 (1978).

R. C. Smith, K. S. Baker, Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

H. R. Gordon, Appl. Opt. 17, 1631 (1978).
[CrossRef] [PubMed]

1977 (1)

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1975 (1)

1974 (2)

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

X. X. Govindjee, R. Govindjee, Sci. Am. xx, 000 (Dec.1974).

1973 (1)

R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
[CrossRef]

1967 (1)

Austin, R. W.

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Baker, K. S.

K. S. Baker, R. C. Smith, Limnol. Oceanogr. 27, 3 (1982).
[CrossRef]

R. C. Smith, K. S. Baker, Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

Bricaud, A.

A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
[CrossRef]

Brown, O. B.

Campbell, J. W.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

W. E. Esaias, J. W. Campbell, Trans. Am. Geophys. Union 63, No. 3, 69 (1982).

Clark, D. K.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

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

Duntley, S. Q.

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Edgerton, C. F.

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Esaias, W. E.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

W. E. Esaias, J. W. Campbell, Trans. Am. Geophys. Union 63, No. 3, 69 (1982).

Goldman, C. R.

R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
[CrossRef]

Gordon, H. R.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

H. R. Gordon, Appl. Opt. 17, 1631 (1978).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
[CrossRef] [PubMed]

Govindjee, R.

X. X. Govindjee, R. Govindjee, Sci. Am. xx, 000 (Dec.1974).

Govindjee, X. X.

X. X. Govindjee, R. Govindjee, Sci. Am. xx, 000 (Dec.1974).

Grew, G. W.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

G. W. Grew, “Real-Time Test of MOCS Algorithm during Superflux 1980,” in NASA CP-2188 (1981), p. 301.

Hoge, F.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

Hoge, F. E.

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

Jacobs, M. M.

Jenkin, K. R.

P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), p. 296.

Jerlov, N. G.

N. G. Jerlov, “Significant Relationships between Optical Properties of the Sea,” in Optical Aspects of Oceanography, N. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), p. 77.

Joyce, T. M.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

Kendall, B.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

Kitchen, J.

J. Kitchen, Proc. Soc. Photo-Opt. Instrum. Eng. 160, 31 (1978).

Moran, S. E.

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Morel, A.

A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
[CrossRef]

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography (Academic, New York, 1974), p. 4.

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

Poole, L. R.

T. A. Talay, L. R. Poole, “Spectral Atmospheric Observations at Nantucket Island, May 7–14, 1981,” NASA Tech. Memo. 83196 (1981).

Preisenderfer, R. W.

R. W. Preisenderfer, Hydrologic Optics, Vol. I, Introduction, PB-259 793 (U.S. Department of Commerce, Washington, D.C., 1976), p. 57.

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
[CrossRef]

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Ramsey, R. C.

P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).

Smith, R. C.

K. S. Baker, R. C. Smith, Limnol. Oceanogr. 27, 3 (1982).
[CrossRef]

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

R. C. Smith, K. S. Baker, Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
[CrossRef]

J. E. Tyler, R. C. Smith, J. Opt. Soc. Am. 57, 595 (1967).
[CrossRef]

R. C. Smith, J. E. Tyler, Photochemistry and Photobiology Reviews, Vol. 1 (Plenum, New York, 1976), p. 147.

Sorkin, M.

P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).

Swift, R.

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

Swift, R. N.

Talay, T. A.

T. A. Talay, L. R. Poole, “Spectral Atmospheric Observations at Nantucket Island, May 7–14, 1981,” NASA Tech. Memo. 83196 (1981).

Tyler, J. E.

R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
[CrossRef]

J. E. Tyler, R. C. Smith, J. Opt. Soc. Am. 57, 595 (1967).
[CrossRef]

R. C. Smith, J. E. Tyler, Photochemistry and Photobiology Reviews, Vol. 1 (Plenum, New York, 1976), p. 147.

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), p. 296.

White, P. G.

P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).

Wilson, W. H.

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Zaneveld, J. R. V.

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (5)

A. Bricaud, A. Morel, L. Prieur, Limnol. Oceanogr. 26, 43 (1981).
[CrossRef]

K. S. Baker, R. C. Smith, Limnol. Oceanogr. 27, 3 (1982).
[CrossRef]

R. C. Smith, K. S. Baker, Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

R. C. Smith, J. E. Tyler, C. R. Goldman, Limnol. Oceanogr. 18, 189 (1973).
[CrossRef]

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

H. R. Gordon, R. C. Smith, J. R. V. Zaneveld, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 14 (1979).

J. Kitchen, Proc. Soc. Photo-Opt. Instrum. Eng. 160, 31 (1978).

Sci. Am. (1)

X. X. Govindjee, R. Govindjee, Sci. Am. xx, 000 (Dec.1974).

Science (1)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

SIO Ref. (1)

R. W. Austin, S. Q. Duntley, W. H. Wilson, C. F. Edgerton, S. E. Moran, “Ocean Color Analysis,” SIO Ref.74–10 (1974).

Trans. Am. Geophys. Union (2)

W. E. Esaias, J. W. Campbell, Trans. Am. Geophys. Union 63, No. 3, 69 (1982).

W. E. Esaias, T. M. Joyce, J. W. Campbell, G. W. Grew, F. Hoge, B. Kendall, R. Swift, Trans. Am. Geophys. Union 63, No. 3, 59 (1982).

Other (11)

R. C. Smith, J. E. Tyler, Photochemistry and Photobiology Reviews, Vol. 1 (Plenum, New York, 1976), p. 147.

G. W. Grew, “Real-Time Test of MOCS Algorithm during Superflux 1980,” in NASA CP-2188 (1981), p. 301.

J. W. Campbell, J. P. Thomas, Eds., “The Chesapeake Bay Plume Study: Superflux 1980,” NASA CP-2188 (1981).

T. A. Talay, L. R. Poole, “Spectral Atmospheric Observations at Nantucket Island, May 7–14, 1981,” NASA Tech. Memo. 83196 (1981).

N. G. Jerlov, “Significant Relationships between Optical Properties of the Sea,” in Optical Aspects of Oceanography, N. Jerlov, E. S. Nielsen, Eds. (Academic, New York, 1974), p. 77.

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

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography (Academic, New York, 1974), p. 4.

R. W. Preisenderfer, Hydrologic Optics, Vol. I, Introduction, PB-259 793 (U.S. Department of Commerce, Washington, D.C., 1976), p. 57.

P. G. White, K. R. Jenkin, R. C. Ramsey, M. Sorkin, “Development and Flight Test of the Multichannel Ocean Color Sensor (MOCS),” NASA CR-2311 (1973).

Since the algorithm is evaluated over the 440–529-nm region, the spectral region where accessory photopigments absorb strongly along with chlorophylls a, b, and c, and their degradation products, use of the term chlorophyll may be somewhat inaccurate. However, we will refer to chlorophyll concentration measured in situ and make the assumption that all blue-absorbing photopigments covary with chlorophyll.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968), p. 296.

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Figures (14)

Fig. 1
Fig. 1

Inflection ratio spectra typical of blue oceanic waters (from Grew1).

Fig. 2
Fig. 2

Normalized inflection spectra (from Grew1) showing effect of chlorophyll-like pigments.

Fig. 3
Fig. 3

Regression of the logarithms of AOL chlorophyll fluorescence and in situ (sea truth) chlorophyll against the logarithm of the MOCS inflection ratio at 490 nm from data taken over Nantucket Shoals on 9 May 1981.

Fig. 4
Fig. 4

Comparison of (a) MOCS-derived chlorophyll computed using 9 May regression coefficients, (b) simultaneously measured AOL chlorophyll fluorescence, and (c) in situ chlorophyll along a transect south of Nantucket Shoals on 14 May 1981.

Fig. 5
Fig. 5

Schematic illustrating geometric interpretation of Δ2f(λ).

Fig. 6
Fig. 6

Influence of chlorophyll-like pigments C on irradiance reflectance R(λ) from Smith and Baker.6

Fig. 7
Fig. 7

Spectral curvature Δ2 logR of irradiance reflectance spectra shown in Fig. 6.

Fig. 8
Fig. 8

Normalized inflection ratio spectra computed for irradiance reflectance given by Smith and Baker model.6

Fig. 9
Fig. 9

Spectral curvature of (a) backscatter coefficient b′ and (b) diffuse attenuation coefficient for irradiance K as computed by Smith and Baker model.6

Fig. 10
Fig. 10

Spectral curvature of irradiance reflectance measured in natural waters.15,16

Fig. 11
Fig. 11

Spectral curvature of diffuse attenuation coefficient for irradiance measured in natural waters.1517

Fig. 12
Fig. 12

Effect of solar altitude on downwelling irradiance at the surface as measured at Nantucket Island, 7–14 May 1981 (from Talay and Poole20).

Fig. 13
Fig. 13

Mean and range in spectral curvature of downwelling irradiances shown in Fig. 12.

Fig. 14
Fig. 14

Effect of varying amounts of (a) dissolved organic matter and (b) nonabsorbing particulates (scat) on regression of spectral) curvature of K at 490 nm against log(C).

Tables (1)

Tables Icon

Table I Characteristics of the Multichannel Ocean Color Sensor from 〈1〉

Equations (41)

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G m , n ( λ i ) = S ( λ i ) 2 S ( λ i m ) S ( λ i + n )
H ( λ i ) = G 2,2 ( λ i ) : green water G 2,2 ( λ i ) : blue water
G 2,2 ( λ 7 ) = S ( 490 ) 2 S ( 460 ) S ( 521 ) ,
log C = a b log G 2,2 ( λ 7 )
Δ ω f ( λ ) = f ( λ + ω 2 ) f ( λ ω 2 ) ·
Δ ω 2 log S ( λ i ) = log G 2,2 ( λ i ) ,
Δ 2 log S ( λ ) = Δ [ log S ( λ + 15 ) log S ( λ 15 ) ] = Δ log S ( λ + 15 ) Δ log S ( λ 15 ) = [ log S ( λ + 30 ) log S ( λ ) ] [ log S ( λ ) log S ( λ 30 ) ] = log S ( λ + 30 ) 2 log S ( λ ) + log S ( λ 30 ) = log [ S ( λ + 30 ) S ( λ 30 ) S ( λ ) 2 ] = log G 2,2 ( λ ) for λ = λ i .
log C = a + b Δ 2 log S ( λ 7 )
Δ 2 log S ( λ ) = log S ( λ + 30 ) 2 log S ( λ ) + log S ( λ 30 )
Δ 2 log S = Δ 2 log S 1 + Δ 2 log S 2 ,
R ( λ ) = 4 b ( λ ) 9 K ( λ ) ,
b ( λ ) = b w ( λ ) + 0.001 C .
E ( λ , z ) = E ( λ , z 0 ) exp [ K ( λ ) ( z z 0 ) ] .
K ( λ ) = k 0 ( λ ) + k 1 ( λ ) C ,
Δ 2 log R ( λ ) = Δ 2 log b ( λ ) Δ 2 log K ( λ ) .
K ( λ ) a ( λ ) + b ( λ ) ,
L ( λ , h s ) = L A ( λ , h s ) + T ( λ , h s ) [ L W ( λ , 0 + ) + L S ( λ ) ] ,
L W ( λ , 0 + ) = t E d ( λ , 0 ) R ( λ , 0 ) m Q ( λ ) ,
Δ 2 log L W = Δ 2 log E d + Δ 2 log R Δ 2 log Q .
Δ f ( λ ) ω d f d λ ,
Δ 2 f ( λ ) ω 2 d 2 f d λ 2 .
Δ 2 log ( S 1 + S 2 ) = ω 2 [ d 2 S 1 d λ 2 + d 2 S 2 d λ 2 ( S 1 + S 2 ) ( d S 1 d λ + d S 2 d λ ) 2 ( S 1 + S 2 ) 2 ] .
Δ 2 log ( S 1 + S 2 ) = ( 1 ρ ) Δ 2 log S 1 + ρ Δ 2 log S 2 + ρ ( 1 ρ ) ( Δ log S 1 Δ log S 2 ) 2 ,
ρ ( λ ) S 2 ( λ ) S 1 ( λ ) + S 2 ( λ ) .
ρ ( 1 ρ ) ( Δ log S 1 Δ log S 2 ) 2
X = b / ( a + b ) ,
a = a w + a c + a x ,
b = b w + b c + b x .
Δ 2 log b = p ( ω 2 / λ 2 ) .
Δ 2 log b 0.015.
Δ 2 log ( a + b ) = Δ 2 log ( S 1 + S 2 ) ,
S 1 = a w + a c + b w + b c ,
S 2 = a x + b x .
S DOM a DOM .
a DOM ( λ ) = a DOM ( λ 0 ) exp [ s ( λ λ 0 ) ] ,
Δ log S 2 = s ω ,
Δ 2 log ( S 1 + S 2 ) = ( 1 ρ ) Δ 2 log S 1 + ρ ( 1 ρ ) ( Δ log S 1 + s ω ) 2
S 2 ( λ ) = b ( λ ) = b ( λ 0 ) ( λ 0 / λ ) ,
Δ log S 2 = ω λ ,
Δ 2 log S 2 = ω 2 λ 2 ·
Δ 2 log ( S 1 + S 2 ) = ( 1 ρ ) Δ 2 log S 1 + ρ ω 2 λ 2 + ρ ( 1 ρ ) ( Δ log S 1 + ω λ ) 2 ·

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