Abstract

Inverse multiple-scattering transport methods, derived from the linear Boltzmann equation, are potentially suitable for remote-sensing studies because they do not require measurements within the scattering medium. The methods investigated yield the single-frequency angular redistribution function for a homogeneous one-dimensional slab medium that is uniformly illuminated. Illustrative numerical results for a monodirectional incident beam show that the methods are ill conditioned with respect to simulated measurement errors, with the severity depending strongly on the direction of the incident beam.

© 1985 Optical Society of America

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  1. C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).
  2. C. E. Siewert, “On establishing a two-term scattering law in the theory of radiative transfer,” Z. Angew. Math. Phys. 30, 522–525 (1979).
    [CrossRef]
  3. C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transfer 22, 441–446 (1979).
    [CrossRef]
  4. N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
    [CrossRef]
  5. C. E. Siewert, W. L. Dunn, “On inverse problems for plane-parallel media with non-uniform surface illumination,” J. Math. Phys. 23, 1376–1378 (1982).
    [CrossRef]
  6. E. W. Larsen, “Solution of multidimensional inverse transport problems,” J. Math. Phys. 25, 131–135 (1984).
    [CrossRef]
  7. R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
    [CrossRef]
  8. C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transfer 30, 523–526 (1983).
    [CrossRef]
  9. N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transfer 30, 527–535 (1983).
    [CrossRef]
  10. N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
    [CrossRef]
  11. R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1982).
    [CrossRef]
  12. C. E. Siewert, “The FN-method for solving radiative transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
    [CrossRef]
  13. R. D. M. Garcia, C. E. Siewert, “Radiative transfer in finite homogeneous plane-parallel atmospheres,” J. Quant. Spectrosc Radiat. Transfer 27, 141–148 (1982).
    [CrossRef]
  14. C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
    [CrossRef]
  15. W. L. Dunn, J. R. Maiorino, “On the numerical characteristics of an inverse solution for three-term radiative transfer,” J. Quant. Spectrosc. Rad. Transfer 24, 203–209 (1980).
    [CrossRef]
  16. N. J. McCormick, “Inverse methods for remote determination of properties of optically thick atmospheres,” Appl. Opt. 22, 2556–2558 (1983).
    [CrossRef] [PubMed]
  17. L. W. Johnson, R. D. Riess, Numerical Analysis (Addison-Wesley, Reading, Mass., 1982).
  18. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  19. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

1984 (1)

E. W. Larsen, “Solution of multidimensional inverse transport problems,” J. Math. Phys. 25, 131–135 (1984).
[CrossRef]

1983 (3)

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transfer 30, 523–526 (1983).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transfer 30, 527–535 (1983).
[CrossRef]

N. J. McCormick, “Inverse methods for remote determination of properties of optically thick atmospheres,” Appl. Opt. 22, 2556–2558 (1983).
[CrossRef] [PubMed]

1982 (4)

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1982).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Radiative transfer in finite homogeneous plane-parallel atmospheres,” J. Quant. Spectrosc Radiat. Transfer 27, 141–148 (1982).
[CrossRef]

C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

C. E. Siewert, W. L. Dunn, “On inverse problems for plane-parallel media with non-uniform surface illumination,” J. Math. Phys. 23, 1376–1378 (1982).
[CrossRef]

1981 (2)

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

1980 (1)

W. L. Dunn, J. R. Maiorino, “On the numerical characteristics of an inverse solution for three-term radiative transfer,” J. Quant. Spectrosc. Rad. Transfer 24, 203–209 (1980).
[CrossRef]

1979 (3)

C. E. Siewert, “On establishing a two-term scattering law in the theory of radiative transfer,” Z. Angew. Math. Phys. 30, 522–525 (1979).
[CrossRef]

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transfer 22, 441–446 (1979).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

1978 (2)

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

C. E. Siewert, “The FN-method for solving radiative transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

Devaux, C.

C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Dunn, W. L.

C. E. Siewert, W. L. Dunn, “On inverse problems for plane-parallel media with non-uniform surface illumination,” J. Math. Phys. 23, 1376–1378 (1982).
[CrossRef]

W. L. Dunn, J. R. Maiorino, “On the numerical characteristics of an inverse solution for three-term radiative transfer,” J. Quant. Spectrosc. Rad. Transfer 24, 203–209 (1980).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “Radiative transfer in finite homogeneous plane-parallel atmospheres,” J. Quant. Spectrosc Radiat. Transfer 27, 141–148 (1982).
[CrossRef]

Johnson, L. W.

L. W. Johnson, R. D. Riess, Numerical Analysis (Addison-Wesley, Reading, Mass., 1982).

Larsen, E. W.

E. W. Larsen, “Solution of multidimensional inverse transport problems,” J. Math. Phys. 25, 131–135 (1984).
[CrossRef]

Maiorino, J. R.

W. L. Dunn, J. R. Maiorino, “On the numerical characteristics of an inverse solution for three-term radiative transfer,” J. Quant. Spectrosc. Rad. Transfer 24, 203–209 (1980).
[CrossRef]

McCormick, N. J.

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transfer 30, 527–535 (1983).
[CrossRef]

N. J. McCormick, “Inverse methods for remote determination of properties of optically thick atmospheres,” Appl. Opt. 22, 2556–2558 (1983).
[CrossRef] [PubMed]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1982).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Riess, R. D.

L. W. Johnson, R. D. Riess, Numerical Analysis (Addison-Wesley, Reading, Mass., 1982).

Sanchez, R.

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transfer 30, 527–535 (1983).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1982).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transfer 30, 523–526 (1983).
[CrossRef]

C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Radiative transfer in finite homogeneous plane-parallel atmospheres,” J. Quant. Spectrosc Radiat. Transfer 27, 141–148 (1982).
[CrossRef]

C. E. Siewert, W. L. Dunn, “On inverse problems for plane-parallel media with non-uniform surface illumination,” J. Math. Phys. 23, 1376–1378 (1982).
[CrossRef]

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transfer 22, 441–446 (1979).
[CrossRef]

C. E. Siewert, “On establishing a two-term scattering law in the theory of radiative transfer,” Z. Angew. Math. Phys. 30, 522–525 (1979).
[CrossRef]

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

C. E. Siewert, “The FN-method for solving radiative transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

Yuan, Y. L.

C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (1)

C. Devaux, C. E. Siewert, Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Astrophys. Space Sci. (1)

C. E. Siewert, “The FN-method for solving radiative transfer problems in plane geometry,” Astrophys. Space Sci. 58, 131–137 (1978).
[CrossRef]

J. Math. Phys. (5)

N. J. McCormick, “Transport scattering coefficients from reflection and transmission measurements,” J. Math. Phys. 20, 1504–1507 (1979).
[CrossRef]

C. E. Siewert, W. L. Dunn, “On inverse problems for plane-parallel media with non-uniform surface illumination,” J. Math. Phys. 23, 1376–1378 (1982).
[CrossRef]

E. W. Larsen, “Solution of multidimensional inverse transport problems,” J. Math. Phys. 25, 131–135 (1984).
[CrossRef]

R. Sanchez, N. J. McCormick, “General solutions to inverse transport problems,” J. Math. Phys. 22, 847–855 (1981).
[CrossRef]

N. J. McCormick, R. Sanchez, “Inverse problem transport calculations for anisotropic scattering coefficients,” J. Math. Phys. 22, 199–208 (1981).
[CrossRef]

J. Quant. Spectrosc Radiat. Transfer (1)

R. D. M. Garcia, C. E. Siewert, “Radiative transfer in finite homogeneous plane-parallel atmospheres,” J. Quant. Spectrosc Radiat. Transfer 27, 141–148 (1982).
[CrossRef]

J. Quant. Spectrosc. Rad. Transfer (1)

W. L. Dunn, J. R. Maiorino, “On the numerical characteristics of an inverse solution for three-term radiative transfer,” J. Quant. Spectrosc. Rad. Transfer 24, 203–209 (1980).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (4)

C. E. Siewert, “On the inverse problem for a three-term phase function,” J. Quant. Spectrosc. Radiat. Transfer 22, 441–446 (1979).
[CrossRef]

R. Sanchez, N. J. McCormick, “Numerical evaluation of optical single-scattering properties using multiple-scattering inverse transport methods,” J. Quant. Spectrosc. Radiat. Transfer 28, 169–184 (1982).
[CrossRef]

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization—I,” J. Quant. Spectrosc. Radiat. Transfer 30, 523–526 (1983).
[CrossRef]

N. J. McCormick, R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization—II,” J. Quant. Spectrosc. Radiat. Transfer 30, 527–535 (1983).
[CrossRef]

Nucl. Sci. Eng. (1)

C. E. Siewert, “The inverse problem for a finite slab,” Nucl. Sci. Eng. 67, 259–260 (1978).

Z. Angew. Math. Phys. (1)

C. E. Siewert, “On establishing a two-term scattering law in the theory of radiative transfer,” Z. Angew. Math. Phys. 30, 522–525 (1979).
[CrossRef]

Other (3)

L. W. Johnson, R. D. Riess, Numerical Analysis (Addison-Wesley, Reading, Mass., 1982).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

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Figures (3)

Fig. 1
Fig. 1

Split-mesh collocation scheme for Nθ = 5 with seven points at −μo. (○ denotes a node for calculation of A, and ● denotes a node for calculation of S at −μo)

Fig. 2
Fig. 2

Ratio RI of the percent error in scattering function to percent error in the intensity versus N*. (Binomial scattering function has α = 8 and f0 = 0.99 with τo = 5.)

Fig. 3
Fig. 3

Ratio RΩ of the percent error in scattering function to the error in direction in degrees. (Binomial scattering function has α = 8 and f0 = 0.99 with τo = 5.)

Tables (4)

Tables Icon

Table 1 Scattering Coefficients and Percent Error in Scattering Functiona

Tables Icon

Table 2 Percent Error in Scattering Function for Methods 0 and 1 versus θmaxa

Tables Icon

Table 3 Percent Error in Scattering Function for Methods 0 and 1 versus Different Combinations of Errors in θ and ϕ for the Incident Illuminationa

Tables Icon

Table 4 Percent Error in Scattering Function for Different Split-Mesh Collocation Schemes versus Percent Random Error in Intensitiesa

Equations (34)

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f ( ω ) = ( 4 π ) 1 n = 0 N ( 2 n + 1 ) f n P n ( ω ) ,
f ( ω ) f ̂ ( ω ) = 4 π ( f ( ω ) f ̂ ( ω ) ) 2 d ω = 1 2 n = 0 N ( 2 + 1 ) ( f n f ̂ n ) 2 ,
f ( ω ) = ( 4 π ) 1 f 0 ( 1 + α ) 2 α ( 1 + ω ) α , α 0.
I ( 0 , μ , ϕ ) = δ ( μ μ o ) δ ( ϕ ) 0 μ , μ o 1 ,
I ( τ o , μ , ϕ ) = 0 , 1 μ 0 .
I ( τ , μ , ϕ ) = I s ( τ , μ , ϕ ) + I c ( τ , μ , ϕ )
I c ( τ , μ , ϕ ) = m = 0 N ( 2 δ m 0 ) I c m ( τ , μ ) cos ( m ϕ ) ,
I s ( τ , μ , ϕ ) = δ ( μ μ o ) δ ( ϕ ) exp ( τ / μ o ) ,
I m = 1 2 π 0 2 π cos ( m ϕ ) I ( τ , μ , ϕ ) d ϕ .
A k F k = S k
n = m N A m n k F n k = S m k , m = 0 to N
F n 0 = f n ,
F n 1 = f n / ( 1 f n ) ,
A m n k = ( 1 ) n m α m n [ 0 2 π d ϕ 1 1 d μ P n m ( μ ) × cos ( m ϕ ) μ k I ( τ , μ , ϕ ) ] 2 0 τ o ,
S m k = 4 0 1 μ 2 k I m ( τ , μ ) I m ( τ , μ ) d μ 0 τ 0 .
α m n = ( 2 n + 1 ) ( n m ) ! / ( n + m ) ! .
S m k = 4 μ o I 2 k c m ( 0 , μ o ) .
n = m N * A m n F n = S m .
F n = m = n N * A n m 1 S m ,
I m ( τ , μ o ) = ( M 1 ) 1 k = 1 M ( 1 δ k 1 / 2 ) × ( 1 δ k M / 2 ) I ( τ , μ o , ϕ k ) cos ( m ϕ k ) ,
θ i = ( i 1 ) θ max / ( N θ 1 ) , i = 1 to N θ , ϕ j = ( j 1 ) π / ( N i 1 ) , j = 1 to N i ,
N i = 1 + integer [ 3 sin ( θ i ) / sin ( θ 2 ) + 1 / 2 ] .
p ( x ) = cos 2 ( π x / 2 ) , | x | 1 ,
δ I = ( 1 / 3 2 / π 2 ) 1 / 2 σ x .
σ n 2 = [ x / ( 1 + x ) ] 2 F n ( x ) d x .
A m n ( 1 ) n m α m n P n m ( μ o ) 2 for large m .
C K A A 1 ,
δ f / f C K δ S / S ,
A 1 > | λ K | ,
C K A | λ K |
1 / λ K = A K K = α K K P K K ( μ o ) 2 .
P K K ( μ o ) = ( 1 μ o 2 ) K / 2 k = 0 K 1 ( 2 k + 1 ) .
C K / C K + 1 | λ K / λ K + 12 | = [ ( 2 K + 3 ) / ( 2 K + 1 ) ] ( 1 μ o 2 ) ,
C K + 1 ~ C K ( 1 μ o 2 ) 1 .

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