Abstract

We adapt the differential method to the study of scattering from randomly rough inhomogeneous films, and we extend the application domain of the surface-integral method to rough surfaces with many embedded scatterers. These methods are compared in the case of geometries in which both volume and surface scattering occur. A good agreement is obtained, and the advantages and drawbacks of each technique are pointed out. The angular scattering from rough inhomogeneous structures corresponding to models of snowcover in the radio-frequency domain or paints in the optical domain is shown.

© 1998 Optical Society of America

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References

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  1. H. Kaplan, “Black coatings are critical in optical design,” Phot. Spec. 31, 48–50 (1997).
  2. H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
    [CrossRef] [PubMed]
  3. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  4. A. Fung, “First-order radiative transfer solution—active sensing,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 2, pp. 49–122.
  5. C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
    [CrossRef]
  6. S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
    [CrossRef]
  7. K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
    [CrossRef]
  8. L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
    [CrossRef]
  9. P. Vincent, “Differential methods,” in Progress in Optics XXII, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
  10. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  11. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  12. M. Saillard, G. Toso, “Electromagnetic scattering from bounded of infinite subsurface bodies,” Radio Sci. 32, 1347–1359 (1997).
    [CrossRef]
  13. K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
    [CrossRef]
  14. J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–363 (1977).
    [CrossRef]
  15. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  16. A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
    [CrossRef]
  17. A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1–8 (1997).
    [CrossRef]
  18. K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
    [CrossRef]
  19. A. Fung, “Scattering and emission models for snow and sea ice,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 9, p. 382.
  20. C. Mätzler, “Applications of the interaction of microwaves with the natural snowcover,” Remote Sens. Rev. 2, 259–392 (1987).
    [CrossRef]
  21. M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).
  22. S. Surdyk, M. Fily, “Results of a stratified snow emissivity model based on the wave approach: application to the Antarctic sheet,” J. Geophys. Res. 100, 8837–8848 (1995).
    [CrossRef]
  23. K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag. 45, 1419–1430 (1997).
    [CrossRef]
  24. H. Giovannini, C. Amra, “Enhanced absorption in very rough overcoated black surfaces,” International Symposium on Optical Science, Engineering, and Instrumentation, San Diego, Proc. SPIE3133, (1997), pp. 110–114.
    [CrossRef]
  25. M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. (Paris) 19, 173–176 (1988).
    [CrossRef]
  26. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–295 (1978).
    [CrossRef]

1997

H. Kaplan, “Black coatings are critical in optical design,” Phot. Spec. 31, 48–50 (1997).

H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
[CrossRef] [PubMed]

M. Saillard, G. Toso, “Electromagnetic scattering from bounded of infinite subsurface bodies,” Radio Sci. 32, 1347–1359 (1997).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface,” J. Opt. Soc. Am. A 14, 1–8 (1997).
[CrossRef]

K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag. 45, 1419–1430 (1997).
[CrossRef]

1996

K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
[CrossRef]

K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

1995

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

S. Surdyk, M. Fily, “Results of a stratified snow emissivity model based on the wave approach: application to the Antarctic sheet,” J. Geophys. Res. 100, 8837–8848 (1995).
[CrossRef]

1994

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

1993

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

1990

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

1988

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. (Paris) 19, 173–176 (1988).
[CrossRef]

1987

C. Mätzler, “Applications of the interaction of microwaves with the natural snowcover,” Remote Sens. Rev. 2, 259–392 (1987).
[CrossRef]

1984

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1978

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–295 (1978).
[CrossRef]

1977

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–363 (1977).
[CrossRef]

Amra, C.

H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
[CrossRef] [PubMed]

C. Amra, “First-order vector theory of bulk scattering in optical multilayers,” J. Opt. Soc. Am. A 10, 365–374 (1993).
[CrossRef]

H. Giovannini, C. Amra, “Enhanced absorption in very rough overcoated black surfaces,” International Symposium on Optical Science, Engineering, and Instrumentation, San Diego, Proc. SPIE3133, (1997), pp. 110–114.
[CrossRef]

Bellemain, P.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

Benallegne, M.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

Chan, C. H.

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Chiu, T.

K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag. 45, 1419–1430 (1997).
[CrossRef]

Dietrich, S.

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Fily, M.

S. Surdyk, M. Fily, “Results of a stratified snow emissivity model based on the wave approach: application to the Antarctic sheet,” J. Geophys. Res. 100, 8837–8848 (1995).
[CrossRef]

Fung, A.

A. Fung, “Scattering and emission models for snow and sea ice,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 9, p. 382.

A. Fung, “First-order radiative transfer solution—active sensing,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 2, pp. 49–122.

Giovannini, H.

H. Giovannini, C. Amra, “Scattering-reduction effect with overcoated rough surfaces: theory and experiment,” Appl. Opt. 36, 5574–5579 (1997).
[CrossRef] [PubMed]

H. Giovannini, C. Amra, “Enhanced absorption in very rough overcoated black surfaces,” International Symposium on Optical Science, Engineering, and Instrumentation, San Diego, Proc. SPIE3133, (1997), pp. 110–114.
[CrossRef]

Haase, A.

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Hallikainen, M. T.

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

Hugonin, J. P.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–363 (1977).
[CrossRef]

Kaplan, H.

H. Kaplan, “Black coatings are critical in optical design,” Phot. Spec. 31, 48–50 (1997).

Li, L.

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Lussky, R. F.

K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
[CrossRef]

Madrazo, A.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

Mätzler, C.

C. Mätzler, “Applications of the interaction of microwaves with the natural snowcover,” Remote Sens. Rev. 2, 259–392 (1987).
[CrossRef]

Maystre, D.

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. (Paris) 19, 173–176 (1988).
[CrossRef]

Mc Gurn, A. R.

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

Montiel, F.

Nevière, M.

Nieto-Vesperinas, M.

Nyfors, E. G.

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

O’Neill, K.

K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
[CrossRef]

Oh, Y.

K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Pak, K.

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Paulsen, K. D.

K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
[CrossRef]

Petit, R.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–363 (1977).
[CrossRef]

Rakotoarivony, L.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

Saillard, M.

M. Saillard, G. Toso, “Electromagnetic scattering from bounded of infinite subsurface bodies,” Radio Sci. 32, 1347–1359 (1997).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. (Paris) 19, 173–176 (1988).
[CrossRef]

Sarabandi, K.

K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag. 45, 1419–1430 (1997).
[CrossRef]

K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Sihvola, A. H.

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

Surdyk, S.

S. Surdyk, M. Fily, “Results of a stratified snow emissivity model based on the wave approach: application to the Antarctic sheet,” J. Geophys. Res. 100, 8837–8848 (1995).
[CrossRef]

Taconet, O.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

Tiuri, M. E.

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

Toso, G.

M. Saillard, G. Toso, “Electromagnetic scattering from bounded of infinite subsurface bodies,” Radio Sci. 32, 1347–1359 (1997).
[CrossRef]

Tsang, L.

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Ulaby, F. T.

K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

Vidal-Madjar, D.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–295 (1978).
[CrossRef]

P. Vincent, “Differential methods,” in Progress in Optics XXII, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.

Ann. Phys. (New York)

A. A. Maradudin, T. Michel, A. R. Mc Gurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (New York) 203, 255–276 (1990).
[CrossRef]

Appl. Opt.

IEEE J. Oceanic Eng.

M. E. Tiuri, A. H. Sihvola, E. G. Nyfors, M. T. Hallikainen, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng. OE-9, 377–382 (1994).

IEEE Trans. Antennas Propag.

K. Sarabandi, T. Chiu, “Electromagnetic scattering from slightly rough surfaces with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag. 45, 1419–1430 (1997).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

K. O’Neill, R. F. Lussky, K. D. Paulsen, “Scattering from a metallic object embedded near the randomly rough surface of a lossy dielectric,” IEEE Trans. Geosci. Remote Sens. 34, 367–376 (1996).
[CrossRef]

K. Sarabandi, Y. Oh, F. T. Ulaby, “A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces,” IEEE Trans. Geosci. Remote Sens. 34, 425–432 (1996).
[CrossRef]

J. Electron. Waves Appl.

L. Rakotoarivony, O. Taconet, D. Vidal-Madjar, P. Bellemain, M. Benallegne, “Radar backscattering over agricultural bare soils,” J. Electron. Waves Appl. 10, 187–210 (1996).
[CrossRef]

J. Geophys. Res.

S. Surdyk, M. Fily, “Results of a stratified snow emissivity model based on the wave approach: application to the Antarctic sheet,” J. Geophys. Res. 100, 8837–8848 (1995).
[CrossRef]

J. Opt. (Paris)

M. Saillard, D. Maystre, “Scattering from random rough surfaces: a beam simulation method,” J. Opt. (Paris) 19, 173–176 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–363 (1977).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–295 (1978).
[CrossRef]

Phot. Spec.

H. Kaplan, “Black coatings are critical in optical design,” Phot. Spec. 31, 48–50 (1997).

Phys. Rep.

S. Dietrich, A. Haase, “Scattering of x-rays and neutrons at interfaces,” Phys. Rep. 260, 1–138 (1995).
[CrossRef]

Phys. Rev. B

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Radio Sci.

M. Saillard, G. Toso, “Electromagnetic scattering from bounded of infinite subsurface bodies,” Radio Sci. 32, 1347–1359 (1997).
[CrossRef]

K. Pak, L. Tsang, L. Li, C. H. Chan, “Combined random rough surface and volume scattering based on Monte Carlo simulations of solutions of Maxwell’s equations,” Radio Sci. 28, 331–338 (1993).
[CrossRef]

Remote Sens. Rev.

C. Mätzler, “Applications of the interaction of microwaves with the natural snowcover,” Remote Sens. Rev. 2, 259–392 (1987).
[CrossRef]

Other

H. Giovannini, C. Amra, “Enhanced absorption in very rough overcoated black surfaces,” International Symposium on Optical Science, Engineering, and Instrumentation, San Diego, Proc. SPIE3133, (1997), pp. 110–114.
[CrossRef]

A. Fung, “First-order radiative transfer solution—active sensing,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 2, pp. 49–122.

P. Vincent, “Differential methods,” in Progress in Optics XXII, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.

A. Fung, “Scattering and emission models for snow and sea ice,” in Microwave Scattering and Emission Models and Their Applications, F. T. Ulaby, ed. (Artech House, Boston, 1994), Chap. 9, p. 382.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem. Throughout the paper the geometries are invariant along the y axis, and they are illuminated under normal incidence by a Gaussian beam of waist w=1 [Eq. (4a)].

Fig. 2
Fig. 2

DRC of a rough surface. δ=0.1, a=1, L=16, ε2=2.25, λ1=1. The waist of the incident beam is w=1; normal incidence for (a) s polarization, (b) p polarization.

Fig. 3
Fig. 3

Geometry of the problems: (a) stack of two rough interfaces, (b) buried cylindrical object under a rough surface. The structures considered are invariant along the y axis.

Fig. 4
Fig. 4

(a) Buried object (cylinder) under a flat surface. (b) Comparison between a boundary finite-element description and the mixed representation. Dashed curve, 5 elements are used to describe the rod; dotted–dashed curve, 15 elements are used to describe the rod. Solid curve, 50 elements are used to describe the rod (convergence reached); dotted curve, mixed representation with one term in the Rayleigh expansion.

Fig. 5
Fig. 5

Buried object (cylinder) under a rough surface. The structures considered are invariant along the y axis.

Fig. 6
Fig. 6

DRC of the structure shown in Fig. 5: δ=0.1, a=1, L=16, ε2=2.25+10-2i, ε3=9. ε is ε2 in the bulk and εs in the scatterer. The waist of incident beam is w=1. Normal incidence for (a) s polarization, (b) p polarization.

Fig. 7
Fig. 7

DRC of a rough inhomogeneous medium with 87 cylindrical scatterers randomly distributed. δ=0.1, a=1, L=16, ε2=1.482. Thickness of the nonhomogeneous region, D=0.6; diameter of the scatterers. ϕ=λ1/20; waist of the incident beam: w=1. Normal incidence and S polarization for (a) εs=3.15, (b) εs=9.

Fig. 8
Fig. 8

Model of the snow layer. Structures considered are invariant along the y axis. Diameter of the scatterers is ϕ=λ1/30. δ1=8.10-2, a1=4.10-1, δ2=3.10-2, a2=1.6.10-1, ε2=5, ε3=1.482, εs=3.15; ε is ε3 in the bulk and εs in the scatterer.

Fig. 9
Fig. 9

DRC of the structure described in Fig. 8 for two densities (in volume) of the scatterers. Scatterers are randomly distributed. Average of 50 realizations. The waist of the incident beam is w=1. Normal incidence for s polarization.

Fig. 10
Fig. 10

Model of the snow layer as a graded index medium. The upper surface and the lower surface are identical. δ=0.1, a=1, L=25; ε2=5; ε(x, z)=1.57+10-1d, where d is the distance of the point (x, z) from the surface.

Fig. 11
Fig. 11

DRC of the structure described in Fig. 10. Average of 200 realizations, plane incident wave. Normal incidence for s polarization.

Equations (39)

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

εt(x, z)=ε1ifz>D,
εt(x, z)=ε(x, z)if0<z<D,
εt(x, z)=ε2ifz<0.
2F+kj2F=0,
Finc(x, z)
=P(α-αinc)exp[iαx-iγ1(α)(z-z0)]dα,
P(α)=w exp(-w2α2/2),
γj(α)=(kj2-α2)1/2,Im(γj)>0,j=1, 2,
αinc=k1 sin θinc.
F(x, z)=Finc(x, z)+f1(α)exp[iαx+iγ1(α)z]dα.
F(x, z)=f2(α)exp[iαx-iγ2(α)z]dα,
Rθs=|γ1(αs)f1(αs)|2Pinc,
Tθs=|γ2(αs)f2(αs)|2Pinc,
Pinc=|P(α-αinc)γ1(α)|2dα.
2E+k2(x, y)E=0,
k2(x, z)=k2(z)+Δk2(x, z),
E(x, z)=e(α, z)exp(iαx)dα,
Δk2(x, z)=Δκ2(α, z)exp(iαx)dα.
2e(α, z)z2+[k2(z)-α2]e(α, z)
+Δκ(α-α, z)e(α, z)dα=0.
eeα(z)=T¯(z, z)eeα(z),
e(z)=e(-NΔα, z)e(NΔα, z).
f1(-NΔα)f1(NΔα)f2(-NΔα)f2(NΔα)=S¯00P(-NΔα-αinc)P(NΔα-αinc).
x1k2Hx+z1k2Hz+H=0.
Hz=k2E˜,
E˜z=-x1k2Hx-H.
1k2(x, z)=1k2(z)+Δ 1k2(x, z),
Δ 1k2(x, z)=0if|x|>L/2.
hz(α, z)=k2(z)e˜(α, z)+Δκ(α-α, z)e˜(α, z)dα,
e˜z(α, z)=h(α, z)α2 1k2(z)-1+ααΔ 1κ2(α-α, z)h(α, z)dα.
F(P)=Finc+C1G1(P, M)ϕ1(M)ds,ifPΩ1,
F(P)=C2G2(P, M)ϕ2(M)ds,ifPΩ2.
12F(M)=C1C2-G(M, M)dFdnΩ(M)+F(M) dGdn(M, M)ds,MC1C2,
FjR(P)=n=-+bn(j)Hn(l)(krj)exp(inθj),
Fjimp(rj, θj)=n=-+an(j)Jn(krj)exp(inθj),
12F(M)=C1-G(M, M)dFdnΩ(M)+F(M) dGdn(M, M)ds+j>1FjR(M).
a0(j)=(F-FjR)(Oj).
b0(j)=S00(j)C1-G(Oj, M)dFdnΩ(M)+F(M) dGdn(Oj, M)ds+S00(j)pjFpR(Oj).
f1(α)=-i4πγ1(α)ϕ1(α)exp[-iαx-iγ1(α)z]ds.

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