Abstract

The multiwavelength Raman lidar technique in combination with sophisticated inversion algorithms has been recognized as a new tool for deriving information about the microphysical properties of atmospheric aerosols. The input optical parameter sets, provided by respective aerosol Raman lidars, are at the theoretical lower limit at which these inversion algorithms work properly. For that reason there is ongoing intense discussion of the accuracy of these inversion methods and the possibility of simultaneous retrieval of the particle size distribution and the complex refractive index. We present results of the eigenvalue analysis, used to study the information content of multiwavelength lidar data with respect to microphysical particle properties. Such an analysis provides, on a rather mathematical basis, more insight into the limitations of these inversion algorithms regarding the accuracy of the retrieved parameters. We show that the effective radius may be retrieved to 50% accuracy and the real and imaginary part of the complex refractive index to ±0.05 and ±0.005i, if the imaginary part is <0.02i. These results are in accordance with the classic approach of simulation studies with synthetic particle size distributions. Major difficulties are found with a particle effective radius of <0.15 μm. In that case the complex refractive index may not be derived with sufficient accuracy. The eigenvalue analysis also shows that the accuracy of the derived parameters degrades if the imaginary part is >0.02i. Furthermore it shows the importance of the simultaneous use of backscatter and extinction coefficients for the retrieval of microphysical parameters.

© 2005 Optical Society of America

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References

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  1. A. N. Tikhonov, V. Y. Arsenin, eds., Solution of Ill-Posed Problems (Wiley, 1977).
  2. S. Twomey, eds., Introduction to the Mathematics of Inversion in Remote Sensing and Direct Measurements (Elsevier, 1977).
  3. V. E. Zuev, I. E. Naats, eds., Inverse Problems of Lidar Sensing of the Atmosphere (Springer-Verlag, 1983).
    [CrossRef]
  4. P. Qing, H. Nakane, Y. Sasano, S. Kitamura, “Numerical simulation of the retrieval of aerosol size distribution from multiwavelength laser radar measurements,” Appl. Opt. 28, 5259–5265 (1989).
    [CrossRef] [PubMed]
  5. D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
    [CrossRef]
  6. D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38, 2346–2357 (1999).
    [CrossRef]
  7. D. Müller, U. Wandinger, A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation,” Appl. Opt. 38, 2358–2368 (1999).
    [CrossRef]
  8. I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, 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. Müller, K. Franke, 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. C. Böckmann, “Hybrid regularization method for ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions,” Appl. Opt. 40, 1329–1342 (2001).
    [CrossRef]
  11. D. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
    [CrossRef]
  12. D. Althausen, D. Müller, A. Ansmann, U. Wandinger, H. Hube, E. Clauder, S. Zörner, “Scanning 6-wavelength 11-channel aerosol lidar,” J. Atmos. Ocean. Technol. 17, 1469–1482 (2000).
    [CrossRef]
  13. I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
    [CrossRef]
  14. D. Müller, U. Wandinger, D. Althausen, M. Fiebig, “Comprehensive particle characterization from three-wavelength Raman lidar observations: case study,” Appl. Opt. 40, 4863–4869 (2001).
    [CrossRef]
  15. C. L. Mateer, “On the information content of Umkehr observations,” J. Atmos. Sci. 22, 370–381 (1965).
    [CrossRef]
  16. A. Ben-David, B. M. Herman, J. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory,” Appl. Opt. 27, 1235–1242 (1988).
    [CrossRef] [PubMed]
  17. D. P. Donovan, A. I. Carswell, “Principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction measurements,” Appl. Opt. 36, 9406–9424 (1997).
    [CrossRef]
  18. M. J. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Ocean. Technol. 13, 863–873 (1996).
    [CrossRef]
  19. C. F. Bohren, D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2004

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

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

2002

2001

2000

D. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

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

1999

1998

1997

1996

M. J. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Ocean. Technol. 13, 863–873 (1996).
[CrossRef]

1989

1988

1965

C. L. Mateer, “On the information content of Umkehr observations,” J. Atmos. Sci. 22, 370–381 (1965).
[CrossRef]

Althausen, D.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

D. Müller, 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. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

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

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

Ansmann, A.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

D. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

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

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

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

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

Ben-David, A.

Böckmann, C.

Bohren, C. F.

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

Carswell, A. I.

Clauder, E.

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

Donovan, D. P.

Fiebig, M.

Franke, K.

Griaznov, V.

Herman, B. M.

Hube, H.

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

Huffmann, D. R.

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

Kitamura, S.

Kolgotin, A.

Mateer, C. L.

C. L. Mateer, “On the information content of Umkehr observations,” J. Atmos. Sci. 22, 370–381 (1965).
[CrossRef]

Mattis, I.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

Müller, D.

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

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, 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]

D. Müller, 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. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

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

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

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

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

Nakane, H.

Post, M. J.

M. J. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Ocean. Technol. 13, 863–873 (1996).
[CrossRef]

Qing, P.

Reagan, J.

Sasano, Y.

Veselovskii, I.

Wagner, F.

D. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

Wandinger, U.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, 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]

D. Müller, 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. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

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

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

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

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

Whiteman, D. N.

Zörner, S.

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

Appl. Opt.

P. Qing, H. Nakane, Y. Sasano, S. Kitamura, “Numerical simulation of the retrieval of aerosol size distribution from multiwavelength laser radar measurements,” Appl. Opt. 28, 5259–5265 (1989).
[CrossRef] [PubMed]

D. Müller, U. Wandinger, D. Althausen, I. Mattis, A. Ansmann, “Retrieval of physical particle properties from lidar observations of extinction and backscatter at multiple wavelengths,” Appl. Opt. 37, 2260–2263 (1998).
[CrossRef]

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

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

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

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

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

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

Geophys. Res. Lett.

D. Müller, F. Wagner, D. Althausen, U. Wandinger, A. Ansmann, “Physical properties of the Indian aerosol plume derived from six-wavelength lidar observation on 25 March 1999 of the Indian Ocean Experiment,” Geophys. Res. Lett. 27, 1403–1406 (2000).
[CrossRef]

J. Atmos. Ocean. Technol.

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

M. J. Post, “A graphical technique for retrieving size distribution parameters from multiple measurements: visualization and error analysis,” J. Atmos. Ocean. Technol. 13, 863–873 (1996).
[CrossRef]

J. Atmos. Sci.

C. L. Mateer, “On the information content of Umkehr observations,” J. Atmos. Sci. 22, 370–381 (1965).
[CrossRef]

J. Geophys. Res.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Multiyear aerosol observations with dual-wavelength Raman lidar in the framework of EARLINET,” J. Geophys. Res. 109, D13203, doi: (2004).
[CrossRef]

Other

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

S. Twomey, eds., Introduction to the Mathematics of Inversion in Remote Sensing and Direct Measurements (Elsevier, 1977).

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

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

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Figures (15)

Fig. 1
Fig. 1

Wavelength dependence of (a) backscattering and (b) extinction coefficients calculated for lognormal size distributions of spherical particles with r0 = 0.1, 1, 2, and 4 μm; ln σ = 0.4, m = 1.33–i0. All α and β are normalized to ∫β(λ)dλ = ∫α(λ)dλ = 1 in the interval considered.

Fig. 2
Fig. 2

Wavelength dependence of (a) backscatter and (b) extinction coefficients for refractive indices: m = 1.33–i0, 1.45–i0, and 1.6–i0 calculated for a lognormal size distribution of spherical particles with r0 = 1 μm and ln σ = 0.4.

Fig. 3
Fig. 3

Extinction and backscatter coefficients as a function of wavelength for a lognormal size distribution with r0 = 1 μm and ln σ = 0.4 calculated at, solid curve, mI = 0; dashed curve, 0.01; dash–dot curve, 0.02; mR = 1.45.

Fig. 4
Fig. 4

Wavelength dependence of the, solid curves, extinction, and, dash–dot curves, backscatter coefficients for m = 1.3–i0 and m = 1.5–i0. Calculations are performed for a lognormal size distribution with mode width ln σ = 0.4 and mode radius r0 = 0.1 μm.

Fig. 5
Fig. 5

Lidar ratio for, solid symbols, m = 1.3–i0 and, open symbols, m = 1.5–i0 calculated for mode radius, circles, r0 = 0.1 μm and, squares, 1 μm.

Fig. 6
Fig. 6

Minimal eigenvalues calculated from squares, α(λ); solid circles, β(λ); and, open circles, their combination for different fixed values of mean radius r0. (a) The real part of the refractive index was varied from mR = 1.3 to mR = 1.6 with a step width of ΔmR = 0.1, while mI was kept fixed at mI = 0. (b) The imaginary part was varied from mI = 0 to 0.02 with a step width of 0.01, while mR was kept fixed at mR = 1.5. The values for β(λ) and α(λ) cover the wavelength range from 0.35 to 1.06 μm, except for the dash–dot curve (squares with crosses) that shows the case for α(λ) in the interval from 0.35 to 0.53 μm.

Fig. 7
Fig. 7

Minimum eigenvalues for different fixed values of mean radius r0 calculated from a combination of α(λ) and β(λ) for mI = 0, 0.01, 0.02. The real part of the refractive index varied from 1.3 to 1.6 with a step width of 0.1: dash–dot curve (open stars), results for mI = 0 and ΔmR = 0.05; horizontal dotted line, level corresponding to a measurement error, δ = 10%.

Fig. 8
Fig. 8

Minimal eigenvalues for different fixed values of mean radius r0 calculated from the combination of αi(λ) and βi(λ). The real part of the refractive index was kept fixed at mR = 1.5, while the imaginary part was varied at intervals: circles, [0, 0.01]; squares, [0, 0.02]; triangles, [0, 0.03] with a step width of, solid symbols, 0.01; open symbols, 0.005.

Fig. 9
Fig. 9

Minimum eigenvalues for different fixed values of mean radius r0 calculated from the combination of αi(λ) and βi(λ). The real part of the refractive index mR is varied from 1.3 to 1.6 with ΔmR = 0.1, while mI is varied in the intervals: circles, [0, 0.01]; triangles, [0, 0.02]; squares, [0, 0.03] with a step width of ΔmI = 0.01. Also shown for comparison are the results for fixed mI = 0, while mR is varied, open stars, and for a fixed mR = 1.5 when mI is varied in the interval, open circles, [0, 0.01]; open squares, [0, 0.03].

Fig. 10
Fig. 10

Minimum eigenvalue as a function of r0,max calculated from, open squares, βi(λ); open circles, αi(λ); and, solid circles, their combination. The calculations are performed in the radius interval [r0min, r0max], r0,min = 0.05 μm; r0,max is varied by a step width of 0.1 and 0.5 μm. The complex refractive index is m = 1.5–i0; solid stars, result obtained for a combination of βi(λ) and αi(λ) for m = 1.5–i0.01.

Fig. 11
Fig. 11

Minimum eigenvalues calculated from the combination of αi(λ) and βi(λ) when two parameters are varied: solid circles, r0 and mR; open circles, r0 and mI. The real part of the refractive index mR is varied from 1.3 to 1.6 with ΔmR = 0.1, while, solid circles, mI = 0. The imaginary part of the refractive index mI is varied from 0.0 to 0.03 with ΔmI = 0.01, while, open circles, mR = 1.5. The calculations are performed in a radius interval of [r0min, r0max], r0min = 0.05 μm, and a step width of Δr0 = r0/2. For comparison the dash–dot curves show the results when two parameters are fixed: open squares, mR and mI; solid stars, r0 and mI; open stars, r0 and mR.

Fig. 12
Fig. 12

Possible ways to estimate the complex refractive index. The minimum eigenvalue is shown as a function of radius. Comparisons between the elements with different radii but the same values of m are excluded. Calculations are performed, solid symbols, for fixed value mI = 0 and for mI varied in the interval range of, open circles, [0; 0.01] and, open stars, [0; 0.02] with ΔmI = 0.01 and ΔmR = 0.15. The eigenvalues are determined for the radius interval [r0min, r0max] with r0min = 0.05 μm and a step width of Δr0 = r0/2.

Fig. 13
Fig. 13

Estimations of the particles’ size when the refractive index is unknown. The minimum eigenvalue is shown as a function of r0,max for mR varying in the range of 1.3 < mR < 1.6 with ΔmR = 0.15; circles, mI varying in the range of 0 < mI < 0.02 with ΔmI = 0.01. The plot also shows the results for transparent particles (circles, mI = 0) and for the case, squares, when the refractive index of m = 1.5–i0 is assumed known. The eigenvalues are determined at a radius interval of [r0min, r0max], r0min = 0.05 μm, and a step width of Δr0 = r0/2.

Fig. 14
Fig. 14

First four eigenvalues of the covariance matrix calculated for backscatter and extinction coefficients: solid lines, refractive index kept fixed at m = 1.45–i0; dotted lines, case in which mR and mI are varied within the range of 1.3–1.6 and 0–0.02, respectively.

Fig. 15
Fig. 15

Eigenvalues as a function of r0max: circles, third eigenvalue; squares, fourth eigenvalue. Calculations are performed at an interval of [r0min, r0max]; r0min = r0max/5 for three backscattering coefficients (solid circles, 3β) at 355, 532, and 1064 nm wavelengths. The plot also shows the eigenvalues for backscattering at 400 nm (open circles, 4β), 710 nm (circles with crosses, 5β) and 800 nm (squares with crosses, 6β) are added. The refractive index is 1.45–i0. The volume kernels are smoothed with a size distribution with ln σ = 0.4: dotted line, level corresponding to 10% measurements errors.

Equations (18)

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β ( λ ) = r min r max f ( r ) K β ( r ,     m ,     λ ) d r , α ( λ ) = r min r max f ( r ) K α ( r ,     m ,     λ ) d r .
n ( r ) ln r = n t ( 2 π ) 1 / 2 ln σ exp [ - ( ln r - ln r 0 ) 2 2 ( ln σ ) 2 ] ,
β ( λ ) d λ = 1 ,             α ( λ ) d λ = 1
i a i β i .
i a i 2 = 1 ,
q = λ min λ max [ i a i β i ( λ ) ] 2 d λ
[ a 1 a n ] = a ,             [ β 1 β n ] = B ,
q = a * B B * a d λ = a * C β a ,
q = a * C β a = ξ * U * C β U ξ .
q = ξ * Λ ξ - i l β i ξ i 2 .
q = λ min λ max [ i a i β i ( λ ) ] 2 d λ + λ min λ max [ i a i α i ( λ ) ] 2 d λ .
i = 1 M × N - 1 ( M × N - i ) .
λ min λ max β ij 2 ( λ ) d λ ,
( r 1 ,     m 1 ) ( r 2 ,     m 1 ) ( r N ,     m 1 ) ( r 1 ,     m 2 ) ( r 2 ,     m 2 ) ( r N ,     m 2 ) ( r 1 ,     m M ) ( r 2 ,     m M ) ( r N ,     m M ) .
i l i = 1.
i β i 2 ( λ ) d λ .
D = r min r max K i ( r ) K j ( r ) d r ,
r 0 min r 0 max K ˜ i 2 ( r 0 ,     m ) d r 0 = 1 ,

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