Abstract

To compare measurements performed in different geometrical conditions, one must take into account the angular anisotropy of the reflection properties of natural surfaces. As use of the exact boundary conditions in the radiative transfer codes seems prohibitive, a simple but accurate formulation of the problem has been sought. In this paper, two average angular reflectances are defined from which the reflected radiance may be deduced for any distribution of the downward radiance. Calculations made for different atmospheric models show that the solar directionality is partly preserved in the downward radiation field, so that the average reflectances can be written as a linear combination of actual reflectance and spherical albedo of the surface. Finally, the feasibility of detecting directional properties from space measurements is discussed.

© 1983 Optical Society of America

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References

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  1. K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
    [CrossRef]
  2. P. Koepke, K. T. Kriebel, Appl. Opt. 17, 260 (1978).
    [CrossRef] [PubMed]
  3. B. M. Fitch, J. Atmos. Sci. 38, 2717 (1981).
    [CrossRef]
  4. J. E. Chance, E. W. LeMaster, Appl. Opt. 16, 407 (1977).
    [CrossRef] [PubMed]
  5. G. H. Suits, Remote Sensing Environ. 2, 117 (1972).
    [CrossRef]
  6. K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
    [CrossRef]
  7. K. T. Kriebel, Appl. Opt. 17, 253 (1978).
    [CrossRef] [PubMed]
  8. D. Tanre, M. Herman, P. Y. Deschamps, A. de Leffe, Appl. Opt. 18, 3587 (1979).
    [CrossRef] [PubMed]
  9. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).
  10. W. M. Irvine, Icarus 25, 175 (1975).
    [CrossRef]
  11. K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
    [CrossRef]

1981

B. M. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

1979

1978

1977

1975

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

1972

G. H. Suits, Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

1966

K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
[CrossRef]

1965

K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
[CrossRef]

1964

K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
[CrossRef]

Bouricius, G. M.

K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
[CrossRef]

K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
[CrossRef]

Chance, J. E.

Coulson, K. L.

K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
[CrossRef]

K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
[CrossRef]

de Leffe, A.

Deschamps, P. Y.

Fenn, R. W.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

Fitch, B. M.

B. M. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

Garing, J. S.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

Gray, E. L.

K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
[CrossRef]

K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
[CrossRef]

Herman, M.

Irvine, W. M.

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

Koepke, P.

Kondratyev, K. Y.

K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
[CrossRef]

Kriebel, K. T.

LeMaster, E. W.

McClatchey, R. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

Mironova, Z. F.

K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
[CrossRef]

Otto, A. N.

K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
[CrossRef]

Selby, J. E. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

Suits, G. H.

G. H. Suits, Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

Tanre, D.

Voltz, F. E.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

Appl. Opt.

Icarus

K. L. Coulson, E. L. Gray, G. M. Bouricius, Icarus 5, 139 (1966).
[CrossRef]

W. M. Irvine, Icarus 25, 175 (1975).
[CrossRef]

J. Atmos. Sci.

B. M. Fitch, J. Atmos. Sci. 38, 2717 (1981).
[CrossRef]

J. Geophys. Res.

K. L. Coulson, G. M. Bouricius, E. L. Gray, J. Geophys. Res. 70, 4601 (1965).
[CrossRef]

Pure Appl. Geophys.

K. Y. Kondratyev, Z. F. Mironova, A. N. Otto, Pure Appl. Geophys. 59, 207 (1964).
[CrossRef]

Remote Sensing Environ.

G. H. Suits, Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

Other

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL 71-0279, Environmental Research Paper 354 (1971).

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Figures (14)

Fig. 1
Fig. 1

Reflectance of pasture (in polar coordinates) in the principal solar incidence plane for three zenith solar angles at λ = 520 nm. The curves labeled Exp correspond to Kriebel’s measurements.7 The curves labeled Th correspond to the approximate Eq. (2).

Fig. 2
Fig. 2

Same as Fig. 1 but for forest, λ = 520 nm.

Fig. 3
Fig. 3

Same as Fig. 1 but for savanna, λ = 520 nm.

Fig. 4
Fig. 4

Same as Fig. 1 but for pasture at λ = 860 nm.

Fig. 5
Fig. 5

Successive orders of radiation interactions in the coupled ground–atmosphere system.

Fig. 6
Fig. 6

Downward radiances L a (polar coordinates) for two zenith solar angles and two atmospheric models at λ = 450 nm. The radiances are computed for a unit solar irradiance Eo. Radiance L a has not been drawn in the nearly forward scattering directions; the values of L a ( θ s , θ s , O ) and the solar incident direction are given.

Fig. 7
Fig. 7

Same as Fig. 6 but at λ = 850 nm.

Fig. 8
Fig. 8

Average angular reflectance ρ ¯ vs the actual reflectance ρ for pasture at λ = 450 nm. The different points are obtained for different geometrical conditions.

Fig. 9
Fig. 9

Same as Fig. 8 but for forest.

Fig. 10
Fig. 10

Same as Fig. 8 but for savanna.

Fig. 11
Fig. 11

Same as Fig. 8 but for savanna at λ = 850 nm.

Fig. 12
Fig. 12

Average angular reflectance ρ ¯ ¯ vs the actual reflectance ρ for savanna at λ = 450 nm. The different points are obtained for different geometrical conditions.

Fig. 13
Fig. 13

Spectral variations of ρ a ¯ ( λ ) γ ( λ ) 1 T s ( θ s ) T s ( θ υ )(see text) for the three atmospheric models and two zenith solar angles. These values must be compared to the typical relative contrast Δρr (~0.5–1.5) in Figs. 14.

Fig. 14
Fig. 14

Spectral variations of Δ ρ a ( λ 1 ) Δ ρ a ( λ 2 ) γ ( λ 1 ) T s ( λ 1 ) γ ( λ 2 ) T s ( λ 2 )as a function of λ2 (see text) for λ1 = 650 nm for the worst case of the extreme atmospheric model (V5) and two zenith solar angles. The curves labeled without correction correspond to the values plotted in Fig. 13. These values must be compared to the typical relative contrast Δρr (~0.5–1.5) in Figs. 14.

Tables (3)

Tables Icon

Table I Coefficients a and a′ of the Linear Regression of ρ ¯ and ρ ¯ ¯

Tables Icon

Table II Coefficients A and B of the Development of the Diffused Radiation Field

Tables Icon

Table III Comparison Between Coefficient b of the Linear Regression of ρ ¯ and ρ ¯ ¯ and γB Coefficient, Product of the Ground Spherical Albedo γ by the Weight B of the Isotropic Component in the Downward Radiation Field

Equations (42)

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ρ ( θ s , θ υ , φ υ ) = L ( θ υ , φ υ ) cos θ s E o ,
ρ ( θ s , θ υ , φ υ ) = s = 0 ( 2 δ o s ) cos s φ υ l m ρ lm s P l s ( θ s ) P m s ( θ υ ) ,
ρ ¯ ( θ s , θ υ , φ υ ) = 0 2 π 0 1 μ L a ( θ s , θ , φ ) ρ ( θ , θ υ , φ φ υ ) d Ω E ( θ s ) ,
E ( θ s ) = 0 2 π 0 1 μ L a ( θ s , θ , φ ) d Ω .
ρ ¯ ( θ s , θ υ , φ υ ) = 0 2 π 0 1 μ ρ ( θ s , θ , φ ) L a ( θ , θ υ , φ φ υ ) d Ω E ( θ υ ) .
ρ * ( τ , θ s , θ υ , φ υ ) = ρ a ( τ , θ s , θ υ , φ υ ) + ρ ( θ s , θ υ , φ υ ) × exp ( τ / μ s ) exp ( τ / μ υ ) + ρ ¯ ( τ , θ s , θ υ , φ υ ) t d ( θ s ) exp ( τ / μ υ ) + ρ ¯ ( τ , θ υ , θ s , φ υ ) exp ( τ / μ s ) t d ( θ υ ) + ρ ¯ ¯ ( τ , θ s , θ υ , φ υ ) t d ( θ s ) t d ( θ υ ) ,
ρ ¯ ( θ s , θ υ , φ υ ) a ρ ( θ s , θ υ , φ υ ) + b ,
ρ ¯ ¯ ( θ s , θ υ , φ υ ) a ρ ( θ s , θ υ , φ υ ) + b ,
L a ( θ s , θ υ , φ υ ) = cos θ s E ( θ s ) [ A δ ( θ s , θ υ ) + B ] ,
a A ,
a A 2
γ = 0 2 π d φ υ 0 2 π d φ s 0 1 μ υ d μ υ 0 1 μ s d μ s ρ ( θ s , θ υ , φ υ φ s ) 0 2 π d φ υ 0 2 π d φ s 0 1 μ υ d μ υ 0 1 μ s d μ s .
ρ ¯ ( θ s , θ υ , φ υ ) a ρ ( θ s , θ υ , φ υ ) + b ,
ρ ¯ ( θ s , θ υ , φ υ ) = ρ ¯ ( θ υ , θ s , φ υ ) a ρ ( θ υ , θ s , φ υ ) + b a ρ ( θ s , θ υ , φ υ ) + b ,
ρ ¯ ¯ ( θ s , θ υ , φ υ ) a 2 ρ ( θ s , θ υ , φ υ ) + γ B .
ρ * ( τ , θ s , θ υ , φ υ ) = ρ a ( τ , θ s , θ υ , φ υ ) + ρ ( θ s , θ υ , φ υ ) × [ exp ( τ / μ s ) + a t d ( θ s ) ] × [ exp ( τ / μ υ ) + a t d ( θ υ ) ] + b [ T ( θ s ) T ( θ υ ) exp ( τ / μ s ) exp ( τ / μ υ ) ] ,
Δ ρ * = Δ ρ a + Δ ρ [ exp ( τ / μ s ) + a t d ( θ s ) ] [ exp ( τ / μ υ ) + a t d ( θ υ ) ] ,
Δ ρ * = ρ * ( τ , θ s , θ υ , φ υ = 0 ) ρ * ( τ , θ s , θ υ , φ υ = 180 ) , Δ ρ a = ρ a ( τ , θ s , θ υ , φ υ = 0 ) ρ a ( τ , θ s , θ υ , φ υ = 180 ) , Δ ρ = ρ ( θ s , θ υ , φ υ = 0 ) ρ ( θ s , θ υ , φ υ = 180 ) .
Δ ρ a Δ ρ T s ( θ s ) T s ( θ υ ) ,
T s ( θ ) = exp ( τ / μ ) + a t d ( θ ) .
ρ ( λ s , θ s , θ υ , φ υ ) = γ ( λ ) ρ r ( θ s , θ υ , φ υ ) ,
[ ρ * ( λ 1 , θ 1 ) ρ * ( λ 2 , θ 1 ) ] [ ρ * ( λ 1 , θ 2 ) ρ * ( λ 2 , θ 2 ) ] = Δ ρ a ( λ 1 ) Δ ρ a ( λ 2 ) + Δ ρ ( λ 1 ) T s ( λ 1 ) Δ ρ ( λ 2 ) T s ( λ 2 ) .
Δ ρ Δ ρ a ( λ 1 ) Δ ρ a ( λ 2 ) γ ( λ 1 ) T s ( λ 1 ) γ ( λ 2 ) T s ( λ 2 ) .
ρ a ¯ ( λ ) γ ( λ ) 1 T s ( θ s ) T s ( θ υ )
Δ ρ a ( λ 1 ) Δ ρ a ( λ 2 ) γ ( λ 1 ) T s ( λ 1 ) γ ( λ 2 ) T s ( λ 2 )
L ( Ω , Ω ) = P ( Ω , Ω ) μ L ( Ω ) d Ω ,
d L 1 ( Ω s , Ω υ ) = ρ ( Ω , Ω υ ) μ P ( Ω s , Ω ) μ s E s d Ω exp ( τ / μ υ ) ;
L 1 ( Ω s , Ω υ ) = μ s E s exp ( τ / μ υ ) Ω μ ρ ( Ω , Ω υ ) P ( Ω s , Ω ) d Ω .
L 1 ( Ω s , Ω υ ) = μ s E s t d ( θ s ) ρ ¯ ( Ω s , Ω υ ) exp ( τ / μ υ ) ,
t d ( θ s ) = Ω μ P ( Ω s , Ω ) d Ω ,
ρ ¯ ( Ω s , Ω υ ) = Ω μ ρ ( Ω , Ω υ ) P ( Ω s , Ω ) d Ω Ω μ P ( Ω s , Ω ) d Ω .
μ s E s exp ( τ / μ s ) ρ ( Ω s , Ω ) μ d Ω ,
d L 2 ( Ω s , Ω υ ) = μ s E s exp ( τ / μ s ) ρ ( Ω s , Ω ) P ( Ω , Ω υ ) μ d Ω ,
L 2 ( Ω s , Ω υ ) = μ s E s exp ( τ / μ s ) Ω ρ ( Ω s , Ω ) P ( Ω , Ω υ ) μ d Ω .
L 2 ( Ω s , Ω υ ) = μ s E s exp ( τ / μ s ) ρ ¯ ( Ω s , Ω υ ) t d ( θ υ ) ,
t d ( θ υ ) = Ω P ( Ω , Ω υ ) μ d Ω .
ρ ¯ ( Ω s , Ω υ ) = Ω ρ ( Ω s , Ω ) P ( Ω , Ω υ ) μ d Ω Ω P ( Ω , Ω υ ) μ d Ω .
P ( Ω , Ω ) P ( Ω , Ω ) , ρ ( Ω , Ω ) ρ ( Ω , Ω ) ,
ρ ¯ ( Ω s , Ω υ ) ρ ¯ ( Ω υ , Ω s ) .
L ( Ω s , Ω υ ) = μ s E s μ s E s Ω Ω P ( Ω o , Ω ) ρ ( Ω , Ω ) × P ( Ω , Ω υ ) μ d Ω μ d Ω .
L ( Ω s , Ω υ ) = μ s E s t d ( θ s ) ρ ¯ ¯ ( Ω s , Ω υ ) μ s E s t d ( θ υ ) ,
ρ ¯ ¯ ( Ω s , Ω υ ) = Ω Ω P ( Ω o , Ω ) ρ ( Ω , Ω ) P ( Ω , Ω υ ) μ d Ω μ d Ω Ω μ P ( Ω s , Ω ) d Ω Ω μ P ( Ω , Ω υ ) d Ω .

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