Abstract

The validation of the multiresolution model of sea surface infrared optical properties developed at ONERA is investigated in the one-dimensional case by comparison with a reference model, using a submillimeter discretization of the surface. Having expressed the optical properties, we detail the characteristics of each model. A set of numerical tests is made for various wind speeds, resolutions, and realizations of the sea surface. The tests show a good agreement between the results except for grazing angles, where the impact of multiple reflections and the effects of adjacent rough surfaces on shadow have to be investigated.

© 2009 Optical Society of America

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References

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  1. P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
    [CrossRef]
  2. 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] [PubMed]
  3. K. Yoshimori, K. Itoh, and Y. Ichioka, “Thermal radiative and reflective characteristics of a wind-roughened water surface,” J. Opt. Soc. Am. A 11, 1886-1893 (1994).
    [CrossRef]
  4. C. R. Zeisse, “Radiance of the ocean horizon,” J. Opt. Soc. Am. A 12, 2022-2030 (1995).
    [CrossRef]
  5. C. Bourlier, J. Saillard, and G. Berginc, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effects,” in Proceedings of IEEE Conference on Transactions on Geoscience and Remote Sensing (IEEE, 2001), pp. 379-392.
    [CrossRef]
  6. V. Ross, D. Dion, and G. Potvin, “Detailed analytical approach to the Gaussian surface bidirectional reflectance distribution function specular component applied to the sea surface,” J. Opt. Soc. Am. A 22, 2442-2453 (2005).
    [CrossRef]
  7. N. R. Nalli, P. J. Minnett, and P. van 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] [PubMed]
  8. T. Elfouhaily, B. Chapron, and K. Katsaros, “A unified directional spectrum for long and short wind-driven waves,” J. Geophys. Res. 102, 15781-15796 (1997).
    [CrossRef]
  9. B. G. Smith, “Geometrical shadowing of a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1967), pp. 668-671.
    [CrossRef]
  10. 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]
  11. C. Bourlier, “Unpolarized infrared emissivity with shadow from anisotropic rough sea surfaces with non-Gaussian statistics,” Appl. Opt. 44, 4335-4349 (2005).
    [CrossRef] [PubMed]
  12. P. Beckmann, “Shadowing of random rough surfaces,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1965), pp. 384-388.
    [CrossRef]
  13. 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]
  14. 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).
    [PubMed]
  15. K. Masuda, “Infrared sea surface emissivity including multiple reflection effect for isotropic Gaussian slope distribution model,” Remote Sens. Environ. 103, 488-496 (2006).
    [CrossRef]
  16. S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.
  17. J. P. Theiler and B. G. Henderson, “A geometrical constraint on shadowing in rough surfaces,” Proc. SPIE 3122, 271-279 (1997).
    [CrossRef]
  18. F. Poirion and C. Soize, “Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes,” Rech. Aérosp. 1, 41-61 (1989).
  19. A. Fournier and W. T. Reeves, “A simple model of ocean waves,” in Proceedings of ACM SIGGRAPH Computer Graphics (1986), pp. 75-82.
    [CrossRef]
  20. H. U. Sverdrup and W. H. Munk, “Wind, sea and swell: theory of relations for forecasting,” Tech. Rep. H.O.Pub.601 (U.S. Navy Hydrographic Office, 1947).
  21. R. A. Brockelman and T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1966), pp. 621-629.
    [CrossRef]
  22. R. W. Preisendorfer and C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293-1316 (1986).
    [CrossRef]
  23. A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
    [CrossRef]
  24. V. Ross and D. Dion, “Sea surface slope statistics derived from sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112, C09015(2007).
    [CrossRef]

2008 (1)

2007 (2)

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] [PubMed]

V. Ross and D. Dion, “Sea surface slope statistics derived from sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112, C09015(2007).
[CrossRef]

2006 (3)

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(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]

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).
[PubMed]

2005 (2)

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]

1997 (2)

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

J. P. Theiler and B. G. Henderson, “A geometrical constraint on shadowing in rough surfaces,” Proc. SPIE 3122, 271-279 (1997).
[CrossRef]

1995 (2)

C. R. Zeisse, “Radiance of the ocean horizon,” J. Opt. Soc. Am. A 12, 2022-2030 (1995).
[CrossRef]

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
[CrossRef]

1994 (1)

1989 (1)

F. Poirion and C. Soize, “Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes,” Rech. Aérosp. 1, 41-61 (1989).

1986 (1)

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

1954 (1)

Beckmann, P.

P. Beckmann, “Shadowing of random rough surfaces,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1965), pp. 384-388.
[CrossRef]

Berginc, G.

C. Bourlier, J. Saillard, and G. Berginc, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effects,” in Proceedings of IEEE Conference on Transactions on Geoscience and Remote Sensing (IEEE, 2001), pp. 379-392.
[CrossRef]

Bourlier, C.

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] [PubMed]

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).
[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, J. Saillard, and G. Berginc, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effects,” in Proceedings of IEEE Conference on Transactions on Geoscience and Remote Sensing (IEEE, 2001), pp. 379-392.
[CrossRef]

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

Brockelman, R. A.

R. A. Brockelman and T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1966), pp. 621-629.
[CrossRef]

Caillault, K.

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] [PubMed]

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

Chapron, B.

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

Cox, C.

Dion, D.

V. Ross and D. Dion, “Sea surface slope statistics derived from sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112, C09015(2007).
[CrossRef]

V. Ross, D. Dion, and G. Potvin, “Detailed analytical approach to the Gaussian surface bidirectional reflectance distribution function specular component applied to the sea surface,” J. Opt. Soc. Am. A 22, 2442-2453 (2005).
[CrossRef]

Elfouhaily, T.

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

Fauqueux, S.

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] [PubMed]

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

Fournier, A.

A. Fournier and W. T. Reeves, “A simple model of ocean waves,” in Proceedings of ACM SIGGRAPH Computer Graphics (1986), pp. 75-82.
[CrossRef]

Gentili, B.

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
[CrossRef]

Hagfors, T.

R. A. Brockelman and T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1966), pp. 621-629.
[CrossRef]

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]

J. P. Theiler and B. G. Henderson, “A geometrical constraint on shadowing in rough surfaces,” Proc. SPIE 3122, 271-279 (1997).
[CrossRef]

Huet, T.

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

Ichioka, Y.

Itoh, K.

Katsaros, K.

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

Krapez, J. C.

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

Labarre, L.

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] [PubMed]

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

Malherbe, C.

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[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]

Miesch, C.

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

Minnett, P. J.

Mobley, C. D.

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

Morel, A.

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
[CrossRef]

Munk, W.

Munk, W. H.

H. U. Sverdrup and W. H. Munk, “Wind, sea and swell: theory of relations for forecasting,” Tech. Rep. H.O.Pub.601 (U.S. Navy Hydrographic Office, 1947).

Nalli, N. R.

Poirion, F.

F. Poirion and C. Soize, “Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes,” Rech. Aérosp. 1, 41-61 (1989).

Potvin, G.

Preisendorfer, R. W.

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

Reeves, W. T.

A. Fournier and W. T. Reeves, “A simple model of ocean waves,” in Proceedings of ACM SIGGRAPH Computer Graphics (1986), pp. 75-82.
[CrossRef]

Ross, V.

V. Ross and D. Dion, “Sea surface slope statistics derived from sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112, C09015(2007).
[CrossRef]

V. Ross, D. Dion, and G. Potvin, “Detailed analytical approach to the Gaussian surface bidirectional reflectance distribution function specular component applied to the sea surface,” J. Opt. Soc. Am. A 22, 2442-2453 (2005).
[CrossRef]

Saillard, J.

C. Bourlier, J. Saillard, and G. Berginc, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effects,” in Proceedings of IEEE Conference on Transactions on Geoscience and Remote Sensing (IEEE, 2001), pp. 379-392.
[CrossRef]

Simoneau, P.

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] [PubMed]

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1967), pp. 668-671.
[CrossRef]

Soize, C.

F. Poirion and C. Soize, “Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes,” Rech. Aérosp. 1, 41-61 (1989).

Sverdrup, H. U.

H. U. Sverdrup and W. H. Munk, “Wind, sea and swell: theory of relations for forecasting,” Tech. Rep. H.O.Pub.601 (U.S. Navy Hydrographic Office, 1947).

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]

Theiler, J. P.

J. P. Theiler and B. G. Henderson, “A geometrical constraint on shadowing in rough surfaces,” Proc. SPIE 3122, 271-279 (1997).
[CrossRef]

van Delst, P.

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]

Voss, K. J.

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
[CrossRef]

Yoshimori, K.

Zeisse, C. R.

Appl. Opt. (4)

J. Geophys. Res. (3)

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

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters : a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13143-13150(1995).
[CrossRef]

V. Ross and D. Dion, “Sea surface slope statistics derived from sun glint radiance measurements and their apparent dependence on sensor elevation,” J. Geophys. Res. 112, C09015(2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. Oceanogr. (1)

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

Proc. SPIE (2)

P. Simoneau, K. Caillault, S. Fauqueux, T. Huet, J. C. Krapez, L. Labarre, C. Malherbe, and C. Miesch, “MATISSE: Version 1.4 and future developments,” Proc. SPIE 6364, 636409(2006).
[CrossRef]

J. P. Theiler and B. G. Henderson, “A geometrical constraint on shadowing in rough surfaces,” Proc. SPIE 3122, 271-279 (1997).
[CrossRef]

Rech. Aérosp. (1)

F. Poirion and C. Soize, “Simulation numérique de champs vectoriels stochastiques gaussiens homogènes et non homogènes,” Rech. Aérosp. 1, 41-61 (1989).

Remote Sens. Environ. (2)

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]

Other (7)

S. Fauqueux, K. Caillault, C. Bourlier, P. Simoneau, and L. Labarre, “Multi resolution infrared optical properties for Gaussian sea surfaces,” in Proceedings of the 18th IASTED International Conference: Modeling and Simulation, R. Wamkeue, ed. (ACTA Press, 2007), pp. 197-202.

B. G. Smith, “Geometrical shadowing of a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1967), pp. 668-671.
[CrossRef]

A. Fournier and W. T. Reeves, “A simple model of ocean waves,” in Proceedings of ACM SIGGRAPH Computer Graphics (1986), pp. 75-82.
[CrossRef]

H. U. Sverdrup and W. H. Munk, “Wind, sea and swell: theory of relations for forecasting,” Tech. Rep. H.O.Pub.601 (U.S. Navy Hydrographic Office, 1947).

R. A. Brockelman and T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1966), pp. 621-629.
[CrossRef]

C. Bourlier, J. Saillard, and G. Berginc, “Theoretical study on two-dimensional Gaussian rough sea surface emission and reflection in the infrared frequencies with shadowing effects,” in Proceedings of IEEE Conference on Transactions on Geoscience and Remote Sensing (IEEE, 2001), pp. 379-392.
[CrossRef]

P. Beckmann, “Shadowing of random rough surfaces,” in Proceedings of IEEE Transactions on Antennas and Propagation (IEEE, 1965), pp. 384-388.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry definition.

Fig. 2
Fig. 2

Geometrical representation of ψ ( θ obs ; γ ) , d S ε ( γ ) , and d A ε ( θ obs ; γ ) .

Fig. 3
Fig. 3

Geometrical representation of the size L of the orthogonal projection on the ellipsoid defining Earth of the considered part of the sea surface S.

Fig. 4
Fig. 4

For u 10 = 5 m · s 1 , error percentage on Y ana as a function of log 2 ( L ) obtained for the considered seeds (10, diamond; 13, circle; 1, triangle; 49, plus; and 65, star).

Fig. 5
Fig. 5

Same as Fig. 3 for u 10 = 10 m · s 1 .

Fig. 6
Fig. 6

For u 10 = 5 m · s 1 , L = 1 m and seed = 10 : (a) multiresolution analytical CBRDF as functions of θ i (x axis) and θ obs (y axis), (b) reference CBRDF obtained for the n 0 run as functions of θ i (x axis) and θ obs (y axis), (c) absolute deviation between the multiresolution analytical and reference CBRDF as functions of θ i (x axis) and θ obs (y axis), and (d) directional emissivity as a function of θ obs . The error percentage on Y ana is 1.25%.

Fig. 7
Fig. 7

Same as Fig. 6 for u 10 = 5 m · s 1 , L = 8 m and seed = 10 . The error percentage on Y ana is 1.34%.

Fig. 8
Fig. 8

Same as Fig. 6 for u 10 = 5 m · s 1 , L = 1 m and seed = 49 . The error percentage on Y ana is 2.02%.

Fig. 9
Fig. 9

Same as Fig. 6 for u 10 = 5 m · s 1 , L = 8 m and seed = 49 . The error percentage on Y ana is 2.03%.

Fig. 10
Fig. 10

For u 10 = 5 m · s 1 , L = 1 m , and seed = 10 , CBRDF as a function of θ obs obtained by the reference and multiresolution analytical model for θ i = 60 ° (a);  θ i = 0 ° (b);  θ i = 52 ° (c) and θ i = 84 ° (d).

Fig. 11
Fig. 11

Same as Fig. 10 for u 10 = 5 m · s 1 , L = 1 m , and seed = 49 .

Fig. 12
Fig. 12

For u 10 = 5 m · s 1 , L = 256 m , directional emissivity (a) and CBRDF as a function of θ obs obtained by the reference and multiresolution analytical models, for θ i = 0 ° (b);  θ i = 52 ° (c); and θ i = 84 ° (d).

Equations (30)

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

J obs ε ( θ obs ) = + d J obs ε ( θ obs , γ ) P ( γ ) d γ ,
ρ ( θ i ) = 1 2 ( sin 2 ( θ i θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ t ) ) ,
n a sin θ i = n w sin θ t ,
ϵ ( θ i ) = 1 ρ ( θ i ) .
ψ ( θ obs ; γ ) = | θ obs + atan γ | .
d J obs ε ( θ obs ; γ ) = [ 1 ρ ( | θ obs + atan γ | ) ] d A ε ( θ obs ; γ ) × B ( T s ) ,
J obs ε ( θ obs ) = ε ( S , θ obs ) + d A ε ( θ obs ; γ ) P ( γ ) d γ × B ( T s ) .
ε ( S , θ obs ) = + [ 1 ρ ( | θ obs + atan γ | ) ] d A ε ( θ obs ; γ ) P ( γ ) d γ + d A ε ( θ obs ; γ ) P ( γ ) d γ .
g ( θ obs ; γ ) = cos θ obs ( 1 γ tan θ obs ) .
ε ( S , θ obs ) = + [ 1 ρ ( | θ obs + atan γ | ) ] ( 1 γ tan θ obs ) h ε ( θ obs ; γ ) P ( γ ) d γ + ( 1 γ tan θ obs ) h ε ( θ obs ; γ ) P ( γ ) d γ .
J obs ρ ( θ obs ) = + d J obs ρ ( θ obs ; γ ) P ( γ ) d γ .
γ s ( θ obs , θ i ) = tan ( θ obs + θ i 2 ) .
J obs ρ ( θ obs ) = Ω i d J obs ρ [ θ obs ; γ s ( θ obs , θ i ) ] P [ γ s ( θ obs , θ i ) ] 2 cos 2 ( θ obs + θ i 2 ) d θ i .
d J obs ρ [ θ obs , γ s ( θ obs , θ i ) ] = ρ ( | θ obs θ i | 2 ) d A ρ [ θ obs , θ i ; γ s ( θ obs , θ i ) ] × 1 Ω i ( θ i ) × N i ( θ i ) ,
1 Ω i ( θ i ) = { 1 if  θ i Ω i 0 otherwise .
J obs ρ ( θ obs ) = Ω i f ( S , θ i , θ obs ) cos θ i N i ( θ i ) d θ i × + d A ε ( θ obs ; γ ) P ( γ ) d γ .
f ( S , θ i , θ obs ) = ρ ( | θ obs θ i | 2 ) [ 1 γ s ( θ obs , θ i ) tan θ obs ] cos 2 ( θ obs + θ i 2 ) × h ρ [ θ obs , θ i ; γ s ( θ obs , θ i ) ] P [ γ s ( θ obs , θ i ) ] 2 cos θ i + ( 1 γ tan θ obs ) h ε ( θ obs ; γ ) P ( γ ) d γ .
u j = 3 + ( j 1 / 2 ) × Δ u , j = 1 N ;
k j = 10 u j = k j + N , j = 1 N ;
Δ j = k j + 1 k j = Δ j + N , i = j N ;
ξ x = j = 1 2 N Δ j G ( k j ) 2 log χ j cos [ k j x ω ( k j ) t + φ j ] ,
ω 2 ( k ) = ( g k + τ k 3 ρ w ) tanh ( k H ) ,
θ ( γ ; θ i , θ obs ) = acos [ 1 + γ × γ s ( θ obs , θ i ) 1 + γ 2 1 + γ s 2 ( θ obs , θ i ) ] ;
P N f [ γ s ( θ obs , θ i ) ] = 1 N f n = 1 N f 1 2 π σ ˜ exp [ θ 2 ( γ n ; θ i , θ obs ) 2 σ ˜ 2 ] ,
ξ x = j = 1 2 N L Δ j G ( k j ) 2 log χ j cos [ k j x ω ( k j ) t + φ j ] ,
h ε ( θ obs ; γ ) = { 1 ( , cot θ obs ] ( γ ) × [ 1 + Λ m , σ 2 ( θ obs ) ] 1 if     θ obs 0 1 [ cot θ obs , + ) ( γ ) × [ 1 + Λ m , σ 2 ( θ obs ) ] 1 if     θ obs 0 ,
Λ m , σ 2 ( θ obs ) = { cot θ obs + ( γ tan θ obs 1 ) P m , σ 2 ( γ ) d γ if     θ obs 0 cot θ obs ( γ tan θ obs 1 ) P m , σ 2 ( γ ) d γ if     θ obs 0 .
h ρ ( θ obs , θ i ; γ ) = [ 1 + Λ m , σ 2 ( θ obs ) + Λ m , σ 2 ( θ i ) ] 1 .
Y ( S ) = π / 2 π / 2 | ε ( S , θ ) | d θ + π / 2 π / 2 π / 2 π / 2 | f ( S , θ i , θ obs ) cos θ i | d θ i d θ obs .
D ( u 10 , α , L ) = Y ana [ S ( u 10 , α , L ) ] Y ref [ S n 0 ( u 10 , α , L ) ] Y ref [ S n 0 ( u 10 , α , L ) ] × 100.

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