Abstract

We have applied the discrete-ordinate method to solve the radiative-transfer problem pertaining to a system consisting of two strata with different indices of refraction. The refraction and reflection at the interface are taken into account. The relevant changes (as compared with the standard problem with a constant index of refraction throughout the medium) in formulation and solution of the radiative-transfer equation, including the proper application of interface and boundary conditions, are described. Appropriate quadrature points (streams) and weights are chosen for the interface-continuity relations. Examples of radiative transfer in the coupled atmosphere–ocean system are provided. To take into account the region of total reflection in the ocean, additional angular quadrature points are required, compared with those used in the atmosphere and in the refractive region of the ocean that communicates directly with the atmosphere. To verify the model we have tested for energy conservation. We also discuss the effect of the number of streams assigned to the refractive region and the total reflecting region on the convergence. Our results show that the change in the index of refraction between the two strata significantly affects the radiation field. The radiative-transfer model we present is designed for application to the atmosphere–ocean system, but it can be applied to other systems that need to consider the change in the index of refraction between two strata.

© 1994 Optical Society of America

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References

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  1. K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef] [PubMed]
  2. C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
    [CrossRef]
  3. H. Gordon, M. Wang, “Surface-roughness considerations for atmospheric correction of ocean color sensor. I: The Rayleigh-scattering component,” Appl. Opt. 31, 4247–4260 (1992).
    [CrossRef] [PubMed]
  4. G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
    [CrossRef]
  5. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular-scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
    [CrossRef] [PubMed]
  6. J. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Division of Plant Industry Tech. Paper 36 (Commonwealth Scientific and Industrial Research Organization, Canberra, Australia, 1981), p. 16.
  7. T. Nakajima, M. Tananka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
    [CrossRef]
  8. S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
    [CrossRef]
  9. K. Stamnes, “The theory of multiple scattering of radiation in plane-parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
    [CrossRef]
  10. K. Stamnes, R. A. Swanson, “A new look at the discrete-ordinate method for radiative-transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
    [CrossRef]
  11. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960) p. 12.
  12. M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
    [CrossRef]
  13. G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
    [CrossRef] [PubMed]
  14. W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative-flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
    [CrossRef]
  15. R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant-energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489, 62–67 (1984).
  16. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).
  17. R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1981).
    [CrossRef] [PubMed]
  18. A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geol. 5, 335–349 (1982).
  19. C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

1992 (1)

1991 (1)

1990 (1)

S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
[CrossRef]

1989 (2)

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

1988 (1)

1986 (1)

K. Stamnes, “The theory of multiple scattering of radiation in plane-parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[CrossRef]

1983 (1)

T. Nakajima, M. Tananka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

1982 (1)

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geol. 5, 335–349 (1982).

1981 (3)

1977 (2)

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative-flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

Adams, C.

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

Baker, K. S.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960) p. 12.

Fenn, R. W.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

Gallegos, C.

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant-energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489, 62–67 (1984).

Garing, J. S.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

Gentili, B.

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular-scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Gordon, H.

H. Gordon, M. Wang, “Surface-roughness considerations for atmospheric correction of ocean color sensor. I: The Rayleigh-scattering component,” Appl. Opt. 31, 4247–4260 (1992).
[CrossRef] [PubMed]

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Humphreys, T.

Jayaweera, K.

S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
[CrossRef]

K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

Jin, Z.

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Kattawar, G.

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

G. Plass, T. Humphreys, G. Kattawar, “Ocean–atmosphere interface: its influence on radiation,” Appl. Opt. 20, 917–931 (1981).
[CrossRef] [PubMed]

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Kirk, J.

J. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Division of Plant Industry Tech. Paper 36 (Commonwealth Scientific and Industrial Research Organization, Canberra, Australia, 1981), p. 16.

McClatchey, R. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

Mobley, C.

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Morel, A

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Morel, A.

Nakajima, T.

T. Nakajima, M. Tananka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Plass, G.

Reinersman, P.

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Schiebe, R.

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant-energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489, 62–67 (1984).

Selby, J. E. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

Smith, R. C.

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geol. 5, 335–349 (1982).

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

Stamnes, K.

S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
[CrossRef]

K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

K. Stamnes, “The theory of multiple scattering of radiation in plane-parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete-ordinate method for radiative-transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

Stavn, R.

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant-energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489, 62–67 (1984).

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A new look at the discrete-ordinate method for radiative-transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Tanaka, M.

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Tananka, M.

T. Nakajima, M. Tananka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

Tsay, S. C.

S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
[CrossRef]

K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

Volz, F. E.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

Wang, M.

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative-flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

Appl. Opt. (5)

J. Atmos. Sci. (2)

K. Stamnes, R. A. Swanson, “A new look at the discrete-ordinate method for radiative-transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative-flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

T. Nakajima, M. Tananka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

S. C. Tsay, K. Stamnes, K. Jayaweera, “Radiative transfer in stratified atmospheres: development and verification of a unified model,” J. Quant. Spectrosc. Radiat. Transfer 43, 133–148(1990).
[CrossRef]

M. Tanaka, T. Nakajima, “Effects of oceanic turbidity and index of refraction of hydrosols on the flux of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 18, 93–111 (1977).
[CrossRef]

Limnol. Oceanogr. (2)

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere–ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

Mar. Geol. (1)

A. Morel, R. C. Smith, “Terminology and units in optical oceanography,” Mar. Geol. 5, 335–349 (1982).

Rev. Geophys. (1)

K. Stamnes, “The theory of multiple scattering of radiation in plane-parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[CrossRef]

Other (5)

J. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Division of Plant Industry Tech. Paper 36 (Commonwealth Scientific and Industrial Research Organization, Canberra, Australia, 1981), p. 16.

C. Mobley, B. Gentili, H. Gordon, Z. Jin, G. Kattawar, A Morel, P. Reinersman, K. Stamnes, R. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. (to be published).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960) p. 12.

R. Stavn, R. Schiebe, C. Gallegos, “Optical controls on the radiant-energy dynamics of the air/water interface: the average cosine and the absorption coefficient,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 489, 62–67 (1984).

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, J. S. Garing, Rep. AFCRL–72–0497, (Air Force Cambridge Research Laboratories, Bedford, Mass., 1972).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the coupled-radiative-transfer model for the atmosphere–ocean system.

Fig. 2
Fig. 2

Variation of irradiances with the relative index of refraction at several locations for isotropic scattering, incident flux F 0 = 1.0, with incident zenith angle θ0 = 30°, bottom-surface albedo = 1.0, τ a = 1.0, and τ* = 2.0. For the absorption case, the only difference from the conserevative case is in the single-scattering albedo, ω = 0.9, in the lower medium.

Fig. 3
Fig. 3

The effect of stream combinations within the refractive region and the total reflective region (represented by the number ratio) on the convergence for the Henyey–Greenstein scattering-phase function, with asymmetry factor g = 0.7. Other input parameters are the same as those for the conservative case in Fig. 2. Shown in the panels are the upward irradiances at the tops of the slabs and their comparison with the Benchmark irradiance incident on the system.

Fig. 4
Fig. 4

Similar to Fig. 3 but showing the effect of scattering asymmetry on the irradiance computation (for only one group of stream combinations).

Fig. 5
Fig. 5

Spectral distributions of downward irradiances at the top of the atmosphere, just above the ocean surface (+0 m), just below the ocean surface (−0 m), and at several depths in the ocean for a model of a clear midlatitude atmosphere and pure sea water, with a solar zenith angle θ0 = 30°.

Fig. 6
Fig. 6

Similar to Fig. 5 but showing the upward irradiances.

Fig. 7
Fig. 7

Spectral distribution of ocean-surface reflectance for considering refraction and neglecting refraction, and for several solar zenith angles. Also shown are the downward and upward irradiance differences at the ocean surface when refraction is considered and neglected, ΔF = F Refr↓F Noref↓, ΔF = F Refr↑F Noref↑. The same atmospheric and oceanic models used in Fig. 5 are used here.

Fig. 8
Fig. 8

Distributions of the downward and the upward irradiances with height in the atmosphere and depth in the ocean, the results of neglecting refraction, and the relative deviation. 00 = 30°, λ = 500 nm.

Fig. 9
Fig. 9

Distributions of the total mean intensity (total scalar irradiance/4π) with height in the atmosphere and depth in the ocean, the result of neglecting refraction, and the relative deviation. 00 = 30°, λ = 500 nm.

Fig. 10
Fig. 10

Distributions of the azimuthally averaged intensity (radiance) for the refraction case (n = 1.33) and the no-refraction case (n = 1.0); 00 = 30°; λ = 600 nm. (a) Just below the ocean surface, (b) just above the ocean surface.

Equations (39)

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μ d I ( τ , μ , ϕ ) d τ = I ( τ , μ , ϕ ) - S ( τ , μ , ϕ ) ,
S ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π d ϕ - 1 1 p ( τ , μ , ϕ , μ , ϕ ) × I ( τ , μ , ϕ , ) d μ + Q ( τ , μ , ϕ ) ,
Q air ( τ , μ , ϕ ) = ω ( τ ) 4 π F 0 p ( τ , μ , ϕ , - μ 0 , ϕ 0 ) exp ( - τ / μ 0 ) + ω ( τ ) 4 π F 0 R ( - μ 0 , n ) p ( τ , μ , ϕ , μ 0 , ϕ 0 ) × exp [ - ( 2 τ a - τ ) / μ 0 ] ,
Q ocn ( τ , μ , ϕ ) = ω ( τ ) 4 π μ 0 μ 0 n ( μ 0 , n ) F 0 T ( - μ 0 , n ) × p ( τ , μ , ϕ , - μ 0 n , ϕ 0 ) exp ( - τ a / μ 0 ) × exp [ - ( τ - τ a ) / μ 0 n ] ,
μ 0 n ( μ 0 , n ) = 1 - ( 1 - μ 0 2 ) / n 2 .
I ( τ , μ , ϕ ) = m = 0 2 N - 1 I m ( τ , μ ) cos m ( ϕ - ϕ 0 )
p ( τ ; μ , ϕ , μ , ϕ ) p ( τ ; cos Θ ) = l = 0 2 N - 1 ( 2 l + 1 ) g l ( τ ) P l ( cos Θ ) ,
μ d I m ( τ , μ ) d τ = I m ( τ , μ ) - - 1 1 D m ( τ , μ , μ ) I m ( τ , μ ) d μ - Q m ( τ , μ ) ,             m = 0 , 1 , , 2 N - 1 ,
D m ( τ , μ , μ ) = ω ( τ ) 2 l = m 2 N - 1 ( 2 l + 1 ) g l ( τ ) × ( l - m ) ! ( l + m ) ! P l m ( μ ) P l m ( μ ) ,
Q air m ( τ , μ ) = X 0 m ( τ , μ ) exp ( - τ / μ 0 ) + X 01 m ( τ , μ ) exp ( τ / μ 0 ) ,
X 0 m ( τ , μ ) = ω ( τ ) 4 π F 0 ( 2 - δ m 0 ) l = 0 2 N - 1 ( - 1 ) l + m ( 2 l + 1 ) g l ( τ ) × ( l - m ) ! ( l + m ) ! P l m ( μ ) P l m ( μ 0 ) ,
X 01 m ( τ , μ ) = ω ( τ ) 4 π F 0 R ( - μ 0 , n ) exp ( - 2 τ a / μ 0 ) ( 2 - δ m 0 ) × l = 0 2 N - 1 ( 2 l + 1 ) g l ( τ ) ( l - m ) ! ( l + m ) ! × P l m ( μ ) P l m ( μ 0 ) ,
δ m 0 = { 1 , if m = 0 ; 0 , otherwise .
Q ocn m ( τ , μ ) = X 02 m ( τ , μ ) exp ( - τ / μ 0 n ) ,
X 02 m ( τ , μ ) = ω ( τ ) 4 π μ 0 μ 0 n ( μ 0 , n ) T { ( - μ 0 , n ) F 0 exp [ - τ a ( 1 μ 0 - 1 μ 0 n ) ] } × ( 2 - δ m 0 ) l = 0 2 N - 1 ( - 1 ) l + m ( 2 l + 1 ) g l ( τ ) × ( l - m ) ! ( l + m ) ! P l m ( μ ) P l m ( μ 0 n ) .
μ i a d I m ( τ , μ i a ) d τ = I m ( τ , μ i a ) - j = - N 1 j 0 N 1 w j a D m ( τ , μ i a , μ j a ) × I m ( τ , μ j a ) - Q air m ( τ , μ i a ) ,             i = ± 1 , , ± N 1 ,
μ i o d I m ( τ , μ i o ) d τ = I m ( τ , μ i o ) - j = - N 2 j 0 N 2 w j o D m ( τ , μ i o , μ j o ) × I m ( τ , μ j o ) - Q ocn m ( τ , μ i o ) ,             i = ± 1 , , ± N 2 ,
μ i o = S ( μ i a ) = 1 - [ 1 - ( μ i a ) 2 ] / n 2 ,             i = 1 , 2 , , N 1 ,
w i o = w i a ( d S ( μ a ) d μ a ) μ a = μ i a = μ i a n 2 S ( μ i a ) w i a ,             i = 1 , 2 , , N 1 .
I ( τ , μ i a ) = j = 1 N 1 [ C - j G - j ( μ i a ) exp ( k j a τ ) + C j G j ( μ i a ) exp ( - k j a τ ) ] ,             i = ± 1 , , ± N 1 ,
I ( τ , μ i o ) = j = 1 N 2 [ C - j G - j ( μ i o ) exp ( k j o τ ) + C j G j ( μ i o ) exp ( - k j o τ ) ] ,             i = ± 1 , , ± N 2 ,
U ( τ , μ i a ) = Z 0 ( μ i a ) exp ( - τ / μ 0 ) + Z 01 ( μ i a ) exp ( τ / μ 0 ) ,
j = - N 1 j 0 N 1 [ ( 1 + μ j a μ 0 ) δ i j - w j a D ( τ , μ i a , μ j a ) ] Z 0 ( μ j a ) = X 0 ( τ , μ i a ) ,
j = - N 1 j 0 N 1 [ ( 1 + μ j a μ 0 ) δ i j - w j a D ( τ , μ i a , μ j a ) ] Z 01 ( μ j a ) = X 01 ( τ , μ i a ) .
U ( τ , μ i o ) = Z 02 ( μ i o ) exp [ - τ / μ 0 n ( μ 0 , n ) ] ,
j = - N 2 j 0 N 2 [ ( 1 + μ j o μ 0 n ) δ i j - w j o D ( τ , μ i o , μ j o ) ] Z 02 ( μ j o ) = X 02 ( τ , μ i o ) .
I p ( τ , μ i a ) = j = 1 N 1 [ C - j p G - j p ( μ i a ) exp ( k j p a τ ) + C j p G j p ( μ i a ) exp ( - k j p a τ ) ] + U p ( τ , μ i a ) ,             i = ± 1 , , ± N 1 and p L 1 ,
I p ( τ , μ i o ) = j = 1 N 2 [ C - j p G - j p ( μ i o ) exp ( k j p o τ ) + C j p G j p ( μ i o ) exp ( - k j p o τ ) ] + U p ( τ , μ i o ) ,             i = ± 1 , , ± N 2 and L 1 < p L 1 + L 2 .
I 1 ( 0 , - μ i a ) = I ( - μ i a ) ,             i = 1 , , N 1 ,
I p ( τ p , μ i a ) = I p + 1 ( τ p , μ i a ) ,             i = ± 1 , , ± N 1 , and p = 1 , , L 1 - L 1 ;
I L 1 ( τ a , μ i a ) = I L 1 ( τ a , - μ i a ) R ( - μ i a , n ) + [ I L 1 + 1 ( τ a , μ i o ) / n 2 ] T ( + μ i o , n ) ,             i = 1 , 2 , , N 1 ,
I L 1 + 1 ( τ a , - μ i o ) / n 2 = [ I L 1 + 1 ( τ a , μ i o ) / n 2 ] R ( + μ i o , n ) + I L 1 ( τ a , - μ i a ) T ( - μ i a , n ) ,             i = 1 , 2 , , N 1 ,
I L 1 + 1 ( τ a , - μ i o ) = I L 1 + 1 ( τ a , μ i o ) ,             i = N 1 + 1 , , N 2 ;
I p ( τ p , μ i o ) = I p + 1 ( τ p , μ i o ) ,             i = ± 1 , , ± N 2 ,             p = L 1 + 1 , , L 1 + L 2 - 1 ;
I L 1 + L 2 ( τ * , μ i o ) = I g ( μ i o ) ,             i = 1 , 2 , , N 2 .
R ( - μ i a , n ) = 1 2 [ ( μ i a - n μ i o μ i a + n μ i o ) 2 + ( μ i o - n μ i a μ i o + n μ i a ) 2 ] ,
R ( + μ i o , n ) = R ( - μ i a , n ) ,
T ( - μ i a , n ) = 2 n μ i a μ i o [ ( 1 μ i a + n μ i o ) 2 + ( 1 μ i o + n μ i a ) 2 ]
T ( + μ i o , n ) = T ( - μ i a , n ) .

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