Abstract

The hybrid regularization technique developed at the Institute of Mathematics of Potsdam University (IMP) is used to derive microphysical properties such as effective radius, surface-area concentration, and volume concentration, as well as the single-scattering albedo and a mean complex refractive index, from multiwavelength lidar measurements. We present the continuation of investigations of the IMP method. Theoretical studies of the degree of ill-posedness of the underlying model, simulation results with respect to the analysis of the retrieval error of microphysical particle properties from multiwavelength lidar data, and a comparison of results for different numbers of backscatter and extinction coefficients are presented. Our analysis shows that the backscatter operator has a smaller degree of ill-posedness than the operator for extinction. This fact underlines the importance of backscatter data. Moreover, the degree of ill-posedness increases with increasing particle absorption, i.e., depends on the imaginary part of the refractive index and does not depend significantly on the real part. Furthermore, an extensive simulation study was carried out for logarithmic-normal size distributions with different median radii, mode widths, and real and imaginary parts of refractive indices. The errors of the retrieved particle properties obtained from the inversion of three backscatter (355, 532, and 1064 nm) and two extinction (355 and 532 nm) coefficients were compared with the uncertainties for the case of six backscatter (400, 710, 800 nm, additionally) and the same two extinction coefficients. For known complex refractive index and up to 20% normally distributed noise, we found that the retrieval errors for effective radius, surface-area concentration, and volume concentration stay below approximately 15% in both cases. Simulations were also made with unknown complex refractive index. In that case the integrated parameters stay below approximately 30%, and the imaginary part of the refractive index stays below 35% for input noise up to 10% in both cases. In general, the quality of the retrieved aerosol parameters depends strongly on the imaginary part owing to the degree of ill-posedness. It is shown that under certain constraints a minimum data set of three backscatter coefficients and two extinction coefficients is sufficient for a successful inversion. The IMP algorithm was finally tested for a measurement case.

© 2005 Optical Society of America

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  27. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, 1996).
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    [CrossRef]
  29. R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
    [CrossRef]
  30. G. M. Wing, “Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind,” J. Integral Equ. 9, 191–204 (1985).
  31. P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–875 (1992).
    [CrossRef]
  32. J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
    [CrossRef]
  33. D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
    [CrossRef]

2004 (3)

2003 (1)

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

2002 (3)

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, D. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
[CrossRef] [PubMed]

G. Lesins, P. Chylek, U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. 107, 4094, doi:10.1029/2001JD000973 (2002).
[CrossRef]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

2001 (1)

2000 (2)

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

1999 (2)

1998 (1)

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

1997 (1)

I. Schult, J. Feichter, W. F. Cooke, “Effect of black carbon and sulfate aerosols on the global radiation budget,” J. Geophys. Res. 102, 30107–30117 (1997).
[CrossRef]

1992 (1)

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–875 (1992).
[CrossRef]

1988 (1)

P. C. Hansen, “Computation of the singular value expansion,” Computing 40, 185–199 (1988).
[CrossRef]

1985 (2)

R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

G. M. Wing, “Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind,” J. Integral Equ. 9, 191–204 (1985).

1959 (1)

S.-H. Chang, “A generalization of a theorem of Hille and Tamarkin with applications,” Proc. London Math. Soc. 3, 22–29 (1959).

1933 (1)

W. F. Foshag, “New mineral names,” Am. Mineral. 18, 179–180 (1933).

1908 (1)

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

Allen, R. C.

R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

Althausen, D.

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

Amiridis, V.

Amodeo, A.

Andreae, M. O.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Ansmann, A.

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. de Tomasi, M. Frioud, I. V. Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodriguez, J. Schneider, V. Shcherbakov, M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977–989 (2004).
[CrossRef] [PubMed]

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999).
[CrossRef]

Balin, I.

Balis, D.

Böckmann, C.

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 values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

Boselli, A.

Bösenberg, J.

Bukowiecki, N.

N. Bukowiecki, “Mobile pollutant measurement laboratories—spatial distribution and seasonal variation of aerosol parameters in the Zürich (Switzerland) and Minneapolis (USA) area,” Ph.D. thesis (Swiss Federal Institute of Technology, Zurich, 2003).

Chang, S.-H.

S.-H. Chang, “A generalization of a theorem of Hille and Tamarkin with applications,” Proc. London Math. Soc. 3, 22–29 (1959).

Chaykovski, A.

Chourdakis, G.

Chylek, P.

G. Lesins, P. Chylek, U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. 107, 4094, doi:10.1029/2001JD000973 (2002).
[CrossRef]

Clauder, E.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

Collins, D. R.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Comeron, A.

Cooke, W. F.

I. Schult, J. Feichter, W. F. Cooke, “Effect of black carbon and sulfate aerosols on the global radiation budget,” J. Geophys. Res. 102, 30107–30117 (1997).
[CrossRef]

de Hoog, F. R.

F. R. de Hoog, “Review of Fredholm equations of the first kind,” in The Application and Numerical Solution of Integral Equations, R. S. Anderssen, F. R. de Hoog, M. A. Lukas, eds. (Sijthoff Noordhoff, Leyden, The Netherlands, 1980), pp. 119–134.

de Tomasi, F.

Delaval, A.

Durkee, P. A.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Eckhardt, S.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Eixmann, R.

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, 1996).

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

Faber, V.

R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

Feichter, J.

I. Schult, J. Feichter, W. F. Cooke, “Effect of black carbon and sulfate aerosols on the global radiation budget,” J. Geophys. Res. 102, 30107–30117 (1997).
[CrossRef]

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” (AFGL-TR-79-0214, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass. 01731, 1979).

Flagan, R. C.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Formenti, P.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Foshag, W. F.

W. F. Foshag, “New mineral names,” Am. Mineral. 18, 179–180 (1933).

Freudenthaler, V.

Frioud, M.

Gasso, S.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Gordon, H. R.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Griaznov, V.

Grigorov, I. V.

Hågård, A.

Hanke, M.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, 1996).

Hansen, P. C.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–875 (1992).
[CrossRef]

P. C. Hansen, “Computation of the singular value expansion,” Computing 40, 185–199 (1988).
[CrossRef]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

Hegg, D. A.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Heintzenberg, J.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Hess, M.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Horvat, M.

Horvath, H.

H. Horvath, “Influence of atmospheric aerosols upon the global radiation balance,” in Atmospheric Particles, R. M. Harrison, R. E. van Grieken, eds. (Wiley, New York, 1998), pp. 543–596.

Hube, H.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

Huffman, D. R.

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

Iarlori, M.

Jonsson, H. H.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Koepke, P.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Kolgotin, A.

Komguem, L.

Kreipl, S.

Kress, R.

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

Larcheveque, G.

Lesins, G.

G. Lesins, P. Chylek, U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. 107, 4094, doi:10.1029/2001JD000973 (2002).
[CrossRef]

Livingston, J. M.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Lohmann, U.

G. Lesins, P. Chylek, U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. 107, 4094, doi:10.1029/2001JD000973 (2002).
[CrossRef]

Louis, A. K.

A. K. Louis, Inverse und schlecht gestellte Probleme (Teubner, Stuttgart, Germany, 1989).

Matthey, R.

Matthias, V.

Mattis, I.

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

Mie, G.

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

Müller, D.

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, D. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
[CrossRef] [PubMed]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999).
[CrossRef]

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, 1996).

Noone, K. J.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Ostrom, E.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Pandis, S. N.

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley, New York, 1998).

Papayannis, A.

Pappalardo, G.

Rizi, V.

Rocadenbosch, F.

Rodriguez, J. A.

Russell, P. B.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Schmid, B.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Schneider, J.

Schult, I.

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

I. Schult, J. Feichter, W. F. Cooke, “Effect of black carbon and sulfate aerosols on the global radiation budget,” J. Geophys. Res. 102, 30107–30117 (1997).
[CrossRef]

Seinfeld, J. H.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley, New York, 1998).

Shcherbakov, V.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” (AFGL-TR-79-0214, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass. 01731, 1979).

Stohl, A.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Thomas, T.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Veselovskii, I.

Voss, K. J.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Wandinger, U.

V. Matthias, J. Bösenberg, V. Freudenthaler, A. Amodeo, I. Balin, D. Balis, A. Chaykovski, G. Chourdakis, A. Comeron, A. Delaval, F. de Tomasi, R. Eixmann, A. Hågård, L. Komguem, S. Kreipl, R. Matthey, V. Rizi, J. A. Rodriguez, U. Wandinger, X. Wang, “Aerosol lidar intercomparison in the framework of the EARLINET project. 1. Instruments,” Appl. Opt. 43, 961–976 (2004).
[CrossRef]

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. de Tomasi, M. Frioud, I. V. Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodriguez, J. Schneider, V. Shcherbakov, M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977–989 (2004).
[CrossRef] [PubMed]

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, D. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
[CrossRef] [PubMed]

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
[CrossRef]

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999).
[CrossRef]

Wang, X.

Wehner, B.

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Welton, E. J.

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

Wendisch, M.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Wex, H.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Whiteman, D.

Wiedensohler, A.

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

Wiegner, M.

Wing, G. M.

R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

G. M. Wing, “Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind,” J. Integral Equ. 9, 191–204 (1985).

Zörner, S.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

Am. Mineral. (1)

W. F. Foshag, “New mineral names,” Am. Mineral. 18, 179–180 (1933).

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (6)

D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter data by inversion with 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 with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
[CrossRef]

C. Böckmann, “Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distribution,” Appl. Opt. 40, 1329–1342 (2001).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, D. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
[CrossRef] [PubMed]

V. Matthias, J. Bösenberg, V. Freudenthaler, A. Amodeo, I. Balin, D. Balis, A. Chaykovski, G. Chourdakis, A. Comeron, A. Delaval, F. de Tomasi, R. Eixmann, A. Hågård, L. Komguem, S. Kreipl, R. Matthey, V. Rizi, J. A. Rodriguez, U. Wandinger, X. Wang, “Aerosol lidar intercomparison in the framework of the EARLINET project. 1. Instruments,” Appl. Opt. 43, 961–976 (2004).
[CrossRef]

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. de Tomasi, M. Frioud, I. V. Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodriguez, J. Schneider, V. Shcherbakov, M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977–989 (2004).
[CrossRef] [PubMed]

Bull. Am. Meteorol. Soc. (1)

M. Hess, P. Koepke, I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Meteorol. Soc. 79, 831–844 (1998).
[CrossRef]

Computing (1)

P. C. Hansen, “Computation of the singular value expansion,” Computing 40, 185–199 (1988).
[CrossRef]

Geophys. Res. Lett. (1)

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, doi: (2002).
[CrossRef]

Inverse Probl. (1)

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–875 (1992).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning six-wavelength eleven-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[CrossRef]

J. Geophys. Res. (3)

G. Lesins, P. Chylek, U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. 107, 4094, doi:10.1029/2001JD000973 (2002).
[CrossRef]

I. Schult, J. Feichter, W. F. Cooke, “Effect of black carbon and sulfate aerosols on the global radiation budget,” J. Geophys. Res. 102, 30107–30117 (1997).
[CrossRef]

D. Müller, I. Mattis, A. Ansmann, B. Wehner, D. Althausen, U. Wandinger, “Closure study on optical and microphysical properties of a mixed urban and Arctic haze air mass observed with Raman lidar and Sun photometer,” J. Geophys. Res. 109, D13206, doi: (2004).
[CrossRef]

J. Integral Equ. (1)

G. M. Wing, “Condition numbers of matrices arising from the numerical solution of linear integral equations of the first kind,” J. Integral Equ. 9, 191–204 (1985).

J. Math. Anal. Appl. (1)

R. C. Allen, W. R. Boland, V. Faber, G. M. Wing, “Singular values and condition numbers of Galerkin matrices arising from linear integral equations of the first kind,” J. Math. Anal. Appl. 109, 564–590 (1985).
[CrossRef]

Proc. London Math. Soc. (1)

S.-H. Chang, “A generalization of a theorem of Hille and Tamarkin with applications,” Proc. London Math. Soc. 3, 22–29 (1959).

Tellus, Ser. B (2)

B. Schmid, J. M. Livingston, P. B. Russell, P. A. Durkee, H. H. Jonsson, D. R. Collins, R. C. Flagan, J. H. Seinfeld, S. Gasso, D. A. Hegg, E. Ostrom, K. J. Noone, E. J. Welton, K. J. Voss, H. R. Gordon, P. Formenti, M. O. Andreae, “Clear-sky closure studies of lower tropospheric aerosol and water vapor during ACE-2 using airborne sunphotometer, airborne in-situ, space-borne, and ground-based measurements,” Tellus, Ser. B 52, 568–593 (2000).
[CrossRef]

J. Heintzenberg, T. Thomas, B. Wehner, A. Wiedensohler, H. Wex, A. Ansmann, I. Mattis, D. Müller, M. Wendisch, S. Eckhardt, A. Stohl, “Arctic haze over Central Europa,” Tellus, Ser. B 55, 796–807 (2003).
[CrossRef]

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H. Horvath, “Influence of atmospheric aerosols upon the global radiation balance,” in Atmospheric Particles, R. M. Harrison, R. E. van Grieken, eds. (Wiley, New York, 1998), pp. 543–596.

J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley, New York, 1998).

The Intergovernmental Panel on Climate Change (IPCC), IPCC Third Assessment Report—Climate Change 2001: The Scientific Basis (Cambridge U. Press, Cambridge, UK, 2001).

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

F. R. de Hoog, “Review of Fredholm equations of the first kind,” in The Application and Numerical Solution of Integral Equations, R. S. Anderssen, F. R. de Hoog, M. A. Lukas, eds. (Sijthoff Noordhoff, Leyden, The Netherlands, 1980), pp. 119–134.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, 1996).

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” (AFGL-TR-79-0214, U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass. 01731, 1979).

N. Bukowiecki, “Mobile pollutant measurement laboratories—spatial distribution and seasonal variation of aerosol parameters in the Zürich (Switzerland) and Minneapolis (USA) area,” Ph.D. thesis (Swiss Federal Institute of Technology, Zurich, 2003).

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

A. K. Louis, Inverse und schlecht gestellte Probleme (Teubner, Stuttgart, Germany, 1989).

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1998).

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Figures (11)

Fig. 1
Fig. 1

(a) Backscatter and (b) extinction number kernel functions K π / ext n for ( m 1 = 1.5 + 0.0 i ) and (c), (e) backscatter and (d), (f) extinction volume kernel functions K π / ext v for different refractive indices: (c), (d) without absorption ( m 1 = 1.5 + 0.0 i ) and (e), (f) with strong absorption ( m 2 = 1.5 + 0.5 i ) .

Fig. 2
Fig. 2

Qualitative approximations to six right singular functions v i for i = 1 , 5, 10, 15, 20, 25 of the volume backscatter kernel, (a) m 1 = 1.5 + 0.0 i and (b) m 2 = 1.5 + 0.5 i . We see the typical behavior of increasing oscillations.

Fig. 3
Fig. 3

Approximations to the singular values and to the degree of ill-posedness of the volume backscatter and extinction kernels K π / ext v , (a), (b) without absorption ( m 1 = 1.5 + 0.0 i ) and (c), (d) with strong absorption ( m 2 = 1.5 + 0.5 i ) . The values c ( n ) , n = 10 , 15, 20, 25, 30 are the condition numbers of the resulting coefficient matrices; see Eq. (7). The numbers follow from Galerkin discretization in dependence on the discretization dimension n.

Fig. 4
Fig. 4

Degree of ill-posedness of the backscatter and extinction volume kernels in dependence on the real and imaginary parts of the refractive index. There is (a) no significant influence of the real part but (b) significant influence of the imaginary part.

Fig. 5
Fig. 5

Simulation scheme for the case of unknown refractive index. ssa, single-scattering albedo.

Fig. 6
Fig. 6

Mean relative errors of the particle properties derived with known refractive index for the 3 + 2 case (circles) and the 6 + 2 case (squares). The uncertainty bars correspond to the standard deviation.

Fig. 7
Fig. 7

Dependence of retrieval errors on the imaginary part of the refractive index without input noise.

Fig. 8
Fig. 8

Mean relative errors of the properties inverted with unknown refractive index for the 3 + 2 case (circles) and the 6 + 2 case (squares). The uncertainty bars correspond to the standard deviation.

Fig. 9
Fig. 9

Mean relative errors of the inverted refractive indices for the 3 + 2 case (circles) and the 6 + 2 case (squares). The uncertainty bars correspond to the standard deviation.

Fig. 10
Fig. 10

Solution domain of the refractive index for the measurement case.

Fig. 11
Fig. 11

Screen shot of the developed software. The plot in front shows the result browser with results from a simulation. The plot in the background provides a partial view on the main window.

Tables (2)

Tables Icon

Table 1 Log-Normal Distributions Used for the Simulations

Tables Icon

Table 2 Retrieval Results for the Measurement Case from April 8, 2002, Carried Out with the Raman Lidar at IfT a

Equations (18)

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n ( r ) = N t r 1 2 π ln σ exp - 0.5   ( ln r - ln r med ) 2 ln 2   σ ,
Γ ( λ ,   z ) = r 0 r 1 K π / ext n ( r ,   λ ,   m ,   s ) n ( r ,   z ) d r = r 0 r 1 π r 2 Q π / 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 - b n ) 2 ,
Q ext = 2 k 2 r 2   n = 1 ( 2 n + 1 ) Re ( a n + b n ) ,
Γ ( λ ,   z ) = r 0 r 1 K π / ext v ( r ,   λ ,   m ) v ( r ,   z ) d r = r 0 r 1   3 4 r   Q π / ext ( r ,   λ ,   m ) v ( r ,   z ) d r ,
a t = 3   v ( r ) r   d r , v t = v ( r ) d r , r eff = 3   v t a t .
K ( r ,   λ ) = i = 1 μ i u i ( r ) v i ( λ ) ,
r I r = [ r 0 ,   r 1 ] , λ I λ = [ λ 0 ,   λ 1 ] ,
( A ) ij = ϕ i ,   K ψ j , i , j = 1 , , n .
A = UDV T = i = 1 n σ i u i v i T ,
u j ( r ) = i = 1 n ( U ) ij ψ i ( r ) , j = 1 , , n ,
v j ( λ ) = i = 1 n ( V ) ij ϕ i ( λ ) , j = 1 , , n .
σ i ( n ) σ i ( n + 1 ) μ i ,
0 μ i - σ i ( n ) ( K L 2 2 - A F 2 ) 1 / 2 Δ n ,
i = 1 , , n .
i = 1 n ( μ i - σ i ( n ) ) 2 Δ n 2 .
ψ i ( r ) = h r - 1 / 2 : r I r ( i ) i = 1 , , n 0 : otherwise ,
ϕ i ( λ ) = h λ - 1 / 2 : λ I λ ( i ) , i = 1 , , n 0 : otherwise .

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