Abstract

We consider the problem of retrieving the aerosol extinction coefficient from Raman lidar measurements. This is an ill–posed inverse problem that needs regularization, and we propose to use the Expectation–Maximization (EM) algorithm to provide stable solutions. Indeed, EM is an iterative algorithm that imposes a positivity constraint on the solution, and provides regularization if iterations are stopped early enough. We describe the algorithm and propose a stopping criterion inspired by a statistical principle. We then discuss its properties concerning the spatial resolution. Finally, we validate the proposed approach by using both synthetic data and experimental measurements; we compare the reconstructions obtained by EM with those obtained by the Tikhonov method, by the Levenberg-Marquardt method, as well as those obtained by combining data smoothing and numerical derivation.

© 2016 Optical Society of America

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References

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    [Crossref]
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  23. V. Shcherbakov, “Regularized algorithm for raman lidar data processing,” Appl. Opt. 46, 4879–4889 (2007).
    [Crossref] [PubMed]
  24. P. Pornsawad, C. Böckmann, C. Ritter, and M. Rafler, “Ill-posed retrieval of aerosol extinction coefficient profiles from raman lidar data by regularization,” Appl. Opt. 47, 1649–1661 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  28. H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
    [Crossref]
  29. A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
    [Crossref]
  30. X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
    [Crossref]
  31. A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
    [Crossref]

2015 (1)

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

2014 (2)

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

F. Benvenuto and M. Piana, “Regularization of multiplicative iterative algorithms with nonnegative constraint,” Inverse Probl. 30, 035012 (2014).
[Crossref]

2013 (1)

L. Osterloh, C. Böckmann, D. Nicolae, and A. Nemuc, “Regularized inversion of microphysical atmospheric particle parameters: Theory and application,” J. Comput. Phys. 237, 79–94 (2013).
[Crossref]

2012 (2)

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

P. Pornsawad, G. D’Amico, C. Böckmann, A. Amodeo, and G. Pappalardo, “Retrieval of aerosol extinction coefficient profiles from raman lidar data by inversion method,” Appl. Opt. 51, 2035–2044 (2012).
[Crossref] [PubMed]

2010 (1)

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

2009 (2)

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

2008 (1)

2007 (3)

V. Shcherbakov, “Regularized algorithm for raman lidar data processing,” Appl. Opt. 46, 4879–4889 (2007).
[Crossref] [PubMed]

Y. Wang, S. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[Crossref]

E. Resmerita, H. W. Engl, and A. N. Iusem, “The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis,” Inverse Probl. 23, 2575 (2007).
[Crossref]

2004 (3)

2003 (2)

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

J. Bösenberg and V. Matthias, “Earlinet: A european aerosol research lidar network to establish an aerosol climatology,” Report. Max-Planck-Institut fur Meteorologie 348, 1–191 (2003).

2002 (1)

H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
[Crossref]

1999 (2)

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

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

1997 (1)

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[Crossref]

1992 (2)

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and 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]

1990 (1)

1988 (1)

Agnew, J.

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

Aime, C.

H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
[Crossref]

Amiridis, V.

Amodeo, A.

Ansmann, A.

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

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370–5385 (2004).
[Crossref] [PubMed]

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

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and 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]

A. Ansmann, M. Riebesell, and C. Weitkamp, “Measurement of atmospheric aerosol extinction profiles with a raman lidar,” Opt. Lett. 15, 746–748 (1990).
[Crossref] [PubMed]

Armenante, M.

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

Arsenin, V. I.

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-posed Problems (VH Winston & Sons, 1977).

Baldasano, J. M.

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Balin, I.

Balis, D.

Benvenuto, F.

F. Benvenuto and M. Piana, “Regularization of multiplicative iterative algorithms with nonnegative constraint,” Inverse Probl. 30, 035012 (2014).
[Crossref]

Bertero, M.

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[Crossref]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC, 1998).
[Crossref]

Bhawar, R.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Boccacci, P.

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC, 1998).
[Crossref]

Böckmann, C.

Böhme, D.

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Boselli, A.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

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

Bösenberg, J.

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

V. Matthais, V. Freudenthaler, A. Amodeo, I. Balin, D. Balis, J. Bösenberg, A. Chaikovsky, G. Chourdakis, A. Comeron, A. Delaval, F. De Tomasi, R. Eixmann, A. Hagard, L. Komguem, S. Kreipl, R. Matthey, V. Rizi, J. A. Rodriguez, U. Wandinger, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 1. Instruments,” Appl. Opt. 43, 961–976 (2004).
[Crossref] [PubMed]

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370–5385 (2004).
[Crossref] [PubMed]

J. Bösenberg and V. Matthias, “Earlinet: A european aerosol research lidar network to establish an aerosol climatology,” Report. Max-Planck-Institut fur Meteorologie 348, 1–191 (2003).

Brown, J. C.

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Cacciani, M.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Chaikovsky, A.

Chourdakis, G.

Comeron, A.

D’Amico, G.

D’Avino, L.

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

D’Isidoro, M.

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

De Tomasi, F.

Delaval, A.

Di Girolamo, P.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Di Iorio, T.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Dubovik, O.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Eixmann, R.

Emslie, A. G.

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Engl, H. W.

E. Resmerita, H. W. Engl, and A. N. Iusem, “The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis,” Inverse Probl. 23, 2575 (2007).
[Crossref]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer Science & Business Media, 1996).
[Crossref]

Fan, S.

Y. Wang, S. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[Crossref]

Fang, H.

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

Feng, X.

Y. Wang, S. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[Crossref]

Freudenthaler, V.

Frioud, M.

Grainger, R.

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

Grigorov, I. V.

Hagard, A.

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer Science & Business Media, 1996).
[Crossref]

Horvat, M.

Iarlori, M.

Iarrlori, M.

Iusem, A. N.

E. Resmerita, H. W. Engl, and A. N. Iusem, “The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis,” Inverse Probl. 23, 2575 (2007).
[Crossref]

Kaprielov, B.

Karyampudi, V. M.

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

Kolev, I.

Kolgotin, A.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Komguem, L.

Kontar, E. P.

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Kreipl, S.

Kuhn, H. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (1951), pp. 481–492.

H. W. Kuhn, “Nonlinear Programming: A Historical View,” in Traces and Emergence of Nonlinear Programming (SpringerBasel, 2014), pp. 393–414.
[Crossref]

Lahmann, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

Lanteri, H.

H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
[Crossref]

Larcheveque, G.

Massone, A. M.

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Matthais, V.

Matthey, R.

Matthias, V.

Michaelis, W.

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and 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]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

Müller, D.

Nemuc, A.

L. Osterloh, C. Böckmann, D. Nicolae, and A. Nemuc, “Regularized inversion of microphysical atmospheric particle parameters: Theory and application,” J. Comput. Phys. 237, 79–94 (2013).
[Crossref]

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer Science & Business Media, 1996).
[Crossref]

Nicolae, D.

L. Osterloh, C. Böckmann, D. Nicolae, and A. Nemuc, “Regularized inversion of microphysical atmospheric particle parameters: Theory and application,” J. Comput. Phys. 237, 79–94 (2013).
[Crossref]

Osterloh, L.

L. Osterloh, C. Böckmann, D. Nicolae, and A. Nemuc, “Regularized inversion of microphysical atmospheric particle parameters: Theory and application,” J. Comput. Phys. 237, 79–94 (2013).
[Crossref]

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Palm, S. P.

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

Pan, C.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

Pandolfi, M.

Papayannis, A.

Pappalardo, G.

Parvanov, O.

Pérez, C.

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Peters, D.

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

Piana, M.

F. Benvenuto and M. Piana, “Regularization of multiplicative iterative algorithms with nonnegative constraint,” Inverse Probl. 30, 035012 (2014).
[Crossref]

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[Crossref]

Pisani, G.

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

Pornsawad, P.

Povey, A.

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

Rafler, M.

Reagen, J. A.

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

Resmerita, E.

E. Resmerita, H. W. Engl, and A. N. Iusem, “The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis,” Inverse Probl. 23, 2575 (2007).
[Crossref]

Riebesell, M.

Ritter, C.

Rizi, V.

Rocadenbosch, F.

Roche, M.

H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
[Crossref]

Rodrigues, J. A.

Rodriguez, J. A.

Sannino, A.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

Schneidenbach, L.

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Schneider, J.

Schwartz, R. A.

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Shcherbakov, V.

Song, C.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

Spinelli, N.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

Summa, D.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Talenti, G.

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-posed Problems (VH Winston & Sons, 1977).

Tucker, A. W.

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (1951), pp. 481–492.

Veselovskii, I.

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Vicente, D.

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

Voss, E.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

Wandinger, U.

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370–5385 (2004).
[Crossref] [PubMed]

V. Matthais, V. Freudenthaler, A. Amodeo, I. Balin, D. Balis, J. Bösenberg, A. Chaikovsky, G. Chourdakis, A. Comeron, A. Delaval, F. De Tomasi, R. Eixmann, A. Hagard, L. Komguem, S. Kreipl, R. Matthey, V. Rizi, J. A. Rodriguez, U. Wandinger, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 1. Instruments,” Appl. Opt. 43, 961–976 (2004).
[Crossref] [PubMed]

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

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

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and 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]

Wang, X.

Wang, Y.

Y. Wang, S. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[Crossref]

Weitkamp, C.

Wiegner, M.

Zanella, R.

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

Zanni, L.

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

Zhao, Y.

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

Appl. Opt. (9)

I. Kolev, O. Parvanov, and B. Kaprielov, “Lidar determination of winds by aerosol inhomogeneities: motion velocity in the planetary boundary layer,” Appl. Opt. 27, 2524–2531 (1988).
[Crossref] [PubMed]

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

V. Matthais, V. Freudenthaler, A. Amodeo, I. Balin, D. Balis, J. Bösenberg, A. Chaikovsky, G. Chourdakis, A. Comeron, A. Delaval, F. De Tomasi, R. Eixmann, A. Hagard, L. Komguem, S. Kreipl, R. Matthey, V. Rizi, J. A. Rodriguez, U. Wandinger, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 1. Instruments,” Appl. Opt. 43, 961–976 (2004).
[Crossref] [PubMed]

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

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the earlinet project. 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370–5385 (2004).
[Crossref] [PubMed]

V. Shcherbakov, “Regularized algorithm for raman lidar data processing,” Appl. Opt. 46, 4879–4889 (2007).
[Crossref] [PubMed]

P. Pornsawad, C. Böckmann, C. Ritter, and M. Rafler, “Ill-posed retrieval of aerosol extinction coefficient profiles from raman lidar data by regularization,” Appl. Opt. 47, 1649–1661 (2008).
[Crossref] [PubMed]

P. Pornsawad, G. D’Amico, C. Böckmann, A. Amodeo, and G. Pappalardo, “Retrieval of aerosol extinction coefficient profiles from raman lidar data by inversion method,” Appl. Opt. 51, 2035–2044 (2012).
[Crossref] [PubMed]

Appl. Phys. B (1)

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, and W. Michaelis, “Combined raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[Crossref]

Astrophysical J. Lett. (1)

M. Piana, A. M. Massone, E. P. Kontar, A. G. Emslie, J. C. Brown, and R. A. Schwartz, “Regularized electron flux spectra in the 2002 july 23 solar flare,” Astrophysical J. Lett. 595, L127 (2003).
[Crossref]

Atmos. Environ. (1)

P. Di Girolamo, D. Summa, R. Bhawar, T. Di Iorio, M. Cacciani, I. Veselovskii, O. Dubovik, and A. Kolgotin, “Raman lidar observations of a saharan dust outbreak event: Characterization of the dust optical properties and determination of particle size and microphysical parameters,” Atmos. Environ. 50, 66–78 (2012).
[Crossref]

Atmos. Meas. Tech. (1)

A. Povey, R. Grainger, D. Peters, and J. Agnew, “Retrieval of aerosol backscatter, extinction, and lidar ratio from raman lidar with optimal estimation,” Atmos. Meas. Tech. 7, 757–776 (2014).
[Crossref]

Bound-Lay Meteorol. (1)

A. Boselli, M. Armenante, L. D’Avino, M. D’Isidoro, G. Pisani, N. Spinelli, and X. Wang, “Atmospheric aerosol characterization over Naples during 2000–2003 earlinet project: Planetary boundary-layer evolution and layering,” Bound-Lay Meteorol. 132, 151–165 (2009).
[Crossref]

Bull. Amer. Meteor. Soc. (1)

V. M. Karyampudi, S. P. Palm, J. A. Reagen, and H. Fang, “Validation of the saharan dust plume conceptual model using lidar, meteosat, and ecmwf data,” Bull. Amer. Meteor. Soc. 80, 1045 (1999).
[Crossref]

Comput. Phys. Commun. (1)

L. Osterloh, C. Pérez, D. Böhme, J. M. Baldasano, C. Böckmann, L. Schneidenbach, and D. Vicente, “Parallel software for retrieval of aerosol distribution from lidar data in the framework of earlinet-asos,” Comput. Phys. Commun. 180, 2095–2102 (2009).
[Crossref]

EPJ Web Conf. (1)

X. Wang, A. Boselli, A. Sannino, C. Song, N. Spinelli, Y. Zhao, and C. Pan, “Calibration of multi-wavelength raman polarization lidar,” EPJ Web Conf. 89, 01002 (2015).
[Crossref]

Inverse Probl. (5)

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[Crossref]

H. Lanteri, M. Roche, and C. Aime, “Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms,” Inverse Probl. 18, 1397 (2002).
[Crossref]

M. Bertero, P. Boccacci, G. Talenti, R. Zanella, and L. Zanni, “A discrepancy principle for Poisson data,” Inverse Probl. 26, 105004 (2010).
[Crossref]

E. Resmerita, H. W. Engl, and A. N. Iusem, “The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis,” Inverse Probl. 23, 2575 (2007).
[Crossref]

F. Benvenuto and M. Piana, “Regularization of multiplicative iterative algorithms with nonnegative constraint,” Inverse Probl. 30, 035012 (2014).
[Crossref]

J. Aerosol Sci. (1)

Y. Wang, S. Fan, and X. Feng, “Retrieval of the aerosol particle size distribution function by incorporating a priori information,” J. Aerosol Sci. 38, 885–901 (2007).
[Crossref]

J. Comput. Phys. (1)

L. Osterloh, C. Böckmann, D. Nicolae, and A. Nemuc, “Regularized inversion of microphysical atmospheric particle parameters: Theory and application,” J. Comput. Phys. 237, 79–94 (2013).
[Crossref]

Opt. Lett. (1)

Report. Max-Planck-Institut fur Meteorologie (1)

J. Bösenberg and V. Matthias, “Earlinet: A european aerosol research lidar network to establish an aerosol climatology,” Report. Max-Planck-Institut fur Meteorologie 348, 1–191 (2003).

Other (5)

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC, 1998).
[Crossref]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer Science & Business Media, 1996).
[Crossref]

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-posed Problems (VH Winston & Sons, 1977).

H. W. Kuhn and A. W. Tucker, “Nonlinear programming,” in Proceedings of 2nd Berkeley Symposium (1951), pp. 481–492.

H. W. Kuhn, “Nonlinear Programming: A Historical View,” in Traces and Emergence of Nonlinear Programming (SpringerBasel, 2014), pp. 393–414.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Study of the resolution power of EM: reconstruction of a synthetic aerosol extinction profile made of multiple equispaced peaks in the case of 500,000 EM iterations (top panel) and of 10,000 EM iterations (bottom panel). The zoomed plots clearly show that at convergence the resolution power is virtually ideal.

Fig. 2
Fig. 2

Integral (right panel) of a portion of the multiple δ function profile made of three peaks in the case of 10,000 EM iterations (left panel).

Fig. 3
Fig. 3

Mean reconstructed aerosol extinction profile (blue), together with the confidence band and the true profile (black) obtained from the EARLINET synthetic Raman signal, with K = 3 in the stopping criterion. Left: λ = 355; right: λ = 532.

Fig. 4
Fig. 4

Comparison of performances between EM, Tikhonov, Levenberg-Marquardt and ACR in the case of the EARLINET data with λ = 355; for EM, Levenberg-Marquardt and Tikhonov, regularization is realized by exploiting semi-convergence.

Fig. 5
Fig. 5

Top left: analysis of experimental data. The uncertainties on the ACR reconstruction are shown just in some significative points. Top right: EM reconstructed profile with confidence band. Bottom left: backscatter coefficient β, and a zoom of its peak. Bottom right: residuals (difference between measured and reconstructed signals).

Fig. 6
Fig. 6

Left: analysis of experimental data. The uncertainties on the ACR reconstruction are shown just in some significative points. Right: EM reconstructed profile with confidence band (very thin in this case).

Equations (14)

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

P λ ( z ) = C λ z 2 β λ ( z ) exp ( 2 0 z α λ ( z ) d z ) ,
P μ ( z ) = C μ z 2 ρ ( z ) exp ( 0 z ( α μ ( z ) + α λ ( z ) ) d z ) ,
log ( P ( z ) z 2 C ρ ( z ) ) = 0 z α ( z ) d z ,
y = x
y = H x .
x ^ = arg min x Ω D ( y , x )
P ( y | x ) = A exp ( y H x 2 ) ,
D ( y , x ) = y H x 2 .
Ω = { x n | i = 1 n x i 2 E }
x ^ = arg min x Ω { H x y 2 + η x 2 } ,
P ( y | x ) = i = 1 N exp ( ( H x ) i ) ( H x ) i y i y i ! .
D K L ( y , x ) = 2 N i = 1 N y i log y i ( H x ) i + ( H x ) i y i + y i log ( y i ) .
x k + 1 = x k H T 1 H T y H x k ,
Δ i = 1 i j = 1 i ( P j P ¯ j ) σ j P .

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