Abstract

A veil of atmospheric haze reduces the visibility of all distant objects by decreasing their apparent contrast. In this paper equations are derived which describe the manner in which the apparent contrast of any object depends upon the distance of the observer. The treatment is not limited to horizontal paths of sight, but applies also to the apparent contrast of objects on the ground as seen from the air, and to the apparent contrast of objects aloft as viewed from the ground. The equations are not limited to the case of a homogeneous standard atmosphere; they may be applied to many kinds of non-standard atmospheric conditions. For every path of sight there exists a luminance level which will be transmitted unchanged. The apparent luminance of any receding object approaches this equilibrium level. For many paths of sight the equilibrium luminance is matched by the luminance of some portion of the horizon sky.

© 1948 Optical Society of America

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References

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  1. Summary Technical Report of N.D.R.C. Division 16, Vol. 2.
  2. S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).
  3. H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
    [Crossref] [PubMed]
  4. S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).
  5. S. Q. Duntley, J. Opt. Soc. Am. 36, 713A (1946).
  6. A. Schuster, Phil. Mag. 5, 243 (1903).
    [Crossref]
  7. S. Q. Duntley, J. Opt. Soc. Am. 33, 252 (1943).
    [Crossref]
  8. S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
    [Crossref]
  9. International Commission on Illumination, Neuvième Session, p. 3 (1937).
  10. “Aerial haze and its effect on photography from the air,” Research Laboratory, Eastman Kodak Company, D. Van Nostrand Company, Inc. (1923).
  11. R. Tousey and E. O. Hulburt, J. Opt. Soc. Am. 37, 78 (1947).
    [Crossref]
  12. H. Koschmieder, Beitr. z. Phys. d. freien Atm. 12, 33 (1924), and Beitr. z. Phys. d. freien Atm. 12, 171 (1924).
  13. W. E. K. Middleton, Visibility in Meteorology (University of Toronto Press, Toronto, 1941).
  14. F. Benford, J. Opt. Soc. Am. 36, 524 (1946).
    [Crossref] [PubMed]
  15. C. A. Douglas and L. L. Young, , Civil Aeronautics Administration.
  16. Organization Météorologique Internationale, “Conférence des directeurs à Varsovie 1935,” Vol. 1, No. 29Leyden1936.
  17. W. E. K. Middleton, J. Opt. Soc. Am. 32, 139 (1942).
    [Crossref]
  18. Encyclopaedia Britannica,  3, 129 (1945).

1947 (1)

1946 (5)

H. R. Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
[Crossref] [PubMed]

F. Benford, J. Opt. Soc. Am. 36, 524 (1946).
[Crossref] [PubMed]

S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).

S. Q. Duntley, J. Opt. Soc. Am. 36, 713A (1946).

S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).

1945 (1)

Encyclopaedia Britannica,  3, 129 (1945).

1943 (1)

1942 (2)

1937 (1)

International Commission on Illumination, Neuvième Session, p. 3 (1937).

1924 (1)

H. Koschmieder, Beitr. z. Phys. d. freien Atm. 12, 33 (1924), and Beitr. z. Phys. d. freien Atm. 12, 171 (1924).

1903 (1)

A. Schuster, Phil. Mag. 5, 243 (1903).
[Crossref]

Benford, F.

Blackwell, H. R.

Douglas, C. A.

C. A. Douglas and L. L. Young, , Civil Aeronautics Administration.

Duntley, S. Q.

S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).

S. Q. Duntley, J. Opt. Soc. Am. 36, 713A (1946).

S. Q. Duntley, J. Opt. Soc. Am. 36, 359A (1946).

S. Q. Duntley, J. Opt. Soc. Am. 33, 252 (1943).
[Crossref]

S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
[Crossref]

Hulburt, E. O.

Koschmieder, H.

H. Koschmieder, Beitr. z. Phys. d. freien Atm. 12, 33 (1924), and Beitr. z. Phys. d. freien Atm. 12, 171 (1924).

Middleton, W. E. K.

W. E. K. Middleton, J. Opt. Soc. Am. 32, 139 (1942).
[Crossref]

W. E. K. Middleton, Visibility in Meteorology (University of Toronto Press, Toronto, 1941).

Schuster, A.

A. Schuster, Phil. Mag. 5, 243 (1903).
[Crossref]

Tousey, R.

Young, L. L.

C. A. Douglas and L. L. Young, , Civil Aeronautics Administration.

Beitr. z. Phys. d. freien Atm. (1)

H. Koschmieder, Beitr. z. Phys. d. freien Atm. 12, 33 (1924), and Beitr. z. Phys. d. freien Atm. 12, 171 (1924).

Encyclopaedia Britannica (1)

Encyclopaedia Britannica,  3, 129 (1945).

J. Opt. Soc. Am. (9)

Neuvième Session (1)

International Commission on Illumination, Neuvième Session, p. 3 (1937).

Phil. Mag. (1)

A. Schuster, Phil. Mag. 5, 243 (1903).
[Crossref]

Other (5)

“Aerial haze and its effect on photography from the air,” Research Laboratory, Eastman Kodak Company, D. Van Nostrand Company, Inc. (1923).

Summary Technical Report of N.D.R.C. Division 16, Vol. 2.

W. E. K. Middleton, Visibility in Meteorology (University of Toronto Press, Toronto, 1941).

C. A. Douglas and L. L. Young, , Civil Aeronautics Administration.

Organization Météorologique Internationale, “Conférence des directeurs à Varsovie 1935,” Vol. 1, No. 29Leyden1936.

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Figures (8)

Fig. 2
Fig. 2

Arrows n and n′ indicate the directions in which the luminance of the horizon sky is determined by light scattered from the rays of the sun at the same angle as light scattered downward along the path of sight. Arrows m and m′ indicate the directions in which the luminance of the horizon sky is determined by light scattered from the rays of the sun at the same angle as light scattered upward along the path of sight.

Fig. 3
Fig. 3

Apparent contrast of distant black panels as measured with a photographic telephotometer. From the slope of the line: βo=0.209 per thousand yards; v=18,700 yards

Fig. 4
Fig. 4

Apparent contrast of distant black panels as measured with a photographic telephotometer. From the slope of the line: βo=0.047 per thousand yards: v=83,200 yards

Fig. 5
Fig. 5

The points represent the relative number of scattering particles per unit of volume in an optical standard atmosphere.

Fig. 6
Fig. 6

Optical slant-range diagram for the optical standard atmosphere. Solid curves represent the relation between R ¯ and R for various sight-path elevation angles θ. Broken lines represent loci of equal altitude expressed in feet.

Fig. 7
Fig. 7

Optical slant-range diagram similar to Fig. 4, but adapted to the solution to problems involving short slant ranges.

Fig. 8
Fig. 8

Optical slant-range diagram for θ=25 degrees. Accentuated curve shows relation between R ¯ and R when the ground haze has a sharp upper boundary at 5000 feet, above which the meteorological range is five times greater than it is below the boundary.

Equations (32)

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d t / d r = - μ r r - B r t + l t + B r s + F r I r .
- d s / d r = - μ r s - B r s + l s + B r t + B r I r .
d t / d r = - β r t + τ r q ,
- d s / d r = - β r s + σ r q .
W o W R d t - β o t + τ o q = o R f ( r ) d r ,
W o W R d s - β o s + σ o q = - R o f ( r ) d r .
R ¯ = o R f ( r ) d r .
W R = τ o q β o ( 1 - e - β o R ¯ ) + W o e - β o R ¯ ,
W R = σ o q β o ( 1 - e - β o R ¯ ) + W o e - β o R ¯ .
B R = τ o q β o ( 1 - e - β o R ¯ ) + B o e - β o R ¯ ,
B R = σ o q β o ( 1 - e - β o R ¯ ) + B o e - β o R ¯ ,
( B R - B R ) = ( B o - B o ) e - β o R ¯ ,
C o = B o - B o B o ,
C R = B R - B R B R .
C R = B o B R C o e - β o R ¯ .
B R = σ o q β o ( 1 - e - β o R ¯ o , ) ,
B o = σ o q β o ( 1 - e - β o R ¯ R , ) .
C R = C o e - β o R ¯ [ 1 - e - β o R ¯ R , 1 - e - β o R ¯ o , ] .
C R = C 0 e - β o R .
C R = C o [ 1 - τ o q β o B o ( 1 - e β o R ¯ ) ] - 1 .
C R = C o [ 1 - B m B o ( 1 - e β o R ¯ ) ] - 1 .
C R = C 0 [ 1 - B H B 0 ( 1 - e β 0 R ) ] - 1 .
T R = C R / C o = e - β o R .
T = e - β o .
B R = B H ( 1 - T R ¯ ) - B o T R ¯ ,
C R = C o T R .
d I r / d r = - μ r I r - B r I r - F r I r = - β o f ( r ) I r ,
T R = e - β 0 R ¯ .
v = 3.912 β o = 1.699 log 10 1 T .
N / N o = e - y / 21 , 700 ,
R ¯ = 21 , 700 csc θ [ e - R 1 sin θ / 21 , 700 - e - R 2 sin θ / 21 , 700 ] .
R ¯ = 21 , 700 csc θ [ 1 - e - R sin θ / 21 , 700 ] .