Abstract

The feasibility of using a generalized stochastic inversion methodology to estimate aerosol size distributions accurately by use of spectral extinction, backscatter data, or both is examined. The stochastic method used, inverse Monte Carlo (IMC), is verified with both simulated and experimental data from aerosols composed of spherical dielectrics with a known refractive index. Various levels of noise are superimposed on the data such that the effect of noise on the stability and results of inversion can be determined. Computational results show that the application of the IMC technique to inversion of spectral extinction or backscatter data or both can produce good estimates of aerosol size distributions. Specifically, for inversions for which both spectral extinction and backscatter data are used, the IMC technique was extremely accurate in determining particle size distributions well outside the wavelength range. Also, the IMC inversion results proved to be stable and accurate even when the data had significant noise, with a signal-to-noise ratio of 3.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  2. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  3. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).
  4. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
    [CrossRef]
  5. O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
    [CrossRef]
  6. H. Scnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991); “Simultaneous determination of size distribution and refractive index of colloidal particles from static light-scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
    [CrossRef]
  7. D. A. Ligon, T. W. Chen, J. B. Gillespie, “Determination of aerosol parameters from light-scattering data using an inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
    [CrossRef] [PubMed]
  8. M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, Y. Xu, “Inversion of particle-size distribution from angular light-scattering data with genetic algorithms,” Appl. Opt. 38, 2677–2685 (1999).
    [CrossRef]
  9. D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999); “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
    [CrossRef]
  10. W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.
  11. P. Wang, G. S. Kent, M. P. McCormick, L. W. Thomason, G. K. Yue, “Retrieval analysis of aerosol-size distribution with simulated extinction measurements at SAGE III wavelengths,” Appl. Opt. 35, 433–440 (1996).
    [CrossRef] [PubMed]
  12. F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
    [CrossRef]
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

1999

1996

1991

1987

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

1980

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

1966

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

1955

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Agrawal, Y. C.

Ansmann, A.

Bohren, C. F.

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

Chen, T. W.

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Flannery, B. P.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.

Gillespie, J. B.

Glatter, O.

Hagan, F.

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

Hu, T.

Huffman, D. R.

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

Kent, G. S.

Ligon, D. A.

Lu, Y.

McCormick, M. P.

Müller, D.

Press, W. H.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.

Riley, J. B.

Scnablegger, H.

Shifrin, K. S.

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Teukolsky, S. A.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.

Thomason, L. W.

Tribus, M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Vetterling, W. T.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.

Wandinger, U.

Wang, P.

Wang, S.

Xu, Y.

Ye, M.

Yue, G. K.

Zhu, Z.

Appl. Opt.

J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
[CrossRef] [PubMed]

H. Scnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991); “Simultaneous determination of size distribution and refractive index of colloidal particles from static light-scattering experiments,” J. Colloid Interface Sci. 158, 228–242 (1993).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999); “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
[CrossRef]

M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, Y. Xu, “Inversion of particle-size distribution from angular light-scattering data with genetic algorithms,” Appl. Opt. 38, 2677–2685 (1999).
[CrossRef]

P. Wang, G. S. Kent, M. P. McCormick, L. W. Thomason, G. K. Yue, “Retrieval analysis of aerosol-size distribution with simulated extinction measurements at SAGE III wavelengths,” Appl. Opt. 35, 433–440 (1996).
[CrossRef] [PubMed]

D. A. Ligon, T. W. Chen, J. B. Gillespie, “Determination of aerosol parameters from light-scattering data using an inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
[CrossRef] [PubMed]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys.

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

J. Appl. Crystallogr.

O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” J. Appl. Crystallogr. 13, 7–11 (1980).
[CrossRef]

J. Comput. Phys.

F. Hagan, “On the construction of well-conditioned systems for Fredholm I problems by mesh-adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

J. Phys. Chem.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Part. Charact.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

Other

W. H. Press, W. T. Vetterling, S. A. Teukolsky, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, England, 1992), pp. 684–687.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Synthetically generated spectral extinction and backscatter for three log-normal distributions, defined as Region I {r m = 0.2 µm, σ m = 0.2, N 0 = 106 cm-3}, Region II {r m = 0.6 µm, σ m = 0.085, N 0 = 106 cm-3}, and Region III {r m = 1.0 µm, σ m = 0.05, N 0 = 106 cm-3}. (a) Spectral extinction coefficient, (b) spectral backscatter coefficient.

Fig. 2
Fig. 2

Comparison of exact number distribution and retrieved number distribution by the IMC method for the Region I aerosol with (a) extinction-only data, (b) backscatter-only data, (c) extinction and backscatter data.

Fig. 3
Fig. 3

Comparison of exact number distribution and retrieved number distribution by the IMC method for the Region II aerosol with (a) extinction-only data, (b) backscatter-only data, (c) extinction and backscatter data.

Fig. 4
Fig. 4

Comparison of exact number distribution and retrieved number distribution by the IMC method for the Region III aerosol with (a) extinction-only data, (b) backscatter-only data, (c) extinction and backscatter data.

Fig. 5
Fig. 5

Effects of noise on the retrieved number distribution by IMC method for the Region I aerosol by use of both extinction and backscatter data. The amount of noise used was (a) 1.0%, (b) 10.0%, and (c) 33.0% Gaussian noise.

Fig. 6
Fig. 6

Effects of noise on the retrieved number distribution by the IMC method for the Region II aerosol by use of both extinction and backscatter data. The amount of noise used was (a) 1.0%, (b) 10.0%, and (c) 33.0% Gaussian noise.

Fig. 7
Fig. 7

Effects of noise on the retrieved number distribution by the IMC method for the Region III aerosol by use of both extinction and backscatter data. The amount of noise used was (a) 1.0%, (b) 10.0%, and (c) 33.0% Gaussian noise.

Fig. 8
Fig. 8

Results of IMC inversion for a broadly distributed, bimodal distribution {mode 1, r m = 1.5 µm, σ m = 0.2, N 0 = 1.5 × 106 cm-3; mode 2, r m = 2.5 µm, σ m = 0.25, N 0 = 1. × 106 cm-3} by use of the spectral extinction and backscatter data over the wavelength range 0.45–1.85 µm for 50 evenly separated wavelengths. A noise of 5% was superimposed upon the synthetic data.

Fig. 9
Fig. 9

Results of inversion of spectral extinction measurements of polylatex hydrosol: (a) experimentally measured spectral scattering coefficient, (b) inversion results with the IMC. Stand. Dev., standard deviation.

Fig. 10
Fig. 10

Results of inversion of spectral extinction and backscatter for the Region II aerosol when the refractive index was determined by an iterative scheme.

Equations (16)

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

cλ=0 πr2Qext2πr/λdndrdr,
cλ=N00 Kextr, λgrdr,
cλ=N0i=1M K¯extri, λfri,
K¯extri, λ=1δrri-δr/2ri+δr/2 Kextr, λdr
bλ=N0i=1M K¯backri, λfri,
K¯backri, λ=1δrri-δr/2ri+δr/2 πr2Qback2πr/λdr
cj=N0i=1MK¯extjifi, bj=N0i=1MK¯backjifi.
cj=N0w0i=1MK¯extjifi, bj=N0w0i=1MK¯backjifi,
w0=0 wrgrdr
K¯extri, λ=1δrri-δr/2ri+δr/2Kextr, λwrdr
wr=4/3πr3.
cj0=1nwalknw=1nwalkK¯extjrnnw.
N02=1Jj=1Jcjcj02.
Χ˜2=1Jj=1Jcj-N0cj02σj2,
gr=12πrσmexp-lnr/rm22σm2,
K¯extri, λ=1δrri-δr/2ri+δr/2πr2Qextr, λ-ΩˆdσscatdΩdΩdr,

Metrics