Abstract

A method is proposed that permits one to retrieve physical parameters of tropospheric particle size distributions, e.g., effective radius, volume, surface-area, and number concentrations, as well as the mean complex refractive index on a routine basis from backscatter and extinction coefficients at multiple wavelengths. The optical data in terms of vertical profiles are derived from multiple-wavelength lidar measurements at 355, 400, 532, 710, 800, and 1064 nm for backscatter data and 355 and 532 nm for extinction data. The algorithm is based on the concept of inversion with regularization. Regularization is performed by generalized cross-validation. This method does not require knowledge of the shape of the particle size distribution and can handle measurement errors of the order of 20%. It is shown that at least two extinction data are necessary to retrieve the particle parameters to an acceptable accuracy. Simulations with monomodal and bimodal logarithmic-normal size distributions show that it is possible to derive effective radius, volume, and surface-area concentrations to an accuracy of ±50%, the real part of the complex refractive index to ±0.05, and the imaginary part to ±50%. Number concentrations may have errors larger than ±50%.

© 1999 Optical Society of America

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  1. R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
    [CrossRef] [PubMed]
  2. J. T. Kiel, B. P. Briegleb, “The relative roles of sulfate aerosols and greenhouse gases,” Science 260, 311–314 (1993).
    [CrossRef]
  3. J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
    [CrossRef] [PubMed]
  4. R. J. Charlson, J. Heintzenberg, eds., Aerosol Forcing of Climate (Wiley, Chichester, 1995).
  5. T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
    [CrossRef]
  6. P. V. Hobbs, B. J. Huebert, eds., “Atmospheric aerosols. A new focus of the International Global Atmospheric Chemistry Project (IGAC),” (IGAC Core Project Office, Massachusetts Institute of Technology, Cambridge, Mass., 1996).
  7. J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).
  8. P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).
  9. D. Müller, D. Althausen, U. Wandinger, A. Ansmann, “Multiple-wavelength aerosol lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).
  10. A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
    [CrossRef] [PubMed]
  11. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  12. S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, 95–109 (1965).
    [CrossRef]
  13. A. N. Tikhonov, V. Y. Arsenin, eds., Solution of Ill-Posed Problems (Wiley, New York, 1977).
  14. S. Twomey, ed., Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).
  15. V. E. Zuev, I. E. Naats, eds., Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  16. P. C. Sabatier, “Basic concepts and methods of inverse problems,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Hilger, Bristol, UK, 1987).
  17. A. K. Louis, ed., Inverse und schlecht gestellte Probleme (Teubner, Stuttgart, 1989).
    [CrossRef]
  18. J. Hadamard, “Sur les problémes aux derivees parielies et leur signification physique,” Bull. Univ. Princeton49–52 (1902).
  19. M. A. Box, B. H. J. McKellar, “Analytic inversion of multispectral extinction data in the anomalous diffraction approximation,” Opt. Lett. 3, 91–93 (1978).
    [CrossRef] [PubMed]
  20. A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1: Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).
  21. J. G. McWhirther, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  22. M. A. Box, B. H. J. McKellar, “Relationship between two analytic inversion formulas for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
    [CrossRef] [PubMed]
  23. C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
    [CrossRef] [PubMed]
  24. C. D. Capps, R. L. Henning, G. M. Hess, “Analytic inversion of remote-sensing data,” Appl. Opt. 21, 3581–3587 (1982).
    [CrossRef] [PubMed]
  25. J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
    [CrossRef] [PubMed]
  26. M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
    [CrossRef]
  27. G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using singular function theory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
    [CrossRef] [PubMed]
  28. A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory,” Appl. Opt. 27, 1235–1242 (1988).
    [CrossRef] [PubMed]
  29. A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 2: Use,” Appl. Opt. 27, 1243–1254 (1988).
    [CrossRef] [PubMed]
  30. B. P. Curry, “Constrained eigenfunction method for the inversion of remote sensing data: application to particle size determination from light scattering measurements,” Appl. Opt. 28, 1345–1355 (1989).
    [CrossRef] [PubMed]
  31. G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
    [CrossRef]
  32. B. T. Evans, G. R. Fournier, “Approximations of polydispersed extinction,” Appl. Opt. 35, 3281–3285 (1996).
    [CrossRef] [PubMed]
  33. G. P. Box, “Effects of smoothing and measurement-wavelength range on the accuracy of analytic eigenfunction inversions,” Appl. Opt. 34, 7787–7792 (1995).
    [CrossRef] [PubMed]
  34. D. P. Donovan, A. I. Carswell, “Retrieval of stratospheric aerosol physical properties using multiwavelength lidar backscatter and extinction measurements,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).
  35. 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]
  36. D. P. Donovan, A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. 36, 9406–9424 (1997).
    [CrossRef]
  37. G. Backus, F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 266, 169–205 (1968).
    [CrossRef]
  38. G. Backus, F. Gilbert, “Uniqueness of inaccurate gross earth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 (1970).
    [CrossRef]
  39. E. R. Westwater, A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1341–1348 (1973).
    [CrossRef]
  40. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of linear system produced by quadrature,” Assoc. Comput. Mach. 10, 97–101 (1963).
    [CrossRef]
  41. M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
    [CrossRef]
  42. R. J. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values,” SIAM J. Numer. Anal. 8, 616–622 (1971).
    [CrossRef]
  43. 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]
  44. H. Müller, “Die Bestimmbarkeit der atmosphärischen Aerosolgrößenverteilung mit Hilfe eines 4-Wellenlängen-Lidars,” Ph.D. dissertation (Wissenschaftliche Mitteilung Nr. 44, Universität München, München, Germany, 1981).
  45. E. F. Maher, N. M. Laird, “EM algorithm reconstruction of particle size distributions from diffusion battery data,” J. Aerosol Sci. 16, 557–570 (1985).
    [CrossRef]
  46. P. Rairoux, “Mesures par lidar de la pollution atmospherique et des parametres meteorologiques,” Ph.D. dissertation (These 955, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 1991).
  47. J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
    [CrossRef]
  48. C. Dellago, H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I. Basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
    [CrossRef]
  49. H. Horvath, C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—II. Case studies,” J. Aerosol Sci. 24, 143–154 (1993).
    [CrossRef]
  50. W. von Hoyningen-Huene, M. Wendisch, “Variability of aerosol optical parameters by advective processes,” Atmos. Environ. 28, 923–933 (1994).
    [CrossRef]
  51. G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
    [CrossRef]
  52. G. Feingold, C. J. Grund, “Feasibility of using multiwavelength lidar measurements to measure cloud condensation nuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 (1994).
    [CrossRef]
  53. B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
    [CrossRef]
  54. M. Wendisch, W. von Hoyningen-Huene, “Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements,” Atmos. Environ. 28, 785–792 (1994).
    [CrossRef]
  55. U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
    [CrossRef] [PubMed]
  56. 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]
  57. F. Sun, J. W. L. Lewis, “Simplex deconvolutions of particle-size distribution functions from optical measurements,” Appl. Opt. 34, 8437–8446 (1995).
    [CrossRef] [PubMed]
  58. 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]
  59. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1975).
    [CrossRef]
  60. S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distribution,” J. Comput. Phys. 18, 188–200 (1975).
    [CrossRef]
  61. O. V. Dubovik, T. V. Lapyonok, S. L. Oshchepkov, “Improved technique for data inversion: optical sizing of multicomponent aerosols,” Appl. Opt. 34, 8422–8436 (1995).
    [CrossRef] [PubMed]
  62. R. Lattes, J. L. Lions, eds., The Method of Quasi-reversibility—Applications to Partial Differential Equations (Elsevier, New York, 1969).
  63. M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
    [CrossRef]
  64. V. V. Veretennikov, V. S. Kozlov, I. E. Naats, V. Ya. Fadeev, “Optical studies of smoke aerosols: an inversion method and its applications,” Opt. Lett. 4, 411–413 (1979).
    [CrossRef] [PubMed]
  65. A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).
  66. 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]
  67. J. K. Wolfenbarger, J. H. Seinfeld, “Inversion of aerosol size distribution data,” J. Aerosol Sci. 21, 227–247 (1990).
    [CrossRef]
  68. J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solutions to the aerosol data inversion problem,” SIAM J. Sci. Stat. Comput. 12, 342–361 (1991).
    [CrossRef]
  69. U. Amato, M. F. Carfora, V. Cuomo, C. Serio, “Objective algorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452 (1995).
    [CrossRef] [PubMed]
  70. A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 5, 1035–1038 (1963).
  71. R. J. Hanson, J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equations of the first kind,” Numer. Math. 24, 291–307 (1975).
    [CrossRef]
  72. D. W. Cooper, L. A. Spielmann, “Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors,” Atmos. Environ. 10, 723–729 (1976).
    [CrossRef]
  73. S. K. Friedlander, ed., Smoke, Dust and Haze: Fundamentals of Aerosol Behavior (Wiley, New York, 1977).
  74. E. T. Jaynes, ed., Papers on Probability, Statistics and Statistical Physics (Synthese Library, Kluwer Academic, Dordrecht, The Netherlands, 1983).
  75. E. Yee, “On the interpretation of diffusion battery data,” J. Aerosol Sci. 20, 797–811 (1989).
    [CrossRef]
  76. U. Amato, W. Hughes, “Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Problems 7, 793–808 (1991).
    [CrossRef]
  77. P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
    [CrossRef]
  78. P. Paatero, “Extreme value estimation, a method for regularizing ill-posed inversion problems,” in Ill-posed Problems in Natural Sciences, Proceedings of the International Conference in Moscow, August 1991, A. N. Tikhonov, ed. (VSP, Utrecht, The Netherlands, 1991).
  79. C. Böckmann, J. Niebsch, “A mollifier method for aerosol size distribution,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).
  80. D. Müller, “Entwicklung eines Inversionsalgorithmus zur Bestimmung mikrophysikalischer Partikelparameter des atmosphärischen Aerosols aus koḿbinierten Mehrwellenlängen- und Raman-Lidarmessungen,” Ph.D. dissertation (Universität Leipzig, Leipzig, Germany, 1997).
  81. G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  82. P. Craven, G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 377–403 (1979).
    [CrossRef]
  83. K. S. Shifrin, I. G. Zolotov, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114–2124 (1996).
    [CrossRef] [PubMed]
  84. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  85. F. O’Sullivan, “A statistical perspective on ill-posed inverse problems,” Stat. Sci. 1, 502–527 (1986).
    [CrossRef]
  86. S.-Å. Gustafson, “Regularizing a Volterra integral equation problem by means of convex programming,” (University of Stavanger, Stavanger, Norway, 1991).
  87. D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
    [CrossRef]
  88. V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
    [CrossRef]
  89. D. Lesnic, L. Elliot, D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
    [CrossRef]
  90. R. R. Picard, R. D. Cook, “Cross-validation of regression models,” J. Am. Stat. Assoc. 79, 575–583 (1984).
    [CrossRef]
  91. A. R. Davies, R. S. Anderssen, “Optimization in the regularization of ill-posed problems,” (Centre for Mathematical Analysis, The Australian National University, Canberra, Australia, 1985).
  92. T. Lyche, K. Mørken, “A data reduction strategy for splines with applications to the approximation of functions and data,” J. Numer. Anal. 8, 185–208 (1988).
    [CrossRef]
  93. G. Wahba, Department of Statistics, University of Wisconsin–Madison, Madison, Wisc. 53706-1685 (personal communication, 1996).
  94. P. Paatero, Department of Physics, University of Helsinki, SF-00014 Helsinki, Finland (personal communication, 1996).
  95. D. M. Allen, “The relationship between variable selection and data augmentation and a method for prediction,” Technometrics 16, 125–127 (1974).
    [CrossRef]
  96. A. S. Householder, ed., The Theory of Matrices in Numerical Analysis (Blaisdell, New York, 1964).
  97. 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, Washington, D.C., 18–22 June 1989 (Institute of Electrical and Electronic Engineers, Service Center, Piscataway, N.J., 1989), p. 589.
  98. A. Ishimaru, R. J. Marks, L. Tsang, C. M. Lam, D. C. Park, S. Kitamura, “Particle-size distribution determination using optical sensing and neural networks,” Opt. Lett. 15, 1221–1223 (1990).
    [CrossRef] [PubMed]
  99. 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]
  100. T. Rother, German Remote Sensing Data Center, German Aerospace Research Establishment, D-17235 Neustrelitz, Germany (personal communication, 1998).

1997 (1)

1996 (7)

K. S. Shifrin, I. G. Zolotov, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114–2124 (1996).
[CrossRef] [PubMed]

B. T. Evans, G. R. Fournier, “Approximations of polydispersed extinction,” Appl. Opt. 35, 3281–3285 (1996).
[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]

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. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

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]

1995 (7)

1994 (5)

W. von Hoyningen-Huene, M. Wendisch, “Variability of aerosol optical parameters by advective processes,” Atmos. Environ. 28, 923–933 (1994).
[CrossRef]

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

G. Feingold, C. J. Grund, “Feasibility of using multiwavelength lidar measurements to measure cloud condensation nuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 (1994).
[CrossRef]

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

M. Wendisch, W. von Hoyningen-Huene, “Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements,” Atmos. Environ. 28, 785–792 (1994).
[CrossRef]

1993 (4)

C. Dellago, H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I. Basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[CrossRef]

H. Horvath, C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—II. Case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[CrossRef]

J. T. Kiel, B. P. Briegleb, “The relative roles of sulfate aerosols and greenhouse gases,” Science 260, 311–314 (1993).
[CrossRef]

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

1992 (4)

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
[CrossRef] [PubMed]

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

1991 (3)

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

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[CrossRef]

J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solutions to the aerosol data inversion problem,” SIAM J. Sci. Stat. Comput. 12, 342–361 (1991).
[CrossRef]

1990 (2)

1989 (3)

1988 (4)

T. Lyche, K. Mørken, “A data reduction strategy for splines with applications to the approximation of functions and data,” J. Numer. Anal. 8, 185–208 (1988).
[CrossRef]

A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory,” Appl. Opt. 27, 1235–1242 (1988).
[CrossRef] [PubMed]

A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 2: Use,” Appl. Opt. 27, 1243–1254 (1988).
[CrossRef] [PubMed]

P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
[CrossRef]

1987 (1)

1986 (3)

F. O’Sullivan, “A statistical perspective on ill-posed inverse problems,” Stat. Sci. 1, 502–527 (1986).
[CrossRef]

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

1985 (1)

E. F. Maher, N. M. Laird, “EM algorithm reconstruction of particle size distributions from diffusion battery data,” J. Aerosol Sci. 16, 557–570 (1985).
[CrossRef]

1984 (3)

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

R. R. Picard, R. D. Cook, “Cross-validation of regression models,” J. Am. Stat. Assoc. 79, 575–583 (1984).
[CrossRef]

J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
[CrossRef] [PubMed]

1982 (2)

1981 (1)

1979 (4)

V. V. Veretennikov, V. S. Kozlov, I. E. Naats, V. Ya. Fadeev, “Optical studies of smoke aerosols: an inversion method and its applications,” Opt. Lett. 4, 411–413 (1979).
[CrossRef] [PubMed]

M. A. Box, B. H. J. McKellar, “Relationship between two analytic inversion formulas for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
[CrossRef] [PubMed]

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

P. Craven, G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

1978 (4)

J. G. McWhirther, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

M. A. Box, B. H. J. McKellar, “Analytic inversion of multispectral extinction data in the anomalous diffraction approximation,” Opt. Lett. 3, 91–93 (1978).
[CrossRef] [PubMed]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1: Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).

1976 (1)

D. W. Cooper, L. A. Spielmann, “Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors,” Atmos. Environ. 10, 723–729 (1976).
[CrossRef]

1975 (3)

R. J. Hanson, J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equations of the first kind,” Numer. Math. 24, 291–307 (1975).
[CrossRef]

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1975).
[CrossRef]

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distribution,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

1974 (1)

D. M. Allen, “The relationship between variable selection and data augmentation and a method for prediction,” Technometrics 16, 125–127 (1974).
[CrossRef]

1973 (1)

E. R. Westwater, A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1341–1348 (1973).
[CrossRef]

1971 (1)

R. J. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values,” SIAM J. Numer. Anal. 8, 616–622 (1971).
[CrossRef]

1970 (1)

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

1968 (2)

G. Backus, F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 266, 169–205 (1968).
[CrossRef]

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

1965 (1)

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, 95–109 (1965).
[CrossRef]

1963 (2)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of linear system produced by quadrature,” Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 5, 1035–1038 (1963).

1962 (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

1902 (1)

J. Hadamard, “Sur les problémes aux derivees parielies et leur signification physique,” Bull. Univ. Princeton49–52 (1902).

Allen, D. M.

D. M. Allen, “The relationship between variable selection and data augmentation and a method for prediction,” Technometrics 16, 125–127 (1974).
[CrossRef]

Althausen, D.

D. Müller, D. Althausen, U. Wandinger, A. Ansmann, “Multiple-wavelength aerosol lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

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 Problems 7, 793–808 (1991).
[CrossRef]

Anderson, T. L.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Anderssen, R. S.

A. R. Davies, R. S. Anderssen, “Optimization in the regularization of ill-posed problems,” (Centre for Mathematical Analysis, The Australian National University, Canberra, Australia, 1985).

Ansmann, A.

Backus, G.

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

G. Backus, F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 266, 169–205 (1968).
[CrossRef]

Bashurova, V. S.

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[CrossRef]

Bassini, A.

Bates, T. S.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Ben-David, A.

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

Beyerle, G.

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

Böckmann, C.

C. Böckmann, J. Niebsch, “A mollifier method for aerosol size distribution,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

Bohren, C. F.

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

Box, G. P.

G. P. Box, “Effects of smoothing and measurement-wavelength range on the accuracy of analytic eigenfunction inversions,” Appl. Opt. 34, 7787–7792 (1995).
[CrossRef] [PubMed]

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

Box, M. A.

Briegleb, B. P.

J. T. Kiel, B. P. Briegleb, “The relative roles of sulfate aerosols and greenhouse gases,” Science 260, 311–314 (1993).
[CrossRef]

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

Capps, C. D.

Carfora, M. F.

Carswell, A. I.

D. P. Donovan, A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. 36, 9406–9424 (1997).
[CrossRef]

D. P. Donovan, A. I. Carswell, “Retrieval of stratospheric aerosol physical properties using multiwavelength lidar backscatter and extinction measurements,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

Castagnoli, F.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Cess, R. D.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Chahine, M. T.

Charlson, R. J.

J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Chaykovskiy, A. P.

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

Coakley, J. A.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Cohen, A.

E. R. Westwater, A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1341–1348 (1973).
[CrossRef]

Cook, R. D.

R. R. Picard, R. D. Cook, “Cross-validation of regression models,” J. Am. Stat. Assoc. 79, 575–583 (1984).
[CrossRef]

Cooper, D. W.

D. W. Cooper, L. A. Spielmann, “Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors,” Atmos. Environ. 10, 723–729 (1976).
[CrossRef]

Craven, P.

P. Craven, G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

Cuomo, V.

Curry, B. P.

Davies, A. R.

A. R. Davies, R. S. Anderssen, “Optimization in the regularization of ill-posed problems,” (Centre for Mathematical Analysis, The Australian National University, Canberra, Australia, 1985).

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

Del Guasta, M.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Dellago, C.

C. Dellago, H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I. Basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[CrossRef]

H. Horvath, C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—II. Case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[CrossRef]

Deshler, T.

Dickinson, R. E.

J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
[CrossRef] [PubMed]

Dlugi, R.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Dod, R. L.

P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
[CrossRef]

Donovan, D. P.

D. P. Donovan, A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. 36, 9406–9424 (1997).
[CrossRef]

D. P. Donovan, A. I. Carswell, “Retrieval of stratospheric aerosol physical properties using multiwavelength lidar backscatter and extinction measurements,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

Dubovik, O. V.

Elliot, L.

D. Lesnic, L. Elliot, D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Evans, B. T.

Fadeev, V. Ya.

Feingold, G.

G. Feingold, C. J. Grund, “Feasibility of using multiwavelength lidar measurements to measure cloud condensation nuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 (1994).
[CrossRef]

Ferri, F.

Fournier, G. R.

Fymat, A. L.

Gallo, K. P.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Gilbert, F.

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

G. Backus, F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 266, 169–205 (1968).
[CrossRef]

Goldfarb, S.

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

Golub, G. H.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Graf, H.-F.

J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

Grund, C. J.

G. Feingold, C. J. Grund, “Feasibility of using multiwavelength lidar measurements to measure cloud condensation nuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 (1994).
[CrossRef]

Gustafson, S.-Å.

S.-Å. Gustafson, “Regularizing a Volterra integral equation problem by means of convex programming,” (University of Stavanger, Stavanger, Norway, 1991).

Hadamard, J.

J. Hadamard, “Sur les problémes aux derivees parielies et leur signification physique,” Bull. Univ. Princeton49–52 (1902).

Hales, J. M.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Hallett, F. R.

Hansen, J. E.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Hanson, R. J.

R. J. Hanson, J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equations of the first kind,” Numer. Math. 24, 291–307 (1975).
[CrossRef]

R. J. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values,” SIAM J. Numer. Anal. 8, 616–622 (1971).
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Heintzenberg, J.

J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

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]

Henning, R. L.

Herman, B. M.

Hess, G. M.

Hofmann, D. J.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Horvath, H.

C. Dellago, H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I. Basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[CrossRef]

H. Horvath, C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—II. Case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[CrossRef]

Huffman, D. R.

C. 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 Problems 7, 793–808 (1991).
[CrossRef]

Ingham, D. B.

D. Lesnic, L. Elliot, D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Ishimaru, A.

Ivanov, A. P.

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

Jones, P. D.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Karl, T. R.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Kiel, J. T.

J. T. Kiel, B. P. Briegleb, “The relative roles of sulfate aerosols and greenhouse gases,” Science 260, 311–314 (1993).
[CrossRef]

King, M. D.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[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 networks,” 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, Washington, D.C., 18–22 June 1989 (Institute of Electrical and Electronic Engineers, Service Center, Piscataway, N.J., 1989), p. 589.

Klett, J. D.

Knight, R. W.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Knudsen, B.

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

Kolenda, J.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Koutzenogil, K. P.

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[CrossRef]

Kozlov, V. S.

Kukla, G.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Kulmala, M.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Laird, N. M.

E. F. Maher, N. M. Laird, “EM algorithm reconstruction of particle size distributions from diffusion battery data,” J. Aerosol Sci. 16, 557–570 (1985).
[CrossRef]

Lam, C. M.

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1975).
[CrossRef]

Lapyonok, T. V.

Lesnic, D.

D. Lesnic, L. Elliot, D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

Lewis, J. W. L.

Lindseay, J.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Lyche, T.

T. Lyche, K. Mørken, “A data reduction strategy for splines with applications to the approximation of functions and data,” J. Numer. Anal. 8, 185–208 (1988).
[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]

Maher, E. F.

E. F. Maher, N. M. Laird, “EM algorithm reconstruction of particle size distributions from diffusion battery data,” J. Aerosol Sci. 16, 557–570 (1985).
[CrossRef]

Marks, R. J.

McKellar, B. H. J.

McWhirther, J. G.

J. G. McWhirther, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Michaelis, W.

Mielke, B.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

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]

Morandi, M.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Mørken, K.

T. Lyche, K. Mørken, “A data reduction strategy for splines with applications to the approximation of functions and data,” J. Numer. Anal. 8, 185–208 (1988).
[CrossRef]

Müller, D.

D. Müller, “Entwicklung eines Inversionsalgorithmus zur Bestimmung mikrophysikalischer Partikelparameter des atmosphärischen Aerosols aus koḿbinierten Mehrwellenlängen- und Raman-Lidarmessungen,” Ph.D. dissertation (Universität Leipzig, Leipzig, Germany, 1997).

D. Müller, D. Althausen, U. Wandinger, A. Ansmann, “Multiple-wavelength aerosol lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

Müller, H.

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]

H. Müller, “Die Bestimmbarkeit der atmosphärischen Aerosolgrößenverteilung mit Hilfe eines 4-Wellenlängen-Lidars,” Ph.D. dissertation (Wissenschaftliche Mitteilung Nr. 44, Universität München, München, Germany, 1981).

Naats, I. E.

Nakane, H.

Neuber, R.

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

Niebsch, J.

C. Böckmann, J. Niebsch, “A mollifier method for aerosol size distribution,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

Nychka, D.

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

O’Neill, C. A.

J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
[CrossRef] [PubMed]

O’Sullivan, F.

F. O’Sullivan, “A statistical perspective on ill-posed inverse problems,” Stat. Sci. 1, 502–527 (1986).
[CrossRef]

Oshchepkov, S. L.

Osipenko, F. P.

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

Paatero, P.

P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
[CrossRef]

P. Paatero, “Extreme value estimation, a method for regularizing ill-posed inversion problems,” in Ill-posed Problems in Natural Sciences, Proceedings of the International Conference in Moscow, August 1991, A. N. Tikhonov, ed. (VSP, Utrecht, The Netherlands, 1991).

P. Paatero, Department of Physics, University of Helsinki, SF-00014 Helsinki, Finland (personal communication, 1996).

Paganini, E.

Park, D. C.

Penner, J. E.

J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
[CrossRef] [PubMed]

Peterson, T. C.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Phillips, J. L.

R. J. Hanson, J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equations of the first kind,” Numer. Math. 24, 291–307 (1975).
[CrossRef]

Picard, R. R.

R. R. Picard, R. D. Cook, “Cross-validation of regression models,” J. Am. Stat. Assoc. 79, 575–583 (1984).
[CrossRef]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

J. G. McWhirther, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Plummer, N.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[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]

Pugh, T.

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

Pusep, A. Y.

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[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, Washington, D.C., 18–22 June 1989 (Institute of Electrical and Electronic Engineers, Service Center, Piscataway, N.J., 1989), p. 589.

Quenzel, H.

Quinn, P. K.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Rairoux, P.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

P. Rairoux, “Mesures par lidar de la pollution atmospherique et des parametres meteorologiques,” Ph.D. dissertation (These 955, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 1991).

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Raunemaa, T.

P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
[CrossRef]

Razovayev, V.

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Reagan, J. A.

Reichardt, J.

Riebesell, M.

Rother, T.

T. Rother, German Remote Sensing Data Center, German Aerospace Research Establishment, D-17235 Neustrelitz, Germany (personal communication, 1998).

Russell, P. B.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Sabatier, P. C.

P. C. Sabatier, “Basic concepts and methods of inverse problems,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Hilger, Bristol, UK, 1987).

Sacco, V. M.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Sasano, Y.

Schrems, O.

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

Schwartz, S. E.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Sealey, K. M.

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

Seinfeld, J. H.

J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solutions to the aerosol data inversion problem,” SIAM J. Sci. Stat. Comput. 12, 342–361 (1991).
[CrossRef]

J. K. Wolfenbarger, J. H. Seinfeld, “Inversion of aerosol size distribution data,” J. Aerosol Sci. 21, 227–247 (1990).
[CrossRef]

Serio, C.

Shcherbakov, V. N.

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

Shifrin, K. S.

Shokirev, N. V.

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[CrossRef]

Smith, C. B.

Spielmann, L. A.

D. W. Cooper, L. A. Spielmann, “Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors,” Atmos. Environ. 10, 723–729 (1976).
[CrossRef]

Stefanutti, L.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Stein, B.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Sun, F.

Swietlicki, E.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Thomalla, E.

Tikhonov, A. N.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 5, 1035–1038 (1963).

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.

Twomey, S.

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distribution,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, 95–109 (1965).
[CrossRef]

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of linear system produced by quadrature,” Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

Venturi, V.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Veretennikov, V. V.

Viera, G.

von Hoyningen-Huene, W.

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

M. Wendisch, W. von Hoyningen-Huene, “Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements,” Atmos. Environ. 28, 785–792 (1994).
[CrossRef]

W. von Hoyningen-Huene, M. Wendisch, “Variability of aerosol optical parameters by advective processes,” Atmos. Environ. 28, 923–933 (1994).
[CrossRef]

Wahba, G.

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

P. Craven, G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

G. Wahba, Department of Statistics, University of Wisconsin–Madison, Madison, Wisc. 53706-1685 (personal communication, 1996).

Wandinger, U.

Wang, J.

Warneck, P.

J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

Weidauer, D.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Weitkamp, C.

Wendisch, M.

W. von Hoyningen-Huene, M. Wendisch, “Variability of aerosol optical parameters by advective processes,” Atmos. Environ. 28, 923–933 (1994).
[CrossRef]

M. Wendisch, W. von Hoyningen-Huene, “Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements,” Atmos. Environ. 28, 785–792 (1994).
[CrossRef]

Westwater, E. R.

E. R. Westwater, A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1341–1348 (1973).
[CrossRef]

Wittrock, F.

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

Wolf, J. P.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Wolfenbarger, J. K.

J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solutions to the aerosol data inversion problem,” SIAM J. Sci. Stat. Comput. 12, 342–361 (1991).
[CrossRef]

J. K. Wolfenbarger, J. H. Seinfeld, “Inversion of aerosol size distribution data,” J. Aerosol Sci. 21, 227–247 (1990).
[CrossRef]

Wöste, L.

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Yee, E.

E. Yee, “On the interpretation of diffusion battery data,” J. Aerosol Sci. 20, 797–811 (1989).
[CrossRef]

Zolotov, I. G.

Zuccagnoli, L.

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

Am. J. Math. (1)

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1975).
[CrossRef]

Appl. Opt. (23)

E. R. Westwater, A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol size distributions from optical scattering measurements,” Appl. Opt. 12, 1341–1348 (1973).
[CrossRef]

M. A. Box, B. H. J. McKellar, “Relationship between two analytic inversion formulas for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
[CrossRef] [PubMed]

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]

C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
[CrossRef] [PubMed]

C. D. Capps, R. L. Henning, G. M. Hess, “Analytic inversion of remote-sensing data,” Appl. Opt. 21, 3581–3587 (1982).
[CrossRef] [PubMed]

J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
[CrossRef] [PubMed]

G. Viera, M. A. Box, “Information content analysis of aerosol remote sensing experiments using singular function theory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
[CrossRef] [PubMed]

A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory,” Appl. Opt. 27, 1235–1242 (1988).
[CrossRef] [PubMed]

A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 2: Use,” Appl. Opt. 27, 1243–1254 (1988).
[CrossRef] [PubMed]

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

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131 (1992).
[CrossRef] [PubMed]

D. P. Donovan, A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. 36, 9406–9424 (1997).
[CrossRef]

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

G. P. Box, “Effects of smoothing and measurement-wavelength range on the accuracy of analytic eigenfunction inversions,” Appl. Opt. 34, 7787–7792 (1995).
[CrossRef] [PubMed]

U. Wandinger, A. Ansmann, J. Reichardt, T. Deshler, “Determination of stratospheric aerosol microphysical properties from independent extinction and backscattering measurements with a Raman lidar,” Appl. Opt. 34, 8315–8329 (1995).
[CrossRef] [PubMed]

O. V. Dubovik, T. V. Lapyonok, S. L. Oshchepkov, “Improved technique for data inversion: optical sizing of multicomponent aerosols,” Appl. Opt. 34, 8422–8436 (1995).
[CrossRef] [PubMed]

F. Sun, J. W. L. Lewis, “Simplex deconvolutions of particle-size distribution functions from optical measurements,” Appl. Opt. 34, 8437–8446 (1995).
[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. S. Shifrin, I. G. Zolotov, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114–2124 (1996).
[CrossRef] [PubMed]

B. T. Evans, G. R. Fournier, “Approximations of polydispersed extinction,” Appl. Opt. 35, 3281–3285 (1996).
[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]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 1: Multispectral extinctions in the anomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 (1978).

Assoc. Comput. Mach. (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by inversion of linear system produced by quadrature,” Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

Atmos. Environ. (3)

W. von Hoyningen-Huene, M. Wendisch, “Variability of aerosol optical parameters by advective processes,” Atmos. Environ. 28, 923–933 (1994).
[CrossRef]

M. Wendisch, W. von Hoyningen-Huene, “Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements,” Atmos. Environ. 28, 785–792 (1994).
[CrossRef]

D. W. Cooper, L. A. Spielmann, “Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors,” Atmos. Environ. 10, 723–729 (1976).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer, V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, T. C. Peterson, “A new perspective on recent global warming,” Bull. Am. Meteorol. Soc. 74, 1007–1023 (1993).
[CrossRef]

Bull. Univ. Princeton (1)

J. Hadamard, “Sur les problémes aux derivees parielies et leur signification physique,” Bull. Univ. Princeton49–52 (1902).

Contrib. Atmos. Phys. (2)

J. Heintzenberg, H.-F. Graf, R. J. Charlson, P. Warneck, “Climate forcing and the physico-chemical life cycle of the atmospheric aerosol—Why do we need an integrated interdisciplinary global research programme?” Contrib. Atmos. Phys. 69, No. 2, 261–271 (1996).

P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heintzenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell, E. Swietlicki, “Closure in tropospheric aerosol-climate research: a review and future needs for addressing aerosol direct shortwave radiative forcing,” Contrib. Atmos. Phys. 69, No. 4, 547–577 (1996).

Geophys. J. R. Astron. Soc. (1)

G. Backus, F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 266, 169–205 (1968).
[CrossRef]

Geophys. Res. Lett. (2)

G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, B. Knudsen, “Multiwavelength lidar measurements of stratospheric aerosols above Spitsbergen during winter 1992/93,” Geophys. Res. Lett. 21, 57–60 (1994).
[CrossRef]

B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux, L. Stefanutti, J. P. Wolf, L. Wöste, “Stratospheric aerosol size distribution from multispectral lidar measurements at Sodankylä during EASOE,” Geophys. Res. Lett. 21, 1311–1314 (1994).
[CrossRef]

Inverse Problems (2)

M. Bertero, C. De Mol, E. R. Pike, “Particle size distribution from spectral turbidity: a singular-system analysis,” Inverse Problems 2, 247–258 (1986).
[CrossRef]

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

Izv. Atmos. Oceanic Phys. (1)

A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, V. N. Shcherbakov, “Study of the aerosol optical properties and microstructure by the method of multiwave sounding,” Izv. Atmos. Oceanic Phys. 22, 633–639 (1986).

J. Aerosol Sci. (7)

J. K. Wolfenbarger, J. H. Seinfeld, “Inversion of aerosol size distribution data,” J. Aerosol Sci. 21, 227–247 (1990).
[CrossRef]

P. Paatero, T. Raunemaa, R. L. Dod, “Composition characteristics of carbonaceous particle samples, analyzed by EVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226 (1988).
[CrossRef]

E. Yee, “On the interpretation of diffusion battery data,” J. Aerosol Sci. 20, 797–811 (1989).
[CrossRef]

D. Lesnic, L. Elliot, D. B. Ingham, “An inversion method for the determination of the particle size distribution from diffusion battery measurements,” J. Aerosol Sci. 26, 797–812 (1995).
[CrossRef]

E. F. Maher, N. M. Laird, “EM algorithm reconstruction of particle size distributions from diffusion battery data,” J. Aerosol Sci. 16, 557–570 (1985).
[CrossRef]

C. Dellago, H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I. Basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[CrossRef]

H. Horvath, C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—II. Case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[CrossRef]

J. Aerosol. Sci. (1)

V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, N. V. Shokirev, “Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: mathematical aspects,” J. Aerosol. Sci. 22, 373–388 (1991).
[CrossRef]

J. Am. Stat. Assoc. (1)

R. R. Picard, R. D. Cook, “Cross-validation of regression models,” J. Am. Stat. Assoc. 79, 575–583 (1984).
[CrossRef]

J. Am. Stat. Soc. (1)

D. Nychka, G. Wahba, S. Goldfarb, T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 (1984).
[CrossRef]

J. Assoc. Comput. Mach. (1)

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

J. Atmos. Oceanic Technol. (2)

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]

G. Feingold, C. J. Grund, “Feasibility of using multiwavelength lidar measurements to measure cloud condensation nuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 (1994).
[CrossRef]

J. Atmos. Sci. (2)

G. P. Box, K. M. Sealey, M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, “Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distribution,” J. Comput. Phys. 18, 188–200 (1975).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, 95–109 (1965).
[CrossRef]

J. Numer. Anal. (1)

T. Lyche, K. Mørken, “A data reduction strategy for splines with applications to the approximation of functions and data,” J. Numer. Anal. 8, 185–208 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

J. G. McWhirther, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[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]

Numer. Math. (2)

P. Craven, G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 377–403 (1979).
[CrossRef]

R. J. Hanson, J. L. Phillips, “An adaptive numerical method for solving linear Fredholm integral equations of the first kind,” Numer. Math. 24, 291–307 (1975).
[CrossRef]

Opt. Lett. (3)

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

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

Science (3)

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

J. T. Kiel, B. P. Briegleb, “The relative roles of sulfate aerosols and greenhouse gases,” Science 260, 311–314 (1993).
[CrossRef]

J. E. Penner, R. E. Dickinson, C. A. O’Neill, “Effects of aerosols from biomass burning on the global radiation budget,” Science 256, 1432–1434 (1992).
[CrossRef] [PubMed]

SIAM J. Numer. Anal. (1)

R. J. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values,” SIAM J. Numer. Anal. 8, 616–622 (1971).
[CrossRef]

SIAM J. Sci. Stat. Comput. (1)

J. K. Wolfenbarger, J. H. Seinfeld, “Regularized solutions to the aerosol data inversion problem,” SIAM J. Sci. Stat. Comput. 12, 342–361 (1991).
[CrossRef]

Sov. Math. Dokl. (1)

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 5, 1035–1038 (1963).

Stat. Sci. (1)

F. O’Sullivan, “A statistical perspective on ill-posed inverse problems,” Stat. Sci. 1, 502–527 (1986).
[CrossRef]

Technometrics (2)

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

D. M. Allen, “The relationship between variable selection and data augmentation and a method for prediction,” Technometrics 16, 125–127 (1974).
[CrossRef]

Other (26)

A. S. Householder, ed., The Theory of Matrices in Numerical Analysis (Blaisdell, New York, 1964).

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, Washington, D.C., 18–22 June 1989 (Institute of Electrical and Electronic Engineers, Service Center, Piscataway, N.J., 1989), p. 589.

R. Lattes, J. L. Lions, eds., The Method of Quasi-reversibility—Applications to Partial Differential Equations (Elsevier, New York, 1969).

G. Wahba, Department of Statistics, University of Wisconsin–Madison, Madison, Wisc. 53706-1685 (personal communication, 1996).

P. Paatero, Department of Physics, University of Helsinki, SF-00014 Helsinki, Finland (personal communication, 1996).

T. Rother, German Remote Sensing Data Center, German Aerospace Research Establishment, D-17235 Neustrelitz, Germany (personal communication, 1998).

A. R. Davies, R. S. Anderssen, “Optimization in the regularization of ill-posed problems,” (Centre for Mathematical Analysis, The Australian National University, Canberra, Australia, 1985).

S.-Å. Gustafson, “Regularizing a Volterra integral equation problem by means of convex programming,” (University of Stavanger, Stavanger, Norway, 1991).

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

S. K. Friedlander, ed., Smoke, Dust and Haze: Fundamentals of Aerosol Behavior (Wiley, New York, 1977).

E. T. Jaynes, ed., Papers on Probability, Statistics and Statistical Physics (Synthese Library, Kluwer Academic, Dordrecht, The Netherlands, 1983).

P. Paatero, “Extreme value estimation, a method for regularizing ill-posed inversion problems,” in Ill-posed Problems in Natural Sciences, Proceedings of the International Conference in Moscow, August 1991, A. N. Tikhonov, ed. (VSP, Utrecht, The Netherlands, 1991).

C. Böckmann, J. Niebsch, “A mollifier method for aerosol size distribution,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

D. Müller, “Entwicklung eines Inversionsalgorithmus zur Bestimmung mikrophysikalischer Partikelparameter des atmosphärischen Aerosols aus koḿbinierten Mehrwellenlängen- und Raman-Lidarmessungen,” Ph.D. dissertation (Universität Leipzig, Leipzig, Germany, 1997).

H. Müller, “Die Bestimmbarkeit der atmosphärischen Aerosolgrößenverteilung mit Hilfe eines 4-Wellenlängen-Lidars,” Ph.D. dissertation (Wissenschaftliche Mitteilung Nr. 44, Universität München, München, Germany, 1981).

P. Rairoux, “Mesures par lidar de la pollution atmospherique et des parametres meteorologiques,” Ph.D. dissertation (These 955, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 1991).

J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P. Wolf, L. Wöste, F. Castagnoli, M. Del Guasta, M. Morandi, V. M. Sacco, L. Stefanutti, V. Venturi, L. Zuccagnoli, “Aerosol size distribution measurements using a multispectral lidar-system,” in Lidar for Remote Sensing, R. J. Becherer, R. M. Hardesty, J. P. Meyzonette, eds., Proc. SPIE1714, 208–219 (1992).
[CrossRef]

R. J. Charlson, J. Heintzenberg, eds., Aerosol Forcing of Climate (Wiley, Chichester, 1995).

D. Müller, D. Althausen, U. Wandinger, A. Ansmann, “Multiple-wavelength aerosol lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

P. V. Hobbs, B. J. Huebert, eds., “Atmospheric aerosols. A new focus of the International Global Atmospheric Chemistry Project (IGAC),” (IGAC Core Project Office, Massachusetts Institute of Technology, Cambridge, Mass., 1996).

D. P. Donovan, A. I. Carswell, “Retrieval of stratospheric aerosol physical properties using multiwavelength lidar backscatter and extinction measurements,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1996).

A. N. Tikhonov, V. Y. Arsenin, eds., Solution of Ill-Posed Problems (Wiley, New York, 1977).

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

V. E. Zuev, I. E. Naats, eds., Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, Berlin, 1983).
[CrossRef]

P. C. Sabatier, “Basic concepts and methods of inverse problems,” in Basic Methods of Tomography and Inverse Problems, P. C. Sabatier, ed. (Hilger, Bristol, UK, 1987).

A. K. Louis, ed., Inverse und schlecht gestellte Probleme (Teubner, Stuttgart, 1989).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Inversion window and base functions therein (solid lines). Dotted curve, monomodal, logarithmic-normal size distribution function.

Fig. 2
Fig. 2

Qualitative illustration of the minimization concept: solid curve, penalty function ‖Awg2 of Eq. (14); dashed curve, penalty function Γ(v); dotted curve, term ‖Awg2 + γΓ(v) that has a minimum in the range 0 ≤ γ < ∞.

Fig. 3
Fig. 3

Effect of the value of the Lagrange multiplier γ on the quality of the inversion. Three different inversion results are shown for a given monomodal particle distribution [curve (D)] and Lagrange multipliers with values of (A) γ = 10-2, (B) γ = 102, (C) γ = 105.

Fig. 4
Fig. 4

Qualitative behavior of the GCV parameter depending on the Lagrange multiplier γ for the case of the inversion of a particle size distribution from six backscatter and two extinction coefficients: vertical line, global mimimum; (a) behavior of the GCV parameter in the entire range covered by the Lagrange multiplier; (b) area around the global minimum on a strongly enhanced resolution.

Fig. 5
Fig. 5

Behavior of the GCV parameter depending on the Lagrange multiplier γ in the case of six backscatter and two extinction coefficients if the position of the inversion window, expressed in terms of the positions of the supports [vertical lines in (a) and (b)], is varied: (a) interval division with which the structures of the (dotted curve) sought particle distribution can only barely be reproduced, (b) ideal position of the inversion window as well as the interval division, (c) GCV parameter depending on the Lagrange multiplier for the cases shown in (a) and (b). Vertical lines indicate the global minimum.

Fig. 6
Fig. 6

Behavior of the GCV parameter depending on the Lagrange multiplier γ in the case of 16 base functions for the inversion of six backscatter and two extinction coefficients.

Fig. 7
Fig. 7

Behavior of the GCV parameter depending on the Lagrange multiplier γ in the inversion of (solid curve) six backscatter and two extinction coefficients, (dashed curve) seven backscatter coefficients and one extinction coefficient, and (dotted curve) eight backscatter coefficients. Vertical lines denote the global minimum.

Fig. 8
Fig. 8

Inversion results for (A) a monomodal and (B) a bimodal distribution. The given distributions are shown as solid curves, the distributions from the inversion as dashed curves.

Fig. 9
Fig. 9

Calculation steps for inversion with regularization.

Equations (20)

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βλ=0 Kβr, m, λ, sfrdr,
αλ=0 Kαr, m, λ, sfrdr,
giλ=rminrmax Kir, m, λ, svrdr+iexpλ, i=β, α; λ=0.355,, 1.064 μm,
Kir, m, λ, s=34rQir, m, λ.
gp=rminrmax Kpr, mvrdr+pexp,
Kpr, m=34rQpr, m.
vr=j wjBjr+mathr.
B1r=0r<r11-r-r1r2-r1r1rr20r>r2, Bjr=0rrj-11-rj-rrj-rj-1rj-1<rrj1-r-rjrj+1-rjrj<rrj+10r>rj+1 j=2,,7, B8r=0r<r71-r8-rr8-r7r7rr80r>r8.
gp=j Apjmwj+p.
Apjm=rminrmax Kpr, mBjrdr.
g=Aw+.
w=A-1g + ,
e22=Aw-g2.
e2Aw-g2+γΓv.
Γv=wTHw,
H=1-2100000-25-4100001-46-4100001-46-4100001-46-4100001-46-4100001-45-2000001-21.
w=ATA+γH-1ATg.
PGCVγ=1kI-Mγg21ktraceI-Mγ2  min.
Mγ=AATA+γH-1AT,
γ=γ0 exp1kplnATApp, γ0=10-3,, 105.

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