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

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2001 (1)

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]

1999 (3)

1998 (4)

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]

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]

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]

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]

1997 (1)

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).

1996 (6)

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]

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]

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]

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]

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]

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]

1995 (3)

1994 (1)

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]

1993 (2)

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]

D. S. Covert, J. Heintzenberg, “Size distribution and chemical properties of aerosol at Ny Alesund, Svalbard,” Atmos. Environ. 27A, 2989–2997 (1993).
[CrossRef]

1992 (3)

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Prob. 8, 849–872 (1992).
[CrossRef]

J. T. King, “Multilevel algorithms for ill-posed problems,” Numer. Math. 61, 311–334 (1992).
[CrossRef]

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]

1991 (4)

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]

M. Hanke, “Accelerated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

Q. Yin, Z. Zhang, D. Kuang, “Channel selection of atmospheric remote sensing,” Appl. Opt. 35, 7136–7143 (1991).
[CrossRef]

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Prob. 7, 793–808 (1991).
[CrossRef]

1990 (3)

A. K. Louis, P. Maaß, “A mollifier method for linear operator equations of the first kind,” Inverse Prob. 6, 427–440 (1990).
[CrossRef]

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]

R. Plato, G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math. 57, 63–79 (1990).
[CrossRef]

1989 (1)

1985 (1)

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]

1982 (1)

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]

1981 (1)

1980 (1)

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]

Allen, R. C.

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]

Amato, U.

U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995).
[CrossRef] [PubMed]

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Prob. 7, 793–808 (1991).
[CrossRef]

Ansmann, A.

Bassini, A.

Biele, J.

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]

Böckmann, C.

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]

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]

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).

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.

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.

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.

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.

Bohren, G. F.

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

Boland, W. R.

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]

Box, G. P.

Brakhage, H.

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).

Carfora, M. F.

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]

U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995).
[CrossRef] [PubMed]

Chaikovskii, A. P.

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).

Covert, D. S.

D. S. Covert, J. Heintzenberg, “Size distribution and chemical properties of aerosol at Ny Alesund, Svalbard,” Atmos. Environ. 27A, 2989–2997 (1993).
[CrossRef]

Crump, J. G.

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]

Cuomo, V.

Deuflhard, P.

P. Deuflhard, A. Hohmann, Numerische Mathematik: eine algorithmisch orientierte Einführung (de Gruyter, Berlin, 1991).

Douglass, R. E.

Drabek, P.

P. Drabek, A. Kufner, Integralgleichungen (Teubner, Stuttgart, Germany, 1996).
[CrossRef]

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularisation of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

H. W. Engl, Integralgleichungen (Springer-Verlag, Vienna, 1997).
[CrossRef]

Esposito, F.

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]

Faber, V.

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]

Ferri, F.

Garvey, D. M.

Gillespie, J. B.

Gilyazov, S. F.

S. F. Gilyazov, N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer Academic, Dordrecht, The Netherlands, 2000).
[CrossRef]

Gol’dman, N. L.

S. F. Gilyazov, N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer Academic, Dordrecht, The Netherlands, 2000).
[CrossRef]

Groetsch, C. W.

C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitma, Boston, 1984).

Hallett, F. R.

Hanke, M.

M. Hanke, “Accelerated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

H. W. Engl, M. Hanke, A. Neubauer, Regularisation of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems (Longman Scientific & Technical, Essex, England, 1995).

Hansen, P. C.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Prob. 8, 849–872 (1992).
[CrossRef]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, Philadelphia, 1998).
[CrossRef]

Heintzenberg, J.

Hinds, W. C.

W. C. Hinds, Aerosol Technology (Wiley, New York, 1982).

Hofmann, B.

B. Hofmann, Mathematik Inverser Probleme (Teubner, Stuttgart, Germany, 1999).

Hohmann, A.

P. Deuflhard, A. Hohmann, Numerische Mathematik: eine algorithmisch orientierte Einführung (de Gruyter, Berlin, 1991).

Huffman, D. R.

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

Hughes, W.

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Prob. 7, 793–808 (1991).
[CrossRef]

Hutko, I. S.

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).

Ishimaru, A.

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]

Ivanov, A. P.

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).

Jorge, H. G.

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]

Kandlikar, M.

M. Kandlikar, G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

Kent, G. S.

King, J. T.

J. T. King, “Multilevel algorithms for ill-posed problems,” Numer. Math. 61, 311–334 (1992).
[CrossRef]

Kirsch, A.

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996).
[CrossRef]

Kitamura, S.

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]

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]

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.

Korol, M. M.

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).

Kress, R.

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).
[CrossRef]

Kuang, D.

Kufner, A.

P. Drabek, A. Kufner, Integralgleichungen (Teubner, Stuttgart, Germany, 1996).
[CrossRef]

Lam, C. M.

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]

Lindberg, J. D.

Louis, A. K.

A. K. Louis, P. Maaß, “A mollifier method for linear operator equations of the first kind,” Inverse Prob. 6, 427–440 (1990).
[CrossRef]

A. K. Louis, Inverse and Schlecht Gestellte Probleme (Teubner, Stuttgart, Germany, 1989).
[CrossRef]

Maaß, P.

A. K. Louis, P. Maaß, “A mollifier method for linear operator equations of the first kind,” Inverse Prob. 6, 427–440 (1990).
[CrossRef]

Mackowski, D. W.

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]

Marks, R. J.

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]

McCormick, M. P.

Mekler, A. A.

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).

Mishchenko, M. I.

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]

Müller, D.

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]

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]

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.

Müller, H.

Munkelwitz, H. R.

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]

Nakane, H.

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularisation of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

Neuber, R.

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]

Niebsch, J.

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.

Ogren, J. A.

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]

Ohi, A.

Ohta, K.

Osipenko, F. P.

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).

Paganini, E.

Parameswaran, K.

Park, D. C.

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]

Plato, R.

R. Plato, G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math. 57, 63–79 (1990).
[CrossRef]

Post, M. J.

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]

Qing, P.

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]

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.

Quenzel, H.

Rajeev, K.

Ramachandran, G.

M. Kandlikar, G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

Rother, T.

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]

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.

Sasano, Y.

Schmidt, K.

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.

Seidman, T. I.

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]

Seinfeld, J. H.

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]

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]

Serio, C.

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]

U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995).
[CrossRef] [PubMed]

Shcherbakov, V. N.

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).

Sokolovskaia, N.

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).

Tang, I. N.

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]

Tauroginskaya, S. B.

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).

Thomalla, E.

Thomason, L. W.

Travis, L. D.

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]

Tsang, L.

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]

Twomey, S.

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

Vainikko, G.

R. Plato, G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math. 57, 63–79 (1990).
[CrossRef]

Wandinger, U.

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]

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]

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.

Wang, J.

Wang, P.-H.

Wauer, J.

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]

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.

Wing, G. M.

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]

Wolfenbarger, J. K.

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]

Yin, Q.

Yoshiyama, H.

Yue, G. K.

Zhang, Z.

Aerosol Sci. Technol. (1)

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]

Appl. Opt. (14)

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]

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]

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]

U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995).
[CrossRef] [PubMed]

Q. Yin, Z. Zhang, D. Kuang, “Channel selection of atmospheric remote sensing,” Appl. Opt. 35, 7136–7143 (1991).
[CrossRef]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Atmos. Environ. (1)

D. S. Covert, J. Heintzenberg, “Size distribution and chemical properties of aerosol at Ny Alesund, Svalbard,” Atmos. Environ. 27A, 2989–2997 (1993).
[CrossRef]

Inverse Prob. (2)

A. K. Louis, P. Maaß, “A mollifier method for linear operator equations of the first kind,” Inverse Prob. 6, 427–440 (1990).
[CrossRef]

U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Prob. 7, 793–808 (1991).
[CrossRef]

Inverse Prob. (1)

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Prob. 8, 849–872 (1992).
[CrossRef]

J. Aerosol Sci. (1)

M. Kandlikar, G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999).
[CrossRef]

J. Aerosol Sci. (2)

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]

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]

J. Atmos. Oceanic Technol. (1)

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]

J. Atmos. Sci. (1)

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]

J. Geophys. Res. (1)

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]

J. Math. Anal. Appl. (1)

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]

J. Optim. Theory Appl. (1)

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]

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

Lith. Phys. J. (1)

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).

Numer. Math. (2)

M. Hanke, “Accelerated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

R. Plato, G. Vainikko, “On the regularization of projection methods for solving ill-posed problems,” Numer. Math. 57, 63–79 (1990).
[CrossRef]

Numer. Math. (1)

J. T. King, “Multilevel algorithms for ill-posed problems,” Numer. Math. 61, 311–334 (1992).
[CrossRef]

Opt. Lett. (1)

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]

Pure Appl. Opt. (1)

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]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. (1)

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]

Other (23)

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.

S. F. Gilyazov, N. L. Gol’dman, Regularization of Ill-Posed Problems by Iteration Methods (Kluwer Academic, Dordrecht, The Netherlands, 2000).
[CrossRef]

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.

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.

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996).
[CrossRef]

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).
[CrossRef]

P. Drabek, A. Kufner, Integralgleichungen (Teubner, Stuttgart, Germany, 1996).
[CrossRef]

C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

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).

P. Deuflhard, A. Hohmann, Numerische Mathematik: eine algorithmisch orientierte Einführung (de Gruyter, Berlin, 1991).

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.

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

W. C. Hinds, Aerosol Technology (Wiley, New York, 1982).

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

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.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitma, Boston, 1984).

B. Hofmann, Mathematik Inverser Probleme (Teubner, Stuttgart, Germany, 1999).

A. K. Louis, Inverse and Schlecht Gestellte Probleme (Teubner, Stuttgart, Germany, 1989).
[CrossRef]

H. W. Engl, M. Hanke, A. Neubauer, Regularisation of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

H. W. Engl, Integralgleichungen (Springer-Verlag, Vienna, 1997).
[CrossRef]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, Philadelphia, 1998).
[CrossRef]

M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems (Longman Scientific & Technical, Essex, England, 1995).

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).

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Figures (11)

Fig. 1
Fig. 1

(a) B-spline basis of the order of k = 4 on a nonequidistant grid, (b) favorable subbasis of the hybrid regularization technique.

Fig. 2
Fig. 2

Reconstruction results of the seventh example in Table 2 with 5 + 2 noiseless input data, i.e., the fourth setup case in Table 1, by means of different numbers a of B splines and orders k: (a) a = 4, k = 4, γ = 0; (b) a = 9, k = 6, γ = 0; (c) a = 8, k = 8, γ = 0; (d) a = 10, k = 9, γ = 0.

Fig. 3
Fig. 3

Reconstruction results of the volume distribution for noiseless data (example 7 in Table 2) with (a) 6 + 2 (first setup case in Table 1) and (c) 5 + 2 (third setup case in Table 1) wavelengths and for noisy data (15%) with (b) 6 + 2 and (d) 5 + 2 wavelengths. The regularization parameters are (a) a = 12, k = 4, γ = 0; (b) a = 4, k = 5, γ = 1; (c) a = 7, k = 5, γ = 0, (d) a = 3, k = 5, γ = 0.

Fig. 4
Fig. 4

Reconstruction results of the volume distribution for noiseless data (example 7 in Table 2) with (a) 6 + 0 (second setup case in Table 1) and (c) 3 + 1 (fifth setup case in Table 1) wavelengths and for noisy data (10%) with (b) 6 + 0 and (d) 3 + 1 wavelengths. The regularization parameters are (a) a = 4, k = 5, γ = 1; (b) a = 3, k = 4, γ = 0; (c) a = 6, k = 5, γ = 0, (d) a = 3, k = 4, γ = 0.

Fig. 5
Fig. 5

Example (a) 1 of Table 2, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, all with noiseless input data as well as with 6 + 2 backscatter and extinction coefficients, i.e., the first setup case in Table 1.

Fig. 6
Fig. 6

Example (a) 1 of Table 2, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, all with noiseless input data as well as with 3 + 1 backscatter and extinction coefficients, i.e., the fifth setup in Table 1.

Fig. 7
Fig. 7

(a) Example 1 of Table 3 (top), (b) example 1 of Table 3 (bottom), (c) example 2 of Table 3 (top), (d) example 2 of Table 3 (bottom), (e) example 3 of Table 3 (top), (f) example 3 of Table 3 (bottom), all with noiseless input data as well as with 6 + 2 backscatter and extinction coefficients, i.e., the first setup case in Table 1.

Fig. 8
Fig. 8

Example 7 in Table 2 (but with m exact = 1.5 + 0.0i) and the first setup case in Table 1 with an additional unknown refractive index: (a) reconstruction of a refractive-index range with an error smaller than 1.0% with respect to the backscatter and extinction coefficients, (b) reconstruction of the volume distribution by use of the determined refractive index m = 1.51 + 0.0025i; the regularization triple was determined to a = 6, k = 5, and γ = 0.

Fig. 9
Fig. 9

Example 7 in Table 2 and the first setup case in Table 1 with an additional unknown refractive index: (a) reconstruction of a refractive-index range with an error smaller than 1.2% with respect to the backscatter and extinction coefficients, (b) reconstruction of the volume distribution by use of the determined refractive index m = 1.5 + 0.01i; the regularization triple was determined to a = 12, k = 4, and γ = 0.

Fig. 10
Fig. 10

Example 2 in Table 2 and the first setup case in Table 1 with an additional unknown refractive index: (a) reconstruction of a refractive index range with an error smaller than 1% with respect to the backscatter and extinction coefficients, (b) reconstruction of the volume distribution by use of the determined refractive index m = 1.438 + 0.49i; the regularization triple was determined to a = 18, k = 5, and γ = 1.

Fig. 11
Fig. 11

Example 2 in Table 2 (but with m exact = 1.5 + 0.5i) and the first setup case in Table 1 with an additional unknown refractive index: (a) reconstruction of a refractive index range with an error smaller than 1.14% with respect to the backscatter and extinction coefficients, (b) reconstruction of the volume distribution by use of the determined refractive index m = 1.52 + 0.5i; the regularization triple was determined to a = 15, k = 4, and γ = 0.

Tables (5)

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Table 1 Different Common Setup Cases of Multispectral Lidar Devices

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Table 2 Different Simulation Examples of Monomodal Logarithmic-Normal Distributions and the Inversion Results of Noiseless Input Data, Three Setup Cases of Table 1

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Table 3 Different Simulation Examples of Bimodal Distributions (Top) and Gamma Distributions (Bottom) and the Inversion Results of Noiseless Input Data, the Setup Case of Table 1

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Table 4 Relative Errors (%) of the Inversion Results of Microphysical Parameters from Noiseless Input Data for the First Six Examples in Table 2, each Example with Six Different Refractive Indices, and for Three Setup Cases of Table 1

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Table 5 Relative Errors (%) of the Inversion of Mean Microphysical Parameters of Monomodal Logarithmic-Normal Distributions with Noiseless Input Data for Examples 1, 2, and 4–6 with 18 Different Refractive Indices and with Noisy Data for Example 5 with 5 Different Refractive Indicesa

Equations (34)

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β aer λ ,   z = r min r max   k ˆ π r ,   λ ;   m n r ,   z d r = r min r max   π r 2 Q π r ,   λ ;   m n r ,   z d r ,
α aer λ ,   z = r min r max   k ˆ ext r ,   λ ;   m n r ,   z d r = r min r max   π r 2 Q ext r ,   λ ;   m n r ,   z d r .
Q π = 1 k ˜ 2 r 2 n = 1 2 n + 1 - 1 n a n - c n 2 ,   Q ext = 2 k ˜ 2 r 2 n = 1 2 n + 1 Re a n + c n ,
a n = m ψ n m ω ψ n ω - ψ n ω ψ n m ω m ψ n m ω ζ n ω - ζ n ω ψ n m ω ,
c n = ψ n m ω ψ n ω - m ψ n ω ψ n m ω ψ n m ω ζ n ω - m ζ n ω ψ n m ω , ψ n t = tj n t ,   χ n t = ty n t ,   ζ n t = ψ t + χ t i ,
y λ = r min r max   k ˆ λ ,   r x r d r .
x = j = 1 y ,   u j μ j   v j + φ
K y δ = j = 1 y δ ,   u j μ j   v j .
Ω   :   D X ,   Ω x = min ! ,   x D X ,
x γ δ - x = K γ y δ - y + K γ - K y .
X n = span ϕ 1 ,     ,   ϕ n ,   Y m = span ρ 1 ,     ,   ρ m .
x n = b 1 ϕ 1 + + b n ϕ n
P m Kx n = P m y .
R n y = P n K | X n P n y K y ,   n ,
x n δ - x x n - x + K n P n y - y δ x n - x + K n δ x n - x + δ ν n .
j = 1 n K ϕ j λ i   b j = y λ i ,   i = 1 ,     ,   m .
A ij   : = r min r max   k ˆ λ i ,   r ϕ j r d r .
N i 1 r   : = χ τ i , τ i + 1 r = 1   r τ i ,   τ i + 1 0   otherwise ,
N ik r   : = r - τ i τ i + k - 1 - τ i   N i , k - 1 r + τ i + k - r τ i + k - τ i + 1   N i + 1 , k - 1 r ,
i = 1 a r min r max   k ˆ λ j ,   r N ik r d r   b i γ = y λ j + δ j ,   j = 1 ,     ,   m = N + M ,
A = UD m , a V T = i = 1 min m , a ν i u i v i T ,
b γ = A γ y + δ = V D a , m γ U T y + δ ,
D a , m γ = diag 1 ν 1 ,     ,   1 ν min m , a - γ ,   0 ,     ,   0 ,
1 ν j : = 1 ν j ν j > 0 0 ν j = 0 ,
y λ j = r min r max   K ˜ v r ,   λ ;   m v r d r ,
K ˜ v r ,   λ j ;   m   : = K π v r ,   λ j ;   m λ j Λ l π K ext v r ,   λ j ;   m λ j Λ l ext ,
K π / ext v r ,   λ ;   m = 3 4 r   Q π / ext r ,   λ ;   m .
n r = 1 r 1 2 π ln   σ exp - 0.5   ln   r - ln   r mod 2 ln 2   σ ,
n r = j = 1 2 1 r 1 2 π ln   σ j exp - 0.5   ln   r - ln   r mod j 2 ln 2 σ j ,
n r = p 1 r p 2   exp - p 3 r p 4 ,
r eff =   n r r 3 d r   n r r 2 d r ,
a t = 4 π     n r r 2 d r ,
v t = 4 π 3     n r r 3 d r
n t =   n r d r .

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