Abstract

Sea surface infrared reflectivity is an important parameter in maritime remote sensing. Usually, single reflection by the sea surface is considered. However, a loss of energy is then reported for large zenith observation angles (θ>50°) with a peak of about 4% for θ80°, because of the neglect of the multiple surface reflections. This paper presents calculations for the polarized infrared reflectivity of one-dimensional sea surfaces (2D problems) with two surface reflections, by introducing a bistatic illumination function with two reflections. The results show good agreement with the ones obtained by a Monte Carlo ray-tracing method. It is also shown that the energy conservation criterion is better satisfied after considering two surface reflections.

© 2013 Optical Society of America

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  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]
  2. W. Su, T. P. Charlock, and K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
    [CrossRef]
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  4. K. Caillault, S. Fauqueux, C. Bourlier, P. Simoneau, and L. Labarre, “Multiresolution optical characteristics of rough sea surface in the infrared,” Appl. Opt. 46, 5471–5481 (2007).
    [CrossRef]
  5. V. Ross, D. Dion, and D. St-Germain, “Experimental validation of the modtran 5.3 sea surface radiance model using miramer campaign measurements,” Appl. Opt. 51, 2264–2276 (2012).
    [CrossRef]
  6. D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
    [CrossRef]
  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]
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    [CrossRef]
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    [CrossRef]
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  13. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [CrossRef]
  14. H. Li, N. Pinel, and C. Bourlier, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
    [CrossRef]
  15. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
    [CrossRef]
  16. B. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
    [CrossRef]
  17. M. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
    [CrossRef]
  18. C. Bourlier, J. Saillard, and G. Berginc, “Intrinsic infrared radiation of the sea surface,” Prog. Electromagn. Res. 27, 185–335 (2000).
    [CrossRef]
  19. 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]
  20. K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488–496 (2006).
    [CrossRef]
  21. 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]
  22. N. R. Nalli, P. J. Minnett, and P. Delst, “Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: theoretical development and calculations,” Appl. Opt. 47, 3701–3721 (2008).
    [CrossRef]
  23. 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]
  24. 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]
  25. 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]
  26. C. R. Zeisse, C. P. McGrath, K. M. Littfin, and H. G. Hughes, “Infrared radiance of the wind-ruffled sea,” J. Opt. Soc. Am. A 16, 1439–1452 (1999).
    [CrossRef]
  27. D. E. Freund, R. I. Joseph, D. J. Donohue, and K. T. Constantikes, “Numerical computations of rough sea surface emissivity using the interaction probability density,” J. Opt. Soc. Am. A 14, 1836–1849 (1997).
    [CrossRef]
  28. T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
    [CrossRef]
  29. C. Bourlier, J. Saillard, and G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
    [CrossRef]

2012 (1)

2011 (2)

H. Li, N. Pinel, and C. Bourlier, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
[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]

2009 (1)

2008 (1)

2007 (1)

2006 (2)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[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]

2005 (1)

2002 (2)

W. Su, T. P. Charlock, and K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: II. Multiple scattering,” Waves Random Media 12, 175–200 (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 (2)

C. Bourlier, J. Saillard, and G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, and G. Berginc, “Intrinsic infrared radiation of the sea surface,” Prog. Electromagn. Res. 27, 185–335 (2000).
[CrossRef]

1999 (1)

1997 (3)

1996 (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]

D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
[CrossRef]

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]

1994 (1)

1973 (1)

1970 (1)

P. J. Lynch and R. J. Wagner, “Rough-surface scattering: shadowing, multiple scatter, and energy conservation,” J. Math. Phys. 11, 3032–3042 (1970).
[CrossRef]

1969 (1)

M. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

1967 (2)

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
[CrossRef]

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

1954 (1)

Alexan, K.

D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
[CrossRef]

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: 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]

C. Bourlier, J. Saillard, and G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, and G. Berginc, “Intrinsic infrared radiation of the sea surface,” Prog. Electromagn. Res. 27, 185–335 (2000).
[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]

H. Li, N. Pinel, and C. Bourlier, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
[CrossRef]

K. Caillault, S. Fauqueux, C. Bourlier, P. Simoneau, and L. Labarre, “Multiresolution optical characteristics of rough sea surface in the infrared,” Appl. Opt. 46, 5471–5481 (2007).
[CrossRef]

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: 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]

C. Bourlier, J. Saillard, and G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

C. Bourlier, J. Saillard, and G. Berginc, “Intrinsic infrared radiation of the sea surface,” Prog. Electromagn. Res. 27, 185–335 (2000).
[CrossRef]

P. Schott, N. de Beaucoudrey, and C. Bourlier, “Reflectivity of one-dimensional rough surfaces using the ray tracing technique with multiple reflections,” in Geoscience and Remote Sensing Symposium, 2003. IGARSS ‘03. Proceedings. 2003 IEEE International (2003), vol. 7, pp. 4214–4216.

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.

Chapron, B.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[CrossRef]

Charlock, T. P.

Chenault, D. B.

Constantikes, K. T.

Cox, C.

de Beaucoudrey, N.

P. Schott, N. de Beaucoudrey, and C. Bourlier, “Reflectivity of one-dimensional rough surfaces using the ray tracing technique with multiple reflections,” in Geoscience and Remote Sensing Symposium, 2003. IGARSS ‘03. Proceedings. 2003 IEEE International (2003), vol. 7, pp. 4214–4216.

Delst, P.

Dion, D.

Donohue, D. J.

Elfouhaily, T.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[CrossRef]

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]

Freund, D. E.

Goldstein, D. L.

Hale, G. M.

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]

Hughes, H. G.

Ichioka, Y.

Itoh, K.

Joseph, R. I.

Katsaros, K.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[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.

Lawrence, O. E.

D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
[CrossRef]

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]

H. Li, N. Pinel, and C. Bourlier, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
[CrossRef]

Littfin, K. M.

Lynch, P. J.

P. J. Lynch and R. J. Wagner, “Rough-surface scattering: shadowing, multiple scatter, and energy conservation,” J. Math. Phys. 11, 3032–3042 (1970).
[CrossRef]

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]

McGrath, C. P.

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]

Minnett, P. J.

Munk, W.

Nalli, N. R.

N. R. Nalli, P. J. Minnett, and P. Delst, “Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: theoretical development and calculations,” Appl. Opt. 47, 3701–3721 (2008).
[CrossRef]

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]

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, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
[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]

Potvin, G.

Querry, M. R.

Reid, F.

D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
[CrossRef]

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]

Ross, V.

Rutledge, K.

Saillard, J.

C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface: 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]

C. Bourlier, J. Saillard, and G. Berginc, “Intrinsic infrared radiation of the sea surface,” Prog. Electromagn. Res. 27, 185–335 (2000).
[CrossRef]

C. Bourlier, J. Saillard, and G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing functions,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

Sancer, M.

M. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

Schott, P.

P. Schott, N. de Beaucoudrey, and C. Bourlier, “Reflectivity of one-dimensional rough surfaces using the ray tracing technique with multiple reflections,” in Geoscience and Remote Sensing Symposium, 2003. IGARSS ‘03. Proceedings. 2003 IEEE International (2003), vol. 7, pp. 4214–4216.

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]

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]

St-Germain, D.

Su, W.

Tyo, J. S.

Vaitekunas, D. A.

D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “Shipir/ntcs: a naval ship infrared signature countermeasure and threat engagement simulator,” Proc. SPIE 2744, 411–424 (1996).
[CrossRef]

Vandemark, D.

T. Elfouhaily, B. Chapron, K. Katsaros, and D. Vandemark, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781–15796 (1997).
[CrossRef]

Wagner, R. J.

P. J. Lynch and R. J. Wagner, “Rough-surface scattering: shadowing, multiple scatter, and energy conservation,” J. Math. Phys. 11, 3032–3042 (1970).
[CrossRef]

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
[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.

Yoshimori, K.

Zeisse, C. R.

Appl. Opt. (9)

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]

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]

W. Su, T. P. Charlock, and K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
[CrossRef]

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[CrossRef]

K. Caillault, S. Fauqueux, C. Bourlier, P. Simoneau, and L. Labarre, “Multiresolution optical characteristics of rough sea surface in the infrared,” Appl. Opt. 46, 5471–5481 (2007).
[CrossRef]

N. R. Nalli, P. J. Minnett, and P. Delst, “Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: theoretical development and calculations,” Appl. Opt. 47, 3701–3721 (2008).
[CrossRef]

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]

H. Li, N. Pinel, and C. Bourlier, “Polarized infrared emissivity of one-dimensional Gaussian sea surfaces with surface reflections,” Appl. Opt. 50, 4611–4621 (2011).
[CrossRef]

V. Ross, D. Dion, and D. St-Germain, “Experimental validation of the modtran 5.3 sea surface radiance model using miramer campaign measurements,” Appl. Opt. 51, 2264–2276 (2012).
[CrossRef]

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]

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IEEE Trans. Geosci. Remote Sens. (1)

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Figures (12)

Fig. 1.
Fig. 1.

Three cases of single surface reflection for 1D surfaces. (a) In case 1, the transmitter and the receiver are on different sides (θ0, θi<0), while in (b) case 2 and (c) case 3, they are on the same side (θ0, θi>0), with the receiver being lower (θθi) in (b) and higher (θ<θi) in (c).

Fig. 2.
Fig. 2.

Incidence ray s^i is reflected twice by the surface into the observation direction s^.

Fig. 3.
Fig. 3.

Four configurations of surface reflections (inverse path), with the reflection ray M0(θ01) propagating. (a) Case 1, rightward and downward. (b) Case 2, rightward and upward. (c) Case 3, leftward and upward. (d) Case 4, leftward and downward.

Fig. 4.
Fig. 4.

Average bistatic illumination function with one reflection S¯B1 versus θi for three θ: (a) θ=30°, (b) θ=60°, and (c) θ=80°. The wind speed u12 is 10m/s.

Fig. 5.
Fig. 5.

Hemispherical average bistatic illumination function S¯B1,hemi for surfaces with wind speed u12=10m/s.

Fig. 6.
Fig. 6.

Average bistatic illumination function with two reflections S¯B2 versus θi, for three θ: (a) θ=30°, (b) θ=60°, and (c) θ=80°. The wind speed u12 is 10m/s.

Fig. 7.
Fig. 7.

Hemispherical average bistatic illumination function with two surface reflections for wind speed u12=10m/s.

Fig. 8.
Fig. 8.

Bidirectional reflectivity with one reflection ρ1 versus θi, for three θ: (a) θ=30°, (b) θ=60°, and (c) θ=80°. The wind speed u12 is 10m/s and the wavelength is 10 μm.

Fig. 9.
Fig. 9.

Hemispherical average infrared reflectivity with one reflection ρ1hemi versus θ, for u12=10m/s and λ=10μm.

Fig. 10.
Fig. 10.

Bidirectional reflectivity with two reflections ρ2 versus θi, for three values of θ: (first row) θ=30°, (second row) θ=60°, and (third row) θ=80°, in horizontal polarization on the left, and in vertical polarization on the right. The wind speed u12 is 10m/s and the wavelength is 10 μm.

Fig. 11.
Fig. 11.

Hemispherical average infrared reflectivity with two reflection ρ2hemi with respect to θ for u12=10m/s and λ=10μm.

Fig. 12.
Fig. 12.

Verification of the energy conservation. The sum of surface emissivity and hemispherical average reflectivity is shown: (a) ε0+ρ1hemi is studied, (b) ε1 and is taken into account, and (c) then ρ2hemi is finally considered. The wind speed is u12=10m/s, and the wavelength is λ=10μm.

Equations (34)

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SB1(θ,θi,γ0,ζ0)=p(abc)=p(ac)p(b|ac),
p(b|ac)=δ(θispeθi),
p(ac)p(a)p(c)=SM(θ,γ0,ζ0)SM(θi,γ0,ζ0)=F(ζ0)Λ(μ)+Λ(μi),
p(ac)=p(a)p(c|a)p(a)=F(ζ0)Λ(μ).
p(ac)=p(c)p(a|c)p(c)=F(ζ0)Λ(μi).
S¯B1(θ,θi)=SB1(θ,θi,γ0,ζ0)1,
1=++p(ζ0,γ0)dζ0dγ0
W(θi,θispe,Δθi)={1,for|θispeθi|Δθi0,otherwise.
S¯B1,hemi(θ)=π/2π/2SB1(θ,θi,γ0,ζ0)dθi1,
SB2(θ,θi,γ0,ζ0,γ1,ζ1)=p(abcd)=p(ab)p(c|ab)p(d|abc).
p(ab)=ϒ(μγ0)F(ζ0)Λ(μ)×{1forcases1and4,1F(ζ0)Λ(μ)forcases2and3,
p(c|ab)=δ(θispeθi),
p(d|abc)={[0,90°<θi<θ10,p(d),otherθi,case1,p(d),case2,[1,0°<θi<θ,p(d),otherθi,case3,[1,0°<θi<θ,0,θ10<θi<90°,p(d),otherθi,case4,
p(d)={F(ζ1)Λ(μi),forθi>0,F(ζ1)Λ(μi),forθi>0.
S¯B2(θ,θi)=SB2(θ,θi,γ0,ζ0,γ1,ζ1)2,
2=++++p(γ0,ζ0,γ1,ζ1)dζ1dγ1dζ0dγ0
S¯B2,hemi(θ)=π/2π/2SB2(θ,θi,γ0,ζ0,γ1,ζ1)dθi2,
pγ1(γ1)={ϒ(γ1μ1)μ1+pγ(t)dtpγ(γ1),forcases1and2,ϒ(μ1γ1)μ1pγ(t)dtpγ(γ1),forcases3and4,
cosχ0=n^0·s^=cosθγ0sinθ1+γ02,
ρ1,H,Vlocal(χ0)=|rH,V(χ0)|2,
rH(χ)=cosχncosχcosχ+ncosχ,rV(χ)=ncosχcosχncosχ+cosχ,
sin(χ)=sin(χ)n.
ρ1,H,V(θ,θi)=|rH,V(χ0)|2g0SB11,
g0=1γ0tanθ.
ρ1,H,Vhemi(θ)=π/2+π/2|rH,V(χ0)|2g0SB1dθi1.
ρ2,Hlocal=|rH1(χ1)|2|rH(χ0)|2,ρ2,Vlocal=|rV1(χ1)|2|rV(χ0)|2,
ρ2,H,V(θ,θi)=ρ2,H,Vlocalg0SB22,
ρ2,H,Vhemi(θ)=π/2+π/2ρ2,H,Vlocalg0SB2dθi1.
ε(θ)+ρhemi(θ)=1.
pγ(γ)=12πσγexp(γ22σγ2),
σγ2=3.16×103u12.
Lc=2σζ/σγ,
ρ1,H,VMC=1Nsi=1N1|rH,V(χ0,i)|2g0,i,ρ2,H,VMC=1Nsi=1N2|rH,V(χ0,i)|2|rH,V(χ1,i)|2g0,i,
g0=1γ0,itanθ

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