94 METEOROLOGICAL OPTICS 
two cases, but it is simpler not to think of contrast when 
dealing with a point source of light. 
Dealing first with the visual range of an object seen 
against the horizon sky, the problem is simply to put 
the proper values of Co and ¢ in equation (8) and solve 
for r. For a given object, which looks smaller as it 
becomes more distant, ¢ is unfortunately an empirical 
function of r. 
The standard procedure, used by all writers until 
very recently, is to restrict the problem to “large” 
objects in full daylight; that is, to the approximately 
vertical portions of the curves at the extreme left of 
Fig. 2, so that « may be considered constant. Meteoro- 
logical writers have adopted a quantity variously called 
the standard visibility [53], Luftlichtweite [32] and, more 
recently, the meteorological range [20], calculated on the 
assumption that « = --0.02. Let us call this V,,, and 
note that if it is referred to a black object (Co = —1) 
against the horizon sky, it is Just a convenient sub- 
stitute for oo; because if we write (8) 
—0.02 = —e78¥m (9) 
and take natural logarithms, we obtain 
Va = 3912/00 - (10) 
The ‘meteorological range”’ is defined as “‘that distance 
for which the contrast transmittance of the atmosphere 
is two per cent” [20, p. 238]. 
If now we break away from the restrictions of a 
“large object” and full daylight, we immediately run 
into the difficulty that equation (8) can be solved only 
by a process of successive approximations. The awk- 
wardness of this led the workers at the Tiffany Founda- 
tion to devise a remarkable series of nomograms [20], 
prepared for various levels of field luminance from over- 
cast starlight to full daylight, from which the visual 
range of an object of any area may be read directly if 
its inherent contrast and the meteorological range are 
known. These nomograms were based on the actual 
Tiffany data referred to previously, and therefore cor- 
respond to a 50 per cent probability of detection. It 
is hoped that further nomograms will be forthcoming, 
based on a probability of detection much nearer unity, 
though the existing ones may be used for many purposes 
by dividing the inherent contrast of the object by 2 
before entering the nomogram [20, p. 249]. 
The estimation of the inherent contrast of the object 
remains a stumbling block, except for the ideal black 
object. If we had a grey object of luminance factor 
B standing vertically under a sky of unzform luminance 
B,, its inherent luminance would be By) = B,@/2 and 
its contrast Cy = B/2 — 1. A white object, for example, 
would have a contrast of — 14, and this quantity has 
been used in a great deal of theory under the mistaken 
assumption that a densely overcast sky is uniform. 
Unfortunately its luminance is about three times as 
great at the zenith as at the horizon [389] and it has been 
shown [86] that, for a vertical white object, this results 
in values of Co between 0 and +1, depending on the 
reflectance of the ground. The complications naturally 
increase when the sun is shining. 
Turning now to oblique paths of sight, we have the 
remarkably ingenious theory of Duntley [19] and the 
nomograms based on it [20]. The reader is asked to 
consult the two papers concerned, especially the first, 
which is far too long to summarize here, and to make up 
his own mind as to the practical utility of the theory. 
It may be that some “operational” research is indi- 
cated. 
The visual range of lights at night presents a simpler 
problem. The illuminance from a source of J candle- 
power at a distance r in an atmosphere of extinction 
coefficient oo is 
=) = 
i, = lif eo, 
(11) 
and the visual range is then given by introducing the 
threshold illuminance £;: 
Rs IAG”. (12) 
A suitable nomogram may easily be constructed for 
the solution of (12). If oblique vision is involved, the 
problem of measuring or estimating co remains. 
Instruments for the Measurement of the Visual Range 
Before we can calculate the visual range of any 
particular object or light signal, we must have a know]- 
edge of o or of V,,. A large number of different instru- 
ments have been designed for the measurement of these 
quantities, some requiring photometric settings on the 
part of the observer, others completely objective or even 
automatic, but none of them seem to be in use at any 
considerable number of meteorological stations. 
Nearly all these mstruments fall into four classes: 
(1) devices which measure the transmittance of a 
more-or-less extended sample of the atmosphere; (2) 
instruments which measure the reduction of known 
contrasts; (8) instruments which measure the light 
scattered from a small sample of air at one or many 
angles; and (4) empirical meters of various types which 
do not measure o directly. 
Transmittance meters form a fairly numerous class, 
which may be divided into two subclasses depending 
on whether they are usable only at night or at all 
times. Those for use only at night are generally visual 
telephotometers and may make use of a distant light, 
as for instance that of Collier and Taylor [13], or of a 
beam of light projected from the mstrument and re- 
turned by a distant mirror, as that of Foitzik (22, 23]. 
For a very excellent discussion of visual telephotom- 
eters and their limitations the reader may be referred 
to a paper by Collier [12]. 
A similar duality of optical systems is found in 
photoelectric transmittance meters, which are often 
usable throughout the day and night. Those with a 
distant mirror, represented by that of Bergmann [3], 
can easily make use of a modulated light beam and a 
tuned amplifier to make them insensitive to daylight, 
and a null method of measurement to reduce the effect 
of fluctuations in the supply voltage. They have the 
disadvantage of complexity and very high cost. Simple 
photometry of a distant projector [17] is cheaper. 
