Abstract

A stochastic inverse technique based on a genetic algorithm (GA) to invert particle-size distribution from angular light-scattering data is developed. This inverse technique is independent of any given a priori information of particle-size distribution. Numerical tests show that this technique can be successfully applied to inverse problems with high stability in the presence of random noise and low susceptibility to the shape of distributions. It has also been shown that the GA-based inverse technique is more efficient in use of computing time than the inverse Monte Carlo method recently developed by Ligon et al. [Appl. Opt. 35, 4297 (1996)].

© 1999 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. 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]
  4. G. F. Miller, “Fredholm equation of the first kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. Walsh, eds. (Clarendon, Oxford, 1974), pp. 175–188.
  5. G. M. Quist, P. J. Wyatt, “Empirical solution to inverse-scattering problem by optical strip-map technique,” J. Opt. Soc. Am. A 2, 1979–1985 (1985).
    [CrossRef]
  6. M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
    [CrossRef] [PubMed]
  7. J. H. Koo, E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31, 2130–2140 (1992).
    [CrossRef] [PubMed]
  8. L. P. Baryvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
    [CrossRef]
  9. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Part. Syst. Charact. 4, 128–133 (1988).
    [CrossRef]
  10. L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds,” Appl. Opt. 15, 378–383 (1976).
    [CrossRef] [PubMed]
  11. E. R. Westwater, A. Cohen, “Application of Backus–Gibert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1340–1344 (1973).
    [CrossRef] [PubMed]
  12. F. Ferri, A. Bassini, E. Paganin, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995).
    [CrossRef] [PubMed]
  13. S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
    [CrossRef]
  14. J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
    [CrossRef]
  15. B. P. Curry, “Constrained eigenfunction method for the inversion of remote sensing data: application to particle size determination from light scattering measurement,” Appl. Opt. 28, 1345–1355 (1989).
    [CrossRef] [PubMed]
  16. J. H. Holland, Adaptation in Natural and Artificial System (U. Michigan Press, Ann Arbor, Mich., 1975).
  17. J. D. Bagley, “The behavior of adaptive systems which employ genetic and correlation algorithms,” Dissertation Abstr. Int. 28, 5106B (1967).
  18. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  19. Y. Liu, L. Kang, Y. Chen, Genetic Algorithms (Academic, Beijing, 1995; in Chinese).
  20. R. V. Davalos, B. Rubinsky, “An evolutionary-genetic approach to heat transfer analysis,” J. Heat Transfer 118, 528–531 (1997).
    [CrossRef]
  21. 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]
  22. L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).
  23. D. Whitley, “The genitor algorithm and selection pressure: why rank-based allocation of reproductive trials is best,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 116–121.
  24. L. J. Eshelman, R. A. Caruana, J. D. Schaffer, “Biases in the crossover landscape,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 10–19.
  25. N. Johnson, S. Kotz, Continuous Univariate Distributions—1 and 2 (Houghton Mifflin, New York, 1970).
  26. A. B. Yu, N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
    [CrossRef]
  27. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4813 (1991).
    [CrossRef] [PubMed]

1997 (1)

R. V. Davalos, B. Rubinsky, “An evolutionary-genetic approach to heat transfer analysis,” J. Heat Transfer 118, 528–531 (1997).
[CrossRef]

1996 (3)

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]

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

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]

1995 (1)

1994 (1)

1992 (1)

1991 (1)

1990 (1)

A. B. Yu, N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

1989 (2)

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

B. P. Curry, “Constrained eigenfunction method for the inversion of remote sensing data: application to particle size determination from light scattering measurement,” Appl. Opt. 28, 1345–1355 (1989).
[CrossRef] [PubMed]

1988 (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Part. Syst. Charact. 4, 128–133 (1988).
[CrossRef]

1985 (1)

1976 (1)

1973 (1)

1967 (1)

J. D. Bagley, “The behavior of adaptive systems which employ genetic and correlation algorithms,” Dissertation Abstr. Int. 28, 5106B (1967).

Agrawal, Y. C.

Arridge, S.

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

Bagley, J. D.

J. D. Bagley, “The behavior of adaptive systems which employ genetic and correlation algorithms,” Dissertation Abstr. Int. 28, 5106B (1967).

Baryvel, L. P.

L. P. Baryvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Bassini, A.

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]

M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
[CrossRef] [PubMed]

Caruana, R. A.

L. J. Eshelman, R. A. Caruana, J. D. Schaffer, “Biases in the crossover landscape,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 10–19.

Chen, T. W.

Chen, Y.

Y. Liu, L. Kang, Y. Chen, Genetic Algorithms (Academic, Beijing, 1995; in Chinese).

Cheng, J.

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

Chow, L. C.

Cohen, A.

Cope, M.

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

Curry, B. P.

Davalos, R. V.

R. V. Davalos, B. Rubinsky, “An evolutionary-genetic approach to heat transfer analysis,” J. Heat Transfer 118, 528–531 (1997).
[CrossRef]

Davis, L.

L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).

Delpy, D. T.

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

Eshelman, L. J.

L. J. Eshelman, R. A. Caruana, J. D. Schaffer, “Biases in the crossover landscape,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 10–19.

Ferri, F.

Gillespie, J. B.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

He, J.

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

Hirleman, E. D.

J. H. Koo, E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31, 2130–2140 (1992).
[CrossRef] [PubMed]

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Part. Syst. Charact. 4, 128–133 (1988).
[CrossRef]

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial System (U. Michigan Press, Ann Arbor, Mich., 1975).

Johnson, N.

N. Johnson, S. Kotz, Continuous Univariate Distributions—1 and 2 (Houghton Mifflin, New York, 1970).

Jones, A. R.

L. P. Baryvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

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]

M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
[CrossRef] [PubMed]

Kang, L.

Y. Liu, L. Kang, Y. Chen, Genetic Algorithms (Academic, Beijing, 1995; in Chinese).

Kerker, M.

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

Koo, J. H.

Kotz, S.

N. Johnson, S. Kotz, Continuous Univariate Distributions—1 and 2 (Houghton Mifflin, New York, 1970).

Leong, K. H.

Ligon, D. A.

Liu, Y.

Y. Liu, L. Kang, Y. Chen, Genetic Algorithms (Academic, Beijing, 1995; in Chinese).

Miller, G. F.

G. F. Miller, “Fredholm equation of the first kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. Walsh, eds. (Clarendon, Oxford, 1974), pp. 175–188.

Paganin, E.

Quist, G. M.

Riley, J. B.

Rubinsky, B.

R. V. Davalos, B. Rubinsky, “An evolutionary-genetic approach to heat transfer analysis,” J. Heat Transfer 118, 528–531 (1997).
[CrossRef]

Schaffer, J. D.

L. J. Eshelman, R. A. Caruana, J. D. Schaffer, “Biases in the crossover landscape,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 10–19.

Standish, N.

A. B. Yu, N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Tien, C. L.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).

van der Zee, P.

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

Wang, S.

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

Westwater, E. R.

Whitley, D.

D. Whitley, “The genitor algorithm and selection pressure: why rank-based allocation of reproductive trials is best,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 116–121.

Wyatt, P. J.

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]

Yu, A. B.

A. B. Yu, N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Zhang, S.

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

Appl. Opt. (8)

E. R. Westwater, A. Cohen, “Application of Backus–Gibert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1340–1344 (1973).
[CrossRef] [PubMed]

L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds,” Appl. Opt. 15, 378–383 (1976).
[CrossRef] [PubMed]

B. P. Curry, “Constrained eigenfunction method for the inversion of remote sensing data: application to particle size determination from light scattering measurement,” Appl. Opt. 28, 1345–1355 (1989).
[CrossRef] [PubMed]

J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4813 (1991).
[CrossRef] [PubMed]

J. H. Koo, E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31, 2130–2140 (1992).
[CrossRef] [PubMed]

M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
[CrossRef] [PubMed]

F. Ferri, A. Bassini, E. Paganin, “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 an inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
[CrossRef] [PubMed]

Dissertation Abstr. Int. (1)

J. D. Bagley, “The behavior of adaptive systems which employ genetic and correlation algorithms,” Dissertation Abstr. Int. 28, 5106B (1967).

Inverse Probl. (2)

S. Arridge, P. van der Zee, D. T. Delpy, M. Cope, “Particle sizing in the Mie scattering region: singular-value analysis,” Inverse Probl. 5, 671–689 (1989).
[CrossRef]

J. He, S. Wang, J. Cheng, S. Zhang, “Inversion of particle size distribution from light scattering spectrum,” Inverse Probl. 12, 633–639 (1996).
[CrossRef]

J. Heat Transfer (1)

R. V. Davalos, B. Rubinsky, “An evolutionary-genetic approach to heat transfer analysis,” J. Heat Transfer 118, 528–531 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Thermophys. Heat Transfer (1)

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]

Part. Part. Syst. Charact. (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Part. Syst. Charact. 4, 128–133 (1988).
[CrossRef]

Powder Technol. (1)

A. B. Yu, N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Other (11)

L. Davis, Handbook of Genetic Algorithms (Van Nostrand Reinhold, New York, 1991).

D. Whitley, “The genitor algorithm and selection pressure: why rank-based allocation of reproductive trials is best,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 116–121.

L. J. Eshelman, R. A. Caruana, J. D. Schaffer, “Biases in the crossover landscape,” in Proceedings of the Third International Conference on Genetic Algorithms and Their Applications, J. D. Schaffer, ed. (Morgan Kaufmann, Los Altos, Calif., 1989), pp. 10–19.

N. Johnson, S. Kotz, Continuous Univariate Distributions—1 and 2 (Houghton Mifflin, New York, 1970).

J. H. Holland, Adaptation in Natural and Artificial System (U. Michigan Press, Ann Arbor, Mich., 1975).

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Y. Liu, L. Kang, Y. Chen, Genetic Algorithms (Academic, Beijing, 1995; in Chinese).

H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).

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

G. F. Miller, “Fredholm equation of the first kind,” in Numerical Solution of Integral Equations, L. M. Delves, J. Walsh, eds. (Clarendon, Oxford, 1974), pp. 175–188.

L. P. Baryvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Inversion results for asymmetrical distributions. Results are retrieved from noise-free measured data and from noisy data with NSR = 5%. (a), (c) Small-angle region; (b), (d) full-angle region. The exact distribution for (a) and (b) has parameters σ = 1.8 and μ = 3.7 and for (c) and (d) has parameters σ = 2.3 and μ = -1.8. The refractive index of the particles is m = 1.57 - i0.56.

Fig. 2
Fig. 2

Particle-size distribution functions chosen for exact distribution. Function 1 is a broad distribution with σ = 1.1 and μ = 0.0. Function 2 is a narrow distribution with σ = 7.8 and μ = 0.0.

Fig. 3
Fig. 3

Inversion results for the broad distributions. Results are retrieved from the noise-free measured data and from noisy data with NSR = 5%. (a) Small-angle region, (b) full-angle region. The exact distribution has parameters σ = 1.1 and μ = 0.0. The refractive index of the particles is m = 1.57 - i0.56.

Fig. 4
Fig. 4

Inversion results for narrow distributions. Results are retrieved from the noise-free measured data and from noisy data for the NSR values listed. (a) Small-angle region, (b) full-angle region. The exact distribution has parameters σ = 7.8 and μ = 0.0. The refractive index of the particles is m = 1.57 - i0.56.

Fig. 5
Fig. 5

Inversion results for unimodal distributions with different random-noise levels. The results were retrieved from the measured data for the NSR values listed. (a) Small-angle region, (b) for full-angle region. The exact distribution has parameters σ = 2.5 and μ = 2.0. The refractive index of the particles is m = 1.57 - i0.56.

Fig. 6
Fig. 6

Inversion results for multimodal distributions. The exact distribution is bimodal, the two peaks of which are characterized by σ1 = 2.5, μ1 = 2.0 and σ2 = 1.8, μ2 = -3.7, respectively, for (a)–(c) the small-angle region and (d)–(f) the full-angle region. The results in (a) and (d) were retrieved from noise-free measured data; those in (b) and (e), from noisy measured data with NSR = 5%. The results in (c) and (f) were retrieved from noisy measured data with NSR = 10%. The refractive index of the particles is m = 1.57 - i0.56.

Fig. 7
Fig. 7

Relative error of inverted results and exact distributions for several wavelengths. The results were retrieved from noisy measured data with NSR = 1%. (a) Small-angle region, (b) full-angle region. The refractive index of the particles is m = 1.33.

Fig. 8
Fig. 8

Inversion results for unimodal distributions from the measured data obtained with three refractive indices of particles. (a) Small-angle region, (b) full-angle region. The exact distribution has parameters σ = 2.3 and μ = -1.8. The exact refractive index of the particles is 1.33.

Fig. 9
Fig. 9

Comparison of computing time for the GA-based inverse technique and the IMC technique. The conditions of the numerical tests are listed in Table 1. The relative errors in (a), (b), (c), and (d) are 1.0-3, 1.0-4, 1.0-5, and 1.0-6, respectively. The triangles with crossed centers represent failed numerical tests of the IMC method in which the computing time was longer than 3600 s.

Tables (1)

Tables Icon

Table 1 Conditions of the Numerical Tests for Comparison of Computing Time

Equations (15)

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

Iθ=N 0αmax iθ, αfαdα,
iθ, α=i1θ, α, m+i2θ, α, m8π2 λ2I0,
N j=1M ik,jfjωj-Ik=0,  k=1, 2,, K,
Fk=j=1Mik,jI1-i1,jIkfjωjI1j=1M ik,jfjωj=0, k=1, 2,, K,
N=1Kk=1KIkj=1M ik,jfjωj.
φf=k=1K Fk2,  f=f1, f2,, fMT.
minfRM φf=k=1K Fk2, f0,  j=1M fj=1.
(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 1 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1),
fj=xj/65535,
pi=1-φf-φminfφmaxf-φminf,
β=Np2a-1a-a2-4Ra-11/2,
1 1 1 1 1 1 1 1 1 11 1 1 1 1,0 0 0 0 0 0 0 0 00 0 0 0 0,
1 1 1 1 1 1 0 0 0 00 0 0 0 0,0 0 0 0 0 0 1 1 1 11 1 1 1 1.
fα=σ2πtt-1-1 exp-0.5μ+σ lntt-12.
Ĩθ=Iθ1+εNSR,

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