Abstract

The inversion technique of Backus and Gilbert is applied to the determination of size distributions of spherical particles from optical scattering measurements. The spatial resolution inherent in a set of multiwavelength measurements is studied as a function of number of measurements, measurement noise level, and radius. The inversion technique is then applied to computer simulated intensity data to recover size distributions. These examples indicate that the distribution can be recovered at selected points without using a priori assumptions about the shape of the distribution.

© 1973 Optical Society of America

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References

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  1. A. C. Holland, G. Gagné, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]
  2. R. Eiden, Appl. Opt. 5, 569 (1966).
    [CrossRef] [PubMed]
  3. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]
  4. E. de Bary, B. Braun, K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Special Rept. 33, AFCRL-G5-710, Cambridge, Mass. (1965).
  5. M. P. McCormick, J. D. Lawrence, F. R. Crowfield, Appl. Opt. 7, 2424 (1968).
    [CrossRef] [PubMed]
  6. E. W. Barrett, O. Ben-Dov, J. Appl. Meteor. 6, 505 (1967).
    [CrossRef]
  7. J. V. Dave, Appl. Opt. 8, 1161 (1969).
    [CrossRef] [PubMed]
  8. J. V. Dave, Appl. Opt. 10, 2035 (1971).
    [CrossRef] [PubMed]
  9. B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
    [CrossRef]
  10. H. Grassl, Appl. Opt. 10, 2534 (1971).
    [CrossRef] [PubMed]
  11. G. Backus, F. Gilbert, Philos. Trans. Roy. Soc. London 266, 123 (1970).
    [CrossRef]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  13. T. S. Fahlen, H. C. Bryant, J. Opt. Soc. Am. 58, 304 (1968).
    [CrossRef]
  14. A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
    [PubMed]
  15. B. J. Mason, The Physics of Clouds (Clarendon Press, Oxford, 1971).

1971 (3)

1970 (2)

G. Backus, F. Gilbert, Philos. Trans. Roy. Soc. London 266, 123 (1970).
[CrossRef]

A. C. Holland, G. Gagné, Appl. Opt. 9, 1113 (1970).
[CrossRef] [PubMed]

1969 (1)

1968 (2)

1967 (1)

E. W. Barrett, O. Ben-Dov, J. Appl. Meteor. 6, 505 (1967).
[CrossRef]

1966 (1)

1964 (1)

Backus, G.

G. Backus, F. Gilbert, Philos. Trans. Roy. Soc. London 266, 123 (1970).
[CrossRef]

Barrett, E. W.

E. W. Barrett, O. Ben-Dov, J. Appl. Meteor. 6, 505 (1967).
[CrossRef]

Ben-Dov, O.

E. W. Barrett, O. Ben-Dov, J. Appl. Meteor. 6, 505 (1967).
[CrossRef]

Braun, B.

E. de Bary, B. Braun, K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Special Rept. 33, AFCRL-G5-710, Cambridge, Mass. (1965).

Browning, S. R.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Bryant, H. C.

Bullrich, K.

E. de Bary, B. Braun, K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Special Rept. 33, AFCRL-G5-710, Cambridge, Mass. (1965).

Cohen, A.

A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
[PubMed]

Crowfield, F. R.

Cupp, R. E.

A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
[PubMed]

Dave, J. V.

de Bary, E.

E. de Bary, B. Braun, K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Special Rept. 33, AFCRL-G5-710, Cambridge, Mass. (1965).

Deirmendjian, D.

Derr, V. E.

A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
[PubMed]

Eiden, R.

Fahlen, T. S.

Gagné, G.

Gilbert, F.

G. Backus, F. Gilbert, Philos. Trans. Roy. Soc. London 266, 123 (1970).
[CrossRef]

Grassl, H.

Herman, B. M.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

Holland, A. C.

Lawrence, J. D.

Mason, B. J.

B. J. Mason, The Physics of Clouds (Clarendon Press, Oxford, 1971).

McCormick, M. P.

McNice, G. T.

A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
[PubMed]

Reagan, J. A.

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

van de Hulst, H. C.

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

Appl. Opt. (7)

J. Appl. Meteor. (1)

E. W. Barrett, O. Ben-Dov, J. Appl. Meteor. 6, 505 (1967).
[CrossRef]

J. Atmos. Sci. (1)

B. M. Herman, S. R. Browning, J. A. Reagan, J. Atmos. Sci. 28, 763 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

Philos. Trans. Roy. Soc. London (1)

G. Backus, F. Gilbert, Philos. Trans. Roy. Soc. London 266, 123 (1970).
[CrossRef]

Other (4)

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

A. Cohen, V. E. Derr, G. T. McNice, R. E. Cupp, The Measurement of Mie Scattering Intensities from Monodispersed Spherical Particles as a Function of Wavelength, to be published in Appl. Opt. (1973).
[PubMed]

B. J. Mason, The Physics of Clouds (Clarendon Press, Oxford, 1971).

E. de Bary, B. Braun, K. Bullrich, Tables Related to Light Scattering in a Turbid Atmosphere, Special Rept. 33, AFCRL-G5-710, Cambridge, Mass. (1965).

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Figures (9)

Fig. 1
Fig. 1

MSI curves at radii of 4.5 μm and 2.5 μm for m = 1.33 and m = f(λ). i(n) is the scattering intensity of the component normal to the scattering plane, for a scattering angle of 90°. +—m = 1.33; 0—m = f(λ).

Fig. 2
Fig. 2

Weighting functions for multispectral 90° scattering in relative units.

Fig. 3
Fig. 3

Averaging kernels constructed from twenty-one multispectral weighting functions. The wavelength interval was 400–700 nm.

Fig. 4
Fig. 4

Averaging kernels at r0 = 0.9 μm constructed from fourteen and twenty-one multispectral weighting functions. The wavelength intervals were 400–595 nm and 400–700 nm. The two curves on the right have greater than minimum spread.

Fig. 5
Fig. 5

Error-spread trade-off curves for fourteen and twenty-one measurements. The units are such that multiplying the abscissas’ value by 7.8, 11.9, and 8.5 gives the percentage error in inferring the size distribution at r0 = 0.9 μm for distributions I, II, and III of Sec. IV. The curves are labeled with the assumed measurement errors.

Fig. 6
Fig. 6

Averaging kernels at r0 = 0.75 μm constructed from twenty-one weighting functions over the range 0.5–1.0 μm and 0.5–0.77 μm.

Fig. 7
Fig. 7

Approximation of rectangular function at 0.75 μm with total width = 0.1 μm by a linear combination of twenty-one weighting functions.

Fig. 8
Fig. 8

Scattering intensities relative to I (λ = 550 nm) for distributions I, II, and III. The lines drawn between points are not meant to suggest values for intermediate points.

Fig. 9
Fig. 9

Inferred size distribution function at six radii from twenty-one multispectral simulated measurements.

Tables (1)

Tables Icon

Table I Minimum Spread as a Function of Radius for Twenty-one Measurements

Equations (18)

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g i = a b K i ( r ) f ( r ) d r , i = 1 , 2 , , n ,
S = E { T } ,
f , A = a b A ( r ) f ( r ) d r ,
a b A ( r ) d r = 1.
A ( r ) = i = 1 n a i K i ( r ) .
( a ) = a b [ A ( r ) - δ ( r - r 0 ) ] 2 d r = min
f ( r ) r 0 = a b A ( r 0 , r ) f ( r ) d r = i = 1 n a i ( r 0 ) g i .
s ( r 0 ) = 12 a b ( r - r 0 ) 2 A 2 ( r ) d r ,
= a T S a ,
S i j = 12 a b ( r - r 0 ) 2 K i ( r ) K j ( r ) d r ,
a = [ a 1 , a 2 , , a n ] T .
f ( r ) r 0 , V [ f ( r ) r 0 ] is given by V [ f ( r ) r 0 ] = a T S a .
a ( θ ) = [ u T W ( θ ) - 1 u ] - 1 W ( θ ) - 1 u ,
( u ) i = a b K i ( r ) d r ,
W ( θ ) = S cos θ + w S sin θ ,
F ^ ( r ) = i = 1 n K i ( r ) a i .
a b [ F ^ ( r ) - F ( r ) ] 2 d r
a b F ^ ( r ) d r = 1.

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