Abstract

The information content of a set of optical data with respect to the particle size distribution is discussed in a numerical study. We show how the kernels of the integral equation relating size distribution and optical properties can be used to determine the particle size range in which an inversion of the size distribution is possible. We present an iterative least squares fit algorithm allowing the inversion of optical data to yield a histogram distribution for the particle size distribution. We discuss the uniqueness and stability of the solutions in relation to the range of radii and in relation to the number of histogram columns by means of synthetic data calculated via Mie theory.

© 1981 Optical Society of America

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References

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  1. A. Fymat, in “Inversion Methods in Atmospheric Remote Sounding,” NASA CP-004 (1976).
  2. H. Quenzel, H. Müller, “Optical Properties of Single Mie Particles: Diagrams of Intensity-, Extinction-, Scattering- and Absorption Efficiencies,” Wissenschaftliche Mitteilung Nr. 34., U. München, Meteorologisches Institut (1978).
  3. C. Junge, Ber. Dtsch. Wetterdienstes 35, 261, 1952.
  4. K. T. Whitby, G. M. Sverdrup, in The Character and Origins of Smog Aerosols, G. M. Hidy, P. K. Mueller, Eds. (Wiley, New York, 1980).
  5. J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).
  6. H. Quenzel, J. Geophys. Res. 75, 2915 (1970).
    [CrossRef]

1978

J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).

1970

H. Quenzel, J. Geophys. Res. 75, 2915 (1970).
[CrossRef]

1952

C. Junge, Ber. Dtsch. Wetterdienstes 35, 261, 1952.

Fymat, A.

A. Fymat, in “Inversion Methods in Atmospheric Remote Sounding,” NASA CP-004 (1976).

Heintzenberg, J.

J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).

Junge, C.

C. Junge, Ber. Dtsch. Wetterdienstes 35, 261, 1952.

Müller, H.

H. Quenzel, H. Müller, “Optical Properties of Single Mie Particles: Diagrams of Intensity-, Extinction-, Scattering- and Absorption Efficiencies,” Wissenschaftliche Mitteilung Nr. 34., U. München, Meteorologisches Institut (1978).

Quenzel, H.

H. Quenzel, J. Geophys. Res. 75, 2915 (1970).
[CrossRef]

H. Quenzel, H. Müller, “Optical Properties of Single Mie Particles: Diagrams of Intensity-, Extinction-, Scattering- and Absorption Efficiencies,” Wissenschaftliche Mitteilung Nr. 34., U. München, Meteorologisches Institut (1978).

Sverdrup, G. M.

K. T. Whitby, G. M. Sverdrup, in The Character and Origins of Smog Aerosols, G. M. Hidy, P. K. Mueller, Eds. (Wiley, New York, 1980).

Whitby, K. T.

K. T. Whitby, G. M. Sverdrup, in The Character and Origins of Smog Aerosols, G. M. Hidy, P. K. Mueller, Eds. (Wiley, New York, 1980).

Ber. Dtsch. Wetterdienstes

C. Junge, Ber. Dtsch. Wetterdienstes 35, 261, 1952.

Contrib. Atmos. Phys.

J. Heintzenberg, Contrib. Atmos. Phys. 51, 91 (1978).

J. Geophys. Res.

H. Quenzel, J. Geophys. Res. 75, 2915 (1970).
[CrossRef]

Other

K. T. Whitby, G. M. Sverdrup, in The Character and Origins of Smog Aerosols, G. M. Hidy, P. K. Mueller, Eds. (Wiley, New York, 1980).

A. Fymat, in “Inversion Methods in Atmospheric Remote Sounding,” NASA CP-004 (1976).

H. Quenzel, H. Müller, “Optical Properties of Single Mie Particles: Diagrams of Intensity-, Extinction-, Scattering- and Absorption Efficiencies,” Wissenschaftliche Mitteilung Nr. 34., U. München, Meteorologisches Institut (1978).

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Figures (9)

Fig. 1
Fig. 1

Kernel functions K(m,r,λ) as functions of the particle radius r for λ = 0.2 μm (—) and λ = 12.8 μm (-·-); refractive index m = 1.5 − 0.02i.

Fig. 2
Fig. 2

Ratios of kernel functions for different pairs of spectral channels as function of the particle radius r; refractive index m = 1.5 − 0.02i.

Fig. 3
Fig. 3

Flowchart of the inversion algorithm. See Sec. III for explanation of symbols.

Fig. 4
Fig. 4

Histogram size distributions inverted from synthetic optical data calculated for four different models. The given model distributions are drawn as full smooth lines; shaded areas show the scatter of the individual inversion results. The optimum seven-column subrange was used in the inversions.

Fig. 5
Fig. 5

As Fig. 4, but with the number of columns increased from seven to ten. The range of scatter for the individual solutions is shown for model A only. An arrowed lower limit of the shaded area for a histogram column marks the range of scatter extending down to zero.

Fig. 6
Fig. 6

As Fig. 4, but with the number of columns increased to forty.

Fig. 7
Fig. 7

As Fig. 4, but the size limits for the inversion have increases by a factor of three.

Fig. 8
Fig. 8

As Fig. 4, but the size limits for the inversion decreased by a factor of three. In between two unconnected dotted lines the volume of the inverted size distribution is zero.

Fig. 9
Fig. 9

Histogram size distributions inverted from synthetic optical data with 5% random noise. The five different inversion results plotted for each model distribution result from five sets of data with different random noise but the same average error.

Tables (1)

Tables Icon

Table I Results of Fig. 5 as Relative Volume Deviations of the Single Columns in %a

Equations (9)

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σ s ( λ ) = + Q s ( m , r , λ ) · π r 2 d N ( r ) d log r d log r .
σ s ( λ ) = + Q s ( m , r , λ ) · π r 2 4 / 3 · π r 3 d V ( r ) d log r d log r .
K ( m , r , λ ) = 3 Q s ( m , r , λ ) 4 r .
S j e = + K j ( r ) d V ( r ) d log r d log r ,
S j c = l = 1 L K ¯ j l · υ ¯ l
K ¯ j l = Δ log r l K j ( r ) d log r .
D opt = 1 J j = 1 J ( S j c S j e S j e ) 2 .
S j c = a · K ¯ j r · υ ¯ r + b · l = 1 l r L K ¯ j l · υ ¯ l .
D ¯ opt 1 N n = 1 N D opt ( n ) .

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