Abstract

The Twomey-Chahine inversion algorithm is applied to experimental atmospheric transmittance data in the 0.4–2.4-μm wavelength range and atmospheric aerosol size distributions deduced. The conditions for successful inversion of transmittance data are investigated in numerical experiments, and it is shown that too small a wavelength range results in a Junge-type distribution in all cases and that noise in the measurements in excess of 4–5% results in inversion artifacts.

© 1982 Optical Society of America

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  6. A. L. Fymat, Appl. Opt. 17, 1675 (1978).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1980 (3)

C. Frohlich, Proceedings, IRS Radiation Commission of IAMAP, Fort Collins, Colo. (1980), p. 201.

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

1979 (2)

N. Wolfson, J. H. Joseph, Y. Mekler, J. Appl. Meteorol 18, 543 (1979).
[CrossRef]

Z. Levin, J. D. Lindberg, J. Geophys. Res. 64, 6941 (1979).
[CrossRef]

1978 (4)

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

M. A. Box, B. K. J. McKellar, Opt. Lett. 3, 91 (1978).
[CrossRef] [PubMed]

A. L. Fymat, Appl. Opt. 17, 1675 (1978).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

1969 (1)

1968 (1)

1963 (1)

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. USSR 15, 362 (1963).

1961 (1)

Abreu, L. W.Abreu

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Ben-Shalom, A.

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

Box, M. A.

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Cabib, D.

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

Callery, W. O.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Chahine, M. T.

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Curcio, J. A.

Devir, A. D.

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

Fenn, R. W.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

E. P. Shettle, R. W. Fenn, “Models of the Aerosols for the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” AFGL-TR-79-0214 (Sept.1979).

Fernandez, G.

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Frohlich, C.

C. Frohlich, Proceedings, IRS Radiation Commission of IAMAP, Fort Collins, Colo. (1980), p. 201.

Fymat, A. L.

Goldschmidt, D.

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

Goody, R. M.

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), p. 191.

Herman, B. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Hoidale, G. B.

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Hoihjelle, D. L.

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Joseph, J. H.

N. Wolfson, J. H. Joseph, Y. Mekler, J. Appl. Meteorol 18, 543 (1979).
[CrossRef]

King, M. D.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Levin, Z.

Z. Levin, J. D. Lindberg, J. Geophys. Res. 64, 6941 (1979).
[CrossRef]

Lindberg, J. D.

Z. Levin, J. D. Lindberg, J. Geophys. Res. 64, 6941 (1979).
[CrossRef]

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Lipson, S. G.

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

McClatchey, R. A.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

McKellar, B. K. J.

Mekler, Y.

N. Wolfson, J. H. Joseph, Y. Mekler, J. Appl. Meteorol 18, 543 (1979).
[CrossRef]

Middleton, W. E. K.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952).

Oppenheim, U. P.

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

Perelman, A. Ya.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. USSR 15, 362 (1963).

Pinnick, G.

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

E. P. Shettle, R. W. Fenn, “Models of the Aerosols for the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” AFGL-TR-79-0214 (Sept.1979).

Shifrin, K. S.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. USSR 15, 362 (1963).

Stenmark, E. B.

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

Tanaka, M.

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).

Wolfson, N.

N. Wolfson, J. H. Joseph, Y. Mekler, J. Appl. Meteorol 18, 543 (1979).
[CrossRef]

Yamamoto, G.

Appl. Opt. (2)

Infrared Phys. (1)

A. Ben-Shalom, D. Cabib, A. D. Devir, D. Goldschmidt, S. G. Lipson, U. P. Oppenheim, Infrared Phys. 20, 165 (1980).
[CrossRef]

J. Appl. Meteorol (1)

N. Wolfson, J. H. Joseph, Y. Mekler, J. Appl. Meteorol 18, 543 (1979).
[CrossRef]

J. Atmos. Sci. (2)

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

G. Pinnick, D. L. Hoihjelle, G. Fernandez, E. B. Stenmark, J. D. Lindberg, G. B. Hoidale, J. Atmos. Sci. 35, 2020 (1978).
[CrossRef]

J. Geophys. Res. (1)

Z. Levin, J. D. Lindberg, J. Geophys. Res. 64, 6941 (1979).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Opt. Spectrosc. USSR (1)

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. USSR 15, 362 (1963).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

A. Ben-Shalom, A. D. Devir, S. G. Lipson, U. P. Oppenheim, Proc. Soc. Photo-Opt. Instrum. Eng. 253, 261 (1980).

Proceedings, IRS Radiation Commission of IAMAP, Fort Collins, Colo (1)

C. Frohlich, Proceedings, IRS Radiation Commission of IAMAP, Fort Collins, Colo. (1980), p. 201.

Other (6)

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W.Abreu Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

E. P. Shettle, R. W. Fenn, “Models of the Aerosols for the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” AFGL-TR-79-0214 (Sept.1979).

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952).

R. M. Goody, Atmospheric Radiation (Clarendon, Oxford, 1964), p. 191.

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

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

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Figures (8)

Fig. 1
Fig. 1

Kernel of the integral (6) for a, m = 1.5 and b, m = 1.5–0.1i.

Fig. 2
Fig. 2

Inversion results as a function of the first-guess distribution shape (parameter R2): A, exact size distribution (R1 = 0.01 μm, R2 = 0.12 μm, R3 = 5 μm); B, inverted size distribution with R2 = 0.52 μm; C, inverted size distribution with R2 = 0.02 μm; D, inverted size distribution with R2 = 0.52 μm from only three measured wavelengths: 0.55, 0.63, 1.06 μm.

Fig. 3
Fig. 3

Inversion results as a function of refractive indices: a, exact size distribution with R1 = 0.03 μm, R2 = 0.1 μm, R3 = 5 μm; b, initial guess (β = 3.5); c, inverted size distribution m = 1.5–0.01i (rural type, RH = 70%); d, inverted size distribution m = 1.45–0.001i (rural type, RH = 80%); e, inverted size distribution m = 1.55–0.1i (urban type, RH = 80%).

Fig. 4
Fig. 4

Inversion results as a function of added noise: a, exact size distribution with R1 = 0.03 μm, R2 = 0.1 μm, R3 = 5 μm; b, 1% noise; c, 3% noise; d, 11% noise.

Fig. 5
Fig. 5

Convergence of the average residuals in the presence of noise: a, 1% noise; b, 3% noise (better first guess); c,11% noise. Comparison between b and c also shows independence of the rms residual on the first guess.

Fig. 6
Fig. 6

Estimated visibility compared with the measured value.

Fig. 7
Fig. 7

Inverted Junge size distribution a compared with the distribution measured by the Knollenberg instrument b.

Fig. 8
Fig. 8

Inverted size distributions for desert aerosols: A, along 11-km path with R2 = 0.12 μm; B, along 11-km path with R2 = 0.52 μm; C, along 4.5-km path with R2 = 0.52 μm; D, along 4.5-km path with R2 = 0.12 μm.

Tables (1)

Tables Icon

Table I Experimentally Measured Extinction Coefficients Used for the Inversion

Equations (16)

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T ( λ ) = exp - { [ γ A ( λ ) + γ M ( λ ) ] · R } ,
γ M ( λ ) = ν 4 / ( 9.27 × 10 18 - 1.07 × 10 9 ν 2 ) ,
γ A ( λ ) = π r 2 Q ( α , m ) d N d r d r ,
Q = 2 - 4 exp ( - ρ · tan β ) · cos β / ρ - 4 exp ( - ρ tan β ) ( cos β / ρ ) 2 · cos ( ρ - 2 β ) + 4 · ( cos β / ρ ) 2 · cos 2 β ,
d N / d r = f ( r ) · r - β
γ A ( λ ) = π ( λ 2 π ) 3 - β α 1 α 2 Q ( α , m ) α · [ f ( α ) · α 3 - β ] d α .
α = 2 π r λ max to α = 2 π r λ min .
γ A = K λ - q ,
d N ( n ) d r ( r ) = d N ( n - 1 ) d r ( r ) · i = 1 N [ 1 - i ( n - 1 ) · K i ( λ i , r , m ) ] ,
d N d r = { C · ( R 2 ) - 4 for R 1 r R 2 , C · r - 4 for R 2 < r R 3 ,
V = 3.91 γ ( 0.55 ) .
V = V obs · ( 1.3 ± 0.3 ) .
( n ) ( λ ) = 1 - Γ ( λ ) ± Δ ( λ ) γ A ( n ) ( λ ) ,
D = lim n 1 N - 1 i = 1 N ( n ) 2 ( λ i ) = lim n 1 N - 1 i = 1 N [ 1 - Γ ( λ i ) ± Δ ( λ i ) γ A ( n ) ( λ i ) ] 2 .
lim n γ A ( n ) ( λ i ) = Γ ( λ i ) Γ ( λ i ) .
D = 1 N - 1 i = 1 N [ 2 Δ ( λ i ) Γ ( λ i ) ] 2 [ 1 1 Δ ( λ i ) Γ ( λ i ) ] 2 1 N - 1 i = 1 N { 2 Δ ( λ i ) Γ ( λ i ) [ 1 ± Δ ( λ i ) Γ ( λ i ) ] } 2 1 N - 1 i = 1 N [ 2 Δ ( λ i ) Γ ( λ i ) ] 2 = rms relative measurement error .

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