Abstract

The classical inversion of forward scattered light to obtain the size distribution of spherical particles is considered. Using the diffraction approximation, the measurement points are selected so as to reduce the influence of the particle refractive index and to make the Chahine inversion scheme usable with a good convergence rate. The algorithm efficiency, in the presence of refractive-index errors and of experimental noise, is studied. An interesting behavior of the Chahine inversion scheme, in the presence of very large experimental errors, is outlined.

© 1983 Optical Society of America

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References

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  1. D. Deirmendjian, Rev. Geophys. Space Phys. 18, 341 (1980).
    [CrossRef]
  2. D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  3. S. Twomey, J. Comput. Phys. 18, 188 (1975).
    [CrossRef]
  4. K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).
  5. M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
    [CrossRef]
  6. J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  8. D. Deirmendjian, “Scattering and Polarization Properties of Polydispersed Suspensions with Partial Absorption,” in Electromagnetic Scattering. International Series of Monographs on Electromagnetic Waves, Vol. 5, M. Kerker, Ed. (Pergamon, New York, 1963).
  9. J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
    [CrossRef]
  10. R. Zerull, R. H. Giese, in Planets, Stars, and Nebulae Studies with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), pp. 901–904.
  11. A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]

1980 (1)

D. Deirmendjian, Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

1975 (2)

S. Twomey, J. Comput. Phys. 18, 188 (1975).
[CrossRef]

J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

1972 (1)

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

1970 (2)

1962 (1)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Chahine, M. T.

M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

D. Deirmendjian, “Scattering and Polarization Properties of Polydispersed Suspensions with Partial Absorption,” in Electromagnetic Scattering. International Series of Monographs on Electromagnetic Waves, Vol. 5, M. Kerker, Ed. (Pergamon, New York, 1963).

Gagne, G.

Gashko, V. A.

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

Giese, R. H.

R. Zerull, R. H. Giese, in Planets, Stars, and Nebulae Studies with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), pp. 901–904.

Hansen, J. E.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Holland, A. C.

Phillips, D. L.

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

Travis, L. D.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Turchin, V. F.

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

Turoviseva, L. S.

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

Twitty, J. T.

J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
[CrossRef]

Twomey, S.

S. Twomey, J. Comput. Phys. 18, 188 (1975).
[CrossRef]

van de Hulst, H. C.

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

Zerull, R.

R. Zerull, R. H. Giese, in Planets, Stars, and Nebulae Studies with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), pp. 901–904.

Appl. Opt. (1)

Izv. Acad. Sci. USSR Atmos Oceanic Phys. (1)

K. S. Shifrin, V. F. Turchin, L. S. Turoviseva, V. A. Gashko, Izv. Acad. Sci. USSR Atmos Oceanic Phys. 8, 1266 (1972).

J. Assoc. Comput. Mach. (1)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

J. Atmos. Sci. (2)

M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
[CrossRef]

J. T. Twitty, J. Atmos. Sci. 32, 584 (1975).
[CrossRef]

J. Comput. Phys. (1)

S. Twomey, J. Comput. Phys. 18, 188 (1975).
[CrossRef]

Rev. Geophys. Space Phys. (1)

D. Deirmendjian, Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Other (3)

R. Zerull, R. H. Giese, in Planets, Stars, and Nebulae Studies with Photopolarimetry, T. Gehrels, Ed. (U. Arizona Press, Tucson, 1974), pp. 901–904.

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

D. Deirmendjian, “Scattering and Polarization Properties of Polydispersed Suspensions with Partial Absorption,” in Electromagnetic Scattering. International Series of Monographs on Electromagnetic Waves, Vol. 5, M. Kerker, Ed. (Pergamon, New York, 1963).

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Figures (14)

Fig. 1
Fig. 1

Value of the normalized phase function P(θ) at the given scattering angle θ = 3°. The results are for spherical nonabsorbing particles with radius r. The P(θ) is given as a function of r/λ for two values of the particle real refractive index: m = 1.33 and m = 1.55. Solid lines: diffraction approximation.

Fig. 2
Fig. 2

Same as Fig. 1, but for θ = 5.5°.

Fig. 3
Fig. 3

Convergence of the residuals σp as a function of the order of iteration p. Full curves correspond to inversion with the correct particle refractive index m = 1.45 of four measurement samples with different values of the rms experimental error σs. Points correspond to the inversion of a noiseless measurement sample, but with an incorrect value m = 1.33.

Fig. 4
Fig. 4

Convergence of the residuals σp for the case of model C. The results are for noiseless measurements, but for three different guesses for the initial values n(0)(rj): (a) n(0)(r) ∼ r−2; (b) n(0)(r) ∼ r−1; (c) n(0)(r) = cst.

Fig. 5
Fig. 5

Exact and inverted functions n(r) for the case of model C. Logarithmic scale. Observation wavelength: λ = 1 μm. The results are for a noiseless measurement sample and with the correct guess (m = 1.45) for the particle refractive index.

Fig. 6
Fig. 6

Same as Fig. 5, but for the bimodal gamma distribution (see text). Linear scale.

Fig. 7
Fig. 7

Same as Fig. 5, but the three inverted results correspond to the three different guesses for n(0)(rj), as labeled in Fig. 4.

Fig. 8
Fig. 8

Effect of an incorrect choice for the particle refractive index on the inverted result. Case of model C. Actual particle refractive index: m = 1.45, λ = 1 μm. The three-inverted results correspond to a noiseless measurement sample, but with three different guesses for m(1.33, 1.45, 1, 55) in the calculation of the inverting kernel.

Fig. 9
Fig. 9

Same as Fig. 8, but for the gamma bimodal distribution.

Fig. 10
Fig. 10

Effect of the measurement errors on the inverted results. Case of model C. λ = 1 μm. Inversions are for the actual refractive index. Four inverted results are presented, which correspond to four different measurement samples, with the same rms error σs ≃ 3.5%.

Fig. 11
Fig. 11

Same as Fig. 10, but for the gamma bimodal distribution and with σs ≃ 1.5%.

Fig. 12
Fig. 12

The rms error σn(rjs) in the inverted result n(rj) as a function of the rms experimental error σs in the measurement sample (see text). Results are for the gamma bimodal distribution for three different classes of particle.

Fig. 13
Fig. 13

Variation as a function of P of the rms relative error, σ n ¯ ( P , r j , σ s ) in n ¯ ( P , r j , σ s ), which is the average of P independent inverted results. Case of the gamma bimodal distribution, rj = 2.6 μm; noise level in the measurement samples: σs = 30%. Points give the ratio [ σ n ¯ ( P , r j , σ s ) / σ n ( r j , σ s ) ] 2 as a function of P [see text, Eq. (24)].

Fig. 14
Fig. 14

Full line: average value of twenty-five inverted results, from independent samples with rms experimental error σs = 18%. Case of the gamma bimodal distribution. Dashed line: for comparison, inverted result using as input data the average of the twenty-five measurement samples.

Tables (2)

Tables Icon

Table I Dimensional Resolution

Tables Icon

Table II Case of the Gamma Bimodal Distribution; Distribution of the Relative Errors in n(rj) as a function of the Gaussian Noise Level σs, for three Different Particle Classes

Equations (29)

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p ( θ ) = P ( θ ) P ( θ 0 ) = 0 π r 2 Q ( m , r / λ ) P ( m , r / λ , θ ) N ( r ) d r 0 π r 2 Q ( m , r / λ ) P ( m , r / λ , θ 0 ) N ( r ) d r ,
p ( θ ) = 0 n ( r ) P ( m , r / λ , θ ) d r ,
n ( r ) = π r 2 Q ( m , r / λ ) N ( r ) 0 π r 2 Q ( m , r / λ ) P ( m , r / λ , θ 0 N ) ( r ) d r
0 n ( r ) d r = 1 P ¯ ( θ 0 ) .
P diff ( r / λ , θ ) = 4 α 2 [ J 1 ( α sin θ ) α sin θ ] 2 ,
Δ ( r λ ) 1 2 ( m 1 ) ,
[ P diff ( r / λ , θ i ) r ] r = r i = 0 ,
r i λ = 1.84 2 π sin θ i 17 θ i ( deg ) .
r max 17 λ θ min ( deg ) .
P ¯ ( m , r i / λ , θ j ) = r i Δ r i / 2 r i + Δ r i / 2 P ( m , r / λ , θ j ) π r 2 Q ( m , r / λ ) d r r i Δ r i / 2 r i + Δ r i / 2 π r 2 Q ( m , r / λ ) d r .
p ( p ) ( θ i ) = j = 1 N P ¯ ( m , r j / λ , θ i ) n ( p ) ( r j ) Δ r j ,
n ( p + 1 ) ( r i ) = n ( p ) ( r i ) p mes ( θ i ) p ( p ) ( θ i ) ,
σ p = { 1 N i = 1 N [ p mes ( θ i ) p ( p ) ( θ i ) p mes ( θ i ) ] 2 } 1 / 2
Δ r / λ = 0.4 Δ r / λ = 0.5 Δ r / λ = 1 for 0.4 r / λ 4 , for 4 r / λ 7 , for 7 r / λ 10 .
N ( r ) = N 0 r ( 1 / υ eff 3 ) exp [ r / ( r eff , υ eff ] ,
[ j = 1 N P ¯ ( m , r j / λ , θ i ) n ( r j ) Δ r j ] / [ j = 1 N n ( r j ) Δ r j ]
j = 1 N n ( 0 ) ( r j ) Δ r j = n ( 0 ) j = 1 N Δ r j = 1
N ( r ) = N 0 , for r r 0 , for r r 0 N ( r ) = N 0 ( r r 0 ) 4 , with r 0 = 0.1 μ m ;
r 1 eff = 1 μ m ; r 2 eff = 3 μ m ; υ 1 eff = υ 2 eff = 0.07 .
Δ n m ( r j , σ s ) n ( r j ) = n m ( r j , σ s ) n ( r j ) n ( r j )
σ n ( r j , σ s ) = { 1 M m = 1 M [ Δ n m ( r j , σ s ) n ( r j ) ] 2 } 1 / 2
| Δ n ( r j , σ s ) n ( r j ) | σ n ( r j , σ s / P ) ,
| Δ n ( r j , σ s ) n ( r j ) | σ n ( r j , σ s ) P ,
σ n ( σ s ) P < σ n ( σ s P )
n ¯ ( P , r j , σ s ) = 1 P p = 1 P n p ( r j , σ s ) .
σ n ¯ ( P , r j , σ s ) 1 P σ n ( r j , σ s )
| Δ n n | σ n
| Δ n n | 2 σ n
| Δ n n | 2 Δ n ¯ n

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