OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 40, Iss. 9 — Mar. 20, 2001
  • pp: 1329–1342

Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions

Christine Böckmann  »View Author Affiliations


Applied Optics, Vol. 40, Issue 9, pp. 1329-1342 (2001)
http://dx.doi.org/10.1364/AO.40.001329


View Full Text Article

Enhanced HTML    Acrobat PDF (385 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A specially developed method is proposed to retrieve the particle volume distribution, the mean refractive index, and other important physical parameters, e.g., the effective radius, volume, surface area, and number concentrations of tropospheric and stratospheric aerosols, from optical data by use of multiple wavelengths. This algorithm requires neither a priori knowledge of the analytical shape of the distribution nor an initial guess of the distribution. As a result, even bimodal and multimodal distributions can be retrieved without any advance knowledge of the number of modes. The nonlinear ill-posed inversion is achieved by means of a hybrid method combining regularization by discretization, variable higher-order B-spline functions and a truncated singular-value decomposition. The method can be used to handle different lidar devices that work with various values and numbers of wavelengths. It is shown, to my knowledge for the first time, that only one extinction and three backscatter coefficients are sufficient for the solution. Moreover, measurement errors up to 20% are allowed. This result could be achieved by a judicious fusion of different properties of three suitable regularization parameters. Finally, numerical results with an additional unknown refractive index show the possibility of successfully recovering both unknowns simultaneously from the lidar data: the aerosol volume distribution and the refractive index.

© 2001 Optical Society of America

OCIS Codes
(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics
(010.1110) Atmospheric and oceanic optics : Aerosols
(100.0100) Image processing : Image processing
(100.3190) Image processing : Inverse problems
(280.0280) Remote sensing and sensors : Remote sensing and sensors
(290.0290) Scattering : Scattering

History
Original Manuscript: April 4, 2000
Revised Manuscript: August 31, 2000
Published: March 20, 2001

Citation
Christine Böckmann, "Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions," Appl. Opt. 40, 1329-1342 (2001)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-9-1329


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. C. Böckmann, U. Wandinger, D. Müller, “Inversion of aerosol particle properties from multiwavelength lidar measurements,” in Proceedings of the Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Florence, Italy, 1999), pp. 218–226.
  3. H. G. Jorge, J. A. Ogren, “Sensitivity of Retrieved Aerosol Properties to Assumptions in the Inversion of Spectral Optical Depths,” J. Atmos. Sci. 53, 3669–3683 (1996). [CrossRef]
  4. H. Yoshiyama, A. Ohi, K. Ohta, “Derivation of the aerosol size distribution from a bistatic system of a multiwavelength laser with the singular value decomposition method,” Appl. Opt. 35, 2642–2648 (1996). [CrossRef] [PubMed]
  5. U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995). [CrossRef] [PubMed]
  6. M. F. Carfora, F. Esposito, C. Serio, “Numerical methods for retrieving aerosol size distributions from optical measurements of solar radiation,” J. Aerosol Sci. 29, 1225–1236 (1998). [CrossRef]
  7. M. Hanke, “Accelerated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991). [CrossRef]
  8. Q. Yin, Z. Zhang, D. Kuang, “Channel selection of atmospheric remote sensing,” Appl. Opt. 35, 7136–7143 (1991). [CrossRef]
  9. M. J. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Oceanic Technol. 13, 863–873 (1996). [CrossRef]
  10. A. P. Chaikovskii, A. P. Ivanov, F. P. Osipenko, V. N. Shcherbakov, I. S. Hutko, M. M. Korol, S. B. Tauroginskaya, “Multi-wavelength lidar measurements of background aerosol and aerosol pollution,” Lith. Phys. J. 37, 348–356 (1997).
  11. G. P. Box, “Effects of smoothing and measurement-wavelength range on the accuracy of analytic eigenfunction inversions,” Appl. Opt. 34, 7787–7791 (1995). [CrossRef] [PubMed]
  12. J. Wang, F. R. Hallett, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–197 (1996). [CrossRef] [PubMed]
  13. M. Kandlikar, G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999). [CrossRef]
  14. J. Heintzenberg, H. Müller, H. Quenzel, E. Thomalla, “Information content of optical data with respect to aerosol properties: numerical studies with a randomized minimization-search-technique inversion algorithm,” Appl. Opt. 20, 1308–1315 (1981). [CrossRef] [PubMed]
  15. 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]
  16. K. Rajeev, K. Parameswaran, “Iterative method for the inversion of multiwavelength lidar signals to determine aerosol size distribution,” Appl. Opt. 37, 4690–4700 (1998). [CrossRef]
  17. P.-H. Wang, G. S. Kent, M. P. McCormick, L. W. Thomason, G. K. Yue, “Retrieval analysis of aerosol-size distribution with simulated extinction measurements at SAGE III wavelengths,” Appl. Opt. 35, 433–440 (1996). [CrossRef] [PubMed]
  18. P. Qing, H. Nakane, Y. Sasano, S. Kitamura, “Numerical simulation of the retrieval of aerosol size distribution from multiwavelength laser radar measurements,” Appl. Opt. 28, 5259–5265 (1989). [CrossRef] [PubMed]
  19. D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion through regularization: theory,” Appl. Opt. 38, 2346–2357 (1999). [CrossRef]
  20. D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion through regularization: simulation,” Appl. Opt. 38, 2358–2367 (1999). [CrossRef]
  21. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitma, Boston, 1984).
  22. B. Hofmann, Mathematik Inverser Probleme (Teubner, Stuttgart, Germany, 1999).
  23. A. K. Louis, Inverse and Schlecht Gestellte Probleme (Teubner, Stuttgart, Germany, 1989). [CrossRef]
  24. H. W. Engl, M. Hanke, A. Neubauer, Regularisation of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996). [CrossRef]
  25. H. W. Engl, Integralgleichungen (Springer-Verlag, Vienna, 1997). [CrossRef]
  26. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, Philadelphia, 1998). [CrossRef]
  27. M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems (Longman Scientific & Technical, Essex, England, 1995).
  28. H. Brakhage, “On ill-posed problems and the method of conjugate gradients,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, Boston, 1986).
  29. A. K. Louis, P. Maaß, “A mollifier method for linear operator equations of the first kind,” Inverse Prob. 6, 427–440 (1990). [CrossRef]
  30. C. Böckmann, J. Biele, R. Neuber, “Analysis of multi-wavelength lidar data by inversion with mollifier method,” Pure Appl. Opt. 7, 827–836 (1998). [CrossRef]
  31. C. Böckmann, J. Niebsch, “Mollifier methods for aerosol size distribution,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, New York, 1996), pp. 67–70.
  32. U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Prob. 7, 793–808 (1991). [CrossRef]
  33. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996). [CrossRef]
  34. R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989). [CrossRef]
  35. P. Drabek, A. Kufner, Integralgleichungen (Teubner, Stuttgart, Germany, 1996). [CrossRef]
  36. J. T. King, “Multilevel algorithms for ill-posed problems,” Numer. Math. 61, 311–334 (1992). [CrossRef]
  37. C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).
  38. T. I. Seidman, “Nonconvergence result for the application of least-squares estimation to ill-posed problems,” J. Optim. Theory Appl. 30, 535–547 (1980). [CrossRef]
  39. R. Plato, G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math. 57, 63–79 (1990). [CrossRef]
  40. A. A. Mekler, C. Böckmann, N. Sokolovskaia, “Particle distribution from Mie-scattering: kernel representation and singular-value spectrum,” Universität Potsdam, Potsdam, 2000 (Nonlinear Dynamics Preprint ISSN 1432-2935).
  41. R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular value and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985). [CrossRef]
  42. P. Deuflhard, A. Hohmann, Numerische Mathematik: eine algorithmisch orientierte Einführung (de Gruyter, Berlin, 1991).
  43. C. Böckmann, “Projection method for lidar measurements,” in Advanced Mathematical Tools in Metrology III, P. Ciarlini, M. G. Cox, F. Pavese, D. Richter, eds., Vol. 45 of Series on Advances in Mathematics for Applied Sciences (World Scientific, Singapore, 1997), pp. 239–240.
  44. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  45. W. C. Hinds, Aerosol Technology (Wiley, New York, 1982).
  46. J. B. Gillespie, J. D. Lindberg, “Ultraviolet and visible imaginary refractive index of strongly absorbing atmospheric particulate matter,” Appl. Opt. 31, 2112–2115 (1992). [CrossRef] [PubMed]
  47. I. N. Tang, H. R. Munkelwitz, “Water activities, densities, and refractive indices of aqueous sulfates and sodium nitrate droplets of atmospheric importance,” J. Geophys. Res. 99, 18801–18808 (1994). [CrossRef]
  48. J. D. Lindberg, R. E. Douglass, D. M. Garvey, “Carbon and the optical properties of atmospheric dust,” Appl. Opt. 32, 6077–6086 (1993). [CrossRef] [PubMed]
  49. D. S. Covert, J. Heintzenberg, “Size distribution and chemical properties of aerosol at Ny Alesund, Svalbard,” Atmos. Environ. 27A, 2989–2997 (1993). [CrossRef]
  50. P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Prob. 8, 849–872 (1992). [CrossRef]
  51. C. Böckmann, J. Wauer, “Algorithms for the inversion of light scattering data from uniform and non-uniform particles,” J. Aerosol Sci. 32, 49–61 (2001). [CrossRef]
  52. J. Wauer, T. Rother, K. Schmidt, C. Böckmann, “A numerical package to calculate light scattering on non-spherical particles and its application in LIDAR inversion,” in Proceedings of Tenth International Workshop on Multiple Scattering Lidar Experiments, P. Bruscaglioni, ed. (Department of Physics, University of Florence, Florence, Italy, 1999), pp. 237–238.
  53. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996). [CrossRef]
  54. T. Rother, “Generalization of the separation of variables method for non-spherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer 60, 335–353 (1998). [CrossRef]
  55. J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solution to the aerosol data inversion problem,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 12, 342–361 (1991). [CrossRef]
  56. J. G. Crump, J. H. Seinfeld, “Further results on inversion of aerosol size distribution data: higher-order Sobolev spaces and constraints,” Aerosol Sci. Technol. 1, 363–369 (1982). [CrossRef]
  57. S. F. Gilyazov, N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer Academic, Dordrecht, The Netherlands, 2000). [CrossRef]
  58. S. Kitamura, P. Qing, “Neural network application to solve Fredholm integral equations of the first kind,” in Proceedings of International Joint Conference on Neural Networks (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1989), p. 589.
  59. A. Ishimaru, R. J. Marks, L. Tsang, C. M. Lam, D. C. Park, S. Kitamura, “Particle-size distribution determination using optical sensing and neural network,” Opt. Lett. 15, 1221–1223 (1990). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited