Abstract

The emissivity from a stationary random rough surface is derived by taking into account the multiple reflections and the shadowing effect. The model is applied to the ocean surface. The geometric optics approximation is assumed to be valid, which means that the rough surface is modeled as a collection of facets reflecting locally the light in the specular direction. In particular, the emissivity with zero, single, and double reflections are analytically calculated, and each contribution is studied numerically by considering a 1D sea surface observed in the near infrared band. The model is also compared with results computed from a Monte Carlo ray-tracing method.

© 2006 Optical Society of America

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References

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  1. X. Wü and L. Smith, "Emissivity of rough sea surface for 8-13 μm: modeling and verification," Appl. Opt. 36, 2609-2619 (1997).
    [CrossRef] [PubMed]
  2. J. A. Shaw and C. Marston, "Polarized infrared emissivity for a rough water surface," Opt. Exp. 7, 375-380 (2000).
    [CrossRef]
  3. P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
    [CrossRef]
  4. K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
    [CrossRef]
  5. K. Yoshimori, K. Itoh, and Y. Ichioka, "Optical characteristics of a wind-roughened water surface: a two-dimensional theory," Appl. Opt. 34, 6236-6247 (1995).
    [CrossRef] [PubMed]
  6. C. Bourlier, "Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics," Appl. Opt. 44, 4335-4349 (2005).
    [CrossRef] [PubMed]
  7. C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
    [CrossRef]
  8. C. R. Zeiss, C. P. MacGrath, K. M. Littfin, and H. G. Hughes, "Infrared radiance of the wind-ruffled sea," J. Opt. Soc. Am. A 16, 1439-1452 (1999).
    [CrossRef]
  9. J. A. Shaw, "Polarimetric measurements of long-wave infrared spectral radiance from water," Appl. Opt. 38, 379-392 (1999).
  10. D. E. Freund, R. J. Joseph, D. J. Donohue, and K. T. Constantikes, "Numerical computations of rough sea surface emissivity using the interaction probability," J. Opt. Soc. Am. A 14, 1836-1849 (1997).
    [CrossRef]
  11. B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
    [CrossRef]
  12. C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
    [CrossRef]
  13. C. Bourlier and G. Berginc, "Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian, and sea correlations," Waves Random Media 13, 27-58 (2003).
    [CrossRef]
  14. C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
    [CrossRef]
  15. A. G. Pavel'yev, Double Scattering by a Random Irregular Surface (Scripta Publishing, 1984), pp. 5-15.
  16. G. M. Hale and M. R. Querry, "Optical constants of water in the 200-nm to 200-mm wavelength region," Appl. Opt. 12, 555-563 (1973).
    [CrossRef] [PubMed]
  17. C. Cox and W. Munk, "Measurement of the roughness of the sea surface from photographs of the suns glitter," J. Opt. Soc. Am. 44, 838-850 (1954).
    [CrossRef]

2005 (1)

2003 (2)

B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
[CrossRef]

C. Bourlier and G. Berginc, "Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian, and sea correlations," Waves Random Media 13, 27-58 (2003).
[CrossRef]

2002 (2)

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
[CrossRef]

2001 (1)

C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
[CrossRef]

2000 (1)

J. A. Shaw and C. Marston, "Polarized infrared emissivity for a rough water surface," Opt. Exp. 7, 375-380 (2000).
[CrossRef]

1999 (2)

J. A. Shaw, "Polarimetric measurements of long-wave infrared spectral radiance from water," Appl. Opt. 38, 379-392 (1999).

C. R. Zeiss, C. P. MacGrath, K. M. Littfin, and H. G. Hughes, "Infrared radiance of the wind-ruffled sea," J. Opt. Soc. Am. A 16, 1439-1452 (1999).
[CrossRef]

1997 (2)

1996 (1)

P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
[CrossRef]

1995 (1)

1988 (1)

K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
[CrossRef]

1973 (1)

1954 (1)

Allen, M. R.

P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
[CrossRef]

Berginc, G.

C. Bourlier and G. Berginc, "Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian, and sea correlations," Waves Random Media 13, 27-58 (2003).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
[CrossRef]

Bourlier, C.

C. Bourlier, "Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics," Appl. Opt. 44, 4335-4349 (2005).
[CrossRef] [PubMed]

C. Bourlier and G. Berginc, "Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian, and sea correlations," Waves Random Media 13, 27-58 (2003).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
[CrossRef]

Constantikes, K. T.

Cox, C.

Donohue, D. J.

Freund, D. E.

Hale, G. M.

Henderson, B. G.

B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
[CrossRef]

Hughes, H. G.

Ichioka, Y.

Itoh, K.

Joseph, R. J.

Littfin, K. M.

MacGrath, C. P.

Marston, C.

J. A. Shaw and C. Marston, "Polarized infrared emissivity for a rough water surface," Opt. Exp. 7, 375-380 (2000).
[CrossRef]

Masuda, K.

K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
[CrossRef]

Munk, W.

Nightingale, T. J.

P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
[CrossRef]

Pavel'yev, A. G.

A. G. Pavel'yev, Double Scattering by a Random Irregular Surface (Scripta Publishing, 1984), pp. 5-15.

Querry, M. R.

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
[CrossRef]

Shaw, J. A.

J. A. Shaw and C. Marston, "Polarized infrared emissivity for a rough water surface," Opt. Exp. 7, 375-380 (2000).
[CrossRef]

J. A. Shaw, "Polarimetric measurements of long-wave infrared spectral radiance from water," Appl. Opt. 38, 379-392 (1999).

Smith, L.

Takashima, T.

K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
[CrossRef]

Takayama, Y.

K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
[CrossRef]

Theiler, J.

B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
[CrossRef]

Villeneuve, P.

B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
[CrossRef]

Watts, P. D.

P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
[CrossRef]

Wü, X.

Yoshimori, K.

Zeiss, C. R.

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (1)

C. Bourlier, G. Berginc, and J. Saillard, "Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effect," IEEE Trans. Geosci. Remote Sens. 39, 379-392 (2001).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

P. D. Watts, M. R. Allen, and T. J. Nightingale, "Wind speed effects on sea surface emission and reflection for the along track scanning radiometer," J. Atmos. Ocean. Technol. 13, 126-141 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Exp. (1)

J. A. Shaw and C. Marston, "Polarized infrared emissivity for a rough water surface," Opt. Exp. 7, 375-380 (2000).
[CrossRef]

Remote Sens. Environ. (2)

K. Masuda, T. Takashima, and Y. Takayama, "Emissivity of pure and sea waters for the model sea surface in the infrared window regions," Remote Sens. Environ. 24, 313-329 (1988).
[CrossRef]

B. G. Henderson, J. Theiler, and P. Villeneuve, "The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model," Remote Sens. Environ. 88, 453-457 (2003).
[CrossRef]

Waves Random Media (3)

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part I. Single scattering," Waves Random Media 12, 145-174 (2002).
[CrossRef]

C. Bourlier and G. Berginc, "Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian, and sea correlations," Waves Random Media 13, 27-58 (2003).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, "Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length: part II. Multiple scattering," Waves Random Media 12, 175-200 (2002).
[CrossRef]

Other (1)

A. G. Pavel'yev, Double Scattering by a Random Irregular Surface (Scripta Publishing, 1984), pp. 5-15.

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Figures (9)

Fig. 1
Fig. 1

Geometry used to calculate the zero- and the first-order emissivities. In the calculation of the emissivity, one sets θ 0 = θ and ϕ 0 = ϕ .

Fig. 2
Fig. 2

Illustration of the first-order illumination function. At the top (upward case), θ 1 [ 0 ; π / 2 ] s = sgn ( cos θ 1 ) = + 1 and z 1 z 0 . At the bottom (downward case), θ 1 [ π / 2 ; π ] s = sgn ( cos θ 1 ) = 1 and z 1 z 0 . z i stands for the height of the point M i .

Fig. 3
Fig. 3

(Color online) Zero-order emissivities ε 0 , AN and ε 0 , MC computed from an analytical approach [(Eq. 40)] and a Monte Carlo ray-tracing method versus the emission angle θ. The wind speed u 12 = 5 m / s and the wavelength λ = 4   μm .

Fig. 4
Fig. 4

(Color online) First- and second-order emissivities { ε 1,AN , ε 2 , AN , ε 1 , MC } computed from analytical approaches [Eqs. (43) and (45)] and from a Monte Carlo method versus the emission angle θ. The results of Ref. 11 are also plotted. The wind speed u 12 = 5 m / s and the wavelength λ = 4   μm .

Fig. 5
Fig. 5

(Color online) Zero-order average illumination function S ¯ ¯ 0 ( θ ) computed from a Monte Carlo method, from Eq. (48) (without correlation) and when the correlation is taken into account versus the emission angle θ. The wind speed u 12 = 5 m / s .

Fig. 6
Fig. 6

(Color online) First-order average illumination function S ¯ ¯ 1 ( θ ) computed from a Monte Carlo method, from Eq. (50) (without correlation) and when the correlation is taken into account versus the emission angle θ. The wind speed u 12 = 5 m / s .

Fig. 7
Fig. 7

(Color online) First-order emissivities computed from analytical ε 1 , AN , Monte Carlo ε 1 , MC and empirical ε 1 , EM approaches versus the emission angle θ. The results of Ref. 11 are also reported. The wind speeds u 12 = { 5 ,   10 ,   15 ,   20 } m / s and the wavelength λ = 4   μm .

Fig. 8
Fig. 8

(Color online) Absolute relative error | ε 0 , MC + ε 1 , MC ( ε 0 , EM + ε 1 , EM ) | / ( ε 0 , MC + ε 1 , MC ) in percent versus the emission angle θ. The wind speeds u 12 = { 5 ,   10 ,   15 ,   20 } m / s and the wavelengths λ = { 4 , 10 }   μm .

Fig. 9
Fig. 9

(Color online) Parameters m 1 ( top ) and σ 1 ( bottom ) in degrees of the empirical function f versus the rms slope σ s . The label “Fit” in the legend cor-responds to the linear regression of { m 1 ,   σ 1 } . The wavelengths λ = { 4 ,   10 }   μm .

Equations (70)

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ε ( θ , ϕ ) = n = 0 n = N ε n ( θ , ϕ ) .
r V ( ψ 0 ) = n cos ψ 0 cos ψ 0 n cos ψ 1 + cos ψ 0 ,
r H ( ψ 0 ) = cos ψ 0 n cos ψ 0 cos ψ 1 + n cos ψ 0 ,
cos [ ψ 0 ( θ 0 ,   ϕ 0 ; γ x , 0 ,   γ y , 0 ) ] = n ^ 0 m ^ 0 = cos θ 0 ( γ x , 0 cos ϕ 0 + γ y , 0 sin ϕ 0 ) sin θ 0 ( 1 + γ x , 0         2 + γ y , 0         2 ) 1 / 2 .
ϵ l 0 ( θ ,   ϕ ; γ x , 0 ,   γ y , 0 ) = [ 1 | r ( | ψ 0 | ) | 2 ] g 0 ,
g 0 = cos ψ 0 ( 1 + γ x , 0         2 + γ y , 0         2 ) 1 / 2 cos θ 0 = 1 ( γ x , 0 cos ϕ 0 + γ y , 0 sin ϕ 0 ) tan θ 0 .
ε 0 ( θ , ϕ ) = [ 1 | r ( | ψ 0 | ) | 2 ] × g 0 × S ¯ 0 0 ,
0 = + + ( ) p s ( γ x , 0 , γ y , 0 ) d γ x , 0 d γ y , 0 .
ϵ 1 ( θ , ϕ ) = [ 1 | r ( | ψ 1 | ) | 2 ] × | r ( | ψ 0 | ) | 2 × g 0 × S ¯ 1 1 ,
1 = + + ( ) p s ( γ 0 ) p s ( γ 1 ) d γ 0 d γ 1 ,
m ^ 1 = m ^ 0 2 ( n ^ 0 m ^ 0 ) n ^ 0 = m ^ 0 2 ( cos ψ 0 ) n ^ 0 ,
m ^ 1 = ( m x , 0 + G 0 γ x , 0 , m y , 0 + G 0 γ y , 0 , m z , 0 G 0 ) ,
G 0 = 2 g 0 cos θ 0 1 + γ x , 0         2 + γ y , 0         2 = 2 [ cos θ 0 ( γ x , 0 cos ϕ 0 + γ y , 0 sin ϕ 0 ) sin θ 0 ] 1 + γ x , 0         2 + γ y , 0         2 .
ϵ N ( θ , ϕ ) = [ 1 | r ( | ψ N | ) | 2 ] × | r ( | ψ N 1 | ) | 2 × × | r ( | ψ 0 | ) | 2 × g 0 × S ¯ N N .
N = n = 0 n = N + ( ) p s ( γ n ) d γ n .
m ^ n = ( m x , n 1 + G n 1 γ x , n 1 ,   m y , n 1 + G n 1 γ y , n 1 , m z , n 1 G n 1 ) ,
S 0 [ θ ,   ϕ ; M 0 ( γ X , 0 ,   z 0 ) ] = F ( z 0 ) Λ ( θ , ϕ ) .
F ( z 0 ) = z 0 p z ( z ) d z ,
Λ ( θ , ϕ ) = 1 μ + μ + ( γ X , 0 μ ) p s ( γ X , 0 ) d γ X , 0 ,
p s ( γ X , 0 ) = + p s ( γ X , 0 ,   γ Y , 0 ) d γ Y , 0 ,
γ X , 0 = γ x , 0 cos ϕ + γ y , 0 sin ϕ , γ Y , 0 = - γ x , 0 sin ϕ + γ y , 0 cos ϕ .
p s ( γ x , 0 ,   γ y , 0 ) = 1 2 π σ s x σ s y exp ( γ x , 0         2 2 σ s x       2 γ y , 0         2 2 σ s y       2 ) ,   
p s ( γ X , 0 ) = 1 2 π σ s X ( ϕ ) exp [ γ X , 0         2 2 σ s X         2 ( ϕ ) ] ,
Λ ( v ) = exp ( v 2 ) v π   erfc ( v ) 2 v π v = v ( θ , ϕ ) = | cot θ | 2 σ s X ( ϕ ) .
S ¯ 0 ( θ ,   ϕ ; γ X , 0 ) = + S 0 p z ( z 0 ) d z 0 = [ 1 + Λ ( θ ,   ϕ ) ] 1 .
ε 0 ( θ , ϕ ) = [ 1 + Λ ( θ ,   ϕ ) ] 1 [ 1 | r ( | ψ 0 | ) | 2 ] × g 0 0 ,
      0 = + μ d γ X , 0 + ( ) p s ( γ X , 0 ,   γ Y , 0 ) d γ Y , 0 .
S 1 ( θ ,   ϕ ,   θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) = S ˜ 0 b ( θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) × S 0 b ( θ ,   ϕ ; M 0 ,   M ) .    
S 0 b ( θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) = [ F ( z 1 ) / F ( z 0 ) ] s Λ s ( θ 1 , ϕ 1 ) s = sign ( cos   θ 1 ) .
Λ ( θ ,   ϕ ) = 1 μ μ + ( γ X , 0 + μ ) p s ( γ X , 0 ) d γ X , 0 1.
S 1 ( θ ,   ϕ ,   θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) = { 1 [ F ( z 1 ) F ( z 0 ) ] s Λ s ( θ 1 , ϕ 1 ) } × F ( z 0 ) Λ ( θ , ϕ ) .
S ¯ 1 ( θ ,   ϕ ,   θ 1 ,   ϕ 1 ) = + p x ( z 0 ) × [ z l     s z u     s S 1 ( θ ,   ϕ ,   θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) p z ( z 1 ) ] d z 0 ,
z u     + = z 0 z l     + =   for   s = + 1, z u = + z l     = z 0   for   s = 1.
S ¯ 1 ( θ ,   ϕ ,   θ 1 ,   ϕ 1 ) = { Λ 1 + [ ( 1 + Λ 1 + ) ( 2 + Λ ) ] 1 s = + 1 , Λ 1 [ ( 1 + Λ + Λ 1 ) ( 1 + Λ ) ( 2 + Λ ) 1 ] s = 1 ,
1 = + μ γ 1 l γ 1 u + + ( ) p s ( Γ 0 ) p s ( Γ 1 ) × d γ Y , 1 d γ Y , 0 d γ X , 1 d γ X , 0 ,
S N ( Φ ,   Θ ; Z ,   Γ X ) = S 0 b ( θ ,   ϕ ; M 0 ,   M ) ϒ ( μ γ X , 0 ) × n = 1 n = N ϒ ( μ n s n γ X , n ) × S ˜ 0 b ( θ n ,   ϕ n ; M n 1 ,   M n ) ,
S ˜ N ( Φ ,   Θ ; Γ X ) = S ¯ N ( Φ ,   Θ ) n = 0 n = N ϒ ( μ n s n γ X , n ) ,
S ¯ N ( Φ ,   Θ ) = + p z ( z 0 ) z l , 1       s 1 z u , 1       s 1 p z ( z 1 ) z l , N         s N z u , N         s N p z ( z N ) × [ F ( z 0 ) ] Λ ( θ , ϕ ) n = 1 n = N { 1 [ F ( z n ) F ( z n 1 ) ] θ n Λ s n ( θ n , ϕ n ) } d z 0 d z 1 ,   ,   d z N ,
z u , n       + = z n 1 z l , n    + =   for   s n = + 1 , z u , n       = + z l , n    − = z n - 1   for   s n = 1.
Λ 1 + Λ 2 + 2 ( 3 + Λ ) ( 2 + Λ 1 + ) ( 1 + Λ 2 + ) s 1 = + 1 s 2 = + 1 , S ˜ ¯ 2 + lim S ˜ ¯ 2 Λ 1 + + Λ 2 + lim S ˜ ¯ 2 Λ 1 + + lim S ˜ ¯ 2 Λ 2 + s 1 = + 1 s 2 = 1 , Λ 1 Λ 2 + ( 1 + Λ ) ( 3 + Λ ) ( 1 + Λ + Λ 1 ) ( 1 + Λ 2 + ) s 1 = 1 s 2 = + 1 , Λ 1 Λ 2 ( 1 + Λ ) ( 2 + Λ ) ( 3 + Λ ) ( 1 + Λ + Λ 1 ) ( 2 + Λ + Λ 2 ) s 1 = 1 s 2 = 1.
S ˜ ¯ 2 = 4 + Λ + Λ 1 + + Λ 2 ( 3 + Λ ) ( 2 + Λ 1 + ) ( 2 + Λ + Λ 2 ) ( 1 + Λ 1 + + Λ 2 ) .
N = n = 0 n = N + + ( ) p s ( γ X , n ,   γ Y , n ) × ϒ ( μ n s n γ X , n ) d γ X , n d γ Y , n .
ε 0 ( θ ) = 1 1 + Λ ( θ ) + μ [ 1 | r ( | ψ 0 | ) | 2 ] p s ( γ 0 ) g 0 d γ 0 ,
g 0 = 1 γ 0 tan θ 0 ,
cos [ ψ 0 ( θ 0 ; γ 0 ) ] = g 0 cos θ 0 ( 1 + γ 0     2 ) 1 / 2 .
ε 1 ( θ ) = + μ d γ 0 γ 1 l γ 1 u [ 1 | r ( | ψ 1 | ) | 2 ] × | r ( | ψ 0 | ) | 2 S ¯ 1 ( θ 1 ,   θ 2 ) p s ( γ 1 ) p s ( γ 0 ) g 0 d γ 1 .
cos θ 1 = cos θ 0 [ 1 2 g 0 ( 1 + γ 0     2 ) 1 ] .
ε 2 ( θ ) = + μ d γ 0 γ 1 l γ 1 u d γ 1 γ 2 l γ 2 u [ 1 | r ( | ψ 2 | ) | 2 ] × | r ( | ψ 1 | ) | 2 | r ( | ψ 0 | ) | 2 × S ¯ 2 ( θ ,   θ 1 ,   θ 2 ) p s ( γ 2 ) p s ( γ 1 ) p s ( γ 0 ) g 0 d γ 2 ,
cos θ 2 = cos θ 1 [ 1 2 g 0 ( 1 + γ 0     2 ) 1 ] [ 1 2 g 1 ( 1 + γ 1     2 ) 1 ] ,
S ¯ ¯ 0 ( θ ) = { + [ F ( z 0 ) ] Λ p z ( z 0 ) d z 0 } + μ p s ( γ 0 ) d γ 0 = 1 1 + Λ + μ p s ( γ 0 ) d γ 0 .
S ¯ ¯ 0 ( θ ) = [ 1 + erf ( v ) ] { 2 [ 1 + Λ ( v ) ] } 1 .
S ¯ ¯ 1 ( θ ) = + + S ¯ 1 ( θ , θ 1 ) ϒ ( μ γ 0 ) × ϒ ( μ 1 s γ 1 ) p s ( γ 0 ) p s ( γ 1 ) d γ 0 d γ 1 ,
S ¯ ¯ 1 ( θ ) = 1 2 2 π σ s + μ S ¯ 1 [ θ ,   θ 1 ( γ 0 ) ] [ 1 + erf ( v 1 ) ] × exp ( γ 0     2 2 σ s     2 ) d γ 0 .
S ¯ ¯ 1 ( θ ) = + μ d γ 0 + d γ 1 + d z 0 z l     s z u     s d z 1 [ 1 S 0 b ( θ 1 ; M 1 ,   M 0 ) ] S 0 b ( θ ; M 1 ,   M 0 ) ϒ ( μ 1 s γ 1 ) p z ( z 0 ) p z ( z 1 ) p s ( γ 0 ) p s ( γ 1 ) .
f ( θ ) = A exp [ ( θ m 1 ) 2 σ 1     2 ] .
ε 1 , EM = f × ε 1 , AN .
S 0 b ( θ 1 ,   ϕ 1 ; M 1 ,   M 0 ) = { exp ( 0 u L g d u ) if   0 h 0 h 1 η v 1 = u L u t exp ( 0 u L g d u ) G ( v 1 ; h 1 ) otherwise ,
g ( v 1 ; h 1 ,   ζ 1 ,   h 0 ,   u ) = η π f 11 f 33 f 13     2 f 33 × e A B h 1     2 ζ 1     2 [ 1 κ π e κ 2   erfc ( κ ) ] e B 1     2 C 1 [ 1 + erf ( A 1 h 0 + B 1 ) ] .
A = [ f 33 v 1     2 + 2 v 1 ( f 34 ζ 1 + f 14 h 1 f 13 h 0 ) ] f M       1 ,
B = f 11 ( h 1     2 + h 0     2 ) + 2 f 12 h 1 h 0 + 2 ζ 1 ( f 13 h 1 f 14 h 0 ) + f 33 h 1     2 f M h 1     2 ζ 1     2 ,
κ = f 14 h 1 f 13 h 0 + f 34 ζ 1 + f 33 v 1 f 33 f M .
             A 1 = ( f 11 f 33 f 13     2 ) / ( f 33 f M ) > 0 ,
B 1 = h 1 ( f 12 f 33 + f 13 f 14 ) + ζ 1 ( f 13 f 34 f 14 f 33 ) f 33 f M ( f 11 f 33 f 13       2 ) ,
C 1 = h 1     2 f 11 f 33 f 14       2 f 33 f M + ζ 1     2 f 33       2 f 34       2 f 33 f M + 2 h 1 ζ 1 f 11 f 33 f 14 f 34 f 33 f M .
f 11 = 1 f 2     2 f 1     2 , f 33 = 1 f 0     2 f 1     2 , f 12 = f 0 f 2     2 + f 2 f 1     2 f 0 , f 34 = f 2 f 0     2 + f 0 f 1     2 f 2 , f 13 = f 1 ( f 0 f 2 ) , f 14 = f 1 ( 1 f 1     2 f 0 f 2 ) , f M = ( f 33       2 f 34       2 ) / ( 1 f 0     2 ) ,
f 0 = C 0 / σ h     2 ,   f 1 = C / ( σ h σ s ) ,   f 2 = C 2 / σ s     2 .
σ s = C 2 ( 0 ) = 2 σ h / L c ,     η = 2 ,   f 0 = exp ( u 2 ) ,   f 1 = u 2 exp ( u 2 ) ,   f 2 = ( 1 2 u 2 ) exp ( u 2 ) .
u = x / L c , h 0 , 1 = z 0 , 1 / ( 2 σ h ) , ζ 0 = γ 0 / ( 2 σ s ) ,
g = 2 η v 1 Λ 1 e h 0     2 π [ 1 + erf ( h 0 ) ] = g ( v 1 ; h 0 )   for   u > u t .
G ( v 1 ; h 1 ) = [ 1 + erf ( h 1 + u t v 1 η ) 2 ] Λ 1   for   u > u t .

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