Abstract

The Kubelka–Munk hyperbolic equations for reflectance and transmittance of a turbid, isotropically scattering medium have been found formally identical with the solution for reflected and transmitted fluxes of Chandrasekhar’s radiative-transfer equation for isotropically highly scattering media. The absorption and scattering coefficients of the two theories have been related through numerical coefficients. The Kubelka–Munk absorption coefficient is nearly proportional to the true absorption coefficient in the range of reflectance values between 0.6 and 1 but deviations from proportionality up to a factor of 2 occur for lower reflectance values.

© 1972 Optical Society of America

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References

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  1. P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  2. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 9.
  3. Reference 2, p. 15.
  4. Reference 2, p. 19.
  5. G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969), p. 121.
  6. H. H. Theissing, J. Opt. Soc. Am. 40, 232 (1950).
    [Crossref]
  7. Reference 5, p. 99.

1950 (1)

1948 (1)

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 9.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969), p. 121.

Kubelka, P.

Theissing, H. H.

J. Opt. Soc. Am. (2)

Other (5)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), p. 9.

Reference 2, p. 15.

Reference 2, p. 19.

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969), p. 121.

Reference 5, p. 99.

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Figures (1)

Fig. 1
Fig. 1

The values of coefficients χ, η, and their ratio χ/η for the isotropic phase function p(0) = ω ¯ 0. The corresponding values of the Kubelka–Munk function K/S = (1−R)2/2R are given on the upper scale.

Tables (1)

Tables Icon

Table I Values of coefficients χ, η and their ratio χ/η in terms of albedo ω ¯ 0 and roots ξ, for phase function p(0) = ω ¯ 0.

Equations (28)

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- μ d I ν ( τ , μ ) d τ = I ν ( τ , μ ) - 1 2 - 1 1 p ( 0 ) ( μ , μ ) I ν ( τ , μ ) d μ ,
τ = κ ν ρ d z
I ν ( τ , μ ) = exp ( ± ξ κ ν ρ z ) 1 ± ξ μ ,
ω ¯ 0 = 2 ξ ln [ ( 1 + ξ ) / ( 1 - ξ ) ] .
ν + ( τ ) = 0 1 μ I ν ( τ , μ ) d μ
ν - ( τ ) = - - 1 0 μ I ν ( τ , μ ) d μ ,
R = ν - ( 0 ) / ν + ( 0 )
T = ν + ( τ Z ) / ν + ( 0 ) ,
I ν ( τ , μ ) = L 1 exp ( ξ κ ν ρ z ) 1 + ξ μ + L 2 exp ( - ξ κ ν ρ z ) 1 - ξ μ ,
ν - ( τ Z ) / ν + ( τ Z ) = R g ,
L 1 L 2 = exp ( - 2 Y ) ( 1 + φ R g ) ( φ + R g ) ,
Y = ξ κ ν ρ Z
φ = ξ + ln ( 1 - ξ ) ξ - ln ( 1 + ξ ) .
R = 1 + R g ( b coth Y - a ) a + b coth Y - R g
T = b b cosh Y + a sinh Y ,
a = - ( 1 + φ 2 ) 2 φ             and             b = ( a 2 - 1 ) .
K / S = a - 1 ,             S b Z = Y ,
S = ( d R d Z ) Z 0 , R g = 0
K = - ( d T d Z ) Z 0 - S .
α ν = η K ,
η = ( φ - 1 ) ( 1 - ω ¯ 0 ) ξ ( φ + 1 ) ,
σ ν = χ S ,
χ = - ω ¯ 0 2 ξ ( φ - φ - 1 ) .
K S = ( 1 - R ) 2 2 R = ( χ η ) α ν σ ν = ( χ η ) ( 1 - ω ¯ 0 ) ω ¯ 0 ,
F ( R ) = ( 1 - R ) 2 / 2 R
[ η ] ξ 0 = 1 2 , [ χ ] ξ 0 = 4 3 , [ η ] ξ 1 = 1 ξ 1 , [ χ ] ξ 1 = 1 ξ - ln ( 1 + ξ ) 1 1 - ln 2 ,
R ξ 0 = 3 4 Z ( α ν + σ ν ) 1 + 3 4 Z ( α ν + σ ν )             for R g = 0 , R ξ 1 = σ ν ( 1 - ln 2 ) α ν { 1 + coth [ Z ( α ν + σ ν ) ] }             for R g = 0 , T ξ 0 = 1 1 + 3 4 Z ( α ν + σ ν ) , T ξ 1 = e - Z ( α ν + σ ν ) ,
F ( R ) = ( 1 - R ) 2 / 2 R