Abstract

The problem of the reconstruction of the spectrum of a dispersed system from data on its spectral attenuation is studied. The numerical algorithm for obtaining the particle size distribution by the use of the concept of regularization is thoroughly treated. The applicability of this method to the reconstruction of the particle size distribution of a typical marine aerosol is tested. A method of choosing the regularization parameter of the solution for the inverse problem based on an objective estimate of the validity of the obtained solution is proposed. Results are presented for a set of numerical experiments in which the radius interval for which the distribution function can be obtained with a satisfactory accuracy is estimated. The validity of solutions is estimated depending on the measuring spectral range for the attenuation, the radius interval, and the number and position of points within this interval. The possibility of extending the radius interval for which the distribution function can be obtained by the use of extrapolation of the distribution function tail is discussed.

© 1996 Optical Society of America

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References

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  1. K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” in Vol. 34 of Advances in Geophysics (Academic, New York, 1993), pp. 175–252.
    [Crossref]
  2. E. P. Shettle, R. W. Fenn, “Models for aerosols in the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1979), p. 94.
  3. S. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 57–62 (1983).
  4. R. McClatchey, M. Bolle, K. Kondratyev, “A preliminary cloudless standard atmosphere for radiation computation,” WMO/TD-NO24 (World Meteorological Organization, Geneva, 1986).
  5. G. d’Almeida, P. Koepke, E. Shettle, Atmospheric Aerosols (Deepak, Hampton, Va., 1991).
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  7. E. C. Titchmarsh, An Introduction to the Theory of the Fourier Integral (Clarendon, Oxford, 1937).
  8. A. Ya. Perelman, K. S. Shifrin, “Improvements to the spectral transparency method for determining particle-size distribution,” Appl. Opt. 19, 1787–1793 (1980).
    [Crossref] [PubMed]
  9. K. S. Shifrin, Physical Optics of Ocean Water, American Institute of Physics Translation Series (American Institute of Physics, New York, 1988), p. 285.
  10. K. S. Shifrin, I. G. Zolotov, “Information content of the spectral transmittance of the marine atmospheric boundary layer,” submitted to Appl. Opt.
  11. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).
  12. P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
    [Crossref] [PubMed]
  13. D. L. Phillips, “A technique for numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [Crossref]
  14. A. N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

1983 (1)

S. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 57–62 (1983).

1980 (2)

1962 (1)

D. L. Phillips, “A technique for numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Arsenin, V. Ya.

A. N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Bolle, M.

R. McClatchey, M. Bolle, K. Kondratyev, “A preliminary cloudless standard atmosphere for radiation computation,” WMO/TD-NO24 (World Meteorological Organization, Geneva, 1986).

d’Almeida, G.

G. d’Almeida, P. Koepke, E. Shettle, Atmospheric Aerosols (Deepak, Hampton, Va., 1991).

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for aerosols in the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1979), p. 94.

Gathman, S.

S. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 57–62 (1983).

Koepke, P.

G. d’Almeida, P. Koepke, E. Shettle, Atmospheric Aerosols (Deepak, Hampton, Va., 1991).

Kondratyev, K.

R. McClatchey, M. Bolle, K. Kondratyev, “A preliminary cloudless standard atmosphere for radiation computation,” WMO/TD-NO24 (World Meteorological Organization, Geneva, 1986).

McClatchey, R.

R. McClatchey, M. Bolle, K. Kondratyev, “A preliminary cloudless standard atmosphere for radiation computation,” WMO/TD-NO24 (World Meteorological Organization, Geneva, 1986).

Perelman, A. Ya.

Phillips, D. L.

D. L. Phillips, “A technique for numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Shettle, E.

G. d’Almeida, P. Koepke, E. Shettle, Atmospheric Aerosols (Deepak, Hampton, Va., 1991).

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for aerosols in the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1979), p. 94.

Shifrin, K. S.

A. Ya. Perelman, K. S. Shifrin, “Improvements to the spectral transparency method for determining particle-size distribution,” Appl. Opt. 19, 1787–1793 (1980).
[Crossref] [PubMed]

K. S. Shifrin, Physical Optics of Ocean Water, American Institute of Physics Translation Series (American Institute of Physics, New York, 1988), p. 285.

K. S. Shifrin, I. G. Zolotov, “Information content of the spectral transmittance of the marine atmospheric boundary layer,” submitted to Appl. Opt.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” in Vol. 34 of Advances in Geophysics (Academic, New York, 1993), pp. 175–252.
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Titchmarsh, E. C.

E. C. Titchmarsh, An Introduction to the Theory of the Fourier Integral (Clarendon, Oxford, 1937).

Tonna, G.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” in Vol. 34 of Advances in Geophysics (Academic, New York, 1993), pp. 175–252.
[Crossref]

Twomey, S.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Walters, P. T.

Zolotov, I. G.

K. S. Shifrin, I. G. Zolotov, “Information content of the spectral transmittance of the marine atmospheric boundary layer,” submitted to Appl. Opt.

Appl. Opt. (2)

J. Assoc. Comput. Mach. (1)

D. L. Phillips, “A technique for numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Opt. Eng. (1)

S. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 57–62 (1983).

Other (10)

R. McClatchey, M. Bolle, K. Kondratyev, “A preliminary cloudless standard atmosphere for radiation computation,” WMO/TD-NO24 (World Meteorological Organization, Geneva, 1986).

G. d’Almeida, P. Koepke, E. Shettle, Atmospheric Aerosols (Deepak, Hampton, Va., 1991).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

E. C. Titchmarsh, An Introduction to the Theory of the Fourier Integral (Clarendon, Oxford, 1937).

K. S. Shifrin, Physical Optics of Ocean Water, American Institute of Physics Translation Series (American Institute of Physics, New York, 1988), p. 285.

K. S. Shifrin, I. G. Zolotov, “Information content of the spectral transmittance of the marine atmospheric boundary layer,” submitted to Appl. Opt.

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

A. N. Tikhonov, V. Ya. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” in Vol. 34 of Advances in Geophysics (Academic, New York, 1993), pp. 175–252.
[Crossref]

E. P. Shettle, R. W. Fenn, “Models for aerosols in the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratory, Hanscomb Air Force Base, Mass., 1979), p. 94.

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Figures (10)

Fig. 1
Fig. 1

Variants of the sought-for function X(r) of the integral Eq. (1): 1, distribution function (number density) of a typical aerosol X(r) = f(r) in inverse cubic centimeters times inverse micrometers; 2, function X(r) = πr 2 f(r) in inverse cubic centimeters times micrometers. The set of points where f(r) has to be determined is shown by the fitted circles.

Fig. 2
Fig. 2

Spectral attenuation σ(λ) of the aerosol at different indices of refraction n and absorption k. 1, aerosol with n and k of the fine and the large components; 2, n = 1.35 and k = 0; 3, n = 1.35 and k of the fine and the large components; 4, compound n and k; 5, 6, curves of n and k of the fine component; 7, 8, curves of n and k of the large component.

Fig. 3
Fig. 3

Isolines of the relative accumulated spectral attenuation σ r (λ)/σ(λ).

Fig. 4
Fig. 4

Smoothed kernels of the integral Eq. (1) as functions of r for different λ. The smoothed efficiencies for the extinction Q ext(r) are 1, λ = 0.4 μm; 2, λ = 0.8 μm; 3, λ = 1.0 μm; 4, λ = 2.0 μm; 5, λ = 3.0 μm; 6, λ = 5.0 μm. Curve 7 is nonsmoothed Q ext(r) for λ = 0.4 μm. B is a bundle of extinction cross sections πr 2 Q ext for the same values of λ.

Fig. 5
Fig. 5

Example of the dependence of the estimates of the validity of the solutions on the regularization parameter γ (experiment A3). 1, δ a ; 2, δ ¯ a ; 3, δ d ; 4, δ f ; 5, δ f 1; 6, δ n .

Fig. 6
Fig. 6

Spectral attenuation σ(λ) as a function of λ; 1, initial σ(λ) (the set of points where the attenuation is given is shown); 2, σ(λ) for experiment C2; 3, 4, difference between σ(λ) and the attenuation caused by the tail of the distribution function for experiments A t 1 and A t 2, respectively; 5, 6, σ(λ) containing random errors (experiments D1 and D2).

Fig. 7
Fig. 7

Distribution functions (number densities) for different experiments. 1, initial distribution function; 2, A1; 3, A2; 4, A3; 5, B2; 6, B3; 7, B4. The tails of the distribution functions for the experiments are 8, A t 1; 9, A t 2; and 10, B t .

Fig. 8
Fig. 8

Distribution functions (number densities) for experiments E and E t : 1, initial distribution function; 2, E; 3, E t ; 4, extrapolated tail of the distribution function.

Fig. 9
Fig. 9

Distribution functions (number densities) for experiments F and F t : 1, initial distribution function; 2, F; 3, F t ; 4, extrapolated tail of the distribution function.

Fig. 10
Fig. 10

Distribution functions (number densities) reconstructed from the attenuation in the spectral range 0.4–1.0 μm: 1, initial distribution function; 2, V1; 3, V2; 4, V3; 5–7, extrapolated tails of the distribution functions for V t 1, V t 2, and V t 3, respectively.

Tables (2)

Tables Icon

Table 1 Estimate of the Error of the Numerical Experiments in which the Extrapolation is Not Used

Tables Icon

Table 2 Estimate of the Error of the Numerical Experiments in which the Extrapolation is Used

Equations (10)

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σ ( λ ) = π 0 Q ext ( λ , r ) f ( r ) r 2 d r ,
X ( r ) = π r 2 f ( r )
A X = g .
a i k = - r k - 1 r k - r k - 1 P k ( λ i ) + 1 r k - r k - 1 R k ( λ i ) + r k + 1 r k + 1 - r k P k + 1 ( λ i ) - 1 r k + 1 - r k R k ( λ i ) ,             k = 1 , N - 1 , a i N = - r N - 1 r N - r N - 1 P N ( λ i ) + 1 r N - r N - 1 R N ( λ i ) ,
P k ( λ i ) = r k - 1 r k Q ext ( λ i , r ) d r , R k ( λ i ) = r k - 1 r k r Q ext ( λ i , r ) d r ,
( A T A ) X = A T g ,
( A T A + γ H ) X = A T g ,
K = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 - 3 3 - 1 0 0 0 0 0 1 - 3 3 - 1 0 0 0 0 0 1 - 3 3 - 1 0 0 0 0 0 0 1 - 3 3 - 1 ] .
δ a = ( 1 / M ) i . A X - g i / g i
δ d = 1 / ( N - 2 ) × i log f i - 1 - 2 log f i + log f i + 1 / log f i .

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