Abstract

A method was developed for determining the spatial distribution of a source from downward and upward irradiance measurements at a single wavelength in seawater of known optical properties. The algorithm uses measurements at two depths located an arbitrary distance apart and solves two nonlinear equations for two parameters that fit a globally exponential or linear source shape. Complex spatially dependent source shapes can be estimated from an irradiance profile by piecing together estimates from neighboring measurement pairs. Numerical tests illustrate the sensitivity of the algorithm to depth, measurement spacing, chlorophyll concentration, sensor noise, and uncertainty in the a priori assumed inherent optical properties. The algorithm works well with widely spaced measurements, moderate sensor noise, and uncertainties in the optical properties regardless of whether the assumed and true profiles are the same shape.

© 1998 Optical Society of America

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References

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  1. A. Morel, L. Prieur, “Analysis of variations in ocean colour,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  2. J. G. Morin, “Coastal bioluminescence: patterns and functions,” Bull. Mar. Sci. 33, 787–817 (1993).
  3. Y. Ge, H. R. Gordon, K. J. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the ocean: variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
    [PubMed]
  4. J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).
  5. H. Y. Li, M. N. Özışik, “Estimation of the radiation source term with a conjugate–gradient method of inverse analysis,” J. Quant. Spectrosc. Radiat. Transfer 48, 237–244 (1992).
    [CrossRef]
  6. C. E. Siewert, “An inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 50, 603–609 (1993).
    [CrossRef]
  7. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  8. N. J. McCormick, C. E. Siewert, “Particular solutions for the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 46, 519–522 (1991).
    [CrossRef]
  9. R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
    [CrossRef] [PubMed]
  10. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.
  11. 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,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  12. H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.
  13. C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.
  14. P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
    [CrossRef]
  15. R. D. O’Dell, “Revised user’s manual for onedant: a code package for ONE-dimensional, Diffusion-Accelerated, Neutral-particle Transport,” Los Alamos National Laboratory LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).
  16. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  17. I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
    [CrossRef]
  18. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  19. R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
    [CrossRef]
  20. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 796.

1994 (2)

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

1993 (4)

Y. Ge, H. R. Gordon, K. J. Voss, “Simulation of inelastic-scattering contributions to the irradiance field in the ocean: variation in Fraunhofer line depths,” Appl. Opt. 32, 4028–4036 (1993).
[PubMed]

J. G. Morin, “Coastal bioluminescence: patterns and functions,” Bull. Mar. Sci. 33, 787–817 (1993).

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

C. E. Siewert, “An inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 50, 603–609 (1993).
[CrossRef]

1992 (2)

H. Y. Li, M. N. Özışik, “Estimation of the radiation source term with a conjugate–gradient method of inverse analysis,” J. Quant. Spectrosc. Radiat. Transfer 48, 237–244 (1992).
[CrossRef]

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

1991 (2)

N. J. McCormick, C. E. Siewert, “Particular solutions for the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 46, 519–522 (1991).
[CrossRef]

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

1990 (1)

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

1981 (2)

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

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,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

1977 (1)

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

Baker, K. S.

Bernstein, S.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Case, J. F.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Case, K. M.

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

Ferer, K.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Francisco, P. W.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

Ge, Y.

Geiger, M.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Gordon, H.

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

Gordon, H. R.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 796.

Kušcer, I.

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

Lapota, D.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Latz, M. I.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Li, H. Y.

H. Y. Li, M. N. Özışik, “Estimation of the radiation source term with a conjugate–gradient method of inverse analysis,” J. Quant. Spectrosc. Radiat. Transfer 48, 237–244 (1992).
[CrossRef]

McCormick, N. J.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

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

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

N. J. McCormick, C. E. Siewert, “Particular solutions for the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 46, 519–522 (1991).
[CrossRef]

Mobley, C. D.

C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.

Morel, A.

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

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

Morin, J. G.

J. G. Morin, “Coastal bioluminescence: patterns and functions,” Bull. Mar. Sci. 33, 787–817 (1993).

O’Dell, R. D.

R. D. O’Dell, “Revised user’s manual for onedant: a code package for ONE-dimensional, Diffusion-Accelerated, Neutral-particle Transport,” Los Alamos National Laboratory LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).

Özisik, M. N.

H. Y. Li, M. N. Özışik, “Estimation of the radiation source term with a conjugate–gradient method of inverse analysis,” J. Quant. Spectrosc. Radiat. Transfer 48, 237–244 (1992).
[CrossRef]

Prieur, L.

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,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 796.

Sanchez, R.

Sathyendranath, S.

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,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “An inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 50, 603–609 (1993).
[CrossRef]

N. J. McCormick, C. E. Siewert, “Particular solutions for the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 46, 519–522 (1991).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

Smith, R. C.

Tao, Z.

Voss, K. J.

Widder, E. A.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Young, D.

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Zweifel, P. F.

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

Appl. Opt. (3)

Bull. Mar. Sci. (1)

J. G. Morin, “Coastal bioluminescence: patterns and functions,” Bull. Mar. Sci. 33, 787–817 (1993).

J. Quant. Spectrosc. Radiat. Transfer (4)

N. J. McCormick, C. E. Siewert, “Particular solutions for the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 46, 519–522 (1991).
[CrossRef]

H. Y. Li, M. N. Özışik, “Estimation of the radiation source term with a conjugate–gradient method of inverse analysis,” J. Quant. Spectrosc. Radiat. Transfer 48, 237–244 (1992).
[CrossRef]

C. E. Siewert, “An inverse source problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 50, 603–609 (1993).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “On computing the Chandrasekhar polynomials in high order and high degree,” J. Quant. Spectrosc. Radiat. Transfer 43, 201–205 (1990).
[CrossRef]

Limnol. Oceanogr. (4)

A. Morel, L. Prieur, “Analysis of variations in ocean colour,” Limnol. Oceanogr. 22, 709–722 (1977).
[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,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

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

Nav. Res. Rev. (1)

J. F. Case, E. A. Widder, S. Bernstein, K. Ferer, D. Young, M. I. Latz, M. Geiger, D. Lapota, “Assessment of marine bioluminescence,” Nav. Res. Rev. 45, 31–41 (1993).

Transp. Theory Stat. Phys. (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

Other (6)

R. D. O’Dell, “Revised user’s manual for onedant: a code package for ONE-dimensional, Diffusion-Accelerated, Neutral-particle Transport,” Los Alamos National Laboratory LA-9184-M (Los Alamos National Laboratory, Los Alamos, New Mexico, 1989).

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

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983), pp. 49–53.

C. D. Mobley, “The optical properties of water,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995), pp. 43.1–43.56.

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 796.

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Figures (8)

Fig. 1
Fig. 1

Comparison of actual source and linear model with widely spaced measurements in water with τmax = 50 and no sensor noise; C = 1 mg/m3.

Fig. 2
Fig. 2

Comparison of actual source and linear model for various measurement separations in water with τmax = 50 and no sensor noise; C = 1 mg/m3.

Fig. 3
Fig. 3

Comparison of actual source and exponential model for various measurement separations in water with τmax = 50 and no sensor noise; C = 1 mg/m3.

Fig. 4
Fig. 4

Effect of uncertainty in the single-scattering albedo on the linear model for Δτ = 2 measurement separations in water with τmax = 50 and no sensor noise; C = 1 mg/m3.

Fig. 5
Fig. 5

Effect of uncertainty in the shape of the VSF on the linear model for Δτ = 2 measurement separations in water with τmax = 50 and no sensor noise; C = 1 mg/m3.

Fig. 6
Fig. 6

Effect of sensor noise on the linear model source estimates for Δτ = 2 measurement separations in water with τmax = 50 and with sensor noise. Error bars show 95% confidence bounds on the source estimates. Results were averaged over 25,000 runs with randomly sampled, normally distributed noise with a 5% standard deviation; C = 1 mg/m3.

Fig. 7
Fig. 7

Effect of sensor noise on the exponential model source estimates for Δτ = 2 measurement separations in water with τmax = 50 and no sensor noise. Error bars show 95% confidence bounds on the source estimates. Results were averaged over 25,000 runs with randomly sampled, normally distributed noise with a 5% standard deviation; C = 1 mg/m3.

Fig. 8
Fig. 8

Effect of surface illumination on the exponential model for Δτ = 1 measurement separations in water with τmax = 50 and no sensor noise. Surface illumination is isotropic with a magnitude of 10,000 Wm-2 sr-1; C = 1 mg/m3.

Tables (1)

Tables Icon

Table 1 Theoretical Optical Properties for Numerical Tests at 685 nm, with a Delta–Eddington Phase Function Cutoff Angle of 7.94°

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

μ   τ   L τ ,   μ + L τ ,   μ = ω 0 2 - 1 1   β ˜ μ ,   μ L τ ,   μ d μ + 1 2   S τ .
L τ ,   μ = L ˜ τ ,   μ + L asy τ ,   μ ,
L asy τ ,   μ = A ν 1 ϕ ν 1 μ exp - τ / ν 1 + A - ν 1 ϕ - ν 1 μ exp τ / ν 1 .
E ± τ = 0 ± 1   μ L τ ,   μ d μ .
E ± τ = E ˜ ± τ + E asy , ± τ ,
E ˜ ± τ = E t , ± τ + E p , ± τ ,
E asy , ± τ = A ν 1 g ± ν 1 exp - τ / ν 1 + A - ν 1 g ± ν 1 exp τ / ν 1 ,
g ± ν 1 = 0 ± 1   μ ϕ ν 1 μ d μ .
g + ν 1 E ± τ - g - ν 1 E ± τ = g + ν 1 E ˜ ± τ - g - ν 1 E ˜ ± τ + A ± ν 1 g + 2 ν 1 - g - 2 ν 1 exp ± τ / ν 1 .
± E ,   τ 1 ,   τ 2 = ± E ˜ ,   τ 1 ,   τ 2 ,
± E ,   τ 1 ,   τ 2 = E + τ 2 - β ± 1 E - τ 2 - exp ± Δ τ / ν 1 × E + τ 1 - β ± 1 E - τ 1 ,
β = g + ν 1 g - ν 1 .
± E ,   τ 1 ,   τ 2 ± E p ,   τ 1 ,   τ 2 ,
S τ = Q 1 - α τ - τ a ,   τ a τ τ b ,   otherwise   = 0   linear , = Q   exp - α τ   exponential ,
L p τ ,   μ = σ + U τ ,   ν ϕ ν μ + V τ ,   μ ϕ - ν μ d ν N ν ,
U τ ,   ν = 0 τ   S t exp - τ - t / ν d t , V τ ,   ν = τ τ max   S t exp τ - t / ν d t .
g N + 1 ξ j = 0 ,   j = 1   to   N + 1 ,
σ +   ϕ ν μ f ν d ν N ν   j = 1 J   w j μ f ξ j ,
w j μ = C j ξ j l = 1 N 2 l + 1 2   g l ξ j P l μ ,
C j = k = 1 J   h 2 k - 2 g 2 k - 2 2 ξ j - 1 .
L p τ ,   μ j = 1 J w j μ U τ ,   ξ j + w j - μ V τ ,   - ξ j .
E p , ± τ j = 1 J w j e U τ ,   ξ j + V τ ,   ξ j ± w j o U τ ,   ξ j - V τ ,   ξ j ,
a mix λ = a water + 0.06 C 0.602 a C λ a C 440   nm
b mix λ = b water + C 0.62 b C 550   nm 550 λ n ,
S t = 0 ,   τ τ 1 ,   τ τ 3 , = d 1 + d 2 τ + d 3 τ 2 ,   τ 1 < τ < τ 2 , = d 4 exp - α τ ,   τ 2 < τ < τ 3 .
S τ 1 = 0 , S τ 2 + = S τ 2 - , d S / d τ | τ 2 + = d S / d τ | τ 2 - , max   S τ = 100 .
μ   τ   L τ ,   μ + cL τ ,   μ = b 2 n = 0 2 n + 1 f n P n μ × - 1 1   P n μ L τ ,   μ d μ + c 2   S τ ,
ϕ ν j μ = ω 2 ν j ν j - μ   g ν j ,   μ ,   for   ν j     - 1 ,   1 ,
g ν ,   μ = n = 0 N 2 n + 1 f n g n ν P n μ .
g n ν = - 1 1   P n μ ϕ ν μ d μ .
n + 1 g n + 1 ν - 2 n + 1 1 - ω f n ν g n ν + ng n - 1 ν = 0 ,   n 0 .
L τ ,   μ = σ + A ν ϕ ν μ exp - τ / ν + A - ν ϕ - ν μ exp τ / ν d ν ,
E ± τ = E p , ± τ
E ± τ i E p , ± τ i
U τ ,   ξ = 0 ,   τ < τ a , = Q 1 + α ξ 1 - exp - τ - τ a / ξ - α τ - τ a ,   τ a < τ < τ b , = Q   exp - τ - τ b / ξ 1 + α ξ × 1 - exp - τ b - τ a / ξ - α τ b - τ a ,   τ > τ b , V τ ,   ξ = Q   exp - τ a - τ / ξ × 1 - α τ b - τ a + ξ × 1 - exp - τ b - τ a / ξ + α τ b - τ a ,   τ < τ a , = Q 1 - α τ b - τ a + ξ 1 - exp - τ b - τ / ξ + α τ b - τ ,   τ a < τ < τ b , = 0 ,   τ > τ b .
U τ ,   ξ = Q   exp - α τ 1 - exp - 1 - α ξ τ / ξ 1 - α ξ , V τ ,   ξ = Q   exp - α τ 1 - exp - 1 + α ξ τ b - τ / ξ 1 + α ξ .
1 - exp - δ a δ a   k = 0 K - δ a k k + 1 ! .
E p , ± τ j = 1 J w j , ± U τ ,   ξ j + w j , ± V τ ,   ξ j ,
w j , ± = 1 ± 1   w j μ μ d μ , = C j l = 0 N 2 l + 1 2 - 1 ± l g l ξ j 0 1   P l μ μ d μ , = w j e ± w j o ,
w j e = C j n = 0 , n   even N 2 n + 1 2   g n ν j 0 1   P n μ μ d μ , w j o = C j g 1 ξ j / 2 .
0 1   P ν x x σ d x = π 2 - σ - 1 Γ 1 + σ Γ 1 + σ 2 - ν 2 Γ σ 2 + ν 2 + 3 2 ,
0 1   P n + 2 μ μ d μ 0 1   P n μ μ d μ = - n - 1 n + 4 ,   n 2 ,   n   even .

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