Abstract

Retrieval of the aerosol size distribution from optical measurements at ground level is well known to be a difficult problem. Nowadays objective techniques that can give a solution without the intervention of the researcher do not exist. We propose several objective methods that are well based in the mathematical and physical points of view. Their accuracy is evaluated and the top performance of the objective inversion techniques is presented. Moreover physical and experimental suggestions can be drawn to improve the accuracy. Inversions with experimental optical depths are also shown.

© 1995 Optical Society of America

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References

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  1. G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988−993 (1979).
    [Crossref] [PubMed]
  2. M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
    [Crossref]
  3. M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
    [Crossref]
  4. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory. 1: extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
    [Crossref] [PubMed]
  5. A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: theory,” Appl. Opt. 27, 1235–1242 (1988).
    [Crossref] [PubMed]
  6. A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 2: use,” Appl. Opt. 27, 1243–1254 (1988).
    [Crossref] [PubMed]
  7. B. P. Curry, “Constrained eigenfunction method for the inversion of remote-sensing data: application to particle-size determination from light-scattering measurements,” Appl. Opt. 28, 1345–1355 (1989).
    [Crossref] [PubMed]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1975).
  9. A. N. Thikonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  10. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  11. M. A. Lukas, “Regularization,” in The Application and Numerical Solution of Integral Equations, R. S. Anderssen, F. R. de Hoog, M. A. Lukas, eds. (Sijthoff and Noordhoff, The Netherlands, 1980).
    [Crossref]
  12. U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Probl. 7, 793–808 (1991).
    [Crossref]
  13. G. Wahba, “Practical approximate solutions to linear operators equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
    [Crossref]
  14. World Meteorological Organization, “A preliminary cloudless standard atmosphere for radiation computation,” Research Rep. 112 in the World Climate Programme Series, USA (World Meteorological Organization, Case Postale 2300, Geneva 20, Switzerland CH–1211, 1984).
  15. G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).
  16. G. E. Shaw, “Error analysis of multiwavelength Sun photometry,” Pure Appl. Geophys. 114, 1–13 (1976).
    [Crossref]
  17. E. P. Shettle, R. W. Fenn, “Models of atmospheric aerosols and their optical properties,” paper presented at the Electromagnetic Wave Propagation Panel on Advisory Group for Aerospace Research and Development, 22nd Technical Meeting, The Technical University of Denmark, Copenhagen, Denmark, 1975.

1991 (1)

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Probl. 7, 793–808 (1991).
[Crossref]

1989 (1)

1988 (2)

1987 (1)

1986 (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
[Crossref]

1982 (1)

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[Crossref]

1979 (1)

1977 (1)

G. Wahba, “Practical approximate solutions to linear operators equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[Crossref]

1976 (1)

G. E. Shaw, “Error analysis of multiwavelength Sun photometry,” Pure Appl. Geophys. 114, 1–13 (1976).
[Crossref]

Amato, U.

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Probl. 7, 793–808 (1991).
[Crossref]

Arsenin, V. Y.

A. N. Thikonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Ben-David, A.

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
[Crossref]

Box, M. A.

Curry, B. P.

d'Almeida, G. A.

G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
[Crossref]

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models of atmospheric aerosols and their optical properties,” paper presented at the Electromagnetic Wave Propagation Panel on Advisory Group for Aerospace Research and Development, 22nd Technical Meeting, The Technical University of Denmark, Copenhagen, Denmark, 1975.

Herman, B. M.

Hughes, W.

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Probl. 7, 793–808 (1991).
[Crossref]

King, M. D.

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[Crossref]

Koepke, P.

G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

Lukas, M. A.

M. A. Lukas, “Regularization,” in The Application and Numerical Solution of Integral Equations, R. S. Anderssen, F. R. de Hoog, M. A. Lukas, eds. (Sijthoff and Noordhoff, The Netherlands, 1980).
[Crossref]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
[Crossref]

Reagan, J. A.

Shaw, G. E.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models of atmospheric aerosols and their optical properties,” paper presented at the Electromagnetic Wave Propagation Panel on Advisory Group for Aerospace Research and Development, 22nd Technical Meeting, The Technical University of Denmark, Copenhagen, Denmark, 1975.

G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

Thikonov, A. N.

A. N. Thikonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Twomey, S.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1975).

Viera, G.

Wahba, G.

G. Wahba, “Practical approximate solutions to linear operators equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[Crossref]

Appl. Opt. (5)

Inverse Probl. (2)

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Probl. 7, 793–808 (1991).
[Crossref]

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Probl. 2, 247 (1986).
[Crossref]

J. Atmos. Sci. (1)

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[Crossref]

Pure Appl. Geophys. (1)

G. E. Shaw, “Error analysis of multiwavelength Sun photometry,” Pure Appl. Geophys. 114, 1–13 (1976).
[Crossref]

SIAM J. Numer. Anal. (1)

G. Wahba, “Practical approximate solutions to linear operators equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[Crossref]

Other (7)

World Meteorological Organization, “A preliminary cloudless standard atmosphere for radiation computation,” Research Rep. 112 in the World Climate Programme Series, USA (World Meteorological Organization, Case Postale 2300, Geneva 20, Switzerland CH–1211, 1984).

G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics (Deepak, Hampton, Va., 1991).

E. P. Shettle, R. W. Fenn, “Models of atmospheric aerosols and their optical properties,” paper presented at the Electromagnetic Wave Propagation Panel on Advisory Group for Aerospace Research and Development, 22nd Technical Meeting, The Technical University of Denmark, Copenhagen, Denmark, 1975.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1975).

A. N. Thikonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

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

M. A. Lukas, “Regularization,” in The Application and Numerical Solution of Integral Equations, R. S. Anderssen, F. R. de Hoog, M. A. Lukas, eds. (Sijthoff and Noordhoff, The Netherlands, 1980).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Condition number γ of the parametric technique as a function of the number of distributions N.

Fig. 2
Fig. 2

Average error of retrieval I as a function of ε ̂ for the retrieval techniques considered in this paper for a maritime model.

Fig. 3
Fig. 3

Average error of retrieval I as a function of ε ̂ of the LSC retrieval technique for the aerosol models considered in this paper.

Fig. 4
Fig. 4

Results of the inversions on 28 June 1993 at Tito, Italy, obtained with the LSC method at 1000, 1400, and 1800 hours.

Fig. 5
Fig. 5

Optical depths measured at Tito, Italy, on 28 June 1993 at 1000 (dotted curve), 1400 (dashed curve), and 1800 (dotted-dashed curve) hours (15° Eastern Meridian Standard time) and the corresponding fitted regressions from the aerosol size distributions shown in Fig. 4.

Fig. 6
Fig. 6

Relative humidity versus time on 28 June 1993.

Fig. 7
Fig. 7

Set of log-normal functions chosen for the inversions.

Fig. 8
Fig. 8

Results of the inversions on 28 June 1993 as obtained through King's method2 at 1000, 1400, and 1800 hours.

Equations (40)

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+ K ( λ , log r ) n ( log r ) d log r = τ ( λ ) ,
n ε ( log r ) n ( log r ) 2 n ( log r ) 2 γ ( τ ε τ 2 τ 2 + K ε K 2 K 2 ) ,
min n ( log r ) L ( n ) , L ( n ) = i = 1 m [ + K ( λ i , log r ) n ( log r ) d log r τ ( λ i ) ] 2 + Λ S [ n ( log r ) ] ,
S [ n ( log r ) ] = + | d p n ( log r ) ( d log r ) p | 2 d log r , p = 0 , 1 , 2 , ,
S [ n ( log r ) ] = + n ( log r ) log n ( log r ) d log r .
n ( log r ) = i = 1 N n i ( log r ) ,
n ( log r ) = i = 1 N c i n i ( log r ) ,
min c 1 , , c N L ( n ) , L ( n ) = i = 1 m [ + j = 1 N K j ( λ i , log r ) c j n j ( log r ) d log r τ ( λ i ) ] 2 + Λ S [ j = 1 N c j n j ( log r ) ] .
S [ j = 1 N c j n j ( log r ) ] = + [ j = 1 N c j n j ( p ) ( log r ) ] 2 d log r ,
n j ( p ) ( log r ) = d p n j ( log r ) d log p r .
i = 1 N c i { l = 1 m [ + K i ( λ l , log r ) n i ( log r ) d log r × + K j ( λ l , log r ) n j ( log r ) d log r ] + Λ + n i ( p ) ( log r ) n j ( p ) ( log r ) d log r } = l = 1 m τ j + K j ( λ l , log r ) n j ( log r ) d log r , j = 1 , , N .
( B ) i j = + K j ( λ i , log r ) n j ( log r ) d log r ,
[ N ( p ) ] i j = + n i ( p ) ( log r ) n j ( p ) ( log r ) d log r ,
[ B T B + Λ N ( p ) ] c = B T τ ,
S ( n ) = + [ j = 1 N c j n j ( log r ) ] log [ j = 1 N c j n j ( log r ) ] d log r .
V ( Λ ) = 1 m [ I M ( Λ ) ] τ 2 2 { 1 m Tr [ I M ( Λ ) ] } 2 ,
M ( Λ ) = B [ B T B + Λ N ( p ) ] 1 B T
( B T B + Λ S T S ) c = B T τ ,
S T [ ( S T ) 1 B T B S 1 + Λ I ] S c = B T τ ,
B S 1 = G , S c = d
( G T G + Λ I ) d = G T τ ,
M ( Λ ) = G ( G T G + Λ I ) 1 G T .
G = U D V T ,
M ( Λ ) = U D ( D T D + Λ I ) 1 D T U T ,
V ( Λ ) = 1 m [ i = 1 N ( Λ h i σ i 2 + Λ ) 2 + i = N + 1 m h i 2 ] 1 m 2 ( m i = 1 N σ i 2 σ i 2 + Λ ) ,
( S c λ ) i = k = 1 m τ k j = 1 N u i j u k j σ j 2 σ j 2 + Λ .
n 1 ( log r ) = N 1 log σ 1 2 π exp [ ( log r log r 1 ) 2 2 ( log σ 1 ) 2 ] ,
n 2 ( log r ) = N 2 log σ 2 2 π exp [ ( log r log r 2 ) 2 2 ( log σ 2 ) 2 ] ,
n 3 ( log r ) = N 3 log σ 3 2 π exp [ ( log r ) log r 3 ) 2 2 ( log σ 3 ) 2 ] ,
n 4 ( log r ) = N 4 A 4 r 2 exp ( b 4 r ) ,
n 5 ( log r ) = N 5 log σ 5 2 π exp [ ( log r log r 5 ) 2 2 ( log σ 5 ) 2 ] ,
n 6 ( log r ) = N 6 A 6 r 2 exp ( b 6 r ) ,
I = { + [ n ( log r ) n test ( log r ) ] 2 d log r + [ n test ( log r ) ] 2 d log r } 1 / 2 ,
ε ̂ = 100 σ τ ¯ m ,
i = 1 N c i { 2 j = 1 m [ + K i ( λ j , r ) n i ( log r ) d log r + K h ( λ j , r ) n h ( log r ) d log r , ] } + Λ + n h ( log r ) log [ l = 1 N c l n l ( log r ) ] d log r = 2 j = 1 m τ j + K h ( λ j , r ) n h ( log r ) d log r + Λ + n h ( log r ) d log r , h = 1 , , N .
[ B T B + Λ I ( c ( v ) ) ] c ( v + 1 ) = B T τ Λ I ( c ( v ) ) , v 1 ,
[ I ( c ) ] i = + n i ( log r ) log [ i = 1 N c j n j ( log r ) ] d log r ,
[ I ( c ) ] i j = + n i ( log r ) n j ( log r ) k = 1 N c k n k ( log r ) d log r .
( W ) r , s 2 S ( n ) c r c s = 2 j = 1 N + K r ( λ j , log r ) n r ( log r ) d log r × + K s ( λ j , log r ) n s ( log r ) d log r + Λ + n r ( log r ) n s ( log r ) l = 1 n c l n l ( log r ) d log r .
γ T W γ = 2 j = 1 N h = 1 m [ + K h ( λ j , log r ) γ r n r ( log r ) d log r ] 2 + Λ + h = 1 m γ h n h ( log r ) h = 1 m c h n h ( log r ) d log r > 0

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