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trio. 12

TRANSMISSION LOSS IN

RADIO PROPAGATION - I

BY KENNETH A. NORTON

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical <

^f\ote,

12

June, 1959

Transmission Loss in Radio Propagation: U

by

Kenneth A. Norton

NBS Technical Notes are designed to supplement the Bureau' s regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price $ 3.00
This Technical Note was originally given limited distribution

as NBS Report No. 5092, dated July 25, 1957. Since that time parts

of this report have been published in the following references:

(1) K. A. Norton, "Low and medium frequency radio

propagation", Proc. of the International Congress on the

Propagation of Radio Waves at Liege, Belgium, October,

1958, to be published by the Academic Press.

(2) K. A. Norton, "System loss in radio wave propagation",

J. Research, NBS, 63D, pp. 53-73, July-August, 1959.

(3) K. A. Norton, "System loss in radio wave propagation11 , Letter to the Editor, Proc. I. R. E. , to be published.

All of the material in reference (1) is included in this

Technical Note. Some of the material in references (2) and (3) is

new, particularly the definitions of the new terms "system loss" and

"propagation loss". The transmission loss concept was adopted by

the C.C.I. R. at its IXth Plenary Assembly in Los Angeles as is

discussed more fully in reference (3) above.
TRANSMISSION LOSS IN RADIO PROPAGATION: II

by

Kenneth A. Norton

SUMMARY

In an earlier report with this title the concept of transmission loss was defined and its advantages explained. In

this report a survey will be made of the transmission losses

expected for a wide range of conditions, i. e. , for distances

from 10 to 10, 000 statute miles; for radio frequencies from 10 kc

to 100, 000 Mc; for vertical or horizontal polarization; for ground

waves, ionospheric waves, and tropospheric waves; over sea

water or over land which may be either rough or smooth; and for

various geographical and climatological regions.

Note: Tbr attention of the reader is called to additional terms,

discussed in appendix III, which must be added to the transmission losses shown in this report when the antennas are near the

surface. These terms arise from changes in the antenna radiation resistances which occur when the antennas are near the

surface, and represent important corrections to the transmission

loss, particularly at the lower frequencies where the antennas,

assumed to be 30 feet above the surface for many of the calculations, are only very small fractions of a wavelength above the

surface.

TRANSMISSION LOSS IN RADIO PROPAGATION: II

by

Kenneth A. Norton

1. Transmission Loss in Radio Propagation

We will be concerned primarily with the transmission loss

encountered in the propagation of radio energy between a transmitting and a receiving antenna. Simple methods will be given for

determining the magnitude of this transmission loss and its variation in space and time (fading) for any frequency in the presentlyused portion of the radio spectrum and for any kind of transmission

path likely to be encountered in practice. In addition, methods will

be given for estimating radio noise and interference levels. When

combined, these two methods make possible the estimation of the

transmitter power and antenna gain required for satisfactory communication, navigation, or other specific uses of the transmissions.

The transmission loss in a radio system involving propagation between antennas is simply the ratio of the radio frequency

power, pr

, radiated from the transmitting antenna divided by the

resulting radio frequency power, pa

, available from an equivalent

loss -free receiving antenna; thus the system transmission loss =

(p /p )• We see that the transmission loss of a system is a

r a

dimensionless number greater than unity, and that it will often be

convenient to express this in decibels; the transmission loss, L*

expressed in decibels, is thus always positive:

L = ioiog

10 (pr

/Pa) = p

r -p

a

a>

t

*

f

See references 1, 2 and 3.

Throughout this report capital letters will be used to

denote the ratios, expressed in decibels, of the corresponding

quantities designated with lower-case type; e.g. , P =10 log p , r 10 r
This particular choice of definition excludes from the transmission

loss the transmitting and receiving antenna circuit losses'" and any

loss which occurs in any transmission lines which may be used

between the transmitter and the transmitting antenna or between the

receiving antenna and the receiver. This exclusion has the advantage that it results in a measure of loss which is attributable solely

to the transmission medium including the path antenna gain, G_ , which arises from the directivities of the transmitting and receiving

antennas. In addition to the actual transmission loss, L, of the

system, it is also convenient to define the basic transmission loss,

L^, to be the transmission loss expected if the actual antennas were

replaced by isotropic antennas; • this also serves to define the path

antenna gain:

G

p = Lb

- L (2)

Consider first an idealized isotropic transmitting antenna

in free space radiating a power, pr

, expressed in watts. Such an

antenna produces a field intensity of p /4ud watts per square mile

at a distance d expressed in miles provided d > > \, The absorbing

area of a perfectly conducting, isotropic receiving antenna in free

space is equal to \ /4ir where X is the free -space wavelength expressed in miles; the resulting radio frequency power available from

such a receiving antenna when placed at a distance d > > \ from the

isotropic transmitting antenna is thus pa

= pr

(\/4ird) . Thus we find

Antenna circuit loss includes the ground losses arising

from the induction field of the antenna, but excludes losses

occurring in the radiation field.

' In some of the past literature on radio wave propagation,

the intensities of the expected fields have been given in terms of E,

the field strength expressed in decibels above one microvolt per

meter for one kilowatt effective power radiated from a half -wave

dipole. It can be shown that Lm and E are simply and precisely

related by LK = 139. 367 + 20 log,„f - E. D 10 mc
that the basic transmission loss, L^f* for isotropic antennas in free

space ' is given by:

Lbf = 10 log

1Q (p r /p a

) = 10 log

1() (4TTd /\)

2

.36.58 + 20 log^d + 20 log

1Q

fMc

(3)

In the above f, denotes the radio frequency expressed in megacycles. Mc

Fig.l shows this basic transmission loss for isotropic antennas in

free space. For d - 2\, Ly = 28 db and thus (3) is only approximate

when the indicated values of Lw are less than, say, 30 db.

In fact, whenever the calculated transmission loss is less

than, say 30 db, we must consider that the problem involves a

transfer of an appreciable portion of the power between the transmitting and receiving antennas by other than radiation. For example,

there is a direct coupling between the antennas via their induction and

electi*ostatic fields, and this is a negligible factor in the calculation

of the transmission loss only when L. > 30 db. When high gain

antennas are used, their separation must be much greater than 2\ in

order to maintain the condition L > 30 db.

For an actual radio transmission system there will always be

some path antenna gain so that the transmission loss L = Lt_ - G

will be less than the basic transmission loss. In some systems the

free space gains G^ and Gr

of both the transmitting and receiving

antennas, respectively, will be fully realized so that G = Gf + Gr . For example, with half-wave dipoles having a common equatorial

plane and separated by a distance d > > \ in free space

Gj- = Gr

= 2.15 db so that G = 4. 30 db; the transmission loss for such

a system is thus just 4. 3 dl) less than that given by (3) and shown on

Fig.l. Similarly, electrically short dipoles have gains

In some of the past literature on radio wave propagation, the

intensities of the expected fields have been given in terms of

A = L^ - L Df» the attenuation relative to that expected for propagation

in free space; in the case of surface wave propagation with vertical

polarization, the attenuation has usually been expressed relative to an

inverse distance field which is twice the free space field and thus

A1 = L, - Li r + 6. 021 in this case.
BASIC TRANSMISSION LOSS IN FREE SPACE

ISOTROPIC ANTENNAS AT BOTH TERMINALS

L bf = 36.58 + 20 log 10 D + 20log lo f Mc

Frequency in Megacyles

20 30 50 TO 3,000 5,000 7,0

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Frequency in Kilocycles

Figure I
G^ = Gr

- 1. 76 db so that G = 3. 52 db for propagation between

appropriately oriented electrically short dipoles in free space.

The free space gain of a large receiving antenna with an

effective absorbing area of ae square meters will increase with increasing frequency at sufficiently high frequencies:

G

r

= 10 1og

10 ae+2 1og10

fMc -38.54 (4)

(Fori, >100/ «s/a ) Mc e For example, a large parabolic antenna will have an effective absorbing area a between 50 and 70 per cent of its actual area.

2. Transmission Loss in Free Space

Before considering the additional influences of the earth's

surface and of its atmosphere on the propagation and transmission

loss of the radio waves, it is instructive to consider first the characteristics of the transmission loss in free space for three kinds of

systems which are typical of most of the applications encountered

in practice.

Consider first a broadcast type of system in which essentially

non-directional antennas are used at both terminals of the transmission path. For example, if half wave dipoles were used we have

already Been that the system transmission loss, L, will be just 4. 3 db

less than that given by (3) and shown on Fig. 1. For such systems

we see that the loss increases rapidly with increasing frequency

because of the decreasing absorbing area of the receiving antenna.

For this reason such systems should, in general, use the lowest

available frequencies.

Consider next a type of broadcast service in' which a directional

array may be used at one end of the path: television is an example

since the televiewers in remote areas consistently use high gain receiving antennas. If we assume that a half -wave dipole is used at

the other terminal, we may combine (3) and (4) and obtain for the

system transmission loss:

L

f = 72. 97 + 20 log

1() d - 10 log

1() *

q

(5)
Note that the free space transmission loss in this case is independent

of frequency. In this case again, because of the additional loss

arising from the effects of irregular terrain'1" which increase with

increasing frequency, it is generally desirable to keep this kind of

broadcasting service at the lowest available frequencies.

Finally consider a point-to-point type of service in which

two identical high gain (and thus highly directional) antennas are used

at each terminal of the transmission path. For such a system the

free space transmission loss may be obtained from:

L

f = 113. 67+20 log

1() d - 20 to^ f^ - 20 log^ %

(6)

For services of this type it is clear that the highest frequencies free

from the effects of atmospheric absorption are likely to be the most

efficient. The above formula is applicable only to line -of -sight

systems with first Fresnel-zone clearance over terrain which appears

rough to the radio waves, and we will consider later within-line -of

sight smooth-terrain systems and beyond-the -horizon systems

employing tropospheric scatter.

Rayleigh's criterion of the roughness may be used to determine

whether a surface appears to the radio waves to be rough or smooth:

4tt <r sin ib R —\

(7)

In the above equation a, denotes the standard deviation of the terrain

h

heights relative to a smoothed mean height (see Fig. 2), ib = ib = if* T R

denotes the grazing angle with the smoothed mean surface and

X is the wavelength expressed in the same units as ov. When R is

less than 0.1, there will be a well defined specular reflection from

the ground, but when R > 10, the reflected wave will be substantially

weaker and will usually have a very small magnitude.!

The concept of first Fresnel-zone clearance provides a means

of determining when the effects of the ground may be neglected so

that the simple formula (6) may be used for determining the expected

See reference 2.

See references 4, 5 and 6.
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transmission loss to a first approximation. Fig. 2 illustrates the

first Fresnel zone concept. When the terrain along ths path just

touches the elliptical first Fresnel-zone defined by the locus of points

such that a + b = d + \/2, the path is considered to have first Fresnelzone clearance for a system with wavelength X. On Fig. 2, T and R

represent the locations of the transmitting and receiving antennas.

The presence of the ground will have only a small effect on the propagation, provided the antennas are sufficiently elevated so that none of

the terrain lies within the first Fresnel zone and if, in addition,

R > 10 so that the surface appears rough to the radio waves.

3. Transmission Loss for Ground Wave Propagation

The ground wave is that component of the total received field

which has not been reflected (or scattered) from either the ionosphere or the troposphere. It is convenient to divide the ground

wave into two components :' a space wave and a surface wave.

The space wave is the sum of a direct wave and a ground-reflected

wave. Figs. 3 and 4 give examples of space wave propagation.?

Near the radio horizon the ground -reflected wave is out of phase with

the direct wave, and the received fields are quite weak; as the receiving antenna is raised, the relative phase increases until finally

the direct and ground-reflected waves are in phase—at the lobe

maxima shown on Figs. 3 and 4. At still higher heights the relative

See references 7 and 8.

See references 3, 9, 10, 11, 12, and 13.

The term Norton surface wave has been used in several

recent papers in order to distinguish this component of the ground

wave from the Zenneck surface wave with which it has sometimes

been confused; the latter does not exist in practice as shown by Wise

in reference 14. A recent discussion of surface waves by Wait in

reference 15 further clarifies the physical nature of this and other

surface wave components.

See references 16 and 17 for a further discussion of air-toground propagation.

8
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phase continues to increase, lobe minima and maxima occurring

where the direct and ground reflected waves are out-of-phase and

in-phase. Figs, 3 and 4 may be used to show approximately what

happens at some other radio frequency, fj^jc, and ground antenna

height, h, if we modify the transmission losses indicated on these

figures by adding 20 log (fj^/328) and, at the same time, determine

hfor Fig. 3 by (h/35) = (328 /fMc) and for Fig. 4 by (h/115) = (328/fMc).

Figs. 3 and 4 correspond to smooth earth conditions, i. e.

,

for R < < 0. 1. In this case the space wave field strength may be

represented approximately by:

F = 2F sin(2ir h sin ip/X) (8) o

For example, the above equation represents very accurately the

expected field strength for propagation from a horizontal dipole over

a smooth, flat, perfectly conducting surface, where F is the field

strength in free space, F is the expected field strength at a receiving

point corresponding to a grazing angle i|j, and h is the height of the

ground terminal antenna above the smooth surface. The maximum

of the first lobe (F = 2FQ

) occurs when (2-rrh sin i^/M = t/2; according

to Rayleigh's criterion (4tt ov sin i^A) must be less than 0.1 for the

surface to appear smooth to the radio waves. Combining these two

results we find that cr-^ must be less than h/lOir , independent of the

frequency, if we are to expect (8) to apply at the angle i|j corresponding to the maximum of the first lobe. At still higher angles the

requirements for smoothness of the terrain are correspondingly more

stringent. At lower grazing angles, however, the terrain may be

correspondingly rougher; for example, at the angle below the first

lobe maximum where F = 0. 2 FQ corresponding to a transmission

loss 20 db greater than at the maximum of the lobe, cr^ must be less

than 0. 5h for the earth to be considered sufficiently smooth for (8)

to apply and where F = 0. 02 FQ corresponding to a transmission loss

40 db greater than at the lobe maxima, a

-, may be as large as 5h.

Thus we see that the large reductions in the received field below the

maximum of the first lobe as shown on Figs. 3 and 4 and indicated by

(8) are expected to occur even over comparatively rough terrain.

For propagation conditions such that R is large, i. e. , high

frequencies, very rough terrain, or large grazing angles, the ground

11
reflected wave may be described statistically. It has been found''" that

the Rayleigh distribution is appropriate for this purpose when R is

very large, say R > 100, and that a combination of a constant specular

component plus a random Rayleigh component is required foi*

0. 01 < R < 100. Fig. 5 shows theoretical probability distributions

for this case with the parameter K increasing from (- co) for R<0. 01

to values of K greater than 20 for R > 100. Here K is the level in

decibels of the mean power in the random, Rayleigh distributed,

component relative to that of the steady component. As an example

of the use of probability distributions of this kind for describing

space wave propagation conditions, suppose we have an air-to-air

communication system operating at 328 Mc. As we fly over irregular

terrain at a fixed high altitude away from another aircraft at the same

altitude (See Fig. 6), the grazing angle fy decreases from a comparatively large value to zero on the radio horizon, and this corresponds

to a decrease of R from a very large value to zero on the radio

horizon. Thus at short distances the ground-reflected wave will

fluctuate in magnitude over a range indicated by the K = 20 curve on

Fig. 5, while at larger ranges these fluctuations will occur over

smaller and smaller ranges corresponding to the smaller values of

K.

The above statistical description of space wave propagation

over rough terrain is appropriate for propagation paths with Fresnelzone clearance. For still smaller antenna heights involving propagation very near to or just below grazing incidence, the received

space wave is log normally distributed. ' For example, a study r of the fields received on over -land paths from television stations in

the frequency range from 50 to 220 Mc and on receiving antennas with

heights in the range from 12 to 30 feet indicates that the standard

deviation of the received fields is of the order of 6 to 10 db about mean

values of the order of magnitude expected for propagation over a

smooth surface.

Finally, when the transmitting and receiving antennas are

both actually on the surface, the received ground wave is a surface

wave. Furthermore, when the transmitting and receiving antennas

See references 4, 6 and 18.

See references 2 and 19.

See reference 19.

12
-13-

DISTRIBUTION OF THE RESULTANT AMPLITUDE

OF A CONSTANT VECTOR PLUS A RAYLEIGH DISTRIBUTED VECTOR

Power in Random Component is K Decibels

Relative to Power in Constant Component

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Probability that the Ordinate Value will be Exceeded

Figure 5
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are both only a small fraction of a wavelength above the surface, the

received ground wave is still primarily a surface wave together with

a small space-wave component. The transmission loss in surface

wave propagation 1' is very much influenced by the electrical constants

of the ground, especially its conductivity. Although efforts have been

made to correlate these ground constants with soil types so that predictions of the effective ground conductivity could be made, such

studies have not been very successful so far. However, a publication

of the National Bureau of Standards is available T which gives the

measured values of effective ground conductivity for various propagation paths in the United States. For propagation over average land

one may use an effective ground conductivity of 5 milli-mhos per

meter and an effective dielectric constant of 15 although individual

over land paths may have substantially different ground constants,

while over the sea the effective ground conductivity is of the order of

5 mhos per meter with an effective dielectric constant of 80.

Figs. 7, 8, 9, and 10 show the basic transmission loss expected for ground wave propagation over a smooth spherical earth

with the transmitting and receiving antennas both at a height of

30 feet, for either vertical or horizontal polarization and with ground

constants typical of over land and over sea water paths. At frequencies less than 10 Mc, the antenna heights are less than a wavelength

and the ground waves shown for vertical polarization are primarily

surface waves, whereas for frequencies greater than 100 Mc the

ground waves with these antenna heights are primarily space waves

with only a small surface wave component. Note that the proximity

of the earth at low frequencies doubles the received fields for

vertically polarized waves, but suppresses the propagation of horizontally polarized waves: i. e. , horizontally polarized surface waves

are highly attenuated. However, at the higher frequencies involving

primarily space wave propagation, the expected transmission loss

becomes independent of the polarization used.

On frequencies above 10, 000 Mc the radio waves are

appreciably absorbed by the oxygen and water vapor in the atmosphere. Fig. 11 shows the total gaseous atmospheric absorption

near the surface at Washington, D. C. The absorption shown on

Fig. 11 is the median value; for small percentages of the time the

absorption will be considerably greater as a result of absorption

'fi See references 9, 10, 11, and 12,

See reference 20.

15
BASIC TRANSMISSION LOSS EXPECTED FOR GROUND WAVES

PROPAGATED OVER A SMOOTH SPHERICAL EARTH

OVER LAND: a = 0.005 MHOS/METER, € = 15 POLARIZATION: VERTICAL

TRANSMITTING AND RECEIVING ANTENNAS BOTH 30 FEET ABOVE THE SURFACE

20 30 50 70 100 200 300 500 700 1000

Distance in Statute Miles

2000 3000 5000 7000 10000

Figure 7
•17-

BASIC TRANSMISSION LOSS EXPECTED FOR GROUND WAVES

PROPAGATED OVER A SMOOTH SPHERICAL EARTH

OVER SEAWATER: a = 5 MHOS/METER, € = 80

POLARIZATION: VERTICAL

TRANSMITTING AND RECEIVING ANTENNAS BOTH 30 FEET ABOVE THE SURFACE

30

40

50

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03

BASIC TRANSMISSION LOSS EXPECTED FOR GROUND WAVES

PROPAGATED OVER A SMOOTH SPHERICAL EARTH

OVER LAND: <r = 0.005 MHOS/METER, € = 15 POLARIZATION: HORIZONTAL

TRANSMITTING AND RECEIVING ANTENNAS BOTH 30 FEET ABOVE THE SURFACE

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PROPAGATED OVER A SMOOTH SPHERICAL EARTH

OVER SEAWATER: a = 5 MHOS/METER, € = 80

POLARIZATION: HORIZONTAL

TRANSMITTING AND RECEIVING ANTENNAS BOTH 30 FEET ABOVE THE SURFACE

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Figure 10
-20-

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NEAR THE GROUND AT WASHINGTON, D.C.

Meteorological Data From the Ratner Report

July II, 1957

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Frequency in Megacycles

Figure II

80,000 90,000 I00JXX)
by rain. The transmission losses shown on Figs. 7, 8, 9» and 10

for frequencies above 10, 000 Mc were estimated by using the August

absorption shown on Fig. 11.

The heights of the antennas are very important in ground wave

propagation at the higher frequencies. This is illustrated on Fig. 12

which shows for a frequency of 50 Mc the great reduction in transmission loss expected when the antenna height at one terminal is

increased from zero up to 10, 000 feet while the other antenna height

is increased from zero up to 30 feet.

4. Transmission Loss for Ionospheric Propagation

Radio waves with frequencies less than the maximum usable

frequency for a given transmission path are reflected by the ionized

regions of the upper atmosphere with sufficient intensity so that they

often provide a mode of transmission with less loss than that involved

in ground wave propagation. The maximum frequency usable on a

given ionospheric transmission path depends upon the length of the

path, its geographical location, the time of day, the season of the

year, and the phase of the sunspot cycle. Predictions of these

maximum usable frequencies are published regularly three months

in advance by the Central Radio Propagation Laboratory. T Fig. 13

is an example of these predictions showing how the maximum usable

frequencies vary with local time and with geographical location for a

path 4, 000 kilometers long. This particular chart is for February^

1957, a period near sunspot maximum as may be seen on Fig. 14

which shows the smoothed Zurich sunspot numbers from 1750 to 1957.

The sunspot numbers shown on Fig. 14 are averaged over a period of

13 months, but the author has shown in unpublished work that the

sunspot numbers obtained by averaging over a period of three months

are just as well correlated with ionospheric propagation conditions

and thus provide a more useful index for prediction purposes. Fig. 15

shows a typical correlation between the observed maximum usable

frequencies and these three -months -smoothed sunspot numbers.

See references 21, 22 and 23.

f See references 24 and 25.

21
-22-

BASIC TRANSMISSION LOSS EXPECTED IN PROPAGATION

OVER A SMOOTH SPHERICAL EARTH AT 50 MEGACYCLES

HORIZONTAL POLARIZATION; a =0.005 MHOS/ METER; 6=15

FOR THE TERMINAL ANTENNA HEIGHTS INDICATED

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Figure 12
23-

NORTH LATITUDE SOUTH

FIG.I3 MEDIAN F2-4000-MUF, IN Mc, W ZONE, PREDICTED FOR FEBRUARY 1957
-24-

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-25-

CORRELATION OF THE MONTHLY MEDIAN WASHINGTON 2PM.

MAXIMUM USABLE FREQUENCIES FOR FEBRUARY WITH THE

THREE-MONTHS-SMOOTHED ZURICH SUNSPOT NUMBERS

o °> a o

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20 40 60 80 100 120 140 160

Three-Months -Smoothed Zurich Sunspot Numbers

160 200

Figure 15
4.1 Very Low Frequency Ionospheric Propagation

At the very low frequencies below 30 kc, the ionosphere

reflects the waves at relatively low heights, about 70 km in the daytime and 90 km at night. At these low heights the ionization gradients

are sufficiently large so that the ionosphere behaves as a sharp

boundary, and it is convenient to use wave guide theory for determining the phase and amplitude of the received waves; good discussions of this theory are presented in the June, 1957 issue of the

Proceedings of the Institute of Radio Engineers. Figures 16, 17, 18,

and 19 give examples for this frequency range of the transmission loss

expected in propagation between vertical electric dipoles over land

and over sea and for day and night conditions. The values shown on

these four figures -were computed by the methods described by

Wait; _' zJJ the minima and maxima shown are caused by interference between the ground wave and ionospheric wave modes at

distances less than about 1, 000 miles, and are caused by interference

between the several ionospheric modes at the larger distances. At

these long wavelengths the fading of the received waves is caused

by a gradual shift from midday to midnight conditions and consequently

has a very long period; thus at certain distances the received field

may remain weak throughout the day or the night. The comparison

between the calculated and observed locations and magnitudes of such

anomalies provides a useful means for determining the effective

constants of the ionosphere. The dimensionless constant L/H provides

a measure of the effective conductivity of the ionospheric boundary,

and the values of this constant assumed in these examples were

determined by a comparison with observations of transmission loss.

It is expected that L/H will also vary somewhat with the geomagnetic

latitude of the receiving point, but such variations are not expected

to have a large influence on the transmission loss.

It should be noted on Figs. 16 - 19 that values are shown for

the transmission loss expected at distances beyond the antipode of the

transmitter (about 12, 500 miles), and at these larger distances a

stronger signal would be expected from the shorter great circle path

corresponding to transmission in the opposite direction; when short

pulses are transmitted, these signals traveling in opposite directions z~J

will interfere with each other, and the results given on these figures

should be useful in determining the magnitude of this multipath

problem.

26
-27-

TRANSMISSION LOSS EXPECTED BETWEEN

SHORT VERTICAL ELECTRIC DIPOLE ANTENNAS

Day Over Land cr = 0.00477 Mhos /Meter;

Ionospheric Constant L/H=0.l- h = 70km

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Figure 16

I0,C 20,000 40,000
-28-

TRANSMISSION LOSS EXPECTED BETWEEN

SHORT VERTICAL ELECTRIC DIPOLE ANTENNAS

Day Over Sea cr = oo Ionospheric Constant L/H = O.I-, h=70km

40,000 60,000 Kilometers

400 600 1,000 2,000 4,C Statute Miles

Figure 17
-29-

TRANSMISSION LOSS EXPECTED BETWEEN

SHORT VERTICAL ELECTRIC DIPOLE ANTENNAS

Night Over Land cr = 0.0037l Mhos/ Meter

Ionospheric Constant L/H = 0.05 ; h = 90km

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Figure 18
-30-

TRANSMISSION LOSS EXPECTED BETWEEN

SHORT VERTICAL ELECTRIC DIPOLE ANTENNAS

Night Over Sea <x = oo Ionospheric Constant L/H=0.05^ h = 90km

to

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250 400 2,000 4,000 6,000 10,000 20,000 40,000 Kilometers

200 2,000 4,000 6,000 10,000 20,000 40,000

Statute Miles

Figure 19
We will see in a later section how convergence at thecu:*ved surface of the ionosphere is expected to affect the transmission loss; in

calculating the losses shown on Figs. 16-19, allowance was made onlyfor

the convergence expected in the vertical plane and on the as sumption that

the ionosphere is smooth. Allowance for convergence in the horizontal

plane leads to large additional reductions in the transmission loss expected

near the antipode-12, 441 miles -and near 24,881 miles and 37,322 miles.

For a smooth, spherical ionosphere, we should subtract the horizontal

plane convergence C-. - 10 log, [d(miles)/3960 sin {d(miles)/3960 }] from

the transmission losses shown; at distances, p, from the antipodal

points less than 100 miles but greater than 0.1\, we may write

0^= 40.948 -lOlog-^Q p(miles); the value right at the antipodal points is

given by C^= 34.210+ 10 log^o^kc* but this latter includes both long and

short great circle path energy.

The available experimental data indicate that the parameters of

the ionosphere chosen for these calculations lead to about the right conclusions over the range of frequencies from 10 to 20 kc, but at 8 kc the

calculated losses are somewhat greater- than those observed.

Finally, it should be noted that mixtures of day and night and

land and sea conditions are to be expected over these long paths, and

suitable methods of calculation have yet to be developed for such mixed

paths. It will likely be possible, however, to develop empirical methods

for combining the results given here to obtain good estimates of the

transmission loss expected on such mixed paths inmuch the same manner

as has been used for estimating the transmission loss expected in ground

wave propagation over mixed paths. zZ/ izJ

4. 2 Low and Medium Frequency Ionospheric Propagation

As we increase the radio frequency well above 30 kc, the ionosphere behaves much less as a sharp boundary and instead gradually

refracts the waves back to the receiving point only after they have penetrated many kilometers into it, this penetration being greater the higher

the radio frequency. The available evidence appears to indicate that the

D and E regions of the ionosphere, which extend from 70 to 110 km, are

turbulent, consisting of "blobs" of ionization which drift with the mean

wind with velocities often in excess of 100 miles per hour. Aninteresting

discussion of these irregular ionospheric motions is given in a recent

article by Gautier. _/ The radio waves will travel along many different

paths through this turbulent ionized medium, the received field being the

resultant vector sum of the waves received after propagation along these

different paths. At sufficiently high frequencies, the relative phases of

these waves will be random, and the resultant received field will have

a Rayleigh distributed amplitude as shown on Fig. 5; on this figure

K represents the ratio in decibels between the field intensity of the

random ionospheric waves and a steady ground wave or, in the case

of a single ionospheric mode, K represents the ratio in decibels

between the field intensity of the random ionospheric waves and the

steady, specularly-reflected component. Thus it becomes convenient,

particularly for frequencies above 30 kc, to determine the transmission

31
loss separately for the ionospheric and ground wave modes of propagation. Figs. 20 and 21 show the transmission loss expected at 100 kc

between short vertical electric dipole antennas for the ground wave

and several ionospheric wave modes of propagation over land and

over the sea, and for day and night conditions. The method of

calculation used in determining the results shown on Figs. 20 and 21

involves a combination of ray and wave theory. Fig. 22 illustrates

the geometry of our model and some of the assumptions made in the

calculations. The waves are refracted in the troposphere down

towards the earth and, as a consequence, the distance, dp traveled

for a given ray angle of elevation, vJj , before the waves arrive back

at the earth is substantially larger than if there were no atmosphere.

We have idealized our problem by assuming for all points

along the path that the ionosphere has the same height, h, and the

same reflection coefficient, while the ground is assumed to have the

same electrical constants even for propagation all the way to the

antipode at a distance of about 12, 500 miles. The actual ionosphere

and ground reflection conditions over particular propagation paths

are obviously much different from these idealized paths, but our

present model seems better for expository purposes. The principle

of stationary phase (essentially the same as Fermat's principle) leads

to the conclusion that the received waves may be considered to travel

along several discrete ray paths between the transmitter and the

receiver. All of these paths are great circle paths, the shortest

corresponding to the ground wave mode of propagation. The other

paths involve m reflections at the ionosphere, and the waves propagated

along these other paths arrive at the receiving point at successively

later times. By transmitting short pulses, it is possible to observe

these several modes independently at a distant receiving point, and

in this way their physical reality has been verified. The term

"mode of propagation" here, and in the remainder of the ionospheric

propagation discussions, refers to the waves propagated along one of

these ray paths, and has a distinctly different meaning from the usage

in the previous section where the modes of propagation were the

wave guide modes which are simply the successive terms in a mathematical expression for the field. The use of short pulses to make

possible the separate reception of each of these modes is a very

useful device for radio navigation and, in this connection, the estimation of the time of arrival of the successive modes becomes of great

practical importance. These time delays have been studied both

theoretically W III H/ and experimentally 15/ Ikl 11/ 15/ and the

reader is referred to the references for information of this kind; here

we will be primarily interested only in their transmission losses.

32
-33-

MEDIAN TRANSMISSION LOSS OVER LAND AT 100 kc

tr =0.005 Mhos/meter; e = l5 Day h = 70 km ; Night h =90 km

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Figure 20
34

IEDIAN TRANSMISSION LOSS OVER SEA AT 100 kc

cr = 5 Mhos/meter; € = 80

Day h= 70 km; Night h = 90km

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3,000 4,000 5,000 6,000 7,000 8,000 DISTANCE IN STATUTE MILES

10,000

Figure 21
-35-

Case (a) \p Positive and m =3

Case (b) \\> Negative and m = 2 5 k = l dvj/

Figure 22
Consider the phase of a radio wave for a given ray path,

Q{i\i) = 2-rrR/X, where X is the wavelength and R is the total length of

the ray path between the transmitter and the receiver. Now consider

the variation of this phase for all possible adjacent paths between the

transmitter and the receiver as we vary the angle of elevation, ip,

and azimuth, x> i* should be clear (a) that Q{*\>) will be a minimum with

respect to variations in x for the set of rays lying in the great circle

plane, i. e. , for x = an<i (b)» °f this se* there will bem+1 points of

stationary phase, fi t (ib) = 0, for the ray representing the ground wave

and for the m rays reflected at the ionosphere in such a way that the

angle of incidence, <j>, at the ionosphere (for example, at b, d, and f

on Fig. 22) is equal to the angle of reflection, and also (for m > 1)

that the angle of incidence, (90° - ip), at the ground (for example, at

a, c, e, and g on Fig. 22) is equal to the angle of reflection at the

ground. When the angle of elevation, ip, is positive, the waves may

be considered to travel both along the direct ray path, tb, from the

transmitter to the ionosphere and along the ground-reflected ray path,

tab; the reflection points b and f at the ionosphere and c and e at the

ground will be very slightly different for the direct and ground-reflected

ray paths, but this small difference is ignored in our calculations. Note

that the angle of elevation, ip, can be negative as is illustrated on

Fig. 22, case (b).

The following formula may be used to calculate the median transmission loss of an ionospheric mode of propagation involving m reflections

at the ionosphere and a ray path of length, R:

Lm

= L

bf ( R)+ A

t

(

lJJ )+ A

r(^)+ (m - 1) AJip) - C_(R,0.5)+P+mA(c|>,0.5)

g m

(9)

Each of the terms in the above is expressed in decibels; L (R)

bf

denotes the basic free space transmission loss (Set d = R in (3) in

Section 1) for the ray distance, R. We see by Fig. 22 that we may

calculate R as follows:

R & d + d^ + 2m R (ip > 0) (10)

R s d + d + 2m (R -kaip) (ib < 0) (11)

t r om ' ' '

d

t = v(ka tan ip) +2kah-ka tan ty (12)

36
The distance, d , is determined by a formula similar to (12) with h

r t replaced by h ; in this equation ka is the effective earth's radius,

r and k has been chosen equal to 4/3 in our ionospheric examples.

Methods for estimating k as a function of time and geographical

location are given in a later section. It is convenient to choose

several values of i|j at conveniently spaced intervals and then to calculate all of the remaining factors at these particular values of ty.

The space wave radiation factors, A (iM, and A (iM include the

t r gains of the transmitting and receiving antennas, respectively,

relative to that of an isotropic antenna in free space, and allow for

the radiation patterns of the antennas and the loss arising from the

proximity of the antennas to the curved earth. The magnitude of

A (\\t) can be determined from AAty) = L.C^) - L, f (R- + d ), where

t t 1 bf o t L.^) denotes the transmission loss expected for the ground wave mode

propagated between the actual transmitting antenna and an isotropic

receiving antenna placed at the first point of reflection in the

ionosphere, while L, r (R + d ) is the corresponding basic free space

bf o t transmission loss at this distance. Figs. 23 and 24 give typical

values of A (40 expected for short vertical electric dipoles 30 feet

above the ground; in this case we may express A (ip) as follows:

A

t

(i|i) = 20 log

1()

| F| - 1. 761 - 20 log

1() cos i|i - 20 log

1()£(q) (13)

In the above |F| is a "cut-back" factor. When ty is large and positive,

|f| is just jl + R (i|j)| where R is the complex Fresnel reflection

coefficient for plane vertically polarized waves incident on the ground

at the grazing angle ty; when i}j is small or negative, the curvature of

the earth becomes important and the values of | F | have then been

determined by formulas recently developed by Wait. zJj _L~/ The term

1. 761 is just the gain of the short dipole; the term 20 log cos ip allows for the cosine pattern of the dipole; and finally f(q) is the

height gain factor given by equation (19) in reference (11) which allows

for the effect of the height, h of the antenna above the surface. The

, , t "cut -back" factor jF | was calculated for a spherical surface of radius,

a = ka, with k = 4/3; this provides approximately for the effect of

air refraction.

The factor (m - 1)A M) allows for loss on reflection at the

g ground, for example at c and e on Fig. 22 in Case (a) and at c in

Case (b). The amount of this loss will depend on the polarization of

37
\

-38-

oo

Q. CO

<D ±

o

cc

OJc<

o

Q.O

o

co

o

o

CD>D

<D 01

-QT3

O

EFFECTIVE SPACE WAVE RADIATION ATA LARGE DISTANCE FROM

A SHORT VERTICAL DIPOLE 30 FEET ABOVE A SPHERICAL EARTH

Earth's radius 4/3 actual value to allow for the bending near the surface in a standard atmosphere; ground constants o~=0.005 mhos/meter, € = 15

2/3

h > (5/fMc ) kilometers

-10 m

40

35

30

25

20

-10 20 30 40 50 60

Angle of Elevation, \j/, in Degrees

70

-10

Figure 23
-39

O

o

Q. CO

O

c

_a> <

O

T3

"OO

CE

EFFECTIVE SPACE WAVE RADIATION ATA LARGE DISTANCE FROM

A SHORT VERTICAL DIPOLE 30 FEET ABOVE A SPHERICAL EARTH

Earth's radius 4/3 actual value to allow for the bending near the surface in a standard atmosphere; ground constants (J- 5 mhos/ meter, e - 80

r 2/3

h > (5/f,. ) kilometers 'Mc

40

35

30

25

20

20 30 40 50 6(

Angle of Elevation, \p , in Degrees

TO

Figure 24
the downcoming waves. Since the waves reflected from the ionosphere

will have both vertically and horizontally polarized components, even

when the incident waves are linearly polarized, it becomes necessary

to know the relative amounts of energy associated with each polarization in the downcoming waves. This problem is not easy to solve

precisely, and we have obtained an estimate for A (4>) by assuming,

quite arbitrarily, that the energy in the downcoming waves is equally

divided between the two polarizations. Thus, for angles fy > 2°, we

have:

A (i|i) & - 10 log

10 [( |R* | + |k£ |)/2] i|i > 2° (14)

where R and R^ are the complex Fresnel reflection coefficients for

vertical and horizontal polarization, respectively. Since the value

determined by '(14) represents only a few decibels, the use of the

above approximate expression will not lead to serious errors.

It has been shown by Rice 11/ and by Focki^/ that A (0)= 6. 021 db

when ijj equals zero (in the limit as R /X is very large, i. e. , for

2 i-i om

h > (5/f ' ) kilometers) regardless of the polarization or ground

constants; their results can also be used to compute A (if>) for other

values of i{j, but we have instead assumed that A (4>) can be calculated

for tp < by the following approximate formula:

A (iW £ 6. 021 - 2 AJO) + 2 AM) (i|> < 0) (15)

g t t

For values of ^ between and 2°, it is easy to sketch in a smooth

curve between the results given by (14) and (15).

We turn next to a consideration of the convergence factor

C (R, p) which provides a measure of the focusing of the energy on

reflection at the curved surface of the ionosphere exceeded with

probability p. Fig. 25 is a geometrical construction which demonstrates

the nature of this focusing of rays in the vertical plane for ip near zero.

At the antipode of the transmitter, half way around the earth, the rays

are also focused in the horizontal plane. A detailed discussion of this

phenomenon is given in Appendix I. For rays leaving the earth's surface

at grazing incidence (iJj < 0) and at the antipode (m = 90°), it is

necessary to use a wave treatment of the problem, the amount of the

focusing then being a function of the frequency. At points substantially

removed from these caustics, geometrical optics leads to the following

40
-41-

UJ

cj

-z.

LjJ CD

orUJ>

-z. o

UJ L±J X CD

h- 2

cd UJ

z. o

\- o < 2 or2

CD2 o M ^ <

LU or Q CD ^ QT O < H UJ o z 3

or CO

coz 5

or o u_ o o

_J<

CJ

or

h-UJoUJ

CD

!^ <:

Ci <o

S>-

<l> &o

^> ^

CO S ^-«^. ^ c> ^ 1 n C; C* & <0 < ^ <o ^ fc _Q

1th div * «

£3? £

inCO

Z3CD
formula for C (R) which provides for a smooth ionosphere a measure

m

in decibels of the expected increase in the received field due to this

focusing.

r R -« rZmR COS^-k

C (R) = 10 1ogln c = 10 1og_X ; ° m 6 , JQIJtt [• \ r-^ 5-f" 10 m e

10 L

- a sin i|i (d9/dip) J |_ a sin 2m 9 J

(16)

d0

= sin (9 : t) dr

(1?

. dtp cos ip cos. <J> dip

In the above t denotes the total bending of a radio wave in passing

through the troposphere, and methods of calculating T are given in a

later section. The two factors in (16) correspond to the focusing in

the vertical and horizontal planes, respectively. Equation (16) may

be used except near the caustics (ty < 9) and (m 9 = 99°). When vb <

and m 9 ^ 99 #

, we may use:

Cm(R) = 29 log

1() [Rom

- 92.153 i|i (degrees)]+ (19/3) log^f^

- 19 logln [sin 2m (9 - 4/3 M± (49/3)loglrtm - 69. 694 (18)

10 m 10

When ip > 9 and we are at a distance in wavelengths (p/X) from the

antipode, we may use:

Cm(p) - 20 log

10<V**> + " to«M { .,fa"So/d») }

' 10 l°h0 m

+ 10 log

1() f

kc + 10 log

10f J

Q

(2ir cos ty p/X)]

2

+ 36. 172 (i|i > 9)

(19)

The above may be used for m >9 at night since ip is greater than zero

for these modes with h = 99 km. For m < 8, ip < 9 near the antipode

at night, and we may then use the following formula:

Cm(p) = 29 log

1() (Ra

Ara) - (29/3) log^m + (4°/3>l°g

10

f

kc

+ 191og. n [j (2ir p/X)]

2 +44.422 (4*^0)

10 o

(29)

42
In (19) and (20), R denotes the ray distance to the antipode. If a horizontal magnetic dipole is used for reception of the field radiated

from the vertical electric dipole, then the J in (19) and (20) is to be

replaced by Jij here JQ and Jj are Bessel functions. Very near ip = 0,

the values of C (R) determined by (16) will exceed those given by (18),

particularly at the lower frequencies, and in this region a smooth

curve may be drawn between the values for iJj > > and the values

given by (18); similarly, at the antipode the values of Cm(p) given by

(19) will exceed those given by (20), and a smooth curve may be drawn

between the values for 4* > > and those given by (20). Fig. 26 gives

examples of Cm(R) calculated in this way form = 1, 2 and 9 for daytime propagation (h = 70 km), for a smooth ionosphere, and typical

refraction conditions.

It is clear from Fig. 25 that this focusing will be fully realized

in practice only to the extent that the ionosphere presents a smooth

surface to the radio waves. A discussion is presented in Appendix I which indicates how allowance may be made for ionospheric roughness.

It is shown that an individual ionospheric mode of propagation consists

of a steady specularly-reflected component plus a random Rayleigh

distributed component. If we let k denote the ratio of the power in

the random component relative to that in the specularly-reflected

component, then we may estimate the convergence C (R, p) exceeded

100 p% of the time in terms of the values of k (1 - p) exceeded 100(1 - p)%

of the time:

Cm

(R, p) = 10 log

1()

r

c + k (1 - m p)

- 1 + k (1 - p)

(21)

Here c (see 16) denotes the ratio of the received power with and m

without focusing at a smooth ionosphere. As the probability varies

from to 1, k (1 - p) will vary from zero for a smooth ionosphere to

oo for a perfectly rough ionosphere, and Cm(R» p) will vary from the

values given by (16), (18), (19), and (20) for a smooth ionosphere to

zero for a perfectly rough ionosphere. The transmission losses given

in this report correspond to median values, i. e. , p = 0. 5. The random

variable k^ depends upon the radio frequency, angle of incidence, § t and time of day.

As an illustration of the effects of focusing near the antipode

and of the influence of ionospheric roughness, Fig. 27 shows the

43
-44-

CONVERGENCE FACTOR IN IONOSPHERIC PROPAGATION

h = 70 kilometers; N s

= 313 and h s = and k

2

(l)=0

<DO

CD Q

e

o

a// 'i / /

'>/ # //

/// / v/

V / (W i

/ \/

/ V

'1 /

// 7

/x/ /

/ oc V / /

^7 //

/

A/ X

200 300 500 700 1,000 2,000 3,000 5,000 7,000 10,

Distance, d, in Statute Miles

Figure 26
-45-

TRANSMISSION LOSS EXPECTED NEAR THE ANTIPODE

AT NIGHT FOR 100 kc

Over Land = a = 0.005 Mhos/meter, e = 15

L±J Q- Q- 5

X

UJ o

(_> UJ s <

CO £

o o

CO LlI < ^

LT II- UJCD

9,000 10,000 11,000 12,000 13,000 14,000 15,000 I6,(

DISTANCE, d, IN STATUTE MILES

Figure 27
transmission loss expected between vertical electric dipoles in the

range from 8, 000 to 18, 000 miles for 100 kc at night over land. Three

values are shown for each of the modes m = 6, 8, 10, 12, 14, and 16

2 corresponding (a) to a smooth ionosphere (p = 0, k (1) = 0), (b) to an

ionosphere of median roughness (p =0. 5), and (c) to a perfectly rough

ionosphere (p = 1, k (0) = co).

In the immediate vicinity of the antipode, i. e. , within a few

wavelengths, the focusing for a smooth concentric ionosphere is

very large. Thus Fig. 28 shows the value of Cm(Ra , p) expected

right at the antipode at night for a smooth, concentric ionosphere

and for a rough, concentric ionosphere; the values expected for

h = 70 km during the daytime would be only slightly different; Fig. 29

shows the rapid decrease of the focusing as we leave the antipode

and, at distances greater than 100\, the envelope will be just 6 db above

the values of transmission loss shown on Fig. 27, i. e. , the values

shown on Fig. 27 correspond to the single wave expected from a

directive transmitting antenna with an infinite front -to -back ratio.

We see on Fig. 29 that the field expected from the non-directive

dipole oscillates with increasing distance from the antipode; this

oscillation is caused by the interference between the waves arriving

at the receiving point along the short and long great circle paths.

Thus there will be concentric rings around the antipode at which the

expected field will be equal to zero. The radii of these concentric

rings are the same for a concentric ionosphere, regardless of the

number, m, of ionospheric reflections, and are determined by the

zeros of the Bessel functions; for the electric field, the first two

such rings have radii equal to 0. 38X and 0. 88\.

The actual ionosphere will never be concentric with the

surface of the earth. In practice, as the sun rises and sets, or as

the geomagnetic latitude of the reflection point is varied, the surface

of the ionosphere will undoubtedly change in such a way that its

radius of curvature and slope relative to a tangent plane on the earth

will vary over appreciable ranges, and this will cause Cm(R) to vary

up and down relative to the values expected on the basis of the above

analysis. However, except near the antipode, it seems plausible to

assume that the median values of C (R) may not be much influenced

by such changes. The magnitude of the antipodal anomaly will be

substantially reduced by these macroscopic perturbations of the

spherical concentric shell model assumed for our calculations. Also,

for the actual non-concentric ionosphere, the geographical location

of the antipode may be expected to vary with time over a fairly large

46
47-

EXPECTED FOCUSING AT THE ANTIPODE

FOR A CONCENTRIC IONOSPHERE

Ld

CDO

L±J Q

o

rrEo

h = 90KI N s =324

75

70

65

60

55

50

45

40

35

30

25

20

Smooth Ionosphere; p=0

Median Rough Ionosphere; p=0.5

20 50 100 200 500 1000 2000 5000 FREQUENCY IN KILOCYCLES

Figure 28
-48-

THE BEHAVIOR NEAR THE ANTIPODE OF THE

CONVERGENCE FACTORS FOR THE RADIATION FROM A

VERTICAL ELECTRIC DIPOLE FOR A CONCENTRIC

IONOSPHERIC SHELL AROUND A SPHERICAL EARTH

O

<D Q

0.1 0.2

Cm (yo,p)-C m (Ra ,p) =IOIog |0 [j (2 7T/5/X)]

(for reception by a vertical electric dipole) Cm

(/),p)-C m

(R

a

, p) = 10 log|0 [J, [Zirp/\)f

(for reception by a horizontal magnetic dipole) Cm (

yo,p)-Cm (RQ,p) = -15.964-10 log

)0 (/5/X)

(value expected for transmission in one direction around the earth;O.I X< p < 100 miles)

\ /Envelope = -9. 943 -10 loglo (p/X ) for 2\<p < 100 miles

0.5 I 2 5 10 20

Distance from the Antipode in Wavelengths {p/\)

Figure 29
area, and this should result in a net increase in the fading range in

this region. Furthermore, the antipodal locations may be expected

at any given time to be different for the different modes, and thus

the zeros predicted by (19) and (20) and shown on Fig. 29 are not

likely to be observable unless some means is used to exploit the different times of arrival of the individual modes.

Consider next the loss, P, arising from the polarization

characteristics of the downcoming ionospheric waves. An incident

linearly-polarized wave will be reflected as two component waves,

an ordinary and an extraordinary wave, each of which will be elliptically polarized. These two component waves, which will have roughly

the same amplitudes except on frequencies near the gyrofrequency

(near 1. 5 Mc in the United States) will mutually interfere, and this

causes the rapidly varying polarization characteristics of the observed

downcoming waves. The polarization loss, P, arises from the fact

that typical receiving antennas will respond to only one polarization.

The amount of this loss will depend principally upon the transmission

frequency, the penetration frequency for the layer involved, and the

intensity and direction of the earth's magnetic field relative to the

path; it can be calculated if/ 34/ 43/ with some accuracy when these

parameters are known. For more than one reflection at the ionosphere,

the polarization loss becomes a very complex function of the reflection

coefficients for the parallel and perpendicular components of the

incident fields and is difficult to separate from the absorption loss,

A((J>, 1 - p). All of the low frequency examples of ionospheric wave

propagation in this report have been obtained using the empirical

estimates described below for P + A(4>» 1 - p) which thus includes the

polarization loss P; consequently we have calculated the total reflection

loss as m{P + A(cJ>, 1 - p) } - (m - 1)P for m > 1. The calculations at

f = 100 kc have been made in this report by setting P = 3. 01 db, but

this estimate is now believed to be substantially too large, except near

vertical incidence, The calculations for all of the other frequencies

from 20 kc to 1, 000 kc were calculated with P = for all values of m B although this assumption probably leads to somewhat more transmission loss than would be expected for m > 1 since the appropriate value

of P probably lies between and 3 db, approaching the latter value

near vertical incidence; however, we are usually more interested in

the values near oblique incidence for our applications, and this latter

assumption should yield more nearly correct results for the solution

of these problems.

Finally we will consider the loss, A((j>, 1 - p), on reflection

at the ionosphere, exceeded for 100(1 - p)% of the time; this depends

49
-50-

00O

q

<

Q_

MEDIAN LOSS FOR ONE REFLECTION

AT THE IONOSPHERE

cos </>

Figure 30
on the angle of incidence, <}>, at the ionosphere, the radio frequency,

time of day, season of the year, phase of the sunspot cycle and the

geomagnetic latitude of the reflection point. For the examples

developed in this report, we have used some empirical evaluations

of P + A(<}>, 1 - p) made by Belrose 121 on transmission paths between

stations in England, the Scandinavian countries, and Germany.

Using the results in his doctoral thesis, we may express the median

values of P + A((j>, 0. 5) as follows:

P +A(<|>, 0.5) = 17.2 log, n

(f cos <M- 12.4 (70 < f < f ) (22)

10 kc kc D

(NIGHT)

P +A(<j>, 0.5) = 30.8 log, n

(f cos <}>) - 22.6 (70 < f < f ) (23) 10 kc kc D

(FEB. , NOON, SUNSPOT MINIMUM)

P +A(<|>t 0.5) = 33.6 log (f cos <f>) - 22.6 (70 < f <f ) (24)

JLU iCC ICC \-J (FEB. , NOON, SUNSPOT MAXIMUM)

P + A(<|>, 0. 5) = 77. 3 log (f cos (f>) - 64. (70 < f < f ) (25)

1U ICC ICC Ay

(AUG., NOON)

The above formulas were determined empirically from data extending

only over the range of frequencies from 70 - 250 kc and the range of

distances from 350 to 900 miles; a recent analysis of lower frequency

data by Watt, Maxwell and Whelan Z2/ indicates that the absorption

is greater at frequencies less than 70 kc than would be predicted by

the above formulas. Consequently, as shown on Fig. 30, we

have used the theoretical results of Wait and Murphy _' at 20 kc and

then interpolated linearly on a logarithmic frequency scale to obtain

values for intermediate frequencies; the same ionospheric parameters

L/H = 0. 1 by day (i. e. , w/w = 0. 467) and L/H = 0. 05 at night (i. e.

,

u/u) = 0. 3002) were used in these calculations at f = 20 kc as for

r those leading to Figs. 16 to 19, but the index t for the earth's

magnetic field was set equal to 60° in the present calculations^

whereas t was set equal to zero in the calculations leading to

Figs„ 16 to 19.

Figs. 31 to 36 give the median transmission loss expected

in accordance with the above methods of calculation at 20, 50, and

200 kc in over-land and over-sea propagation and for day and night

conditions.

53
54-

MEDIAN TRANSMISSION LOSS OVER LAND AT 50 kc

cr = 0.005 Mhos/meter; £ = 15; h

f = hr =30feet

Day h = 70 km ; Night h = 90 km

CO

UJ

_Jo

Q.a

o

rrh-O

UJ

<O

hLTUJ>

Ia:o

UJ

LU

LU

00Q

UJ

h-OUJ

a.x

UJCO

COo

co

co

<

DC<

3,000 4,000 5,000 6,000 7,000 8,000 DISTANCE IN STATUTE MILES

Figure 33
-55-

MEDIAN TRANSMISSION LOSS OVER SEA AT 50 kc cr= 5 Mhos/meter; € = 80; h t = h r =30feet

Day h=70km; Night h = 90km

co

UJ

_io

Q.Q

O

orh-o

UJ

_lUJ

_l<o

H

oruj>

oro

x

co

UJ

UJ

UJ

00Q

UJ

h-u

U)

0.X

UJco

COo

CO

CO

CO<

o

U)

3,000 4,000 5,000 6,000 7,000 DISTANCE IN STATUTE MILES

10,000

Figure 34
56-

MEDIAN TRANSMISSION LOSS OVER LAND AT 200kc

0"= 0.005 Mhos /meter, € = 15; h t =h r = 30feet

Day h= 70 km; Night h =90 km

CO

Ul

_lo

Q.Q

O

rr

h-o

Ul

_l

UJ

_l<o

H

a:UJ>o

X

in

UJ

ui

Ul

CDQ

Ulo

Ul

Q.X

Ulen

enQ

CO

CO

<

<

a

ui

1,000 2,000 3,000 4,000 5,000 6,000

DISTANCE IN STATUTE Ml

7,000

LES

1,000 9,000 10,000

Figure 35
57-

MEDIAN TRANSMISSION LOSS OVER SEA AT 200 kc

cr = 5 Mhos /meter; € = 80; h

t = hr = 30 feet Day h = 70km; Night h = 90km

en

UJ

_1o

Q.Q

O

CCHo

UJ

<o

IrrUJ>

rro

UJ

UJ

UJ

DDQ

UJ

I-o

Ul

0.X

UlCO

eno

o

en

en

<

or<

3,000 4,000 5,000 6,000 7,000 DISTANCE IN STATUTE MILES

9,000 10,000

Figure 36
Figs. 31 and 32 show the large decrease in the transmission

loss near the antipode, and illustrate the fact that each mode of

propagation has two important branches at points somewhat removed

from the antipode, corresponding to propagation via the short and long

great circle paths, respectively; actually there are still other branches

for each mode corresponding to propagation more than once around the

earth, but these are not shown* The decrease in transmission loss

shown at the antipode is the value for an idealized concentric ionosphere and will, in practice, undoubtedly be somewhat smaller.

Transmission loss curves for the separate modes are not given at

20 kc for sea water since they differ so little from those for overland. It should be noted that the curves on Figs. 16 and 17 for 20 kc

will be more useful for most applications than those on Fig. 31 since

they combine the separate modes with proper relative phases.

Throughout this section the formulas and graphs refer to the

median values of transmission loss for individual modes of propagation. This form of presentation was used since it is more useful in

applications such as the design of navigation systems or of systems

to avoid multipath distortion. To determine the expected median

transmission loss for a continuous wave transmission, it is necessary

to convert the transmission losses for the individual modes to power

ratios, and then add these power ratios; at 500 kc, and possibly even

as low as 50 kc, it is reasonable to assume, that the several ionospheric

modes will have random relative phases so that the median power of

the resultant will be equal to the sum of the median powers of the

individual modes. For example, if there were two modes with equal

median transmission losses, the median transmission loss for the sum

of these two modes would be 3 db less, and for three equal modes the

sum would have 4. 77 db less transmission loss than each individual

mode.

The author has studied the behavior of P + A( <f>, 0. 5) at night

in the United States for frequencies in the standard broadcast band

from 500 to 1, 500 kc over a very wide range of distances, and has

found the following semi -empirical formula:

P t AU, 0. 5) = 26cOS

^

4

(26)

(f COS (j>) mc

(NIGHT)

58
Note that (22) indicates an increasing loss with increasing frequency,

presumably because of a deeper penetration of the D layer as the

frequency is increased, while (26) indicates that the loss decreases

with increasing frequency. At the higher frequencies where (26) was

established, the waves penetrated the D layer and, as shown by

MartynJLi/ this behavior of P + ($, 0. 5) with frequency and angle of

incidence is to be expected. By virtue of the method used for its

determination, (26) includes the polarization loss P; since the extraordinary waves are much weaker than the ordinary waves in this

frequency range, there will be additional polarization loss at each

reflection from the ionosphere. With the above discussion in mind,

it seems appropriate to assume that the penetration frequency of the

D layer at night is effectively defined by the following relation:

P + A(«>, 0. 5)

6) = P + A(4>, 0. 5) (27)

(At the D layer penetration frequency, f , at night)

As determined in this way, the D layer penetration frequency at night

varies from about 500 kc at vertical incidence to about 250 kc with

cos <}> = 0.164, the minimum value expected for a 90 km layer height;

the anomalous behavior of this penetration frequency suggests that

neither of our empirical absorption formulas are very dependable in

this intermediate range of frequencies. Since nothing better is readily

available, it was decided to calculate the transmission loss at night at

frequencies greater than the above -defined D layer penetration frequency by using m 26 cos cb/(f cos (j>) °* ** as the total loss on Mc

reflection; the reflection height was assumed to be 110 km at night, but

the value of cos <(> to be used in the absorption equation was determined

on the assumption that the absorption takes place at a height of 100 km,

Figs. 37 to 40 give the median transmission loss expected

between short vertical electric dipoles at 500 kc and at 1, 000 kc in

over-land and over-sea propagation and for day and night conditions.

The absorption at night was determined by (26) as described above,

but, .in the daytime, (24) and (2 5) were used since radio waves in

this frequency range are then presumably reflected and absorbed by

the D layer at an assumed height of 70 km,

At still higher frequencies during the daytime, the radio waves

will penetrate the D layer and be reflected by the E layer at a height

of about 110 km. The ionospheric absorption is so great during the

daytime in the range of frequencies from, say 500 kc to 4 Mc, and

59
-60-

MEDIAN TRANSMISSION LOSS OVER LAND AT 500 kc

cr = 0.005 Mhos/meter , € = 15 ; h t = h r = 30 feet Day h = 70 km ; Night h =1 10 km

CO

UJ

_1o

a.Q

o

cc

h-oLlI _1

UJ_l<

O

H

CC

LxJ >oXCO

LU

UJm

Q

LUH

o

Ld

Q_X

LU

CO

COo

o

CO

CO

CO

-z. <

cc<

a

Ld

1,000 2,000 3,000 4,000 5,000 6,000 7,C DISTANCE IN STATUTE MILES

Figure 37
-61-

MEDIAN TRANSMISSION LOSS OVER SEA AT500kc

0"=5 Mhos/meter, 6=80,- h =h =30 feet Day h = 70 km , Night h = 110 km

CO

LU_JoQ-Q

O

cr

LU_l

LU

UJ>o

LU

UJ

ssLU

CDQ

U

I-o

UJ

a.x

UJ

COCOo

o

CO

CO

CO

-z. <

cr<

Q

UJ

3,000 4,000 5,000 6,000 7,C DISTANCE IN STATUTE MILES

9,000 10,000

Figure 38
-62-

MEDIAN TRANSMISSION LOSS OVER LAND AT IOOO kc

cr = 0.005 Mhos/meter, e=l5 ; h

t = h r =30 feet Day h= 70 km , Night h = 1 10 km

o

Q.Q

O

rrH

O

u

_l

UJ

_l<o

h-UJ>

H

CCo

x

en

UJ

UJ

UJ

CDQ

UJH

o

UJQ.X

CO

COo

o

CO

CO

<

ir<

Q

UJ

3,000 4,000 5,000 6,000 7,0 DISTANCE IN STATUTE MILES

8,000 9,000 10,000

Figure 39
63-

MEDIAN TRANSMISSION LOSS OVER SEA AT IOOOkc cr= 5 Mhos/meter; €=80, ht = h r =30feet

Day h= 70km; Night h=IIOkm

CO

UJ

_lO

Q-Q

O

rrI-o

UJ

<o

H

trUJ>

oro

x

in

UJ

UJ

UJ

CDo

UJH

CJ

UJ

0.X

UJCO

COo

CO

CO

<

<

3,000 4,000 5,000 6,000 7,000 DISTANCE IN STATUTE MILES

10,000

Figure 40
the reflection phenomena so complex, with reflections taking place

sometimes at the D layer, sometimes at the E layer and sometimes

at the F layer, that useful, simple absorption formulas are not

available. For this reason the calculations of transmission loss

given in this report for this range of frequencies have been made by

extrapolating (23), (24) and (25) to higher frequencies, and by extrapolating to lower frequencies the absorption formulas applicable to

the high frequency band as discussed in the next section.

Figs. 41 to 45 give transmission losses for propagation over

land, based on the above -de scribed methods of computation and on the

methods described in following sections, and are designed to show more

clearly the effect of radio frequency for day and night, for two seasons

and for minimum and maximum sunspot conditions. Only one set of

curves are presented for propagation at night since the seasonal and

sunspot cycle effects on the transmission loss are comparatively small

at night. The curves on these figures give the transmission loss

expected between short electric dipole antennas, oriented vertically at

frequencies less than 5 Mc and horizontally for frequencies greater

than 5 Mc, for the ground wave and for the particular sky wave mode

with a minimum transmission loss at the distances 200, 500, 1, 000,

2, 000, 5, 000 and 10, 000 miles. At each distance we have shown only

the value expected for the single sky wave mode with the minimum

transmission loss; with continuous wave transmission, the losses would

be several db less than these values, particularly at the larger distances

where several sky wave modes with comparable intensities are expected.

Note that ground proximity losses L and L , as discussed in Appendix III, have been omitted in calculating the values shown on Figs. 41

to 45.

64
65-

MEDIAN TRANSMISSION LOSS FOR MIDNIGHT AT THE RECEIVING ANTENNA

February j Sunspot Maximum, a" =0.005 Mhos/meter,€=l5j hf=hr = 30feet

West to East Transmission Path with Washington, DC. at the Midpoint

co

UJ_i

oQ_O

or

\-

o

UJt-O

X

CO

UJ

Ul

CDa

ui

t-oUl

0_X

Ulto

coQ

O

co

CO

CO

z.<

or<

Q

ui

1

I M'TUTT D layer MUF

E layer MUF

E layer penetration freq. for F2 F2 layer MUF

Change of number of hops

05 I 2

MEGACYCLES

Figure 41
-66-

MEDIAN TRANSMISSION LOSS FOR NOON AT THE RECEIVING ANTENNA

February; Sunspot Minimum; O" =0.005 Mhos/meter; € = 15; hf=h r = 30feet

West to East Transmission Path with Washington, DC. at the Midpoint

CO

LJ

_l

o

Q.O

o

ce

h-o

LlI _lUJ

h-

(ro

x

CO

UJ

UJ

DQQ

UJO

LU

a.x

UJCO

COo

o

CO

CO

CO<

or

h-z

<

Q

UJ

50 100
-67-

MEDIAN TRANSMISSION LOSS FOR NOON AT THE RECEIVING ANTENNA

February ; Sunspot Maximium 5 C= 0.005 Mhos /meter , € = 15 ; h

t = h

r = 30 feet

West to East Transmission Path with Washington, D.C. at the Midpoint

COLU

_Jo

g_a

o

or

I-o

UJ

_l

LUH

rro

x

CO

LULU

LU

00oLU

h-o

LU

Q_X

LU

CO

COO

O

CO

CO

CO<

DC<

Q

LU

iiiiir, 'ii mm —: Ground Wave — Ionospheric Wave

•- Ionospheric Scatter

ii i '. ! i,i' mam »,i,| hi, iiiiiiiiiii mini D layer MUF

E layer MUF

E layer penetration freq. for F2 F2 layer MUF

Change of number of hops
-68-

MEDIAN TRANSMISSION LOSS FOR NOON AT THE RECEIVING ANTENNA

August; Sunspot Minimum; O" =0.005 Mhos /meter ; 6 = 15 5 h

t = h

r = 30 feet

West to East Transmission Path with Washington, DC. at the Midpoint

' 'i'".ri'r i i"' ' .. I I. :

. l

' I

</) UJ

_l

e

Q

O

cnt-o

UJ_J

UJ

hrro

x

Ul

UJ00a

X

C/) o

o

in

-z. <

or<

Q

ui

— Ground Wave

Ionospheric Wave

I' M"' I I H I ' .IN. ..I.I IITTtTTtUT D layer MUF

x E layer MUF

® E layer penetration freq. for F2 o F2 layer MUF

|A Change of number of hops

10 20 50 100

MEGACYCLES

Figure 44
-69-

MEDIAN TRANSMISSION LOSS FOR NOON AT THE RECEIVING ANTENNA

August; Sunspot Maximum; o~=0.005 Mhos/meter; €=15; h

t =h r =30feet

West to East Transmission Path with Washington, D.C. at the Midpoint

CO

Ul

_l

o

Q.O

O

CC

h-O

LU

_l

LjJ \-

o

X

CO

LU

UJ

CDQ

UJ

h-o

Ul

Q.X

UlCO

COo

CO

CO

CO<

a

UJ
4. 3 High Frequency Ionospheric Propagation

We see on Figs. 41 to 45 that the variation of transmission

loss -with frequency changes as the various layers of the ionosphere

are penetrated. The method of determining the D layer penetration

frequency was described in the preceding section, and the penetration

frequencies for the higher layers were determined by the methods

described in references 24 and 25; for this purpose the propagation

path was assumed to be from West to East with its midpoint at

Washington, D. C. , and with either noon or midnight at the eastern

end of the path.

During the daytime the polarization and absorption losses in

the high frequency band have been calculated by the methods described

in a Signal Corps report. Z~J Thus the constant attenuation of 8. 9 db

found in their analysis has been somewhat arbitrarily attributed to a polarization loss P, and the following semi-empirical formula used

for calculating the daytime absorption:

A(+, 0.5)^ 615- 5 {cos (0.881 x)}

1^^ 0.0037 s) (DAy) (2g)

< fMc + V COS *

In this formula x denotes the zenith angle of the sun at the reflection

point, and s denotes the smoothed Zurich sunspot number; for sunspot

minimum, s was set equal to 10, and for sunspot maximum, s was

set equal to 150. The gyrofrequency, f = 1. 5 Mc, on the average in

the United States. The angle of incidence, (J), to be used in (28) refers

to the value this angle will have at the absorption level, and this angle

will be systematically larger, for a given angle of elevation iJj , than

the angles of incidence at the higher layers where the reflections take

place. The Signal Corps analysis was based on the assumption that

the absorption takes place at a height of 100 km and this same height

was used in our calculations. In the Signal Corps report, convenient

graphical methods are given for determining many of the factors

involved in calculating (28). Prof. A. Kazantsev sZJ zR/ has proposed

a method for calculating A(cj), 0. 5) which, in essence, involves the

replacement of the numerator of (28) by a constant times the square

of the penetration frequency of the E layer at vertical incidence; this

method appears to have considerable merit, but a critical determination of its accuracy compared to that of the Signal Corps method has

not yet been published.

70
Thus, during the daytime we have used (9) for calculating the

median transmission loss in the high frequency band with P = 8.9 db

and A(fj), 0. 5) determined by (28). Note that the focusing is probably

negligible in this frequency range since the ionosphere will very likely

appear rough to these radio waves most of the time; in the absence

of quantitative information on ionospheric roughness at these higher

frequencies, we have arbitrarily set C (R, 0. 5) = at all distances

m

for f > 2 Mc both day and night.

At night the absorption in the high frequency band is quite

small; it has been estimated in this report by means of (26). We have

already noted, however, that (26) includes a polarization loss, P, and

the Signal Corps report indicates, in effect, that the absorption plus

the polarization loss at night is equal to 8.9 db; thus it appears to be

appropriate to use (9) for calculating the transmission loss with

P + A((}), 0. 5) calculated by (26) for the lower frequencies where

m{p+A(<t>, 0. 5)} > 8.9 db and to set m {P + A(4>, 0. 5)} = 8. 9 db for

all higher frequencies. This is the method used for the examples

presented in this report.

4. 4 Ionospheric Scatter Propagation

At frequencies above the penetration frequency of the E layer t the radio waves are scattered forward with sufficient intensity to be

usable for communications over distances of the order of 600 to

1,400 miles, zll —' —-J The transmission losses shown on Figs. 41 to

45 for this mode of propagation are based on the measurements

reported by Bailey, Bateman and Kirby_' for the Fargo, North Dakota

to Churchill, Manitoba path as extrapolated to other distances and

frequencies by means of a theory developed by Wheelon. _/ _' z2l

Thus Wheelon attributes the scattering to turbulence in the D and E

regions of the ionosphere and, on the assumptions (1) that the spectrum

of this turbulence may be determined by the mixing in gradient

hypothesis and (2) that the viscosity cut-off has a characteristic scale

I = 1. 5 meters, is able to develop a formula for the transmission

loss expected with this mode of propagation.

Wheelon' s analysis leads directly to the spectrum of the

turbulence, but the turbulence may also be characterized in the range

of wave numbers smaller than the viscosity cut-off by the correlation

function (r/l )K.(r/i ) which describes the degree of correlation in

o \ o

the fluctuations in electron density at points a distance r apart; K_

71
denotes the modified Bessel function of the second kind and t is a

o characteristic scale of the turbulence, set equal to 100 meters in our

subsequent analysis. It is interesting to note that this same correlation function is applicable for describing tropospheric turbulence as

well. 57/58/

The following formula gives the median transmission loss

expected for the ionospheric scatter mode of propagation, i.e., on

frequencies above the effective maximum usable frequency, f , of

the scattering region:

L

ms

= L

bf(R) + A

t

(4;) + A

r

(+) " S( °- 5) " 10 1Og

10

S6C *

+ B(k

2

, I , I ) + P + A(4>. 0. 5) (f > f ) (29) o s Mc MUF

In the above, L. (R) denotes the free space transmission loss for waves

bi

traveling a distance corresponding to an average scatter path of

length R; this path length has been determined in this report on the

assumption that the mean layer height h = 87 km both day and night.

For horizontally polarized waves the following formula may be used to

estimate the median value for the sum of the space wave radiation

factors:

Am + A (4») = - 20 log1ft [2 sin (2ir h sin i|j A)]

t r 10 t

- 20 log [2 sin(2ir h sin ^/M] - G (0. 5) (Horizontal polarization)

10 r p

(30)

In the above, h and h denote the heights of the transmitting and

receiving antennas above the local terrain, and G (0. 5) denotes the

p median path antenna gain. For the high gain antennas normally used

for communication by scatter, there will usually be a substantial

"loss in gain" relative to the value G would be expected to have for

P communication between similar antennas in free space. For example,

on the Fargo to Churchill path, G as determined for successive halfhour periods of time, was found to be a random approximately normally

distributed variable with a median value G (0. 5) = 25. 7 db and a standard

deviation of 5. 85 db; the sum of the free space gains in this case was

about 40 db. It has been found that some of this "loss in gain" can be

recovered by directing the antenna towards the better scattering

regions. The values of transmission loss shown on Figs. 41 to 45 are

72
for propagation between short horizontal electric dipoles, and in

this case G = 3. 52 db. The antenna heights, h and h , were taken

p , t r to be 30 feet except at the higher frequencies where somewhat lower

values of h and h were chosen so that 4 h sin ijJ/X. = 1; this choice

t r t, r of h and h effectively minimized the loss and in this case

t r

y A {$>) + A (i|j) = - 12. 041 - G . Near the maximum effective range for

r co/ p

ionospheric scatter, zz! the transmission loss increases so rapidly

with increasing distance that it is useful in some cases to use very

large antenna heights so as to increase the range slightly by the amounts

indicated by (12).

The factor S involves the intensity and scales of the turbulence

and, together with G , exhibits most of the variability of the transmission loss. The median value S(0. 5) undoubtedly varies somewhat

diurnally, seasonally, and with the sunspot cycle but, since such

changes are not large with a probable extreme range of the monthly

medians at a given time of day of less than 20 db, we have calculated

all of the examples in this report by setting S(0. 5) = - 8.4 db, the

value obtained for the Fargo-Churchill path. An analysis is presented

in Appendix II which shows that this value of S(0. 5) is not inconsistent

with what is presently known about ionospheric turbulence.

The factor 10 log sec <j> provides a measure of the size of the

effective scattering volume for transmission paths of various lengths.

2

The transmission loss factor B(k , £ , i. ) may be expressed?

o s B(k

2

, I , I ) = 25 log [1 + k

2

|

2

] + 20 log [1 + (k

2

i 2

) 2/3 OS ] 1U O 1U S

+ (40/3 1og

1() [l + (k

2

/)

2

] (31)

Although the characteristic scale lengths, i and I , are likely to be

o s somewhat variable diurnally and seasonally, we have, for the purpose

of the calculations in this report, taken them to be equal to the constant

values I - 100 meters and 1 = 1. 5 meters; k is defined as follows:

o s

2

k s —47r cos 9

f2
2

When f is equal to the maximum usable frequency, f, .„_» k = 0, MUF

the wavelength in the medium increases without limit, and the scattering is no longer directed forward, but occurs uniformly in all directions.

In this limiting case, B(k , i . I ) - and (29) indicates a scatter

o s loss exceeding that predicted by (9) for normal E layer propagation

at the MUF by only (8.4 - 10 log sec <J>) decibels. For the calculations

in this report, we have somewhat arbitrarily used the median E layer

MUF as a measure of the MUF of the scattering region.

For ionospheric scatter, we have taken P = 3 db for both day

and night propagation conditions. During the day, the absorption term

A(4>, 0. 5) was computed by (28), but at night P + A(<}>, 0. 5) was

determined by (26) up to frequencies for which the resulting value is

greater than 3 db, and at higher frequencies P + A(c(), 0. 5) is set

equal to 3 db.

There is no present evidence for the existence of F layer

ionospheric scatter or for multi-hop E layer scatter and, for this

reason, the transmission loss curves for d = 2, 000, 5, 000, and

10, 000 miles stop abruptly at the MUF; the transmission loss is

expected to increase very rapidly indeed at frequencies just above

the F layer MUF.

5. The Bending of Radio Waves by the Troposphere

Since the density as well as the absolute humidity of the

air decrease with the height, h, above sea level, the refractive index,

n, also usually decreases with h, and this causes radio waves leaving

an antenna at a given angle, i{j, to bend down towards the earth, the

amount of this bending being larger, the smaller the value of i|i. This

is illustrated on Fig. 46 which shows the total bending, t , of a radio

wave traveling entirely through the troposphere and subsequently being

reflected at an ionospheric layer.

Since the refractive index, n, departs from unity by only a

few parts in 10" , it is convenient to describe n in terms of the

refractivity, N, which is defined:

N = (n - 1) X 10 (33)

If the value of N were known as a function of time at every point in

the atmosphere between two radio antennas, it should be possible, in

principle, to predict the instantaneous behavior of the transmission

74
75-

<

^.c

i /

«&

<£.

*\

%#

a = 3960miles

ftr/,U

°">e,&/?/ Of

/onos.

'°ot'°s< lOA•er/c »4W/Jo 'tyer

R,/2

*#.

Figure 46
loss in propagation between these antennas. Actually, of course,

this is not feasible because of the complexity of the solution of such

an electromagnetic problem. Furthermore, even if the engineer could

be provided with this instantaneous information, he would normally be

forced to describe it in some statistical terms before he could use it

effectively in the design or use of radio systems. Consequently, we

are led to the description of N in statistical terms, with the hope that

these statistical characteristics of N may be used for the prediction of

the more important statistical parameters describing the transmission

loss. At a given instant N will vary considerably with height above the

surface, and, to a lesser extent, with distance along the path. If,

however, we average the values of N over a period of an hour, then N

will, on typical propagation paths, be more nearly constant along the

path at a given height, but will normally decrease monotonically with

increasing height above the surface. For the solution of most radio

prediction problems, it is permissible and, indeed, desirable to

average N over still longer periods of time in order to obtain mean

conditions useful in radio systems design. Thus, for predicting the

diurnal, seasonal, or geographical variations of the median transmission loss, we may further average the values of N as determined from

day to day over a period of many years for a particular hour of the

day, month of the year and geographical location. The resulting values

N will vary still less along the path and, on most paths, N, for a given

time of day and season of the year, may be taken to be a function only

of the height, h - h , above the earth's surface where h represents the

height, expressed in kilometers, above sea level, while hg represents

the height of the surface above sea level. There will be some paths

for which N will also vary appreciably in the horizontal plane; examples

are paths with one terminal over land and the other over the sea, and

these will undoubtedly require special treatment. Since the average

values N tend for most paths to be very nearly horizontally homogeneous,

it should only be necessary to know the vertical profile of N at one point

along the path for a successful prediction of the median transmission loss

at a particular time of day and season of the year. For very long paths

on which N does vary appreciably along the path, we may base our radio

predictions on the two N profiles at the intersections of the two radio

horizons with the great circle path.

From the above discussion it appears to be desirable to study

these N profiles, and it will be convenient in the following analysis to

omit the superscripts and simply let N(h) denote these long-term average

values. The most generally reliable single parameter for the description of the profile as it affects radio propagation is the difference, AN,

76
in the values, Ni, at a height of one kilometer above the surface and

Nc the value at the surface:

s AN = N(hs + 1) - N(hs

) = Nx

- Ns (34)

Note that AN is a negative quantity. Most of the diurnal, seasonal,

and geographical variations in propagation between antennas at heights

of less than one kilometer above the surface may be predicted on the

assumption that N(h) decreases linearly with height above the surface

up to a height of one kilometer:

N(h) = N + AN(h - h ) fh <h<h +1] (35)

s s s s

For radio propagation predictions at the higher frequencies

above, say, 50 Mc, the above assumption of linearity for the initial

decrease of N(h) with height is not adequate for some times of the

day or for some geographical locations; in some of these special cases

there may be ducting with a resulting substantial increase ~21 in the

transmission loss for paths just short of the radio horizon, and a very

large decrease !l!l/ in the transmission loss on paths just beyond the

radio horizon. Since these appreciably non-linear profiles occur in

only a very small percentage of all cases, zll we will not consider them

further in this survey report. Furthermore, although the principles of

duct propagation are well understood, zzJ zdl 2z/ there are nevertheless

no very satisfactory formulas for predicting the transmission loss for

the large variety of non-linear profiles typically encountered in practice,

The assumption of a linear profile makes possible the

introduction of a great simplification in radio propagation predictions.

Thus it has been shown z2J that the behavior of radio waves in an

atmosphere with a linear gradient is the same as that expected with

no atmosphere for an earth with effective radius a = ka' where a

1 denotes the actual earth's radius, expressed in kilometers, and a is

defined by:

1 -l l AN , a" ka7

=

a"' + =F~ (36) a ka a

(1 + N .10

b

) s

Bean and Meaney —' demonstrate that there is a high correlation

between the monthly median transmission loss and the monthly median

values of AN, and give maps of the monthly median values of AN for

the United States for several months of the year. The values of AN

77
must be determined from radio-sonde observations and are, as a consequence, not as readily available at all hours of the day nor for as many geographical locations as the surface values, N s

. Fortunately,

AN may be predicted zJJ with quite good accuracy from N and, in the

absence of observations of AN, the following empirical formula may

be used to determine the predicted value AN':

AN1 = - 7. 32 exp{0. 005577 Ng

} (37)

If now we combine (35) and (37), we have the following expression for

the initial behavior of N(h) in what will be referred to as the CRPL

Standard Radio Refractivity Atmosphere as recently proposed and

studied by Bean and Thayer: zJJ

N(h) = N - (h - h ) 7. 32 exp{0. 005577 N } [h < h < h +1] (38)

s s s s s Note that the only parameters in (38) are the surface refractivity, N » and the height, hg

, of the surface above sea level.

Note that (37) may also be used to predict N in terms of known

values of AN:

N' = 412.87 logirt (-AN) - 356.93 (39)

s 10

When values of AN and of N are both available, and when the actual

b

value of Ng differs from the value predicted by (39), it is better to

use N' for predictions when i|j < 3 ° rather than the actual value of Ng

; in other words, AN is slightly better than N as a predictor of propagation conditions for small values of i|^. On the other hand, it is at

present easier and usually also more accurate to predict N for some

particular time of day, season of the year and geographical location,

and then use (37) for determining AN', than it is to use the available

maps —' directly for the prediction of AN. It should also be noted that,

even when AN is available, N_ is a better predictor zli of the bending

at high elevation angles: iji > 3°. It is expected that the recent study

program —I proposed by the International Radio Consultative Committee (CCIR) will tend to expedite the gathering of the data on AN

required for the development of suitable prediction methods; however,

it is unfortunate that emphasis was given in that proposal to the

gathering of data at only two hours of the day, 0200 and 1400 U. T. , since it is precisely the large diurnal variation of AN, occurring at

many locations, which is most difficult at present to predict with

adequate accuracy.

78
The effect on Ns of the height of the surface above sea level

may be determined from the relation:

Nso= N exp(-c h ) (40)

s s

where c

g = 0. 1057 /kilometer = 0. 1701 /statute mile = 0. 03222 /thousand

feet. Bean and Horn zJJ give maps which may be used to estimate the

value of N averaged throughout the day for the months of February

and August at any geographical location in the world, together with a

map of the annual range of Ns

. A comprehensive climatological study

of N8 for the United States is in preparation at C. R. P. L. ; this study

will make available charts useful for the prediction of NQ (and thus

of N by means of (40) above) at 0200, 0800, 1400 and 2000 for

s February, May, August and November and, in addition, gives the

detailed statistical characteristics of N at several representative

weather stations in the United States.

Summarizing the above, we see that the average value of N

up to a height of one kilometer above the surface may be predicted

preferably, when ip is small, in terms of a measured mean gradient,

AN, or, alternatively and with only slightly less accuracy, in terms

of the mean value of the surface refractivity, N

g

. Above one kilometer,

N decreases exponentially with height, and Bean and Thayer —/ give

the following formulas for N(h) in this range

:

N(h) = N- exp[-c. (h - h -1)] (h +1 < h< 9km) (41)

1 i s s

C

i = TT1T l0g

e <V105> < 42 > s

N(h) = 105 exp[- 0.1424 (h - 9)] h > 9 km (43)

Since the constant c depends only on N and h (see Table 5.1 below),

it appears that the mean atmosphere may be described in most cases

in terms of these two parameters or, alternatively, in terms of h

and Ng when AN is known. Note that hg influences the description of

the atmosphere [see (42)] only in the range from one kilometer above

the surface to 9 km above sea level, and then only slightly for the

range of values of h normally encountered in practice; consequently,

we may, for practical purposes, consider that the atmosphere is well

defined by the single parameter N . The success of tide model in

79
Table 5.1

Constants for the CRPL Reference Atmospheres

h

s a' a

c.

l N

s feet MILES -AN' k MILES per kilometer

3960 1 3960.00

200 10, 000 3961.8939 22. 3318 1. 16599 4619. 53 0. 106211

250 5, 000 3960.9470 29. 5124 1. 23165 4878. 50 0. 114559

301 1, 000 3960. 1894 39.2320 1. 33327 5280.00 0.118710

313 700 3960. 1324 41.9388 1. 36479 5404. 57 0. 121796

350 3960 51. 5530 1. 48905 5896. 66 0.130579

400 3960 68. 1295 1. 76684 6996.67 0. 143848

450 3960 90. 0406 2. 34506 9286.44 0. 154004

predicting the bending of radio waves has been examined by Bean

and Thayer zll and leaves little to be desired except in the small

percentage of cases involving non -linear profiles.

For some mathematical analyses of radio propagation, the

above -described CRPL Model Radio Refractivity Atmospheres have

the undesirable characteristic of having discontinuities in the gradient

at one kilometer above the surface and at 9 km above sea level. The

following exponential model is free of this defect and, although it does

not fit the meteorological data above one kilometer as well as the CRPL

Model Radio Refractivity Atmospheres, it does nevertheless provide a

representation useful for many applications:

N(h) = N exp[-c (h - h )1 s e s (44)

The CRPL Model Radio Refractivity Atmospheres have arbitrary

values of N

g

, h

g

, and AN; in the CRPL Standard Radio Refractivity

Atmospheres N_ and AN are related by (37) and (39), but h is arbitrary;

s s and in the CRPL Reference Radio Refractivity Atmospheres N

g

, AN,

and h

g

have the values given in Table 5.1.

80
The constant c is defined in terms of AN and N : e b

exp(-c ) = 1+ |r (45)

e N

8

Table 5, 2 gives the values of c for several values of AN for the

particular case in which N' is determined by (39)} the value N = 313

represents the average of the observed values of Ng in the United

States.

Table 5.2

Typical Constants c for CRPL

e Standard Exponential Radio Refractivity Atmospheres

N(h) = N exp[-c (h - h )]

c

e AN N' per kilometer

s

22.3318 200 0.118400

29.5124 250.0 0.125625

30 252.9 0.126255

39.2320 301.0 0.139632

41.9388 313.0 0.143859

50 344.5 0.156805

51.5530 350.0 0.159336

60 377.2 0.173233

68.1295 400.0 0.186720

70 404.9 0.189829

90.0406 450.0 0.223256

81
-82-

UJoz

UJ

a:

LU 00

U. ID UJ

uj a> cc

cr t UJ 1 X

-1 CO Q_

Q. UJ </)

<Z (Z

O

O UJ 2

"T 1-

UJ Q_ <

•- o 1- zS cr — H <

CO< UJ

or tz

ro >

u_

— UJ

ISON

EFRACT l-<

cr Q- <

Q-Soo * o o o O o o * o CD CO 00 <sf CO C\J TM

133d JO SQNVSD0H1 Nl S

M-M
Fig. 47 compares the paths followed by radio rays leaving the

earth at several selected elevation angles, v}j , for several of the CRPL»

reference atmospheres with the paths expected in a four -thirds earth

atmosphere. Thus, the graph paper used for tracing these rays was

so designed that they are straight lines for the linear gradient atmosphere corresponding to an effective earth's radius of 5,280 statute

miles, i.e., (4/3)» 3, 960 miles. Note the very large departures at

large heights of the rays in all of these representative atmospheres

from the rays in the usually assumed four -thirds' earth atmosphere.

Note that there are large departures from the four-thirds' earth

atmosphere at large heights for N = 301, even though the bending in

this atmosphere is correct for heights h - h_ less than one kilometer.

s An appropriate allowance for this difference in bending is made in the

tropospheric curves presented in subsequent sections of this report,

but no such allowance was made in preparing Figs. 3, 4 and 6. A good

estimate of the correction which should be made to the altitudes of the

contours on Figs. 3 and 4 is simply the difference in heights, at the

appropriate range, between the 4/3 earth rays and the N

g atmosphere

rays. For a detailed discussion of such corrections with appropriate

graphs, see a recent report by Rice, Longley and Norton. ±zJ

The Bean and Thayer report gives the elevation angle error

€ = vji - ifj as a function of electrical path length, R , for rays in the

o

TM reference atmospheres as well as AR = R - R . (Rj/2)

R

e

= \ndR=ct (46)

where c is the velocity of light in a vacuum and teis the time of transit

along the ray path.

5.1 The Influence of Tropospheric Bending on

Ionospheric Propagation

The bending of the radio waves by the troposphere has the

effect of extending the range of ionospheric propagation and, although

the effect is small, it is not negligible near oblique incidence. Table 5.3

shows the influence of N on R-, d_, and cos <f> for ionosphere heights

h a 70, 90, 110, 22 5, 350, and 475 km. The calculations were made by

tracing rays in the CRPL reference atmospheres with N8 = 0, 301 and

400. Since the variations with N are comparatively small, it would

83
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I WJ V_t I—I vu VI 1^/ I—i |— M_J |— |— m coHui\uo(n<-^^i lHln^oon4''0^^coN^ v a)rvj^^o(nmHHvuinin^iv (M^uinr-H(\irirvjo^[ti'-iLrniaj>r'-'iNi(, i o 4- mHco>o M-j^coH^ivOfM^HOocOHvOJt cocOf-icor--4- Oi-HOcMvOc\j(^ooomm^H o so to ^^H^I^^(<,llnc^JtnHcnH(^JHt^tH(\lHHO^HCDH^HOcoln^ln>Oln4'^cocommMH(^lcoHm 1

92
appear to be reasonable for most applications to use only the single

value N = 301 in ionospheric calculations, and the remaining tables in

this section give the convergence factor c for several angles, 4* » and

for m = 1 to 20. The values of c are given since they involve the

geometry in a rather complex way; however, since the ionosphere

probably appears rough to the radio waves reflected at the higher

layers, c is given only for h = 70, 90, and 110 km. m

The results presented in this section were all obtained by ray

tracing methods, and such methods yield reliable results only when the

following two conditions are satisfied: (a) the index of refraction, n,

must not change appreciably in a distance equal to a wavelength, and

(b) the fractional change in the spacing between neighboring rays in a

wavelength along the ray must be small compared with unity. Both of

these conditions require that resort must be made to wave solutions of

the problem at the lower frequencies. Condition (b) above is always

violated at a caustic, and we have shown in Appendix I how such cases

may be treated. It might be supposed, since n changes only from

about 1. 0003 to 1 for a 70 km change in h, that condition (a) would be

well satisfied at frequencies even as low as 5 kc; however, it must be

remembered that this small change in n actually causes appreciable

bending when vji is small, and we should, instead, require that

AN/N < 0. 1(2tt/\) = 0. 002 f if we are to expect ray tracing to apply.

This more stringent requirement is met for f > 65 kc for the Ng = 301

atmosphere, and it appears that a wave solution will be required at

lower frequencies for a precise treatment of the bending. Until an

adequate wave solution becomes available, it would seem that the ray

tracing solution here given should be used even for frequencies as low

as 10 kc since the alternate assumption of N

g = would undoubtedly

yield an even poorer approximation to the actual bending.

5. 2 The Total Bending

Above a height of about 70 km, the troposphere no longer bends

the radio waves appreciably, and the designation t has been given to

the total bending of radio waves passing entirely through it. Table 5.5

gives t , the critical range, R , and R - (R /2) as a function of 4* for

the CRPL Reference Radio Reiractivity Atmospheres. The results

provide a convenient means for determining the true elevation angle i}i and true range R of a satellite at very high heights, say h > 70 km ,

These results may actually be used without appreciable error whenever Re > 2 Rc

.

93
Table 5.5

The Total Bending, t, Critical Range, Reel , and R - (R,/2) for the C.R. P.L. Reference Radio Refractivity Atmospheres

N = 200.0

s h = 10, 000s N =250.0

s h = 5,000'

s N = 301.0

s h = 1,000's + T R

c R -(R./2)e 1 T R

c R -<R,/2)e 1 T R

c R -(R./2)e 1 m. r. m. r. km. meters m. r. km. meters m. r. km. meters

0.0 7.1845 200.400 62.2 9.4504 198.752 78.1 12. 1522 196. 588 95.40

0.5 7. 1022 199.007 61.5 9.3356 197.274 77. 1 11.9871 195.049 94.1

1.0 7.0211 197.632 60.8 9.2226 195.817 76.2 11.8252 193.537 92.9 2.0 6.8630 194.937 59.4 9.0025 192.966 74.3 11.5109 190.592 90.5

3.0 6.7100 192.309 58. 1 8.7899 190. 195 72.5 11.2090 187.744 88.3 4.0 6. 5622 189.743 56.7 8. 5848 187.498 70.8 10.9194 184.987 86.1

5.0 6.4193 187.235 55.4 8.3869 184.870 69.1 10.6416 182.313 83.9

6.0 6.2813 184.782 54.3 8.1962 182.308 67.6 10.3754 179.717 82.0

7.0 6. 1480 182.381 53.1 8.0124 179.807 66.1 10.1202 177. 190 80.0

8.0 6.0194 180.027 51.9 7.8353 177.363 64.7 9.8757 174.730 78.1 9.0 5.8951 177.719 50.8 7.6645 174.973 63.2 9.6413 172.331 76.3

10.0 5.7751 175.454 49.8 7.4999 172.635 61.9 9.4165 169.988 74.6

12.0 5. 5470 171.046 47.8 7. 1882 168. 104 59.3 8.9940 165.461 71.4

15.0 5.2326 164. 720 45.0 6. 7602 161.641 55.7 8.4207 159.027 67.0 20.0 4. 7721 154.868 40.97 6. 1379 151.664 50. 58 7. 5998 149. 131 60.57 25.0 4.3771 145.812 37.44 5.6086 142. 576 46. 11 6.9125 140. 141 55.19 30.0 4.0352 137.488 34. 54 5.1539 134.283 42.31 6.3292 131.954 50.52 40.0 3.4745 122.821 29.49 4.4155 119.781 36.33 5.3943 117.660 43. 17 50.0 3.0364 110.443 25.80 3.8448 107.623 31.50 4.6808 105.694 37.39 65.0 2.5379 95.332 21.46 3.2016 92.847 26.22 3.8847 91.165 31. 17 80.0 2.1689 83.422 18.30 2.7295 81.232 22.29 3.3051 79. 749 26. 50 100.0 1.8076 71. 142 15.30 2.2702 69.270 18.55 2.7442 67.997 21.96

150.0 1.2606 51.401 10.68 1.5794 50. 052 13.02 1.9053 49. 122 15.39 200.0 0.9586 39.975 8.19 1.2000 38.928 9.97 1.4464 38.201 11.80 300.0 0.6379 27.600 5. 59 0. 7978 26.879 6.80 0.9611 26.375 8.03 400.0 0.4693 21.156 4.26 0. 5868 20.604 5.19 0. 7068 20.217 6. 12 600.0 0.2912 14.699 2.94 0.3641 14.316 3.59 0.4384 14. 047 4.25 900.0 0. 1584 10.630 2.13 0. 1980 10.354 2.59 0.2384 10. 158 3.06

94
Table 5.5

The Total Bending, t, Critical Range, R , and R - (R,/2) for the C. R c el P. L. Reference Radio Refractivity Atmospheres

N = 350.

s h = 0's N = 400.3 h = 0' s N = 450.0

s h = 0's

4- T R

c R - (R,/2)e 1 T R

c R

e -(R1 /2) T Rc R - (R./2)e 1 m. r. m. r. km meters m. r. km meters m. r. km meters

0.0 15.6236 192. 552 111. 10 20.4292 189.878 128. 70 27. 5747 191.865 152.6

0. 5 15.3818 190.840 109.40 20. 0507 187.842 126.60 26.9134 189. 128 149. 3 1.0 15. 1452 189. 165 107.80 19. 6822 185.861 124.30 26.2746 186.497 146. 1 2.0 14.6878 185.921 104.80 18.9747 182.063 120.20 25.0635 181. 540 140.2

3. 14.2512 182.809 101.80 18. 3060 178.467 116.40 23.9391 176.959 134.6

4. 13.8349 179.818 99. 00 17. 6748 175.055 112. 70 22.8975 172. 716 129.4

5.0 13.4383 176.939 96. 3 17. 0798 171.811 109.3 21.9340 168.775 124. 7 6.0 13.0605 174. 163 93.8 16. 5190 168. 717 106.0 21.0429 165.099 120.3

7. 12.7008 171.479 91.4 15.9905 165. 759 103.0 20.2186 161.655 116.2

8. 12.3583 168.881 89. 1 15.4923 162.923 100. 1 19.4552 158.414 112.4 9.0 12.0320 166.361 86.9 15.0224 160. 199 97.2 18. 7472 155.350 108.8

10. 11. 7210 163.914 84.8 14. 5787 157. 574 94.6 18. 0892 152.443 105. 5 12.0 11. 1413 159.218 80. 7 13. 7626 152. 593 89.8 16.9049 147. 026 99.3

15. 10. 3651 152.608 75.4 12.6912 145.688 83.3 15.3978 139. 701 91.4

20. 9.2736 142. 573 67.9 11.2232 135.398 74.3 13.4115 129. 072 80. 7

25. 8. 3768 133. 571 61.49 10. 0480 126.325 66.93 11.8776 119.907 72. 17

30. 7.6271 125.452 56.08 9.0849 118.249 60.80 10.6525 111.868 65. 12 40.0 6.4460 111.434 47.70 7.6000 104. 501 51.28 8.8124 98. 383 54.48 50.0 5. 5604 99.827 41.23 6. 5103 93.278 44. 12 7.4944 87. 524 46.62

65.0 4. 5872 85.861 34. 05 5.3338 79.931 36. 33 6.0979 74. 749 38.22 80.0 3.8877 74.969 29. 02 4. 5005 69.621 30.82 5. 1232 64.964 32.30

100.0 3.2172 63.818 24.05 3. 7104 59. 138 25. 51 4.2089 55.077 26.79

150.0 2.2247 46. 007 16.76 2. 5542 42. 519 17.71 2.8853 39.506 18. 50 200.0 1.6860 35. 747 12.83 1.9321 32.998 13.6 2. 1789 30.628 14. 13 300.0 1. 1188 24.662 8. 73 1.2803 22.745 9.21 1.4419 21.093 9.60 400.0 0.8224 18.899 6.65 0.9405 17.423 7.03 1. 0588 16. 153 7. 33 600.0 0. 5099 13. 128 4.62 0. 5830 12. 100 4.86 0.6560 11.216 5. 06 900.0 0.2772 9.494 3.33 0. 3169 8. 749 3. 51 0. 3566 8. 108 3.66

95
in terms of its observed elevation angle iJj , its observed radio range

R_ and the surface value, N_, of refractivity at the observing point:

i|i = i|i - t[1 - (R /R )] (47) o c e

R = (R./2) - R

o 1 c

1 c

3 " 2R

e -

J

2

sin t (48)

6. Tropospheric Scatter

The distance d to the radio horizon of a transmitting

J-it antenna of height, h, , above a smooth spherical earth of radius a

1 and for a linear gradient atmosphere may be determined from:

dT L = N/2ka' h (49) Lt t

where k is defined by (36). For the particular CRPL Reference

N = 301 atmosphere, ka1 = 5, 280 miles and, if h. is expressed in

feet and d, , in miles: d. , = n/Z h, . When h^ is greater than one

kilometer, (49) no longer applies, and reference should then be made

to the preceding section or to references 67 and 70; in particular,

since the horizon is defined by the ray corresponding to i|i = 0, Fig. 47

shows the relation over a smooth earth between dj . and h, at large

heights and for several values of N . If we let h and d, denote the

height and distance to the radio horizon for the receiving antenna, then

receiving antennas at a distance d > dT

. + d, from the transmitting

antenna lie below the horizon ray of the transmitting antenna, and it

becomes convenient to calculate the transmission loss at such distances

in terms of the angular distance, 0. zl Over a smooth, spherical earth

and in a linear gradient atmosphere, may be determined by:

d-d - d

= r—f radians (50)

The angular distance, 9, is a particularly convenient parameter for

making appropriate allowance for the effects of irregularities in the

terrain, and a detailed explanation of methods for calculating the

cumulative distribution of transmission loss in propagation over irregular terrain and for a wide range of atmospheric conditions is given in

96
a recent report by Rice, Longley, and Norton.—/ Thus it is shown in

that report that the received field on these beyond-the -horizon paths

may be considered to consist of diffracted and scattered components.

The scattered component may be explained quantitatively for the

winter afternoon hours in terms of scatter by a turbulent atmosphere

using the mixing-in -gradient hypothesis as the basis for describing the

turbulence, i.e., the (r/i Q )K^(r/i ) correlation function may be used

to describe the correlation in the variations of refractive index at

points a distance r apart in the atmosphere. 22J Some direct experimental evidence for this description of atmospheric turbulence is given

in a recent paper by the author. £°/ The extension of these estimates

of the transmission loss for winter afternoons to all-day, all-year

values is then done empirically, using the angular distance as a

parameter in this empirical analysis.

Fig. 48 shows the values of median basic transmission loss

separately for the diffracted wave and for tropospheric scatter as

calculated in the manner described in reference 70 for the range of

frequencies from 10 to 10, 000 Mc and for transmitting and receiving

antennas both at a height of 30 feet. If we assume that the short term

variations in the scatter fields are Rayleigh distributed, and that the

diffracted waves are relatively steady, then we may determine the

expected combined median basic transmission loss, L, , in terms of

the diffracted wave transmission loss Li j and the median basic scatter

transmission loss, L, as follows: ' bms

L

bm = Lbd- R(0 - 5) (51 >

where K = Lj - L + 1. 592 is the ratio in decibels of the average bd bms

scattered power to the diffracted wave power, and R(0. 5) is given

graphically and in tables in reference 71. When K is less than -16. 5 db, L^ differs from L^ by less than 0.1 db, and when K is greater than

19. 5 db, L, differs from L, by less than 0.1 db.

' bm bms }

Finally, to determine the expected values, L, (p), of basic

transmission loss exceeded by (100 - p) per cent of the hourly medians

during a year, we may simply subtract V(p, 0) as given on Fig. 49

from L, as calculated above from the values shown on Fig. 48.

Fig. 5 shows the influence on the median basic transmission

loss at 100 Mc of changing one antenna height while keeping the other

antenna height fixed at 30 feet. The values given are for a smooth

97
-98-

MEDIAN BASIC TRANSMISSION LOSS FOR

THE GROUND WAVE AND TROPOSPHERIC SCATTER

MODES OF PROPAGATION OVER A SMOOTH SPHERICAL EARTH

Over Land o~ - 0.005 mhos/meter e = 15 Polarization: Horizontal

Transmitting and Receiving Antennas Both 30 Feet Above the Surface

oO

D

00

IUU \

i i i ii 1 1

1

i

Tropospheric Scatt ci it - —

V< no

r? § S

-Ci " Qv> ^

: o'0 o --Q d=! Ci

120 3 mi ^_ -^ = i. — ^ *- — V

1 ^- —<§

130 IO m/ "

*—

»

\ -/-A

\ -/—

'

/ 140 /

/ / ! /

1 rr —w~ L__

=|=J=—^^^^ -/i-

?C V i?

150 s %, 3 It \ 44-

\ rr

"^-^^ H— 4 -\— 4rtt

±t

±t

IfiO " —

_i v -H -\[ —VJ pjit

1

:

~~— ri I z M ),>h- :<-o T)i- *v

"~N 170 -U- v 4-M

•*+. 1

"v ^.tu I

~* \ *±>n

**Y- >,"/

;K

v ,\ V \\ ^>* v \ > V O V

180 N ^> s V Sl)%o H. v \

h\£fr- -,. s ^ N, ' 4$ rv9 Vi^ V >,• Vso

s, N X ^x^ ~~~--~. s. V

:, :^~Nr ^ A s

s. \

-\ ' i

IbU —r^- —*^-v -^o

<s. Hv-^:v>>: sV—

N

"^=u—

V

' ^x i =i=.--\~---^ X i- ^^ -^ \ \ p- j-

,*»"""

\

v. -\ ^. V

\ \--^—V- 5 I200 ^ \4

V s V ! i-«« ^n1^H =ttfr=H V" k

\ A-^ -V-i ^

s -\==^v^ i ,0 'l x >

1

' \ \ 1.a\i

s A v S \ V \ > \

v \ \ A ^ \ , /

V. x \

' \ \ \j - ^E ^- -

:==j*~^-V-

* , \ -v210 Vr^" *F\\4

^

^^ - V^> V W^ - A —I vAVA -p —V-\—

——r

i--tmflV- w& \ -V \ \ ^F --'—v^—

'

; t —

*

iff

-v :=E%. \- - \\^ai: - ' v \ VA - Ti~ "\

\—

v

-»- Vf—

o - \ : S \ \\ 1 s ^ 11 - J 220 \

> v \> S 1 \ i ' \ v "W<-^ \ "\ \\\ /- -\ ^^ ' ^JliF A^ I -r =Y\\fvv- " \ \\ \ —f- \l^!n \ .j s \ \\ i ii i V. \ \ «^i \ \\ ' ' » II II Li. \ _^^ 1 i i %

V \

«V \ \\ \ i ' \, \\ n II \ k°r \ , \ \ 230

;- - V " 4- VV4t: = VvfcX - -v-

-s —\-4

1

\

\Htm

44-

i =* 4t

\ =#::^JU_!_ \ \ Ml- 1 V \

i V' \ \ > n /

L . "*< \ '

i \ 1 311

B

> \

* \ \ 1

210 ^^

~-^ ^

FR

\ N

S :t±\5^

\

* \- 1 i : V+i\j—

-A--h yy7i w3-

\- %

T

- -s n\ —

r

A 1

1

A:

"V. f!n ' \—%- \FT

- \ \ \ \-f- i4

,. ::r=^^%- l \"\ i,

- \^s- - - \\ \ u-. mmip m ^ -A 1 ~^ -W -A s \ A — V-i-Y- Ml \ Mil Hi III ftf

IO 20 50 100 200 500 1,000 2j000 5,000

Frequency in Megacycles

20,000 50,000 100,000
99-

co

coo

_i ^o o h-<

CO CD

CO— <n "> o

00 QC 2 Q_

<ror

O

h- cr

Li_ LU O X

LUO<

Q_

COO

Q_O

cr cr

<r h- > 2

LUX cgac

S|8qi09Q Ul (0'd)A
-100-

UJ

u.

X

X

UJ

MEDIAN BASIC TRANSMISSION LOSS AT 100 MC

Smooth Spherical Earth and a CRPL Reference Ns=30l Atmosphere

Horizontal Polarization Over Land

One Antenna at 30 Feet and the Other Antenna at the Heights Indicated

° Location of Maximum Fields (minimum transmission losses) (Lbd) Ground Wave

(I_bms) Tropospheric Scatter for Winter Afternoon Hours

300 280 260 240 220 200 180 160 140 120 MEDIAN BASIC TRANSMISSION LOSS IN DECIBELS

100 90

Figure 50
earth and a CRPL Reference Ng = 301 atmosphere. The first two

oscillations of the field are shown for d = 10 and 20 miles, but for the

other distances only one oscillation is shown. The six points of field

maxima are shown for all of the distances as circled points. Note

that the total number of maxima to be expected (as a function of range

at a given height or as a function of height at a given range) for a

particular antenna height is equal to the number of half wavelengths

contained in this height; in the present case of 100 Mc, 30 feet represents

6 half -wavelengths; in this connection, see (8), page 11.

The scatter curves on Fig. 50 correspond to the winter afternoon hours, and the reader is referred to reference 70 for curves

suitable for translating these values to transmission losses exceeded

for several percentages of various periods of time. The scatter loss

predictions on Fig. 50 are shown only up to heights just short of the

radio horizon since the method of estimation given in reference 70 is

not applicable to line -of- sight paths.

7. Point -to -Point Radio Relaying by Tropospheric Scatter

As an example of the method of using transmission loss in

systems design, we will consider the problem of estimating the

effective maximum range of a radio relay system using tropospheric

scatter. As an illustration of typical ranges to be expected, we will

assume that the terrain is smooth, and will base our predictions on

a CRPL Reference Radio Refractivity Atmosphere with Ng = 301. "We

will assume that either two 28-foot or two 60-foot parabolic antennas

are used at both ends of the path, with their centers 30 feet above the

ground and connected in a quadruple diversity system. With these

assumptions, we may use the methods described in reference 70 to

determine the transmission loss, L(99), which we would expect one

per cent of the actual hourly median transmission losses to exceed

throughout a period of one year; the use of these one per cent losses

implies that the specified service will be available for 99% of the

hours. Tables 7.1 and 7.2 give for the 28' and 60' antennas the free

space gains G, + G , and the path antenna gains as a function of

frequency and distance, while Tables 7.3 and 7.4 give L(99) as a function of frequency and distance.

The power required to provide a specified type and grade of

service for 99% of the hours may now be obtained from the equation:

P = L + L(99) + R + F + B - 204 (52)

101
Table 7. 1

Path Antenna Gain in Decibels for 28-Foot Parabolic Antennas 30 Feet

Above a Smooth Spherical Earth with a CRPL Model Radio Refractivity

Atmosphere Corresponding to N

g = 301

fMc G +G

t r G

P

in Decibels

db

d = 100

mi.

150 200 300 500 700 1000

100 33.02 33.02 33.02 33.02 32.92 32.82 32.72 32.67

150 40.07 40.07 39.97 39.97 39.87 39.57 39.47 39.37

200 45.06 45.06 44.96 44.86 44.66 44.36 44. 16 44.06

300 52. 11 52.03 51.91 51.71 51.31 50.81 50. 51 50.41

500 60.98 60.75 60.38 60.08 59.38 58.48 58. 18 58. 18

700 66.83 66.35 65.83 65.23 64.33 63.23 62.83 62.93

1000 73.02 72.12 71.22 70.42 69.22 67.92 67.32 67.22

1500 80.07 78.32 76.57 75.87 74.27 72. 57 71.67 71.57

2000 85.06 82.48 80.66 79.26 77.16 75.46 74.36 73.86

3000 92. 11 87.71 84.21 83.41 80.91 78. 51 77.61 77.31

5000 100.98 93.28 90.18 87.68 84.68 82. 18 81.28 80.98

7000 106.83 96.53 92.83 90.03 86.83 84. 12 83.33 82.93

10000 113.02 99.32 95.22 92.52 89.02 86.02 85.22 84.92

102
Table 7.2

Path Antenna Gain in Decibels for 60 -Foot Parabolic Antennas 30 Feet

Above a Smooth Spherical Earth with a CRPL Model Radio Refractivity

Atmosphere Corresponding to N = 301

Mc G +G

t r G : P

.n Decibe:1s

db

d = 100

mi.

150 200 300 500 700 1000

100 46.26 46. 16 46. 16 46.06 45.86 45.46 45.26 45. 16

150 53.31 53.21 53. 11 52.91 52. 51 52. 11 51. 56 51.48

200 58. 30 58. 10 57.90 57.60 57.00 56.30 55.90 55.86

300 65.35 64.95 64.45 63.95 62.85 62.05 61.70 61.90

500 74.22 73.22 72.67 71.02 70.22 68.72 68. 12 68. 12

700 80.07 78.37 77. 07 75. 87 74. 17 72. 57 71.67 71. 57

1000 86.26 83.46 81.46 79.96 77.86 76.26 75.01 74.66

1500 93.31 88. 51 85.91 84. 01 81. 51 79. 11 78.21 77.81

2000 98.30 91.70 88.80 86.60 83.70 81. 10 80.20 79. 80

3000 105.35 95.75 92.25 89.65 86.35 83. 55 82.85 82.45

5000 114.22 100.02 95.62 92.92 89.52 86.52 85.62 85.42

7000 120.07 102.37 97.87 94.97 91. 17 88.22 87. 57 87. 07

10000 126.26 104. 56 99.96 96.66 93.06 90.06 89.26 88.76

103
Table 7.3

Transmission Loss L(99) (Corresponding to Fields Exceeded 99% of

the time) Expected Between Two 28 -Foot Parabolic Antennas at a

Height of 30 Feet Above a Smooth Spherical Earth with a CRPL Model

Radio Refractivity Atmosphere Corresponding to N = 301

L(99) in Decibels

fMc d = 100

mi.

150 200 300 500 700 1000

100 160.21 164. 33 166.95 182. 31 207. 79 233.29 276.57

150 154.20 158.56 161.57 176.52 200.95 225.91 269.52

200 150.24 154. 72 157.92 173.08 197. 38 221. 76 265.56

300 145. 32 150.27 153.55 169. 33 192.88 217.45 260.36

500 140. 10 145.87 149.64 165.94 189. 74 214. 13 255.99

700 137. 39 143.81 147.98 164. 83 188.55 213. 08 254.05

1000 135. 11 142. 31 146.98 164. 07 188. 08 212. 72 251.88

1500 133.32 141.88 146.32 164.50 188.91 213.41 254.20

2000 132.54 141. 36 146.76 165. 62 190.08 214.46 255.50

3000 132. 70 143.69 148.71 168. 01 193.39 217.50 257.99

5000 134, 79 145.77 152.99 173.37 199. 19 223.38 264. 01

7000 137,37 149.86 158. 13 179. 75 206.98 231.32 272. 13

10000 144.68 160. 12 170.68 196. 11 223.53 247.87 288.69

104
Table 7.4

Transmission Loss L(99) (Corresponding to Fields Exceeded 99% of

the time) Expected Between Two 60 -Foot Parabolic Antennas at a

Height of 30 Feet Above a Smooth Spherical Earth with a CRPL Model

Radio Refractivity Atmosphere Corresponding to N = 301

s

L(99) in Decibels

f d = 100 150 200 300 500 700 1000 Mc mi.

100 147.07 151.19 153.91 169.37 195.15 220.75 264.08

150 141.06 145.52 148.63 163.88 188.41 213.83 257.41

200 137.20 141.78 145.19 160.74 185.44 210.02 253.76

300 132.40 137.73 141.31 157.79 181.64 206.26 248.87

500 127.63 133.58 138.70 155.10 179.50 204.19 246.05

700 123.67 132.57 137.34 154.99 179.21 204.24 245.41

1000 123.77 132.07 137.44 155.43 179.74 205.03 244.44

1500 123.13 132.54 138.18 157.26 182.37 206.87 247.96

2000 123.32 133.22 139.43 159.08 184.45 208.62 249.56

3000 124.66 135.65 142.47 162.57 188.35 212.26 252.67

5000 128.05 140.34 147.75 168.53 194.85 219.04 259.57

7000 131.53 144.82 153.19 175.41 202.89 227.18 267.79

10000 139.44 155.38 166.54 192.07 219.49 243.83 284.85

105
Each of the terms in (52) is expressed in decibels; P^ is the transmitter power expressed in decibels above one -watt; L. is the loss in

the transmitting antenna circuit and the transmitting antenna transmission line (this term is set equal to one db for the calculations in this

report); R is the median pre -detection signal -to-r. m. s. noise ratio

required for the specified grade of service; F is the effective receiver

noise figure and includes the effects of the antenna noise as well as

the receiver noise together with the receiving antenna circuit and

transmission line loss; U it is assumed that the receiver incorporates gain adequate to ensure that the first circuit noise is detectable;

B = 10 l°gio(b "*" k ) i s the effective receiver bandwidth factor with

bQ and b expressed in cycles per second; b allows for the drift

between the transmitter and receiver oscillators,7 while bm allows for

the band occupied by the modulation; the constant term (-204) is

10 log-|QkT where k is Boltzmann's constant and the reference temperature is taken to be 288.44' Kelvin; this is just the noise power in a one

cycle per second bandwidth in db relative to one watt.

For the calculations in this report, the transmitter and

receiver oscillators were each assumed to have a stability of one

part in 10 and to vary independently so that b = n/2~ f ° 10" . Table 7. 5 gives the values of b assumed for the various types of

service considered. The effective receiver noise figure has been

estimated as F = 5 logiQf - 5. Table 7. 5 also gives the values of

R for the various kinds of service on the assumption that quadruple

diversity is used. The value of R for the FM Multichannel system

is expected to provide a service with less than an 0. 01% teletype

character error rate. The FM Multichannel System consists of 36

voice channels, each of which can accommodate sixteen 60 words per

minute teletype circuits. The values of R given in Table 7. 5 were

determined by methods given in a recent report by Watt. ±~J The value

of R for the FM Multichannel system corresponds to typical fading

encountered at 1000 Mc, and this value of R may change by a few db

with frequency as the fading changes, but such changes have so far not

been evaluated quantitatively; furthermore, R will also change as the

fading changes from hour to hour.

Table 7. 6 gives as a function of frequency the maximum

permissible hourly median transmission loss for a transmitter power

of 10 kw: L = 204 + P.-L.-R-F-B corresponding to the kinds

of service described above. By combining the information in Tables

7. 3, 7.4, and 7.6, we can estimate the maximum range for a quadruple

diversity system with 10 kw transmitters. These ranges are shown on

Fig. 51 as a function of the radio frequency.

106
Table 7. 5

Type of Service

bm

cycles /sec.

R*

decibels

Signal

Bandwidth

cycles/sec.

Post Detection

Signal -to

noise ratio

decibels

Transmis sion

Loss Measurement

0# -

FM Multichannel System3,750,000 9.5 36 Voice

channels each

capable of use

for sixteen 60

words per min.

teletype circuits

0. 01%

teletype

character

error rate

FM Music 150,000 26.5 15,000 50**

U. S. Standard

Television

3, 750, 000 32.7 3, 750,000 30**

##

Ratio between the median intermediate frequency Rayleigh

distributed signal and the r.m. s. Rayleigh distributed noise.

This ratio will be exceeded with a quadruple diversity system

for 99% of each hour for which the corresponding value of R is

maintained in each receiver.

Diversity reception not involved in this case.

107
Table 7. 6

Maximum Permissible Transmission Loss L, , for 10 KW Transmitters M

Using the Parameters of Table 7. 5

Transmis sion U. S.

loss FM FM Standard

fMc Measurement Multichannel Music Television

100 236.50 162.76 159.74 139.56

150 233.85 161.88 158.86 138.68

200 231.98 161.26 158.23 138.06

300 229.34 160.37 157.35 137. 17

500 226.01 159.27 156.25 136.07

700 223.82 158.54 155.51 135.34

1000 221.50 157.76 154. 74 134.56

1500 218.85 156.88 153.86 133.68

2000 216.98 156.25 153.23 133.05

3000 214.34 155.37 152.35 132. 17

5000 211.01 154.27 151.25 131.07

7000 208.82 153.53 150.51 130.33

10000 206.50 152.76 149.74 129.56

108
— 109

—

co

Q_

O

LU oz

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LJ 3 yj O O > > X Q

Pi UJ

LU =r LU

CO f _] h ^s < O o^ 3

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Op co<

Ld LU rr ucot z

CO ^£ 1 ^ X Q < z

<»— O

ro

ii

<DOo

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8.1

CD

LU Ld Q II-

_o

LU cD o cr o

-5.°

COCo o e- o o c (/> CO <o"^_CD -C

a.

(/> oE oo

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CO

cr

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CO

LU

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m

S31IIAI 3iniVlS Ni 39NVd lAIHIAIIXVlAI
Acknowledgements: The author has had extensive assistance in

preparing this report from many members of the Radio Propagation

Engineering Division Staff. Acknowledgement of most of this assistance

is given in the text or by reference, but in addition the following contributions should be mentioned. Don Watt assisted extensively with

the theory of ionospheric roughness outlined in Appendix I. Anita

Longley was largely responsible for the sky wave computations for

Figs. 31 to 40, inclusive, while Peter Ratcliffe did most of the

calculations for Figs. 41 to 45, inclusive. Lew Vogler calculated

the diffracted fields shown on Fig. 50, while Anita Longley calculated

the scattered fields. Ralph Johler calculated the values given in

Table 1-2. Gertrude Ovale typed the report, and John Harman drafted

most of the figures.

110
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Ill
Kenneth A. Norton, "Space and Surface Waves in Radio

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15. James R. Wait, "Excitation of Surface Waves on Conducting,

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112
18. Kenneth A. Norton and J. W. Herbstreit, "Wright Field

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the Federal Communications Commission Ad Hoc Committee

for the Evaluation of the Radio Propagation Factors Concerning the Television and Frequency Modulation Broadcasting Services in the Frequency Range between 50 and 250 Mc,

May 1949.

20. R. S. Kirby, J. C. Harman, F. M. Capps, andR. N. Jones,

"Effective Radio Ground -Conductivity Measurements in the

United States, " National Bureau of Standards Circular 546,

Feb. 1954.

21. H. E. Bussey, "Microwave Attenuation Statistics Estimated

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.

22. Bradford R. Bean, "Some Meteorological Effects on

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23. B. R. Bean and R. L. Abbott, "Oxygen and Water Vapor

Absorption of Radio Waves in the Atmosphere, " Geofis. Pura

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24. "Instructions for the Use of Basic Radio Propagation Predictions, " NBS Circular 465, 1947, Superintendent of Documents,

Government Printing Office, Washington 25, D. C.

25. "Basic Radio Propagation Predictions - Three Months in

Advance, " CRPL Series D, Superintendent of Documents,

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26. J. R. Wait, "The Mode Theory of VLF Ionospheric Propagation

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113
27. J. R. Wait, "The Attenuation vs Frequency Characteristics

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(1957).

28. Jack N. Brown, "Round -the -World Signals at Very LowFrequency, " J. Geophys. Research, 54, 367-372, (1949)

(1949).

29- G. Millington, "Ground Wave Propagation Over an Inhomogeneous Smooth Earth; Part 1," Proc. Inst. Elec. Engrs., 96,

53, (1949). ~~

30. James R. Wait and James Householder, "Mixed -path Ground - Wave Propagation; Part 1 - Short Distances, " J. Research

NBS, 57, 1-15, (1956); Part 2 - "Larger Distances,"

J. Research NBS, _5_9, 19-26, (1957).

31. T. N. Gautier, "The Ionosphere, " Scientific American,

193 (3), 126-137, (1955).

32. J. R. Johler, W. J. Kellar, andL. C. Walters, "Phase of

the Low Radio Frequency Ground Wave, " National Bureau

of Standards Circular 573, June 26, 1956.

33. J. R. Wait and H. H. Howe, "Amplitude and Phase Curves

for Ground Wave Propagation in the Band 200 Cycles Per

Second to 500 Kilocycles, " National Bureau of Standards

Circular 574, (1956).

34. J. R. Wait and A. Murphy, "Multiple Reflections Between

the Earth and the Ionosphere in VLF Propagation, " Geofis . PuraeAppl. (Milano ) 35, (III), 61-72, (1956)

35. B. G. Pressy, G. E. Ashwell, and C . S. Fowler, "The

Measurement of the Phase Velocity of Ground Wave Propagation at Low Frequencies Over a Land Path, " Proc. Inst.

Elec. Engrs., 100 (Pt. HI ) , 73-84, (1953).

36. G. Hefley, C. England, W. Kellar and R. Dyer, "Skywave

Delays at 100 kc and Noise Signal Interference in the 100 kc

Navigation Band, " Private Communication, June, 195 3.

114
37. A. B. Schneider, "Phase Variations With Range of the Ground

Wave Signal from C. W. Transmitters in the 70-130 kc Band, " Jour. British Inst. Radio Engrs. , 12 (3) , 181-194, (1955).

38. R. H. Doherty, "Pulsed Sky Wave Phenomena Observed at

100 kc, " Private Communication, Feb. 1957.

39. J. R. Wait, "Pattern of a Flush-Mounted Microwave Antenna,"

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40. J. R. Wait, and Alyce M. Conda, "Pattern of an Antenna on

a Curved Lossy Surface, " I. R. E. Trans. , AP-6 , 348-360,

(1958).

41. S. O. Rice, "Diffraction of Plane Radio Waves by a Parabolic

Cylinder, " Bell System Tech. Jour., 3_3, 417-504, (1954).

42. V. A. Fock, "Fresnel Diffraction from Convex Bodies, " Uspekhi Fizicheskikh Nauk (Progress of Physical Sciences),

43, 587-599, (1951) also, Paper No. IX, Astia Document

No. AD 117276: "A series of translations of 13 papers by

Fock, " U. S. Department of Commerce, Office of Technical

Services, Washington 25, D. C., June, 1957.

43. H. Bremmer, "Terrestrial Radio Waves, " Elsevier Publishing Co., 1949, Chapter XI.

(44) Omitted in Revision.

45. J. S. Belrose, "Some Investigations of the Lowest Ionosphere, " a dissertation submitted for the degree of Doctor of Philosophy

in the University of Cambridge in England, December, 1956.

46. A. D. Watt, E. L. Maxwell and E. H. Whelan, "Low Frequency

Propagation Paths in Arctic Areas, " J. Research NBS, 63-D

(Radio Propagation), 99 , (1959).

47. D. F. Martyn, "The Propagation of Medium Radio Waves in

the Ionosphere, " Proc. Phys. Soc . , 47, 323-339, (1935).

115
48. Paul O. Laitinen and George W . Haydon, "Analysis and Prediction of Sky -Wave Field Intensities in the High Frequency Band, " Technical Report No. 9, Signal Corps Radio Propagation Agency,

Ft. Monmouth, New Jersey.

49. A. N. Kazantsev, "Theoretical Calculations of Ionospheric

Absorption in the Different Areas of Change of Ionization as

a Function of Height, " Proceedings of the Academy of Sciences

of the U. S. S. R., 1956, No. 9.

50. A.N. Kazantsev, "Absorption of Short Radio Waves in the

Ionosphere and the Intensity of the Electric Field at the Point

of Reception, " Proceedings of the Academy of Sciences of the

U. S. S. R., 1947, No. 9-

51. D. K. Bailey, R. Bateman, and R. C. Kirby, "Radio Transmission at VHF by Scattering and Other Processes in the Lower

Ionosphere," Proc. IRE, 43, 1181-1230, (1955).

52. R. C. Kirby, "VHF Propagation by Ionospheric Scattering -A

Survey of Experimental Results," IRE Trans. Commun. Syst.

CS-4 (1) , 17-27, (1956).

53. R. C. Kirby, "Extreme Useful Range of VHF Transmission by

Scatter from the Lower Ionosphere, " IRE Convention Record,

1958.

54. A. D. Wheelon, "Radio Frequency and Scattering Angle

Dependence of Ionospheric Scatter Propagation at VHF, " J.

Geophys. Research, 62, 93-112, (1957).

55. A. D. Wheelon, "Diurnal Variations of Signal Level and Scattering Heights for VHF Propagation," J. Geophys. Research, 62,

255-266, (1957).

56. A. D. Wheelon, "Refractive Corrections to Scatter Propagation,"

J. Geophys. Research, 62 , 343-349, (1957).

57. K. A. Norton, "Point -to -Point Radio Relaying via the Scatter

Mode of Tropospheric Propagation," IRE Trans. Commun. Syst.

CS-4 (1), 39-49, (1956).

116
58. K. A. Norton, "Recent Experimental Evidence Favoring the

pK^ (p) Correlation Function for Describing the Turbulence of

Refractivity in the Troposphere and Stratosphere, " Proc. Joint

Commission on Radio Meteorology held under the auspices of

the International Scientific Radio Union at New York University,

August, 1957 (in press).

59. B. R. Bean, "Prolonged Space -Wave Fadeouts at 1046 Mc

Observed in Cheyenne Mt. Propagation Program, " Proc.

IRE, 42 (5) , 848-853, (1954).

60. M. Katzin, R. W. Bauchman, and W . Binnian, "3 and 9

Centimeter Propagation in Low Ocean Ducts, " Proc. IRE,

35 (9) , 891-905, (1947).

6.1. B. R. Bean, "The Atmospheric Bending of Radio Waves,"

Proc. International Conference on Electromagnetic Wave

Propagation, Liege, Belgium, October, 1958, (in press).

62. H. G. Booker and W. Walkinshaw, "The Mode Theory of

Tropospheric Refraction and Its Relation to Wave -Guides and

Diffraction, " Meteorological Factors in Radio-Wave

Propagation - a report of a conference held on 8 April 1946

at the Royal Institution, London, by the Physical Society and

the Royal Meteorological Society; published by the Physical

Society of London, pp. 80-127.

63. J. E. Freehafer, W. T. Fishback, W. H. Furry and D. E.

Kerr, "Theory of Propagation in a Horizontally Stratified

Atmosphere, " Chapter 2, "Propagation of Short Radio Waves, " Vol. 13, Radiation Laboratory Series, McGraw-Hill Book Co.,

1951.

64. C. L. Pekeris, "Wave Theoretical Interpretation of Propagation of 10 -centimeter and 3 -centimeter Waves in Low-Level

Ocean Ducts," Proc. IRE, 35 (5) , 453-462, (1947).

65. J. C. Schelling, C. R. Burrows, and E . B. Ferrell, "UltraShort -Wave Propagation," Proc. IRE, 21(3), 427-463, (1933).

117
66. B. R. Bean and F. M. Meaney, "Some Applications of the

Monthly Median Refractivity Gradient in Tropospheric

Propagation," Proc. IRE, 43, 1419-1431, (1955).

67. B. R. Bean and G. Thayer, "Models of the Atmospheric Radio

Refractive Index, " Proc. IRE, 47, 740-755, (1959).

68. Study Programme No. 90, "Tropospheric Wave Propagation,"

Documents of the VHIth Plenary Assembly, Warsaw, 1956,

vol. 1, pp. 538-539, Published by the International Telecommunication Union, Geneva, 1957.

69- B. R. Bean and J. D. Horn, "The Radio Refractive Index

Climate Near the Ground," J. Research NBS, 63-D, (Radio

Propagation), Sept. (1959).

70. P. L. Rice, A. G. Longley, and K. A. Norton, "Prediction

of the Cumulative Distribution with Time of Ground Wave and

Tropospheric Wave Transmission Loss, " Private Communication, June 1958.

71. K. A. Norton, L. E. Vogler, W. V. Mansfield, and P. J.

Short, "The Probability Distribution of the Amplitude of a

Constant Vector Plus a Rayleigh-Distributed Vector, " Proc.

IRE. 43 (10) , 1354-1361, (1955)

72. CCIR Report No. 65; "Report on Revision of Atmospheric

Radio Noise Data, " Warsaw, 1956; available as a separate

document from the International Telecommunications Union,

Geneva, or in Vol. I of the Documents of the VHIth Plenary

Assembly of the CCIR (Warsaw, 1956).

73. R. W. Plush, A. D. Watt, andE. F. Florman, "Carrier -to

Noise Requirements for Teletype Communication via a Fading

UHF Carrier," Private Communication, July, 1958.

118
Appendix I

The Attenuation of Radio Waves Propagated Between

a Perfectly Reflecting Spherical Ionospheric Layer

and a Spherical Earth

The attenuation with which we are here concerning ourselves

is that due to the spreading of the energy over larger and larger areas

as it progresses further and further from the transmitting antenna.

For the sake of clarity in presentation, several simpler

problems will be solved first in order to illustrate the principles

involved. Consider first the attenuation of waves emanating from an

isotropic radiator in free space as in Fig. 1-1. The total energy

passing through the differential elements of area, dA, and, dA,

1 normal to the radius vector and at the unit of distance R and at R,

respectively, will be equal.

2 dA = R dip r dx = R cos vJj dvp dx (1-1)

dA = R

2

cos Oi ddy d x (1-2)

Now, if we let p and p represent the energy density per unit area at

the distances R and R, we obtain:

P1

dA

1 = p2

dA (1-3)

(P^Pj) = (dA^dA) = R

2

/R

2

(1-4)

Thus we see that the field intensity (i. e. , the energy density)

is inversely proportional in free space to the square of the distance
-2-

Isotropic

Radiator

Figure I -1

Ionosphere

Figure 1-2
from the source. In terms of transmission loss, this result may be

expressed:

L

bf

(R) = L

bf

(R

l

} + Z ° lo8

10 < R/R

i)

(1-5)

Consider next — see Fig. 1-2—the attenuation of waves reflected

from a plane perfectly conducting ionosphere at a height, h, above a

plane earth and with no atmospheric refraction; in this case we have:

dA = - r d x sin if* dr = R cos ij> dtp d\ (1-6)

In this case again we find that the attenuation of waves reflected from

a plane ionosphere is the same as in (1-5).

Finally consider —see Fig. 1 3—the attenuation of waves

reflected from a perfectly conducting spherical ionosphere and a

perfectly conducting spherical earth with the effects of atmospheric

refraction included:

dA = - y d x sin ty dr

dr = 2ad9

y = a sin 2 e

dA = - 2a sin 2 sin vp d 8 dx

(1-7)

(1-8)

(1-9)

(1-10)

By Snell's law:

cos (i[» + 6 - t)

cos ^

{1 + N • 10~ exp(- c h )}(a + h ) C = 2_ ' S-S L (1-11)

{1 + N • 10" exp(-c h)}(a + h) o s

C = cos (8 - t) - tan ^ sin ( - t) d-12)
-4-

Figure I~3
Since C is a constant, independent of 9, t and *\>, we find:

Thu8 lMm ..ft,«>}. {%,*}* (I .14)

In the particular case when i|> = 0:

(d9 /d^ = Q

= - 1 + (dT /d.|i) = q

= - k (1-16) *

MR) = i^k,) + io log

10( gj-1 = LM( R) " C1< R> <I -17>

Thus, if we substitute (X-10) and (1-1) in the above and solve for C^R)^

we obtain:

x R. dA ' l -2a sin 2 9(d0/dijj) J

(1-18)

The function C (R) is a convergence factor, expressed in decibels,

which measures how much stronger the field intensity at the receiving

point is for one reflection at a spherical ionosphere than it would be

if it were plane.

The generalization of this expression to m reflections at the

ionosphere may be obtained by noting that the only changes required

in the above analysis are:

* Note that k here refers to the ratio between the effective and actual

radii of the earth.

5
dr = 2mad0 (1-19)

y = a sin 2 m 9 (1-20)

Thus the convergence factor for m ionospheric reflections may be

written:

C (R> = 10 1og-[ — . R

, ..-..,.

')•.( '.°;M (1-21) m &10 L

- 2ma sin i|i (d0/dilj) J [a sm2mGL '

The above expression has been divided into two factors with subscripts

v and h so that the convergence of the rays in the vertical and horizontal planes, respectively, can be considered separately.

If we introduce the approximate expression

R ~ 2 m a sin 9/sin <j> into (1-21) above, we obtain: o ~ /«% >* > sin 9 If 2 m sin cos »1» "| ,„ „« . C (R) ^ 10 log,.^ : t :—rTJETnT r

* 1 ~—i—n—of (i-21a) m 10 t

- sm <|> sin iji (d9/dip) J I sin <j> sin 2 m 6 J

' o v ° h

The above expression indicates that the convergence in the vertical

plane at points far removed from a caustic is independent of the

number of ionospheric reflections for a given angle, v|j.

Note that the convergence in the vertical plane becomes

infinite when vp approaches zero; this infinite convergence is demonstrated on Fig. 25. Similarly, the convergence in the horizontal

plane becomes infinite at the antipode of the transmitter where

2m9 = ir, Actually, of course, the received energy is finite at these

points, and we may use Airy's integral to evaluate the convergence in

the vertical plane when ty < 0. It can be shown -* by the solution of

the two dimensional wave equation that:

v

[o ho{o^ C_( R) = 10 log, „ -j -^L

1

(1-22 ) v

where Q - 2irR/\ is the phase of. the waves at the receiving point. If
we multiply the numerator and denominator of the first term in

(1-21) by 2tt/X and compare the results with (1-22), we find:

n " (ip) = 4 it m (a /X) sin 4# (dO/dip)

Now we see that the above second derivative of the phase is equal to

zero when the convergence factor becomes infinite; this is the

definition of a caustic and the convergence at this caustic may be

evaluated by means of the third derivative at this point:

O '" (i|j) = - 47rmk(a/X) (i|i< 0)

Thus, at the caustic in the vertical plane, C (R) may be expressed:.!/

C (R) = 10 1 ogio{

9,2* {A (0)}

n m }

(1-23)

In the above A. (0) is the Airy integral—'n with argument zero:

2tt (A.(0)}

2 = 0.79196357.

From the above results we obtain the following expression

for C (R) at the caustic in the vertical plane: m

[ 2Trmk(a/\)J \

R

a sin 2m(0 - kip) m 'h

}(<M 0)

(1-24)

The convergence at the antipode of the transmitter is of a

somewhat different nature. Note, in particular, that there is no

point of stationary phase with respect to variations in the azimuth

angle, x» since the waves appear to be arriving from all directions

at this particular point. The following treatment of this problem

is due to J. R. Wait. —
Using a cylindrical coordinate system centered at the antipode

(i. e. , p, x» z )» W3 may obtain the following axially symmetric

solution of the wave equation, for a time factor exp(iu>t):

E =Aexp[-ik(R + z sin i|i)l J (k p cos 40 (1-25)

z a o

H = Bexp[-ik(R + z sin ty) J (k p cos 4»> (I-25a)

X a 1

where k = 2tr/X.» R is the distance along the ray path to the antipode*

a

A and B are constants, and J and J, denote Bessel functions. For

o 1 the ground wave and for those ionospheric modes for which m ia

sufficiently small so that ip is negative at the antipode, ^ should be

set equal' to zero in (1-25) and (1-2 5a). Note that, in addition to the

oscillations with time, the magnitudes of E and H oscillate with the

z x

distance p from the antipode, E having its maximum value at the

. z antipode while H is equal to zero at the antipode. Note also that the

variation with p is the same, independent of the azimuth angle, x» this would be expected since we have assumed that our source radiates

uniformly in all directions. When kp cos vj; > > 1, we may replace the

Bessel functions by the first terms in their asymptotic expansions

and obtain:

exp{i[k(p cosiJ> - R ) - tt/4]} + exp{-i[k(p cos +R )-tt/4]K

E

z

= A exp( - ikz sin M - — - V

n/ 2tt k p cos ty 4 J

(1-26)

r exp \i[K:^p cos vy - n. f - it/'* j/ - exp \-ij,JK:vp cos ty + R J-tt/^JK

in*| 2

^ * 1

Kl 7 it 1r n f o a ill J

exp{i[k(p cosvJj - R ) - tt/4]} - exp{-i[k(p cos i}> + R^)-tt/4]}

H = - i B exp(- ikz sinvjj-!

(b km - -

1 h] 2 irk p cos ijj

(I-26a)

The two exponential terms in the above may be identified with waves

arriving from opposite directions at a receiving point at a distance p

from the antipode along great circle paths of lengths R - p cos ip and

a

7a
R + p cos ifr, respectively. It is the interference between these

a two waves which causes the oscillations in the magnitude of the field

near the antipode.

To complete our solution we need only evaluate the constants

A and B. Rather than doing this directly, we note by (1-21) that the

geometrical theory indicates that the focusing in the horizontal plane

not too near the antipode is given by:

c.=

R

.C

n°2m9 (2m8<„) (1-27)

n a sin c. m U

and if we multiply (1-27) by the square of the ratio of J E j as given

z by (1-25) and by the first term in (1-26), we obtain the following

expression for c which must be used instead of (1-27) at points very

h

near the antipode:

S,

=

a *\^?TM ft *£ JJ

k P c ° 9 M Zwkpcosiji (1-28)

n a sin c, m w o When we note that p = a sin(ir - 2m 9), the above reduces to:

2 2

c, = 2-rrk R cos vJj [J (k p cos 4*)1 (1-29) h o

For the ground wave R = ira and vji = 0; thus, at the antipode

2

c, = 2u ka and C, = 10 log1rt c, = 34. 210 + 10 log,„ f, , and this

h h

6

10 h

6

10 kc

clearly represents an extremely large focusing effect for the ground

wave at and near this point. The focusing in the horizontal plane

for the sky wave modes is only slightly different, but we must add to

this the focusing in the vertical plane to obtain the total focusing for

these modes. (1-29) is for a vertical electric dipole receiving

antenna; if a horizontal magnetic dipole were used for reception, then

the J should be replaced by J,, o 1

Note that cos \\i > 0. 995 for m < 16, and cos i|> = 1 for m < 8 when

h = 90 km.

7b
Using a cylindrical coordinate system centered at the antipode

(i.e., p, x» z )» W3 may obtain the following axially symmetric

solution of the wave equation, for a time factor exp(icot):

E =Aexp[-ik(R + z sin M J (k p cos 4>) (1-25)

z a o

H = Bexp[-ik(R + z sin «J/) J (k p cos i|>) (I-25a)

X a 1

where k = 2ir/X., R is the distance along the ray path to the antipode„

A and B are constants, and J and J, denote Bessel functions. For

o 1 the ground wave and for those ionospheric modes for which m is

sufficiently small so that 4* is negative at the antipode, v|; should be

set equal* to zero in (1-2 5) and (1-2 5a). Note that, in addition to the

oscillations with time, the magnitudes of E and H oscillate with the

z x

distance p from the antipode, .25E having its maximum value at the

antipode while H is equal to zero at the antipode. Note also that the

variation with p is the same, independent of the azimuth angle, \'>

this would be expected since we have assumed that our source radiates

uniformly in all directions. When kp cos i|j > > 1, we may replace the

Bessel functions by the first terms in their asymptotic expansions

and obtain:

{exp{i[k(pcosi|j-R ) - ir/4]} + exp{-i[k(p cos +R )-ir/4]}-» * a L

>J 2-n k p cos ip

(1-26)

{exp{i[k(pcos*|'-R ) - tt/4]} - exp{-i[k(p cos vp + R )-tt/4]K

V 2irkp cos i|j *

(I-26a)

The two exponential terms in the above may be identified with waves

arriving from opposite directions at a receiving point at a distance p

from the antipode along great circle paths of lengths R - p cos 4" and

a

7a
R + p cos i|*, respectively. It is the interference between these

9. two waves which causes the oscillations in the magnitude of the field

near the antipode.

To complete our solution we need only evaluate the constants

A and B. Rather than doing this directly, we note by (1-21) that the

geometrical theory indicates that the focusing in the horizontal plane

not too near the antipode is given by:

R cos i|> h a sin 2 m 9 (2m 9 < it) (1-27)

and if we multiply (1-27) by the square of the ratio of J E | as given

z by (1-25) and by the first term in (1-26), we obtain the following

expression for c which must be used instead of (1-27) at points very

h

near the antipode:

13 f* f\ a iff X

c, = :—- tt »[J (k p cos 4*) 1 2irkpcosvji (1-28) h a sin 2 m 9 o

r

When we note that p = a sin(ir - 2m 9), the above reduces to:

2 2

c, = 2irk R cos »|i [J (k p cos ip)l (1-29) h o

For the ground wave R = ira and vh = 0; thus, at the antipode

2

c, a 2tt ka and C, = 10 log, rt ct = 34.210 + 10 log,„ f, , and this h h

6

10 h

6

10 kc

clearly represents an extremely large focusing effect for the ground

wave at and near this point. The focusing in the horizontal plane

for the sky wave modes is only slightly different, but we must add to

this the focusing in the vertical plane to obtain the total focusing for

these modes. (1-29) is for a vertical electric dipole receiving

antenna; if a horizontal magnetic dipole were used for reception, then

the J should be replaced by J,. o 1

Note that cos v|v > 0. 995 for m < 16 8 and cos 4* = 1 for m < 8 when

h = 90 km.

7b
All of the above discussion applies to the case when the

effective reflecting surface of the ionosphere is smooth and concentric

with the surface of the earth. In practice, as the sun rises and sets,

or as the geomagnetic latitude of the reflection point is varied, the

surface of the ionosphere will undoubtedly change in such a way that

its radius of curvature and slope relative to a tangent plane on the

earth will vary over appreciable ranges, and this will cause C (R) m

to vary up and down relative to the values expected on the basis of

the above analysis. However, except near the antipode, it seems

plausible to assume that the median values of C (R) may not be m

much influenced by such changes. The magnitude of the antipodal

anomaly will be substantially reduced by these macroscopic

perturbations of the spherical concentric shell model assumed for

our calculations. Note also that there will be concentric rings around

the antipode at which the expected field will be equal to zero. The

radii of these concentric rings are the same, regardless of the

number, m, of ionospheric reflections, and are determined by the

zeros of the Bessel functions; for the electric field, the first two

such rings have radii equal to 0. 38\ and 0. 88\. Note, however, that

the geographical location of the centers of the antipodal anomalies

may be expected to be somewhat different for the different modes for

the actual non-concentric ionosphere, and thus these zeros are not

likely to be observable unless some means is used to exploit their

different times of arrival. The shifts in the geographical locations

of the anomalies caused by these macroscopic changes in the ionosphere would be expected to result in a net increase in the fading

range in the neighborhood of the antipode.

In addition to these systematic macroscopic changes in the ionosphere, the reflecting surface of the ionosphere will be locally rough,

and we will see in the following analysis how this local roughness may
be expected to reduce the median values of convergence as computed

above for a smooth concentric ionosphere.

We have seen above that the convergence depends, at a given

receiving point, upon the smooth variation of the phase of the received

waves with changes in elevation angle and azimuth. If we let a denote the standard deviation of the phase of the waves received via

m hops, then we may use Rayleigh's criterion of ionospheric roughness (see Section 2 for a discussion of Rayleigh's criterion as applied

2

to ground roughness) to calculate o_ in terms of the variance, o\ , of U h

the local effective reflection heights, h, of the ionosphere at the

points of. stationary phase.

720° a cos <() N/m

"n

= "J f 1

" 30'

2

Note that the variance, tr , of phase consists of components arising

from (a) a drift of a fixed pattern of ionospheric irregularities

relative to a reference great-circle-smooth-concentric-ionosphere

path, (b) changes in the shape of these irregularities with time, and

(c) changes in the locations of the reflection points in the horizontal

plane.

3/ Brennan and Phillips — find that variations with time of the intensity

and phase of a one-hop transmission at 543 kc over a 380 mile path

at night indicate rather conclusively that they may be described

adequately most of the time by assuming that the received waves

consist of a steady component with constant phase and approximately

constant amplitude plus a random Rayleigh distributed component of

2

relative intensity k and random relative phase; the amplitude

distribution expected in this case is given on Fig. 5 with

2

K = 10 log k . The fixed component may be identified with a
specular reflection expected from the reference smooth surface

while the random component arises from the surface roughness. It now becomes clear how ionospheric roughness may be expected to

affect the convergence; the specular component will be increased

C (R) db whereas the random component, since its phase is random, m

will not be increased at all. If we Write C (R) s 10 log.,^ c , then m 10 m

we find the following expression for the convergence factor C (R, p)

j m

exceeded 100 p % of the time in terms of the values of k (1 - p)

exceeded 100(1 - p)% of the time:

, c + k (1 - p) > C (R, p) = 10 log 4 ~^-5

I (1*31) 10L l + k

2

(l-p) J

2

Note that k (1 - p) approaches zero as p approaches zero, and thus

C (R, p) approaches C (R), the value expected for a smooth m m _ ionosphere, as p approaches zero. On the other hand, k (1 - p)

approaches co for a perfectly rough ionosphere, and in this case

there will be no convergence and C (R, p) approaches zero. m

2

The values of k to be used in (1-31) may, in principle, be

determined from observations of the variations in either the amplitude

or the phase of the waves corresponding to a single mode of propagation. However, it is ordinarily better in practice to use the variations

in phase as an index to k since the amplitudes of the received waves

also vary with ionospheric absorption, and it is sometimes difficult

to separate out these absorption variations from the amplitude

variations arising from surface roughness alone. In Table 1-1 are

tabulated some experimental measurements of <r . In some cases

the required variance was observed directly, but in other cases it

had to be estimated from phase difference measurements made on

paths with one common terminal, but with their other terminals

10
11

CD rC

+->

Phu

CD

tno

fl I—I ao

<d

en ^ .fi m ^ 3

«j <d

•H>

a;

tn w

b

I—1 g

CMoooCM

i—1

insO

CMo

COoo

1—I

CM

r-o

COin

COo mo

00

oomo

COomo oo

COoi-H o mm

CMo

(Moo

i-H CMO

in00oo

00oo

-ecnou

sOo o

^Ho

ri—i o o

CMO

rr—C o

vDO

CMO o

1—1 o

t—i o1—1 o

1—1 o

in•—i o

oo•—i o

oom

^Ho o

i-H

COo

CO

COo WPo £ CO CO - i-H 1—1 i-H CO CO CO CO CO i-H b O

oo

1—1 o CO CM

LT1

CO

CO

I-H

•—I oo m 00

CM

CO

CO

r-H CM sO

CO o

I—1

CMo

CO

cm

Q.i i—i oo oo oo oo oo oo oo oo

i-H oo OO1—1

CO

00o

CO NOo

CO ^ o1—1 o

i-H o

1—1 o

r—1 o1—i oi-H oi-H oi-H

A ^

^H

r-H i-H

co

CO

m

00

i-H

in

,£>O

b

en

CD<uu

bfl

Cl) P o

00 .—i

1—1

cri—(

COO^

CMLOCO

CO

1—1

1—( oo1—1

in

m 00

rvj

ooo

i—i

CO

CO CM

m CO i-H

XHo

«'w

w

H

Phw

10

H

PhW

CO z< z< Z J Z J

UWP Uwp

UwpPi

5!

XU

Pi2 XO

Pi<2 Xu

Pi

<!2 oz $ < Ph

Pi >\

w

>*3 H

1—1 Z 3 HK

I-H

>*3 HX

1—1 H

I-H

HX

t—i z hX

I-H

HX

i—i z P hl-H z HXI-H z HX

»-H z Wo

PiO

co Z H

Pi ^ w s

Pi

wuw1—1 Wua

Ph

mHZWOPw

in3Pw

inHWPW

Pi

mH&WPw

Pi

mwpw

Pi

in

rHZWpw

Pi

mHWPw

Pi

inHZwpw

Pi

wu

Piw1—1

Ph

NOH

PiW§P Z<o <

r-<

-H

COz<zzw

Pi

ffl

inzwspw

oi

CDU

en

•iH P

cu

w 3 i—( o

no

Om

(MCO o o o o 0000

CO

00oo

CO

00oo

CO

oo00

CO o

COfM

CO om

CO

CM o

oo o

00 o

CM

i—t o

00

CO

in

cr

CD

u

-O vO

CM CMrCM CM oo oo 00 oo o

sD oo

1—1

m

i—i ^H

in.—i

CO

i-H COm

inm

i—i

CM co ^ UO vO r~ oo o-^ o - CM co •tf in vO

i—i
separated by S wavelengths. In this latter case the observed phase

2

difference was related to o as follows:

'ob.. =

2(1 - ^

'0 (I - 32)

In the above p is the correlation between the phase variations along

the two independent paths. The correlation will vary from p = 1 for

S = to p =0 for S > 40X.. Although direct measurements of p do not

appear to be available in the literature, it seems reasonable to

assume that p will be of the same order of magnitude as the correlation between the amplitude variations on paths separated a distance

S at one end. Measurements of the latter correlation were reported

in reference 3, and these data constituted the basis for the estimates

in Table 1-1.

Using (1-30) estimates of cr can also be made, and these

are also given in Table 1-1 and shown on Fig. 1-4. Although the data

are quite scattered, the curved lines labelled day and night, respectively,

represent the estimates used in this report for calculating the median

values of cr using (1-30). It should be noted that or is itself a random

variable which changes over wide ranges from hour to hour and from

day to day. For example, an analysis of the data in reference 3 shows

that the observed phase differences, cr %

. ranged from 5. 3° to more

obs.

than 180°, 10% of the values exceeded 118°, 50% exceeded 34° and 90%

exceeded 14°. We will see below that the maximum value of cr , to

obs.

be expected in practice is 103.9 x n/2 = 147°, which corresponds to

2

k = co, and only 5% of their observed values exceeded this value.

2

The relation between cr and k has been obtained on the

8/ assumption that the data fit the Rice -^ distribution of a constant vector

plus a Rayleigh distributed vector. By integrating over the joint

12
-13-

CO

o_UJ

hLlJ

MEDIAN EFFECTIVE IONOSPHERIC ROUGHNESS PARAMETER, 0"

h

OBTAINED FROM OBSERVATIONS OF 0^

0"

n

( DEGREES) = 2.4 f Rc cos <£ a~ h (km)

L — TE!:^:;m;E" TT , TF ++H—[

T- T-W^--""" ffl- 1

1 i ii j_

S:::::::Bt:::::::::::: it flnu ^ < >r ttt. uu y T

L Tt T "Tjj O Night "

1 -t>« V tt --4+ 1

i—

j

•

1 V^ \ 1 . . -.:.:±.if : &_s it i \" \ i TT 21 V I iii _.::: _ __:

U.I ~ " jl :»ir. it. ::. _ jj —3,1--Si. — it Tr ====== ~ =trrrrF :ffi

:TSJ-^TTt :: = ±±I^EEEEz = :x± zl. ~ —

;

_ 0.5 --pfp^ \ "TTT" s -H r

'

' fjE-:c:::fi: ;: ":_::S: -\--m s %— ft --- ±F- ----# — "P" fe" Tm ^ ~ *"

" ~l u"" .jj

03 - Nl It ——

'

I-- eeeeeeeeej|eeeee eeee; tt === EEE|==fc== ==EEE

Th"'^

hi _ rt —TF 02 rfc 1

- ffr ^ ::5; : -i— -*--£-

4ff - ^ - j-4- *t — s : "---5^ 31: ~~ —H~ T|H s ^

1 S _

" TT" __L__.1LL. ^.. J!

B=

~««zrlt" "i:::::|NIGH":::

n i s Ttt 1 -- .4. __l_-

U.I | U-- :::::^::m-+ ----- Bj\ ::::::k

(\ f!7 --T-ffi- Ie^eeiieeeeeeee ... ..5 .. _ U-Ul :::::::::itt:::::::::::: W - +V $

"— —ft

- J i. .hi Ji|. __ ^v

&1ffiM±l% ^T-i EEEEEE DAY EEEEE

0.05 ±=^E = :::: === = = :=:::

i

1

1

it

0.03 jj--==

iSff|ii|l-eeee====eeee||||||| iliilii

1 _

0.02 ^_ —- :::::::::ffi:::::::::l: tt

_

: i Ti 1 I I |

1 1 1 1 |

i | 1 J III 1 1 i 1 i T 1 1 1

1

20 30 50 70 100 200 300 500 TOO 1000 2000 FREQUENCY IN KILOCYCLES

Figure 1-4
probability distribution given by Rice for all of the variables except

Q , we obtain the following expression for the probability density

2

function for Q with k as a parameter:

2irp{Q) = {1 + n/tT z exp (z ) [1 + erf (z)]} exp (-1/k ) (1-33)

c o s

£"2 where z = —-— . Fig. 1-5 shows p(Q) is symmetrically distributed

k

2 2

about zero for all values of k . When k is very small, sin Q is

distributed approximately normally about zero with variance

2 2 2 c . = k /2. When k approaches infinity, Q is uniformly distributed

sin s? between - 180° and +180° and cr approaches 103.923 degrees. Fig. 1-6

gives the cumulative distribution defined by:

Q(P)

P[0(t)> ft(P)] = 1 - \ p(^)dn (t-34)

- ir

i-i 2

The mean absolute value | n| and variance o_ are also of interest:

TT

I ST2 1 = 2 \ Qp(n) dQ (1-35)

o

TT 2 C 2

o- = 2 \ n p(^)dS7 (1-36)

For many applications the cumulative distribution of the absolute

value of |f}(t)| is of greater interest:—

P' [ \Q (t)| > Q (P')j = 2P[n(t) > (P)] (1-37)

The distribution of P' is given in a recent paper '-* and several of its

percentage points, together with | Q | and <r are shown on Fig. 1-7

as a function of k . By using the median values of <r determined

4 It from Fig. 1-4, we may use the results shown on Fig. 1-7 to determine

2

the median values of k (o. 5) required for the evaluation of C (R, 0. 5). m

14
-15-

THE PROBABILITY DENSITY FUNCTION p(ft)

pto)= ^{|+y¥

^

exp (^)[|+erf (^)]} exp( .X)

0.02 0.05 0.1 0.2 0.5 a IN RADIANS

2 it

0.5 5 10 20 a IN DEGREES

50 200

180

Figure 1-5
-16-

THE CUMULATIVE DISTRIBUTION OF a (t

)

en

UJ

uj

(r

CD

Ixl Q

Q_

<3

90

80

70

60

50

30

20

10

0.0001 0.005 0.01 0.02 0.05 0.1 P[Xitt)>ft(P)]

Figure I~6

0.2 0.3 0.4 0.5
- 17-

QTATiq-nrS OF THE DISTRIBUTION OF THE PHASE

OF A C—vSPLUS A RAYLEIGH DISTRIBUTED VECTOR

100

10 k'

Figure 1-7
Table 1-2 below gives the values of |S7J and of tr for several

values of k .

Table 1-2

k

2

i m *n

\a\

radians radians degrees degrees

0. 01 0.056514 0.070890 3.2380 4.0617

0. 02 0.080059 0. 10051 4.5871 5.7590

0. 05 0. 12726 0. 16023 7.2915 9.1803

0. 1 0.18172 0.23013 10.412 13.185

0. 2 0.26330 0. 34032 15.086 19.499

0. 5 0.44605 0.60664 25.557 34.758

1 0.64346 0.87134 36.868 49.924

2 0.85196 1. 1175 48.813 64.031

5 1.0876 1.3661 62.313 78.270

10 1.2217 1.4972 69.999 85.785

20 1.3212 1.5907 75.701 91.142

50 1.4119 1.6735 80.896 95.884

100 1.4582 1.7149 83.547 98.259

200 1.4911 1.7441 85.431 99.929

500 1.5203 1.7698 87.107 101.40

1000 1.5351 1.7827 87.953 102.14

oo tt/2 tt/>JT 90.000 103.92

18
An analysis was made in reference 3 of both the amplitude

variations on single paths as well as the phase differences between

paths separated by 3. 39 wavelengths at one end. Using their observed

median standard deviation of <r , = 34°, the median value o>» = 41°

obs. U

given in Table 1-1 was estimated; this value corresponds by Fig. 1-7

2

to k (0. 5) = 0. 68. The analysis of their amplitude variations gives

2 ~ z directly the estimate k (0. 5) = 2/a = 0.854, and this latter estimate

is somewhat larger, as might have been expected, since the amplitude

variations are biased by changes in absorption.

19
References to Appendix I

1. L. H. Doherty, "Geometrical optics and the field at a caustic

with applications to radio wave propagation between aircraft, " Cornell University Report EE 138.

2. British Association Mathematical Tables, Part Volume B; British

University Press, Cambridge, 1946.

3. D. G. Brennan and M. Lindeman Phillips, "Phase and

amplitude variability in medium -frequency ionospheric transmission, " Technical Report No. 93, Massachusetts Institute

of Technology Lincoln Laboratory, September, 19 57; note that

the a in this report, which is used as a parameter describing

the Rice distribution of a constant plus a Rayleigh distributed

vector, is equal to Nf2~ /k where k denotes the relative intensity

of the random component as used in this appendix.

4. John A, Pierce, "Intercontinental frequency comparison by

very-low-frequency radio transmission, " Proc. IRE, vol. 45, no. 6, pp. 794-803, June, 1957.

5. P. G. Redgment and D. W. Watson, "Phase -correlation of

medium and very-low-frequency waves using a baseline of

several wavelengths, " Admiralty Signal and Radar Est. , Lythe Hill House, Haslemere, Surrey, England, Monograph

No. 836, October, 1948.

6.

R. H. Doherty, "Pulse Sky Wave Phenomena Observed

at 100 kc, " Private Communication, Feb. 6, 1957

7. Private communication from E. F. Flo rman of the Boulder

Laboratories of the National Bureau of Standards.

8. S* O. Rice, "Properties of a sine wave plus random noise, " Bell System Technical Journal, vol. 27, pp. 109-157, Jan., 1948;

the joint probability distribution is given by equation (4. 6).

9. K. A. Norton, E. L. Shultz, and H. Yarbrough, "The probability

distribution of the phase of the resultant vector sum of a constant

vector plus a Rayleigh-distributed vector, " Jour. Appl. Phys. „ vol. 23, pp. 137-141; January, 1952. Note that the k in this

reference is a power ratio rather than a voltage ratio and that

the formulas and graphs give the distribution of j 4>j.

10. Private communication from J. R. Wait of the National Bureau

of Standards, Boulder Laboratories.

20
Appendix II

The Physics of Ionospheric Scatter Propagation

Wheelon —' —-» —' gives the following formula for the

ed power, p , relative to the power,

s gation over the same distance in free space:

scattered power, p , relative to the power, p , expected for propa-

(p /pr

) = 4ir b sec <j> cr(k ) (II-l)

s i

(r(k

2

) = constant r

2 < [d N /dh] * 5 > f(k ) (1 1-2)

e e o

2 2 2

~ 5 ^ 2

2 2 2/^ ~ 22 2 2

~ 4 ' 3

f(k^> = [i + k^ ri [Hik'rr3

] [i + <k y] m-3)

The dimensionless constant in (1 1-2) is of the order of unity, and will

be set equal to one in the subsequent analysis; k is defined by (32).

[d N /dh] is the gradient of the electron density expressed in

electrons /cubic meter /meter; b is the effective thickness of the

scattering layer expressed in meters; the classical electron radius,

r =2.817*

e in meters.

-15 r = 2.81785 X 10 meters and I is the scale of turbulence expressed

e o

2

Note that when f = f we have k = and f(0) = 1. The constant

S(0. 5) = - 8.4 db determined from the radio data may be readily

identified with:

S(0. 5) = 10 log 4u b o-(O) = - 8. 4 (II-4)

Consequently it follows that:

4irb r

2 < [d N /dh]

2

|

5 > = 0.1445 <H-5)

e e o

II-l
If we let b = 10, 000 meters, we obtain:

< fd N /dhl

2

i 5 > = 14. 48 X 1Q

22

(H-&)

e o

Note that the average indicated by < > is taken over the scattering

volume. In the troposphere it has been found that I is a random

variable with respect to time at a fixed point and with respect to

location at a fixed time; more specifically, L = 10 log i has been

found to be normally distributed about its median value I with

om

cr = 5 db. It seems not unreasonable to assume a similar variability

2

for I in the ionosphere. Similarly we may assume that [d N /dh] is

log -normally distributed about its median value [d N /dhl with a

e m

similar standard deviation, i.e. , about 5 db. On these assumptions

it can be shown —' by simple statistical analysis that:

< [d N /dh]

2

i

5 > = [d N /dh]

2

I 5

exp[0. 02651 o- 2

] (II-7)

e o em om

In the above, (r denotes the standard deviation, expressed in decibels,

2 5

of 10 log {[d N /dh] " i }; if we neglect any correlation between the

e °

2 2 2

variations of [d N /dh] and of I , then <r s (5) -»- (25) = 650 and

2

e 7°

exp [0. 02651 <r ] = 3. 045 X 10 . If we combine (II-7) and (II-6) and

set I = 100 meters, we obtain [d N /dh] = 690 electrons /c. c/ om e m

kilometer. If this value is compared with the value 3, 800 electrons/

2

c. c. /kilometer expected with (r = 0, we see the importance of allowing

for this statistical correction; the actual value probably lies somewhere

between these two estimates, and can be estimated more precisely only

when more adequate information becomes available relative to the

variances of these variables.

The above analysis refers to the scatter expected for frequencies

just above the E layer MUF. The forward scatter on the higher

1 1-2
frequencies where S was actually evaluated becomes independent of J? 22-5/2-5-5 22

since fl + kfl "= k I when k I > > 1. In this case the

o o °

2

correction factor should be determined for <r = 25, i.e.

»

exp[0. 0265 <r ] = 1. 940 and the expected median gradient on these

assumptions is then 2700 electrons /c. c. /km.

It appears from the above analysis that S may increase with

decreasing frequency because of the increasing importance of the

variance of i at these lower frequencies. This statistical factor should

o

not be ignored in analyses of ionospheric scatter data. However, it

was suppressed in the present analysis because of the lack of definitive

data on <r .

Note that scale lengths of the order of I - 100 meters and

om

electron density gradients of the order of 11 000 electrons /c. c. /km.

are not unreasonable values to assume for the lower ionosphere, and

We conclude that Wheelon's theory provides a useful description of

ionospheric turbulence which is not inconsistent with our knowledge

of the ionosphere

II-3

Appendix III

An Additional Height-Gain Factor in Transmission Loss

In free space the field strength e, expressed in volts per meter

at a distance d, expressed in meters, from an isotropic transmitting

antenna radiating p watts may be determined from the relation:

ez

4ir d

(III-l)

(Radiation from an isotropic

antenna in free space)

-7

where z = 4tt c • 10 = impedance of free space expressed in ohms,

o and c = 2.997925* 10 meters per second = velocity of light in free

space.

Now consider the intensity of the radiation field of a short

vertical electric dipole antenna of length 1 and at a height h above a

perfectly conducting plane. By re-distributing the field in the space

above the plane, the radiation resistance is modified by the presence

of the surface as follows:

R = 5ulL [1+ a j e 3\

Z

(m-2)

a

(2k h ) a

sin (2k h ) a

2k h

cos (2k h ) a

(HI-3)*

* These relations are derived by S. A. Schelkunoff in Chapters VI and IX

of the book "Electromagnetic Waves, " D. Van Nostrand Company, 1943.

m-i
In the above k = 2tt/\ = Zrr f /c, i. e. , X is the wavelength in free space.

Note that A approaches zero at large heights above the surface, and

R approaches its free space value. On the other hand A = 1 for

h = 0, and the radiation resistance is then just twice its free space

value. Using (III-2) we find that the field intensity of the short dipole

over the perfectly conducting plane surface may be expressed:

,2

2 p (3/2) [2 cos ty cos (k h sin ty)] e r a

(HI-4)

Z

4ir d

2

[l + A ]

St

(Radiation from a short vertical electric

dipole over a perfectly conducting surface)

Note that the factor (3/2) is just the free space gain of the short dipole

antenna. Since A =1 for h = 0, the field intensity is 3 db greater

a a

when ip = for a dipole on the surface of a perfectly conducting plane

than for a short dipole in free space. In more familiar units (III-4)

with h s t(i s may be expressed:

a

e(uv /meter) = 299,896.2 N/plkwTR (h =i|j=0) (III-5) km a

Furthermore, the effective absorbing area of a short vertical

c dipole antenna a1

plane may be expressed:

electric dipole antenna at a height h above a perfectly conducting

D

a = v2< 3

/

2

> cos> (m-6)

e 4u[l + A

b

]

where A, is defined by (III-3) with h replaced by K, ,

Combining (III-4) and (III-6) we may express the transmission

loss in decibels between short vertical electric dipole s at heights h and

1l above a perfectly conducting plane as follows:

HI-2
= Lir - G - G + A (III-7)

bf t r

A = - 20 log1n [2 cos^>|> cos (k h sin \\>)] + L + L, (IIt-8) "10" a a b

Note that L» is the basic transmission loss expected in free space,

G - G = 1. 761 db, and that the transmission loss, A, relative to free

t r space contains two height gain factors which are not ordinarily considered in field strength calculations. No allowance was made in the

calculations in this report for the additional losses L. and L. which

a b

arise from the redistribution of the field intensity in space which, in

turn, is associated with the proximity of the antennas to the ground.

Thus the transmission losses shown on Figs. 7, 8, 16, 17, 18, 19, 20,

21, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, etc. are too small by

an amount ranging from about 6 db at very low frequencies and low

antenna heights to zero at the higher frequencies. Fig. HI-1 shows

this additional loss as a function of antenna height (h/X), expressed in

wavelengths, for the case of a perfectly reflecting surface, and this

should also represent a good approximation in those cases where the

antennas are erected over large ground screens.

It is of interest, although not surprising, to note that the

transmission loss between vertical electric dipoles on the surface of

a perfectly conducting plane (h = h, = iJj = 0) is the same as if the - a b

dipoles were in free space, even though the field intensity at the surface

is 3 db greater.

It should be noted that Schelkunoff identified the factors in

(III-2) somewhat differently; thus he considered the ground to be an

integral part of the antennas, and set G = 10 log,

n {(3/2)o2/[l -f A ]},

III3
G = 10 log {(3/2)«2/[l + A, ]} and A = - 20 log, J cos i|j cos(kh sinl|i)]. T 10 D 1U 3. It seems to this writer that the terms 10 log [ 1 + A ] should be

10 a, b

excluded from G and G . Thus, according to this approach, G and

t r t G are the free space gains of the antennas, independently of their

r location, and the path antenna gain G , when measured by replacing

P

the actual antennas by isotropic antennas^ will still be approximately

equal to G, + G . t r Suppose now that we use small loop antennas of area S, with

their axes normal to the plane of propagation, parallel to the perfectly

conducting surface and at heights h and h , respectively. In this

a b

case:

R = *djsf

m [1+A] 3X

(IH-10)

^ = (3/2) to-

. v sin (2k h, ) 1 \ b

+

cos (2k h, ) -i D

(2k h

b

) 2 ni J

, ,2 J 2k h, (2k h«) ' b

b

(ni-ii)

e

" 4tt [1 + A' ] (ni-12)

A = - 20 log [2 cos (k h sin i|i)] + LJ 4- L« (111-13)

1U 9, 3-D

Note that At, approaches zero at large heights and A' = 1 £°r \_ - 0. b b b

Consider next the transmission loss between two small loop

antennas at heights h and h , respectively, above a perfectly conduct3. D

ing surface with their axes normal to this surface. In this case:

R = m

3 2

8u z S

3X

4 -A] (in-H)

in-4
A = - 20 log r2 cos

2

4> sin (k h sin ty)] 4- L" + L» (111-15)

10 a a b

L

"a,b = 101

°8lo[

1 - A

a.b] (m-16)

The factor L" is also shown as a function of (h/X.) on Fig. III-l.

Finally consider the transmission loss between two short

horizontal electric dipoles of length I, normal to the plane of propagation and at heights h and h , respectively, above a perfectly conducting

a b

plane surface. In this case:

r = £l_£-L fi . A.] (ni-17)

3\

2

A = - 20 log [2 sin (k h sin i|/)] + L,M + L«" (HI-18)

J. u el 3. D

^b-w^o^-^b' <ra-19)

Note that Lm and Ln both approach (- co) as h approaches zero, but the

radiation resistance R simultaneously approaches zero, and it would be

difficult in practice to keep the radiated power constant as the antennas

are brought ° nearer and nearer to the surface. When h and h, are

a b

both much less than a wavelength, A, as defined by (111-15) for horizontal

loops becomes independent of these heights and equal to A=20 log (kd/5);

similarly A, as defined by (111-18) for horizontal electric dipoles approaches

20 log (2 kd/5) for h and h much less than a wavelength.

Since the factors L m and L,,M were omitted in calculating the

a b

transmission losses shown on Figs. 9 and 10, these values are much

too large at the lower frequencies since the 30-foot antennas are in this

case only a small fraction of a wavelength above the surface.

m-5
All of the above results refer to the case of a perfectly

conducting plane surface and to distances d > > X. For a finitely

conducting ground, the factor 2 cos (k h sin *\>) in (III-8) and (111-13)

a

or the factor 2 sin (k h sin v(j) in (111-15) and (111-18) must be replaced

a by the appropriate attenuation factor, 2 W, relative to the free space

field. For example, for electric dipoles over a flat earth of finite

conductivity and with h = h = 0:

a b

W = 1 1 + i NfiFp" exp ( - p) erfc (- i N^p" | (HI-20)

Here p denotes Sommerfeld's numerical distance as defined in reference

13 where a comprehensive discussion is given of the radiation fields of

electric and magnetic dipoles over a finitely conducting plane earth.

Furthermore, A and A' will be modified when the antennas are located

over a finite ground, J J J J but this difference will often largely be

cancelled in practice if a large ground screen is used under the

antennas. Although (111-14) indicates that R approaches zero as the m

vertical magnetic dipoles approach the perfectly conducting surface,

.1/ Wait —' has shown that R becomes very large when such loops are m

brought near a finitely conducting ground.

It is sometimes convenient to be able to relate the basic transmission loss, I , to the field strength er b

p /! = p = (e

2

/z).(\

2

/4Tr) (HI-21)

a b a

(Isotropic antennas in free space)

Expressed in decibels, we obtain from (111-21):

L^ = 77. 216 + 20 log1rt f. + P - E^ (111-22)

b 10 kc r b

(Isotropic antennas in free space)

III -6
In the above, P is the radiated power expressed in db above

r one kilowatt, and E is the field strength in db above one microvolt per

b

meter. If antennas with free space gains G and G are used, we find

t r that E = E, + G , and the transmission loss between these antennas in

t b t free space may be expressed:

L = L -G -G = 77.216 + 20 Wf +P - E - G (111-23)

b t r B10 kc r t r

(Antennas with gains G and G in free space)

t r ve dipole transmitting antenna G =2.15 db, a from the above the relation given at the bottom of page two of the report.

For a half wave dipole transmitting antenna G =2.15 db, and we obtain

References

1. James R. Wait, "Radiation resistance of a small circular loop

in the presence of a conducting ground, " Jour, of App. Phys. t vol. 24, no. 5, 646-649, May, 1953.

2. James R. Wait and Walter J. Surtees, "Impedance of a toploaded antenna of arbitrary length over a circular grounded

screen," Jour, of Applied Phys. , vol. 25, no. 5, 553-555,

May, 1954.

3. J. R. Wait and W. A, Pope, "Input resistance of L.F. unipole

aerials, " Wireless Engineer, vol. 32, 131-138, May, 1955.

4. J. R. Wait, "Effect of the ground screen on the field radiated

from a monopole, " IRE Trans, on Antennas and Propagation^

vol. AP-4, no. 2, April, 1956.

Ill -7
nr-8-

TRANSMISSION LOSS ARISING FROM A

CHANGE IN THE RADIATION RESISTANCE

OF SHORT DIPOLE ANTENNAS NEAR A

PERFECTLY CONDUCTING SURFACE

C/) _l

UJ

CD

O

UJQ

(h/X) HEIGHT OF ANTENNA IN WAVELENGTHS

ABOVE THE SURFACE

Figure 3H-I

NBS





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