Abstract

We investigate the assessment of uncertainty in the inference of aerosol size distributions from backscatter and extinction measurements that can be obtained from a modern elastic/Raman lidar system with a Nd:YAG laser transmitter. To calculate the uncertainty, an analytic formula for the correlated probability density function (PDF) describing the error for an optical coefficient ratio is derived based on a normally distributed fractional error in the optical coefficients. Assuming a monomodal lognormal particle size distribution of spherical, homogeneous particles with a known index of refraction, we compare the assessment of uncertainty using a more conventional forward Monte Carlo method with that obtained from a Bayesian posterior PDF assuming a uniform prior PDF and show that substantial differences between the two methods exist. In addition, we use the posterior PDF formalism, which was extended to include an unknown refractive index, to find credible sets for a variety of optical measurement scenarios. We find the uncertainty is greatly reduced with the addition of suitable extinction measurements in contrast to the inclusion of extra backscatter coefficients, which we show to have a minimal effect and strengthens similar observations based on numerical regularization methods.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, and D. J. Hofmann, “Climate forcing by antropogenic aerosol,” Science 255, 423-430 (1992).
    [CrossRef] [PubMed]
  2. J. A. Coakley and R. D. Cess, “Response of the NCAR community climate model to the radiative forcing by the naturally occurring tropospheric aerosols,” J. Atmos. Sci. 42, 1677-1692(1985).
    [CrossRef]
  3. G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988-993 (1979).
    [CrossRef] [PubMed]
  4. N. T. O'Neill and J. R. Miller, “Combined solar aureole and solar beam extinction measurements: 2. Studies of the inferred aerosol size distributions,” Appl. Opt. 23, 3697-3703(1984).
    [CrossRef] [PubMed]
  5. D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
    [CrossRef]
  6. G. Tonna, T. Nakajima, and R. Rao, “Aerosol features retrieved from solar aureole data: a simulation study concerning a turbid atmosphere,” Appl. Opt. 34, 4486-4499 (1995).
    [CrossRef] [PubMed]
  7. T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, and B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672-2686 (1996).
    [CrossRef] [PubMed]
  8. M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, “Aerosol size distributions obtained by inversion of spectral optical depth measurements,” J. Atmos. Sci. 35, 2153-2167(1978).
    [CrossRef]
  9. R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
    [CrossRef]
  10. R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
    [CrossRef]
  11. D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
    [CrossRef]
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  13. B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
    [CrossRef] [PubMed]
  14. B. R. Lienert, J. N. Porter, and S. K. Sharma, “Repetitive genetic inversion of optical extinction data,” Appl. Opt. 40, 3476-3482 (2001).
    [CrossRef]
  15. M. D. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Oceanic Technol. 13, 863-869(1996).
    [CrossRef]
  16. C. Böckmann, “Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions,” Appl. Opt. 40, 1329-1342(2001).
    [CrossRef]
  17. 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]
  18. D. Müller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358-2368 (1999).
    [CrossRef]
  19. M. Pahlow, D. Müller, M. Tesche, H. Eichler, G. Feingold, W. L. Eberhard, and Y.-F. Cheng, “Retrieval of aerosol properties from combined multiwavelength lidar and sunphotometer measurements,” Appl. Opt. 45, 7429-7442 (2006).
    [CrossRef] [PubMed]
  20. I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and D. N. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180-1195 (2004).
    [CrossRef] [PubMed]
  21. D. A. Ligon, J. B. Gillespie, and P. Pellegrino, “Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method,” Appl. Opt. 39, 4402-4410(2000).
    [CrossRef]
  22. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer-Verlag, 2004).
  23. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  24. D. Müller, U. Wandinger, D. Althausen, and M. Fiebig, “Comprehensive particle characterization from three-wavelength Raman-lidar observations: case study,” Appl. Opt. 40, 4863-4869 (2001).
    [CrossRef]
  25. B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
    [CrossRef]
  26. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, eds., Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).
  27. J. Tamminen, “Validation of nonlinear inverse algorithms with Markov chain Monte Carlo method,” J. Geophys. Res. 109, D19303 (2004).
    [CrossRef]

2006 (1)

2005 (1)

2004 (2)

2003 (1)

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

2001 (3)

2000 (2)

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

D. A. Ligon, J. B. Gillespie, and P. Pellegrino, “Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method,” Appl. Opt. 39, 4402-4410(2000).
[CrossRef]

1999 (2)

1998 (1)

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

1996 (2)

M. D. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Oceanic Technol. 13, 863-869(1996).
[CrossRef]

T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, and B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672-2686 (1996).
[CrossRef] [PubMed]

1995 (1)

1992 (2)

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

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

1988 (1)

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

1985 (1)

J. A. Coakley and R. D. Cess, “Response of the NCAR community climate model to the radiative forcing by the naturally occurring tropospheric aerosols,” J. Atmos. Sci. 42, 1677-1692(1985).
[CrossRef]

1984 (1)

1979 (1)

1978 (1)

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

Ahmed, S.

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
[CrossRef] [PubMed]

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

Althausen, D.

D. Müller, U. Wandinger, D. Althausen, and M. Fiebig, “Comprehensive particle characterization from three-wavelength Raman-lidar observations: case study,” Appl. Opt. 40, 4863-4869 (2001).
[CrossRef]

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

Ansmann, A.

Böckmann, C.

Bohren, C. F.

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

Boi, P.

Byrne, D. M.

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

Cess, R. D.

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

J. A. Coakley and R. D. Cess, “Response of the NCAR community climate model to the radiative forcing by the naturally occurring tropospheric aerosols,” J. Atmos. Sci. 42, 1677-1692(1985).
[CrossRef]

Charlson, R. J.

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

Cheng, Y.-F.

Clauder, E.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

Coakley, J. A.

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

J. A. Coakley and R. D. Cess, “Response of the NCAR community climate model to the radiative forcing by the naturally occurring tropospheric aerosols,” J. Atmos. Sci. 42, 1677-1692(1985).
[CrossRef]

Devaux, C.

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

Eberhard, W. L.

Eichler, H.

Evans, K. D.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

Feingold, G.

Ferrare, R. A.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

Fiebig, M.

Franke, K.

Gac, J. Y

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

Gillespie, J. B.

Griaznov, V.

Gross, B.

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
[CrossRef] [PubMed]

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

Hales, J. M.

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

Hansen, J. E.

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

Herman, B. M.

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

Herman, B. R.

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
[CrossRef] [PubMed]

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

Herman, M.

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

Hofmann, D. J.

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

Holben, B.

Hube, H.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

Huffman, D. R.

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

Kaipio, J.

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer-Verlag, 2004).

Kaufman, Y.

Kaufman, Y. J.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

King, M. D.

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

Kolgotin, A.

Lienert, B. R.

Ligon, D. A.

Melfi, S. H.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

Miller, J. R.

Moshary, F.

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
[CrossRef] [PubMed]

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

Müller, D.

Nakajima, T.

O'Neill, N. T.

Pahlow, M.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Pellegrino, P.

Poellot, M.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

Porter, J. N.

Post, M. D.

M. D. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Oceanic Technol. 13, 863-869(1996).
[CrossRef]

Rao, R.

Reagan, J. A.

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

Santer, R.

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

Schwartz, S. E.

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

Sharma, S. K.

Shaw, G. E.

Somersalo, E.

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer-Verlag, 2004).

Tamminen, J.

J. Tamminen, “Validation of nonlinear inverse algorithms with Markov chain Monte Carlo method,” J. Geophys. Res. 109, D19303 (2004).
[CrossRef]

Tanre, D.

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

Tesche, M.

Tonna, G.

Veselovskii, I.

Wandinger, U.

Whiteman, D. N.

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and D. N. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180-1195 (2004).
[CrossRef] [PubMed]

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

Zörner, S.

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

Appl. Opt. (13)

G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988-993 (1979).
[CrossRef] [PubMed]

N. T. O'Neill and J. R. Miller, “Combined solar aureole and solar beam extinction measurements: 2. Studies of the inferred aerosol size distributions,” Appl. Opt. 23, 3697-3703(1984).
[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]

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

G. Tonna, T. Nakajima, and R. Rao, “Aerosol features retrieved from solar aureole data: a simulation study concerning a turbid atmosphere,” Appl. Opt. 34, 4486-4499 (1995).
[CrossRef] [PubMed]

T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, and B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672-2686 (1996).
[CrossRef] [PubMed]

D. A. Ligon, J. B. Gillespie, and P. Pellegrino, “Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method,” Appl. Opt. 39, 4402-4410(2000).
[CrossRef]

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

B. R. Lienert, J. N. Porter, and S. K. Sharma, “Repetitive genetic inversion of optical extinction data,” Appl. Opt. 40, 3476-3482 (2001).
[CrossRef]

D. Müller, U. Wandinger, D. Althausen, and M. Fiebig, “Comprehensive particle characterization from three-wavelength Raman-lidar observations: case study,” Appl. Opt. 40, 4863-4869 (2001).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and D. N. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180-1195 (2004).
[CrossRef] [PubMed]

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Extension of the graphical technique for estimation of particle size distribution parameters for the consistent intercomparison of diverse sets of multiwavelength lidar derived optical coefficients,” Appl. Opt. 44, 6462-6473 (2005).
[CrossRef] [PubMed]

M. Pahlow, D. Müller, M. Tesche, H. Eichler, G. Feingold, W. L. Eberhard, and Y.-F. Cheng, “Retrieval of aerosol properties from combined multiwavelength lidar and sunphotometer measurements,” Appl. Opt. 45, 7429-7442 (2006).
[CrossRef] [PubMed]

Geophys. Res. Lett. (1)

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, and K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November-December 1991,” Geophys. Res. Lett. 19, 1599-1602 (1992).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, and S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469-1482(2000).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

M. D. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Oceanic Technol. 13, 863-869(1996).
[CrossRef]

J. Atmos. Sci. (2)

J. A. Coakley and R. D. Cess, “Response of the NCAR community climate model to the radiative forcing by the naturally occurring tropospheric aerosols,” J. Atmos. Sci. 42, 1677-1692(1985).
[CrossRef]

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

J. Geophys. Res. (3)

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, M. Poellot, and Y. J. Kaufman, “Raman lidar measurements of aerosol extinction and backscattering 2. Derivation of aerosol real refractive index, single-scattering albedo, and humidification factor using Raman lidar and aircraft size distribution measurements,” J. Geophys. Res. 103, 19673-19690(1998).
[CrossRef]

D. Tanre, C. Devaux, M. Herman, R. Santer, and J. Y Gac, “Radiative properties of desert aerosols by optical ground based measurements at solar wavelengths,” J. Geophys. Res. 93, 14223-14231 (1988).
[CrossRef]

J. Tamminen, “Validation of nonlinear inverse algorithms with Markov chain Monte Carlo method,” J. Geophys. Res. 109, D19303 (2004).
[CrossRef]

Proc. SPIE (1)

B. R. Herman, B. Gross, F. Moshary, and S. Ahmed, “Methods of assessing uncertainty in determining particle size distribution parameters from optical backscatter and extinction measurements,” Proc. SPIE 5154, 208-215 (2003).
[CrossRef]

Science (1)

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

Other (4)

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

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, eds., Markov Chain Monte Carlo in Practice (Chapman & Hall, 1996).

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer-Verlag, 2004).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(a)–(d) CDFs of retrieved median radius and (e)–(h) GSD with modeled true distribution parameters ( r ¯ , σ ) of: (a), (e) ( 0.6 μm , 1.6 ) ; (b), (f) ( 0.6 μm , 2 ) ; (c), (g) ( 0.3 μm , 1.6 ) ; (d), (h) ( 0.3 μm , 2 ) . The dotted curves represents the Bayesian posterior CDF. The thin black curves represent the estimated CDF using MPPDF retrievals. The thick gray curves represent the estimated CDF using MMD retrievals.

Fig. 2
Fig. 2

Scatterplots of forward Monte Carlo outcomes with contours of Bayesian PDFs. Outcomes using MMD retrievals are shown in (a) and (b). MPPDF retrievals are shown in (c) and (d). The modeled true distribution parameters, ( r ¯ , σ ) , are (a), (c) ( 0.6 μm , 1.6 ) , [corresponding to Figs. 1a, 1e], and (b), (d) ( 0.3 μm , 2 ) , [corresponding to Figs. 1d, 1h].

Fig. 3
Fig. 3

(a), (b) Median radius and (c), (d) GSD CDF deviation from the Bayesian CDF as formulated in (29) using the (a), (c) MPPDF and (b) (d) MMD retrieval methods.

Fig. 4
Fig. 4

A p A m versus A m of sets with 90% credibility for five measurement groups.

Fig. 5
Fig. 5

Individual (dark gray area) and simultaneous (dark gray + light gray area) sets of 90% credibility for measurement groups (a), (f) 5, (b), (g) 4, (c), (h) 3, (d) (i) 2, and (e), (j) 1.

Tables (2)

Tables Icon

Table 1 CDF Deviation Comparison

Tables Icon

Table 2 Measurement Groups

Equations (42)

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

y i = 0 K i ( r ) n ( r ) d r ,
f post ( θ | y ^ ) = f li ( y ^ | θ ) f pri ( θ ) / f marg ( y ^ ) ,
f marg ( y ^ ) = f li ( y ^ | θ ) f pri ( θ ) d θ .
y i = 0 K i ( r ) N 0 n lognormal ( r , p ) d r = T i ( p ) ,
n lognormal ( r , p ) = 1 2 π ln ( σ ) r exp ( - ln ( r / r ¯ ) 2 2 ln ( σ ) 2 ) ,
y ^ i = y i ( 1 + δ i ) .
R ^ k = y i ( 1 + δ i ) y j ( 1 + δ j ) = R k ( 1 + 1 + δ i 1 + δ j - 1 ) = R k ( 1 + ε k ) ,
ε k = ( 1 + δ i ) / ( 1 + δ j ) - 1.
f li ( R ^ | p ) = f ε [ ε ( R ^ , p ) ] J ε / R ^ ( R ^ , p ) ,
J ε / R ^ = | det ( ε R ^ ) | ,
ε R ^ = ( ε 1 / R ^ 1 ε 1 / R ^ N ε N / R ^ 1 ε N / R ^ N ) .
ε i = R ^ i R i ( p ) - 1 ,
J ε / R ^ ( R ^ , p ) = i 1 R i ( p ) .
R i = y i / y N , i { 1 , ... , N - 1 } .
ε i = { 1 + δ i 1 + δ N 1 , i N δ N , i = N ,
δ i = { ε i + ε N + ε i ε N , i N ε N , i = N .
f ε ( ε ) = f δ ( δ ( ε ) ) J δ / ε ( ε ) ,
δ ε = ( 1 + ε N 0 0 1 + ε 1 0 1 + ε N 0 1 + ε 2 0 0 1 + ε N 1 + ε N - 1 0 0 0 1 ) ,
J δ / ε ( ε ) = | ( 1 + ε N ) N - 1 | .
f ε ( ε ) = 1 ( 2 π ) N / 2 i = 1 N σ i exp ( - 1 2 [ i = 1 N - 1 ( ε i + ε N + ε i ε N ) 2 σ i 2 + ε N 2 σ N 2 ] ) | ( 1 + ε N ) N - 1 | .
f ε ( ε ) = 1 ( 2 π ) N / 2 i = 1 N σ i exp ( - 1 2 C ) - exp ( - 1 2 A u 2 ) | ( 1 - B + u ) N - 1 | d u ,
A = 1 σ N 2 + i = 1 N - 1 1 + 2 ε i + ε i 2 σ i 2 ,
B = i = 1 N - 1 ( ε i + ε i 2 ) / σ i 2 A ,
C = i = 1 N - 1 ε i 2 σ i 2 - A B 2 .
f ε ( ε ) = exp ( - 1 2 C ) k = 0 N - 1 ( 2 A ) k + 1 ( 1 - B ) N - 1 - k ( N - 1 k ) [ I k ( ) + I k ( - ) - 2 I k ( A 2 ( B - 1 ) ) ] ( 2 π ) N / 2 i = 1 N σ i ,
I k ( u ) = 0 u x k exp ( - x 2 ) d x ,
I k + 2 ( u ) = 1 2 [ ( k + 1 ) I k ( u ) - u k + 1 exp ( - u 2 ) ] ,
I 0 ( u ) = 1 2 π erf ( u ) ,
I 1 ( u ) = 1 2 [ 1 - exp ( - u 2 ) ] .
Δ ( p ) = max i { | R i ( p ) R ^ i - 1 | }
F X ( x ) = X min x Y min Y max f X , Y ( x , y ) d y d x ,
F ^ X ( x ) = N { X x } / N total ,
( dev   X ) meth = | F X ( x ) - F ^ X , meth ( x ) | d x ,
f ε , θ ( ε , θ ) = f ε ( ε ) f θ ( θ ) ,
f ε , y ^ ( ε , y ^ ) = f ε ( ε ) f θ { T - 1 [ M - 1 ( y ^ , ε ) ] } J θ / y [ M - 1 ( y ^ , ε ) ] J y / y ^ ( y ^ , ε ) .
S ( L ) = { p : f ( p ) L } ,
C ( L ) = S ( L ) f ( p ) d p ,
S p ( L p ) = { p : f p ( p ) L p } ,
S m ( L m ) = { m : f m ( m ) L m } ,
f p ( p ) = Ω m f p , m ( p , m ) d m ,
f m ( m ) = Ω p f p , m ( p , m ) d p ,
C ( L p , L m ) = S p ( L p ) × S m ( L m ) f p , m ( p , m ) d p d m

Metrics