Computational Methods for the Estimation of the Aerosol Size Distributions

  • A. Voutilainen
  • V. Kolehmainen
  • F. Stratmann
  • J. P. Kaipio


Aerosol particles play an important role in many physical and chemical processes in the atmosphere1. Physical and chemical behaviour of aerosol particles is strongly dependent on particle size and thus the size cannot be ignored in the evaluation and theoretical prediction of the effects caused by airborne particles. Since the particle diameter d p can range from few nanometers to about 100 micrometers, a size distribution function is used to describe how certain property, e.g. number, surface area or mass, of particles per unit gas volume is distributed on different particle sizes. The determination of the size distribution function is a very important fundamental task in aerosol research. However, the size distribution cannot be measured directly but it has to be reconstructed on the basis of indirect observations using computational methods. From the mathematical point of view the determination of the size distribution function is an ill-posed problem since the problem does not have a unique solution. The purpose of this chapter is to describe the problem and give a brief review on some computational methods proposed for the reconstruction of particle size distributions.


Markov Chain Monte Carlo Posterior Density Tikhonov Regularization Observation Error Markov Chain Monte Carlo Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. J.H. Seinfeld and S.N. Pendis, Atmospheric Physics and Chemistry. From Air Pollution to Climate Change (John Wiley & Sons, 1998).Google Scholar
  2. 2.
    M. Kandlikar and G. Ramachandran, Inverse methods for analysing aerosol spectrometer measurements: a critical review, J Aerosol Sci. 30, 413–437 (1999).CrossRefGoogle Scholar
  3. 3.
    T.G. Dzubay and H. Hasan, Fitting multimodal lognormal size distributions to cascade impactor data, Aerosol Sci. Tech. 13, 144–150 (1990).CrossRefGoogle Scholar
  4. 4.
    O.G. Raabe, A general method for fitting size distributions to multicomponent aerosol data using weighted least-squares, Env. Sci. Tech. 12, 1162–1167 (1978).CrossRefGoogle Scholar
  5. 5.
    A. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996 ).Google Scholar
  6. 6.
    U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, and F. Romano, Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution, J. Geophys. Res. D 101, 19267–19275 (1996).CrossRefGoogle Scholar
  7. 7.
    J.K. Wolfenbarger and J.H. Seinfeld, Inversion of aerosol size distribution data, J. Aerosol Sci. 21, 227–247 (1990).CrossRefGoogle Scholar
  8. 8.
    V.S. Bashurova, K.P. Koutzenogil, A.Y. Pusep, and N.V. Shokhirev, Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects, J. Aerosol Sci. 22, 373–388 (1991).CrossRefGoogle Scholar
  9. 9.
    U. Amato, M.R. Carfora, V. Cuomo, and C. Serio, Objective algorithms for the aerosol problem, Appl. Opt. 34, 5442–5452 (1995).Google Scholar
  10. 10.
    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).CrossRefGoogle Scholar
  11. 11.
    M.T. Chahine, Determination of the temperature profile in an atmosphere form its outgoing radiance, J. Opt. Soc. Am. 58, 1634–1637 (1968).CrossRefGoogle Scholar
  12. 12.
    G. Ramachandran and M. Kandlikar, Bayesian analysis for inversion of aerosol size distribution data, J. Aerosol Sci. 27, 1099–1112 (1996).CrossRefGoogle Scholar
  13. 13.
    E.F. Maher and N.M. Laird, EM algorithm reconstruction of particle size distributions from diffusion battery data, J. Aerosol Sci. 16, 557–570 (1985).CrossRefGoogle Scholar
  14. 14.
    P. Paatero, The Extreme Value Estimation Deconvolution Method with Applications in Aerosol Research, Technical Report No. HU-P-250, University of Helsinki, Department of Physics (1990).Google Scholar
  15. 15.
    A.Voutilainen, V. Kolehmainen, and J.P. Kaipio, Statistical inversion of aerosol size measurement data, Inv. Probl. Eng. (2001), in press.Google Scholar
  16. 16.
    A. Voutilainen, F. Stratmann, and J.P. Kaipio, A non-homogeneous regularization method for the estimation of narrow aerosol size distributions, J. Aerosol Sci. 31, 1433–1445 (2000).CrossRefGoogle Scholar
  17. 17.
    A. Voutilainen and J.P. Kaipio, Estimation of non-stationary aerosol size distributions using the state-space approach, J. Aerosol Sci. (2001), in press.Google Scholar
  18. 18.
    W. Winklmayr, G.P. Reischl, A.O. Lindner, and A. Berner, A new electromobility spectrometer for the measurement of aerosol size distributions in the size range from 1 to 1000 nm, J. Aerosol Sci. 22, 289–296 (1991).CrossRefGoogle Scholar
  19. 19.
    TSI Inc. (St. Paul, MN, USA, January 10, 2001 );
  20. 20.
    A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques (SIAM, 1990 ).CrossRefGoogle Scholar
  21. 21.
    W.R. Gilks, S. Rickhardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996 ).Google Scholar
  22. 22.
    C.K. Chui and G. Chen, Kalman Filtering (Springer-Verlag, 1987 ).Google Scholar
  23. 23.
    B.D.O Anderson and J.B. Moore, Optimal Filtering (Prentice-Hall, 1979 ).Google Scholar
  24. 24.
    J. Kaipio and E. Somersalo, Nonstationary inverse problems and state estimation, J. Inv. Ill-Posed Problems 7, 273–282 (1999).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • A. Voutilainen
    • 1
  • V. Kolehmainen
    • 1
  • F. Stratmann
    • 2
  • J. P. Kaipio
    • 1
  1. 1.Departament of Applied PhysicsUniversity of KuipioKuipioFinland
  2. 2.Institute for Tropospheric ResearchLeipzigGermany

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