Abstract

The emissivity of two-dimensional anisotropic rough sea surfaces with non-Gaussian statistics is investigated. The emissivity derivation is of importance for retrieval of the sea-surface temperature or equivalent temperature of a rough sea surface by infrared thermal imaging. The well-known Cox–Munk slope probability-density function, considered non-Gaussian, is used for the emissivity derivation, in which the skewness and the kurtosis (related to the third- and fourth-order statistics, respectively) are included. The shadowing effect, which is significant for grazing angles, is also taken into account. 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 addition, multiple reflections are ignored. Numerical results of the emissivity are presented for Gaussian and non-Gaussian statistics, for moderate wind speeds, for near-infrared wavelengths, for emission angles ranging from 0° (nadir) to 90° (horizon), and according to the wind direction. In addition, the emissivity is compared with both measurements and a Monte Carlo ray-tracing method.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. X. Wu, W. 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, C. Marston, “Polarized infrared emissivity for a rough water surface,” Opt. Express 7, 375–380 (2000).
    [CrossRef] [PubMed]
  3. P. D. Watts, M. R. Allen, 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, 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. P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
    [CrossRef]
  6. K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
    [CrossRef]
  7. K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
    [CrossRef] [PubMed]
  8. C. Bourlier, J. Saillard, G. Berginc, “Effect of the observation length on the two-dimensional shadowing function of the sea surface: application on infrared 3–13 µm emissivity,” Appl. Opt. 39, 3433–3442 (2000).
    [CrossRef]
  9. C. Bourlier, G. Berginc, 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]
  10. C. R. Zeiss, C. P. MacGrath, K. M. Littfin, H. G. Hughes, “Infrared radiance of the wind-ruffled sea,” J. Opt. Soc. Am. A 16, 1439–1452 (1999).
    [CrossRef]
  11. J. A. Shaw, “Polarimetric measurements of long-wave infrared spectral radiance from water,” Appl. Opt. 38, 379–392 (1999).
    [CrossRef]
  12. D. E. Freund, R. J. Joseph, D. J. Donohue, K. T. Constantikes, “Numerical computations of rough sea surface emissivity using the interaction probability,” J. Opt. Soc. Am. A 14, 1836–1849 (1997).
    [CrossRef]
  13. C. Cox, 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]
  14. P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93(C11), 13,903–13,912 (1988).
    [CrossRef]
  15. J. Wu, “Effects of atmospheric stability on ocean ripples: a comparison between optical and microwave measurements,” J. Geophys. Res. 96(C4), 7265–7269 (1991).
    [CrossRef]
  16. J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
    [CrossRef] [PubMed]
  17. R. Niclòs, E. Valor, V. Caselles, C. Coll, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
    [CrossRef]
  18. W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]
  19. B. G. Henderson, J. Theiler, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
    [CrossRef]
  20. F. G. Bass, I. M. Fuks, “Calculation of shadowing for wave scattering from a statistically rough surface,” Sov. Radiophys. 7, 101–112 (1964).
  21. F. G. Bass, I. M. Fuks, “Wave Scattering from statistically rough surfaces,” in International Series in Natural Philosophy, C. B. Vesecky, J. F. Vesecky, eds. (Pergamon, 1979).
  22. P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).
  23. C. Bourlier, G. Berginc, J. Saillard, “Monostatic and bi-static statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length. I. Single scattering,” Waves Random Media 12, 145–174 (2002).
    [CrossRef]
  24. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).
  25. B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
    [CrossRef]
  26. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 5, 668–671 (1967).
    [CrossRef]
  27. P. Beckman, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
    [CrossRef]
  28. C. Bourlier, 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]
  29. R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–627 (1967).
    [CrossRef]
  30. J. P. Theiler, B. G. Henderson, “Geometrical constraint on shadowing in rough surfaces,” in Infrared Spaceborne Remote Sensing V, M. Strojnik, B. F. Andresen, eds., SPIE3122, 271–279 (1997).
    [CrossRef]
  31. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-mm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  32. D. Friedman, “Infrared characteristics of ocean water (1.515μ),” Appl. Opt. 8, 2073–2078 (1969).
    [CrossRef] [PubMed]
  33. L. Pontier, C. Dechambenoy, “Détermination des constantes optiques de l’eau liquide entre 1 et 40 μ. Application au calcul de son pouvoir réflecteur et de son émissivté,” Ann. Geophys. 22, 633641 (1966).
  34. D. J. Segelstein, “The complex refractive index of water,” M.S. thesis (University of Missouri, 1981).
  35. D. M. Wieliczka, S. Weng, M. R. Querry, “Wedge shaped cell for highly absorbent liquids: infrared optical constants of water,” Appl. Opt. 28, 1714–1719 (1989).
    [CrossRef] [PubMed]
  36. R. W. Preisendorfer, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
    [CrossRef]
  37. C. Bourlier, G. Berginc, J. Saillard, “Monostatic and bistatic statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length. II. Multiple scattering,” Waves Random Media 12, 175–200 (2002).
    [CrossRef]
  38. A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran7,” (U.S. Air Force Geophysics Laboratory, 1989).

2003 (2)

B. G. Henderson, J. Theiler, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
[CrossRef]

C. Bourlier, 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, J. Saillard, “Monostatic and bi-static statistical shadowing functions from one-dimensional stationary randomly rough surface according to the observation length. I. Single scattering,” Waves Random Media 12, 145–174 (2002).
[CrossRef]

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

2001 (1)

C. Bourlier, G. Berginc, 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 (2)

1999 (2)

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

J. A. Shaw, “Polarimetric measurements of long-wave infrared spectral radiance from water,” Appl. Opt. 38, 379–392 (1999).
[CrossRef]

1997 (3)

1996 (2)

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, 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)

1994 (1)

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

1991 (1)

J. Wu, “Effects of atmospheric stability on ocean ripples: a comparison between optical and microwave measurements,” J. Geophys. Res. 96(C4), 7265–7269 (1991).
[CrossRef]

1989 (1)

1988 (2)

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93(C11), 13,903–13,912 (1988).
[CrossRef]

K. Masuda, T. Takashima, 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]

1986 (1)

R. W. Preisendorfer, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

1973 (1)

1969 (1)

1967 (4)

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–627 (1967).
[CrossRef]

B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 5, 668–671 (1967).
[CrossRef]

1966 (2)

L. Pontier, C. Dechambenoy, “Détermination des constantes optiques de l’eau liquide entre 1 et 40 μ. Application au calcul de son pouvoir réflecteur et de son émissivté,” Ann. Geophys. 22, 633641 (1966).

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

1965 (1)

P. Beckman, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

1964 (1)

F. G. Bass, I. M. Fuks, “Calculation of shadowing for wave scattering from a statistically rough surface,” Sov. Radiophys. 7, 101–112 (1964).

1954 (2)

P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).

C. Cox, 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]

Allen, M. R.

P. D. Watts, M. R. Allen, 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]

Bass, F. G.

F. G. Bass, I. M. Fuks, “Calculation of shadowing for wave scattering from a statistically rough surface,” Sov. Radiophys. 7, 101–112 (1964).

F. G. Bass, I. M. Fuks, “Wave Scattering from statistically rough surfaces,” in International Series in Natural Philosophy, C. B. Vesecky, J. F. Vesecky, eds. (Pergamon, 1979).

Beckman, P.

P. Beckman, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

Berginc, G.

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

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

C. Bourlier, G. Berginc, 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]

C. Bourlier, J. Saillard, G. Berginc, “Effect of the observation length on the two-dimensional shadowing function of the sea surface: application on infrared 3–13 µm emissivity,” Appl. Opt. 39, 3433–3442 (2000).
[CrossRef]

Berk, A.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran7,” (U.S. Air Force Geophysics Laboratory, 1989).

Bernstein, L. S.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran7,” (U.S. Air Force Geophysics Laboratory, 1989).

Bourlier, C.

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

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

C. Bourlier, G. Berginc, 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]

C. Bourlier, J. Saillard, G. Berginc, “Effect of the observation length on the two-dimensional shadowing function of the sea surface: application on infrared 3–13 µm emissivity,” Appl. Opt. 39, 3433–3442 (2000).
[CrossRef]

Brokelman, R. A.

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–627 (1967).
[CrossRef]

Brown, J.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Caselles, V.

R. Niclòs, E. Valor, V. Caselles, C. Coll, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
[CrossRef]

Churnside, J. H.

Coll, C.

R. Niclòs, E. Valor, V. Caselles, C. Coll, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
[CrossRef]

Constantikes, K. T.

Cox, C.

Dechambenoy, C.

L. Pontier, C. Dechambenoy, “Détermination des constantes optiques de l’eau liquide entre 1 et 40 μ. Application au calcul de son pouvoir réflecteur et de son émissivté,” Ann. Geophys. 22, 633641 (1966).

Donohue, D. J.

Feltz, W.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Freund, D. E.

Friedman, D.

Fuks, I. M.

F. G. Bass, I. M. Fuks, “Calculation of shadowing for wave scattering from a statistically rough surface,” Sov. Radiophys. 7, 101–112 (1964).

F. G. Bass, I. M. Fuks, “Wave Scattering from statistically rough surfaces,” in International Series in Natural Philosophy, C. B. Vesecky, J. F. Vesecky, eds. (Pergamon, 1979).

Hagfors, T.

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–627 (1967).
[CrossRef]

Hale, G. M.

Henderson, B. G.

B. G. Henderson, J. Theiler, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
[CrossRef]

J. P. Theiler, B. G. Henderson, “Geometrical constraint on shadowing in rough surfaces,” in Infrared Spaceborne Remote Sensing V, M. Strojnik, B. F. Andresen, eds., SPIE3122, 271–279 (1997).
[CrossRef]

Howell, H. B.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Hughes, H. G.

Hwang, P. A.

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93(C11), 13,903–13,912 (1988).
[CrossRef]

Ichioka, Y.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Itoh, K.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Joseph, R. J.

Knuteson, R. O.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Kuznetsov, P. I.

P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).

Littfin, K. M.

MacGrath, C. P.

Marston, C.

Masuda, K.

K. Masuda, T. Takashima, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
[CrossRef]

Nightingale, T. J.

P. D. Watts, M. R. Allen, 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]

Pontier, L.

L. Pontier, C. Dechambenoy, “Détermination des constantes optiques de l’eau liquide entre 1 et 40 μ. Application au calcul de son pouvoir réflecteur et de son émissivté,” Ann. Geophys. 22, 633641 (1966).

Preisendorfer, R. W.

R. W. Preisendorfer, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Querry, M. R.

Revercomb, H. E.

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Robertson, D. C.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran7,” (U.S. Air Force Geophysics Laboratory, 1989).

Saillard, J.

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

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

C. Bourlier, G. Berginc, 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]

C. Bourlier, J. Saillard, G. Berginc, “Effect of the observation length on the two-dimensional shadowing function of the sea surface: application on infrared 3–13 µm emissivity,” Appl. Opt. 39, 3433–3442 (2000).
[CrossRef]

Saunders, P. M.

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

Segelstein, D. J.

D. J. Segelstein, “The complex refractive index of water,” M.S. thesis (University of Missouri, 1981).

Shaw, J. A.

Shemdin, O. H.

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93(C11), 13,903–13,912 (1988).
[CrossRef]

Smith, B. G.

B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 5, 668–671 (1967).
[CrossRef]

Smith, W. L.

X. Wu, 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, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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]

Stratonovich, V. L.

P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).

Takashima, T.

K. Masuda, T. Takashima, 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, 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, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
[CrossRef]

Theiler, J. P.

J. P. Theiler, B. G. Henderson, “Geometrical constraint on shadowing in rough surfaces,” in Infrared Spaceborne Remote Sensing V, M. Strojnik, B. F. Andresen, eds., SPIE3122, 271–279 (1997).
[CrossRef]

Tikhonov, V. I.

P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).

Valor, E.

R. Niclòs, E. Valor, V. Caselles, C. Coll, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
[CrossRef]

Villeneuve, P.

B. G. Henderson, J. Theiler, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
[CrossRef]

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

Watts, P. D.

P. D. Watts, M. R. Allen, 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]

Weng, S.

Wieliczka, D. M.

Wu, J.

J. Wu, “Effects of atmospheric stability on ocean ripples: a comparison between optical and microwave measurements,” J. Geophys. Res. 96(C4), 7265–7269 (1991).
[CrossRef]

Wu, X.

Yoshimori, K.

K. Yoshimori, K. Itoh, Y. Ichioka, “Optical characteristics of a wind-roughened water surface: a two-dimensional theory,” Appl. Opt. 34, 6236–6247 (1995).
[CrossRef] [PubMed]

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

Zeiss, C. R.

Ann. Geophys. (1)

L. Pontier, C. Dechambenoy, “Détermination des constantes optiques de l’eau liquide entre 1 et 40 μ. Application au calcul de son pouvoir réflecteur et de son émissivté,” Ann. Geophys. 22, 633641 (1966).

Appl. Opt. (8)

Bull. Am. Meteorol. Soc. (1)

W. L. Smith, R. O. Knuteson, H. E. Revercomb, W. Feltz, H. B. Howell, W. P. Menzel, N. R. Nalli, O. Brown, J. Brown, P. Minnett, 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. (3)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 5, 668–671 (1967).
[CrossRef]

P. Beckman, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. 13, 384–388 (1965).
[CrossRef]

R. A. Brokelman, T. Hagfors, “Note of the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. 14, 621–627 (1967).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

C. Bourlier, G. Berginc, 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, 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. Geophys. Res. (4)

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72, 4643–4649 (1967).
[CrossRef]

B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93(C11), 13,903–13,912 (1988).
[CrossRef]

J. Wu, “Effects of atmospheric stability on ocean ripples: a comparison between optical and microwave measurements,” J. Geophys. Res. 96(C4), 7265–7269 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

C. Cox, 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]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

K. Yoshimori, K. Itoh, Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. 11, 1886–1893 (1994).
[CrossRef]

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

J. Phys. Oceanogr. (1)

R. W. Preisendorfer, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Opt. Express (1)

Remote Sens. Environ. (2)

K. Masuda, T. Takashima, 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, P. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte Carlo model,” Remote Sens. Environ. 88, 453457 (2003).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

P. I. Kuznetsov, V. L. Stratonovich, V. I. Tikhonov, “The duration of random function overshoots,” Sov. Phys. Tech. Phys. 24, 103 (1954).

Sov. Radiophys. (1)

F. G. Bass, I. M. Fuks, “Calculation of shadowing for wave scattering from a statistically rough surface,” Sov. Radiophys. 7, 101–112 (1964).

Waves Random Media (3)

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

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

Other (5)

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran: a moderate resolution model for lowtran7,” (U.S. Air Force Geophysics Laboratory, 1989).

D. J. Segelstein, “The complex refractive index of water,” M.S. thesis (University of Missouri, 1981).

J. P. Theiler, B. G. Henderson, “Geometrical constraint on shadowing in rough surfaces,” in Infrared Spaceborne Remote Sensing V, M. Strojnik, B. F. Andresen, eds., SPIE3122, 271–279 (1997).
[CrossRef]

R. Niclòs, E. Valor, V. Caselles, C. Coll, “Sea surface emissivity angular measurements. Comparison with theoretical models,” in Remote Sensing of the Ocean and Sea Ice 2003, C. R. Bostates, R. Santolen eds., SPIE5233, 348–356 (2003).
[CrossRef]

F. G. Bass, I. M. Fuks, “Wave Scattering from statistically rough surfaces,” in International Series in Natural Philosophy, C. B. Vesecky, J. F. Vesecky, eds. (Pergamon, 1979).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Coordinate system used to derive the shadowing function and the emissivity. θ is the emission angle defined with respect to (Oz), and ϕ is the wind direction defined in the plane (Ox, Oy).

Fig. 2
Fig. 2

Illustration of the basis used to calculate the marginal slope PDF. The surface slopes {γX, γY} defined in basis (OX, OY) are obtained from surface slopes {γx γy} defined in basis (OX, OY), by rotation of angle ϕ, where ϕ is the wind direction.

Fig. 3
Fig. 3

Definition of the shadowing function F is an arbitrary point on the surface of height z and of slopes {γx γy} defined in basis (OX, OY) (see Fig. 2).

Fig. 4
Fig. 4

Marginal slope PDFpsX) versus slope γX. Wind speed, u12 = 10 m/s. Wind direction ϕ is equal to 0°, 90°, and 180° in (a), (b), and (c), respectively. G means that the statistics are Gaussian, GS means that the second- and the third-order statistics are included, and GSK means that the second-, third-, and fourth-order statistics are included.

Fig. 5
Fig. 5

Average shadowing function Ω(Λ + 1) versus emission angle θ. Wind speed, u12 = 10 m/s. Wind direction ϕ is equal to 0°, 90°, and 180° in (a), (b), (c), respectively. G means that the statistics are Gaussian, GS means that the second- and third-order statistics are included, and GSK means that the second-, third-, and fourth-order statistics are included.

Fig. 6
Fig. 6

Unpolarized emissivity versus emission angle θ for wind direction ϕ = 0 and for wind speed u12 = 10 m/s. (a) Wavelength λ = 4 µm; (b) λ = 10 µm. G means that the statistics are Gaussian, GS means that the second- and the third-order statistics are included, and GSK means that the second-, third-, and fourth-order statistics are included. In addition, Δε denotes the emissivity difference, defined as Δε = ε(0, ϕ) − ε(π/2, ϕ).

Fig. 7
Fig. 7

Limit angle θpl versus wind speed u12, below which the sea’s surface can be considered plane.

Fig. 8
Fig. 8

Unpolarized emissivity versus wind direction ϕ for emission angle θ = 80° for wind speed u12 = 5 m/s and for wavelengths 4 and 10 µm. G means that the statistics are Gaussian, GS means that the second- and the third-order statistics are included, and GSK means that the second-, third-, and fourth-order statistics are included. In addition, Exp represents the expansion of the emissivity as a cosine series [cos(nϕ] truncated up to the second order.

Fig. 9
Fig. 9

Same variation as in Fig. 8 for wind speed u12 = 15 m/s.

Fig. 10
Fig. 10

Δε(θ) plotted versus emission angle θ for non-Gaussian statistics and for wind speeds u12 = {5, 15} m/s and wavelengths 4 and 10 µm. Δε(θ) = maxϕ∈[0;π] ε(θ, ϕ) − minϕ∈[0;π] ε(θ, ϕ).

Fig. 11
Fig. 11

Limit emission angle, θsh, below which the shadowing effect can be ignored for non-Gaussian statistics, versus wind speed u12, for wind directions ϕ = {0°, 90°, 180°} and for wavelengths 4 and 10 µm.

Fig. 12
Fig. 12

Limit emission angle below which Gaussian statistics can be considered, θst, versus wind speed u12, for wind directions ϕ = {0°, 90°, 180°} and for wavelengths 4 and 10 µm.

Fig. 13
Fig. 13

(a), (c) Comparison of the averaged unpolarized emissivity, εG (superscript G, for Gaussian statistics), with numerical results |εNum1, εNum2| obtained from a Monte Carlo ray-tracing method19 versus the emission angle for wind direction ϕ = 0, wavelength λ = 4 µm, and u12 = 5 m/s. 8Num1 neglects the multiple reflections, whereas for the computation of εNum2 the multiple reflections are taken into account. The wind speeds are u12 = {5, 15} m/s for (a), (b) and (c), (d), respectively, (b), (d) Emissivity differences {|εNum1 − εG| |εNum2 − εG|} versus emission angle.

Fig. 14
Fig. 14

Comparison of averaged unpolarized emissivity with measurements17 versus emission angle. (a)–(c) Wind speed, a12 = 4.5 m/s; (d)–(f) u12 = 10.3 m/s. The unpolarized emissivity is averaged over the ranges (a), (d) 8.2–9.2 µm; (b), (e) 10.5–11.5 µm; and (c), (f) 11.5–12.5 µm. The wind direction is ϕ = 284°. In addition, in (c) and (f), for the curve denoted Adjus the sea’s refractive index is calculated from the adjustment of Friedman.32

Fig. 15
Fig. 15

Comparison of the unpolarized emissivity spectrum with measurements18 of wave number 1 in inverse centimeters. The wind speed is u12 = 5 m/s, and the wind direction is ϕ = 0°.

Tables (1)

Tables Icon

Table 1 Coefficients {ε0,1,2(π)} of the Emissivity Expansiona and the Angle ϕminb (deg), Giving the Minimum of ε(θ, ϕ) for ϕ ∈ [0; π] and for a Given θ

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

S ( θ , ϕ | γ X ; z ) = ϒ ( μ γ X ) × [ P h ( z ) P h ( ) ] Λ ( θ , ϕ ) ,
P h ( z ) = p h ( z ) d z , Λ ( θ , ϕ ) = 1 μ μ ( γ X μ ) p s ( γ X ) d γ X , μ = cot θ .
p s ( γ X ) = + p s ( γ X , γ Y ) d γ Y ,
γ x = γ X cos ϕ γ X sin ϕ , γ y = γ X sin ϕ + γ Y cos ϕ .
S ( θ , ϕ | γ X ) = + S ( θ , ϕ | γ X , z ) × p h ( z ) d z = ϒ ( μ γ X ) Λ + 1
p s ( γ x , γ y ) = 1 2 π σ s x σ s y exp ( γ x 2 2 σ s x 2 γ y 2 2 σ s y 2 ) [ 1 + c 21 2 × ( Γ y 2 1 ) Γ x + c 03 6 ( Γ x 2 3 ) Γ x + c 22 4 × ( Γ x 2 1 ) ( Γ y 2 1 ) + c 40 24 ( Γ y 4 6 Γ y 2 + 3 ) + c 04 24 ( Γ x 4 6 Γ x 2 + 3 ) ] ,
Γ x , y = γ x , y σ s x , s y , σ s x 2 = ( 3.16 u 12 ± 4 ) 10 3 , σ s y 2 = ( 1.92 u 12 + 3 ± 4 ) 10 3 ,
c 21 = ( 0.86 u 12 1 ± 3 ) 10 2 0 , c 03 = ( 3.3 u 12 4 ± 12 ) 10 2 0 ; c 04 = 0.23 ± 0.41 , c 40 = 0.40 ± 0.23 , c 22 = 0.12 ± 0.06 .
p s ( γ X ) = 1 σ s X 2 π exp ( γ X 2 2 σ s X 2 ) [ 1 + α K ( 1 2 γ X 2 σ s X 2 + γ X 4 3 σ s X 4 ) + α S ( γ X σ s X γ X 3 3 σ s X 3 ) ] ,
α S ( ϕ ) = σ s x cos ϕ 2 σ s X 3 [ c 03 ( σ s x cos ϕ ) 2 + 3 c 21 ( σ s y sin ϕ ) 2 ] ,
α K ( ϕ ) = 1 8 σ s X 4 [ c 04 ( σ s x cos ϕ ) 4 + c 40 ( σ s y sin ϕ ) 4 + 3 2 c 22 σ s x 2 σ s y 2 sin 2 ( 2 ϕ ) ] ,
σ s X 2 ( ϕ ) = ( σ s x cos ϕ ) 2 + ( σ s y sin ϕ ) 2 .
Λ ( υ ) = Λ G ( υ ) + α S Λ S ( υ ) + α K Λ K ( υ ) ,
Λ G ( υ ) = exp ( υ 2 ) υ π erfc ( υ ) 2 υ π ,
Λ S ( υ ) = exp ( υ ) 2 3 2 π ,
Λ K ( υ ) = ( 2 υ 2 1 ) exp ( υ ) 2 6 υ π ,
υ ( θ , ϕ ) = μ σ s X 2 = cot θ { 2 [ ( σ s x cos ϕ ) 2 + ( σ s y sin ϕ ) 2 ] } 1 / 2 .
S ( θ , ϕ ) = + S ( θ , ϕ | γ X ) × p s ( γ X ) d γ X = Ω ( υ ) / [ Λ ( υ ) + 1 ] ,
Ω ( υ ) = Ω G ( υ ) + α S Ω S ( υ ) + α K Ω K ( υ ) ,
Ω G ( υ ) = [ 1 + erf ( υ ) ] / 2 ,
Ω S ( υ ) = ( 2 υ 2 1 ) exp ( υ 2 ) 3 2 π ,
Ω K ( υ ) = υ ( 2 υ 2 3 ) exp ( υ 2 ) 3 π .
ɛ ( θ , ϕ ) = + + [ 1 | r ( | ψ | ) | 2 ] p s ( γ x , γ y ) × g × S d γ x d γ y ,
g ( θ , ϕ ; γ x , γ y ) = 1 ( γ x cos ϕ + γ y sin ϕ ) tan θ .
r V ( ψ ) = n cos ψ cos ψ n cos ψ + cos ψ , r H ( ψ ) = cos ψ n cos ψ cos ψ + n cos ψ ,
cos [ ψ ( θ , ϕ ; γ x , γ y ) ] = g × cos θ ( 1 + γ x 2 + γ y 2 ) 1 / 2 ,
ɛ ( θ , ϕ ) = 1 1 + Λ ( θ , ϕ ) μ d γ X × + [ 1 | r ( | ψ | ) | 2 ] p s ( γ x , γ y ) ( 1 γ X μ ) d γ Y ,
cos [ ψ ( θ ; γ X , γ Y ) ] = 1 ( γ X / μ ) cos θ ( 1 + γ X 2 + γ Y 2 ) 1 / 2 .
lim θ π / 2 tan θ 1 + Λ ( θ , ϕ ) = [ 0 + γ X p s ( γ X ) d γ X ] 1 .
p s ( γ x , γ y ) = 1 2 π σ s x σ s y exp ( γ x 2 2 σ s x 2 γ y 2 2 σ s y 2 ) ,
ɛ ( 0 , ϕ ) [ 1 | r ( 0 ) | 2 ] + + p s ( γ s , γ y ) d γ X d γ Y = 1 | r ( 0 ) | 2 = 1 | n 1 n + 1 | 2 .
exp ( γ x 2 2 σ s x 2 γ y 2 2 σ s y 2 ) ,
a = α + β cos ( 2 ϕ ) 2 ( α 2 β 2 ) , b = β sin ( 2 ϕ ) 2 ( α 2 β 2 ) , c = α β cos ( 2 ϕ ) 2 ( α 2 β 2 ) ,
Δ ɛ ɛ d ɛ ɛ = Δ T C 2 λ T 2 e x e x 1 ,
ɛ ( θ , ϕ ) 1 1 + Λ ( θ , ϕ ) μ [ 1 | r ( | ψ | ) | 2 ] × p s ( γ X ) ( 1 γ X μ ) d γ X .
ɛ ( θ , ϕ ) ɛ 0 ( θ ) + ɛ 1 ( θ ) cos ( ϕ ) + ɛ 2 ( θ ) cos ( 2 ϕ ) ,
ɛ 0 ( θ ) = [ ɛ ( θ , 0 ) + ɛ ( θ , π ) + 2 ɛ ( θ , π / 2 ) ] / 4 , ɛ 1 ( θ ) = [ ɛ ( θ , 0 ) ɛ ( θ , π ) ] / 2 , ɛ 2 ( θ ) = [ ɛ ( θ , 0 ) + ɛ ( θ , π ) 2 ɛ ( θ , π / 2 ) ] / 4 .
Δ ɛ ( θ ) = max ϕ [ 0 ; π ] ɛ ( θ , ϕ ) min ϕ [ 0 ; π ] ɛ ( θ , ϕ ) .

Metrics