Abstract

We solve an inverse problem of ocean optics for estimating spatially dependent absorption and scattering coefficients and for determining sources such as fluorescence, bioluminescence, or Raman scattering. The solution requires in situ measurement of the downward and upward plane irradiances and scalar irradiances and a priori estimation of the angular shape of the volume scattering function. Both an explicit algorithm and an implicit one are developed from new two-stream radiative-transfer equations that utilize an asymptotic radiance approximation to close the set of equations. A comparison of numerical tests for the two algorithms is given.

© 1994 Optical Society of America

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  1. K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
    [CrossRef]
  2. E. J. Buskey, E. Swift, “An encounter model to predict natural planktonic bioluminescence,” Limnol. Oceanogr. 35, 1469–1485 (1990).
    [CrossRef]
  3. E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).
  4. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Scripps Institution of Oceanography Publ. 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972), pp. 1–78.
  5. N. J. McCormick, R. Sanchez, H. C. Yi, “Marine bioluminescent estimation algorithms for in situ irradiance measurements,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1302, 38–48 (1990).
  6. H. C. Yi, R. Sanchez, N. J. McCormick, “Bioluminescence estimation from ocean in situ irradiances,” Appl. Opt. 31, 822–830 (1992).
    [CrossRef] [PubMed]
  7. A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
  8. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  9. R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
    [CrossRef]
  10. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 58–76, 87–93, and 200–203.
  11. N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.
  12. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  13. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.3.
  14. Z. Tao, N. J. McCormick, “Bioluminescence estimation using explicit and implicit algorithms,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1750, 126–137 (1992).
  15. M. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, New York, 1980), p. 81.
    [CrossRef]
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.
  17. P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
    [CrossRef]
  18. N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
    [CrossRef] [PubMed]
  19. W. K. W. Li, “Bivariate and trivariate analysis in flow cytometry: phytoplankton size and fluorescence,” Limnol. Oceanogr. 35, 1356–1368 (1990).
    [CrossRef]

1992 (2)

1990 (2)

E. J. Buskey, E. Swift, “An encounter model to predict natural planktonic bioluminescence,” Limnol. Oceanogr. 35, 1469–1485 (1990).
[CrossRef]

W. K. W. Li, “Bivariate and trivariate analysis in flow cytometry: phytoplankton size and fluorescence,” Limnol. Oceanogr. 35, 1356–1368 (1990).
[CrossRef]

1989 (2)

N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
[CrossRef] [PubMed]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

1987 (1)

1986 (1)

K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
[CrossRef]

1984 (1)

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

1939 (1)

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Aas, E.

Arneson, C.

K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
[CrossRef]

Batchelder, H. P.

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

Booth, C. R.

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

Buskey, E. J.

E. J. Buskey, E. Swift, “An encounter model to predict natural planktonic bioluminescence,” Limnol. Oceanogr. 35, 1469–1485 (1990).
[CrossRef]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 58–76, 87–93, and 200–203.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.3.

Feng, D. F.

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.

Gershun, A. A.

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Greenblatt, P. R.

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

Hestenes, M.

M. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, New York, 1980), p. 81.
[CrossRef]

Huber, M. E.

K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
[CrossRef]

Kušcer, I.

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

Li, C. P.

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

Li, W. K. W.

W. K. W. Li, “Bivariate and trivariate analysis in flow cytometry: phytoplankton size and fluorescence,” Limnol. Oceanogr. 35, 1356–1368 (1990).
[CrossRef]

Losee, J. R.

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

McCormick, N. J.

H. C. Yi, R. Sanchez, N. J. McCormick, “Bioluminescence estimation from ocean in situ irradiances,” Appl. Opt. 31, 822–830 (1992).
[CrossRef] [PubMed]

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

N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
[CrossRef] [PubMed]

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

Z. Tao, N. J. McCormick, “Bioluminescence estimation using explicit and implicit algorithms,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1750, 126–137 (1992).

N. J. McCormick, R. Sanchez, H. C. Yi, “Marine bioluminescent estimation algorithms for in situ irradiance measurements,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1302, 38–48 (1990).

Nealson, K. H.

K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
[CrossRef]

Petzold, T. J.

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

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.

Rinaldi, G. E.

Sanchez, R.

H. C. Yi, R. Sanchez, N. J. McCormick, “Bioluminescence estimation from ocean in situ irradiances,” Appl. Opt. 31, 822–830 (1992).
[CrossRef] [PubMed]

N. J. McCormick, R. Sanchez, H. C. Yi, “Marine bioluminescent estimation algorithms for in situ irradiance measurements,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1302, 38–48 (1990).

Stavn, R. H.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Swift, E.

E. J. Buskey, E. Swift, “An encounter model to predict natural planktonic bioluminescence,” Limnol. Oceanogr. 35, 1469–1485 (1990).
[CrossRef]

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

Tao, Z.

Z. Tao, N. J. McCormick, “Bioluminescence estimation using explicit and implicit algorithms,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1750, 126–137 (1992).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.

Van Keuren, J.

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Yi, H. C.

H. C. Yi, R. Sanchez, N. J. McCormick, “Bioluminescence estimation from ocean in situ irradiances,” Appl. Opt. 31, 822–830 (1992).
[CrossRef] [PubMed]

N. J. McCormick, R. Sanchez, H. C. Yi, “Marine bioluminescent estimation algorithms for in situ irradiance measurements,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1302, 38–48 (1990).

Zirino, A.

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 58–76, 87–93, and 200–203.

Appl. Opt. (3)

J. Math. Phys. (1)

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Limnol. Oceanogr. (4)

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

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

E. J. Buskey, E. Swift, “An encounter model to predict natural planktonic bioluminescence,” Limnol. Oceanogr. 35, 1469–1485 (1990).
[CrossRef]

W. K. W. Li, “Bivariate and trivariate analysis in flow cytometry: phytoplankton size and fluorescence,” Limnol. Oceanogr. 35, 1356–1368 (1990).
[CrossRef]

Mar. Biol. (2)

K. H. Nealson, C. Arneson, M. E. Huber, “Identification of marine organisms using kinetic and spectral properties of their bioluminescence,” Mar. Biol. 91, 77–83 (1986).
[CrossRef]

P. R. Greenblatt, D. F. Feng, A. Zirino, J. R. Losee, “Observations of planktonic bioluminescence in the euphotic zone of the California Current,” Mar. Biol. 84, 75–82 (1984).
[CrossRef]

Other (9)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Sec. 48.3.

Z. Tao, N. J. McCormick, “Bioluminescence estimation using explicit and implicit algorithms,” in Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1750, 126–137 (1992).

M. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, New York, 1980), p. 81.
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986), Secs. 10.5–10.7.

E. Swift, J. Van Keuren, H. P. Batchelder, C. R. Booth, C. P. Li, “A moored instrument to measure stimulated and natural oceanic bioluminescence,” in Ocean Optics IX, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.925, 76–86 (1988).

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

N. J. McCormick, R. Sanchez, H. C. Yi, “Marine bioluminescent estimation algorithms for in situ irradiance measurements,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrumen. Eng.1302, 38–48 (1990).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 58–76, 87–93, and 200–203.

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

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Figures (14)

Fig. 1
Fig. 1

Eigenvalue ν0 versus b*/c* for the delta–Eddington approximation to San Diego Harbor sea water.

Fig. 2
Fig. 2

Asymptotic two-stream parameter λ versus b*/c* for the delta–Eddington approximation to San Diego Harbor sea water when any existing source is spatially constant and isotropic.

Fig. 3
Fig. 3

Numerically simulated downward and upward irradiances and scalar irradiances for an assumed source profile.

Fig. 4
Fig. 4

Absorption-coefficient profile estimated from the data shown in Fig. 3. The dashed line is the true value.

Fig. 5
Fig. 5

Single-scattering-albedo profile estimated from the data shown in Fig. 3. The dashed line is the true value.

Fig. 6
Fig. 6

Scattering-coefficient profile estimated from the data shown in Fig. 3. The dashed line is the true value.

Fig. 7
Fig. 7

Source profile estimated from the data shown in Fig. 3.

Fig. 8
Fig. 8

95% confidence bounds, as well as two random results, for the absorption coefficient estimated from the data shown in Fig. 3 with random flucutations. The dashed line is the true value.

Fig. 9
Fig. 9

95% confidence bounds, as well as two random results, for the source estimated from the data shown in Fig. 3 with random fluctuations.

Fig. 10
Fig. 10

Numerically simulated downward and upward irradiances and scalar irradiances with a bioluminescent source present.

Fig. 11
Fig. 11

Absorption-coefficient profile estimated from the data shown in Fig. 10. The dashed line is the true value.

Fig. 12
Fig. 12

Single-scattering-albedo profile estimated from the data shown in Fig. 10. The dashed line is the true value.

Fig. 13
Fig. 13

Scattering-coefficient profile estimated from the data shown in Fig. 10. The dashed line is the true value.

Fig. 14
Fig. 14

Bioluminescent-source profile estimated from the data shown in Fig. 10.

Tables (1)

Tables Icon

Table 1 Asymptotic Mean Cosines μ ¯ + and μ ¯ and Shape Factors r+ and r Versus Arbitrary Single-Scattering Albedo ωa

Equations (75)

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E n + ( z ) = 0 1 P n ( μ ) L ( z , μ ) d μ ,
E n ( z ) = 1 0 P n ( μ ) L ( z , μ ) d μ ,
L ( z , μ ) = 0 2 π L ( z , μ , ϕ ) d ϕ .
E n ( z ) = 1 1 P n ( μ ) L ( z , μ ) d μ , = E n + ( z ) + E n ( z ) ,
μ z L ( z , μ ) + c ( z ) L ( z , μ ) = b ( z ) 1 1 β ( μ , μ ) L ( z , μ ) d μ + Q ( z ) / 2 , 0 z z 0 ,
β ( μ , μ ) = 1 2 n = 0 N ( 2 n + 1 ) f n P n ( μ ) P n ( μ ) , f 0 = 1 .
μ z L ( z , μ ) + c ( z ) L ( z , μ ) = b ( z ) 2 n = 0 N ( 2 n + 1 ) f n P n ( μ ) E n ( z ) + Q ( z ) / 2 .
d z E 1 ± + c E 0 ± = b 2 [ E 0 ± n 1 n odd ( 2 n + 1 ) f n α n E n ] + 1 2 Q ,
α n = 0 1 P n ( μ ) d μ , n odd ,
α n / α n 2 = ( n 2 ) / ( n + 1 ) , n odd , n 3 ,
d z E 1 + a E 0 = Q ,
d z 1 + c 0 = b n 1 n odd ( 2 n + 1 ) f n α n E n ,
n = E n + E n .
d z [ ( n + 1 ) E n + 1 + n E n 1 ] + c h n E n = Q δ n 0 .
h n = ( 2 n + 1 ) ( 1 ω f n ) ,
L ( z , μ ) L asy ( z , μ ) = A + ϕ ( ν 0 , μ ) e c z / ν 0 + A ϕ ( ν 0 , μ ) e c z / ν 0 + L p ( z , μ ) ,
E n g n ( ν 0 ) [ A + e c z / ν 0 + ( 1 ) n A e c z / ν 0 ] + E p n ,
g n ( ν 0 ) = 1 1 P n ( μ ) ϕ ( ν 0 , μ ) d μ .
( n + 1 ) g n + 1 ( ν 0 ) h n ν 0 g n ( ν 0 ) + n g n 1 ( ν 0 ) = 0 , n 0 ,
g 1 ( ν 0 ) = h 0 ν 0 , g 2 ( ν 0 ) = 1 2 ( h 0 h 1 ν 0 2 1 ) .
E n E p n + [ g n ( ν 0 ) / g 1 ( ν 0 ) ] ( E 1 E p 1 ) , n odd , E p n + g n ( ν 0 ) ( E 0 E p 0 ) , n even .
ν 0 2 ( 1 ω ) d z ( E 0 E p 0 ) + c ( E 1 E p 1 ) 0 ,
d z E p 1 + c h 0 E p 0 = Q , d z ( 2 E p 2 + E p 0 ) + c h 1 E p 1 = 0 ,
d z E 1 + a E 0 = Q ,
d z 1 + c 0 b ( 3 f 1 2 + λ ) E 1 ,
ν 0 2 ( 1 ω ) d z E 0 + c E 1 ν 0 2 ( 1 ω ) d z E p 0 + c E p 1 .
λ = n 3 n odd α n ( 2 n + 1 ) f n g n ( ν 0 ) g 1 ( ν 0 ) [ 1 E p 1 E 1 + E p n E 1 g 1 ( ν 0 ) g n ( ν 0 ) ] .
Q ( z ) Q c + z Q l ,
L p ( z , μ ) = 1 2 a [ Q c + Q l ( z μ c b f 1 ) ] .
E p 0 = ( Q c + z Q l ) / a ,
E p 1 = Q l / a c h 1 ,
E p n = 0 , n > 1 .
d z E 1 + a E 0 = Q c + z Q l ,
d z 1 + c 0 b ( 3 f 1 2 + λ ) E 1 ,
ν 0 2 ( 1 ω ) d z E 0 + c E 1 2 Q l g 2 ( ν 0 ) / a h 1 ,
λ = n 3 n odd α n ( 2 n + 1 ) f n g n ( ν 0 ) g 1 ( ν 0 ) ( 1 + Q l a c h 1 E 1 ) .
Δ z i = z i + 1 z i , z ¯ i = ( z i + 1 + z i ) / 2 , Δ E n i = E n ( z i + 1 ) E n ( z i ) , E ¯ n i = [ E n ( z i + 1 ) + E n ( z i ) ] / 2 , Δ ni = n ( z i + 1 ) n ( z i ) , ε ¯ n i = [ ε n ( z i + 1 ) + ε n ( z i ) ] / 2 .
z i z i + 1 E n ( z ) d z Δ z i E ¯ n i
Δ E 1 i Δ z i + a ¯ i E ¯ 0 i = Q ¯ c i + z ¯ i Q ¯ l i ,
Δ 1 i Δ z i + c ¯ i ¯ 0 i b ¯ i ( 3 f 1 2 + λ ¯ i ) E ¯ 1 i ,
ν ¯ 0 i 2 ( 1 ω ¯ i ) Δ E 0 i Δ z i + c ¯ i E ¯ 1 i 2 Q ¯ l i g 2 ( ν ¯ 0 i ) a ¯ i h ¯ 1 i ,
λ ¯ i = n 3 n odd α n ( 2 n + 1 ) f n g n ( ν ¯ 0 i ) g 1 ( ν ¯ 0 i ) ( 1 + Q ¯ l i a ¯ i c ¯ i h ¯ 1 i E ¯ 1 i ) .
Q ¯ c , i 1 + z i Q ¯ l , i 1 = Q ¯ c i + z i Q ¯ l i ,
Q ¯ c i = Q ¯ c , i 1 + z i [ Q ¯ l , i 1 Q ¯ l i ] .
Q ¯ l i = a ¯ i E ¯ 0 i + Δ E 1 i Δ z i z i Q ¯ l , i 1 Q ¯ c , i 1 z ¯ i z i ,
Q ¯ c 0 = a ¯ 0 E ¯ 00 z ¯ 0 Q ¯ l 0 + Δ E 10 Δ z 0 ,
Q ¯ l 0 = ( a ¯ 1 E ¯ 01 a ¯ 0 E ¯ 00 + Δ E 11 Δ z 1 Δ E 10 Δ z 0 ) z ¯ 1 z ¯ 0 .
δ ( d z E 1 ) + E 0 δ a + a δ E 0 = Q ,
F ( u ¯ ) = i = 1 N W i [ 1 i c ( u ¯ ) 1 i m ] 2 + W a i = 0 N 1 ( a ¯ i a ¯ i 1 Δ z i ) 2 + W Q i = 0 N 1 ( Q ¯ l , i 1 c Q ¯ l , i 1 a ) 2 .
1 . i + 1 c ( u ( k ) ) = 1 i m + [ ω ¯ i ( 3 2 f 1 + λ ¯ ) E ¯ 1 i m ¯ 0 i m ] a ¯ i Δ z i 1 ω ¯ i .
d k = g k + β k d k 1 .
β k = g k T g k g k 1 T g k 1 ,
F ( u ¯ ( k ) + α k d k ) = min α F ( u ¯ ( k ) + α d k ) .
u ¯ ( k + 1 ) = u ¯ ( k ) + α k d k ,
β ( μ , μ ) = αδ ( μ μ ) + ( 1 α ) β* ( μ , μ ) , 0 α 1 ,
β* ( μ , μ ) = 1 2 n = 0 N * ( 2 n + 1 ) f n * P n ( μ ) P n ( μ ) .
μ z * L ( z * , μ ) + L ( z * , μ ) = ( b * / c * ) 1 1 β* ( μ , μ ) L ( z * , μ ) d μ + Q ( z * ) / 2 c * ,
ω = b * / c * 1 α ( 1 b * / c * ) .
L ( 0 , μ ) = L 0 δ ( μ μ 0 ) ,
L ( 0 , μ ) = L 0 ,
γ = E + ( 0 ) / 0 Q ( z ) d z .
d z E 1 ± = c ± E 1 ± b E 1 + 1 2 Q ,
a ± = a / μ ¯ ± , b ± = r ± b b / μ ¯ ± , c ± = a ± + b ± .
μ ¯ ± = ± E 1 ± / E 0 ± ,
r + b b / b = 1 E 0 + 0 1 d μ 1 0 d μβ ( μ , μ ) L ( z , μ ) , r b b / b = 1 E 0 1 0 d μ 0 1 d μβ ( μ , μ ) L ( z , μ ) .
r ± b b / b 1 2 [ 1 ( 3 f 1 2 + λ ) μ ¯ ± ] .
b b / b = 1 0 β ( μ , 1 ) d μ , = 1 2 ( 1 λ b ) ,
λ b = n 1 n odd α n ( 2 n + 1 ) f n ,
r ± [ 1 ( 3 f 1 2 + λ ) μ ¯ ± ] ( 1 λ b ) .
μ ¯ ± = A + g ˜ 1 ( ± ν 0 ) e c z / ν 0 + A g ˜ 1 ( ν 0 ) e c z / ν 0 A + g ˜ 0 ( ± ν 0 ) e c z / ν 0 + A g ˜ 0 ( ν 0 ) e c z / ν 0 ,
g ˜ n ( ± ν 0 ) = 0 1 P n ( μ ) ϕ ( ± ν 0 , μ ) d μ ,
ϕ ( ± ν j , μ ) = ϕ ( ν j , ± μ ) = ν j 2 g ( ± ν j , μ ) ν j μ .
Λ ( ± ν j ) = 0 ,
Λ ( x ) = 1 x 2 1 1 g ( x , μ ) x μ d μ , = 1 x 2 1 1 g ( μ , μ ) x μ d μ .
g ( ν j , μ ) = n = 0 N ( 2 n + 1 ) f n g n ( ν j ) P n ( μ ) .

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