Abstract

A new diffusion model is developed for radiative transfer in particulate media. It includes the effects of higher Legendre moments while avoiding the mathematical complexities of solving multiple coupled differential moment equations and satisfying higher-order boundary conditions. The method accurately extrapolates the conventional Eddington approximation to problems involving large absorption. Although the simplifying assumptions limit the model to nonbeam incidence and spatial homogeneity of the scattering material, they are nonrestrictive with regard to the size and shape of the medium, the character of background reflections, and the type of phase function.

© 1979 Optical Society of America

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References

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  1. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
    [CrossRef]
  2. P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).
  3. P. S. Mudgett, L. W. Richards, Appl. Opt. 10, 1485 (1971).
    [CrossRef] [PubMed]
  4. P. S. Mudgett, L. W. Richards, J. Colloid Interface Sci. 39, 551 (1972).
    [CrossRef]
  5. P. S. Mudgett, L. W. Richards, J. Paint Technol. 45, 44 (1973).
  6. J. F. Potter, J. Atmos. Sci. 27, 943 (1970).
    [CrossRef]
  7. B. J. Brinkworth, Appl. Opt. 11, 1434 (1972).
    [CrossRef] [PubMed]
  8. L. F. Gate, J. Phys. D. 4, 1049 (1971).
    [CrossRef]
  9. K. Klier, J. Opt. Soc. Am. 62, 882 (1972).
    [CrossRef]

1976

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

1973

P. S. Mudgett, L. W. Richards, J. Paint Technol. 45, 44 (1973).

1972

1971

1970

J. F. Potter, J. Atmos. Sci. 27, 943 (1970).
[CrossRef]

1931

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Brinkworth, B. J.

Gate, L. F.

L. F. Gate, J. Phys. D. 4, 1049 (1971).
[CrossRef]

Joseph, J. H.

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

Klier, K.

Kubelka, P.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Mudgett, P. S.

P. S. Mudgett, L. W. Richards, J. Paint Technol. 45, 44 (1973).

P. S. Mudgett, L. W. Richards, J. Colloid Interface Sci. 39, 551 (1972).
[CrossRef]

P. S. Mudgett, L. W. Richards, Appl. Opt. 10, 1485 (1971).
[CrossRef] [PubMed]

Munk, F.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

Potter, J. F.

J. F. Potter, J. Atmos. Sci. 27, 943 (1970).
[CrossRef]

Richards, L. W.

P. S. Mudgett, L. W. Richards, J. Paint Technol. 45, 44 (1973).

P. S. Mudgett, L. W. Richards, J. Colloid Interface Sci. 39, 551 (1972).
[CrossRef]

P. S. Mudgett, L. W. Richards, Appl. Opt. 10, 1485 (1971).
[CrossRef] [PubMed]

Weinman, J. A.

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

Wiscombe, W. J.

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

Appl. Opt.

J. Atmos. Sci.

J. F. Potter, J. Atmos. Sci. 27, 943 (1970).
[CrossRef]

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

J. Colloid Interface Sci.

P. S. Mudgett, L. W. Richards, J. Colloid Interface Sci. 39, 551 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Paint Technol.

P. S. Mudgett, L. W. Richards, J. Paint Technol. 45, 44 (1973).

J. Phys. D.

L. F. Gate, J. Phys. D. 4, 1049 (1971).
[CrossRef]

Z. Tech. Phys.

P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).

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Figures (2)

Fig. 1
Fig. 1

Kubelka-Munk absorption parameter η as a function of the ratio of the absorption and extinction coefficients.

Fig. 2
Fig. 2

Kubelka-Munk scattering parameter σ as a function of the ratio of the absorption and extinction coefficients.

Tables (1)

Tables Icon

Table I Comparisons of Kubelka-Munk and Diffusion Coefficients for Isotropic Scattering as Determined by Simple Diffusion Models ( ≪ 1)

Equations (38)

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p ( μ , μ ) = 1 2 π 0 2 π p ( μ , ϕ ; μ , ϕ ) d ϕ = ω 0 i = 0 N ( 2 i + 1 ) ( g i - f ) P i ( μ ) P i ( μ ) + 2 ω 0 f δ ( μ - μ ) .
g i = 1 2 ω 0 - 1 1 P i ( μ ) p ( μ , 1 ) d μ .
- 1 1 p ( μ , μ ) d μ = 2 ω 0 ,
I ( z , μ ) = 0 2 π I ( z , μ , ϕ ) d ϕ = F exp ( - γ z / μ 0 ) δ ( μ - μ 0 ) + I d ( z , μ ) ,
I d ( z , μ ) = 1 2 i = 0 N ( 2 i + 1 ) G i ( z ) P i ( μ ) .
γ = k + s ( 1 - f ) .
G i ( z ) = - 1 1 P i ( μ ) I d ( z , μ ) d μ .
μ d I d ( z , μ ) d z = - γ I d ( z , μ ) + s 2 i = 0 3 ( 2 i + 1 ) ( g i - f ) P i ( μ ) [ G i ( z ) + F P i ( μ 0 ) exp ( - γ z / μ 0 ) ] + Q ( z ) 2 ,
d G 1 d z = - k G 0 + F s ( 1 - f ) exp ( - γ z / μ 0 ) + Q ,
d ( G 0 + 2 G 2 ) d z = - 3 [ k + s ( 1 - g ) ] G 1 + 3 F s μ 0 ( g - f ) exp ( - γ z / μ 0 ) ,
d ( 2 G 1 + 3 G 3 ) d z = - 5 [ k + s ( 1 - g 2 ) ] G 2 + 5 2 F s ( 3 μ 0 2 - 1 ) ( g 2 - f ) exp ( - γ z / μ 0 ) ,
3 d G 2 d z = - 7 [ k + s ( 1 - g 3 ) ] G 3 + 7 2 F s μ 0 ( 5 μ 0 2 - 3 ) ( g 3 - f ) exp ( - γ z / μ 0 ) .
F f , r ( z ) = 0 1 μ I d ( z ± μ ) d μ
G 2 = a 2 ( z ) ( F f + F r ) ;             G 3 = a 3 ( z ) ( F f - F r ) .
d ( F f - F r ) d z - F s ( 1 - f ) exp ( - γ z / μ 0 ) - Q = - k 4 ( 8 - 5 a 2 ) ( F f + F r ) = - K ( F f + F r ) ,
d ( F f + F r ) d z - F s μ 0 16 [ 24 g + 21 g 3 - 45 f - 35 ( g 3 - f ) μ 0 2 ] exp ( - γ z / μ 0 ) = - 1 8 { 12 [ k + s ( 1 - g ) ] - 7 a 3 [ k + s ( 1 - g 3 ) ] } ( F f - F r ) = - ( K + 2 S ) ( F f - F r ) ,
{ 20 a 2 [ k + s ( 1 - g 2 ) ] - k ( 8 - 5 a 2 ) ( 2 + 3 a 3 ) } ( F f + F r ) = - 12 ( F f - F r ) d a 3 d z - 2 F s [ 2 ( 1 - f ) ( 2 + 3 a 3 ) - 5 ( 3 μ 0 2 - 1 ) ( g 2 - f ) ] exp ( - γ z / μ 0 ) - 4 ( 2 + 3 a 3 ) Q ,
2 { 36 a 2 [ k + s ( 1 - g ) ] - 7 a 3 ( 8 + 3 a 2 ) [ k + s ( 1 - g 3 ) ] } ( F f - F r ) = 48 ( F f + F r ) d a 2 d z + F s μ 0 [ 72 a 2 ( g - f ) - 7 ( 8 + 3 a 2 ) ( g 3 - f ) ( 5 μ 0 2 - 3 ) ] exp ( - γ z / μ 0 ) .
d ( F f - F r ) d z - F s ( 1 - f ) exp ( - γ z / μ 0 ) - Q = - 2 k ( F f + F r ) = - K ( F f + F r ) ,
d ( F f + F r ) d z - 3 F s μ 0 2 ( g - f ) exp ( - γ z / μ 0 ) = - 3 2 [ k + s ( 1 - g ) ] ( F f - F r ) = - ( K + 2 S ) ( F f - F r ) .
20 a 2 [ k + s ( 1 - g 2 ) ] = k ( 8 - 5 a 2 ) ( 2 + 3 a 3 ) ,
36 a 2 [ k + s ( 1 - g ) ] = 7 a 3 ( 8 + 3 a 2 ) [ k + s ( 1 - g 3 ) ] .
· J = - c k n = - 4 K c n ( 8 - 5 a 2 ) - 1 ,
4 c 8 - 5 a 2 n = - 1 8 { 12 [ k + s ( 1 - g ) ] - 7 a 3 [ k + s ( 1 - g 3 ) ] } J = - ( K + 2 S ) J ,
G 2 i = a 2 i ( F f + F r ) ; G 2 i + 1 = a 2 i + 1 ( F f - F r )
d F f d z = - ( K + S ) F f + S F r = - ( K f + S f ) F f + S r F r = - ( K f + S ) F f + S F r ,
- d F r d z = S F f - ( K + S ) F r = S f F f - ( K r + S r ) F r = S F f - ( K r + S ) F r ,
K f , r = k μ f , r - 1 = k F f , r 0 1 I ( z , ± μ ) d μ ,
4 K f F f = k ( 7 F f + F r ) ;             4 K r F r = k ( F f + 7 F r ) ,
( S f , r ) / s = [ ( K f , r ) / 2 k ] - ρ ,
K ( F f + F r ) = K f F f + K r F r ,
( K + 2 S ) ( F f - F r ) = ( K f + 2 S f ) F f - ( K r + 2 S r ) F r .
= 1 - ω 0 = k / ( k + s ) ,
K = k / η ;             S = s / σ ,
224 η = 77 + 55 + 35 ( 1 + 2 35 + 121 2 49 ) 1 / 2 ,
σ = 2 η ( 1 - ) ( 16 η - 3 ) 15 η 2 - ( 16 η - 3 ) ,
η = 0.5 ;             σ = [ 4 ( 1 - ) ] / [ 3 - ( 3 + a ) ] ,
D = ( c η 2 σ ) / { ( k + s ) [ 2 η - ( 2 η - σ ) ] }

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