Abstract

Aerosols interact with optical radiation in a complicated way, but in a way that can be described through the methods of electromagnetic theory. By applying constrained linear inversion methods one can recover an estimate for the size spectrum of aerosols in the micron and submicron size range from measured optical data. The amount of information that can be secured is increased by combining measurements of spectral extinction with measurements of optical scattering. From the combined measurements one can obtain about seven pieces of information about the particle size spectrum, provided optical parameters are measured to 2.5% accuracy. The application of linear constrained inversion on simulated noise-degraded data successfully recovers major features, such as the presence of modes, in the aerosol size distribution function.

© 1979 Optical Society of America

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References

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  1. S. Twomey, J. Franklin Inst. 279, 2 (1965).
    [CrossRef]
  2. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  3. J. V. Dave, Appl. Opt. 10, 2035 (1971).
    [CrossRef] [PubMed]
  4. G. Yamamoto, M. Tanaka, Appl. Opt. 8, 447 (1969).
    [CrossRef] [PubMed]
  5. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969), Chap. 3.
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  7. C. E. Junge, “Air Chemistry and Radioactivity,” in International Geophysics Series, J. V. Mieghem, Ed. (Academic, New York, 1963).
  8. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report 320–3237 (IBM Scientific Center, Palo Alto, Calif., 1968).
  9. S. Twomey, H. Howell, Appl. Opt. 6, 2125 (1969).
    [CrossRef]

1971 (1)

1969 (2)

1965 (1)

S. Twomey, J. Franklin Inst. 279, 2 (1965).
[CrossRef]

Dave, J. V.

J. V. Dave, Appl. Opt. 10, 2035 (1971).
[CrossRef] [PubMed]

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report 320–3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969), Chap. 3.

Howell, H.

Junge, C. E.

C. E. Junge, “Air Chemistry and Radioactivity,” in International Geophysics Series, J. V. Mieghem, Ed. (Academic, New York, 1963).

Tanaka, M.

Twomey, S.

S. Twomey, H. Howell, Appl. Opt. 6, 2125 (1969).
[CrossRef]

S. Twomey, J. Franklin Inst. 279, 2 (1965).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Yamamoto, G.

Appl. Opt. (3)

J. Franklin Inst. (1)

S. Twomey, J. Franklin Inst. 279, 2 (1965).
[CrossRef]

Other (5)

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969), Chap. 3.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. E. Junge, “Air Chemistry and Radioactivity,” in International Geophysics Series, J. V. Mieghem, Ed. (Academic, New York, 1963).

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” IBM Report 320–3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

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Figures (5)

Fig. 1
Fig. 1

Kernel functions in the integral Eq. (10): f = 1, H = 105 cm, Γ = 50 × 10−12 g cm−3, ρ = 1 g cm−3, and λ0 = 700 nm.

Fig. 2
Fig. 2

Examples of the effect of varying the constraining smoothing parameter γ on the recovered aerosol size distribution function. The optimum γ for the modeled experiment (2.5% noise) is about 10−4. Too small a γ underconstrains the solution and causes oscillations to appear, while γ being too large results in oversmoothing and the suppression of high frequency structure.

Fig. 3
Fig. 3

Examples of recovered aerosol size distribution functions. The optical vector was degraded with 2.5% rms noise before being inverted. Solid line indicates the actual function f; dotted line indicates f ¯ derived by inversion (The upper left hand block shows two inverted noisy vectors).

Fig. 4
Fig. 4

The rms deviations ( E f 2 ¯ ) 1 / 2 = ( f - f ¯ ¯ ) 1 / 2 between actual and inverted size distribution vectors as a function of the smoothing parameter γ. The error bars show the range of rms deviations recovered when twenty independent noise-degraded (2.5% rms noise) optical vectors were inverted. The norm of the vector f is 2.7. Most of the deviation in the recovered solutions occurred in the small particle size range r < 0.1 μm, where the information content is poor. Values of γ near 10−4 provide the most accurate inversions.

Fig. 5
Fig. 5

Recovered aerosol size spectra for three independent noisy input vectors with 2.5% random noise. The error bars are the standard deviations for fj computed from twenty inversions of noise-degraded measurements.

Tables (2)

Tables Icon

Table I Parameters Calculated for Inversion of the Aerosol Size Spectrum (fj = 1, j = 1 … 20)

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Table II Dominant Eigenvalues of the Matrix A*A

Equations (14)

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g ( x ) = K ( x , r ) f ( r ) d r ,
g i + ɛ i = j = 1 n w i j K i j f j ,
[ g ] + [ ɛ ] = [ A ]             [ f ] ( m x 1 ) ( m x n ) ( n x 1 ) ,
[ f ] = [ A ] - 1 [ g ] .
[ f ] = [ A * A ] - 1 A * g ,
Q ˜ = f * H f = i j h i j f i f j .
i = 1 m ɛ i 2 E 2 .
f = ( A * A + γ H ¯ ¯ ) - 1 A * g .
Q ˜ min = 2 n - 1 ( f i - 1 - 2 f i + f i + 1 ) 2 ,
H ¯ ¯ = | 1 - 2 1 0 . . . . - 1 5 - 4 1 0 . . . 1 - 4 6 - 4 1 0 0 . . 1 - 4 + 6 - 4 1 0 0 . 1 . - 2 1 | .
β ( λ ) = r 1 r 2 π r 2 Q ( r , λ ) ( d N d r ) d r , I ( θ ) = ( λ 0 2 π ) 2 r 1 r 2 [ M 1 ( r , θ ) + M 2 ( r , θ ) 2 ] ( d N d r ) d r .
( d N / d r ) = [ f ( r ) ] / ( r 4 ) .
λ min i = 1 m ɛ i 2 j = 1 n f j 2 = RE 2 · Σ g i 2 Σ f j 2 .
α K ¯ = ( 1 + γ λ K ) - 1 α K ,

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