454 Table 160 (continued) 



HORIZONTAL VISIBILITY 



Tables 160-B1 and 160-B2. — Extinction coefficient as a function of visual range of black 

 objects of standard angular dimensions viewed against the horizon sky in daytime 



Assumptions and Computations: 



According to Koschmieder, 2 and others '" 4 ' 6 under certain conditions, the apparent 

 luminance (photometric brightness), B r , of a black object at a distance R from the 

 observer when viewed horizontally against the horizon sky is given by the equation 



Br=(l-e-° R )B b (A) (4) 



where c = extinction coefficient as previously defined ; 



Bb{A) = luminance of the horizon background of the black object. 



The (A) is inserted to indicate that the luminance is a function of the azimuth re- 

 ferred to a vertical through the sun. It will, of course, be understood that the azimuth 

 of the background is the same as that of the object. 



Equation (4) is not reliable in any azimuth near that of the rising or setting sun. 



Duntley 4 has derived a more general relationship which, for achromatic objects and 

 backgrounds, expresses the fact that the difference of luminance between object and 

 background attenuates exponentially with distance. Equation (5) gives this relationship 

 written in a form representative for a horizontal line of sight in an atmosphere which 

 is homogeneous in its lighting and composition (i.e., scattering, absorption, and attenu- 

 ation coefficients are constant along the path of sight) : 



B r -B b {A) 



Bo-B bo (A)~ e K) 



where B r = apparent luminance of the object at range R; 



Bo = inherent luminance of the object (i.e., at zero range); 

 Bb(A) = apparent luminance of the background of the object at range R; 

 Buo(A) = apparent luminance of the background of the object at zero range 

 (i.e., at the location of the object) ; 

 R = distance between the object and the point of observation. 



The last equation reduces to equation (4) when one applies the condition for a black 

 object 5o = 0, and when B b (A) = Bbo(A). The latter relation is rigorously valid for 

 a horizon background or for a background at infinity. 



Middleton 3 ' " has derived theoretical relationships applicable to the visual range of 

 colored objects. 



From equation (4) one obtains 



When the distance R is equal to the visual range of the object, the apparent contrast 

 of luminance represented by the left-hand member of equation (6) becomes equal to 

 the negative of e, where e = threshold of luminance (brightness) contrast. (The nega- 

 tive sign is indicative of the condition that the object appears darker than the horizon.) 



2 Koschmieder, H., Theorie der horizontalen Sichtweite, Beitr. Phys. freien Atmos., vol. 12, pp. 

 33-53, 171-181, 1924. 



3 Middleton, W. E. K., Visibility in meteorology, 2d ed., Toronto, 1941. 



* (a) Duntley, S. Q., The reduction of apparent contrast by the atmosphere, Journ. Opt. Soc. Amer., 

 vol. 38, pp. 179-191, 1948. 



(b) Duntley, S. Q., The visibility of distant objects, Journ. Opt. Soc. Amer., vol. 38, pp. 237-249, 

 1948. 



5 (a) Coleman, H. S., Morris, F. J., Rosenberger, H. E., and Walker, M. J., A photo-electric 

 method of measuring the atmospheric attenuation of brightness contrast along a horizontal path for 

 the visible region of the spectrum, Journ. Opt. Soc. Amer., vol. 39, pp. 515-521, 1949. 



(b) Coleman, H. S., and Rosenberger, H. E., A comparison of photographic and photo-electric 

 measurements of atmospheric attenuation of brightness contrast, Journ. Opt. Soc. Amer., vol. 39, 

 pp. 990-993, 1949. 



8 (a) Middleton, W. E. K., On the colours of distant objects, and the visual range of colored 

 objects, Trans. Roy. Soc. Canada, Sect. Ill, vol. 29, pp. 127-154, 1935. 



(b) Middleton, W. E. K., The colors of distant objects, Journ. Opt. Soc. Amer., vol. 40, pp. 373- 

 376, 1950. 



(continued) 



SMITHSONIAN METEOROLOGICAL TABLES 



