Abstract

An improved inversion procedure for the retrieval of aerosol size distributions using aureole data is presented. The inversion procedure involved a modified Twomey–Chahine algorithm using differential kernels with subsequent averaging and smoothing of the inverted size distributions. The numerical simulations were performed by calculating the angular scattering coefficients between 1.08° and 10° for the wavelength of 0.2537 μm using four different aerosol models, including a realistic noise estimate of 4%. It is demonstrated that the differential kernel method is superior to the conventional direct kernel approach.

© 1985 Optical Society of America

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References

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  1. R. Santer, M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294 (1983).
    [CrossRef] [PubMed]
  2. A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194 (1981).
    [CrossRef] [PubMed]
  3. D. Deirmendjian, “A Survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341 (1980).
    [CrossRef]
  4. T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the optical properties of aerosols from aureole and extinction data,” Appl. Opt. 22, 1951 (1983).
    [CrossRef]
  5. N. T. O’Neill, J. R. Miller, “Combined solar aureole and solar beam extinction measurements. Parts 1 and 2,” Appl. Opt. 23, 3691 (1984).
    [CrossRef]
  6. M. Z. Hansen, “Atmosphric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441 (1980).
    [CrossRef] [PubMed]
  7. E. Trakhovsky, U. Oppenheim, “Determination of aerosol size distribution from observation of the aureole around a point source. 1: Theoretical,” Appl. Opt. 23, 1003 (1984).
    [CrossRef] [PubMed]
  8. E. Trakhovsky, U. Oppenheim, “Determination of aerosol size distribution from observation of the aureole around a point source. 2: Experimental,” Appl. Opt. 23, 1848 (1984).
    [CrossRef] [PubMed]
  9. S. Twomey, “Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).
  10. K. T. Whitby, “The physical characteristics of sulfur aerosols,” Atmos. Environ. 12, 135 (1978).
    [CrossRef]
  11. E. T. Thomalla, H. Quenzel, “Information content of aerosol optical properties with respect to their size disribution,” Appl. Opt. 21, 3170 (1982).
    [CrossRef] [PubMed]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  13. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490 (1980).
    [CrossRef]
  14. E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (National Technical Information Service, Springfield, Va., 1979).
  15. J. I. F. King, “Differential inversion,” Preprint Abstract Volume, Ninth Conference on Aerospace and Aeronautical Meteorology (American Meteorological Society, Boston, Mass., 1983).
  16. J. I. F. King, “Theory and application of differential inversion to remote sensing,” presented at Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., November 1984.

1984 (3)

1983 (2)

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the optical properties of aerosols from aureole and extinction data,” Appl. Opt. 22, 1951 (1983).
[CrossRef]

R. Santer, M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294 (1983).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

1980 (3)

M. Z. Hansen, “Atmosphric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441 (1980).
[CrossRef] [PubMed]

D. Deirmendjian, “A Survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490 (1980).
[CrossRef]

1978 (1)

K. T. Whitby, “The physical characteristics of sulfur aerosols,” Atmos. Environ. 12, 135 (1978).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, “A Survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (National Technical Information Service, Springfield, Va., 1979).

Fymat, A. L.

Hansen, M. Z.

M. Z. Hansen, “Atmosphric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441 (1980).
[CrossRef] [PubMed]

Herman, M.

King, J. I. F.

J. I. F. King, “Differential inversion,” Preprint Abstract Volume, Ninth Conference on Aerospace and Aeronautical Meteorology (American Meteorological Society, Boston, Mass., 1983).

J. I. F. King, “Theory and application of differential inversion to remote sensing,” presented at Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., November 1984.

Mease, K. D.

Miller, J. R.

Nakajima, T.

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the optical properties of aerosols from aureole and extinction data,” Appl. Opt. 22, 1951 (1983).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490 (1980).
[CrossRef]

O’Neill, N. T.

Oppenheim, U.

Quenzel, H.

Santer, R.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (National Technical Information Service, Springfield, Va., 1979).

Tanaka, M.

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the optical properties of aerosols from aureole and extinction data,” Appl. Opt. 22, 1951 (1983).
[CrossRef]

Thomalla, E. T.

Trakhovsky, E.

Twomey, S.

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

van de Hulst, H. C.

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

Whitby, K. T.

K. T. Whitby, “The physical characteristics of sulfur aerosols,” Atmos. Environ. 12, 135 (1978).
[CrossRef]

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490 (1980).
[CrossRef]

Yamauchi, T.

T. Nakajima, M. Tanaka, T. Yamauchi, “Retrieval of the optical properties of aerosols from aureole and extinction data,” Appl. Opt. 22, 1951 (1983).
[CrossRef]

Appl. Opt. (1)

M. Z. Hansen, “Atmosphric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441 (1980).
[CrossRef] [PubMed]

Appl. Opt. (7)

Atmos. Environ. (1)

K. T. Whitby, “The physical characteristics of sulfur aerosols,” Atmos. Environ. 12, 135 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490 (1980).
[CrossRef]

Rev. Geophys. Space Phys. (1)

D. Deirmendjian, “A Survey of light scattering techniques used in the remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341 (1980).
[CrossRef]

Other (5)

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

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” AFGL-TR-79-0214 (National Technical Information Service, Springfield, Va., 1979).

J. I. F. King, “Differential inversion,” Preprint Abstract Volume, Ninth Conference on Aerospace and Aeronautical Meteorology (American Meteorological Society, Boston, Mass., 1983).

J. I. F. King, “Theory and application of differential inversion to remote sensing,” presented at Workshop on Advances in Remote Sensing Retrieval Methods, Williamsburg, Va., November 1984.

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

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Figures (11)

Fig. 1
Fig. 1

A comparison of the diffraction and modified diffraction kernels with the exact Mie kernel for angles of 1.08° and 8.89°. λ = 0.2537 μm.

Fig. 2
Fig. 2

The differential kernels calculated using the Mie, diffraction, and modified diffraction kernels for the pair of 1.08° and 1.29° λ = 0.2537 μm.

Fig. 3
Fig. 3

The differential kernels calculated using the modified diffraction kernels for six pairs of angles. λ = 0.2537 μm.

Fig. 4
Fig. 4

The Mie and modified diffraction kernels integrated over the angular interval between 1.08° and 10°. Also shown is the simplified rectangular approximation to the integrated kernel used to scale the initial size distribution; see Eq. (14) and expression (15). λ = 0.2537 μm.

Fig. 5
Fig. 5

Top, inverted rural aerosol size distributions using angular sets 1, 2, and 3, with the differential kernels. Bottom, averaged and smoothed inverted size distributions compared with the true rural size distributions.

Fig. 6
Fig. 6

Top, inverted fog aerosol size distributions using angular sets 1, 2, and 3, with the differential kernels. Bottom, averaged and smoothed inverted size distributions compared with the true fog size distribution.

Fig. 7
Fig. 7

A comparison of the inversion methods using differential and direct kernels in the case of a rural model size distribution.

Fig. 8
Fig. 8

A comparison of the inversion methods using differential and direct kernels in the case of a radiation fog model size distribution.

Fig. 9
Fig. 9

A comparison of the inversion methods using differential and direct kernels in the case of a truncated power-law size distribution.

Fig. 10
Fig. 10

A comparison of the inversion methods using differential and direct kernels in the case of a maritime model size distribution.

Fig. 11
Fig. 11

A comparison of the inversion methods using aureole phase function starting from 0.65° and aureole phase function starting from 1.08°.

Tables (3)

Tables Icon

Table 1 Angles Used for the Inversion with the Differential Kernels

Tables Icon

Table 2 Rural Model Simulation

Tables Icon

Table 3 Fog Model Simulation

Equations (32)

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β λ ( θ ) = K m , λ ( θ , r ) d V d log r d log r ,
K ( θ , r ) = C i ( θ , r ) k 2 V ,
i d ( θ , r ) = x 2 J 1 2 ( x sin θ ) ( sin 2 θ ) ,
i s ( θ , r ) = i d ( θ , r ) ( Q ext 2 4 ) ,
Q ext 2 + 2 x 2 / 3 ,
i M ( θ , r ) = [ ( 1 + x 2 / 3 ) x J 1 sin J sin θ ] 2 .
Δ ( log x ) = Δ ln x ln 10 = 1 2 Δ r / r ln 10 = 1 2 Δ λ / λ ln 10 = 0.013 ,
K ( θ , r ) = 3 C k 4 π r [ ( 1 + x 2 / 3 ) J 1 ( x sin θ ) sin θ ] 2 .
K ( θ , θ , r ) K ( θ , r ) K ( θ , r ) .
β ( θ ) β ( θ ) δ β ,
υ ( n ) ( r ) = υ ( n 1 ) ( r ) i = 1 N [ 1 + i ( n 1 ) K * ( θ i , r ) ] ,
K * ( θ i , r ) = K ( θ i , r ) K max ( θ i )
i ( n 1 ) = B ( θ i ) β ( n 1 ) ( θ i ) 1 .
υ ( 0 ) ( r ) = C 1 ,
β ( θ ) d ln θ = [ K ( θ , r ) d ln θ ] υ ( r ) log r .
K ( θ , r ) d ln θ { 0.053 , 0.2 < r < 3.5 μ m 0 , otherwise .
C 1 15.2 β ( θ ) d ln θ .
υ ( n ) ( r ) = υ ( n 1 ) ( r ) i = 1 N [ 1 + i ( n 1 ) K * ( θ , θ , r ) ] ,
i ( n 1 ) = B ( θ , θ ) β ( n 1 ) ( θ , θ ) 1 ,
β ( θ , θ ) = β ( θ ) β ( θ ) B ( θ , θ ) = B ( θ ) B ( θ ) K * ( θ , θ , r ) = K ( θ , θ , r ) / K max ( θ , θ ) ,
V new ( r ) = V old ( r ) ( 1 + i = 1 N i ( n 1 ) / N ) .
υ ( n ) ( r ) = υ ( n 1 ) ( r ) ( 1 + i = 1 N i ( n 1 ) / N ) × i = 1 N [ 1 + 1 ( n 1 ) K * ( θ i , θ i , r ) ] .
d N d r = { 100 , 0.1 r 1 μ m 100 r 4.5 1 r 10 μ m ,
υ ¯ ( r j ) = 1 1024 k = 5 + 5 w k υ ( r j + k ) , j = 1 , 2 , , 75 ,
w k = ( 10 k + 5 ) = 10 ! ( k + 5 ) ! ( 5 k ) ! .
( n ) ¯ = [ i = 1 N ( i ( n ) ) 2 / N ] 1 / 2 ,
D ( n ) ¯ = { i = 1 75 [ 1 υ ( n ) ( r j ) υ ( r j ) ] 2 / 75 } 1 / 2 .
( n 1 ) ¯ / ( n ) ¯ < ρ ,
( n ) ¯ ( % )
D ( n ) ¯ ( % ) , 0.2 r 6 μ m
( n ) ¯ ( % )
D ( n ) ¯ ( % ) , 1 μ m < r < 6 μ m

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