OZONE IN THE ATMOSPHERE 277 
as z continues to increase, however, the function 7/2’ 
reaches a minimum at about 2 = 85°, and shows an 
inversion (Umkehr) [386] at z > 85°, that is, 7/2’ in- 
creases again at very low elevations of the sun. This 
experimental result (Fig. 2) is explained on the basis of 
Z ( DEGREES) 
{e) 56.3 66.9 74.0 79.5 845 88.0 90.0 
0.200 CM 03 
LOG (i'/i)+ CONST 
| 2 3 4 5 6 
OT a2 
Fig. 2—Umkehr curves. 
the distribution of ozone in the scattering atmosphere. 
Zenith light is the sum of the components scattered 
at various altitudes corresponding to the prevailing air 
density. Because of its smaller density, the atmosphere 
above the ozone layer scatters to a lesser extent than 
the atmosphere below this layer. However, the short- 
wave radiation scattered vertically downward traverses 
the attenuating ozone layer by the shortest path and, 
as the zenith distance increases, it attains ever more sig- 
nificance in comparison with the radiation scattered 
below the ozone layer; in this case scattermg power 
would indeed be greater, but the sunlight which is to 
be scattered has been considerably weakened by tra- 
versing the ozone layer by the long, oblique path. The 
stronger the absorption, the greater the tendency for 
the sky radiation of the shortest wave length to find 
the shortest path directly from above through the ozone 
layer. The variation of the Umkehr function 7/2’ with 
z depends on the type of stratification, that is, the 
vertical distribution of ozone. It thus provides a means 
for measuring the vertical distribution. 
We divide the atmosphere into a number of shells 
within each of which we assume the ozone content 
(expressed in centimeters of ozone per kilometer) to be 
constant. We might assume the following shells [389, 45]: 
Altitude Ozone amount 
65-50 km 0, but seattering not zero 
50-35 km ai 
35-20 km Le 
20- 5 km a — (x; + x + w) since x is known. 
5- 0 km u, where uw is assumed to he known. 
Thus the characterization of the vertical distribution 
involves only two unknowns, 2 and x, which is de- 
sirable in order that the numerical solution of the equa- 
tions may not be too difficult. The unknowns 2, and 2» 
enter into the expression for the thickness / em of an 
ozone layer which is traversed by a sun ray incident 
at a zenith distance z. The ray is scattered at poimt A 
where the barometric pressure is 6 and reaches the 
instrument at the point B. The path length in each 
ozone shell is determined trigonometrically; it suffices 
to tabulate the distance s of Fig. 3 as a function of the 
Fie. 3—Subdivision of the atmosphere into layers. 
quantity h — r (7.e., independently of r). Of significance 
for the Rayleigh scattering is the air mass L traversed 
by the ray, which is easily found to be 
L = 1 + (6/760) (m — 1). (7) 
We sum the light scattered by each kilometer A from 
the ground to an altitude of 65 km and find the intensity 
of zenith radiation 7 to be 
i = 210 °"b10 “const (8) 
and similarly for 2’ and 7/2’ or 2’/7. The corresponding 
expression for 2’/7 is the same as the result of observa- 
tions (Fig. 2). It is to be noted that the undetermined 
constant in the observed curve may be determined by 
calculating 2’/7 for z = 0, since the vertical distribution 
of ozone has no effect for z = 0. The two unknowns 
x, and 2x2 require two equations, such as an equation 
for z = 90° and one for z = 80°. The solution is obtained 
graphically by plotting x2 as a function of 2 for both 
cases and determining the intersection of the two curves. 
Choice of a third value of z, such as 2 = 86.5°, provides 
a convenient check. The values of x; and x2 then char- 
acterize the vertical distribution stepwise. 
In addition to this analytical method A, use has also 
been made of a more synthetic method B, which con- 
sists of starting with an assumed vertical ozone dis- 
tribution and varying it until calculated and measured 
Umkehr curves are in agreement. In both methods sec- 
ondary scattering should be further considered. 
In order to obtain a more detailed ozone distribution 
curve it is, of course, desirable to choose our shells as 
thin as possible. It has been verified that essentially 
no ozone is present above 50 km. Between 50 and 35 
km the ozone may be assumed constant as will be seen 
