Abstract

Our aim is to present the application of the hybrid method presented in part I to an inverse procedure to determine particle size and concentration under multiple-scattering conditions. The hybrid method is introduced as a combination of the four-flux method with coefficients obtained from Monte Carlo statistical simulations to take into account the actual three-dimensional geometry. Then an inversion scheme is expanded to enable the application of the hybrid method to particle size and concentration determination. We present the inversion method as well as exemplifying results of spectrum inversions.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier Scientific, Amsterdam, 1977).
  2. G. Backus, F. Gilbert, “Uniqueness in the inversion of gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
    [CrossRef]
  3. M. Czerwiński, J. Mroczka, D. Wysoczański, “Inverse problem in multiple light scattering—hybrid method application,” in Proceedings of MUSCLE, the 10 Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Italy, 1999), pp. 257–263.
  4. M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).
  5. R. P. Brend, Algorithms for Minimisation without Derivatives, Prentice-Hall, Englewood Cliffs, N.J., 1973) Chaps. 3 and 4.
  6. G. E. Forsythe, M. A. Malcolm, B. C. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.2.

1970 (1)

G. Backus, F. Gilbert, “Uniqueness in the inversion of gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Backus, G.

G. Backus, F. Gilbert, “Uniqueness in the inversion of gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Brend, R. P.

R. P. Brend, Algorithms for Minimisation without Derivatives, Prentice-Hall, Englewood Cliffs, N.J., 1973) Chaps. 3 and 4.

Czerwinski, M.

M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).

M. Czerwiński, J. Mroczka, D. Wysoczański, “Inverse problem in multiple light scattering—hybrid method application,” in Proceedings of MUSCLE, the 10 Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Italy, 1999), pp. 257–263.

Forsythe, G. E.

G. E. Forsythe, M. A. Malcolm, B. C. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.2.

Gilbert, F.

G. Backus, F. Gilbert, “Uniqueness in the inversion of gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Malcolm, M. A.

G. E. Forsythe, M. A. Malcolm, B. C. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.2.

Moler, B. C.

G. E. Forsythe, M. A. Malcolm, B. C. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.2.

Mroczka, J.

M. Czerwiński, J. Mroczka, D. Wysoczański, “Inverse problem in multiple light scattering—hybrid method application,” in Proceedings of MUSCLE, the 10 Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Italy, 1999), pp. 257–263.

Twomey, S.

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

Wysoczanski, D.

M. Czerwiński, J. Mroczka, D. Wysoczański, “Inverse problem in multiple light scattering—hybrid method application,” in Proceedings of MUSCLE, the 10 Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Italy, 1999), pp. 257–263.

Philos. Trans. R. Soc. London Ser. A (1)

G. Backus, F. Gilbert, “Uniqueness in the inversion of gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
[CrossRef]

Other (5)

M. Czerwiński, J. Mroczka, D. Wysoczański, “Inverse problem in multiple light scattering—hybrid method application,” in Proceedings of MUSCLE, the 10 Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Italy, 1999), pp. 257–263.

M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).

R. P. Brend, Algorithms for Minimisation without Derivatives, Prentice-Hall, Englewood Cliffs, N.J., 1973) Chaps. 3 and 4.

G. E. Forsythe, M. A. Malcolm, B. C. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.2.

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

Cited By

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

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Comparison of the transmittance spectra for particles with a diameter of 0.4 µm and a concentration of c = 8 × 1014 particles/m3 calculated with the Bouguer–Beer–Lambert method and the Monte Carlo method.

Fig. 2
Fig. 2

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1014 particles/m3; input particle diameter, d = 0.4 × 10-6 m.

Fig. 3
Fig. 3

Comparison of the transmittance spectra for particles with a diameter of 0.4 µm and a concentration of c = 1015 particles/m3 calculated with the Bouguer–Beer–Lambert method and the Monte Carlo method.

Fig. 4
Fig. 4

Results of Monte Carlo spectra inversion. Input concentration, c = 1015 particles/m3; input particle diameter, d = 0.4 × 10-6 m.

Fig. 5
Fig. 5

Comparison of the transmittance spectra for particles with a diameter of 0.9 µm and a concentration of c = 4 × 1013 particles/m3 calculated with the Bouguer–Beer–Lambert method and the Monte Carlo method.

Fig. 6
Fig. 6

Results of Monte Carlo spectra inversion. Input concentration, c = 4 × 1013 particles/m3; input particle diameter, d = 0.9 × 10-6 m.

Fig. 7
Fig. 7

Comparison of the transmittance spectra for particles with a diameter of 0.9 µm and a concentration of c = 8 × 1013 particles/m3 calculated with the Bouguer–Beer–Lambert method and the Monte Carlo method.

Fig. 8
Fig. 8

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1013 particles/m3; input particle diameter, d = 0.9 × 10-6 m.

Fig. 9
Fig. 9

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1014 particles/m3.

Fig. 10
Fig. 10

Results of Monte Carlo spectra inversion. Input concentration, c = 1015 particles/m3.

Fig. 11
Fig. 11

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1014 particles/m3.

Fig. 12
Fig. 12

Results of Monte Carlo spectra inversion. Input concentration, c = 1015 particles/m3.

Fig. 13
Fig. 13

Results of Monte Carlo spectra inversion. Input concentration, c = 4 × 1013 particles/m3.

Fig. 14
Fig. 14

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1013 particles/m3.

Fig. 15
Fig. 15

Results of Monte Carlo spectra inversion. Input concentration, c = 4 × 1013 particles/m3.

Fig. 16
Fig. 16

Results of Monte Carlo spectra inversion. Input concentration, c = 8 × 1013 particles/m3.

Equations (2)

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

Pλ=τmeas-τhyb2,
P=λminλmaxτmeasλi-τhybλi2,

Metrics