66 METEOROLOGICAL OPTICS 
that of the earth’s surface, which ordinarily is several 
times larger, and total reflection (mirages) may occur. 
The expansion of the horizon and the decrease of its 
angular depression is a usual consequence of terrestrial 
refraction. In Fig. 5, an observer at A whose horizontal 
plane is AH would, in case of rectilinear light rays, see 
the horizon at D, where his line of sight is tangent to 
the earth’s surface and the geodetic depression is 6. 
Because of terrestrial refraction he normally sees the 
horizon at D’ in the direction AD” (tangent to the 
curved ray AD’), that is, at the depression 6’. The 
difference 6 — 6’ represents the terrestrial refraction, 
and equations (4) and (5) apply to this case also. How- 
ever, 6 is generally determined by direct observation 
rather than computed from meteorological data and the 
known geometric quantities, because temperature and 
lapse rate in the air layer below the observer vary with 
the nature and contour of the underlying surface. The 
effects of these variables for air layers not in immediate 
contact with the ground were investigated by Brocks 
[2, 3, 4] who found that the terrestrial refraction can be 
quite accurately computed from meteorological data 
and that, in turn, the average lapse rate can be deter- 
mined from the observed curvature of light rays. For 
a ray-path length of 30 km, a change in zenith distance 
of 1” corresponds to a lapse-rate change of 0.04C/100 
m. By means of mutual sighting from both ends of a 
ray path the absolute values of the lapse rate can be 
determined. However, this method is of limited prac- 
tical value, because for steep lapse rates, with their at- 
tendant convection currents, the image fluctuations of 
the light beam would considerably decrease the accuracy 
of measurement. 
Phenomena Due to Special Density Gradients. When 
the decrease in density upwards is greater than normal, 
a condition which may be caused by smaller than nor- 
mal lapse rates, terrestrial refraction is increased and 
objects that are usually beyond the horizon come into 
view. This excessive extension of the normal horizon is 
called looming. The opposite phenomenon of sinking is 
due to an abnormally small vertical density gradient. 
As can be seen from application of equation (5) to 
the average conditions between observer and horizon 
point, the curvature 1/rz of the light ray (A .D’ in Fig. 5) 
becomes smaller with increase in lapse rate y; for 
y = 7’, the curvature becomes zero and the ob- 
server sees horizon at D;for y > y’, his horizon would 
further shrink and end at a point between D and £, 
while the curvature becomes negative (ray convex to- 
ward surface). In this case it is not necessary that 
7 > vy’ over the whole range, although this often occurs 
over strongly heated surfaces; it is sufficient that the 
isopycnic surfaces be inclined upwards toward the ob- 
server, so that the air density is greater there than at 
the distant point on the horizon [42]. The great in- 
creases in, or the reversals of, the normal vertical den- 
sity gradients are generally confined to the air layers 
near the ground. 
When the light rays from the upper portion of a 
distant object LU in Fig. 6 (A) have a different curva- 
ture than those from the lower portion, the geometric 
angle s, that the object subtends at the observer O, 
appears changed. Exner [42] has shown that a linear 
change in refractive index n with height cannot lead to a 
noticeable change in the angle of subtense. When the 
decrease of n with height is slower than it would be ac- 
cording to a linear function as shown by case (A), the 
curvature of the ray LO is greater than that of UO, and 
the apparent angle of subtense s’ between the tangents 
to the respective rays becomes smaller. This phenom-, 
enon, in which the object also appears elevated, is 
called stooping. An enlargement of s’ combined with an 
apparent lifting takes place, when the decrease of n 
with height is more rapid than according to a linear 
function, as shown by case (B) which represents tower- 
ing. Whereas these phenomena result from vertical 
density gradients (drop in density per unit height) that 
decrease (A) and increase (B), respectively, with height, 
case (C) shows a negative density gradient that de- 
creases, and case (D) one that increases with height, 
causing s’ to be enlarged and diminished, respectively. 
U bead : 2 
Cy 
L Lee 4 
U 
sf SS ° 
=D) 
L 
Fic. 6.—Effect of abnormal density gradients on the 
curvature of light rays. 
Because of the increase in density with height, the rays 
are convex toward the ground, and the objects appear 
depressed. The mathematical theory for these phenom- 
ena, as well as for the corresponding ones involving 
horizontal objects, was developed by Exner [42]. 
In case the density distribution in the lower layers is 
such that the rays from an object reach the observer 
along two or more different paths, so that he sees one 
or more images of the object, we speak of swperzor, 
inferior, or lateral mirages, depending on whether the 
image appears above, below, or to the side of the ob- 
ject. The latter case can occur only when the isopycnic 
surfaces are vertical or nearly so, for example, in prox- 
imity to strongly heated walls. This phenomenon was 
theoretically treated by Hillers [17]. In case of a com- 
plicated density distribution in the lower layers, com- 
plex distorted images of distant objects, the Fata Mor- 
gana, may appear. General theories of mirages were 
developed by Nélke, A. Wegener, and others [c. 42], and 
