Abstract

This paper presents two conclusions concerning the scattered light distribution method for particle sizing: (1) a large difference in vector values is one of the primary causes of an ill-conditioned kernel matrix, and (2) under narrow band and high resolution particle size conditions, the forward scattering region would introduce a larger error into the particle size distribution. Numerical simulations verify these results.

© 2012 Optical Society of America

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  1. F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).
  2. I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
    [CrossRef]
  3. X. S. Cai and M. X. Su, Particle Sizing Technology and the Application (Chemical Industry, 2010).
  4. Y. F. Wang, “An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function,” J. Aerosol Sci. 39, 305–322 (2008).
    [CrossRef]
  5. J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
    [CrossRef]
  6. D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
    [CrossRef]
  7. Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
    [CrossRef]
  8. S. Lekhtmakher and M. Shapiro, “About randomness of aerosol size distributions,” J. Aerosol Sci. 36, 1459–1467 (2005).
    [CrossRef]
  9. X. S. Lu, Z. D. Feng, and J. M. Liu, Analysis Theory of Ill-Conditioned System with Application in Surveying (Publishing House of Surveying and Mapping, 2007).
  10. D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
    [CrossRef]
  11. H. Y. Zuo and J. G. Yang, “Retrieving of aerosol size distribution based on the measurement of aerosol optical depth,” Acta Phys. Sin-Ch. Ed. 56, 6132–6136 (2007).
  12. X. Q. He and W. Q. Liu, Application of the Regression Analysis (China Renmin, 2001).
  13. D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).
  14. C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
    [CrossRef]

2010 (1)

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

2008 (2)

Y. F. Wang, “An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function,” J. Aerosol Sci. 39, 305–322 (2008).
[CrossRef]

J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
[CrossRef]

2007 (3)

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[CrossRef]

H. Y. Zuo and J. G. Yang, “Retrieving of aerosol size distribution based on the measurement of aerosol optical depth,” Acta Phys. Sin-Ch. Ed. 56, 6132–6136 (2007).

D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).

2005 (1)

S. Lekhtmakher and M. Shapiro, “About randomness of aerosol size distributions,” J. Aerosol Sci. 36, 1459–1467 (2005).
[CrossRef]

2004 (3)

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).

I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
[CrossRef]

1995 (1)

D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Abraham, A.

D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).

Blumthaler, M.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

Buras, R.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

Cai, X. S.

F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).

X. S. Cai and M. X. Su, Particle Sizing Technology and the Application (Chemical Industry, 2010).

Chachisvilis, M.

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

Cho, J. H.

D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).

Cunningham, A.

I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
[CrossRef]

Diver, J.

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

Elliott, L.

D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Emde, C.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

Fan, S. F.

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[CrossRef]

Feng, X.

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[CrossRef]

Feng, Z. D.

X. S. Lu, Z. D. Feng, and J. M. Liu, Analysis Theory of Ill-Conditioned System with Application in Surveying (Publishing House of Surveying and Mapping, 2007).

Hagen, N.

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

He, X. Q.

X. Q. He and W. Q. Liu, Application of the Regression Analysis (China Renmin, 2001).

Ingham, D. B.

D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Kim, D. H.

D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).

Kulkarni, P.

J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
[CrossRef]

Lekhtmakher, S.

S. Lekhtmakher and M. Shapiro, “About randomness of aerosol size distributions,” J. Aerosol Sci. 36, 1459–1467 (2005).
[CrossRef]

Lesnic, D.

D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Liu, J. M.

X. S. Lu, Z. D. Feng, and J. M. Liu, Analysis Theory of Ill-Conditioned System with Application in Surveying (Publishing House of Surveying and Mapping, 2007).

Liu, W. Q.

X. Q. He and W. Q. Liu, Application of the Regression Analysis (China Renmin, 2001).

Lu, X. S.

X. S. Lu, Z. D. Feng, and J. M. Liu, Analysis Theory of Ill-Conditioned System with Application in Surveying (Publishing House of Surveying and Mapping, 2007).

MacCallum, I.

I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
[CrossRef]

Marchand, P.

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

Mayer, B.

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

McKee, D.

I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
[CrossRef]

Olfert, J. S.

J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
[CrossRef]

Shapiro, M.

S. Lekhtmakher and M. Shapiro, “About randomness of aerosol size distributions,” J. Aerosol Sci. 36, 1459–1467 (2005).
[CrossRef]

Su, M. X.

F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).

X. S. Cai and M. X. Su, Particle Sizing Technology and the Application (Chemical Industry, 2010).

Wang, J.

J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
[CrossRef]

Wang, Y. F.

Y. F. Wang, “An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function,” J. Aerosol Sci. 39, 305–322 (2008).
[CrossRef]

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[CrossRef]

Watson, D.

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

Xu, F.

F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).

Yang, J. G.

H. Y. Zuo and J. G. Yang, “Retrieving of aerosol size distribution based on the measurement of aerosol optical depth,” Acta Phys. Sin-Ch. Ed. 56, 6132–6136 (2007).

Zuo, H. Y.

H. Y. Zuo and J. G. Yang, “Retrieving of aerosol size distribution based on the measurement of aerosol optical depth,” Acta Phys. Sin-Ch. Ed. 56, 6132–6136 (2007).

Acta Phys. Sin-Ch. Ed. (1)

H. Y. Zuo and J. G. Yang, “Retrieving of aerosol size distribution based on the measurement of aerosol optical depth,” Acta Phys. Sin-Ch. Ed. 56, 6132–6136 (2007).

Atmos. Chem. Phys. (1)

C. Emde, R. Buras, B. Mayer, and M. Blumthaler, “The impact of aerosols on polarized sky radiance: model development, validation, and applications,” Atmos. Chem. Phys. 10, 383–396 (2010).
[CrossRef]

Biophys. J. (1)

D. Watson, N. Hagen, J. Diver, P. Marchand, and M. Chachisvilis, “Elastic light scattering from single cells: orientational dynamics in optical trap,” Biophys. J. 87, 1298–1306 (2004).
[CrossRef]

Chin. J. Laser (1)

F. Xu, X. S. Cai, and M. X. Su, “Study of independent model algorithm for determination of particle size distribution,” Chin. J. Laser 31, 223–228 (2004).

Inform. Sci. (1)

D. H. Kim, A. Abraham, and J. H. Cho, “A hybrid genetic algorithm and bacterial foraging approach for global optimization,” Inform. Sci. 177, 3918–3937 (2007).

J. Aerosol Sci. (5)

Y. F. Wang, “An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function,” J. Aerosol Sci. 39, 305–322 (2008).
[CrossRef]

J. S. Olfert, P. Kulkarni, and J. Wang, “Measuring aerosol size distributions with the fast integrated mobility spectrometer,” J. Aerosol Sci. 39, 940–956 (2008).
[CrossRef]

D. Lesnic, L. Elliott, and D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Y. F. Wang, S. F. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[CrossRef]

S. Lekhtmakher and M. Shapiro, “About randomness of aerosol size distributions,” J. Aerosol Sci. 36, 1459–1467 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

I. MacCallum, A. Cunningham, and D. McKee, “The measurement and modelling of light scattering by phytoplankton cells at narrow forward angles,” J. Opt. A: Pure Appl. Opt. 6, 698–702 (2004).
[CrossRef]

Other (3)

X. S. Cai and M. X. Su, Particle Sizing Technology and the Application (Chemical Industry, 2010).

X. Q. He and W. Q. Liu, Application of the Regression Analysis (China Renmin, 2001).

X. S. Lu, Z. D. Feng, and J. M. Liu, Analysis Theory of Ill-Conditioned System with Application in Surveying (Publishing House of Surveying and Mapping, 2007).

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Figures (11)

Fig. 1.
Fig. 1.

Diagram depicting the scattering of particles.

Fig. 2.
Fig. 2.

Size-distribution (C0.2C2.0C4.0=131) and the corresponding F0.6(θ) plot. (a) Size distribution. (b) Corresponding F0.6(θ) plot.

Fig. 3.
Fig. 3.

Size-distribution (C0.2C2.0C4.0=132) and the corresponding F0.6(θ) plot. (a) Size distribution. (b) Corresponding F0.6(θ) plot (rings line).

Fig. 4.
Fig. 4.

Size-distribution (C0.2C2.0C4.0=231) and the corresponding F0.6(θ) plot. (a) Size distribution. (b) Corresponding F0.6(θ) plot (rings line).

Fig. 5.
Fig. 5.

Curves of the column vectors of the kernel matrix.

Fig. 6.
Fig. 6.

Basic structure of a particle size analyzer.

Fig. 7.
Fig. 7.

Curve of R0.6(θ).

Fig. 8.
Fig. 8.

Standard size distributions: (a) Junge distribution, (b) bimodal distribution, (c) triple peak distribution, and (d) four-peaks distribution.

Fig. 9.
Fig. 9.

Correlation coefficients between the retrieval results and the standard size distributions (sorted in ascending order): (a) Junge distribution, (b) bimodal distribution, (c) triple-peak distribution, and (d) four-peaks distribution.

Fig. 10.
Fig. 10.

Optimized angles (every dot represents an angle and the ordinate has no real meaning).

Fig. 11.
Fig. 11.

Correlation coefficients between the retrieval results and the standard size distributions (sorted in ascending order): (a) Junge distribution, (b) bimodal distribution, (c) triple-peak distribution, and (d) four-peaks distribution.

Tables (1)

Tables Icon

Table 1. Comparison of the CNs of the Kernel Matrix for Different Angle Rangesa

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

FΛinp(θ)=rminrmaxPΛinp(r,θ)n(r)dr,
(FΛinp(θ1)FΛinp(θ2)FΛinp(θs))=(PΛinp(r1,θ1)PΛinp(r2,θ1)PΛinp(rM,θ1)PΛinp(r1,θ2)PΛinp(r2,θ2)PΛinp(rM,θ2)PΛinp(r1,θs)PΛinp(r2,θs)PΛinp(rM,θs))(n1n2nM).
R0.6(θ)=lgP0.6(5,θ)/P0.6(0.1,θ),
Fitness=min{cond(P)},

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