Abstract

The bistatic polarized scattering by bare soil samples of a CO2 laser beam at 10.6 μm has been experimentally studied. Large differences between HH and VV curves are usually observed, particularly in the forward plane. A simple phenomenological parameterization is proposed, based on the assumption of totally incoherent scattering by a rough medium. The normalized function F(θ)/F(0) accounting for slope distribution and shadowing is found from angular backscatter to be of the form cosm(θ), with m = 5.24 for all samples. This result is generalized to account for the bistatic case. The index of refraction of the medium is obtained from the ratio of HH and VV curves in the forward plane. Good agreement is found between experimental and calculated curves in the case of sand. The directional reflectivity and emissivity are calculated and compare well with experimental data. The calculated emissivity at nadir, for λ = 10.6 μm, is within 0.5% of the value directly measured from emitted radiation. The backscattered peak has not yet been addressed in detail, therefore preventing relating in a semiquantitative manner the intensity of the backscattered light and the emissivity.

© 1991 Optical Society of America

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References

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  1. F. Becker, “Angular reflectivity and emissivity of natural media,” in Proceedings of the Second International Colloquium on Spectral Signatures of Objects in Remote Sensing, Avignon, France,G. Guyot, A. Verbrugghe, eds. (INRA, Versailles, 1981).
  2. P. Ramanantsizehena, “Etude expérimental et théorique de la réflectivité de sols dans l'infrarouge thermique pour l'interprétation des données de télédétection spatiale,” thesis (University of Strasbourg, Strasbourg, France, 1985).
  3. B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy: I. Theory,” J. Geophys. Res. 86, 3034–3055 (1981).
    [CrossRef]
  4. B. Hapke, “Bidirectional reflectance spectroscopy. III. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
    [CrossRef]
  5. B. Hapke, “Bidirectional reflectance spectroscopy. II. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
    [CrossRef]
  6. K. D. Cooper, J. A. Smith, “A Monte Carlo reflectance model for soil surfaces with three dimensional structure,” IEEE Trans. Geosci. Remote Sensing GE-23, 668–673 (1985).
    [CrossRef]
  7. C. L. Walthell, J. M. Norman, J. M. Wells, G. Campbell, B. L. Blad, “Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces,” Appl. Opt. 24, 383–387 (1985).
    [CrossRef]
  8. B. Pinty, D. Ramond, “A physical model for predicting bidirectional reflectance over bare soils,” Remote Sensing Environ. 27, 273–288 (1989).
    [CrossRef]
  9. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1982), Vol. 2, Chap. 12.
  10. F. Becker, P. Ramanantsizehena, M. P. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
    [CrossRef] [PubMed]
  11. J. C. Leader, “An analysis of the spatial coherence of laser light scattered from a surface with two scales of roughness,” J. Opt. Soc. Am. 66, 536–546 (1976).
    [CrossRef]
  12. J. C. Leader, “Incoherent backscatter from rough surface; the two scale model reexamined,” Radio Sci. 13, 441–457 (1978).
    [CrossRef]
  13. J. C. Leader, “Analysis and prediction of laser scattering from rough surface materials,” J. Opt. Soc. Am. 69, 610–628 (1979).
    [CrossRef]
  14. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
    [CrossRef]
  15. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  16. W. G. Spitzer, D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1334 (1961).
    [CrossRef]
  17. T. Takashima, K. Masuda, “Emissivities of quartz and Sahara dust powders in the infrared region (7–17 μm),” Remote Sensing Environ. 23, 51–63 (1987).
    [CrossRef]
  18. F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
    [CrossRef]
  19. J. P. C. Southall, ed., Helmholtz's Treatise on Physiological Optics (Optical Society of America, Washington, D.C., 1924), Vol. I, p. 231.
  20. W. Egan, T. Hilgeman, “Retroreflectance measurements of photometric standards and coatings,” Appl. Opt. 15, 1845–1849 (1976).
    [CrossRef] [PubMed]
  21. S. A. W. Gerstl, “The angular reflectance signature of the canopy hot spot in the optical regime,” in Proceedings of the International Colloquium on Spectral Signatures of Objects in Remote Sensing, ESA SP-287 (European Space Agency, Nordwijk, The Netherlands, 1988), pp. 129–132.
  22. B. W. Hapke, “Bidirectional reflectance spectroscopy. 4: The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
    [CrossRef]
  23. D. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
    [CrossRef]
  24. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [CrossRef]
  25. R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987).
    [CrossRef] [PubMed]
  26. J. Labed, M. P. Stoll, “Angular variation of land surface spectral emissivity in the thermal infrared band: laboratory investigations on bare soils,” Int. J. Remote Sensing (to be published).

1990

F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
[CrossRef]

1989

B. Pinty, D. Ramond, “A physical model for predicting bidirectional reflectance over bare soils,” Remote Sensing Environ. 27, 273–288 (1989).
[CrossRef]

1987

T. Takashima, K. Masuda, “Emissivities of quartz and Sahara dust powders in the infrared region (7–17 μm),” Remote Sensing Environ. 23, 51–63 (1987).
[CrossRef]

R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987).
[CrossRef] [PubMed]

1986

B. W. Hapke, “Bidirectional reflectance spectroscopy. 4: The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

1985

1984

B. Hapke, “Bidirectional reflectance spectroscopy. III. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

1981

B. Hapke, “Bidirectional reflectance spectroscopy. II. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy: I. Theory,” J. Geophys. Res. 86, 3034–3055 (1981).
[CrossRef]

1979

1978

J. C. Leader, “Incoherent backscatter from rough surface; the two scale model reexamined,” Radio Sci. 13, 441–457 (1978).
[CrossRef]

1976

1968

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

1967

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

1961

W. G. Spitzer, D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1334 (1961).
[CrossRef]

1951

D. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

Becker, F.

F. Becker, P. Ramanantsizehena, M. P. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
[CrossRef] [PubMed]

F. Becker, “Angular reflectivity and emissivity of natural media,” in Proceedings of the Second International Colloquium on Spectral Signatures of Objects in Remote Sensing, Avignon, France,G. Guyot, A. Verbrugghe, eds. (INRA, Versailles, 1981).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Blad, B. L.

Campbell, G.

Cooper, K. D.

K. D. Cooper, J. A. Smith, “A Monte Carlo reflectance model for soil surfaces with three dimensional structure,” IEEE Trans. Geosci. Remote Sensing GE-23, 668–673 (1985).
[CrossRef]

Egan, W.

Fung, A. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1982), Vol. 2, Chap. 12.

Gerstl, S. A. W.

S. A. W. Gerstl, “The angular reflectance signature of the canopy hot spot in the optical regime,” in Proceedings of the International Colloquium on Spectral Signatures of Objects in Remote Sensing, ESA SP-287 (European Space Agency, Nordwijk, The Netherlands, 1988), pp. 129–132.

Hapke, B.

B. Hapke, “Bidirectional reflectance spectroscopy. III. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy: I. Theory,” J. Geophys. Res. 86, 3034–3055 (1981).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy. II. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

Hapke, B. W.

B. W. Hapke, “Bidirectional reflectance spectroscopy. 4: The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

Hilgeman, T.

Kleinmann, D. A.

W. G. Spitzer, D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1334 (1961).
[CrossRef]

Labed, J.

F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
[CrossRef]

J. Labed, M. P. Stoll, “Angular variation of land surface spectral emissivity in the thermal infrared band: laboratory investigations on bare soils,” Int. J. Remote Sensing (to be published).

Leader, J. C.

Masuda, K.

T. Takashima, K. Masuda, “Emissivities of quartz and Sahara dust powders in the infrared region (7–17 μm),” Remote Sensing Environ. 23, 51–63 (1987).
[CrossRef]

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1982), Vol. 2, Chap. 12.

Nerry, F.

F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
[CrossRef]

Norman, J. M.

Pinty, B.

B. Pinty, D. Ramond, “A physical model for predicting bidirectional reflectance over bare soils,” Remote Sensing Environ. 27, 273–288 (1989).
[CrossRef]

Ramanantsizehena, P.

F. Becker, P. Ramanantsizehena, M. P. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
[CrossRef] [PubMed]

P. Ramanantsizehena, “Etude expérimental et théorique de la réflectivité de sols dans l'infrarouge thermique pour l'interprétation des données de télédétection spatiale,” thesis (University of Strasbourg, Strasbourg, France, 1985).

Ramond, D.

B. Pinty, D. Ramond, “A physical model for predicting bidirectional reflectance over bare soils,” Remote Sensing Environ. 27, 273–288 (1989).
[CrossRef]

Rice, D. O.

D. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

Schiffer, R.

Smith, J. A.

K. D. Cooper, J. A. Smith, “A Monte Carlo reflectance model for soil surfaces with three dimensional structure,” IEEE Trans. Geosci. Remote Sensing GE-23, 668–673 (1985).
[CrossRef]

Spitzer, W. G.

W. G. Spitzer, D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1334 (1961).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Stoll, M. P.

F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
[CrossRef]

F. Becker, P. Ramanantsizehena, M. P. Stoll, “Angular variation of the bidirectional reflectance of bare soils in the thermal infrared band,” Appl. Opt. 24, 365–375 (1985).
[CrossRef] [PubMed]

J. Labed, M. P. Stoll, “Angular variation of land surface spectral emissivity in the thermal infrared band: laboratory investigations on bare soils,” Int. J. Remote Sensing (to be published).

Takashima, T.

T. Takashima, K. Masuda, “Emissivities of quartz and Sahara dust powders in the infrared region (7–17 μm),” Remote Sensing Environ. 23, 51–63 (1987).
[CrossRef]

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1982), Vol. 2, Chap. 12.

Valenzuela, G. R.

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

Walthell, C. L.

Wells, E.

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy: I. Theory,” J. Geophys. Res. 86, 3034–3055 (1981).
[CrossRef]

Wells, J. M.

Appl. Opt.

Commun. Pure Appl. Math.

D. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 361–378 (1951).
[CrossRef]

Icarus

B. Hapke, “Bidirectional reflectance spectroscopy. III. Correction for macroscopic roughness,” Icarus 59, 41–59 (1984).
[CrossRef]

B. W. Hapke, “Bidirectional reflectance spectroscopy. 4: The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

IEEE Trans. Antennas Propag.

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

K. D. Cooper, J. A. Smith, “A Monte Carlo reflectance model for soil surfaces with three dimensional structure,” IEEE Trans. Geosci. Remote Sensing GE-23, 668–673 (1985).
[CrossRef]

J. Geophys. Res.

B. Hapke, E. Wells, “Bidirectional reflectance spectroscopy: I. Theory,” J. Geophys. Res. 86, 3034–3055 (1981).
[CrossRef]

B. Hapke, “Bidirectional reflectance spectroscopy. II. Experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

F. Nerry, J. Labed, M. P. Stoll, “Spectral properties of land surfaces in the thermal infrared. Part I: Laboratory measurements of absolute spectral emissivity and reflectivity signatures,” J. Geophys. Res. 95, 7045–7054 (1990).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev.

W. G. Spitzer, D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1334 (1961).
[CrossRef]

Radio Sci.

J. C. Leader, “Incoherent backscatter from rough surface; the two scale model reexamined,” Radio Sci. 13, 441–457 (1978).
[CrossRef]

Remote Sensing Environ.

B. Pinty, D. Ramond, “A physical model for predicting bidirectional reflectance over bare soils,” Remote Sensing Environ. 27, 273–288 (1989).
[CrossRef]

T. Takashima, K. Masuda, “Emissivities of quartz and Sahara dust powders in the infrared region (7–17 μm),” Remote Sensing Environ. 23, 51–63 (1987).
[CrossRef]

Other

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

J. P. C. Southall, ed., Helmholtz's Treatise on Physiological Optics (Optical Society of America, Washington, D.C., 1924), Vol. I, p. 231.

S. A. W. Gerstl, “The angular reflectance signature of the canopy hot spot in the optical regime,” in Proceedings of the International Colloquium on Spectral Signatures of Objects in Remote Sensing, ESA SP-287 (European Space Agency, Nordwijk, The Netherlands, 1988), pp. 129–132.

J. Labed, M. P. Stoll, “Angular variation of land surface spectral emissivity in the thermal infrared band: laboratory investigations on bare soils,” Int. J. Remote Sensing (to be published).

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1982), Vol. 2, Chap. 12.

F. Becker, “Angular reflectivity and emissivity of natural media,” in Proceedings of the Second International Colloquium on Spectral Signatures of Objects in Remote Sensing, Avignon, France,G. Guyot, A. Verbrugghe, eds. (INRA, Versailles, 1981).

P. Ramanantsizehena, “Etude expérimental et théorique de la réflectivité de sols dans l'infrarouge thermique pour l'interprétation des données de télédétection spatiale,” thesis (University of Strasbourg, Strasbourg, France, 1985).

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Figures (16)

Fig. 1
Fig. 1

Diagram of the laboratory experimental setup for bidirectional CO2 polarized laser scattering.

Fig. 2
Fig. 2

Angular variation of the backscattered coefficient for several bare soils for both HH and VV polarizations.

Fig. 3
Fig. 3

Bidirectional scattering coefficient vs output angle θ for HH, VV, HV, and VH polarizations in the principal plane (ϕ = 0, π) and in the perpendicular plane (ϕ = π0/2). Incident angles are (a) θi = 5°; (b) θi = 28°; (c) θi = 44.5°. The sample is SiO2 sand.

Fig. 4
Fig. 4

Bidirectional scattering coefficient vs output angle θ for HH, VV, HV, and VH polarizations in the principal plane (ϕ = 0, π) and in the perpendicular plane (ϕ = π/2). Incident angles are (a) θi = 5°; (b) θi = 28°; (c) θi = 44.5°. The sample is soil P40 from Portugal.

Fig. 5
Fig. 5

Bidirectional scattering coefficient vs output angle θ for HH, VV, HV, and VH polarizations in the principal plane ϕ = 0, π) and in the perpendicular plane (ϕ = π/2). Incident angles are (a) θi = 5°; (b) θi = 28°; (c) θi = 44.5°. The sample is lehm, agricultural soil from northeast France.

Fig. 6
Fig. 6

Bidirectional scattering coefficient vs output angle θ for HH, VV, HV, and VH polarizations in the principal plane (ϕ = 0, π) and in the perpendicular plane (ϕ = π/2). The incident angle is θi = 44.5°. The sample is kaolinite (small pieces).

Fig. 7
Fig. 7

Bidirectional scattering coefficient vs output angle θ for HH, VV, HV, and VH polarizations in the principal plane (ϕ = 0, π) and in the perpendicular plane (ϕ = π/2). The incident angle is θi = 44.5°. The sample is ground sand.

Fig. 8
Fig. 8

Angular variation of the experimental normalized slope function F(θ)/F(0) for several samples deduced from the backscattered measurements.

Fig. 9
Fig. 9

The In plot of the normalized slope function F(θ)/F(0) vs In cos θ.

Fig. 10
Fig. 10

Polarization indicatrix for sand: angle of incidence θi = 51°; direction of input linear polarization with respect to V plane αi = 60°.

Fig. 11
Fig. 11

Comparison between the calculated bidirectional scattering coefficient and experimental results for SiO2 sand. Incident angles are (a) θi = 5°; (b) θi = 28°; (c) θi = 44.5°.

Fig. 12
Fig. 12

Comparison between the calculated bidirectional scattering represented by Fslope(γ) cos(θ1) and experimental results in the principal plane (ϕ = 0, π) for (a) sand, (b) kaolinite, and (c) lehm. Angle of incidence θi = 44.5°.

Fig. 13
Fig. 13

Calculated angular variation of the directional emissivity of sand at λ = 10.6 μm for H and V polarizations. Comparison with experimental data for nonplarized directional emissivity.

Fig. 14
Fig. 14

Calculated angular variation of the directional emissivity of kaolinite at λ = 10.6 μm for H and V polarizations. Comparison with experimental data for nonpolarized directional emissivity.

Fig. 15
Fig. 15

Calculated angular variation of the directional emissivity of sand for H and V polarizations in the reststrahlen band (λ = 9.25 μm).

Fig. 16
Fig. 16

Comparison between the calculated spectral nadir emissivity signature of sand and the experimentally measured signature in the 8–14-μm band.

Tables (2)

Tables Icon

Table I Index of Refraction n of SIO2 Sand (250 ≤ ϕ ≤ 500 μm) from the Ratio of the Polarized Scattering Coefficients σHH(θi, θi, π)/σVV(θi, θi, π)

Tables Icon

Table II Index of Refraction n of Various Samples from the Ratio of the Polarized Scattering Coefficients σHH(θi, θi, π)/σVV(θi, θi, π), θi = 44.5°

Equations (30)

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L ° ( T B i ) λ i = ɛ L λ i ° λ i ( T s i ) + ( 1 ɛ λ i ) Ra λ i ,
ɛ ( θ ) = 1 ρ h ( θ ) ,
σ ° ( θ i , θ , ϕ ) = 4 π ρ b ( θ i , θ , ϕ ) cos ( θ ) cos ( θ i ) ,
σ ° p p ( θ , θ ) = | R p ( 0 ) | 2 cos 4 ( θ ) F slope ( θ ) , σ ° p q ( θ , θ ) = 0 .
cos 4 ( θ ) σ ° p p ( θ ) σ ° p p ( 0 ) = F slope ( θ ) F slope ( 0 ) .
σ ° p q ( θ i , θ , ϕ ) = 1 cos 4 ( γ ) ( C p q R + C p q R ) 2 ( a 2 + b 2 ) 2 F slope ( θ i , θ , ϕ ) .
R = R V = n cos ( θ i ) cos ( θ t ) n cos ( θ i ) + cos ( θ t ) ,
R = R H = cos ( θ i ) n cos ( θ t ) cos ( θ i ) + n cos ( θ t ) ,
a = sin ( θ i ) cos ( θ ) cos ( ϕ ) + cos ( θ i ) sin ( θ ) , b = sin ( θ i ) sin ( ϕ ) , a = sin ( θ ) sin ( ϕ ) , b = sin ( θ ) cos ( θ i ) cos ( ϕ ) sin ( θ i ) cos ( ϕ ) ,
C HH = a b , C VV = a b , C VH = a a , C HV = b b , C HH = a b , C VV = a b , C VH = a b , C HV = a a .
1 cos 4 ( γ ) = 4 [ 1 + sin ( θ i ) sin ( θ ) cos ( ϕ ) + cos ( θ i ) cos ( θ ) ] 2 [ cos ( θ i ) + cos ( θ ) ] 2 .
cos ( θ 1 ) = 1 / 2 [ 1 + sin ( θ i ) sin ( θ ) cos ( ϕ ) + cos ( θ i ) cos ( θ ) ] 1 / 2 .
σ ° HV = σ ° VH = 0 ( no depolarization ) ,
σ ° HH | R H ( θ 1 ) | 2 , σ ° VV | R V ( θ 1 ) | 2 .
σ ° HH | R H ( θ i ) | 2 , σ ° VV | R V ( θ i ) | 2 .
S ( α i , α ) [ cos ( α ) cos ( α i ) R V + sin ( α ) sin ( α i ) R H ] 2 .
A H / A V = P tan ( α i ) , P = R H / R V ,
S ( α i , α ) A V 2 [ cos ( α ) P tan ( α i ) sin ( α ) ] 2 .
θ t = tan 1 ( P 1 ) ( P + 1 ) 1 tan ( θ i ) , n = sin ( θ i ) / sin ( θ t ) = 2.29 ± 0.15 .
σ ° HH ( θ i , θ , π ) / σ ° VV ( θ i , θ , π ) = [ R H ( θ 1 ) / R V ( θ 1 ) ] 2 .
σ ° ( θ i , θ , ϕ ) = σ ° ( θ , θ i , ϕ ) .
cos ( ξ ) = cos ( θ i ) cos ( θ ) + cos ( ϕ ) sin ( θ ) sin ( θ i ) ,
U ( ξ ) = 1 + A exp [ ( 1 / p ) tan ( ξ / 2 ) ] ,
σ ° p q ( θ i , θ , ϕ ) = 1 cos 4 ( γ ) [ C p q R + C p q R ] 2 [ a 2 + b 2 ] 2 cos m ( γ ) cos ( θ i ) U ( ξ ) .
ρ p ( θ i ) = 1 4 π cos ( θ i ) [ σ ° p p ( θ i , θ , ϕ ) + σ q p ( θ i , θ , ϕ ) ] d Ω ,
ρ p ( θ s ) = 1 4 π cos ( θ s ) [ σ ° p p ( θ i , θ s , ϕ ) + σ ° q p ( θ i , θ s , ϕ ) ] d Ω ,
ɛ p ( θ ) = 1 ρ p ( θ ) ,
ɛ ( θ ) = 1 / 2 [ ɛ H ( θ ) + ɛ V ( θ ) ] .
ɛ exp ( 0 ) = 0.955 ± 0.005 .
lehm ɛ model = 0.9516 , ɛ exp = 0.966 ± 0.005 , kaolinite ɛ model = 0.9747 , ɛ exp = 0.985 ± 0.005.

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