Abstract

A synthetic model for the scattering phase function is used to develop simple algebraic equations, valid for any water type, for evaluating the ratio of the backscattering to absorption coefficients of spatially uniform, very deep waters with data from upward and downward planar irradiances and the remotely sensed reflectance. The phase function is a variable combination of a forward-directed Dirac delta function plus isotropic scattering, which is an elementary model for strongly forward scattering such as that encountered in oceanic optics applications. The incident illumination at the surface is taken to be diffuse plus a collimated beam. The algorithms are compared with other analytic correlations that were previously derived from extensive numerical simulations, and they are also numerically tested with forward problem results computed with a modified F N method.

© 2006 Optical Society of America

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References

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  1. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  2. H. R. Gordon, "Inverse methods in hydrologic optics," Oceanologia 44, 9-58 (2002).
  3. Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, "Hyperspectral remote sensing for shallow waters. 1. A semianalytical model," Appl. Opt. 37, 6329-6337 (1998).
    [CrossRef]
  4. Z. Lee, K. L. Carder, and K. Du, "Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance," Appl. Opt. 43, 4957-4964 (2004).
    [CrossRef] [PubMed]
  5. C. D. Mobley, L. K. Sundman, C. O. Davis, J. H. Bowles, T. V. Downes, R. A. Leathers, M. J. Montes, W. P. Bissett, D. D. R. Kohler, R. P. Reid, E. M. Louchard, and A. Gleason, "Interpretation of hyperspectral remote-sensing imagery by spectrum matching and look-up tables," Appl. Opt. 44, 3576-3592 (2005).
    [CrossRef] [PubMed]
  6. E. S. Chalhoub and H. F. Campos Velho, "Estimation of the optical properties of seawater from measurements of exit radiance," J. Quant. Spectrosc. Radiat. Transfer 72, 551-565 (2002).
    [CrossRef]
  7. H. Gordon, O. B. Brown, and M. M. Jacobs, "Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean," Appl. Opt. 14, 417-427 (1975).
    [CrossRef] [PubMed]
  8. A. Morel and L. Prieur, "Analysis of variations in ocean color," Limnol. Oceanogr. 22, 709-722 (1977).
    [CrossRef]
  9. H. R. Gordon and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery (Springer-Verlag, 1983).
  10. J. T. O. Kirk, "Volume scattering function, average cosines, and the underwater light field," Limnol. Oceanogr. 36, 455-467 (1991).
    [CrossRef]
  11. R. E. Walker, Marine Light Field Statistics (Wiley, 1994).
  12. A. Morel and B. Gentili, "Diffuse reflectance of oceanic waters. II. Bidirectional aspects," Appl. Opt. 32, 6864-6879 (1993).
    [CrossRef] [PubMed]
  13. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
    [CrossRef]
  14. M. M. R. Williams, "A synthetic scattering kernel for particle transport in absorbing media with anisotropic scattering," J. Phys. D 11, 2455-2463 (1978).
    [CrossRef]
  15. L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
    [CrossRef]
  16. V. I. Haltrin, "One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater," Appl. Opt. 41, 1022-1028 (2002).
    [CrossRef] [PubMed]
  17. C. D. Mobley, L. K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
    [CrossRef] [PubMed]
  18. B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).
  19. C. E. Siewert, "On the inverse problem for a three-term phase function," J. Quant. Spectrosc. Radiat. Transfer 22, 441-446 (1979), Eq. (31).
    [CrossRef]
  20. N. J. McCormick, "Transport scattering coefficients from reflection and transmission measurements," J. Math. Phys. 20, 1504-1507 (1979).
    [CrossRef]
  21. N. J. McCormick, "Analytic inverse radiative transfer equations for atmospheric and hydrologic optics," J. Opt. Soc. Am. A 21, 1009-1017 (2004).
    [CrossRef]
  22. C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
    [CrossRef]
  23. R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
    [CrossRef]
  24. C. E. Siewert, "The FN method for solving radiative-transfer problems in plane geometry," Astrophys. Space Sci. 58, 131-137 (1978).
    [CrossRef]
  25. R. D. M. Garcia and C. E. Siewert, "Benchmark results in radiative transfer," Transp. Theory Stat. Phys. 14, 437-483 (1985).
    [CrossRef]
  26. R. D. M. Garcia, "A review of the Facile (FN) method in particle transport theory," Transp. Theory Stat. Phys. 14, 391-435 (1985).
    [CrossRef]
  27. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  28. N. J. McCormick, "Asymptotic optical attenuation," Limnol. Oceanogr. 37, 1570-1578 (1992).
    [CrossRef]

2005 (1)

2004 (3)

2003 (1)

C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
[CrossRef]

2002 (4)

V. I. Haltrin, "One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater," Appl. Opt. 41, 1022-1028 (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).

E. S. Chalhoub and H. F. Campos Velho, "Estimation of the optical properties of seawater from measurements of exit radiance," J. Quant. Spectrosc. Radiat. Transfer 72, 551-565 (2002).
[CrossRef]

1998 (1)

1993 (1)

1992 (1)

N. J. McCormick, "Asymptotic optical attenuation," Limnol. Oceanogr. 37, 1570-1578 (1992).
[CrossRef]

1991 (1)

J. T. O. Kirk, "Volume scattering function, average cosines, and the underwater light field," Limnol. Oceanogr. 36, 455-467 (1991).
[CrossRef]

1988 (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

1985 (2)

R. D. M. Garcia and C. E. Siewert, "Benchmark results in radiative transfer," Transp. Theory Stat. Phys. 14, 437-483 (1985).
[CrossRef]

R. D. M. Garcia, "A review of the Facile (FN) method in particle transport theory," Transp. Theory Stat. Phys. 14, 391-435 (1985).
[CrossRef]

1979 (2)

C. E. Siewert, "On the inverse problem for a three-term phase function," J. Quant. Spectrosc. Radiat. Transfer 22, 441-446 (1979), Eq. (31).
[CrossRef]

N. J. McCormick, "Transport scattering coefficients from reflection and transmission measurements," J. Math. Phys. 20, 1504-1507 (1979).
[CrossRef]

1978 (2)

M. M. R. Williams, "A synthetic scattering kernel for particle transport in absorbing media with anisotropic scattering," J. Phys. D 11, 2455-2463 (1978).
[CrossRef]

C. E. Siewert, "The FN method for solving radiative-transfer problems in plane geometry," Astrophys. Space Sci. 58, 131-137 (1978).
[CrossRef]

1977 (1)

A. Morel and L. Prieur, "Analysis of variations in ocean color," Limnol. Oceanogr. 22, 709-722 (1977).
[CrossRef]

1975 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

Bissett, W. P.

Boss, E.

Bowles, J. H.

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

H. Gordon, O. B. Brown, and M. M. Jacobs, "Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean," Appl. Opt. 14, 417-427 (1975).
[CrossRef] [PubMed]

Campos Velho, H. F.

E. S. Chalhoub and H. F. Campos Velho, "Estimation of the optical properties of seawater from measurements of exit radiance," J. Quant. Spectrosc. Radiat. Transfer 72, 551-565 (2002).
[CrossRef]

Carder, K. L.

Case, K. M.

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

Chalhoub, E. S.

E. S. Chalhoub and H. F. Campos Velho, "Estimation of the optical properties of seawater from measurements of exit radiance," J. Quant. Spectrosc. Radiat. Transfer 72, 551-565 (2002).
[CrossRef]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

Davis, C. O.

Davison, B.

B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).

Downes, T. V.

Du, K.

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, "A review of the Facile (FN) method in particle transport theory," Transp. Theory Stat. Phys. 14, 391-435 (1985).
[CrossRef]

R. D. M. Garcia and C. E. Siewert, "Benchmark results in radiative transfer," Transp. Theory Stat. Phys. 14, 437-483 (1985).
[CrossRef]

Gentili, B.

Gleason, A.

Gordon, H.

Gordon, H. R.

H. R. Gordon, "Inverse methods in hydrologic optics," Oceanologia 44, 9-58 (2002).

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

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

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Güleçyüz, M. Ç.

R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
[CrossRef]

C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
[CrossRef]

Haltrin, V. I.

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Jacobs, M. M.

Kaskas, A.

R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
[CrossRef]

C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, "Volume scattering function, average cosines, and the underwater light field," Limnol. Oceanogr. 36, 455-467 (1991).
[CrossRef]

Kohler, D. D. R.

Leathers, R. A.

Lee, Z.

Louchard, E. M.

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]

N. J. McCormick, "Asymptotic optical attenuation," Limnol. Oceanogr. 37, 1570-1578 (1992).
[CrossRef]

N. J. McCormick, "Transport scattering coefficients from reflection and transmission measurements," J. Math. Phys. 20, 1504-1507 (1979).
[CrossRef]

Mobley, C. D.

Montes, M. J.

Morel, A.

Morel, A. Y.

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

Patch, J. S.

Prieur, L.

A. Morel and L. Prieur, "Analysis of variations in ocean color," Limnol. Oceanogr. 22, 709-722 (1977).
[CrossRef]

Reid, R. P.

Siewert, C. E.

R. D. M. Garcia and C. E. Siewert, "Benchmark results in radiative transfer," Transp. Theory Stat. Phys. 14, 437-483 (1985).
[CrossRef]

C. E. Siewert, "On the inverse problem for a three-term phase function," J. Quant. Spectrosc. Radiat. Transfer 22, 441-446 (1979), Eq. (31).
[CrossRef]

C. E. Siewert, "The FN method for solving radiative-transfer problems in plane geometry," Astrophys. Space Sci. 58, 131-137 (1978).
[CrossRef]

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

Steward, R. G.

Sundman, L. K.

Tezcan, C.

R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
[CrossRef]

C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
[CrossRef]

Türeci, R. G.

R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
[CrossRef]

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

Williams, M. M. R.

M. M. R. Williams, "A synthetic scattering kernel for particle transport in absorbing media with anisotropic scattering," J. Phys. D 11, 2455-2463 (1978).
[CrossRef]

Zweifel, P. F.

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

Appl. Opt. (7)

Astrophys. J. (1)

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Astrophys. Space Sci. (1)

C. E. Siewert, "The FN method for solving radiative-transfer problems in plane geometry," Astrophys. Space Sci. 58, 131-137 (1978).
[CrossRef]

J. Geophys. Res. (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10909-10924 (1988).
[CrossRef]

J. Math. Phys. (1)

N. J. McCormick, "Transport scattering coefficients from reflection and transmission measurements," J. Math. Phys. 20, 1504-1507 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

M. M. R. Williams, "A synthetic scattering kernel for particle transport in absorbing media with anisotropic scattering," J. Phys. D 11, 2455-2463 (1978).
[CrossRef]

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

C. Tezcan, A. Kaskas, and M. Ç. Güleçyüz, "The HN method for solving linear transport equation: theory and applications," J. Quant. Spectrosc. Radiat. Transfer 78, 243-254 (2003).
[CrossRef]

R. G. Türeci, M. Ç. Güleçyüz, A. Kaskas, and C. Tezcan, "Application of the HN method to the critical slab problem for reflecting boundary conditions," J. Quant. Spectrosc. Radiat. Transfer 88, 499-517 (2004).
[CrossRef]

C. E. Siewert, "On the inverse problem for a three-term phase function," J. Quant. Spectrosc. Radiat. Transfer 22, 441-446 (1979), Eq. (31).
[CrossRef]

E. S. Chalhoub and H. F. Campos Velho, "Estimation of the optical properties of seawater from measurements of exit radiance," J. Quant. Spectrosc. Radiat. Transfer 72, 551-565 (2002).
[CrossRef]

Limnol. Oceanogr. (3)

N. J. McCormick, "Asymptotic optical attenuation," Limnol. Oceanogr. 37, 1570-1578 (1992).
[CrossRef]

J. T. O. Kirk, "Volume scattering function, average cosines, and the underwater light field," Limnol. Oceanogr. 36, 455-467 (1991).
[CrossRef]

A. Morel and L. Prieur, "Analysis of variations in ocean color," Limnol. Oceanogr. 22, 709-722 (1977).
[CrossRef]

Oceanologia (1)

H. R. Gordon, "Inverse methods in hydrologic optics," Oceanologia 44, 9-58 (2002).

Transp. Theory Stat. Phys. (2)

R. D. M. Garcia and C. E. Siewert, "Benchmark results in radiative transfer," Transp. Theory Stat. Phys. 14, 437-483 (1985).
[CrossRef]

R. D. M. Garcia, "A review of the Facile (FN) method in particle transport theory," Transp. Theory Stat. Phys. 14, 391-435 (1985).
[CrossRef]

Other (5)

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

B. Davison, Neutron Transport Theory (Oxford U. Press, 1957).

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

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

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

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Tables (4)

Tables Icon

Table 1 Tests for ϖ from the Radiance Algorithms of Eqs. (20) and (21) and Irradiance Algorithms of Approximations (28) and (29) with Diffuse Illumination ( f = 0)

Tables Icon

Table 2 Tests for ϖ from the Radiance Algorithms of Eqs. (20) and (21) and Irradiance Algorithms of Approximations (28) and (29) with Collimated, Perpendicular Illumination ( f = 1, μ 0 = 1)

Tables Icon

Table 3 Tests for ϖ from the Radiance Algorithms of Eqs. (20) and (21) and Irradiance Algorithms of Approximations (28) and (29) with General Illumination (0 < f < 1)

Tables Icon

Table 4 More Tests for ϖ from the Radiance Algorithms of Eqs. (20) and (21) and Irradiance Algorithms of Approximations (28) and (29) with General Illumination (0 < f < 1)

Equations (59)

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μ x L ( x , μ , φ ) + L ( x , μ , φ ) = ω 0 2 π 1 1 β ˜ ( μ ˜ ) L ( x , μ , φ ) , x 0 ,
L ( 0 , μ , φ ) = E d [ ( f / μ 0 ) δ ( μ μ 0 ) δ ( φ φ 0 ) + π 1 ( 1 f ) ] , 0 μ 1 , 0 φ 2 π .
L ( x , μ ) = 0 2 π L ( x , μ , φ ) d φ
μ x L ( x , μ ) + L ( x , μ ) = ω 1 1 β ˜ ( μ , μ ) × L ( x , μ ) d μ , x   ≥   0 ,
L ( 0 , μ ) = E d [ ( f / μ 0 ) δ ( μ μ 0 ) + 2 ( 1 f ) ] , 0 μ 1 .
β ˜ ( μ , μ ) = ( 1 / 2 ) [ 1 g + 2 g δ ( μ μ ) ] ,
b ˜ b = 2 π β ˜ ( μ ˜ ) d μ ˜ 4 π β ˜ ( μ ˜ ) d μ ˜ = 1 g 2 ,
b ˜ b = 1 g 2 g ( 1 + g 1 + g 2 1 ) .
μ x L ( x , μ ) + L ( x , μ ) = ω ( 1 g ) 2 1 1 L ( x , μ ) d μ + ω g L ( x , μ ) , x 0.
τ = x ( 1 ω g ) ,
ϖ = ω ( 1 g ) 1 ω g ,
μ τ L ( τ , μ ) + L ( τ , μ ) = ϖ 2 1 1 L ( τ , μ ) d μ , τ 0.
ϖ = 2 b b / a 1 + 2 b b / a ,
ϖ 1 ϖ = ω ( 1 g ) 1 ω = 2 b b a .
ϖ = 2 G ( 1 + G ) 1 2 G ( 1 G + G 2 ) ,
ϖ 1 ϖ = 2 G ( 1 G ) 1 2 G ( 1 + G + G 2 + ) .
ϖ [ 1 1 L ( 0 , μ ) d μ ] 2 = 4 0 1 L ( 0 , μ ) L ( 0 , −μ ) d μ .
1 1 L ( τ , μ ) d μ = ( 1 ϖ ) 1 d d τ 1 1 μ L ( τ , μ ) d μ ,
ϖ 1 ϖ [ 1 1 μ L ( 0 , μ ) d μ ] 2 = 4 0 1 μ 2 L ( 0 , μ ) L ( 0 , μ ) d μ .
E d = 0 1 μ L ( 0 , μ ) d μ , E u = 0 1 μ L ( 0 , −μ ) d μ ,
E 0 u = 0 1 L ( 0 , −μ ) d μ ,
R = E u / E d , R 0 = E 0 u / E d , R μ 0 = L ( 0 , −μ 0 ) / E d ,
ϖ = 4 [ ( f / μ 0 ) R μ 0 + 2 ( 1 f ) R 0 ] [ 2 ( 1 f ) + ( f / μ 0 ) + R 0 ] 2 ,
ϖ 1 ϖ = 4 [ f μ 0 R μ 0 + 2 ( 1 f ) E d - 1 × 0 1 μ 2 L ( 0 , −μ ) d μ ] / ( 1 R ) 2 .
L ( 0 , −μ ) = n = 0 N a n μ n , 0 μ 1 ,
R = E d - 1 n = 0 N a n n + 2 , R 0 = E d - 1 n = 0 N a n n + 1 ,
R μ 0 = E d - 1 n = 0 N a n μ 0 n .
ϖ [ 2 ( 1 f ) + ( f / μ 0 ) + E d - 1 n = 0 N a n n + 1 ] 2 = 4 E d - 1 n = 0 N a n [ f μ 0 n - 1 + 2 ( 1 f ) n + 1 ] ,
ϖ 1 ϖ [ 1 E d - 1 n = 0 N a n n + 2 ] 2 = 4 E d - 1 n = 0 N a n [ f μ 0 n + 1 + 2 ( 1 f ) n + 3 ] .
R r s = L u ( 0 , −1 ) / E d ,
R μ 0 = 1 = 2 π R r s .
R 0 = E d - 1 0 1 L ( 0 , −μ ) d μ 2 E d - 1 0 1 μ L ( 0 , −μ ) d μ = 2 R ,
E d - 1 0 1 μ 2 L ( 0 , −μ ) d μ
≈  ( 2 / 3 ) E d - 1 0 1 μ L ( 0 , −μ ) d μ = ( 2 / 3 ) R ,
ϖ 2 G ( 1 G ) 4 [ ( f / μ 0 ) R μ 0 + 4 ( 1 f ) R ] [ 2 ( 1 f ) + ( f / μ 0 ) + 2 R ] 2 ,
ϖ 1 ϖ 2 G ( 1 + G ) 4 [ f μ 0 R μ 0 + ( 4 / 3 ) ( 1 f ) R ] ( 1 R ) 2 .
R r s ( 4 π ) 1 ( 1 + 2 R ) 2 G ,
R r s ( 4 π ) 1 ( 1 R ) 2 G ,
R r s = E u E d E u L u ( 0 , −1 ) R Q = 0.0949 G ,
R 3 8 ( 1 R ) 2 G .
R G 2 ( 1 + R ) 2 ,
Δ L j     N = | 0 1 μ j [ L N + 1 ( 0 , −μ ) L N ( 0 , −μ ) ] d μ | ,
j = 0 , 1 , 2 ,   .
L ( τ , μ ) = A ( ν 0 ) ϕ ( ν 0 , μ ) exp ( - τ / ν 0 ) + A ( ν 0 )
× ϕ ( ν 0 , μ ) exp ( τ / ν 0 ) + 1 1 A ( ν ) ϕ ( ν , μ ) × exp ( - τ / ν ) d ν , −1 μ 1 ,
ϕ ( ν 0 , μ ) = ϖ ν 0 2 1 ν 0 + μ ,
ϕ ( ν , μ ) = ϖν 2 1 ν + μ , 0 μ 1 , 0 ν 1 ,
1 ( ϖ ν 0 / 2 ) ln [ ( ν 0 + 1 ) / ( ν 0 1 ) ] = 0.
A ( ξ ) N ( ξ ) = 0 1 μ ϕ ( ξ , μ ) L ( 0 , μ ) d μ 0 1 μ ϕ ( ξ , μ ) L ( 0 , −μ ) d μ
n = 0 N a n 0 1 μ n + 1 ϕ ( ξ , μ ) d μ = 0 1 μ ϕ ( ξ , μ ) L ( 0 , μ ) d μ
n = 0 N a n 0 1 μ n + 1 ϕ ˜ m ( −μ ) d μ = 0 1 μ ϕ ˜ m ( μ ) L ( 0 , μ ) d μ , m = 0   to   N ,
ϕ ˜ m ( μ ) = σ ξ m ϕ ( ξ , μ ) d ξ = ν 0     m ϕ ( ν 0 , μ ) + 0 1 ν m ϕ ( ν , μ ) d ν .
n = 0 N a n A n m = S m , m = 0   to   N .
ϕ ˜ m ( μ ) = ϖ 2 [ ν 0     m + 1 m + 1 ] μ ϕ ˜ m 1 ( μ ) ,
ϕ ˜ 0 ( μ ) = ϖ 2 [ ν 0 ν 0 + μ + 1 μ   ln ( 1 + 1 μ ) ] .
A n m = ϖ 2 ( n + 2 ) ( ν 0     m + 1 m + 1 ) A n + 1 , m 1 ,
A j 0 = ϖ 2 0 1 μ j + 1 [ ν 0 ν 0 + μ + 1 μ   ln ( 1 + 1 μ ) ] d μ ,
j = 0 , 1.
S m = E d [ f ϕ ˜ m ( μ 0 ) + 2 ( 1 f ) A 0 m ] .

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