Abstract

A new approach to calculating a light field with multiple scattering in media such as clouds, mists, and ocean water is given. It examines all characteristic properties of the real phase function and is applicable to a scattering medium of any optical thickness with an arbitrary single-scattering albedo. The phase function is represented as a sum of more simple functions. The radiance is given as a sum of appropriate components for which the equations, which can be solved by using a known method within domains in which they work best are obtained. The analytic solution of the problem of sunlight propagation in clouds and mists is given on this basis. It describes the complex multimodal angular radiance distribution. A comparison with different numerical calculations shows a fairly satisfactory accuracy of our analytic formulas.

© 1993 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  2. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1990).
  3. E. P. Zege, I. L. Katsev, “Generalized multiple scattering theory parameters and applicability of approximating a solution,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 27, 172–181 (1991).
  4. L. S. Dolin, “Light beam scattering in the turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).
  5. B. Devison, Neutron Transfer Theory (Clarendon, Oxford, 1958).
  6. A. Zardecki, S. A. W. Gerstl, R. E. Dekinder, “Two- and three-dimensional radiative transfer in the diffusion approximation,” Appl. Opt. 25, 3508–3514 (1986).
    [CrossRef] [PubMed]
  7. E. P. Zege, I. L. Katsev, I. N. Polonsky, “A new approach to solve the radiative transfer equation with an arbitrary phase function,” in Proceedings of the Fourth All-Union Conference on Laser Propagation in a Dispersed Medium (Institute of Experimental Meteorology, Ohninsky USSR, 1988), pp. 172–174.
  8. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).
  9. E. P. Zege, I. L. Katsev, I. N. Polonsky, “New approximate approaches to laser beam propagation and laser return in clouds and mists,” in Proceedings of the Fourth International Workshop on Multiple Scattering Lidar Experiments (Institute of Atmospheric Optics, Tomsk, USSR, 1990), pp. 106–112.
  10. H. C. van de Hulst, Multiple Light Scattering (Tables, Formulae and Application) (Academic, New York, 1980), Vols. 1 and 2.
  11. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
    [CrossRef]
  12. W. J. Wiscombe, “The delta-M method. Rapid yet accurate radiative flux calculations for strongly asymmetric phase function,” J. Atmos. Sci. 32, 577–583 (1977).
  13. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [CrossRef] [PubMed]
  14. W. C. Tam, A. Zardecki, “Off-axis propagation of a laser beam in low visibility weather conditions,” Appl. Opt. 19, 2822–2827 (1980).
    [CrossRef] [PubMed]
  15. J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. (Deepak, Hampton, Va., 1985).
  16. T. A. Sushkevich, S. A. Strelkov, A. A. Yoltukhovski, Method of Characteristics in the Atmospheric Optics Problems (Nauka, Moscow, 1990).
  17. D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsevier, New York, 1962).
  18. A. Zardecki, S. A. W. Gerstl, “Multi-Gassian phase function model for off-axis laser beam scattering,” Appl. Opt. 26, 3000–3004 (1987).
    [CrossRef] [PubMed]
  19. E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).
  20. E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).
  21. L. S. Dolin, V. A. Saveliev, “Backscattering signal characteristics at pulse narrow beam illumination of a turbid medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 505–510 (1971).
  22. O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).
  23. G. N. Plass, G. W. Kattawar, “Influence of single scattering albedo on reflected and transmitted light from clouds,” Appl. Opt. 7, 361–367 (1968).
    [CrossRef] [PubMed]

1992

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).

1991

E. P. Zege, I. L. Katsev, “Generalized multiple scattering theory parameters and applicability of approximating a solution,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 27, 172–181 (1991).

1988

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).

1987

A. Zardecki, S. A. W. Gerstl, “Multi-Gassian phase function model for off-axis laser beam scattering,” Appl. Opt. 26, 3000–3004 (1987).
[CrossRef] [PubMed]

E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).

1986

1980

1977

W. J. Wiscombe, “The delta-M method. Rapid yet accurate radiative flux calculations for strongly asymmetric phase function,” J. Atmos. Sci. 32, 577–583 (1977).

1976

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

1973

1972

O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).

1971

L. S. Dolin, V. A. Saveliev, “Backscattering signal characteristics at pulse narrow beam illumination of a turbid medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 505–510 (1971).

1968

1964

L. S. Dolin, “Light beam scattering in the turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

Bushmakova, O. V.

O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).

Chaikovskaya, L. I.

E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).

Deirmendjan, D.

D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsevier, New York, 1962).

Dekinder, R. E.

Devison, B.

B. Devison, Neutron Transfer Theory (Clarendon, Oxford, 1958).

Dolin, L. S.

L. S. Dolin, V. A. Saveliev, “Backscattering signal characteristics at pulse narrow beam illumination of a turbid medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 505–510 (1971).

L. S. Dolin, “Light beam scattering in the turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

Gerstl, S. A. W.

Gordon, H. R.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1990).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Katsev, I. L.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).

E. P. Zege, I. L. Katsev, “Generalized multiple scattering theory parameters and applicability of approximating a solution,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 27, 172–181 (1991).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).

O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1990).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “New approximate approaches to laser beam propagation and laser return in clouds and mists,” in Proceedings of the Fourth International Workshop on Multiple Scattering Lidar Experiments (Institute of Atmospheric Optics, Tomsk, USSR, 1990), pp. 106–112.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A new approach to solve the radiative transfer equation with an arbitrary phase function,” in Proceedings of the Fourth All-Union Conference on Laser Propagation in a Dispersed Medium (Institute of Experimental Meteorology, Ohninsky USSR, 1988), pp. 172–174.

Kattawar, G. W.

Plass, G. N.

Polonsky, I. N.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).

E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “New approximate approaches to laser beam propagation and laser return in clouds and mists,” in Proceedings of the Fourth International Workshop on Multiple Scattering Lidar Experiments (Institute of Atmospheric Optics, Tomsk, USSR, 1990), pp. 106–112.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A new approach to solve the radiative transfer equation with an arbitrary phase function,” in Proceedings of the Fourth All-Union Conference on Laser Propagation in a Dispersed Medium (Institute of Experimental Meteorology, Ohninsky USSR, 1988), pp. 172–174.

Saveliev, V. A.

L. S. Dolin, V. A. Saveliev, “Backscattering signal characteristics at pulse narrow beam illumination of a turbid medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 505–510 (1971).

Strelkov, S. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Yoltukhovski, Method of Characteristics in the Atmospheric Optics Problems (Nauka, Moscow, 1990).

Sushkevich, T. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Yoltukhovski, Method of Characteristics in the Atmospheric Optics Problems (Nauka, Moscow, 1990).

Tam, W. C.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Tables, Formulae and Application) (Academic, New York, 1980), Vols. 1 and 2.

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method. Rapid yet accurate radiative flux calculations for strongly asymmetric phase function,” J. Atmos. Sci. 32, 577–583 (1977).

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Yoltukhovski, A. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Yoltukhovski, Method of Characteristics in the Atmospheric Optics Problems (Nauka, Moscow, 1990).

Zardecki, A.

Zege, E. P.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).

E. P. Zege, I. L. Katsev, “Generalized multiple scattering theory parameters and applicability of approximating a solution,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 27, 172–181 (1991).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).

E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).

O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1990).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A new approach to solve the radiative transfer equation with an arbitrary phase function,” in Proceedings of the Fourth All-Union Conference on Laser Propagation in a Dispersed Medium (Institute of Experimental Meteorology, Ohninsky USSR, 1988), pp. 172–174.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “New approximate approaches to laser beam propagation and laser return in clouds and mists,” in Proceedings of the Fourth International Workshop on Multiple Scattering Lidar Experiments (Institute of Atmospheric Optics, Tomsk, USSR, 1990), pp. 106–112.

Appl. Opt.

Atmos. Opt.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A modified small-angle diffusion approximation taking into account the specific form of the scattering phase function at small angles,” Atmos. Opt. 1, 19–27 (1988).

Izv. Akad. Nauk SSSR

E. P. Zege, I. N. Polonsky, L. I. Chaikovskaya, “Features of radiation beam propagation at oblilque illumination of absorbing anisotropically scattering medium,” Izv. Akad. Nauk SSSR 23, 486–492 (1987).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution of the radiation transfer equation for strongly anisotropic scattering medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 28, 588–598 (1992).

E. P. Zege, I. L. Katsev, “Generalized multiple scattering theory parameters and applicability of approximating a solution,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 27, 172–181 (1991).

L. S. Dolin, V. A. Saveliev, “Backscattering signal characteristics at pulse narrow beam illumination of a turbid medium,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 7, 505–510 (1971).

O. V. Bushmakova, E. P. Zege, I. L. Katsev, “The distribution of the density in the scattering media from a localized source,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, 711–719 (1972).

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

L. S. Dolin, “Light beam scattering in the turbid medium layer,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 380–382 (1964).

J. Atmos. Sci.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

W. J. Wiscombe, “The delta-M method. Rapid yet accurate radiative flux calculations for strongly asymmetric phase function,” J. Atmos. Sci. 32, 577–583 (1977).

Other

J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures. (Deepak, Hampton, Va., 1985).

T. A. Sushkevich, S. A. Strelkov, A. A. Yoltukhovski, Method of Characteristics in the Atmospheric Optics Problems (Nauka, Moscow, 1990).

D. Deirmendjan, Electromagnetic Scattering on Spherical Polydispersion (Elsevier, New York, 1962).

B. Devison, Neutron Transfer Theory (Clarendon, Oxford, 1958).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1990).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “New approximate approaches to laser beam propagation and laser return in clouds and mists,” in Proceedings of the Fourth International Workshop on Multiple Scattering Lidar Experiments (Institute of Atmospheric Optics, Tomsk, USSR, 1990), pp. 106–112.

H. C. van de Hulst, Multiple Light Scattering (Tables, Formulae and Application) (Academic, New York, 1980), Vols. 1 and 2.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “A new approach to solve the radiative transfer equation with an arbitrary phase function,” in Proceedings of the Fourth All-Union Conference on Laser Propagation in a Dispersed Medium (Institute of Experimental Meteorology, Ohninsky USSR, 1988), pp. 172–174.

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Figures (5)

Fig. 1
Fig. 1

Cloud phase function (Cloud C.1 model from Ref. 17 at λ = 0.7 μm, solid curve), its approximation from Eqs. (7) at N = 5 (asterisks), and the component p3(β) (dashed curve).

Fig. 2
Fig. 2

Scattering light radiance, both transmitted (θ < 90°) and reflected (θ > 90°) by a slab cloud with optical thickness (a) τ0 = 65, and within the cloud at optical thickness τ: (b) τ = 3.2, (c) τ = 12.8, and (d) τ = 32 for normal incidence (μ0 = 1). Circles, Monte Carlo data15; asterisks, spherical harmonics method15; squares, asymptotic solution15; and rhombuses, layer adding method.

Fig. 3
Fig. 3

Radiance transmitted through the slab cloud with optical thickness τ0 for μ0 = 0.5: τ0 = 0.1 (solid curve) and τ0 = 10 (dashed curve). Histogram indicates the Monte Carlo data.23

Fig. 4
Fig. 4

Radiance transmitted through the slab cloud with optical thickness τ0 for μ0 = 0.5: (a) τ0 = 1, (b) τ0 = 3, (c) τ0 = 6, (d)τ0 = 10. Rhombuses and triangles show layer adding data in two azimuthal planes, (0, π) and (π/2, 3π/2), respectively, the solid and the long-dashed curves indicate the proposed method with I3(τ, n) calculated by the DA and tho medium-dashed and the short-dashed curves indicate the single-scattering approximation.

Fig. 5
Fig. 5

Mean angle 〈θ〉 (dashed curve) and angle θmax of the scattering radiance spread function Is(z0, n) as a function of optical thickness τ0 at μ0 = 0.5.

Tables (1)

Tables Icon

Table 1 Phase Function Approximation Parameters for Cloud of the Cloud C.1 Model

Equations (47)

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α = ( σ ) / [ σ ( 1 g ) ] 0.5 ; ζ = σ z 0 ( 1 g ) > 1 .
α 1 , ζ 0.5 .
p ( β ) = k = 1 N a k p k ( β ) .
p 2 ( β ) = n = 1 M x n P n ( β ) ,
D I s ( r , n ) = L I s ( r , n ) + L I 0 ( r , n ) , D = n r + , L = σ 4 π p ( n , n ) d n ,
1 2 0 π p ( β ) sin β d β = 1 , g = 1 2 0 π cos β p ( β ) sin β d β .
p ( β ) = k = 1 N a k p k ( β ) , k = 1 N a k = 1 ,
g k = 1 2 0 π cos β p k ( β ) sin β d β , k = 1 N a k g k = g .
I s ( r , n ) = k = 1 N I k ( r , n ) ,
D I k ( r , n ) = i = 1 k a i L i I k ( r , n ) + a k L k i = 0 k 1 I i ( r , n ) , L i I k ( r , n ) = σ 4 π 4 π p i ( n , n ) I k ( r , n ) d n .
ω k = ω i = 1 k a i , ω = σ / , p k ( β ) = i = 1 k a i p i ( β ) / i = 1 k a i .
1 g k 1 1 g k , 2 k M .
p k ( β ) = exp [ ( β β k ) 2 / ( 2 υ k ) ] , k = 1 , 4 , 5 , p 2 ( β ) = exp [ β ( 3 / υ 2 ) 1 / 2 ] ,
( 1 g 1 ) / ( 1 g 2 ) = 0.007 , ( 1 g 2 ) / ( 1 g 3 ) = 0.088 .
D I k ( r , n ) = i = 1 3 a i L i I k ( r , n ) + a k L k i = 0 3 I i ( r , n ) , k = 4 , 5 .
a 3 L 3 I k ( r , n ) i = 1 2 a i L i I k ( r , n )
D I k ( r , n ) = i = 1 2 a i L i I k ( r , n ) + a k L k i = 0 2 I i ( r , n ) , k = 4 , 5 .
p j ( β ) = a 1 p 1 ( β ) + a 2 p 2 ( β ) a 1 + a 2 .
D 1 I 1 ( r , n ) = a 1 L 1 I 1 ( r , n ) + a 1 L 1 I 0 ( r , n ) ,
D 2 I 2 ( r , n ) = a 2 L 2 I 2 ( r , n ) + a 2 L 2 I 0 , 2 ( r , n ) ,
I 0 , 2 ( r , n ) = exp ( 2 z / μ 0 ) [ I 0 ( r , n ) exp ( z / μ 0 ) ]
D I 3 ( r , n ) = L I 3 ( r , n ) + a 3 L 3 i = 0 2 I i ( r , n ) .
I 0 ( τ , n ) = exp ( τ / μ 0 ) δ ( μ μ 0 ) δ ( ψ ) .
I s ( 0 , n ) = 0 , at μ > 0 ; I s ( τ 0 , n ) = 0 , at μ < 0 .
I k ( 0 , n ) = 0 , at μ > 0 ; I k ( τ 0 , n ) = 0 , at μ < 0 .
I k ( τ , n ) = S k 2 π ( V xk V yk ) 1 / 2 exp [ ( θ x b k ) 2 2 V xk θ y 2 2 V yk ] , ( k = 1 , 2 ) ,
tan θ x = tan θ cos ψ , sin θ y = sin θ sin ψ .
S k = exp [ ( 1 ω k ) ξ k / s k ρ k / 2 ] / ( cosh ξ k ) 1 / 2 exp ( τ k / μ 0 ) . b k = ( Θ k θ 0 ) / T k + θ 0 , V xk = V k α s V k / T k ( b k θ 0 ) 2 ( 1 T k ) , V yk = V k α s tanh ( ξ k ) / T k ,
T k = 1 exp ( τ k / μ 0 ) ( cosh ζ k ) 1 / 2 ,
V k = { ( 1 + sin 2 Θ k sin 2 θ 0 ) [ 1 exp ( 2 ρ k ) ] 4 ρ k sin 2 Θ k } 1 / 2 / cos 2 Θ k ) ,
sin Θ k = sin θ 0 exp ( ρ k ) , s k = [ ( 1 ω k ) ω k ( 1 g k ) ] 1 / 2 , V k α s = s k / ( 1 ω k ) , ζ k = 0 ρ k d ρ k V k .
s k τ k = 0 ρ k d ρ k cos Θ k V k .
ρ k = ln cosh ξ k , V k = tanh ξ k , ξ k = s k τ k + θ 0 2 tanh ( ξ k ) / 2 ;
i = 0 2 I i ( r , n ) = exp ( 3 z / μ 0 ) δ ( μ μ 0 ) δ ( ψ ) .
i = 0 2 I i ( r , n ) = exp ( ϰ τ / μ 0 ) δ ( μ μ 0 ) δ ( ψ ) ,
I 3 ( τ , n ) = 1 4 π { w ( τ ) 3 q n [ k d w ( τ ) d τ 3 B 1 ( τ ) ] } ,
w ( τ ) = 4 π I 3 ( τ , n ) d n ,
B 1 ( τ ) = 4 π n B ( τ , n ) d n ,
w ( τ ) = ω a 3 ( 1 + 3 q g 3 ϰ ) 0 τ 0 w 0 ( τ , τ ) exp ( ϰ τ / μ 0 ) d τ 3 ω a 3 q g 3 μ 0 [ w 0 ( τ , τ = 0 ) w 0 ( τ , τ = τ 0 ) exp ( ϰ τ 0 / μ 0 ) ] ,
B 1 ( τ ) = ω a 3 g 3 exp ( ϰ τ / μ 0 ) n 0 [ U ( τ ) U ( τ τ 0 ) ] ,
w 0 ( τ , τ ) = 1 q γ [ sinh γ ( τ + 2 q ) sinh γ ( τ 0 + 2 q τ ) sinh γ ( τ 0 + 4 q ) U ( τ τ ) sinh γ ( τ τ ) ] ,
w ( τ ) = C sinh 4 q γ sinh γ ( τ 0 + 4 q ) × [ ( m 2 q ϰ / μ 0 ) exp ( ϰ τ 0 / μ 0 ) sinh γ ( τ + 2 q ) m exp ( ϰ τ / μ 0 ) sinh γ ( τ 0 + 4 q ) + ( m + 2 q ϰ / μ 0 ) sinh γ ( τ 0 τ + 2 q ) ] ,
m = 1 + 3 q ϰ g 3 , C = a 3 4 γ q 2 ( ϰ / μ 0 ) 2 .
I ( τ , n ) = A μ 0 u 0 ( μ ) u 0 ( μ 0 ) π sinh 4 q γ sinh γ ( τ 0 + 4 q ) ,
A = a 3 ( 1 + 3 q ϰ g 3 ) μ 0 + 2 q ϰ q ϰ 2 ( 2 + 3 μ 0 ) ,
ϰ = a 3 × ( 2 + 3 μ 0 g 3 ) + [ ( 2 + 3 μ 0 g 3 ) 2 + 4 μ 0 ( 2 + 3 μ 0 ) / q a 3 ] 1 / 2 2 ( 2 + 3 μ 0 ) .
θ = θ I s ( τ , n ) d n / I s ( τ , n ) d n .

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