Abstract

A modified version of the nonlinear iterative Chahine algorithm is presented and applied to the inversion of spectral extinction data for particle sizing. Simulated data were generated in a λ range of 0.2–2 μm, and particle-size distributions were recovered with radii in the range of 0.14–1.4 μm. Our results show that distributions and sample concentrations can be recovered to a high degree of accuracy when the indices of refraction of the sample and of the solvent are known. The inversion method needs no a priori assumptions and no constraints on the particle distributions. Compared with the algorithm originally proposed by Chahine, our method is much more stable with respect to random noise, permits a better quality of the retrieved distributions, and improves the overall reliability of the fitting. The accuracy and resolution of the method as functions of noise were investigated and showed that the retrieved distributions are quite reliable up to noise levels of several rms percent in the data. The sensitivity to errors in the real and imaginary parts of the refraction index of the particles was also examined.

© 1995 Optical Society of America

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 7.
  2. G. Gouesbet, G. Gréhan, eds., Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
  3. E. D. Hirleman, ed., Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State University Printing Services, Tempe, Ariz., 1990), pp. 169–435.
  4. M. Maeda, S. Nakae, M. Ikegami, eds., Proceedings of the Third International Congress on Optical Particle Sizing (n.p., 1993).
  5. S. Twomey, H. B. Howell, “Some aspects of the optical estimation of microstructure in fog and cloud,” Appl. Opt. 6, 2125–2131 (1967).
    [CrossRef] [PubMed]
  6. A. Ångström, “On the atmospheric transmission of sun radiation and on dust in the air,” Geogr. Ann. 11, 156–166 (1929).
    [CrossRef]
  7. J. A. Curcio, “Evaluation of atmospheric aerosol particle size distribution from scattering measurements in the visible and infrared,” J. Opt. Soc. Am. 51, 548–551 (1961).
    [CrossRef]
  8. S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
    [CrossRef]
  9. S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, No. 2, 95–109 (1965).
    [CrossRef]
  10. E. E. Uthe, “Particle-size evaluations using multiwavelength extinction measurements,” Appl. Opt. 21, 454–459 (1982).
    [CrossRef] [PubMed]
  11. R. L. Zollars, “Turbidimetric method for online determination of latex particle number and particle-size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
    [CrossRef]
  12. H. Quenzel, “Determination of size distribution of atmospheric aerosol particles from spectral solar radiation measurements,” J. Geophys. Res. 75, 2915–2921 (1970).
    [CrossRef]
  13. G. E. Shaw, “Intercomparison of equatorial and polar multiwavelength atmospheric optical depths and sky radiance,” Bull. Am. Meteorol. Soc. 54, 1073–1080 (1973).
  14. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  15. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
    [CrossRef]
  16. G. E. Backus, J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problems,” Geophys. J. R. Astron. Soc. 13, 247–276 (1967).
    [CrossRef]
  17. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977), Chap. 7, p. 179.
  18. N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
    [CrossRef]
  19. N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
    [CrossRef]
  20. G. Yamamoto, M. Tanaka, “Determination of aerosol size distribution by spectral attenuation measurements,” Appl. Opt. 8, 447–453 (1969).
    [CrossRef] [PubMed]
  21. M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
    [CrossRef]
  22. P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
    [CrossRef] [PubMed]
  23. J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  24. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
    [CrossRef] [PubMed]
  25. M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
    [CrossRef]
  26. M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Ref. 2, pp. 55–61.
  27. G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
    [CrossRef]
  28. M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
    [CrossRef]
  29. M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
    [CrossRef]
  30. H. Grassl, “Determination of aerosol size distributions from spectral attenuation measurements,” Appl. Opt. 10, 2534–2538 (1971).
    [CrossRef] [PubMed]
  31. R. Santer, M. Herman, “Particle size distribution from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294–2301 (1983).
    [CrossRef] [PubMed]
  32. F. Ferri, M. Giglio, U. Perini, “Inversion of light scattered data from fractals by means of the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
    [CrossRef] [PubMed]
  33. A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
    [CrossRef]
  34. E. Trakhovsky, S. G. Lipson, A. D. Devir, “Atmospheric aerosols investigated by inversion of experimental transmittance data,” Appl. Opt. 21, 3005–3010 (1982).
    [CrossRef] [PubMed]
  35. J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
  36. C. F. Bohren, D. R. Hufman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 77.
  37. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.
  38. W. H. Richardson, “Bayesian based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  39. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]

1992 (2)

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

1989 (1)

1986 (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

1985 (1)

1983 (1)

1982 (3)

1980 (2)

R. L. Zollars, “Turbidimetric method for online determination of latex particle number and particle-size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
[CrossRef] [PubMed]

1979 (2)

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

1978 (2)

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

1975 (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1973 (1)

G. E. Shaw, “Intercomparison of equatorial and polar multiwavelength atmospheric optical depths and sky radiance,” Bull. Am. Meteorol. Soc. 54, 1073–1080 (1973).

1972 (1)

1971 (1)

1970 (2)

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

H. Quenzel, “Determination of size distribution of atmospheric aerosol particles from spectral solar radiation measurements,” J. Geophys. Res. 75, 2915–2921 (1970).
[CrossRef]

1969 (1)

1968 (1)

1967 (2)

G. E. Backus, J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problems,” Geophys. J. R. Astron. Soc. 13, 247–276 (1967).
[CrossRef]

S. Twomey, H. B. Howell, “Some aspects of the optical estimation of microstructure in fog and cloud,” Appl. Opt. 6, 2125–2131 (1967).
[CrossRef] [PubMed]

1965 (1)

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, No. 2, 95–109 (1965).
[CrossRef]

1963 (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1962 (1)

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

1961 (1)

1929 (1)

A. Ångström, “On the atmospheric transmission of sun radiation and on dust in the air,” Geogr. Ann. 11, 156–166 (1929).
[CrossRef]

Ångström, A.

A. Ångström, “On the atmospheric transmission of sun radiation and on dust in the air,” Geogr. Ann. 11, 156–166 (1929).
[CrossRef]

Backus, G. E.

G. E. Backus, J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problems,” Geophys. J. R. Astron. Soc. 13, 247–276 (1967).
[CrossRef]

Bassini, A.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Ref. 2, pp. 55–61.

Bohren, C. F.

C. F. Bohren, D. R. Hufman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 77.

Box, G. P.

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

Box, M. A.

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
[CrossRef] [PubMed]

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Chahine, M. T.

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

Crump, J. G.

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).

Curcio, J. A.

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Ref. 2, pp. 55–61.

Devir, A. D.

Ferri, F.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

F. Ferri, M. Giglio, U. Perini, “Inversion of light scattered data from fractals by means of the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
[CrossRef] [PubMed]

Giglio, M.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

F. Ferri, M. Giglio, U. Perini, “Inversion of light scattered data from fractals by means of the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
[CrossRef] [PubMed]

Gilbert, J. F.

G. E. Backus, J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problems,” Geophys. J. R. Astron. Soc. 13, 247–276 (1967).
[CrossRef]

Grassl, H.

Herman, B. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Herman, M.

Howell, H. B.

Hufman, D. R.

C. F. Bohren, D. R. Hufman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 77.

Joseph, J. H.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 7.

King, M. D.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Lipson, S. G.

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

McWhirter, J. G.

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Mekler, Y.

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

Musazzi, S.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

Paganini, E.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

Perini, U.

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

F. Ferri, M. Giglio, U. Perini, “Inversion of light scattered data from fractals by means of the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
[CrossRef] [PubMed]

Phillips, D. L.

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

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Ref. 2, pp. 55–61.

Quenzel, H.

H. Quenzel, “Determination of size distribution of atmospheric aerosol particles from spectral solar radiation measurements,” J. Geophys. Res. 75, 2915–2921 (1970).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Richardson, W. H.

Santer, R.

Sealey, K. M.

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

Seinfeld, J. H.

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).

Shaw, G. E.

G. E. Shaw, “Intercomparison of equatorial and polar multiwavelength atmospheric optical depths and sky radiance,” Bull. Am. Meteorol. Soc. 54, 1073–1080 (1973).

Tanaka, M.

Trakhovsky, E.

Twomey, S.

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

S. Twomey, H. B. Howell, “Some aspects of the optical estimation of microstructure in fog and cloud,” Appl. Opt. 6, 2125–2131 (1967).
[CrossRef] [PubMed]

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, No. 2, 95–109 (1965).
[CrossRef]

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977), Chap. 7, p. 179.

Uthe, E. E.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.

Viera, G.

Walters, P. T.

Wolfson, N.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

Yamamoto, G.

Zollars, R. L.

R. L. Zollars, “Turbidimetric method for online determination of latex particle number and particle-size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

Aerosol Sci. Technol. (1)

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).

Appl. Opt. (9)

S. Twomey, H. B. Howell, “Some aspects of the optical estimation of microstructure in fog and cloud,” Appl. Opt. 6, 2125–2131 (1967).
[CrossRef] [PubMed]

G. Yamamoto, M. Tanaka, “Determination of aerosol size distribution by spectral attenuation measurements,” Appl. Opt. 8, 447–453 (1969).
[CrossRef] [PubMed]

P. T. Walters, “Practical applications of inverting spectral turbidity data to provide aerosol size distributions,” Appl. Opt. 19, 2353–2365 (1980).
[CrossRef] [PubMed]

E. E. Uthe, “Particle-size evaluations using multiwavelength extinction measurements,” Appl. Opt. 21, 454–459 (1982).
[CrossRef] [PubMed]

E. Trakhovsky, S. G. Lipson, A. D. Devir, “Atmospheric aerosols investigated by inversion of experimental transmittance data,” Appl. Opt. 21, 3005–3010 (1982).
[CrossRef] [PubMed]

R. Santer, M. Herman, “Particle size distribution from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294–2301 (1983).
[CrossRef] [PubMed]

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
[CrossRef] [PubMed]

F. Ferri, M. Giglio, U. Perini, “Inversion of light scattered data from fractals by means of the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
[CrossRef] [PubMed]

H. Grassl, “Determination of aerosol size distributions from spectral attenuation measurements,” Appl. Opt. 10, 2534–2538 (1971).
[CrossRef] [PubMed]

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

G. E. Shaw, “Intercomparison of equatorial and polar multiwavelength atmospheric optical depths and sky radiance,” Bull. Am. Meteorol. Soc. 54, 1073–1080 (1973).

Geogr. Ann. (1)

A. Ångström, “On the atmospheric transmission of sun radiation and on dust in the air,” Geogr. Ann. 11, 156–166 (1929).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

G. E. Backus, J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problems,” Geophys. J. R. Astron. Soc. 13, 247–276 (1967).
[CrossRef]

Inverse Problems (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distributions from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

J. Appl. Meteorol. (2)

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative study of inversion techniques. Part I: Accuracy and stability,” J. Appl. Meteorol. 18, 543–555 (1979).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion techniques. Part II: Resolving power, conservation of normalization and superposition principles,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

J. Assoc. Comput. Mach. (2)

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

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

J. Atmos. Sci. (3)

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

M. T. Chahine, “Inverse problems in radiative transfer: determination of atmospheric parameters,” J. Atmos. Sci. 27, 960–967 (1970).
[CrossRef]

J. Colloid Interface Sci. (1)

R. L. Zollars, “Turbidimetric method for online determination of latex particle number and particle-size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, No. 2, 95–109 (1965).
[CrossRef]

J. Geophys. Res. (1)

H. Quenzel, “Determination of size distribution of atmospheric aerosol particles from spectral solar radiation measurements,” J. Geophys. Res. 75, 2915–2921 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A (1)

J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Opt. Eng. (1)

A. Bassini, S. Musazzi, E. Paganini, U. Perini, F. Ferri, M. Giglio, “Optical particle sizer based on the Chahine inversion scheme,” Opt. Eng. 31, 1112–1117 (1992).
[CrossRef]

Other (8)

C. F. Bohren, D. R. Hufman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 77.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 9, p. 127.

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Ref. 2, pp. 55–61.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977), Chap. 7, p. 179.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 7.

G. Gouesbet, G. Gréhan, eds., Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).

E. D. Hirleman, ed., Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State University Printing Services, Tempe, Ariz., 1990), pp. 169–435.

M. Maeda, S. Nakae, M. Ikegami, eds., Proceedings of the Third International Congress on Optical Particle Sizing (n.p., 1993).

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Figures (12)

Fig. 1
Fig. 1

Behavior of the extinction efficiency Q ext(x, m) as a function of the adimensional parameter x = 2πr/λ for different values of the relative refraction index of the particle m.

Fig. 2
Fig. 2

Results of our inversion method in the case of a monodisperse input distribution with r inp = 0.5 μm: (a) Output distributions for the noiseless case (solid lines) and for 10 different samples of 3% rms noise (dotted lines). (b) Input signals (circles) and output signals corresponding to the noiseless case (solid curve).

Fig. 3
Fig. 3

Results of our inversion method in the case of a Gaussian input distribution characterized by r inp = 0.5 μm and σinp = 0.1 μm, c inp = 106 cm−3: (a) Input distribution (solid curve), output distribution for the noiseless case (solid lines), and output distributions for 10 different samples of 3% rms noise (dotted lines). (b) Input signals (circles) and output signals corresponding to the noiseless case (solid curve).

Fig. 4
Fig. 4

Results of the classical Chahine inversion method for the same input distribution of Fig. 3: input distribution (solid curve), output distribution for the noiseless case (solid lines), and a typical output distribution for a 3% rms noise (dotted lines).

Fig. 5
Fig. 5

Comparison between output and input parameters characterizing the same Gaussian distribution shown in Fig. 3: (a) Behavior of the ratios 〈r out〉/r inp, 〈σout〉/σinp, and 〈c out〉/c inp as a function of the noise level. The error bars show the spread of the results obtained over 100 different samples of noise. The lines through the symbols are guides to the eye. (b) Behavior of the stopping value of the 〈rme〉 as a function of the noise level. The line corresponds to 〈rme〉 = rms noise.

Fig. 6
Fig. 6

Behavior of the output ratio 〈σout〉/〈r out〉 as a function of the input ratio σinp/r inp for different levels of noise. Input distributions were Gaussians with r inp = 0.5 μm and σinp varying between 0 and 0.2 μm. The error bars indicate the spread of the different values obtained over 100 different samples of noise.

Fig. 7
Fig. 7

Results of our inversion method in the case of a bi-Gaussian input distribution with the two peaks located near the minimum resolved distance. The two Gaussians are characterized by r inp a = 0.4 μm, σinp a = 0.02 μm, c inp a = 0.5 × 106 cm−3 and by r inp b = 0.6 μm, σinp b = 0.03 μm, c inp b = 0.5 × 106 cm−3. Three percent rms noise was added to the input signals. The input distribution (solid curve), output distribution for the noiseless case (solid lines), and output distributions for 10 different samples of noise (dotted lines) are shown.

Fig. 8
Fig. 8

Results of our inversion method in the case of a tri-Gaussian input distribution characterized by r inp a = 0.3 μm and σinp = 0.03 μm, r inp b = 0.6 μm and σinp b = 0.06 μm, and r inp c = 1.2 μm and σinp c = 0.12 μm. The concentrations were all the same equal to c inp = 0.33 × 106 cm−3. One percent rms noise was added to the input data. The input distribution (solid curve), output distribution for the noiseless case (solid lines), and output distributions for 10 different samples of noise (dotted lines) are shown.

Fig. 9
Fig. 9

Results of our inversion method when Re{m out} ≠ Re{m inp}. The input distribution is a Gaussian with r inp = 0.5 μm, σinp = 0.05 μm, c inp = 106 cm−3, and m inp = 1.50 (solid curve). We retrieve the output distributions supposing that m out = 1.45 (dashed lines), m out = 1.50 (solid lines), and m out = 1.55 (dotted lines). All the tests were done with no noise added to the input signals.

Fig. 10
Fig. 10

Results of our inversion method when Im{m out} ≠ Im{m inp}. The input distribution is a Gaussian with r inp = 0.5 μm, σinp = 0.05 μm, c inp = 106 cm−3, and m inp = 1.50. (a) Input distribution (solid curve) and output distribution retrieved if m out = 1.50 + 0.05i is supposed. (b) Input signals (circles) and output signals (solid curve). The test was done with no noise added to the input signals.

Fig. 11
Fig. 11

Behavior of the ratios r out/r inp, σoutinp, and c out/c inp as a function of Re{m out}. The input distribution was the same as in Fig. 9 with m inp = 1.50. All the tests were done with no noise added to the input data. The lines through the symbols are guides to the eye.

Fig. 12
Fig. 12

Behavior of the ratios r out/r inp, σoutinp, and c out/c inp as a function of Im{m out}. The input distribution was the same as in Fig. 9 with m inp = 1.50. All the tests were done with no noise added to the input data. The lines through the symbols are guides to the eye.

Tables (2)

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Table 1 Output Parameters Corresponding to Three Different Monodisperse Input Distributionsa

Tables Icon

Table 2 Comparison between the Classical Chahine Method and our Method for a Gaussian Input Distribution Characterized by r inp = 0.5 μm, σinp = 0.1 μm, and c inp = 1.00 × 106 cm−3a

Equations (11)

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P T = P 0 exp [ - α ( λ ) L ] ,
α ( λ ) = π r 2 Q ext ( r , λ , m ) N ( r ) d r ,
α ( λ i ) = j N j A i j ,             j = 1 , 2 , , q ,
A i j = r j - 1 r j π r 2 Q ext ( r , λ i , m ) d r ,
r k = r min a k ,             k = 0 , 1 , , q ,
R j = x peak 2 π λ j ,             j = 1 , 2 , , q ,
λ j = λ min a j - 1 .
N j p + 1 = N j p α meas ( λ j ) α calc p ( λ j ) .
N j p + 1 = N j p i = 1 q W i j α meas ( λ i ) α calc p ( λ i ) ,
W i j = A i j i A i j .
rme = { 1 q i = 1 q [ α meas ( λ i ) - α calc ( λ i ) ] 2 [ α calc ( λ i ) ] 2 } 1 / 2 .

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