Abstract

The issue that we examine was at the center of a recent scientific debate. A new model by Sadot and Kopeika [J. Opt. Soc. Am. A 10, 172 (1993)] suggested a practical approach to the imaging process through the atmosphere. A comment on that model by Bissonnette [J. Opt. Soc. Am. A 11, 1175 (1994)] and the response to that comment by Kopeika and Sadot [J. Opt. Soc. Am. A 12, 1017 (1995)] only underlined the open questions in this field of atmospheric blurring effects. We suggest a physical model that describes the relationship between the optical properties of the atmosphere and the characteristics of an imaging system. The model describes how different components of the light that are reaching the imaging system, after passing through the atmosphere, are detected by it. The model includes the effects of the finite size of the detector elements of the imaging system and the dynamic range and the finite field-of-view limits of the imager. After the presentation of the model, theoretical predictions and their comparisons with the experimental data of Kopeika and of Bissonnette are given.

© 1997 Optical Society of America

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    [Crossref]
  11. Y. J. Kaufman, “Atmospheric effect on spatial resolution of surface imagery: errata,” Appl. Opt. 23, 4164–4172 (1984).
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  12. A. Zardecki, S. A. W. Gerstl, F. Embury, “Multiple scattering effects in spatial frequency filtering,” Appl. Opt. 23, 4124–4130 (1984).
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  16. D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground—I: Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
    [Crossref]
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    [Crossref]
  32. P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
    [Crossref]
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    [Crossref]
  34. Y. Kuga, A. Ishimaru, “Modulation transfer function of layered inhomogeneous random media using the small-angle approximation,” Appl. Opt. 25, 4382–4385 (1986).
    [Crossref] [PubMed]
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  36. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  37. F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).
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1995 (2)

1994 (1)

1993 (5)

1992 (1)

L. R. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[Crossref]

1991 (3)

L. de Luca, G. Cardone, “Modulation transfer function cascade model for a sampled IR imaging system,” Appl. Opt. 30, 1659–1664 (1991).
[Crossref] [PubMed]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

1988 (1)

1986 (2)

1985 (2)

Y. Kuga, A. Ishimaru, “Modulation transfer function and image transmission through randomly distributed spherical particles,” J. Opt. Soc. Am. A 2, 2330–2335 (1985).
[Crossref]

D. J. Diner, J. V. Martonchik, “Influence of aerosol scattering on atmospheric blurring of surface features,” IEEE Trans. Geosci. Remote Sens. GE-23, 618–624 (1985).
[Crossref]

1984 (3)

1982 (1)

Y. J. Kaufman, “Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity,” J. Geophys. Res. 87, 4137–4147 (1982).
[Crossref]

1980 (1)

Y. Mekler, Y. J. Kaufman, “The effect of Earth’s atmosphere on contrast reduction for a nonuniform surface albedo and ‘two-halves’ field,” J. Geophys. Res. 85, 4067–4083 (1980).
[Crossref]

1978 (2)

1975 (1)

R. L. Fante, “Electromagnetic beam propagation in a turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[Crossref]

1966 (1)

1964 (1)

Abreu, L. W.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Anderson, G. P.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Ben Dor, B.

B. Ben Dor, P. Bruscaglioni, A. Devir, P. Donelli, A. Ismaelli, “Cloud, fog and aerosol effect on the MTF of optical systems,” in Optics in Atmospheric Propagation and Adaptive Systems, A. Kohule, A. D. Devir, eds., Proc. SPIE2580, 106–114 (1995).
[Crossref]

Bissonnette, L. R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bruscaglioni, P.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Inhomogeneity of turbid medium and its effect on the MTF of an optical system,” Nuovo Cimento D 15, 775–783 (1993).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium,” Appl. Opt. 32, 2813–2824 (1993).
[Crossref] [PubMed]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

B. Ben Dor, P. Bruscaglioni, A. Devir, P. Donelli, A. Ismaelli, “Cloud, fog and aerosol effect on the MTF of optical systems,” in Optics in Atmospheric Propagation and Adaptive Systems, A. Kohule, A. D. Devir, eds., Proc. SPIE2580, 106–114 (1995).
[Crossref]

Cardone, G.

Chang, H. W.

Charnotskii, M. I.

Chetwynd, J. H.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Clough, S. A.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Darbinyan, R. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

de Luca, L.

Devir, A.

B. Ben Dor, P. Bruscaglioni, A. Devir, P. Donelli, A. Ismaelli, “Cloud, fog and aerosol effect on the MTF of optical systems,” in Optics in Atmospheric Propagation and Adaptive Systems, A. Kohule, A. D. Devir, eds., Proc. SPIE2580, 106–114 (1995).
[Crossref]

Diner, D. J.

D. J. Diner, J. V. Martonchik, “Influence of aerosol scattering on atmospheric blurring of surface features,” IEEE Trans. Geosci. Remote Sens. GE-23, 618–624 (1985).
[Crossref]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground—I: Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[Crossref]

Donelli, P.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Inhomogeneity of turbid medium and its effect on the MTF of an optical system,” Nuovo Cimento D 15, 775–783 (1993).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium,” Appl. Opt. 32, 2813–2824 (1993).
[Crossref] [PubMed]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

B. Ben Dor, P. Bruscaglioni, A. Devir, P. Donelli, A. Ismaelli, “Cloud, fog and aerosol effect on the MTF of optical systems,” in Optics in Atmospheric Propagation and Adaptive Systems, A. Kohule, A. D. Devir, eds., Proc. SPIE2580, 106–114 (1995).
[Crossref]

Dror, I.

Elepov, B. S.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

Embury, F.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in a turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[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,” (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1979).

Fried, D. L.

Gallery, W. O.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Gerstl, S. A. W.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hong, Q. H.

A. H. Lettington, Q. H. Hong, “A discrete modulation transfer function for focal plane arrays,” Infrared Phys. 34, 109–114 (1993).
[Crossref]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hufnagel, R. E.

Ishimaru, A.

Ismaelli, A.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Inhomogeneity of turbid medium and its effect on the MTF of an optical system,” Nuovo Cimento D 15, 775–783 (1993).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium,” Appl. Opt. 32, 2813–2824 (1993).
[Crossref] [PubMed]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

B. Ben Dor, P. Bruscaglioni, A. Devir, P. Donelli, A. Ismaelli, “Cloud, fog and aerosol effect on the MTF of optical systems,” in Optics in Atmospheric Propagation and Adaptive Systems, A. Kohule, A. D. Devir, eds., Proc. SPIE2580, 106–114 (1995).
[Crossref]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Karagin, B. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Kaufman, Y. J.

Y. J. Kaufman, “Atmospheric effect on spatial resolution of surface imagery: errata,” Appl. Opt. 23, 4164–4172 (1984).
[Crossref]

Y. J. Kaufman, “Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity,” J. Geophys. Res. 87, 4137–4147 (1982).
[Crossref]

Y. Mekler, Y. J. Kaufman, “The effect of Earth’s atmosphere on contrast reduction for a nonuniform surface albedo and ‘two-halves’ field,” J. Geophys. Res. 85, 4067–4083 (1980).
[Crossref]

Kneizis, F. X.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Kopeika, N. S.

Kuga, Y.

Lettington, A. H.

A. H. Lettington, Q. H. Hong, “A discrete modulation transfer function for focal plane arrays,” Infrared Phys. 34, 109–114 (1993).
[Crossref]

Lutomirski, R. F.

Marchuk, G. I.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

Martonchik, J. V.

D. J. Diner, J. V. Martonchik, “Influence of aerosol scattering on atmospheric blurring of surface features,” IEEE Trans. Geosci. Remote Sens. GE-23, 618–624 (1985).
[Crossref]

D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground—I: Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97–125 (1984).
[Crossref]

Mekler, Y.

Y. Mekler, Y. J. Kaufman, “The effect of Earth’s atmosphere on contrast reduction for a nonuniform surface albedo and ‘two-halves’ field,” J. Geophys. Res. 85, 4067–4083 (1980).
[Crossref]

Mikhailov, G. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

Nazaraliev, M. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Karagin, B. S. Elepov, Monte-Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980).

Pearce, W. A.

W. A. Pearce, “A study of the effects of the atmosphere on Thematic Mapper observations,” (EG&G/Washington Analytical Service Center, Riverdale, Md., 1977).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.

Sadot, D.

Selby, J. E. A.

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

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,” (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1979).

F. X. Kneizis, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, “Atmospheric transmittance/radiance: computer code lowtran 7”, (Air Force Geophysics Laboratory, U.S. Air Force Systems Command, Hanscom Air Force Base, Bedford, Mass., 1988).

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Tsang, L.

Valley, M. T.

M. T. Valley, “Evaluation of equivalent spheres for use in modeling nonspherical aerosol modulation transfer function,” in Atmospheric Propagation and Remote Sensing II, A. Kohnle, W. B. Miller, Proc. SPIE1968, 130–141 (1993).
[Crossref]

Zaccanti, G.

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Inhomogeneity of turbid medium and its effect on the MTF of an optical system,” Nuovo Cimento D 15, 775–783 (1993).
[Crossref]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium,” Appl. Opt. 32, 2813–2824 (1993).
[Crossref] [PubMed]

P. Bruscaglioni, P. Donelli, A. Ismaelli, G. Zaccanti, “A numerical procedure for calculating the effect of a turbid medium on the MTF of an optical system,” J. Mod. Opt. 38, 129–142 (1991).
[Crossref]

P. Donelli, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Experimental validation of a Monte Carlo procedure for the evaluation of the effect of a turbid medium on the point spread function of an optical system,” J. Mod. Opt. 38, 2189–2201 (1991).
[Crossref]

Zardecki, A.

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991).

Appl. Opt. (9)

R. F. Lutomirski, “Atmospheric degradation of electro-optical system performance,” Appl. Opt. 17, 3915–3921 (1978).
[Crossref] [PubMed]

Y. J. Kaufman, “Atmospheric effect on spatial resolution of surface imagery: errata,” Appl. Opt. 23, 4164–4172 (1984).
[Crossref]

A. Zardecki, S. A. W. Gerstl, F. Embury, “Multiple scattering effects in spatial frequency filtering,” Appl. Opt. 23, 4124–4130 (1984).
[Crossref] [PubMed]

Y. Kuga, A. Ishimaru, H. W. Chang, L. Tsang, “Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory,” Appl. Opt. 25, 3803–3805 (1986).
[Crossref] [PubMed]

L. R. Bissonnette, “ Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

IEEE Trans. Geosci. Remote Sens. (1)

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Geometry of an imaging system looking at a point source through the intervening atmosphere.

Fig. 2
Fig. 2

Simulated normalized PSF of an imaging system (the size of the detector element is 15 μm, the number of detector elements is 512, and the optics diameter is 4 cm) operating in the visible wavelength (0.55 μm) for three focal lengths: 1, 10, and 100 cm. The aerosol models used are (a) tropospheric aerosol (visibility of 50 km): τ=0.078 and ω0t=0.97; (b) rural aerosol (visibility of 5 km); τ=0.78 and ω0t=0.96; and (c) radiation fog (visibility of 0.5 km); τ=7.8 and ω0t=1.0. Dotted curve, direct component; dashed curves, atmospheric scattered components; solid curves, sum of them.

Fig. 3
Fig. 3

Simulated normalized PSF of an imaging system (the size of the detector element is 38 μm, the number of detector elements is 256, and the optics diameter is 4 cm) operating in the mid-IR wavelength (4.6 μm) for three focal lengths: 1, 10, and 100 cm. The aerosol models used are (a) tropospheric aerosol (visibility of 50 km): τ=0.17 and ω0t=0.0024; (b) rural aerosol (visibility of 5 km): τ=0.31 and ω0t=0.21, and (c) radiation fog (visibility of 0.5 km): τ=9.3 and ω0t=0.89. Dotted curve, direct component; dashed curves, atmospheric scattered components; solid curves, sum of them.

Fig. 4
Fig. 4

Visualization of the differences between two atmospheric effects: (a) contrast reduction and (b) spatial blurring. In each plot the thin curve represents the original contrast (of a bright object on a dark background) in the object plane, and the thick curve is the scan over the object in the optics plane, which includes the atmospheric effects.

Fig. 5
Fig. 5

Comparison of the PSF measured by Bissonnette with the predicted PSF, according to the presented model. The comparisons are for (a) haze, (b) light radiation fog, and (c) dense advection fog at a range of 531 m (Figs. 6, 7, and 9 of Ref. 38). The measured data (solid squares) appear together with three additional curves: expected direct and system scattered radiation (dashed curves), expected direct and atmospheric scattered radiation (thin solid curves), and the expected signal, which is composed of the direct, system, and atmosphere scattered radiation (thick solid curves).

Fig. 6
Fig. 6

Expected PSF in the experiment of Dror and Kopeika,39 according to the model given in this paper. The simulation presented is for the day of the heaviest aerosol loading in the Dror–Kopeika experiment (July 16, 1991).

Equations (13)

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MTF(νx, νy)=--PSF(x, y)×exp[-2πi(xνx+yνy)]dxdy, 
Iopt(r)=D24r2 J12πDrλf0,
Idir(r)=12πσ2 exp-r22σ2,
IFOV=arctan(σ/f0)σ/f0.
T=exp(-τ)
PSFdir(r)=Idir(r)[exp(-τ)]σ2=exp(-τ)2π exp-r22σ2.
tan θ=r/f0.
PSFsca(r)=Isca(r/f0)(σ/f0)2.
PSFsca(r)=Isca(r/f0) σ2 cos θ(f0/cos θ)2=Isca(r/f0) σ2f02 f0r2+f023.
PSFdet(r)=Isca(r/f0) σ2f02 f0r2+f023+exp(-τ)2π exp-r22σ2.
PSFdet(θ)=Isca(θ)IFOV2 cos3 θ+exp(-τ)2π exp-θ22IFOV2.
ω0t=kst/ket,ket=kea+kem,kst=ksa+ksm,
τ=ket(z)dz.

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