Abstract

A search procedure based on a least-squares method including a regularization scheme constructed from numerical filtering is presented. This method, with the addition of a nephelometer, can be used to determine the particle-size distributions of various scattering media (aerosols, fogs, rocket exhausts, motor plumes) from angular static light-scattering measurements. For retrieval of the distribution function, the experimental data are matched with theoretical patterns derived from Mie theory. The method is numerically investigated with simulated data, and the performance of the inverse procedure is evaluated. The results show that the retrieved distribution function is quite reliable, even for strong levels of noise.

© 2000 Optical Society of America

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  1. N. G. Stanley-Woods, R. W. Lines, Particle Size Analysis (Royal Society of Chemistry, London, 1992).
    [CrossRef]
  2. H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
    [CrossRef]
  3. H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
    [CrossRef] [PubMed]
  4. A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
    [CrossRef]
  5. H. Jiang, J. Pierce, J. Kao, E. Sevick-Muruca, “Measurement of particle-size distribution and volume fraction in concentrated suspensions with photon migration techniques,” Appl. Opt. 36, 3310–3318 (1997).
    [CrossRef] [PubMed]
  6. N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
    [CrossRef]
  7. J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).
  8. F. Ferri, A. Bassini, E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995).
    [CrossRef] [PubMed]
  9. R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
    [CrossRef]
  10. G. Mie, “Beitrage zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  11. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  12. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  13. H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
    [CrossRef] [PubMed]
  14. M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
    [CrossRef]
  15. A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
    [CrossRef]
  16. F. Ferri, M. Giglio, U. Perini, “Inversion of light scattering data from fractals by the Chahine iterative algorithm,” Appl. Opt. 28, 3074–3082 (1989).
    [CrossRef] [PubMed]
  17. D. A. Ligon, T. W. Chen, J. B. Gillespie, “Determination of aerosol parameters from light scattering data using a inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
    [CrossRef] [PubMed]
  18. H. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).
  19. 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]
  20. G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using an analytical eingenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
    [CrossRef]
  21. M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 55–61.
    [CrossRef]
  22. M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Prob. 2, 247–258 (1986).
    [CrossRef]
  23. 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]
  24. M. Bertero, C. De Mol, G. A. Viano, “On the regularization of linear inverse problems in Fourier optics,” in Applied Inverse Problems, by P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1978), pp. 180–199.
  25. M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, by H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
  26. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).
  27. P. L. Butzer, S. Jansche, “A direct approach to the Mellin transform,” Math. Subj. Class. 45A15, 44-02–44-03 (1991).
  28. M. K. Atakishiyeva, N. M. Atakishiyev, “On the Mellin transforms of hypergeometric polynomials,” J. Phys. A 32, 33–41 (1999).
    [CrossRef]
  29. J. G. McWhirter, “A stabilized model-fitting approach to the processing of laser anemometry and the other photon correlation data,” Opt. Acta 27, 83–105 (1980).
    [CrossRef]
  30. H. M. Wadworth, ed., Handbook of Statistical Methods for Engineers and Scientists (McGraw-Hill, New York, 1990).
  31. C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
    [CrossRef]
  32. T. N. E. Greville, Theory and Applications of Spline Functions (Academic, New York, 1969).
  33. 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]
  34. 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]
  35. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  36. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  37. H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, New York, 1981).
  38. M. E. Essawy, A. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
    [CrossRef]
  39. J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.
  40. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  41. E. D. Hirleman, “Optical scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
    [CrossRef]

1999

M. K. Atakishiyeva, N. M. Atakishiyev, “On the Mellin transforms of hypergeometric polynomials,” J. Phys. A 32, 33–41 (1999).
[CrossRef]

1998

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

1997

1996

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

M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
[CrossRef]

1995

1992

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]

1991

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

P. L. Butzer, S. Jansche, “A direct approach to the Mellin transform,” Math. Subj. Class. 45A15, 44-02–44-03 (1991).

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

H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
[CrossRef] [PubMed]

1989

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

H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
[CrossRef]

1988

1987

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

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

1986

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

1985

1980

J. G. McWhirter, “A stabilized model-fitting approach to the processing of laser anemometry and the other photon correlation data,” Opt. Acta 27, 83–105 (1980).
[CrossRef]

M. E. Essawy, A. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

1978

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]

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
[CrossRef]

1977

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

1966

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

1908

G. Mie, “Beitrage zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

1895

H. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

Agrawal, Y. C.

Atakishiyev, N. M.

M. K. Atakishiyeva, N. M. Atakishiyev, “On the Mellin transforms of hypergeometric polynomials,” J. Phys. A 32, 33–41 (1999).
[CrossRef]

Atakishiyeva, M. K.

M. K. Atakishiyeva, N. M. Atakishiyev, “On the Mellin transforms of hypergeometric polynomials,” J. Phys. A 32, 33–41 (1999).
[CrossRef]

Bassini, A.

Bauckhage, K.

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

Beer, J. M.

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Ben-David, A.

Bertero, M.

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

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 55–61.
[CrossRef]

M. Bertero, C. De Mol, G. A. Viano, “On the regularization of linear inverse problems in Fourier optics,” in Applied Inverse Problems, by P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1978), pp. 180–199.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, by H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.

Bohren, C. F.

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

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 analytical eingenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
[CrossRef]

Brewster, M. Q.

M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
[CrossRef]

Butzer, P. L.

P. L. Butzer, S. Jansche, “A direct approach to the Mellin transform,” Math. Subj. Class. 45A15, 44-02–44-03 (1991).

Chae, S. K.

H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
[CrossRef]

Chen, T. W.

Corana, A.

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

De Boor, C.

C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

De Jaeger, N.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

De Mol, C.

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

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 55–61.
[CrossRef]

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, by H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.

M. Bertero, C. De Mol, G. A. Viano, “On the regularization of linear inverse problems in Fourier optics,” in Applied Inverse Problems, by P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1978), pp. 180–199.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Delfour, A.

M. E. Essawy, A. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Demeyere, H.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Doicu, A.

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

Essawy, M. E.

M. E. Essawy, A. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

Ferri, F.

Finsy, R.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Giglio, M.

Gillespie, J. B.

Glatter, O.

Greville, T. N. E.

T. N. E. Greville, Theory and Applications of Spline Functions (Academic, New York, 1969).

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Herman, B. M.

Hirleman, E. D.

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

Huffman, D. R.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
[CrossRef]

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

Hunt, A. J.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
[CrossRef]

Jansche, S.

P. L. Butzer, S. Jansche, “A direct approach to the Mellin transform,” Math. Subj. Class. 45A15, 44-02–44-03 (1991).

Jiang, H.

Jones, M. R.

M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
[CrossRef]

Kao, J.

Köser, J.

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

Kuentzmann, P.

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Lee, H. S.

H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
[CrossRef]

Ligon, D. A.

Lines, R. W.

N. G. Stanley-Woods, R. W. Lines, Particle Size Analysis (Royal Society of Chemistry, London, 1992).
[CrossRef]

Liu, B. Y. H.

H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
[CrossRef]

Marchesi, M.

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

Matini, C.

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

McCreath, C. G.

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

McWhirter, J. G.

J. G. McWhirter, “A stabilized model-fitting approach to the processing of laser anemometry and the other photon correlation data,” Opt. Acta 27, 83–105 (1980).
[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]

Mellin, H.

H. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Paganini, E.

Perini, U.

Perry, R. J.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
[CrossRef]

Pierce, J.

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Prob. 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 Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 55–61.
[CrossRef]

Prévost, M.

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Reagan, J. A.

Ridella, S.

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

Riley, J. B.

Schnablegger, H.

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]

Sevick-Muruca, E.

Shifrin, K. S.

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

Sneyers, K.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Stanley-Woods, N. G.

N. G. Stanley-Woods, R. W. Lines, Particle Size Analysis (Royal Society of Chemistry, London, 1992).
[CrossRef]

Swinthenbank, J.

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Tarrin, P.

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Taylor, D. S.

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Traineau, J.-C.

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

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, 2nd ed. (Dover, New York, 1981).

Van der Beelen, J.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Van der Meeren, P.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Van Laethem, M.

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Viano, G. A.

M. Bertero, C. De Mol, G. A. Viano, “On the regularization of linear inverse problems in Fourier optics,” in Applied Inverse Problems, by P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1978), pp. 180–199.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, by H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.

Viera, G.

Wriedt, T.

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

Yamada, Y.

M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
[CrossRef]

ACM Trans. Math. Software

A. Corana, M. Marchesi, C. Matini, S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Trans. Math. Software 13, 262–280 (1987).
[CrossRef]

Acta Soc. Sci. Fenn.

H. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

AIAA J.

M. E. Essawy, A. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

Ann. Phys. (Leipzig)

G. Mie, “Beitrage zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Appl. Opt.

R. J. Perry, A. J. Hunt, D. R. Huffman, “Experimental determinations of Mueller scattering matrices for nonspherical particles,” Appl. Opt. 17, 2701–2710 (1978).
[CrossRef]

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

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]

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]

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

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

H. Schnablegger, O. Glatter, “Optical sizing of small colloidal particles: an optimized regularization technique,” Appl. Opt. 30, 4889–4896 (1991).
[CrossRef] [PubMed]

H. Jiang, J. Pierce, J. Kao, E. Sevick-Muruca, “Measurement of particle-size distribution and volume fraction in concentrated suspensions with photon migration techniques,” Appl. Opt. 36, 3310–3318 (1997).
[CrossRef] [PubMed]

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

F. Ferri, A. Bassini, E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995).
[CrossRef] [PubMed]

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

Inverse Prob.

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

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

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

J. Atmos. Sci.

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]

J. Part. Charact.

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

J. Phys. A

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. K. Atakishiyeva, N. M. Atakishiyev, “On the Mellin transforms of hypergeometric polynomials,” J. Phys. A 32, 33–41 (1999).
[CrossRef]

J. Thermophys. Heat Transfer

M. R. Jones, M. Q. Brewster, Y. Yamada, “Application of a genetic algorithm to the optical characterization of propellant smoke,” J. Thermophys. Heat Transfer 10, 372–377 (1996).
[CrossRef]

Math. Subj. Class.

P. L. Butzer, S. Jansche, “A direct approach to the Mellin transform,” Math. Subj. Class. 45A15, 44-02–44-03 (1991).

Opt. Acta

J. G. McWhirter, “A stabilized model-fitting approach to the processing of laser anemometry and the other photon correlation data,” Opt. Acta 27, 83–105 (1980).
[CrossRef]

Part. Part. Syst. Charact.

H. S. Lee, S. K. Chae, B. Y. H. Liu, “Size characterization of liquid-borne particles by light scattering counters,” Part. Part. Syst. Charact. 6, 93–99 (1989).
[CrossRef]

A. Doicu, J. Köser, T. Wriedt, K. Bauckhage, “Light scattering simulation and measurement of monodisperse spheroids using a phase Doppler anemometer,” Part. Part. Syst. Charact. 15, 257–262 (1998).
[CrossRef]

N. De Jaeger, H. Demeyere, R. Finsy, K. Sneyers, J. Van der Beelen, P. Van der Meeren, M. Van Laethem, “Particle sizing by photon correlation spectroscopy. I. Monodisperse lattices: Influence of scattering angle and concentration of monodisperse material,” Part. Part. Syst. Charact. 8, 179–186 (1991).
[CrossRef]

Prog. Astronaut. Aeronaut.

J. Swinthenbank, J. M. Beer, D. S. Taylor, C. G. McCreath, “A laser diagnostic technique for the measurement of droplets and particle size distribution. Experimental diagnostics in gas phase combustion systems,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Other

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

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

N. G. Stanley-Woods, R. W. Lines, Particle Size Analysis (Royal Society of Chemistry, London, 1992).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Particle sizing by inversion of extinction data,” in Proceedings of an International Symposium on Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 55–61.
[CrossRef]

M. Bertero, C. De Mol, G. A. Viano, “On the regularization of linear inverse problems in Fourier optics,” in Applied Inverse Problems, by P. C. Sabatier, ed. (Springer-Verlag, Berlin, 1978), pp. 180–199.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, by H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.

H. M. Wadworth, ed., Handbook of Statistical Methods for Engineers and Scientists (McGraw-Hill, New York, 1990).

C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

T. N. E. Greville, Theory and Applications of Spline Functions (Academic, New York, 1969).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

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

H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, New York, 1981).

J.-C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

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Figures (12)

Fig. 1
Fig. 1

Structure of the inverse method: NNLS, nonnegative LS method; LSM, layered synthetic microstructure; other abbreviations defined in text.

Fig. 2
Fig. 2

(a) Main principles of the numerical filtering used in determining the associated cutoff frequencies for each discrete angle. (b) Evolution of cutoff frequency as a function of noise level.

Fig. 3
Fig. 3

Main principles for the construction of the bounding functions J min (D) and J max (D).

Fig. 4
Fig. 4

Comparison of the normalized theoretical LN g theo(D) PSD (solid curves), the optimized smooth PSD g (D) (open circles), and the bounding functions g min (D) (dotted curves) and g max (D) (dashed curves) as functions of various diameters. Data for these cases are listed in Table 1, and the corresponding figures are (a) LN1a; (b) LN1b, (c) LN1c.

Fig. 5
Fig. 5

Comparison of the theoretical renormalized PSD g theo(D), referred to as LN1a (solid curve) and the bounding functions g min (D) (open circles) and g max (D) (dashed curve).

Fig. 6
Fig. 6

Comparison of the normalized theoretical LN g theo(D) PSD (solid curves), the optimized smooth PSD g (D) (open circles), and the bounding functions g min (D) (dotted curves) and g max (D) (dashed curves) as functions of distribution width. Data for these cases are listed in Table 1, and the corresponding figures are (a) LN2a, (b) LN2b, (c) LN2c.

Fig. 7
Fig. 7

Comparison of the theoretical normalized referenced PSD g theo(D), the optimized smooth PSD g (D) (open circles) and the bounding functions g min (D) (dotted curves) and g max (D) (dashed curves). Data for these cases are listed in Table 2, and the corresponding figures are (a) PL2, (b) PL1, (c) C3, (d) C1.

Fig. 8
Fig. 8

Comparison of the theoretical normalized bimodal LN PSD g theo(D) (solid curves), the smooth PSD g (D) (open circles), and its associated confidence intervals defined by g min (D) (dotted curves) and g max (D) (dashed curves). Data for these cases are listed in Table 3, and the corresponding figures are (a) bimo1, (b) zoom on bimo1, (c) bimo2, (d) bimo3, (e) bimo4, (f) bimo5.

Fig. 9
Fig. 9

Comparison of the theoretical normalized LN PSD g theo(D) (solid curves), the smooth PSD g (D) (open circles), and its associated confidence intervals defined by g min (D) (dotted curves) and g max (D) (dashed curves) as functions of mean noise level. Data for LN3 are listed in Table 4, and the corresponding figures are (a) εnoise = 1%, (b) εnoise = 2%, (c) εnoise = 5%, (d) εnoise = 10%.

Fig. 10
Fig. 10

Representation of the theoretical nonnoisy scattering diagram βtheo i ) for the LN PSD g theo(D) for LN4 illustrated in Table 4 as a function of i; the number of discrete scattering directions as defined by θ i = i1.8°, i ∈ [1,9] and θ i = 134° + (i - 10)0.5°, i ∈ [10,32].

Fig. 11
Fig. 11

Left, theoretical normalized LN PSD g theo(D) (solid curves) compared with the smooth PSD g (D) (open circles) obtained for a mean noise level εnoise = 5% and three different samples of errors, denoted (a), (b), and (c). Associated confidence intervals, defined by g min (D) (dotted curves) and g max (D) (dashed curves) are also depicted. Right, samples of errors Δβtheo i , εnoise)/βtheo i ) (solid curves) and the residual differences rd i , εnoise) (open circles) (percent). The case studied, LN4, is defined in Table 6.

Fig. 12
Fig. 12

Comparison of the theoretical normalized LN PSD g theo(D) (solid curves), the smooth distribution g (D) (open circles), and the associated bounding functions g min (D) (dotted curves) and g max (D) (dashed curves) as functions of the set of discrete scattering angles. The cases studied, denoted LN5x, are defined in Table 5 and are represented in the figures as (a) LN5a, (b) LN5b, (c) LN5c, (d) LN5d, (e) LN5e, (f) LN5f.

Tables (8)

Tables Icon

Table 1 Parameters of Monomodal LN Distributionsa

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Table 2 Parameters of Referenced Monomodal Distributionsa

Tables Icon

Table 3 Parameters of Binomodal LN Distributionsa

Tables Icon

Table 4 Monomodal LN Distributions to Test the Method’s Sensitivity against Noise: Mean Level and Samplinga

Tables Icon

Table 5 Monomodal LN Distributions to Test the Method’s Sensitivity against the Set of Discrete Scattering Anglesa

Tables Icon

Table 6 Mean Diameters D¯ and Their Associated Standard Deviations ΔD Compared with the Theoretical Values Dtheo¯ ± ΔDtheo for Various Monomodal Distributions

Tables Icon

Table 7 Mean Diameters D¯ and Their Associated Standard Deviations ΔD Compared with the Theoretical Values Dtheo¯ ± ΔD theo for Different Noise Levels and Samplings

Tables Icon

Table 8 Mean Diameters D¯ and Their Associated Standard Deviations ΔD Compared with the Theoretical Values Dtheo¯ ± ΔDtheo for Different Sets of Scattering Angles

Equations (85)

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Wmeasθ, λ=I0ΔνΔΩ0 fDσD/λ, m*, θdVdΩdD,
Smeasθ, λ=c1×Wmeasθ, λ,
Wmeasθ, λ=c2×βmeasθ, λ,
βmeasθ, λ=0 fDσD/λ, m*λ, θdD.
Smeasθ, λ=c3×βmeasθ, λ,
Kextλ=-1LlnSmeas0, λS0,
Kextλ=0 fDσextD/λ, m*λdD,
NT=0 fDdD,
Cν=0πD36 fDdD.
βmeasθi, λ=0 fDσD/λ, m*λ, θidD, i1, N.
ab Kx, yUnydy=αnUnx
4σθDνπD2=σθνD.
gD=πD24 fD.
IOX=0 σθiνDXDdD,
limυD σθiνD=const.0.
KθνD=σθνDexp-γνD.
hD=gDexpγνD,
βθimeasν=0 hDKθiνDdD,  i1, N.
θminθmaxJ1xmax sin θxmax sin θ2 sin θdθ0.02 0πJ1xmax sin θxmax sin θ2 sin θdθ,
θmin=1.8°, θmax=16.2°θmin=134°, θmax=145°
Dmin0.2λ.
Dmax KθiνDdD  0Dmax KθiνDdD,  i1, N.
0Dmin KθiνDdD  0Dmax KθiνDdD,  i1, N,
βθimeasν=DminDmax hDKθiνDdD,  i1, N.
MTθz=0 xz-1Kθxdx,  z=1/2+iω.
Uω+ν, θD=ReMTθz D-zπ|MTθz|,
Uω-ν, θD=ImMTθz D-zπ|MTθz|,
αω±ν, θ=±αων, θ=±|MTθz|,
hD=0 aω+Uω+Ddω+0 aω-Uω-Ddω,
aω±=0 hDUω±DdD,
βθmeasν=0 aω+αω+Uω+ν, θdω+0 aω-αω-Uω-ν, θdω,
aω±=1αω±0 βθmeasνUω±ν, θdν.
hD=0ωmax aω+Uω+Ddω+0ωmax aω-Uω-Ddω,
βθmeasν=0ωmax aω+αω+Uω+ν, θdω+0ωmax aω-αω-Uω-ν, θdω.
expt/2hexpt=-ωmaxωmax bω expiωtdω,
bω=MTθz2αωπaω++iaω-.
tj=j πωmax,  j1, M.
χp=plnDmax/Dmin.
hlD=j=1Ml cjlSjlD,
tkl=lnDmin+k+1Δtl,
Δtl=lnDmax/DminMl+3.
βθimeasν=j=1Ml uijlcjl or βθ=Ulcl,
uijl=Dj-2lDj+2l KθiνDSjDdD, j1, Ml, i1, N.
h0D=j=1N cj0Sj0D.
aω±0=0 h0DUω±DdD.
β0ν, θi, ωm=0ωm aω+0αω+ν, θiUω+ν, θidω+0ωm aω-0αω-ν, θiUω-ν, θidω.
Δβ0ν, θi, ωm<Δβθimeasν,  ωmωmaxi,
Δβ0ν, θi, ωm=|β0ν, θi, ωm-βθimesν|
Δβ0ν, θi, ωmΔβθimeasν, i1, N, ωm>ωmax,
χ1ωmaxχN+3.
χp0-1<ωmax<χp0,
ωmax0=χp0=p0lnDmax/Dmin.
tk1=lnDmin+k+1ωmax0,  k-1, p0.
M1=p0-3.
h1D=j=1M1 cj1Sj1D.
Δβl=UlΔcql+Δβmeas,
Δβl=βθimeas-βθi, clq,  i1, N, Δcql=cnlq+1-cnlq,  n1, Ml, Δβmeas=Δβθimeas,  i1, N.
minΔclUlΔcl-ΔβlΔβmeas,
hD=k=1M ckSkD.
βθi, ν=DminDmax hDKθiνDdD,  i1, N.
p=minkck>0, q=maxkck>0,  k1, M.
Δχθimeas=1.96 Δβθimeas.
zjn=tp-nΔt+j+1ωmax, j-1, M+2, n1, Nmax,
htraD, n=j=1M ejnCjD, n.
βθi, ν, n=Dmin*Dmax* htraD, nKθiνDdD, i1, N.
βθimeasν-Δχθimeasνβθi, ν, nβθimeasν+Δχθimeasν,  i1, N.
JminD, 1=minhtraD, 1, hD, JminD, n=minJminD, n-1, htraD, n, n2;
JmaxD, 1=maxhtraD, 1, hD, JmaxD, n=maxJmaxD, n-1, htraD, n, n2
zjn=tq+nΔt-M+2-jωmax, j-1, M+2, n1, Nmax.
fLNr=1rexp-12ln r-ln rgln σg2.
fMGr=rp1-1 exp-p2rp3.
fRPLr=r/p1p2-1r1+r/p1p2p3.
βtheoθi=0 ftheoD|F2D, θi|2dD.
Ē=1Ni=1NEi0, σnoise21/2.
Ei0, σnoise¯=Ei0, σnoise×εnoise/Ē
εnoise=1Ni=1NEi0, σnoise¯21/2.
Δβtheoθi, εnoise=Ei0, σnoise¯×βtheoθi.
gtheor=ftheor×πr2.
CS=DminDmax gDdD.
Cν=DminDmax2D3 gDdD=10-6.
gtheoD=gtheoDmaxgDmaxgtheoD.
D¯= DgDdD gDdD,
ΔD= D-D¯2gDdD gDdD1/2.
rdθi, εnoise=βtheoθi-βθiβtheoθi,
rd¯εnoise=1Ni=1N |rdθi, εnoise|.

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