Abstract

A fast analytical model of the transfer of solar radiation in plane-parallel clouds is developed by extending the Eddington approximation to handle an azimuthally dependent radiance field. When compared with the more precise doubling method, the results from this approximate model show a typical accuracy of better than 3% for fluxes and 10% for intensities over a wide range of input parameters. This accuracy may deteriorate somewhat for small cloud thicknesses, large solar zenith angles, viewing angles close to the horizon, or viewing angles close to the solar azimuthal direction. The computational speed of the analytical model is, however, a few hundred times faster than that of the more precise models, which makes it well suited for applications involving iteration over spectral intervals, time intervals, or cloud populations.

© 1980 Optical Society of America

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References

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  1. E. P. Shettle, J. A. Weinman, J. Atmos. Sci. 27, 1048 (1970).
    [CrossRef]
  2. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
    [CrossRef]
  3. R. Davies, J. Atmos. Sci. 35, 1712 (1978).
    [CrossRef]
  4. J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
    [CrossRef]

1978 (1)

R. Davies, J. Atmos. Sci. 35, 1712 (1978).
[CrossRef]

1976 (1)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

1971 (1)

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[CrossRef]

1970 (1)

E. P. Shettle, J. A. Weinman, J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

Davies, R.

R. Davies, J. Atmos. Sci. 35, 1712 (1978).
[CrossRef]

Hansen, J. E.

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[CrossRef]

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Shettle, E. P.

E. P. Shettle, J. A. Weinman, J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

E. P. Shettle, J. A. Weinman, J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

J. Atmos. Sci. (4)

E. P. Shettle, J. A. Weinman, J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

R. Davies, J. Atmos. Sci. 35, 1712 (1978).
[CrossRef]

J. E. Hansen, J. Atmos. Sci. 28, 120 (1971).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Cloud albedo vs a for different τ. μ0 = 1. Dashed lines represent the approximate solution. Solid lines represent the doubling solution.

Fig. 2
Fig. 2

Cloud albedo vs μ0 for different a. τ = 30.

Fig. 3
Fig. 3

Cloud albedo vs τ for different μ0. a = 1.

Fig. 4
Fig. 4

Relative azimuthally averaged radiance vs μ. F0 = 1, a = 1, μ0 = 0.5, τ = 300. Line a, doubling. Line b, I3. Line c, I2. Line d, I1.

Fig. 5
Fig. 5

Relative azimuthally averaged radiance vs μ for different μ0. F0 = 1, a = 1, τ = 30.

Fig. 6
Fig. 6

Relative azimuthally averaged radiance vs μ for different τ. F0 = 1, a = 1, μ0 = 0.5.

Fig. 7
Fig. 7

Relative azimuthally averaged radiance vs μ for different a. F0 = 1, τ = 30, μ0 = 0.5.

Fig. 8
Fig. 8

Relative radiance vs μ for different ϕ. F0 = 1, μ0 = 0.5, τ = 30, a = 1.

Fig. 9
Fig. 9

Relative radiance vs ϕ for different τ. F0 = 1, μ0 = 0.5, μ = −0.5, a = 1.

Fig. 10
Fig. 10

Relative radiance vs ϕ for different a. F0 = 1, μ0 = 0.5, τ = 30, μ = −0.5.

Tables (2)

Tables Icon

Table I Parameter Ranges for the Specified Model Accuracy

Tables Icon

Table II Summary of Comparative Model Timing

Equations (14)

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μ d I d z = - k I + k J 0 + k J ,
I 1 ( z , θ , ϕ ) = I 0 + ν cos ϕ I x + ν sin ϕ I y + μ I z ,
I x = 3 a g 4 π ( 1 - a g ) F 0 sin θ 0 cos ϕ 0 exp ( - k z / μ 0 ) ,
I y = 3 a g 4 π ( 1 - a g ) F 0 sin θ 0 sin ϕ 0 exp ( - k z / μ 0 ) ,
F 1 ± = 0 ± 1 0 2 π I 1 μ d μ d ϕ ,
J 0 + J = a I 0 - μ a g k ( 1 - a g ) d I 0 d z + a 4 π × ( 1 + 3 g cos ψ 1 - a g ) F 0 exp ( - k z / μ 0 ) ,
I 2 = { I ( Z 0 ) exp [ k ( Z 0 - z ) / μ ] - k μ z Z 0 ( J 0 + J ) exp [ + k ( z - z ) / μ ] d z μ < 0 , J 0 + J , μ = 0 , I ( 0 ) exp ( - k z / μ ) + k μ 0 z ( J 0 + J ) exp [ - k ( z - z ) / μ ] d z , μ > 0 ,
I 3 ( μ ) = { I 2 ( μ ) F 1 - / F 2 - μ < 0 , I 2 ( μ ) F 1 + / F 2 + μ > 0 ,
I ss = { α [ exp ( - k Z 0 / μ 0 ) × exp [ k ( Z 0 - z ) / μ ] - exp ( - k z / μ 0 ) ] μ < 0 , α [ exp ( - k z / μ ) - exp ( - k z / μ 0 ) ] μ > 0 ,
α = a 4 π F 0 p ( μ , ϕ ; μ 0 , ϕ 0 ) × μ 0 μ - μ 0 .
I ss = { β [ exp ( - k Z 0 / μ 0 ) × exp [ k ( Z 0 - z ) / μ ] - exp ( - k z / μ 0 ) ] μ < 0 , β [ exp ( - k z / μ ) - exp ( - k z / μ 0 ) ] μ > 0 ,
β = a 4 π F 0 ( 1 - f ) { 1 + 3 g [ μ μ 0 + ν ν 0 cos ( ϕ - ϕ 0 ) ] } × μ 0 μ - μ 0 .
I 3 = ( I 2 - I ss ) × ( F 1 - F ss ) ( F 2 - F ss ) + I ss .
albedo = F 3 - ( z = 0 ) / μ 0 F 0 .

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