Abstract

Equations are derived for estimating the size distribution of particles from measurements of components of the Stokes vector of radiation backscattered from an optically thick, plane-parallel aerosol. A two-step inversion procedure is needed. The inverse (Fredholm) size-distribution procedure utilizes the lowest-order angular moment of the element S44 of the normalized scattering matrix at different wavelengths obtained from an inverse radiative-transfer procedure. This procedure requires time-dependent, backscattered polarized radiance measurements at long and short times after radiation pulses at different wavelengths illuminate the aerosol; the procedure is based on the assumption of azimuthal symmetry in order to obtain an uncoupling of the components of the polarized radiance. If the radiation wavelengths are between 0.3 and 10.6 μm, the procedure will work best if the radii of spherical particles are between 0.04 μm and a few micrometers.

© 1990 Optical Society of America

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References

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  7. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950; Dover, New York, 1960).
  10. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [Crossref]
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  15. C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
    [Crossref]
  16. C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).
  17. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophy. 128, 1–16 (1983).
  18. W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).
  19. C. E. Siewert, F. J. V. Pinheiro, “On the scattering of polarized light,” Z. Angew. Math. Phys. 33, 807–818 (1982).
    [Crossref]
  20. G. C. Pomraning, “The effects of polarization on diffusion descriptions of radiative transfer,” Astron. Astrophys. 2, 410–424 (1969).
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  24. The results of Ref. 23 above incorrectly contain the factor z0exp(−z02/4Dct) that depends on the arbitrary location z0at which the authors assumed all incident particles have their first collision.
  25. I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
    [Crossref]
  26. P. Vestrucci, C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 31, 177–183 (1984).
    [Crossref]
  27. W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” National Center for Atmospheric Research Tech. Note NCAR/TN-140 + STR (National Center for Atmospheric Research, Boulder, Colo., 1979).
  28. G. S. Kent, G. K. Yue, U. O. Farrukh, A. Deepak, “Modeling atmospheric aerosol backscatter at CO2laser wavelengths. 1: aerosol properties, modeling techniques, and associated problems,” Appl. Opt. 22, 1655–1665 (1983).
    [Crossref] [PubMed]

1989 (4)

1988 (2)

1987 (2)

1986 (1)

1984 (2)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

P. Vestrucci, C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 31, 177–183 (1984).
[Crossref]

1983 (2)

G. S. Kent, G. K. Yue, U. O. Farrukh, A. Deepak, “Modeling atmospheric aerosol backscatter at CO2laser wavelengths. 1: aerosol properties, modeling techniques, and associated problems,” Appl. Opt. 22, 1655–1665 (1983).
[Crossref] [PubMed]

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophy. 128, 1–16 (1983).

1982 (3)

C. E. Siewert, F. J. V. Pinheiro, “On the scattering of polarized light,” Z. Angew. Math. Phys. 33, 807–818 (1982).
[Crossref]

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

N. J. McCormick, “Remote characterization of a thick slab target with a pulsed laser,” J. Opt. Soc. Am. 72, 756–759 (1982).
[Crossref]

1981 (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[Crossref]

1980 (1)

1976 (1)

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

1969 (1)

G. C. Pomraning, “The effects of polarization on diffusion descriptions of radiative transfer,” Astron. Astrophys. 2, 410–424 (1969).

1959 (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Anderson, R.

Ben-David, A.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bossert, D. J.

Bronk, B. V.

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950; Dover, New York, 1960).

de Rooij, W. A.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Deepak, A.

Duracz, T.

Elliott, R. A.

Emmons, D. R.

Farrukh, U. O.

Furutsu, K.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Herman, B. M.

Hovenier, J. W.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophy. 128, 1–16 (1983).

Huffman, D. R.

Ito, S.

Kattawar, G. W.

Kavaya, M. J.

Kent, G. S.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kušcer, I.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

McCormick, N. J.

Patterson, M. S.

Pinheiro, F. J. V.

C. E. Siewert, F. J. V. Pinheiro, “On the scattering of polarized light,” Z. Angew. Math. Phys. 33, 807–818 (1982).
[Crossref]

Plass, G. N.

Pomraning, G. C.

G. C. Pomraning, “The effects of polarization on diffusion descriptions of radiative transfer,” Astron. Astrophys. 2, 410–424 (1969).

Reagan, J. A.

Ribaric, M.

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Siewert, C. E.

P. Vestrucci, C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 31, 177–183 (1984).
[Crossref]

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

C. E. Siewert, F. J. V. Pinheiro, “On the scattering of polarized light,” Z. Angew. Math. Phys. 33, 807–818 (1982).
[Crossref]

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[Crossref]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Twomey, S.

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

Van De Merwe, W. P.

van der Mee, C. V. M.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophy. 128, 1–16 (1983).

van der Stap, C. C. A. H.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Vestrucci, P.

P. Vestrucci, C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 31, 177–183 (1984).
[Crossref]

Wilson, B. C.

Wiscombe, W. J.

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” National Center for Atmospheric Research Tech. Note NCAR/TN-140 + STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

Yue, G. K.

Appl. Opt. (7)

Astron. Astrophy. (1)

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophy. 128, 1–16 (1983).

Astron. Astrophys. (3)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

C. E. Siewert, “On the phase matrix basic to the scattering of polarized light,” Astron. Astrophys. 109, 195–200 (1982).

G. C. Pomraning, “The effects of polarization on diffusion descriptions of radiative transfer,” Astron. Astrophys. 2, 410–424 (1969).

Astrophys. J. (1)

C. E. Siewert, “On the equation of transfer relevant to the scattering of polarized light,” Astrophys. J. 245, 1080–1086 (1981).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

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

P. Vestrucci, C. E. Siewert, “A numerical evaluation of an analytical representation of the components in a Fourier decomposition of the phase matrix for the scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 31, 177–183 (1984).
[Crossref]

Opt. Acta (1)

I. Kuščer, M. Ribarič, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[Crossref]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Z. Angew. Math. Phys. (1)

C. E. Siewert, F. J. V. Pinheiro, “On the scattering of polarized light,” Z. Angew. Math. Phys. 33, 807–818 (1982).
[Crossref]

Other (6)

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950; Dover, New York, 1960).

W. J. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” National Center for Atmospheric Research Tech. Note NCAR/TN-140 + STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

The results of Ref. 23 above incorrectly contain the factor z0exp(−z02/4Dct) that depends on the arbitrary location z0at which the authors assumed all incident particles have their first collision.

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Figures (2)

Fig. 1
Fig. 1

δ0 versus size parameter x for nonabsorbing particles with a relative refractive index n.

Fig. 2
Fig. 2

δ0 and β1/3 versus size parameter x for nonabsorbing particles with a relative refractive index n.

Equations (52)

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g ( r eff , λ k ) = G 1 r min r max π r 2 g ( r , λ k ) f ( r ) d r G 1 j = 1 J w j π r j 2 g ( r j , λ k ) f ( r j ) , k = 1 , , K ,
G = r min r max π r 2 f ( r ) d r j = 1 J w j π r j 2 f ( r j ) .
f ( r ) = C r ( ν + 1 ) ,
f ( r ) = C r a 1 exp ( a 2 r a 3 ) ,
f ( r ) = C r a 1 exp ( a 2 r a 3 ) ,
g = Af ,
f [ ( Af g ) T ( A f g ) + γ S ( f ) ] = 0 ,
f = ( A T A + γ H ) 1 A T g .
I ( τ , μ , t ) = 0 2 π I ( τ , μ , ϕ , t ) d ϕ .
[ ( c σ ) 1 t + μ τ + 1 ] I ( τ , μ , t ) = ( ω / 2 ) 1 1 P ( μ , μ ) I ( τ , μ , t ) d μ , τ 0 ,
P ( μ , μ ) = n = 0 N Π n ( μ ) B n Π n ( μ ) ,
Π n ( μ ) = diag { P n ( μ ) , R n ( μ ) , R n ( μ ) , P n ( μ ) }
R n ( μ ) = [ ( n 2 ) ! ( n + 2 ) ! ] 1 / 2 ( 1 μ 2 ) d 2 d μ 2 P n ( μ ) , n 2.
B n = [ β n γ n 0 0 γ n α n 0 0 0 0 ζ n n 0 0 n δ n ] , n = 0 , , N
δ 0 = 1 2 1 1 S 44 ( θ ) d ( cos θ ) ,
[ ( c σ ) 1 t + μ τ + 1 ] Ψ ( τ , μ , t ) = ( ω / 2 ) n = 0 N P n ( μ ) C n × 1 1 P n ( μ ) Ψ ( τ , μ , t ) d μ , τ 0 ,
P n ( μ ) = diag { P n ( μ ) , R n ( μ ) } .
C n = [ β n γ n γ n α n ] ,
C n = [ δ n n n ζ n ] .
Ψ n ( τ , t ) = 1 1 P n ( μ ) Ψ ( τ , μ , t ) d μ , Ψ ˜ n ( τ , t ) = 1 1 μ P n ( μ ) Ψ ( τ , μ , t ) d μ ,
[ ( c σ ) 1 t + 1 ω C n / ( 2 n + 1 ) ] Ψ n ( τ , t ) + τ Ψ ˜ n ( τ , t ) = 0 , n = 0,1,2 , ,
1 1 P n ( μ ) P m ( μ ) d μ = [ 2 / ( 2 n + 1 ) ] diag { 1 , ( 1 δ 0 n ) ( 1 δ 1 n ) } δ n m .
[ ( c σ ) 1 t + 1 ω δ 0 ] V 0 ( τ , t ) + τ V 1 ( τ , t ) = 0 ,
[ ( c σ ) 1 t + 1 ω δ 1 / 3 ] V 1 ( τ , t ) + τ V ˜ 1 ( τ , t ) = 0 ,
V ˜ 0 ( τ , t ) = V 1 ( τ , t ) .
K V ( τ , t ) = V ˜ 1 ( τ , t ) V 0 ( τ , t ) = 1 1 μ 2 V ( τ , t ) d μ 1 1 V ( τ , t ) d μ .
V n ( τ , t ) = exp [ ( 1 ω δ 0 ) c σ t ] ψ V n ( τ , t ) , n = 0,1 ,
( c σ ) 1 t ψ V 0 ( τ , t ) + τ ψ V 1 ( τ , t ) = 0 ,
[ ( c σ ) 1 t + ω ( δ 0 δ 1 / 3 ) ] ψ V 1 ( τ , t ) + τ [ K V ( τ ) ψ V 0 ( τ , t ) ] = 0.
( c σ ) 1 t ψ V 1 ( τ , t ) ω ( δ 0 δ 1 / 3 ) ψ V 1 ( τ , t ) ,
ψ V 1 ( τ , t ) K V [ ω ( δ 0 δ 1 / 3 ) ] 1 τ ψ V 0 ( τ , t ) .
τ ψ V 0 ( τ , t ) = D V τ 2 ψ V 0 ( τ , t ) ,
D V = c σ K V [ ω ( δ 0 δ 1 / 3 ) ] 1 .
V n ( 0 , t ) t 3 / 2 exp [ ( 1 ω δ 0 ) c σ t ] , n = 0,1 ,
I n ( 0 , t ) t 3 / 2 exp [ ( 1 ω ) c σ t ] , n = 0,1.
V ( 0 , 1 , t ) t 3 / 2 exp [ ( 1 ω δ 0 ) c σ t ] ,
I ( 0 , 1 , t ) t 3 / 2 exp [ ( 1 ω ) c σ t ] .
V n ( 0 , r , t ) t 5 / 2 exp [ ( 1 ω δ 0 ) c σ t ] exp ( r 2 / 4 D V t ) , n = 0,1.
( 1 ω δ 0 ) c σ = t ln [ t 3 / 2 V ( 0 , 1 , t ) ] t ln [ t 3 / 2 V ( 0 , 1 , t ) ] t 2 t 2 ,
f ( t ) = K 1 k = 1 K f ( t k ) .
I ( 0 , 1 , t ) exp [ ( 1 + μ 0 1 ) c σ t ] .
( 1 + μ 0 1 ) c σ = t ln [ I ( 0 , 1 , t ) ] t ln [ I ( 0 , 1 , t ) ] t 2 t 2 .
S ( t ) = 0 t A ( t ) I ( 0 , 1 , t t ) d t .
A ( t ) exp [ ( t κ a ) 2 / κ 2 ] , t 2 κ a = 0 , t > 2 κ a ,
S ( t ) exp [ ( 1 + μ 0 1 ) c σ a ( t / a ) ] [ erf κ * erf ( κ * t / a ) ] , t / a 2 κ ,
κ * = κ + 1 2 ( 1 + μ 0 1 ) c σ a .
π 1 / 2 exp [ ( κ * t m / a ) 2 ] = ( κ * κ ) [ erf κ * erf ( κ * t m / a ) ]
S ( t ) ( 1 + μ 0 1 ) c σ t 1 + exp [ ( 1 + μ 0 1 ) c σ t ] .
| Δ δ 0 | | Δ V | | ( δ 0 V ) ω , c σ | + | Δ ω | | ( δ 0 ω ) V , c σ | + | Δ ( c σ ) | | [ δ 0 ( c σ ) ] V , ω | .
| Δ δ 0 | 1 ω c σ t ( | Δ V V | + δ 0 | Δ I I | ) + 1 + δ 0 2 ω δ 0 ω | ( Δ ( c σ ) c σ ) |
δ 0 = 2 x 2 ( n 2 + 2 ) 2 [ 30 ( n 2 + 2 ) + 36 x 2 ( n 2 2 ) ] 1 ,
β 1 / 3 = 1 2 1 1 S 11 cos θ d ( cos θ ) ,

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