Abstract

Air clarity (c) is defined as the distance from an observer to a black ridge, or dark building, or some similar object when it is just barely visible against the horizon sky. The expression is derived for visual range (v) in miles of a small smoke, such as a Forest Service lookout would be expected to detect. Visual range depends upon direction with respect to the sun, clarity, height of smoke layer, binocular power, and Fechner’s constant for the observer. The expression cannot be solved explicitly for v but families of curves are plotted. One interesting result is that under certain conditions, adding smoke to all the air increases the visual range of a small smoke. A visibility meter was constructed to determine clarity (general visibility). The form of meter finally adopted is to be slipped over one objective of a binocular. It consists essentially of a small-angle prism (0.5 diopter) that is moved past a small aperture before the objective. When covering part of the aperture there are two images in the field of view, the added one being due to the prism. As the prism covers more and more of the aperture, the original image grows weaker and the new one grows stronger. When the original image becomes so weak that the boundary line between the object (e.g., ridge or dark building) and the horizon haze just disappears, the distance the observer would have to be from the object for it to be just barely visible may be calculated from the position of the prism and the actual distance of the object. This is c. Curves are drawn to save calculations by the operator. The meter is adapted for use at Forest Service fire lookouts, airports, etc.

© 1940 Optical Society of America

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References

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  1. L. A. Jones, Phil. Mag. 39, 96 (1920).
    [Crossref]
  2. M. G. Bennett, Q. J. R. Met. Soc.56 (1930), No. 233.
  3. G. M. Byram, J. Opt. Soc. Am. 25, 388–392 (1935).
    [Crossref]
  4. Middleton, Visibility in Meteorology (University of Toronto Press, 1935).
  5. Nutting, Bull. Nat. Bur. Stand. 3, 59–64 (1907).
    [Crossref]
  6. Humphreys, Physics of the Air (1929).
  7. Middleton, Trans. Roy. Soc. Can. 3rd series, Sec. 3,  29, 127 (1935).

1935 (2)

G. M. Byram, J. Opt. Soc. Am. 25, 388–392 (1935).
[Crossref]

Middleton, Trans. Roy. Soc. Can. 3rd series, Sec. 3,  29, 127 (1935).

1930 (1)

M. G. Bennett, Q. J. R. Met. Soc.56 (1930), No. 233.

1929 (1)

Humphreys, Physics of the Air (1929).

1920 (1)

L. A. Jones, Phil. Mag. 39, 96 (1920).
[Crossref]

1907 (1)

Nutting, Bull. Nat. Bur. Stand. 3, 59–64 (1907).
[Crossref]

Bennett, M. G.

M. G. Bennett, Q. J. R. Met. Soc.56 (1930), No. 233.

Byram, G. M.

Humphreys,

Humphreys, Physics of the Air (1929).

Jones, L. A.

L. A. Jones, Phil. Mag. 39, 96 (1920).
[Crossref]

Middleton,

Middleton, Trans. Roy. Soc. Can. 3rd series, Sec. 3,  29, 127 (1935).

Middleton, Visibility in Meteorology (University of Toronto Press, 1935).

Nutting,

Nutting, Bull. Nat. Bur. Stand. 3, 59–64 (1907).
[Crossref]

Bull. Nat. Bur. Stand. (1)

Nutting, Bull. Nat. Bur. Stand. 3, 59–64 (1907).
[Crossref]

J. Opt. Soc. Am. (1)

Phil. Mag. (1)

L. A. Jones, Phil. Mag. 39, 96 (1920).
[Crossref]

Physics of the Air (1)

Humphreys, Physics of the Air (1929).

Q. J. R. Met. Soc. (1)

M. G. Bennett, Q. J. R. Met. Soc.56 (1930), No. 233.

Trans. Roy. Soc. Can. 3rd series (1)

Middleton, Trans. Roy. Soc. Can. 3rd series, Sec. 3,  29, 127 (1935).

Other (1)

Middleton, Visibility in Meteorology (University of Toronto Press, 1935).

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Figures (11)

Fig. 1
Fig. 1

Haze brightness relative to that 90° from sun. Full curves, theoretical for large smoke particles; dotted curve, theoretical for air molecules or smoke much smaller than the wave-length of light.

Fig. 2
Fig. 2

A one-foot smoke. Black board just not distinguishable from sky where smoke is one foot in diameter.

Fig. 3
Fig. 3

For a given cross section the number of smoke particles is inversely proportional to diameter of smoke column. Above is a one-foot smoke.

Fig. 4
Fig. 4

One-foot smoke visual range (v) in terms of air clarity (c) for various ratios (r) of shade to sun brightness and various magnifying powers (m) and smoke sizes (w). k=0.032.

Fig. 7
Fig. 7

r in terms of angle to sun and smoke and sun conditions.

Fig. 8
Fig. 8

Binocular-type visibility meter (old form). In diagram at right, 1−f and f are fractional areas.

Fig. 9
Fig. 9

The form of the meter finally adopted by the U. S. Forest Service in 1937.

Fig. 10
Fig. 10

View through the binocular of Figs. 8 and 9.

Fig. 11
Fig. 11

Air clarity (c) vs. fraction (f) of lens uncovered by prism in Fig. 8 when upper ridge line in Fig. 10 of the nearer ridge just disappears. x is distance of nearer ridge and x′ of farther ridge.

Equations (25)

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d i = s d x - a i d x .
i = ( 1 - e - a x ) s / a + i 0 e - a x ,
( i 1 - i 2 ) / i 1 = 0.032 ,
s / a - ( 1 - e - a c ) s / a s / a = k .
c = ( - log c k ) / a .
i = 1 - k x / c + i 0 k x / c .
d = 2 x / m .
1 - k d / d 2 = r .
r = 1 - k m / 2 x = 1 - k m / 2 v ,
r = 1 - k m w / 2 v .
i = 1 - k v / c + i 0 k v / c .
1 - k = 1 - k v / c 1 - k v / c + i 0 k v / c .
c / m w = - ( v / m w ) log k log [ 1 + r ( 1 - k m w / 2 v ) ( 1 - k ) / k ] .
r = brightness of a smoke column brightness of same smoke in sunlight .
e - a x ,
e ( h log k ) / ( c sin α ) .
e ( h log k ) / ( c sin α ) · e b θ + n .
e [ ( h log k ) / ( c sin α ) ] + b θ e [ ( h log k ) / ( c sin α ) ] + b 90 + n .
b = - 0.0291 ;             n = k 0.2 / 17.1.
r = n e [ ( h log k ) / ( c sin α ) ] + b θ + n = 1 1 + 17.1 k [ ( h ) / ( c sin α ) ] - 0.2 · e - 0.0291 θ
r = 1 1 - 34.03 e [ ( - 3.442 h ) / ( c sin α ) ] - 0.0291 θ .
0.968 = 1 - k = ( r f + 1 - f ) / 1 , r = 1 - k / f .
r = 1 - k x / c .
x / c = log k ( k / f ) = 1 + 0.668 log 10 f ,             for             k = 0.032.
1 - k / f = 1 - k x / c 1 - k ( x / x ) ( x / c ) .