Abstract

Surface reflection is an important phenomenon that must be taken into account when studying sea surface infrared emissivity, especially at large observation angles. This paper models analytically the polarized infrared emissivity of one-dimensional sea surfaces with shadowing effect and one surface reflection, by assuming a Gaussian surface slope distribution. A Monte Carlo ray-tracing method is employed as a reference. It is shown that the present model agrees well with the reference method. The emissivity calculated by the present model is then compared with measurements. The comparisons show that agreements are greatly improved by taking one surface reflection into account. The Monte Carlo ray-tracing results of sea surface infrared emissivity with two and three reflections are also determined. Their contributions are shown to be negligible.

© 2011 Optical Society of America

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  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]
  2. C. Bourlier, “Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics,” Appl. Opt. 44, 4335–4349 (2005).
    [CrossRef] [PubMed]
  3. S. Fauqueux, K. Caillault, P. Simoneau, and L. Labarre, “Multiresolution infrared optical properties for Gaussian sea surfaces: Theoretical validation in the one-dimensional case,” Appl. Opt. 48, 5337–5347 (2009).
    [CrossRef] [PubMed]
  4. X. Wu and W. L. Smith, “Emissivity of rough sea surface for 8–13 μm: Modeling and verification,” Appl. Opt. 36, 2609–2619 (1997).
    [CrossRef] [PubMed]
  5. W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
    [CrossRef]
  6. R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
    [CrossRef]
  7. 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]
  8. K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488–496 (2006).
    [CrossRef]
  9. 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, 453–467 (2003).
    [CrossRef]
  10. H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
    [CrossRef]
  11. C. Bourlier, “Unpolarized emissivity with shadow and multiple reflections from random rough surfaces with the geometric optics approximation: Application to Gaussian sea surfaces in the infrared band,” Appl. Opt. 45, 6241–6254 (2006).
    [CrossRef] [PubMed]
  12. J. A. Shaw, “Degree of linear polarization in spectral radiances from water-viewing infrared radiometers,” Appl. Opt. 38, 3157–3165 (1999).
    [CrossRef]
  13. J. Shaw and C. Marston, “Polarized infrared emissivity for a rough water surface,” Opt. Express 7, 375–380 (2000).
    [CrossRef] [PubMed]
  14. N. Pinel and C. Bourlier, “Scattering from very rough layers under the geometric optics approximation: further investigation,” J. Opt. Soc. Am. A 25, 1293–1306 (2008).
    [CrossRef]
  15. B. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [CrossRef]
  16. C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
    [CrossRef]
  17. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [CrossRef]
  18. G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]

2011 (1)

H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
[CrossRef]

2009 (1)

2008 (1)

2006 (2)

C. Bourlier, “Unpolarized emissivity with shadow and multiple reflections from random rough surfaces with the geometric optics approximation: Application to Gaussian sea surfaces in the infrared band,” Appl. Opt. 45, 6241–6254 (2006).
[CrossRef] [PubMed]

K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488–496 (2006).
[CrossRef]

2005 (2)

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

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

2003 (1)

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, 453–467 (2003).
[CrossRef]

2002 (1)

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
[CrossRef]

2000 (1)

1999 (1)

1997 (1)

1996 (2)

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

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]

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)

1967 (1)

B. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

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, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
[CrossRef]

Bourlier, C.

H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
[CrossRef]

N. Pinel and C. Bourlier, “Scattering from very rough layers under the geometric optics approximation: further investigation,” J. Opt. Soc. Am. A 25, 1293–1306 (2008).
[CrossRef]

C. Bourlier, “Unpolarized emissivity with shadow and multiple reflections from random rough surfaces with the geometric optics approximation: Application to Gaussian sea surfaces in the infrared band,” Appl. Opt. 45, 6241–6254 (2006).
[CrossRef] [PubMed]

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, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
[CrossRef]

Brown, J.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Brown, O.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Caillault, K.

Caselles, V.

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

Coll, C.

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

Cox, C.

Fauqueux, S.

Feltz, W.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

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, 453–467 (2003).
[CrossRef]

Howell, H. B.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Knuteson, R. O.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Labarre, L.

Li, H.

H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
[CrossRef]

Marston, C.

Masuda, K.

K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488–496 (2006).
[CrossRef]

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]

McKeown, W.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Menzel, W. P.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Minnett, P.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Munk, W.

Nalli, N. R.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Niclòs, R.

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

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]

Pinel, N.

H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
[CrossRef]

N. Pinel and C. Bourlier, “Scattering from very rough layers under the geometric optics approximation: further investigation,” J. Opt. Soc. Am. A 25, 1293–1306 (2008).
[CrossRef]

Querry, M. R.

Revercomb, H. E.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
[CrossRef]

Sanchez, J. M.

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

Shaw, J.

Shaw, J. A.

Simoneau, P.

Smith, B.

B. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

Smith, W. L.

X. Wu and W. L. Smith, “Emissivity of rough sea surface for 8–13 μm: Modeling and verification,” Appl. Opt. 36, 2609–2619 (1997).
[CrossRef] [PubMed]

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

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, 453–467 (2003).
[CrossRef]

Valor, E.

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[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, 453–467 (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]

Wu, X.

Appl. Opt. (6)

Bull. Am. Meteorol. Soc. (1)

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, N. R. Nalli, H. B. Howell, W. P. Menzel, O. Brown, J. Brown, P. Minnett, and W. McKeown, “Observations of the infrared radiative properties of the ocean implications for the measurement of sea surface temperature via satellite remote sensing,” Bull. Am. Meteorol. Soc. 77, 41–51 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

B. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[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 (1)

Opt. Express (1)

Remote Sens. Environ. (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]

K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488–496 (2006).
[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, 453–467 (2003).
[CrossRef]

R. Niclòs, E. Valor, V. Caselles, C. Coll, and J. M. Sanchez, “In situ angular measurements of thermal infrared sea surface emissivity—validation of models,” Remote Sens. Environ. 94, 83–93 (2005).
[CrossRef]

Waves Random Complex Media (2)

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. single scattering,” Waves Random Complex Media 12, 145–173 (2002).
[CrossRef]

H. Li, N. Pinel, and C. Bourlier, “A monostatic illumination function with surface reflections from one-dimensional rough surfaces,” Waves Random Complex Media 21, 105–134 (2011).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometry of zero-order emissivity (without surface reflection).

Fig. 2
Fig. 2

Geometry of first-order emissivity (with one surface reflection).

Fig. 3
Fig. 3

Four cases of single surface reflection, with (a) case 1, the reflection ray M 0 ( θ 01 ) propagates rightward and downward, (b) case 2, M 0 ( θ 01 ) propagates rightward and upward, (c) case 3, M 0 ( θ 01 ) propagates leftward and upward, and (d) case 4, M 0 ( θ 01 ) propagates leftward and downward.

Fig. 4
Fig. 4

Marginal slope histogram of the (a) first- order illumination function and (b) average first-order illumination function.

Fig. 5
Fig. 5

First-order emissivity for u 12 = 10 m / s , λ = 10 μm [(a) H polarization, (b) V polarization], and u 12 = 5 m / s , λ = 4 μm [(c) H polarization, (d) V polarization].

Fig. 6
Fig. 6

Uncorrelated zero-order emissivity of the surface (solid curve) and the corresponding Monte Carlo result (triangles), and the total emissivity of the surface (dashed curve) and the corresponding Monte Carlo result (wedges) for (a) H polarization and (b) V polarization.

Fig. 7
Fig. 7

(a) DOP of the first-order infrared emissivity of the sea surface and the comparison between the DOPs without surface reflections [zero-order, solid curve in (b)] and with one surface reflection [total, dashed curve in (b)].

Fig. 8
Fig. 8

Comparison with measurements by Smith et al. [5] at observation angles of 36.5 ° (upper), 56.5 ° (intermediate), and 73.5 ° (lower). The measurement data are taken from Fig. 10 of [5]. Simulation is made under a wind speed of 5 m / s .

Fig. 9
Fig. 9

Comparison with measurements by Niclòs et al. at wind speeds of 4.5 m / s [(a)–(c)] and 10.3 m / s [(d)–(f)]. Three wavelength channels are considered: 8.2 9.2 μm [(a) and (d)], 10.5 11.5 μm [(b) and (e)], 11.5 12.5 μm [(c) and (f)].

Equations (30)

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2 π ρ c cos 3 χ 0 λ ,
cos χ 0 = n ^ 0 · m ^ 0 = cos θ γ 0 sin θ ( 1 + γ 0 2 ) 1 / 2 .
ε 0 , V , H = [ 1 | r V , H ( χ 0 ) | 2 ] g 0 S 0 ( μ , γ 0 , ζ 0 ) 0 ,
0 = + + p ( ζ 0 , γ 0 ) d ζ 0 d γ 0 ,
r V ( χ ) = n cos χ cos χ n cos χ + cos χ , r H ( χ ) = cos χ n cos χ cos χ + n cos χ ,
g 0 = 1 γ 0 tan θ .
S 0 ( μ , γ 0 , ζ 0 ) = ϒ ( μ γ 0 ) F ( ζ 0 ) Λ ( μ ) ,
F ( ζ ) = ζ p ζ ( t ) d t .
Λ ( μ ) = 1 μ μ + ( γ μ ) p γ ( γ ) d γ .
p ( ζ , γ , ζ 0 , γ 0 ) = p ζ ( ζ ) p γ ( γ ) p ζ ( ζ 0 ) p γ ( γ 0 ) ,
ε 1 local = [ 1 | r ( χ 1 ) V , H | 2 ] | r ( χ 0 ) V , H | 2 .
{ ε 1 , V local = [ 1 | r V ( χ 1 ) | 2 ] | r V ( χ 0 ) | 2 ε 1 , H local = [ 1 | r H ( χ 1 ) | 2 ] | r H ( χ 0 ) | 2 .
{ ε 1 , V = ε 1 , V local g 0 S 1 1 ε 1 , H = ε 1 , H local g 0 S 1 1 ,
1 = + + + + × p ( ζ 1 , γ 1 , ζ 0 , γ 0 ) d ζ 1 d γ 1 d ζ 0 d γ 0 ,
m ^ 01 = 2 ( n ^ 0 · m ^ 0 ) n ^ 0 m ^ 0 = 2 cos χ 0 n ^ 0 m ^ 0 .
cos θ 01 = m ^ 01 · z ^ = 2 cos θ γ 0 sin θ 1 + γ 0 2 cos θ = cos θ [ 2 g 0 ( 1 + γ 0 2 ) 1 1 ] .
{ t 1 = + 1 , γ 0 < tan ( θ / 2 ) t 1 = 1 , γ 0 > tan ( θ / 2 ) .
θ 01 = t 1 · arccos { cos θ [ 2 g 0 ( 1 + γ 0 2 ) 1 1 ] } ,
{ θ 10 = θ 01 π for     θ 01 > 0 θ 10 = θ 01 + π for     θ 01 < 0 .
cos χ 1 = cos θ 10 γ 1 sin θ 10 ( 1 + γ 1 2 ) 1 / 2 .
S 1 ( μ , ζ 0 , γ 0 ) = { F ( ζ 0 ) Λ ( μ ) 1 & 4 F ( ζ 0 ) Λ ( μ ) [ 1 F ( ζ 0 ) Λ ( μ 01 ) ] 2 & 3 0 otherwise ,
Λ ( μ 01 ) = { 1 μ 01 μ 01 + ( γ μ 01 ) p γ ( γ ) d γ , θ 01 > 0 1 μ 01 μ 01 ( γ μ 01 ) p γ ( γ ) d γ , θ 01 < 0 .
S ¯ 1 ( μ , γ 0 ) = + S 1 ( μ , γ 0 , ζ 0 ) p ( γ 0 , ζ 0 ) d ζ 0 .
S ¯ ¯ 1 ( μ ) = + + S 1 ( μ , γ 0 , ζ 0 ) p ( γ 0 , ζ 0 ) d ζ 0 d γ 0 .
{ γ 1 > μ 01 for     θ 01 > 0 γ 1 < μ 01 for     θ 01 < 0 .
p ( γ 1 ) = { ϒ ( γ 1 μ 01 ) μ 01 + p γ ( γ ) d γ p γ ( γ 1 ) for     θ 01 > 0 ϒ ( μ 01 γ 1 ) μ 01 p γ ( γ ) d γ p γ ( γ 1 ) for     θ 01 < 0 ,
DOP = ε H ε V ε H + ε V .
p γ ( γ ) = 1 2 π σ γ exp ( γ 2 2 σ γ 2 ) ,
σ γ 2 3.16 × 10 3 u 12 ,
{ ε MC , V = 1 N s i = 1 i = N i [ 1 | r V ( χ 1 , i ) | 2 ] | r V ( χ 0 , i ) | 2 g 0 ε MC , H = 1 N s i = 1 i = N i [ 1 | r H ( χ 1 , i ) | 2 ] | r H ( χ 0 , i ) | 2 g 0 .

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