Abstract

In applying the graphical technique to the estimation of the particle size distribution (PSD) parameters, determination of proper bounds surrounding the solution space for a particular confidence level is essential to the consistent intercomparison of diverse multiwavelength lidar optical data sets. The graphical technique utilizes ratios of backscatter and/or extinction coefficients, and it is shown that if the correlation between ratios is not taken into account in calculating the error bounds, the solution space will be overestimated, resulting in relatively larger discrepancies for a larger number of optical coefficients. A method for correcting the bounds, to account for the correlation is developed for various numbers of wavelengths. These improved bounds are then applied, for the case of a monomodal lognormal PSD, with an assumed refractive index, to assess the role additional Raman extinction channels play in improving retrieval capability of a typical three-channel backscatter lidar (1064, 532, and 355 nm) under varying noise levels. Applying the same formalism to underlying bimodal distributions of coarse and fine particles can result in false monomodal solutions. However, when both Raman optical extinction channels are available, no solution is obtained. This can potentially serve as a quick and simple method, prior to a more complex regularization analysis, to differentiate between cases in which the fine mode is dominant versus the cases in which the contribution from the coarse mode is significant.

© 2005 Optical Society of America

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References

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  1. O. Dubovik, M. D. King, A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res. 105, 20673–20696 (2000).
    [CrossRef]
  2. C. W Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, 1984).
  3. K. S. Shifrin, “Simple relationships for the Ångstrom parameter of disperse systems,” Appl. Opt. 34, 4480–4485 (1995).
    [CrossRef] [PubMed]
  4. N. T. O’Neill, O. Dubovik, T. F. Eck, “Modified Ångstrom exponent for the characterization of submicrometer aerosols,” Appl. Opt. 40, 2368–2375 (2001).
    [CrossRef]
  5. D. Althausen, D. Muller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zorner, “Scanning six-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
    [CrossRef]
  6. C. Bockmann, “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]
  7. D. Mueller, U. Wandinger, D. Althausen, M. Fiebig, “Comprehensive particle characterization from three-wavelength Raman-Lidar observations: case study,” Appl. Opt. 40, 4863–4869 (2001).
    [CrossRef]
  8. I. Veselovskii, A. Kolgotin, V. Griaznov, D. Muller, U. Wandinger, D. N. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
    [CrossRef] [PubMed]
  9. I. Veselovskii, A. Kolgotin, V. Griaznov, D. Muller, U. Wandinger, D. N. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180–1195 (2004).
    [CrossRef] [PubMed]
  10. 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]
  11. G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
    [CrossRef]
  12. G. K. Yue, “Retrieval of stratospheric aerosol size distributions and integral properties from simulated lidar backscatter measurements,” Appl. Opt. 39, 5446–5455 (2000).
    [CrossRef]
  13. P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
    [CrossRef]
  14. P. S. S. Devara, P. Ernest Raj, G. Pandithurai, “Aerosol-profile measurements in the lower troposphere with four-wavelength bistatic argon-ion lidar,” Appl. Opt. 34, 4416–4425 (1995).
    [CrossRef] [PubMed]
  15. Numerical Recipes: The art of Scientific Computing, W. H. Press (ed.), section 15.6 (Cambridge U. Press, 2002).
  16. 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]
  17. 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, 129–142 (1993).
    [CrossRef]
  18. A. Papoulius, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill1991), Sec. 6.2.
  19. O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
    [CrossRef]

2004

2002

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

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

2001

2000

G. K. Yue, “Retrieval of stratospheric aerosol size distributions and integral properties from simulated lidar backscatter measurements,” Appl. Opt. 39, 5446–5455 (2000).
[CrossRef]

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

O. Dubovik, M. D. King, A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res. 105, 20673–20696 (2000).
[CrossRef]

1997

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

1996

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]

1995

1994

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]

1993

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, 129–142 (1993).
[CrossRef]

1989

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Althausen, D.

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

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

Anderson, J.

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

Ansmann, A.

D. Althausen, D. Muller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zorner, “Scanning six-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[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]

Bockmann, C.

Clauder, E.

D. Althausen, D. Muller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zorner, “Scanning six-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
[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, 129–142 (1993).
[CrossRef]

Devara, P. S. S.

Dubovik, O.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

N. T. O’Neill, O. Dubovik, T. F. Eck, “Modified Ångstrom exponent for the characterization of submicrometer aerosols,” Appl. Opt. 40, 2368–2375 (2001).
[CrossRef]

O. Dubovik, M. D. King, A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res. 105, 20673–20696 (2000).
[CrossRef]

Eck, T. F.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

N. T. O’Neill, O. Dubovik, T. F. Eck, “Modified Ångstrom exponent for the characterization of submicrometer aerosols,” Appl. Opt. 40, 2368–2375 (2001).
[CrossRef]

Ernest Raj, P.

Fiebig, M.

Fuller, W. H.

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Griaznov, V.

Groetsch, C. W

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

Holben, B. N.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

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, 129–142 (1993).
[CrossRef]

Hube, H.

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

Kaufman, Y. J.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

King, M. D.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

O. Dubovik, M. D. King, A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res. 105, 20673–20696 (2000).
[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]

Kolgotin, A.

Lu, J.

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

McCormick, M. P.

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Mohnen, V. A.

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

Mueller, D.

Muller, D.

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]

O’Neill, N. T.

Osborn, M. T.

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Pandithurai, G.

Papoulius, A.

A. Papoulius, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill1991), Sec. 6.2.

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]

Saxana, V. K.

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

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]

Shifrin, K. S.

Slutsker, I.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Smirnov, A.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Swissler, T. J.

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Tanre, D.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Veselovskii, I.

Wandinger, U.

Wang, P.-H.

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

Wang, P-H.

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Whiteman, D. N.

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]

Yue, G. K.

G. K. Yue, “Retrieval of stratospheric aerosol size distributions and integral properties from simulated lidar backscatter measurements,” Appl. Opt. 39, 5446–5455 (2000).
[CrossRef]

G. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Zorner, S.

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

Appl. Opt.

P. S. S. Devara, P. Ernest Raj, G. Pandithurai, “Aerosol-profile measurements in the lower troposphere with four-wavelength bistatic argon-ion lidar,” Appl. Opt. 34, 4416–4425 (1995).
[CrossRef] [PubMed]

K. S. Shifrin, “Simple relationships for the Ångstrom parameter of disperse systems,” Appl. Opt. 34, 4480–4485 (1995).
[CrossRef] [PubMed]

G. K. Yue, “Retrieval of stratospheric aerosol size distributions and integral properties from simulated lidar backscatter measurements,” Appl. Opt. 39, 5446–5455 (2000).
[CrossRef]

C. Bockmann, “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]

N. T. O’Neill, O. Dubovik, T. F. Eck, “Modified Ångstrom exponent for the characterization of submicrometer aerosols,” Appl. Opt. 40, 2368–2375 (2001).
[CrossRef]

D. Mueller, U. Wandinger, D. Althausen, 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. Muller, U. Wandinger, D. N. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
[CrossRef] [PubMed]

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

Geophys. Res. Lett.

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. K. Yue, J. Lu, V. A. Mohnen, P.-H. Wang, V. K. Saxana, J. Anderson, “Retrieving aerosol optical properties from moments of the particle size distribution,” Geophys. Res. Lett. 24, 651–654 (1997).
[CrossRef]

J Atmos. Sci.

O. Dubovik, B. N. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanre, I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

J. Aerosol Sci.

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, 129–142 (1993).
[CrossRef]

J. Atmos. Ocean. Technol.

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

J. Atmos. Oceanic Technol.

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. Geophys. Res.

O. Dubovik, M. D. King, A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements, J. Geophys. Res. 105, 20673–20696 (2000).
[CrossRef]

P-H. Wang, M. P. McCormick, T. J. Swissler, M. T. Osborn, W. H. Fuller, G. K. Yue, “Inference of stratospheric aerosol composition and size distribution from SAGE II Satellite measurements,” J. Geophys. Res. 94, 8435–8446 (1989).
[CrossRef]

Other

Numerical Recipes: The art of Scientific Computing, W. H. Press (ed.), section 15.6 (Cambridge U. Press, 2002).

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

A. Papoulius, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill1991), Sec. 6.2.

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Figures (13)

Fig. 1
Fig. 1

Example of graphical determination of PSD parameter solution space from three optical coefficients generated from an underlying lognormal distribution with r ¯ 0 = 0.65 μm, σ0 = 1.4 and with the addition of 2% uncertainty for (a) extinction coefficient ratios, (b) backscatter coefficient ratios and (c) S ratios.

Fig. 2
Fig. 2

Verification of PDF distribution function formula [Eq. (4)] with a numerically generated histogram of the ratio statistics for the case of Δd = Δn = 0.1.

Fig. 3
Fig. 3

Comparison of the cumulative probability distribution function for the case of 3, 4 and 5 optical coefficients showing the increased conditional probability for correlated ratios. Solid curves calculated for correlated ratios; while the dashed curves, for uncorrelated ratios.

Fig. 4
Fig. 4

Numerical inversions retrievals relative to graphical bounds at Clevel = 0.65 on ensembles of three channel backscatter coefficients generated from a priori PSD parameters, with equal uncorrelated fractional variances of 10%: (a) r ¯ 0 = 0.7, σ0 = 1.4; (b) r ¯ = 0.4, GSD = 1.4; (c) r ¯ 0 = 0.4, σ0 = 1.6, (d) r ¯ = 0.7, GSD = 1.6.

Fig. 5
Fig. 5

(a) Confidence domain obtained by using optical coefficients generated from an example PSD ( r ¯ 0 = 0.5, σ0 = 1.4, denoted by the +) with an uncertainty of 10%, and confidence level set to 65%. (b) Fractional number density retrieval obtained from the confidence domain by using two different approaches, combining the results from different optical channels.

Fig. 6
Fig. 6

Assessment over the lognormal space of parameter retrieval uncertainty span (maximum deviation) for the case of three backscatter coefficients with 10% coefficient uncertainty and confidence level of 65%. (a) maximum deviation of normalized mode radius Δ r ¯ N, (b) maximum deviation of normalized mode variance ΔσN (c) normalized uncertainty area ΔAN. The gray scale bar to the right is a measure of retrieval uncertainty span (lighter regions have lower retrieval errors).

Fig. 7
Fig. 7

Enhancement of parameter retrieval as extinction coefficients are added using uncorrelated bound definition. (a) E4,3 [(3β + 1α) versus (3β)]. (b) E5,4 [(3β + 2α) versus (3β + 1α)]. Lighter regions have larger enhancement factors.

Fig. 8
Fig. 8

Enhancement of parameter retrieval as extinction coefficients are added using correlated bound definition. (a) E4,3 [(3β + 1α) versus (3β)]. (b) E5,4 [(3β + 2α) versus (3β + 1α)]. Lighter regions have larger enhancement factors.

Fig. 9
Fig. 9

Zoom view of parameter retrieval enhancement to the fine urban aerosol mode regime. Calculations performed using correlated bounds. (a) E4,3 [(3β + 1α) vs (3β)]. (b) E5,4 [(3β + 2α) versus (3β + 1α)]. Lighter regions have larger enhancement factors.

Fig. 10
Fig. 10

Comparisons of normalized parameter retrieval maximum and minimum bounds between different optical coefficient sets as function of uncertainty level for different parameter sizes. All analyses use mr = 1.5, mi = −0.02. (a) r ¯ 0 = 0.8, σ0 = 1.8, large particle size; (b) r ¯ 0 = 0.5, σ0 = 1.6, intermediate size; (c) r ¯ 0 = 0.3, σ0 = 1.4, small (accumulation mode) size.

Fig. 11
Fig. 11

False monomodal retrieval from a set of optical coefficients 3β + 2α generated from an initial bimodal distribution: mode 1: r ¯ 0 = 0.3, σ0 = 1.4, mr = 1.5, mi = −.02; mode 2: r ¯ 0 = 0.7, σ0 = 1.4, mr = 1.5, mi = −.02. Also shown for reference are the retrievals for each individual mode.

Fig. 12
Fig. 12

Representative bimodal aerosol PSD modes obtained from Aeronet over New York City. (a) and (b) are representative of near monomodal distributions, while (c) and (d) were chosen for their significant bimodal characteristics.

Fig. 13
Fig. 13

Graphical retrieval regions in the lognormal PSD parameter space for the different bimodal distributions displayed in Fig. 12. The refractive index was chosen to be fixed at mr = 1.5, mi = −0.02. In (c) and (d), where the coarse aerosol mode is significant, the 3β (dashed curve) and 3β + 1α (dotted curve) coefficient sets can result in false retrievals, while there is no retrieval for the 3β + 2α. (solid curve) set.

Tables (1)

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Table 1 Percentage of numerical Inversions within correlated Graphical Confidence Bounds versus Theoretical value Clevel = 0.65a

Equations (10)

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{ β ( λ i ; p ) α ( λ i ; p ) } = 0 { K β ( r , λ i ) K α ( r , λ i ) } n ( r ) d r ,
n lognormal ( r , p ) = N d 2 π ln ( σ ) r exp ( - ln ( r / r ¯ ) 2 ln ( σ ) 2 ) .
{ R 1 = α 532 α 355 , R 2 = α 1064 α 355 , R 3 = α 1064 α 532 } .
R k - Δ R k R k i , j R k + Δ R k ,             k = 1 - 3.
P ( δ ; Δ n , Δ d ) = 1 2 π Δ n Δ d exp ( - D / 2 ) { 2 A exp [ - A 2 × ( B - 1 ) 2 ] + ( B - 1 ) 2 π A erf [ A 2 × ( B - 1 ) ] } , A = 1 Δ d 2 + ( 1 + δ ) 2 Δ n 2 ,             B = A - 1 δ + δ 2 Δ n 2 , D = δ 2 Δ n 2 - A B 2 ,
C ( δ ) = - δ δ P ( δ ) d δ
C N ( δ ) = - δ δ - δ δ P N ( δ 1 δ N ) d δ 1 d δ N ,
C N uncorr ( δ ) = C level ,
E 4 , 3 = log 10 [ ( Δ A N ) 3 ( Δ A N ) 4 ] ( for 3 β versus 3 β + 1 α ) , E 5 , 4 = log 10 [ ( Δ A N ) 4 ( Δ A N ) 5 ] 3 β + 1 α versus 3 β + 1 α versus 3 β + 2 α ,
d V d log ( r ) = V f 2 π ln ( σ f ) exp [ - ln ( r / r f ) 2 ln ( σ f ) 2 ] + V c 2 π ln ( σ c ) exp [ - ln ( r / r c ) 2 ln ( σ c ) 2 ] .

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