Abstract

The effect of the earth’s atmosphere on the visibility in the nadir direction, as determined by the human eye, is presented. The visibility characteristics can be obtained from the combined eye–atmosphere modulation transfer function. This function is applied to the calculation of visibility thresholds of an area of an albedo different from its surroundings. It is shown that the atmospheric effect of diffusing sharp boundaries between adjacent areas has a significant influence on the visibility threshold. A sensitivity study is carried out to investigate the effects of different atmospheric optical conditions on the visibility thresholds. This model emphasizes the use of a combined eye–atmosphere theory to predict visibility conditions from airplanes.

© 1981 Optical Society of America

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References

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  1. Y. J. Kaufman, J. Geophys. Res. 84, 3165 (1979).
    [CrossRef]
  2. Yu. Mekler, Y. J. Kaufman, J. Geophys. Res. 85, 4067 (1980).
    [CrossRef]
  3. S. Q. Duntley et al., Appl. Opt. 3, 549 (1964).
    [CrossRef]
  4. I. Overington, E. P. Levin, Opt. Acta 18, 341 (1971).
    [CrossRef]
  5. I. Overington, J. Opt. Soc. Am. 63, 1043 (1973).
    [CrossRef] [PubMed]
  6. I. Overington, Vision and Acquisition (Penetch, London, 1976).
  7. W. E. K. Middleton, J. Opt. Soc. Am. 27, 112 (1937).
    [CrossRef]
  8. G. A. Fry, “The Eye and Vision,” in Applied Optics and Optical Engineering, Vol. 2 (Academic, New York, 1965), p. 46.
  9. F. W. Cambell, D. G. Green, J. Physiol. 181, 576 (1965).
  10. F. W. Cambell, J. G. Robson, J. Physiol. 197, 551 (1968).
  11. L. Elterman, “Vertical Attenuation Model with 8 Surface Meteorological Ranges 2 to 13 km,” AFCRL-70-0200 Report, U.S. Air Force.
  12. Y. J. Kaufman, J. H. Joseph, “Evaluation of Surface Albedo and Extinction Characteristics of the Atmosphere from Satellite Images,” submitted to J. Geophys. Res.
  13. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

1980 (1)

Yu. Mekler, Y. J. Kaufman, J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

1979 (1)

Y. J. Kaufman, J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

1973 (1)

1971 (1)

I. Overington, E. P. Levin, Opt. Acta 18, 341 (1971).
[CrossRef]

1968 (1)

F. W. Cambell, J. G. Robson, J. Physiol. 197, 551 (1968).

1965 (1)

F. W. Cambell, D. G. Green, J. Physiol. 181, 576 (1965).

1964 (1)

1937 (1)

Cambell, F. W.

F. W. Cambell, J. G. Robson, J. Physiol. 197, 551 (1968).

F. W. Cambell, D. G. Green, J. Physiol. 181, 576 (1965).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Duntley, S. Q.

Elterman, L.

L. Elterman, “Vertical Attenuation Model with 8 Surface Meteorological Ranges 2 to 13 km,” AFCRL-70-0200 Report, U.S. Air Force.

Fry, G. A.

G. A. Fry, “The Eye and Vision,” in Applied Optics and Optical Engineering, Vol. 2 (Academic, New York, 1965), p. 46.

Green, D. G.

F. W. Cambell, D. G. Green, J. Physiol. 181, 576 (1965).

Joseph, J. H.

Y. J. Kaufman, J. H. Joseph, “Evaluation of Surface Albedo and Extinction Characteristics of the Atmosphere from Satellite Images,” submitted to J. Geophys. Res.

Kaufman, Y. J.

Yu. Mekler, Y. J. Kaufman, J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

Y. J. Kaufman, J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

Y. J. Kaufman, J. H. Joseph, “Evaluation of Surface Albedo and Extinction Characteristics of the Atmosphere from Satellite Images,” submitted to J. Geophys. Res.

Levin, E. P.

I. Overington, E. P. Levin, Opt. Acta 18, 341 (1971).
[CrossRef]

Mekler, Yu.

Yu. Mekler, Y. J. Kaufman, J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

Middleton, W. E. K.

Overington, I.

I. Overington, J. Opt. Soc. Am. 63, 1043 (1973).
[CrossRef] [PubMed]

I. Overington, E. P. Levin, Opt. Acta 18, 341 (1971).
[CrossRef]

I. Overington, Vision and Acquisition (Penetch, London, 1976).

Robson, J. G.

F. W. Cambell, J. G. Robson, J. Physiol. 197, 551 (1968).

Appl. Opt. (1)

J. Geophys. Res. (2)

Y. J. Kaufman, J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

Yu. Mekler, Y. J. Kaufman, J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Physiol. (2)

F. W. Cambell, D. G. Green, J. Physiol. 181, 576 (1965).

F. W. Cambell, J. G. Robson, J. Physiol. 197, 551 (1968).

Opt. Acta (1)

I. Overington, E. P. Levin, Opt. Acta 18, 341 (1971).
[CrossRef]

Other (5)

G. A. Fry, “The Eye and Vision,” in Applied Optics and Optical Engineering, Vol. 2 (Academic, New York, 1965), p. 46.

L. Elterman, “Vertical Attenuation Model with 8 Surface Meteorological Ranges 2 to 13 km,” AFCRL-70-0200 Report, U.S. Air Force.

Y. J. Kaufman, J. H. Joseph, “Evaluation of Surface Albedo and Extinction Characteristics of the Atmosphere from Satellite Images,” submitted to J. Geophys. Res.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

I. Overington, Vision and Acquisition (Penetch, London, 1976).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the relative intensity of two-halves field (dotted line), the degraded intensity by atmospheric scattering (dashed line), and the resultant illumination of the retina (solid line).

Fig. 2
Fig. 2

Contrast sensitivity for sine wave and square wave gratings normalized for normalized maximum.10

Fig. 3
Fig. 3

Atmospheric MTF calculated from the two-halves field case for several optical thicknesses τ0 and two average surface reflectivities (ÃL). Index of refraction is n = 1.5–i0.0. Junge size distribution was taken with β = 3.3 [see Eq. (30)].

Fig. 4
Fig. 4

Graph of the intensity as a function of the distance from the two-halves field border for an Elterman atmosphere for several optical thicknesses of the atmosphere. (πF0 is the incident flux per unit area perpendicular to the incident beam.)

Fig. 5
Fig. 5

Eye–atmosphere MTF for an observer at the 10-km height for 0.55-μm wavelength, sun zenith angle 40°, refractive index = (1.5, 0.0), and β = 3.3.

Fig. 6
Fig. 6

Same as in Fig. 4 but for the close field, emphasizing the linearity of the intensity in small distances from the border.

Fig. 7
Fig. 7

Example of the dependence of δ, η, and η* on the optical thickness for ÃL = 0.4, ΔAL = 0.03 [wavelength = 0.55 μm, sun zenith angle = 20°, refractive index = (1.5, 0.0), β = 3.3)].

Fig. 8
Fig. 8

Dependence of η and η* on the optical thickness for five values of ÃL (ÃL = 0.05, 0.10, 0.20, 0.30, 0.40). ÃL = 0.05 corresponds to the rightmost graphs of η and η* (same parameters as in Fig. 7).

Fig. 9
Fig. 9

Dependence of η* on the optical thickness for four values of ΔALAL = 0.03, 0.10, 0.20, 0.27) corresponding to the four graphs from left to right [wavelength = 0.75 μm, sun zenith angle = 60°, refractive index = (1.5, 0.0) ÃL = 0.3)].

Fig. 10
Fig. 10

Dependence of the threshold visible ground albedo difference on optical thickness for several size distributions. Exponential approximation shows the value of ΔAth for an absorbing atmosphere (no scattering).

Equations (31)

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ln | ( k 2 + k 3 ) · + 1 k 3 · + 1 | = F 1 ( n ) + δ ,
k 2 = I * 2 I * 1 I 2 I 1 .
k 3 = I * 2 I 2 I 1 I 2 .
F 1 ( n ) = k 1 n ( n 1 ) .
= 1 k 2 [ F 1 ( n ) + δ ] .
M E N ( ν ) = exp [ ( 2 π ν ) 2 σ 2 2 ] ,
I ( x ) = F 1 [ M ( ω ) a L ( ω ) ] + I 0 ,
I * ( x ) = F 1 { M E ( ω ) · F [ I ( x ) ] } .
M E ( ω ) = M E 0 M E N ( ω ) .
I * ( x ) = π / 2 π / 2 I ( x ) f ( x x ) d x for x 0 x x 0 ,
f ( x ) = 1 σ 2 π exp ( x 2 / 2 σ 2 ) ,
n θ = n F ( θ + 1 ) 1 / 2 ,
I ( x ) = I 0 + I 0 s S t ( x ) + [ 1 1 + I 1 s S t ( x ) ] x ,
I * ( x ) = I 0 + I 1 x + ( I 0 s + I 1 s x ) erf ( x σ 2 ) 2 π · I 1 s σ exp ( x 2 2 σ 2 ) ·
I * sharp ( x ) = I 0 + I 0 s erf ( x σ 2 ) ·
k 2 deg = D e Δ x + k 2 sharp 2 D e x 0 + 1 ,
deg = 2 D e x 0 + 1 D e Δ x + k 2 sharp [ F 1 ( n ) + δ ] .
η = deg sharp = 2 D e x 0 + 1 D e Δ x k 2 sharp + 1 .
I 1 = I ( a ) I ( + 0 ) I ( a ) + I ( 0 ) 2 a ; I 0 s = I ( + 0 ) I ( 0 ) 2 .
D e = 1 2 a [ I ( a ) I ( a ) I ( + 0 ) I ( 0 ) 1 ] .
η = ρ x 0 · [ 1 + ( ρ x 0 1 ) Δ x / x 0 2 k 2 sharp ] 1 ,
ρ x = 2 × D e + 1 = I ( x ) I ( x ) I ( + 0 ) I ( 0 ) .
deg ( ) = η deg ( x 0 ) = η * sharp .
η = [ I ( + ) I ( ) I ( x 0 ) I ( x 0 ) ] · [ I ( x 0 ) I ( ) ] = ρ ρ x 0 · [ I ( x 0 ) I ( ) ] .
η * = η η = ρ [ I ( x 0 ) I ( ) ] · [ 1 + ( ρ x 0 1 ) Δ x / x 0 2 k 2 sharp ] 1
[ I ( x 0 ) I ( 0 ) ] [ 1 ( ) I ( 0 ) ] .
I A L = I 0 + G A L 1 r A L ,
deg ( ) = I A 1 I A 2 I A 2 = G I A 2 [ Δ A L ( 1 r A ˜ L ) 2 r 2 4 ( Δ A L ) 2 ] ,
δ I ( ) = G [ ( 1 r A ˜ L ) 2 r 2 ( Δ A L ) 2 / 4 ] 1 .
Δ A th = I ( x ) δ I ( 0 ) sharp [ 1 + ( ρ x 0 1 ) Δ x / x 0 2 k 2 sharp ] 1 .
d N ( r ) d log r = Cr β for 0.04 μ m r 10 μ m .

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