Abstract

Reflection from a field of randomly located, thin vertical cylinders is analyzed. The bidirectional reflectivity Rdf for the direct solar beam (first reflection) is

Rdf(s,z,θ0,θ)=rc(-zcosz+sinz)1-exp[-s(tanθ0+tanθ)]4(cotθ0+cotθ),

where z is the azimuth (in radians) measured in a horizontal plane from the antisolar direction, θ0 is the zenith angle of the solar beam, θ is the viewing zenith angle, s is the projection on a vertical plane of cylindrical sections over a unit area, and rc is the Lambert law spectral reflectivity of the cylinders. The bidirectional reflectivity thus tends to rcπ tanθ0/4 at the azimuth π and at very large viewing zenith angles, such that tanθ ≫ tanθ0 and tanθ0 ⩾ 1. A comparison with the bidirectional reflectivities measured over dense vegetation of a Florida swamp indicates that this model describes a highly significant component of this canopy reflection. The possibility of extracting information about the reflectivity of the plants material (rc) and about the canopy structure (s) from the reflectivities measured at various solar elevations is discussed and an approach for such extraction is formulated. The extraction of rc appears practical; that of s, difficult.

© 1984 Optical Society of America

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References

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  1. K. L. Coulson, Appl. Opt. 5, 905 (1966).
    [CrossRef] [PubMed]
  2. V. V. Salomonson, “Anisotropy in the Reflected Solar Radiation,” Ph.D. Dissertation, Department of Atmospheric Science, Colorado State U., Fort Collins (1968), 143 pp.
  3. V. V. Salomonson, W. R. Marlatt, Remote Sensing Environ. 2, 1 (1972).
    [CrossRef]
  4. D. S. Kimes, Appl. Opt. 22, 1364 (1983).
    [CrossRef] [PubMed]
  5. J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publishers, The Hague, 1981).
    [CrossRef]
  6. H. Moldau, “Vegetative Course of Luminance Factor of Leaves of Some Plants,” in Questions on Radiation Regime of Plant Stand, Academy of Science ESSR, Institute of Physics and Astronomy, Tartu (1965), in Russian, pp. 89–95.
  7. J. Otterman, “Albedo of Forest Modeled as a Plane with Dense Protrusions,” J. Climate Appl. Meteorol. 23, 297 (1984).
    [CrossRef]

1984 (1)

J. Otterman, “Albedo of Forest Modeled as a Plane with Dense Protrusions,” J. Climate Appl. Meteorol. 23, 297 (1984).
[CrossRef]

1983 (1)

1972 (1)

V. V. Salomonson, W. R. Marlatt, Remote Sensing Environ. 2, 1 (1972).
[CrossRef]

1966 (1)

Coulson, K. L.

Kimes, D. S.

Marlatt, W. R.

V. V. Salomonson, W. R. Marlatt, Remote Sensing Environ. 2, 1 (1972).
[CrossRef]

Moldau, H.

H. Moldau, “Vegetative Course of Luminance Factor of Leaves of Some Plants,” in Questions on Radiation Regime of Plant Stand, Academy of Science ESSR, Institute of Physics and Astronomy, Tartu (1965), in Russian, pp. 89–95.

Otterman, J.

J. Otterman, “Albedo of Forest Modeled as a Plane with Dense Protrusions,” J. Climate Appl. Meteorol. 23, 297 (1984).
[CrossRef]

Ross, J.

J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publishers, The Hague, 1981).
[CrossRef]

Salomonson, V. V.

V. V. Salomonson, W. R. Marlatt, Remote Sensing Environ. 2, 1 (1972).
[CrossRef]

V. V. Salomonson, “Anisotropy in the Reflected Solar Radiation,” Ph.D. Dissertation, Department of Atmospheric Science, Colorado State U., Fort Collins (1968), 143 pp.

Appl. Opt. (2)

J. Climate Appl. Meteorol. (1)

J. Otterman, “Albedo of Forest Modeled as a Plane with Dense Protrusions,” J. Climate Appl. Meteorol. 23, 297 (1984).
[CrossRef]

Remote Sensing Environ. (1)

V. V. Salomonson, W. R. Marlatt, Remote Sensing Environ. 2, 1 (1972).
[CrossRef]

Other (3)

J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publishers, The Hague, 1981).
[CrossRef]

H. Moldau, “Vegetative Course of Luminance Factor of Leaves of Some Plants,” in Questions on Radiation Regime of Plant Stand, Academy of Science ESSR, Institute of Physics and Astronomy, Tartu (1965), in Russian, pp. 89–95.

V. V. Salomonson, “Anisotropy in the Reflected Solar Radiation,” Ph.D. Dissertation, Department of Atmospheric Science, Colorado State U., Fort Collins (1968), 143 pp.

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Figures (7)

Fig. 1
Fig. 1

Geometry of reflection from a vertical area perpendicular to the principal plane: (a) (b) (c).

Fig. 2
Fig. 2

Bidirectional reflectivity factor Rdf/rc as a function of the azimuth z and the zenith angle θ of the viewing direction, for two values of the protrusion parameter (s = 0.2 on the left; s = 1.0 on the right), for the solar zenith angle of 21.8°.

Fig. 3
Fig. 3

Bidirectional reflectivity factor Rdf/rc as a function of the azimuth z and the zenith angle θ of the viewing direction, for two values of the protrusion parameter (s = 0.2 on the left; s = 1.0 on the right), for the solar zenith angle of 50.2°.

Fig. 4
Fig. 4

Bidirectional reflectivity factor Rdf/rc in the principal plane (z = π) vs the viewing zenith angle θ for various solar zenith angles, solid lines for s ≫ 1 and dotted lines for s = 0.4.

Fig. 5
Fig. 5

Bidirectional reflectivity factor Rdf/rc in the principal plane (z = π) vs tan θ for various solar zenith angles, solid lines for s ≫ 1 and dotted lines for s = 0.4.

Fig. 6
Fig. 6

Comparison of Salomonson’s bidirectional reflectivities measured over a densely vegetated Florida swamp2 and the model, solid lines for s ≫ 1, dotted line for s = 0.4, z = π, θ0 = 71.5°.

Fig. 7
Fig. 7

Comparison of Salomonson’s bidirectional reflectivities measured over a densely vegetated Florida swamp2 and the model, solid line for s ≫ 1, dotted line for s = 0.4, z = π, θ0 = 57.5°.

Equations (11)

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R d f ( s , z , θ 0 , θ ) = r c ( - z cos z + sin z ) 1 - exp [ - s ( tan θ 0 + tan θ ) ] 4 ( cot θ 0 + cot θ ) ,
d L w ( α , θ 0 ) = r c S λ sin θ 0 cos α d W / π ,
d L W ( ζ , θ , θ 0 ) = - r c S λ sin θ 0 cos ζ sin θ d W / π ,             π / 2 ζ 3 π / 2.
- ζ h S λ sin θ 0 cos ζ d ζ ,             π / 2 ζ 3 π / 2.
π L c ( z , θ , θ 0 ) r c S λ sin θ 0 sin θ ρ h = - π / 2 ( π / 2 ) + z cos ζ cos ( ζ - z ) d ζ = - [ cos z π / 2 ( π / 2 ) + z cos 2 ζ d ζ + sin z π / 2 ( π / 2 ) + z cos ζ sin ζ d ζ ] = - 1 2 [ cos z ( ζ + sin 2 ζ 2 ) | π / 2 ( π / 2 ) + z + sin z sin 2 ζ | π / 2 ( π / 2 ) + z ] = 1 2 [ - z cos z + sin z ] ,             0 z π .
L s u ( z , θ , θ 0 ) = r c S λ sin θ 0 tan θ 4 π ( - z cos z + sin z ) s .
L 0 d ( s , z , θ 0 , θ ) = r c S λ sin θ 0 tan θ ( - z cos z + sin z ) × 0 s exp [ - σ ( tan θ 0 + tan θ ) ] d σ / 4 π = r c S λ sin θ 0 tan θ ( - z cos z + sin z ) × 1 - exp [ - s ( tan θ 0 + tan θ ) ] 4 π ( tan θ 0 + tan θ ) ,             0 z π .
R d f ( s , z , θ 0 , θ ) = r c ( - z cos z + sin z ) 1 - exp [ - s ( tan θ 0 + tan θ ) ] 4 ( cot θ 0 + cos θ )
R d f ( s , π , θ 0 , θ ) = r c π 1 - exp [ - s ( tan θ 0 + tan θ ) ] 4 ( cot θ 0 + cot θ ) .
R d f = r c π 4 ( cot θ 0 + cot θ ) ,
S λ r c cos θ 0 4 ( cot θ 0 + cot θ ) = S λ r c 4 ( 1 sin θ 0 + cot θ cos θ 0 ) .

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