Abstract

A critical analysis is given of the applicability of six-beam models to radiative transfer in particulate materials. The method of introducing transverse scattering in these models is shown to cause fundamental difficulties in the case of physically plausible phase functions; in particular, the effective absorptivity is abnormally large and thus results in incorrect reflectances and transmittances. Six-beam calculations for several media are compared with accurate solutions, with Chu-Churchill two-beam results, and with a simple modification to the Eddington approximation, the last being generally superior over a wide range of conditions.

© 1976 Optical Society of America

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References

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  1. C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
    [CrossRef]
  2. A. G. Emslie, J. R. Aronson, Appl. Opt. 12, 2563 (1973).
    [CrossRef] [PubMed]
  3. J. R. Aronson, A. G. Emslie, Appl. Opt. 12, 2573 (1973).
    [CrossRef] [PubMed]
  4. J. R. Aronson, A. G. Emslie, in Infrared and Raman Spectroscopy of Lunar and Terrestrial Minerals, C. Karr, Ed. (Academic, New York, 1975), pp. 143–164.
  5. W. E. Meador, W. R. Weaver, J. Atmos. Sci., accepted for publication (1977).
  6. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950; reprinted by Dover, New York, 1960).
  7. W. M. Irvine, J. Geophys. Res. 74, 2795 (1969).
    [CrossRef]

1973

1969

W. M. Irvine, J. Geophys. Res. 74, 2795 (1969).
[CrossRef]

1955

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

Aronson, J. R.

A. G. Emslie, J. R. Aronson, Appl. Opt. 12, 2563 (1973).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, Appl. Opt. 12, 2573 (1973).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, in Infrared and Raman Spectroscopy of Lunar and Terrestrial Minerals, C. Karr, Ed. (Academic, New York, 1975), pp. 143–164.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950; reprinted by Dover, New York, 1960).

Chu, C. M.

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

Churchill, S. W.

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

Emslie, A. G.

A. G. Emslie, J. R. Aronson, Appl. Opt. 12, 2563 (1973).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, Appl. Opt. 12, 2573 (1973).
[CrossRef] [PubMed]

J. R. Aronson, A. G. Emslie, in Infrared and Raman Spectroscopy of Lunar and Terrestrial Minerals, C. Karr, Ed. (Academic, New York, 1975), pp. 143–164.

Irvine, W. M.

W. M. Irvine, J. Geophys. Res. 74, 2795 (1969).
[CrossRef]

Meador, W. E.

W. E. Meador, W. R. Weaver, J. Atmos. Sci., accepted for publication (1977).

Weaver, W. R.

W. E. Meador, W. R. Weaver, J. Atmos. Sci., accepted for publication (1977).

Appl. Opt.

J. Geophys. Res.

W. M. Irvine, J. Geophys. Res. 74, 2795 (1969).
[CrossRef]

J. Phys. Chem.

C. M. Chu, S. W. Churchill, J. Phys. Chem. 59, 855 (1955).
[CrossRef]

Other

J. R. Aronson, A. G. Emslie, in Infrared and Raman Spectroscopy of Lunar and Terrestrial Minerals, C. Karr, Ed. (Academic, New York, 1975), pp. 143–164.

W. E. Meador, W. R. Weaver, J. Atmos. Sci., accepted for publication (1977).

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, 1950; reprinted by Dover, New York, 1960).

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Figures (1)

Fig. 1
Fig. 1

Reflectance R as a function of single-scattering albedo ω0 for normal incidence on semi-infinite media with isotropic phase functions: (a) Chu-Churchill two-beam method [equivalent in this case to the Aronson-Emslie six-beam model given by Eq. (40)]; (b) hybrid delta-Eddington approximation; (c) exact results; (d) Chu-Churchill six-beam model.

Equations (63)

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μ d I ( τ , μ , ϕ ) d τ = I ( τ , μ , ϕ ) - 1 4 π - 1 1 0 2 π p ( μ , ϕ ; μ , ϕ ) I ( τ , μ , ϕ ) d ϕ d μ .
1 4 π - 1 1 0 2 π p ( μ , ϕ ; μ , ϕ ) d ϕ d μ = ω 0 ,
I ( τ , μ , ϕ ) = π F δ ( μ + μ 0 ) δ ( ϕ - ϕ 0 ) exp ( - τ / μ 0 ) + n = 1 ω 0 n I n ( τ , μ , ϕ ) .
( μ 0 + μ ) G 1 , 2 = p ( ± μ , ϕ ; μ , ϕ ) p ( μ , ϕ ; - μ 0 , ϕ 0 ) ,
I ( 0 , μ , ϕ ) = F μ 0 p ( μ , ϕ ; - μ 0 , ϕ 0 ) 4 ( μ 0 + μ ) { 1 - exp [ - τ ( μ 0 + μ ) / μ 0 μ ] } + F μ 0 2 16 π ( μ 0 + μ ) { 1 - exp [ - τ ( μ 0 + μ ) / μ 0 μ ] } - 1 1 0 2 π G 1 d ϕ d μ - F μ 0 16 π exp ( - τ / μ 0 ) 0 1 0 2 π μ G 1 μ - μ [ exp ( - τ / μ ) - exp ( - τ / μ ) ] d ϕ d μ - F μ 0 16 π exp ( - τ / μ ) × - 1 0 0 2 π μ G 1 μ - μ [ exp ( τ / μ ) - exp ( τ / μ ) ] d ϕ d μ ,
I ( τ , - μ , ϕ ) = π F δ ( μ - μ 0 ) δ ( ϕ - ϕ 0 ) exp ( - τ / μ 0 ) + F μ 0 p ( - μ , ϕ ; - μ 0 , ϕ 0 ) 4 ( μ 0 - μ ) [ exp ( - τ / μ 0 ) - exp ( - τ / μ ) ] + F μ 0 2 16 π ( μ 0 - μ ) [ exp ( - τ / μ 0 ) - exp ( - τ / μ ) ] × - 1 1 0 2 π G 2 d ϕ d μ - F μ 0 16 π exp [ - τ ( μ 0 + μ ) / μ 0 μ ] × 0 1 0 2 π μ G 2 μ + μ [ exp ( τ / μ ) - exp ( - τ / μ ) ] d ϕ d μ - F μ 0 16 π - 1 0 0 2 π μ G 2 μ + μ [ exp ( τ / μ ) - exp ( - τ / μ ) ] d ϕ d μ
μ π F μ 0 I ( 0 , μ , ϕ ) = τ p ( μ , ϕ ; - μ 0 , ϕ 0 ) 4 π μ 0 + τ 16 π 2 lim 0 - 0 2 π G 1 d ϕ d μ
μ π F μ 0 I ( τ , - μ , ϕ ) = ( 1 - τ μ 0 ) δ ( μ - μ 0 ) δ ( ϕ - ϕ 0 ) + τ p ( - μ , ϕ ; - μ 0 , ϕ 0 ) 4 π μ 0 + τ 16 π 2 lim 0 - 0 2 π G 2 d ϕ d μ
R = τ 2 μ 0 lim 0 [ 1 p ( μ , - μ 0 ) d μ + 1 2 1 - p ( μ , μ ) p ( μ , - μ 0 ) d μ d μ ]
T = 1 - τ μ 0 + τ 2 μ 0 lim 0 [ - 1 - p ( μ , - μ 0 ) d μ + 1 2 - 1 - - p ( μ , μ ) p ( μ , - μ 0 ) d μ d μ ] ,
p ( μ , μ ) = 1 2 π 0 2 π p ( μ , ϕ ; μ , ϕ ) d ϕ .
α α = 1 + 1 4 ( 1 - ω 0 ) lim 0 - p ( μ , - μ 0 ) [ 2 ( 1 - ω 0 ) + - p ( μ , μ ) d μ ] d μ .
p ( μ , ϕ ; μ j , ϕ j ) = 4 π ω 0 i = 1 6 S i j δ ( μ - μ i ) δ ( ϕ - ϕ i ) ,
i = 1 6 S i j = 1.
R = ω 0 ( S b + 4 ω 0 S t 2 ) τ ,
T = 1 - τ + ω 0 ( S f + 4 ω 0 S t 2 ) τ ,
α α = 1 + 4 ω 0 S t + 4 ω 0 2 S t ( S f + S b + 2 S t ) ,
S f , b = 1 4 π ω 0 0 1 0 2 π μ 2 p ( ± μ , ϕ ; 1 , ϕ 1 ) d ϕ d μ
R = ω 0 τ 18 ( 3 + 2 ω 0 ) ; T = 1 - τ + ω 0 τ 18 ( 3 + 2 ω 0 ) ; α α = 1 + 2 ω 0 9 ( 3 + 2 ω 0 )
R = ω 0 τ 2 ; T = 1 - τ + ω 0 τ 2 ; α = α .
S f , b = 1 4 π ω 0 0 1 0 2 π p ( ± μ , ϕ ; 1 , ϕ 1 ) d ϕ d μ .
I ( τ , μ , ϕ ) = i = 1 6 I i ( τ ) δ ( μ - μ i ) δ ( ϕ - ϕ i )
μ i d I i d τ = I i - ω 0 j = 1 6 S i j I j
d I + d τ = ( 1 - ω 0 S f ) I + - ω 0 S b I - - 4 ω 0 S t I 3 ,
d I - d τ = ω 0 S b I + - ( 1 - ω 0 S f ) I - + 4 ω 0 S t I 3 ,
0 = [ 1 - ω 0 ( 1 - 2 S t ) ] I 3 - ω 0 S t ( I + + I - ) ,
I ± ( τ ) = 0 1 0 2 π μ I ( τ , ± μ , ϕ ) d ϕ d μ = I 1 , 2 ( τ ) .
d I + d τ = ( 1 + ω 0 S f ) I + - ω 0 S b I -
d I - d τ = ω 0 S b I + - ( 1 - ω 0 S f ) I - ,
S f , b = S f , b + 4 ω 0 S t 2 1 - ω 0 ( 1 - 2 S t )
R = ω 0 S b [ 1 - exp ( - 2 k τ ) ] 1 + k - ω 0 S f - ( 1 - k - ω 0 S f ) exp ( - 2 k τ ) ,
T = 2 k exp ( - k τ ) 1 + k - ω 0 S f - ( 1 - k - ω 0 S f ) exp ( - 2 k τ ) ,
α α = 1 - ω 0 + 6 ω 0 S t 1 - ω 0 + 2 ω 0 S t ,
k = [ ( 1 - ω 0 S f ) 2 - ω 0 2 S b 2 ] 1 / 2 .
R = ω 0 τ 2 ( 3 - 2 ω 0 ) ; T = 1 - τ + ω 0 τ 2 ( 3 - 2 ω 0 ) ; α α = 3 3 - 2 ω 0 .
I ± ( τ ) = 0 1 0 2 π μ I ( τ , ± μ , ϕ ) d ϕ d μ = 3 1 / 2 [ I ( τ ) , J ( τ ) ] .
d I + d τ = 3 1 / 2 ( 1 - ω 0 S f ) I + - 3 1 / 2 ω 0 S b I -
d I - d τ = 3 1 / 2 ω 0 S b I + - 3 1 / 2 ( 1 - ω 0 S f ) I -
S f , b = S f , b + 2 S t ; S f + S b = S f + S b + 4 S t = 1.
R = 3 1 / 2 ω 0 S b [ 1 - exp ( - 2 k τ ) ] k + 3 1 / 2 ( 1 - ω 0 S f ) + [ k - 3 1 / 2 ( 1 - ω 0 S f ) ] exp ( - 2 k τ ) ,
T = 2 k exp ( - k τ ) k + 3 1 / 2 ( 1 - ω 0 S f ) + [ k - 3 1 / 2 ( 1 - ω 0 S f ) ] exp ( - 2 k τ ) ,
α = 3 1 / 2 α ,
k = { 3 [ ( 1 - ω 0 S f ) 2 - ω 0 2 S b 2 ] } 1 / 2 .
R = τ 2 0 1 p ( μ , - 1 ) d μ ; T = 1 - τ ( 1 - ω 0 ) - τ 2 0 1 p ( μ , - 1 ) d μ ; α = α .
I ( τ , μ , ϕ ) = π F δ ( μ + 1 ) δ ( ϕ - ϕ 0 ) exp ( - τ ) + i = 1 6 I i ( τ ) δ ( μ - μ i ) δ ( ϕ - ϕ i )
μ i d I i d τ = I i - ω 0 j = 1 6 S i j I j - π F ω 0 S i 0 exp ( - τ )
S 10 = S 20 = S 30 = 1 6 ω 0 0 1 p ( μ , - 1 ) d μ
R ( 6 - beam ) = ω 0 [ ( 1 - ω 0 + 2 ω 0 S t ) S b + 4 ω 0 S t 2 ] ( 1 + k - ω 0 S f ) ( 1 - ω 0 + 2 ω 0 S t ) - 4 ω 0 2 S t 2
R ( 2 - beam ) = ω 0 S b 1 + k - ω 0 S f ,
S f , b ( 6 - beam ) = 1 2 ω 0 0 1 μ 2 p ( μ , ± 1 ) d μ ,
S t ( 6 - beam ) = 1 8 ω 0 - 1 1 ( 1 - μ 2 ) p ( μ , 1 ) d μ ,
S f , b ( 2 - beam ) = 1 2 ω 0 0 1 p ( μ , ± 1 ) d μ .
I d ( τ , ± μ ) = 0 2 π I d ( τ , ± μ , ϕ ) d ϕ 1 1 - g 2 ( 1 - μ 0 ) { ( 1 - g 2 ) [ ( 1 ± 3 μ 2 ) I d + ( τ ) + ( 1 3 μ 2 ) I d - ( τ ) ] + g 2 δ ( μ - μ 0 ) I d ± ( τ ) } ,
g = 1 2 ω 0 - 1 1 μ p ( μ , 1 ) d μ .
R = ω 0 [ γ 2 + β ( k + γ 1 - γ 2 ) ] ( 1 + k ) ( k + γ 1 ) ,
k = ( γ 1 2 - γ 2 2 ) 1 / 2 ,
γ 1 = 1 4 [ 7 - 3 g 2 - ω 0 ( 4 + 3 g ) + ω 0 g 2 ( 4 β + 3 g ) ] ,
γ 2 = - 1 4 [ 1 - g 2 - ω 0 ( 4 - 3 g ) - ω 0 g 2 ( 4 β + 3 g - 4 ) ] ,
β = 1 2 ω 0 0 1 p ( μ , - 1 ) d μ .
R ( 6 - beam ) = ω 0 { 6 - 5 ω 0 + 2 [ 3 ( 1 - ω 0 ) ( 3 - 2 ω 0 ) ] 1 / 2 } - 1 ,
R ( 2 - beam ) = ω 0 [ 2 - ω 0 + 2 ( 1 - ω 0 ) 1 / 2 ] - 1 ,
R ( Eddington ) = 3 ω 0 { 9 - 6 ω 0 + 5 [ 3 ( 1 - ω 0 ) ] 1 / 2 } - 1 ,
R ( exact ) = 1 - ( 1 - ω 0 ) 1 / 2 H ( μ 0 = 1 , ω 0 ) ,

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