Abstract

A simple, approximate analytical formula is proposed for the reflection function of a semi-infinite, homogeneous particulate layer. It is assumed that the zenith angle of the viewing direction is equal to zero (thus corresponding to the case of nadir observations), whereas the light incidence direction is arbitrary. The formula yields accurate results for incidence–zenith angles less than approximately 85° and can be useful in analyzing satellite nadir observations of optically thick clouds.

© 2002 Optical Society of America

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References

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  1. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).
  2. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).
  3. I. N. Minin, Radiative Transfer Theory in Planetary Atmospheres (Nauka, Moscow, 1988).
  4. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991).
  5. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions (Praxis, Chichester, UK, 2001).
  6. S. Chandrasekhar, Radiative Transfer (Oxford Press, Oxford, UK, 1950).
  7. M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
    [CrossRef]
  8. M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
    [CrossRef]

1999 (1)

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

1987 (1)

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford Press, Oxford, UK, 1950).

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991).

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991).

King, M. D.

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

Kokhanovsky, A.

A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions (Praxis, Chichester, UK, 2001).

Minin, I. N.

I. N. Minin, Radiative Transfer Theory in Planetary Atmospheres (Nauka, Moscow, 1988).

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).

Van de Hulst, H. C.

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

Zakharova, N. T.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991).

J. Atmos. Sci. (1)

M. D. King, “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63, 409–432 (1999).
[CrossRef]

Other (6)

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Nauka, Moscow, 1972).

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas and Applications (Academic, New York, 1980).

I. N. Minin, Radiative Transfer Theory in Planetary Atmospheres (Nauka, Moscow, 1988).

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, New York, 1991).

A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions (Praxis, Chichester, UK, 2001).

S. Chandrasekhar, Radiative Transfer (Oxford Press, Oxford, UK, 1950).

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Figures (2)

Fig. 1
Fig. 1

Dependence D(1ξ) as obtained from Eq. (7) (symbols) and Eq. (9) (solid curve) for A=1.468 and B=7.748.

Fig. 2
Fig. 2

(a) Dependence of the reflection function R(1, ξ) at nadir observation on the incidence angle θ0 at λ=0.65 µm and effective radii of 6 and 16 µm. (b) Dependence of the error of the approximation of Eq. (11) on the incidence angle at λ=0.65 µm and effective radii of 6 and 16 µm.

Equations (20)

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R(θ0, θ, ϕ)=I(θ0, θ, ϕ)I*(θ0),
I*(θ0)=F cos θ0
R(η, ξ, ϕ)=R0(η, ξ, ϕ)-yK0(η)K0(ξ)
R0(η, ξ, ϕ)=Rss(η, ξ, ϕ)+Rms(η, ξ, ϕ).
Rss(η, ξ, ϕ)=p(ϑ)4(η+ξ),
ϑ=arccos[-ηξ+(1-η2)1/2(1-ξ2)1/2 cos ϕ]
Rms(η, ξ)=a+bηξ+c(η+ξ)4(η+ξ).
R0(η, ξ, ϕ)=p(ϑ)4(η+ξ)+a+bηξ+c(η+ξ)4(η+ξ).
D(η, ξ)=4(η+ξ)R0(η, ξ, ϕ)-p(ϑ).
D(η, ξ)=a+bηξ+c(η+ξ).
D(1, ξ)=A+Bξ,
D(1, ξ)=a+c+(c+b)ξ.
R0(θ0)=α+β cos θ01+cos θ0+p(π-θ0)4(1+cos θ0),
α=a/40.37,β=b/41.94.
R0(θ0)=0.37+1.94 cos θ01+cos θ0,
R(θ, θ0, ϕ, τ)=R0(θ, θ0, ϕ)-tK0(cos θ)K0(cos θ0),
t=10.75τ(1-g)+α,
r=2π02πdϕ0πdθ cos θ0πdθ0 cos θ0R(θ0, θ, ϕ, τ)
r=1-R0(θ, θ0, ϕ)-R(θ, θ0, ϕ, τ)K0(cos θ)K0(cos θ0),
τ=3W2ρreff1+1.1xeff2/3,g=0.88-12xeff2/3,

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