136 TECHNICAL SURVEY 
tion of the ground would have if the ground were 
plane. The reflection surface for a spherical surface, 
F, is then equal to pD. 
5. Irregularities of the earth’s surface which affect 
the reflection coefficient. 
If Ey is the magnitude of the direct wave and F 
is the magnitude of the reflection coefficient, then 
the field strength of the reflected ray is FE. 
The phase difference between the direct and re- 
flected fields is given by an angle 6 which is the 
‘sum of: 
1. The phase difference, WY, resulting from the 
difference in path length, R2—R;; 
2. The phase difference, ¢, suffered by the reflected 
wave upon reflection from the ground. 
The amplitude of the resultant field for a non- 
directive antenna is then given by GE», where 
G = V1 + F? + 2F cos 6 (1) 
is the earth gain factor which is illustrated in Figure 2 
Ficure 3. Phase addition of direct and reflected rays. 
A curve drawn to represent the contour of constant 
field strength H=GE) as a function of the range Ri 
and the angle of elevation 6 gives the vertical cov- 
erage diagram for that particular field strength. Cal- 
culation of these diagrams usually requires a con- 
‘siderable amount of detailed and laborious work. 
Consider the simple case of the vertical coverage 
diagram of a horizontal dipole antenna located above 
a plane earth in a homogeneous atmosphere. If the 
plane of Figure 2 is perpendicular to the dipole axis, 
the radiation pattern of the antenna is a circle of unit 
radius. The ratio, F'2, of the magnitude of the re- 
flected wave to that of the incident wave is given by 
the magnitude, p, of the reflection coefficient. For 
‘propagation to distances that are great compared 
with the antenna elevation, the path lengths R, and 
R, are not greatly different, and the attenuation due 
to path length is approximately the same for both 
direct and reflected waves. For this set of conditions 
the resultant field is H=GE , and equation (1) 
reduces to 
G = V1+ p? + 2p cos 5° (2) 
In this form G is the plane earth gain factor and a 
plot of the curves #=GE,)=constant as a function of 
range and angle of elevation gives the coverage dia- 
gram. It depends only upon the magnitude of the re- 
flection coefficient, the phase changes related to re- 
flection and to the difference in path length R.—R,. 
Since radar requires t,.v-way transmission the re- 
ceived field strength is proportional to G?/R?,. Other 
modifying factors must, however, be introduced if the 
antenna and the target have directional radiative, 
properties 
Both the magnitude of the reflection coefficient 
+F and the phase angle by which the reflected 
wave lags behind the incident wave are functions of 
the frequency, the polarization of the radiation, the 
angle of grazing with the surface, the conductivity, 
dielectric constant, and roughness of the ground or 
‘sea surface. Figure 4 illustrates the variation of 
F(=p) and ¢ for reflection from a smooth plane sea 
surface for frequencies of 100 to 3,000 me, for both 
types of polarization, at different grazing angles. It 
may be noted that for horizontal polatization p is 
approximately unity and ¢ nearly 180°, irrespective 
of the frequency and the magnitude of the grazing 
angle. This is the simplest situation to be encountered 
and most nearly approximates the idealized case of 
a perfect reflector with horizontal polarization. For 
this case p is exactly unity, and ¢ ds exactly 180°. 
For vertical polarization over the sea or either type 
of polarization over ground, both p and @ depart 
widely from unity and 180°, respectively. Variations 
ir these quantities greatly complicate the calculation 
of coverage diagrams. 
The reflection coefficient of microwaves is usually 
found to be small over land. This is essentially due to 
irregularities of the land surface. When these irregu- 
larities are sufficiently small, reflection from land is 
found to be considerable. 
Since the receiver, or target, is usually located at a 
distance from the transmitter which is large in com- 
parison to the height above the ground, the direct 
and reflected rays are very nearly parallel, making an 
angle 6 with the horizontal (Figure 2). The reflected 
Tay may be supposed to issue from an image trans- 
mitter 7’, which fs as far below the ground as the 
true transmitter is above it. The path difference be- 
tween the direct and reflected rays is equal to the 
distance T’A. By. the figure this is equal to 2h; sin B, 
where h; is transmitter height. For small values of 8 
this is practically equal to 2:8 if B is measured in 
radians. The corresponding phase shift due to path 
difference is equal to 
Y= Mp. 
At the point of reflection the phase of a ray changes 
discontinuously by the amount ¢, which is the phase 
angle of the reflection coefficient. For horizontal 
polarization, to again take the simplest case, the 
phase shift @ at reflection is practically 180°, or 
m radians. (For vertical polarization, see Figure 4, ¢ 
is more complicated.) Adding the phase change W, 
corresponding to difference in path length, gives the 
