PB I51374

jSoulder laboratories

920.15

PREDICTION OF THE CUMULATIVE

DISTRIBUTION WITH TIME OF GROUND

WAVE AND TROPOSPHERIC WAVE

TRANSMISSION LOSS

PART I - THE PREDICTION FORMULA

BY P. L. RICE , A. G. LONGLEY , AND K. A. NORTON

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical ^iote

15

July, 1959

Prediction of the Cumulative Distribution with Time

of Ground Wave and Tropospheric Wave Transmission Loss

Part 1 - The Prediction Formula

by

P. L. Rice, A. G. Longley and K. 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 $1 .50
This Technical Note was originally given limited distribution as

NBS Report No. 5582, dated June 30, 1958. There have been some

minor corrections and additions, including more graphs of some functions

in order to simplify the calculations. Also added here is a figure showing

deviations of all of the currently available forward scatter data from predicted values; the theory itself was used to eliminate, insofar as this is

feasible, those long-term median values influenced by diffraction.

The following reports which depend upon the prediction methods

described in this Technical Note are now being prepared for publication:

(1) A. P. Bar sis, K. A. Norton, and P. L. Rice, "Predicting

the Performance of Long Distance Tropospheric Communication

Circuits. "

(2) A. P. Barsis, P. L. Rice, and K. A. Norton, "The Use of

Measurements in Predicting the Performance of Tropospheric

Communication Circuits. "

The recent report* of the Television Allocations Study Organization

to the Federal Communications Commission, "Engineering Aspects of

Television Allocations, " includes smooth earth curves of field strength

versus distance calculated using the methods of this Technical Note, as

well as comparisons with data obtained over land at VHF and UHF. This

TASO report alsb includes estimates of long-term variability as a

function of distance, antenna height, and frequency, rather than angular

distance. For angular distances less than 10 milliradians and for lineof-sight distances, the estimates prepared for TASO are better than those

given in this Note.

Within a year it is hoped to submit for publication in the NBS

Journal of Research, Part D, a detailed explanation of the prediction

methods of this Technical Note, greatly simplified without loss of

accuracy, and including a discussion of the theory. At that time, improved estimates of long-term variability will also be made available,

with^an explanation of how they were obtained from the data.

NBS Technical Note No. 12, "Transmission Loss in Radio Propagation - II", by K. A. Norton and the paper "System Loss in Radio Propagation, " by K. A. Norton, J. Research, NBS, 63D, pp. 53-73, July - August, 19 59, both contain examples of the use of the methods of calculation given in this Technical Note.

* This report may be obtained from Dr. G. R. Town, Ames, Iowa at a cost of $10. 00.
Prediction of the Cumulative Distribution with Time

of Ground Wave and Tropospheric Wave Transmission Loss

Part I - The Prediction Formula

by

P. L. Rice, A, G. Longley, K. A. Norton

This report describes a method for predicting the cumulative distribution with time of transmission loss at frequencies above

10 megacycles per second over paths of arbitrary length. The method

makes use of available information about terrain profiles and surface

meteorological data, and is based on the CRPL radio standard atmospheres, in which the radio refractive index decreases linearly with

height for the first kilometer above ground, and then decreases

exponentially with height ±j . Discussion of the theoretical basis for

this formula and a demonstration of its accuracy by comparison with

experimental data are reserved for later parts of this report.

1. Introduction

Terrain irregularities, climate, and weather play principal roles

in determining the strength and fading properties of a tropospheric

signal. Estimates of transmission lossr.' and its long-term and shortterm variability are essential to the design and allocation of military

and commercial communication links and broadcasting and navigational

facilities.

Until fairly recently, distances much beyond the horizon of a

radio transmitting antenna were considered too great for useful communication links at frequencies above 100 megacycles per second. This

particular range of distance , the extra-diffraction region out to about

1, 000 miles, is important in the application of tropospheric forward

scatter techniques, and our prediction formula is especially reliable in

this range. For propagation paths longer than 1, 000 miles, only the

ionosphere will support measurable scatter signals. For distances

well beyond the horizon in the frequency range above 100 Mc, observed

scatter signals are tropospheric. The present usefulness of the
ionospheric scatter mechanism ceases at about 60 Mci.', while tropospheric propagation promises to be useful for communication purposes

for frequencies between 40 Mc and about 10, 000 Mci'.

At these frequencies and out to distances just beyond the radio

horizon of a transmitter, the dominant propagation mechanism is

diffraction. An earlier report by Norton, Rice, and Vogler.2/ gives

a formula for calculating transmission loss due to diffraction over

rough terrain profiles.

At greater distances, the dominant propagation mechanism is

usually forward scatter °/, especially during winter afternoon hours

when strong ducts or elevated layers are rare. During these hours,

observed transmission losses are in excellent agreement with forward

scatter theory.

Often, for other times of day and seasons of the year, the scatter

phenomenon may more properly be called reflection, and sometimes

no distinction can be made between "forward scatter" from a turbulent

atmosphere".' and the addition of "incoherent reflections" from patchy

elevated layers—' . The examination of transmission loss variations

over a particular path during some period for which detailed information about layer heights and intensities is available can be very

illuminating £/. The prediction method described here gives good

estimates of results to be expected over such paths when only surface

meteorological data are available. Median diffraction and scatter

fields estimated for winter afternoons are combined and then converted

empirically to estimates for other periods of time.

2. Definition of Transmission Loss

Transmission loss, L, is defined as the ratio expressed in

decibels of the power, p r

, radiated from a transmitting antenna to the

power, pa

, available from the receiving antenna:

L= 10 log

1() (p r /pa) (2.1)

It is convenient to express the results of propagation measurements and theoretical calculations in terms of the "basic transmission

loss" expected if isotropic transmitting and receiving antennas were

used. This basic transmission loss, Lb, may be determined from the

measured transmission loss, L, by adding the path antenna gain G : Jr
L = L + G (2.2) b P

When G is not known, an estimate of L* may be made for short paths

by assuming that the free space gains of the transmitting and receiving

antennas are realized, i.e. , that Gp

= Gt + Gr

. The corresponding

value of basic transmission loss will be a little too large. For larger

distances, see Section 11 and Reference 9 for methods of estimating G_.

Basic transmission loss L, is related to field strength, E, expressed in decibels above one microvolt per meter for one kilowatt of

effective power radiated from a half-wave dipole:

where

L= 139. 367 + 20 loglft f - E - GT (2.3) b °10 mc L

GT = G + G - G . (2.4)

Li t r p

In determining E from measured data it is implicitly assumed that

the free space antenna gains are realized; i. e. , that GT = 0. The appendix

contains a brief summary of useful relationships between such quantities

as transmission loss, field strength, attenuation relative to free space,

and the voltage or power available at receiver terminals, including relationships useful for the solution of some systems design problems.

3. Cumulative Distribution of Transmission Loss

This paper considers cumulative distributions of hourly median

values of transmission loss. All subsequent discussion and the distributions shown in Figs. 2-12, are concerned with hourly medians of

basic transmission loss and their diurnal and seasonal variability. This

paper is not concerned with estimating short-term (within -the -hour)

variability.

Basic transmission loss L^ expected for the diffraction mode of

propagation during winter afternoons is calculated according to the

formulas in Reference 5, as modified for high antennas in Section 7. For winter afternoons, the median basic transmission loss, Lu , due oms

to forward scatter, is given by the following:

L. = G(ti) + 30 logl „ f - 20 log1rt d bms " s

10 mc 510 mi

+ H -F -F - F(h ,N)+A (3.1) o o s eg s a

-3-
The radio wave frequency, fmc , is in megacycles per second, and the

great circle propagation path distance, d, is in statute miles. Other

terms are explained in later sections.

Reference 4 gives a less complex formula for calculating Lbms'

but (3. 1) is more reliable for propagation paths with unusual profiles

or climatology, for very high antennas, and for the lower frequencies.

Given Lbd and Lbms' corresponding to diffraction and forward

scatter, the resultant median basic transmission loss is

Lbm= L

bd- R(°- 5) ' (3 - 2 »

R (0.5) is given graphically as a function of (Lbd " Lbms) on Fig. 1,

as determined from a paper by Norton, Vogler, Mansfield and Short 10.'

.

Within one -tenth of a decibel, Lbm - ^-"bd when (L-bd " Lbms) ^ s less

than -18. 1 db, and L-bm = ^bms when L«bd ~ Lbms ^ s greater than 17.9 db,

The analysis of transmission loss data was concentrated mainly

on times of day, 1 p.m. to 6 p.m., when great departures of refractive

index profiles from a linear gradient are least likely, in order to avoid

the frequent occurrence of ducting and layer conditions. Field strengths

are higher in summer than in winter on an average, and diurnal trends

are usually most pronounced between May and October in the northern

hemisphere, with minimum fields occurring in the afternoon. To allow

for such effects, data were grouped into eight "Time Blocks", defined

in the following table:

Table I Time Blocks

1. Nov. -Apr. 6A.M. - 1P.M.

2. Nov. -Apr. 1P.M. - 6P.M.

3. Nov. - Apr. 6 P.M. - 12 Mn.

4. May -Oct. 6 A.M. - 1P.M.

5. May - Oct. 1 P. M. - 6 P.M.

6. May - Oct. 6 P. M. - 12 Mn.

7. May - Oct. 12 Mn. - 6 A. M.

8. Nov. - Apr. 12 Mn. - 6 A. M.

Hourly median distributions for winter (Time Blocks 1, 2, 3, 8)

and summer (Time Blocks 4, 5, 6, 7) are shown as Figs. 12a and 12b.
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Empirical curves, given in Figs. 2-12, define a set of time

variability factors V(p, 0) which show, as a function of angular

distance, the difference between median fields and fields for other

percentages of the time within each time block and for all hours in a

year. The angular distance, 9, is the angle between horizon rays

from transmitter and receiver; methods for its determination are

given in Section 5. The field strength E(p) exceeded by p per cent

of the hourly medians in a given time block is given by

E(p) = E (Time Block 2) + V(p, 6) (3.3)

and the corresponding transmission loss L(p) is given by

L(p) = L (Time Block 2) - V(p, G) (3.4) m

where Em and Lm are median field strength and median transmission

loss estimated for winter afternoons, and V(p, G) is the time variability

factor appropriate for the period of time for which E(p) or L(p) is desired.

Summarizing,

L(p) = L

fad - R(0.5) - G - V(p, G) . (3.5)

R(0. 5) as given by Fig. 1 requires (1) the calculation of Lua by methods

explained in Reference 5 and Section 7, and (2) the calculation of Lrums

as given by (3. 1) and explained in Section 9. Formulas for estimating G

are given for the general case in Reference 9, and in Section 1 1 of this

report for the special case where free space antenna gains are the same

for both transmitting and receiving antennas. V(p, G) for all hours of

the year, winter and summer, appropriately combining the time blocks, is

shown in Figs. 12, 12a, 12b. Varying p, (3. 5) provides a prediction

for the entire cumulative distribution of radio transmission loss for all

hours of the year.

A comparison of measured and calculated values for KIXL-FM

in Dallas, Texas, recorded by the University of Texas at Austin is

shown in Fig. 13 for each time block and for all hours of the year.

Several measured cumulative distributions are shown for each calculated

one; the station was monitored all day and all night for three and

a half years. From Fig. 1 3 it may be noted that medians for

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MEDIAN FORWARD SCATTER RADIO TRANSMISSION LOSS

RELATIVE TO FREE SPACE CORRESPONDING TO

WINTER AFTERNOON CONDITIONS

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40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 DISTANCE IN STATUTE MILES720 760

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PREDICTED FOR AN EXPONENTIAL

ATMOSPHERE USING NBS REPORT No. 5582

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( Omittir g 50 minute media is)

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Figure 13a
Explanation of Fig. 13a,

Observed median values of attenuation relative to free space are

plotted versus distance in Fig„ 13a for a large, heterogeneous sample

of data, obtained in most cases with broad beam antennas. No normalization for the effects of frequency, antenna height, loss of antenna

gain, terrain or meteorological parameters is included. Almost all

recordings were made in the wintertime, and correspond to conditions

where tropospheric forward scatter is expected to be the dominant propagation mechanism. Two points are indicated where ionospheric rather

than tropospheric scatter may have been recorded at 100 Mc. No data

are included for cases where diffraction is expected to be more important than forward scatter, such as short paths where the angular distance

is small.

Fig. 13a also shows deviations of data from the point-to-point

transmission loss prediction method of this technical note; these deviations are plotted versus distance, using the same data and the same

scale. The NRL overwater 50-minute medians shown in Fig. 13a were

excluded in calculating the r. m. s. deviation of 6. 75 db from the

empirical curve and the r. m. s. deviation of 5. 77 db from the theoretical

values.

The data include 80 paths from a special experiment conducted

in Ohio with randomly selected receiving sites at 85 and 125 miles.

For these paths, long-term median observed transmission loss is

slightly greater than when antenna sites are carefully selected to be

unobstructed. The prediction method in this report indicates an increase

in transmission loss of about 1 2 db per degree of increase in horizon

elevation angles, and these angles were often large for the randomly

selected "Ohio mobile" receiving sites.

-18b-
Time Block No. 4 vary by as much as three and a half decibels from

year to year, even though each value represents almost 1300 hours

from the same season and time of day each year. Note that our predicted values, for this particular path, depart from the observed

values no more than the observed values depart from each

other from year to year. In a subsequent part of this report a comparison of our predicted values will be made for all of the paths for which

we have both transmission loss and profile information. We conclude

from an examination of these data that it is usually safer to base the

design of a point-to-point radio relaying system on the transmission

losses predicted by our formula than to base this design on a set of

transmission loss measurements made over a period of only a few

months. It should be noted in this connection that path loss tests are

seldom made at the precise locations or with the same antennas as

are used in the final systems, and consequently really provide very little

pertinent information as to the final performance of the system over a

long period of time. Furthermore the prediction formula cannot be used

without precise profile information and, when this is obtained, it often

leads to the choice of an alternate site which the prediction formula

may indicate is somewhat better.

As an example of the use of the formulas in this report, the

detailed calculations required to reproduce the theoretical curves of

Fig. 13 are given in Section 12.

4. Description of Radio Standard Atmospheres

The forward scatter theory which provides a basis for prediction

of radio transmission loss at great distances depends upon work done

by Booker and Gordon to explain scattering from the stratosphere. JJ/

The scattered power is assumed proportional to the square of the mean

gradient, (dn/dh), of the refractive index of the atmosphere. In what

follows we will use the refractivity N = (n - 1) x 10 as a measure of n. Recent work by Bean and Thayer 1/ has led to the definition of a set of

standard atmospheres which agree with average measured refractivity

profiles from the ground to great heights, and which depend upon the

average value of the surface refractivity, N , and the height of the

surface above sea level, h . Each of the new standard atmospheres

has a linear refractive index gradient for the first kilometer above

ground. Refractivity decreases exponentially from this height above

the surface to a value N = 105 at a height h = 9 kilometers above sea

level. For greater heights the standard atmospheres are the same,

with an exponential gradient independent of the surface values of

refractivity.

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20
The average refractivity gradient near the ground, AN /km, is

specified by the following empirical relationship in terms of the average

surface refractivity, N :

AN per kilometer = -1.2,2 exp(0. 005577 N ) (4. la)

"When a good estimate of the average gradient AN is available

from refractivity profiles, the average surface refractivity N to be

5

used in the calculations should be obtained from (4.1b):

N» =412. 87 log | AN /km | - 356.93 (4.1b)

S J.U

rather than using the observed average value of N in case it differs

s appreciably from N' . Transmission loss L(p) is obtained for a given

N_ by linear db interpolation or extrapolation from calculations for

two adjacent values selected from the four reference values, N = 250,

301, 350, and 400.

The refractivity gradient, AN, largely determines radio ray

bending through the atmosphere. For each AN, an effective earth's

radius, a , is defined in such a way that rays drawn above an earth

with this effective radius will not bend until they reach heights exceeding one kilometer. Table II shows for the four reference atmospheres

the values of the effective earth's radius, a, in statute miles, and

the corresponding linear gradient, AN /km, for the first kilometer

above the surface. Also tabulated are the logarithmic gradients, v , appropriate for greater heights where the refractivity N decreases

exponentially with height.

Fig. 14 illustrates for both rough and smooth earth conditions

the geometry of a forward scatter link in a linear gradient atmosphere.

The most important parameters shown are the propagation path

distance, d, and elevation angles of the radio horizons, a and (3 , oo oo

21
-22-

GEOMETRY FOR A LINEAR GRADIENT ATMOSPHERE

Distance to earth s center = a

Distances are measured in statute miles along a great circle arc

Dc #00 ~ "a- + #ot + #or

Figure 14
for a linear gradient atmosphere. Section 5 of this report shows how

the true elevation angles a and (3 are obtained for the longer paths

and for very large antenna heights, using the standard exponential

atmospheres of Table II.

Within the scattering volume pictured in Fig. 14, the relationship

between refractivity, N, and height above ground, h - h , is taken to

be

N(h-h)=N expT-v (h - h )] (4.2)

s e s

where h is the height of a scatterer above sea level and h is the average

s height of the surface above sea level:

h

S = (hLt

+ h

Lr )/2 < 4 ' 3)

Where h cannot be determined, the values assumed in Table II may be used,

The parameters h and h are heights of radio horizon obstacles

J_/t i_ir

above sea level, as illustrated in Fig. 14. Extrapolated refractivity

values, N ,, and the logarithmic gradients, -y per mile, are listed

in Table II for various ranges of the height of the "scattering center"

above ground, h . An approximation to this height has been derived

in terms of the height, h , of the crossover of radio horizon rays above

the reference axis AiA ? from transmitter to receiver:

h =* 6 h + 0.

(4.4)

eg o

Here, h is in thousands of feet and h is in miles; note the difference

eg o

in units. 6 Section 8 contains a further discussion of these parameters.

5. Path Geometry

The steps required to obtain accurate estimates of the geometrical

parameters defined in Fig. 14 for an equivalent earth of radius a are

as follows:

a. Plotting a Terrain Profile

Plot a great circle path terrain profile between transmitter

and receiver locations. If topographic maps are used, denote by

(N XX, W ) and (N , W ) the latitude and longitude of antennas at

Cs La

23
A, and A? on the surface of a spherical earth of radius 3960 miles.

Let (N, W) be the geographical coordinates of any point on the

great circle between Ai and Ai. Then,

A cos (W - W ) cos N + B sin (W - W ) cos N + sin N = (5.1)

is the general equation for a plane through the center of the earth

and a point (x, y, z), where x = cos (W - W ) cos N,

y = sin(W - W.) cos N, z = sin N. This is a cartesian coordinate

system with z in the direction of the earth's axis, x - y the

equatorial plane, and the x-z plane passing through the antenna

location A,. Geographical coordinates of (x, y, z) and (N, W),

and the constants A and B are evaluated by substituting the known

coordinates (N,, W,) and (1NL, W2) in (5.1):

A = - tan N (5.2)

B = tan N cot (W, - W ) - tan N

£

esc (W - WV). (5. 3)

A great circle path is plotted on a contour map by choosing

values for N and solving (5.1) for W, or vice versa. The great

circle distance d between points with coordinates (N, W) and

(N', W) is given by

cos (d/a») = sin N sin N1 + cos N cos Nf cos (W - W')t (5.4)

where a' is the actual earth's radius.

b. Determining Radio Horizons

Having obtained a great circle path terrain profile, the next

step is to determine the location of radio horizon obstacles at each

end of the path. Where a smooth earth is involved, over the sea

for instance, the methods described in subsection (5d) for calculating

aQ and (3 are easier than methods described in subsections (5b) and

(5c). For an antenna less than one kilometer above ground, the

location of the equivalent earth radio horizon is used in the calcu*

lations. The correct horizons maximize the horizon elevation angles

9et and 9er : h

Lt " hts

d

Lt

9et = 5280 d

S

" "2a"

radians > < 5 ' 5 > Lt

24
h - h d

9 - r.^„~ , - -^— radians (5.6)

er 5280 dT 2a x

' Lr

where heights are in feet and distances in miles. Having

approximately located the radio horizons by inspection, the

values of h.-, . and d-r . or h^ and d, corresponding to nearby

high points are substituted into (5. 5) and (5. 6); values which

maximize 6 and fi are the correct ones. If 9 , or 0_ r

is

et er et er

negative, choose the value nearest zero.

As illustrated in Fig. 14, h and h are transmitting and

x s rs receiving antenna heights above sea level, and d, , and d are

J-<t j_,r

distances to radio horizon obstacles from transmitting and

receiving antennas. The effective earth's radius, a, is found

in Table II.

c. Computation of Angular Distance

The angles a , (3 , and = aQ + P for a reference

atmosphere are in general larger than the angles a , (3 , and

8 computed for the corresponding linear gradient atmosphere

if either antenna or the scatter volume is more than one kilometer

above the surface. Accordingly, 6 is computed in two steps;

first, a _ and (3 are calculated for a linear gradient atmosphere,

and then a and (3 are obtained by adding small corrections,

Aa_ and A3 to a and 6 . It will be seen that these corrections

o o oo oo

are either zero or negligible for short paths if |h. - h-, . | and

h „ - hT ,. are less than one kilometer. 1 r s j-ir i

h - h

d n . ts rs ,. ._ _. a = r— + 9 + — radians (5.7)

oo 2a et 5280 d

h - h

P =5-+6 - cm/8 radians (5.8)

oo 2a er 5280 d

paths.

These angles a and P are positive for beyond-the -horizon

oo oo J

d

9^ = 9,+ radians (5.9)

ot et a

'

25
d

Lr =9 + radians (5.10) or er a

These angles may be positive or negative, and over a smooth

spherical earth are zero, as may be seen from Fig. 14.

D = d - dT A - dx (5.11)

s Lt Lr

D

9 = a +6 =— +0 +9 =— + +9 radians (5.12)

oo oo oo a ot or a et er

Over a smooth spherical earth*

d

Lt

=y(a/5280) (2h

te

). (5. 13)

d_ = /(a/5280) (2 h ), (5.14) Lr V

x re

and the angular distance, 8 , is the distance between radio

oo

horizons, Ds , divided by the effective earth's radius, a, as may

be seen using (5.12). Exact expressions relating d, ., d- and

hf , h in a linear gradient atmosphere are given by (7. 3) and

(7.4) in Section 7.

Next, correcting a and (3 to values expected in a reference

atmosphere:

a = a + Aa (d . 6 , N ) (5.15) o oo o st ot s

P = P + AP (d , e , N ) (5.16) o oo o sr or s

The correction Aa in milliradians is plotted in Figs. 15 - 22 for

the four reference atmospheres of Table II, as a function of d

g^., 9

ot, and Ng

. These same figures are used to read AP , given

d in statute miles and 9rt _. in milliradians.

sr or

dp dp

oo

26
-27-

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< 40 —

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50 60 70 80 90 d st IN STATUTE MILES

10 120 130

Figure 15
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h

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Figure 23
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LUo

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IT<

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ANGULAR DISTANCE, 6, OVER A SMOOTH EARTH AS A FUNCTION

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, BETWEEN HORIZONS

140

130 ^:

120

100

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Figure 24
da da

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In practice, the approximate expressions in (5.17) and (5.18) are

used to find distances d o4. or d„_ Ol DA

crossover of radio horizon rays,

used to find distances d 0+ or d„_ from a horizon obstacle to the

In Figs. 15 - 22, when * is negative, replace dg

^. by

| d £ - a ( | QQ^ | ) | and read the curve for + = to obtain Aq . Similarly, if or is negative and A(3 is required, replace dgr

by | dsr - a ( | or | ) | and read the curve for or = 0. In these

formulas O^ and or are expressed in radians.

d. Computation of with Very High Antennas Over a Smooth Earth

If the difference | hts

- h^ | or | hrs - hLr | greatly exceeds

one kilometer, equivalent earth geometry may not locate the

radio horizon correctly. In this case, refraction between the

antenna and its horizon departs from what is expected in a

linear gradient atmosphere. The problem is still simple for a

smooth earth, but becomes quite complex over a very rough

terrain profile. Fig. 23 shows dj^j. plotted versus h^ s

- h^ for

each of the four reference atmospheres of Table II, for the case

®ot = 0. For a smooth earth and h^- s > h^f read d^ from Fig. 23,

compute dj^j., and compute Ds from (5.11). For a smooth earth,

@ot = ^or = 0» ds ^ = dgr = Dg

/ 2, and may be read from Fig. 24

as a function of Ds

/ 2. Then,

<L = d_ + D /2, d. = dT + D /2,

1 Lt s 2 Lr s

s = d,/<L, a =s0/(l + s), p =6/(l + s). (5.19)

C. 1 o o

This subsection (d) should apply for the case of aircraft

flying high over rough terrain, wherever the assumption of an

average smooth earth seems reasonable. Figs. 23 and 24, with

(5.19) make the computation of aQ and (3 easier than it is when

terrain profiles must be plotted.

37
e. Computation of 9 with Very High Antennas over a Rough Earth

Over a rough terrain profile with either | h^ - h.j+\ or

I h - hj^j. [ greater than one kilometer, it is possible to trace

rays carefully in any one of the four reference atmospheres by

using one of Figs. 26 - 29. This was necessary in only one

situation among paths analyzed by CRPL, where one antenna was

located on top of Pike's Peak in Colorado. Fig. 25 shows the

path of a radio ray and the parameters used in Figs. 26 - 29.

Figs. 30 - 33 are helpful for interpolating values from Figs. 26 - 29.

Only extreme situations should require the use of this

latter set of figures, so their use is not explained in detail.

Tracing a ray from the transmitting antenna to its horizon,

identify h. - hj . with h, d-r t with d, and 6 . with a. The

figures are also used to trace a ray from this horizon to the

crossover of horizon rays.

6. Determination of Surface Refractivity, N

Where measured values of N

g

are not available, estimates of

Ng for the U. S. may be obtained for Time Block 2 from Fig. 34.

For other areas, refer to a recent report by Bean and HornJ_£/ . In

Fig. 34, Ns is not plotted directly, but is obtained from the reducedto-sea -level refractivity, N : Nso=N exp (-0.03222 h ) , (6.1)

s where h is the elevation of the ground above mean sea level, in

thousands of feet, as given by (4. 3). A world map of minimum monthly

mean N values is shown as Fig. 34a; these values may be substituted

for Time Block 2 values if the latter are not available.

7. Effective Antenna Heights, h, and h

re

The definition of h. and h given in this section is used in this

paper to calculate L»u m(? > ant^ ^n Reference 5 to calculate I-o-j*

-38-
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a. Antenna Heights hi and h^. e Less than One Kilometer Above the Surface

For antennas elevated more than a hundred feet above

ground, the effective height was defined as the height of the

antenna above a circular arc fitted by least squares to a portion

of the terrain. This was chosen to be eight -tenths of the terrain

along the great circle path extending from the antenna to its

four -thirds earth radio horizon. It is assumed that critical

ground reflection lobery in the lower regions of the scattering

volume is unaffected by terrain near either the antenna or its

radio horizon.

This definition of effective antenna height is not satisfactory for most receiving antennas if their height above ground

is small with respect to average terrain irregularities. Some

effective heights in such cases are an order of magnitude

greater than the structural antenna heights. The structural

height is used, therefore, for receiving antennas, and the more

complicated definition of effective antenna height is used for

antennas higher than 100 feet above ground.

Except for the ambiguity of such a definition, the structural

height of an antenna, sometimes including a building, cliff, or

hill as part of the structure, might have been used to define hj

and h' for both high and low antennas,

re

b. Antennas More than One Kilometer above the Surface

If h* as defined in subsection (a) is less than one kilometer,

te

it is used as the effective transmitting antenna height h. in the

prediction formulas, since radio ray refraction from the antenna

to its horizon is then what is expected in a linear gradient

atmosphere. For higher antennas, Fig. 35 is used to obtain a correction Ah, which reduces h|

e to the value h^. e to be used in

the formulas of Reference 5 and this report. The same comments

apply ^^

to h' and h : J re re

h

te = h

te " AMhte> Ns> < 7 ' D

h

re

= h

re " Ah < hre' Ns> < 7 ' 2 >

Over a smooth earth, the actual antenna heights, h' and

h' correspond to horizon distances dT. and dT in a reference

atmosphere. Effective antenna heights h. and h are defined

49
in such a way as to preserve these correct horizon distances

over a smooth earth in a linear gradient atmosphere. Except

for extremely high antennas, (5.13) and (5.14) show the simple

relationship that holds between antenna height and horizon

distance over a smooth earth in a linear gradient atmosphere;

exact expressions are

h

te = a[sec (dLt

/a) -1] (7.3)

h = a[sec (dT /a) - 1

J

(7.4)

re Lr

8. Parameters Required to Compute Li-bras

Tabulate the following parameters, for use in Section 9. Refer

to Fig. 14 for illustrations of path geometry. Section 9 explains the

individual terms in (3.1), used to calculate L, for each of two

reference atmospheres.

N = surface refractivity, obtained from (4. lb) if possible, or from

direct measurement of winter afternoon conditions, or from

Fig. 34 and (6. 1), or from Reference 12.

a = equivalent earth's radius. Choose two values from Table IE, for

reference atmospheres corresponding to N values above and

below the correct Ng

. Calculate the following parameters for

each reference atmosphere.

= angular distance. (Tabulate two values. )

s = path asymmetry parameter:

s = a /p (8.1)

o o

It is helpful to identify antennas in such a way that s < 1.

h = height of horizon ray crossover above a straight line between the

antennas:

h = sd9/(l + s)

2

= a (3 d/0 = d, a = d , |3 miles (8.2)

o o o 1 o 2 o

where d, d_, d£ are in statute miles, and angles aQ, (3 , are

in radians. (5.19) defines d^ and d^.

50
-51-

REDUCTION Ah OF EFFECTIVE ANTENNA HEIGHT

for very high antennas

h

te

=h'

te

-Ah(h'

te

,N s )

n re

= h re" An( h

re> N

s

)

10'

10

10

10

10 2 fl

10

I llllllMlililllliililililliillliliilllillllllli^'iMlhlillll 310

h

fe

or h

re

IN FEET

Figure 35
K. = height of horizon ray crossover above a straight line between the

radio horizons:

h, = d d 6/D miles (8.3)

1 st sr s

where d , and d as illustrated in Fig. 14 are defined by (5.17)

st sr and (5.18). is in radians in (8.3)

h = height of scattering center above ground, in thousands of feet.

Accuracy of this determination is not important for the formulas

in this paper, so a careful derivation of h is not given. Use

(4.4) from Section 4 when no better estimare of h is available.

y per mile = logarithmic gradient of refractivity with height,

determined from Table II as a function of N and h

s eg

h = average height of a propagation path above mean sea level. Where

terrain profiles are available, use (4. 3); otherwise, use the

values in Table n.

h

s = (hLt + hLr)/2 (4.3)

r\ = normalized height of horizon ray crossover:

s

Tl =4yh , (8.4)

s o

a dimensionless parameter. Use y per mile and h in miles

in (8.4). °

ti = y d 9 (8.5)

This is another dimensionless parameter, the most important one in

the theory. Use y per mile, path distance d in miles, and angular

distance 6 in radians in (8. 5). If the asymmetry parameter s = 1,

then t| = r\ . Note that equal antenna heights do not necessarily imply

a symmetrical path, except over a smooth earth.

Comparing (8.4) and (8.5):

r] = 4sti/(1 + s)

2

(8.6)

s Effects of energy reflected from the ground are a function of

t) and of effective antenna height in wavelengths multiplied by the

elevation angle of the radio horizon:

52
v = 4-irh a /X , (8. 7)

t te o

v = 4ir h 3 /X , (8. 8) r re o

where h^

e and hre are effective antenna heights above the reflecting

surface between each antenna and its radio horizon, as discussed in

Section 7. These heights and the free space radio wavelength, X, are

expressed in feet in this report, and the angles a_ and 6 are in

radians.

Computations described in the next section also require

q = v

r /v

t

, r

} = vt [1 + (1/s)] , r

£

= qs^ = v

r

(1 + s). (8.9)

These parameters are required only when r\ < 1. s

9. The Scatter Prediction Formula

Repeating (3. 1), median basic transmission loss due to forward

scatter in Time Block No. 2, winter afternoons, is given by

U = G(ti) + 30 log,„ f -20 log,„ d . + A

t>ms v " e

10 mc 510 mi a

+ H - F - F - F (h , N ) . (3.1) o o s eg s

In this section, terms of (3. 1) are explained in detail.

a. The Attenuation Function, G(t])

G(-n) shows the decibel attenuation of field strength with

increasing ydG, and is the most important term in (3. 1). At very

great distances, the last four terms are negligible. For 500'

and 30' antennas at 100 Mc, (3. 1) corresponds to an attenuation

rate of about 0. 12 db per mile from two to five hundred miles

over a smooth earth.

G(ti) = G (ti) + AG (9.1) o G (r\) is plotted as Fig. 36, r\ is given by (8. 5), and AG is

tabulated for the reference atmospheres in Table II for various

ranges of the height of the scattering center h above ground,

given in thousands of feet.

-53-
54-

THE ATTENUATION FUNCTION G (T])

250

240

230

220

200

CO

_lUJ

99 iso

o

LlI Q

300

;EEE: 290

280

„ 270

1 co

260

250

240

230

220

99o

LJQ

P~ O

210

002 005 0.1 0.2 0.5 I 2 V

Figure 36

5 10 20 50 100
The Sea Level Correction, F

o The "sea level correction", F , allows for increased fields

expected when h, < h . See Fig. 14. For antenna both at sea

level, h, = hQ and FQ

= 0; in general,

F = 4. 343 v (h - h ) (9.2)

o o 1

where y per mile is tabulated in Table II, h

Q and h. are given in

miles by (8. 2) and (8. 3) . The terms F and F are usually

negligible.

The Asymmetry Correction, F

The "asymmetry correction", F , allows for departures

of the asymmetry parameter s from unity. F(s) is the product

of the functions C(s) and M(n) shown in Figs. 37 and 38. Note that

C(l/s) = C(s).

d. The Frequency Gain Function, H

The "frequency gain function", H , in (3. 1) is a measure

of how much energy is sent well above the crossover of radio

horizon rays by reflection from the earth. Scattering efficiency

is reduced at higher altitudes. For most commonly occurring

antenna heights, the field strength at the horizon ray crossover

should be 10 to 20 decibels below free space ±~J , corresponding

to maximum values for H of 16 to 26 db at great distances. For

low antennas and short paths, H may be much larger.

The computation of HQ

is different for the two special cases,

ti > 1 and ti_ < 1, corresponding to large and small distances,

s s In either case, values H (v.(.) and H (v

r

) are read from Fig. 39

as a function of r\ , v., and v as defined by (8. 4), (8. 7), (8. 8),

and these readings are averaged:

H

o

( % - l ] = I [ Ho

(v

t

} + Ho

{vr^ + AHo

(9 * 3)

AH

q

= 6 (0. 6 - log

10 ns

) log

10

s log

10 q (9. 4)

AH_ is defined only for ri > 1. For r\ < 1, AH is calculated from o J (9.4) with r) set equal to

'S1. 'So

55-
-56-

\

O

II

ii

if) o

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45

40

35

30

15

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 •-V4.

Figure 37
-57-

THE FUNCTION M (77)

THE ASYMMETRY FUNCTION F (s, v

,

-- C(S> M(V

0.01 0.02 V

Figure 38
[ See (8. 1) and (8. 9) for definitions of s and q. ] AH is an

approximate correction factor, valid if s and q are each within

the limits 0. 1 to 10. For values of s or q outside these limits,

use the nearest limit for s or q in (9. 4). The frequency gain,

H , is always positive, so if AH overcorrects, assume H =0.

Also, AHQ must not exceed 1 / 2 [ HQ

(v

t

) + HQ

(v

r

)]

.

Fig. 39 is drawn for the special case where s = 1 and

q = 1, and (9. 3) implicitly assumes that frequency gain effects

at the two ends of a propagation path are independent. This is

nearly true for r\ > 1 (large distances), but at the shorter

distances it is not true. Accordingly, an asymptotic expression

which is exact for y = (no bending of radio rays) is evaluated,

and HQ

(t| < 1) is obtained by linear db interpolation between

H (l) = HQ

(n

s = 1) and HQ(0)^ HQ (^ = 0):

2 2

H (0) = 10 log

2(1

- 8 q > , (9. 5)

r^[h( ri )-h(r 2

)]

where h(r. ) and h(r 2

) are read from Fig. 40, and r, , r£» and q

are defined by (8.9). The function h(r) is also shown on Fig. 40a.

If sq = 1, read HQ

(0) from Fig. 41. For vt > 10, vr > 10,

H (0) = 0.

o '

Then, for r\ < 1,

o s o so o

where H (1) is determined from (9.3) and Fig. 39, with tj = 1,

and H (0) is determined from (9. 5).

For equal effective antenna heights, sq = 1. Graphs of H

versus v., r\ , and s are given in Figs. 39a - 39f. This special

case occurs frequently in system design calculations.

e. Atmospheric Absorption, A

Atmospheric absorption is important at long distances and at

frequencies above 1000 Mc. Table III, Section 10, lists values of

Aa appropriate for the calculation of weak fields, E(99), corresponding to L(p) for p = 99%.

-58-
-59-

THE FREQUENCY GAIN FUNCTION H FOR 17^

I

for a =/3Q and v

f = v

r (s = l,q = l)

v

t = 47rh te a /X , v

r = 47rh re pQ/\

T7S =4/ h has the values indicated on the curves

o

0.00I 0.002 0.005 0.0I 0.02 0.05 0.I 0.2 0.5 I 2 5 10 20 50 200 500 1,000

V t orVr

Figure 39
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f. The Empirical Corrections F(h , eg

N ) s This completes the discussion of terms in the prediction

N This

s>- formula (3.1), except for the last term, F(h , empirical correction, evaluated from radio transmission loss

data, represents the variance of refractivity turbulence. zZJ

The origin of this term -will be further discussed in Part III. F(h , N ) is plotted as Fig. 42 for the reference atmospheres

of Table H.

10. Atmospheric Absorption

For a recent study involving sixty -foot parabolic dishes at each

end of propagation paths from 100 to 1000 miles long, the oxygen and

water vapor absorption^ A , exceeded 1% of the time was estimated, using

methods described by B. R. Bean and R. Abbott in recent papers.

' —

1

Table III lists these values for frequencies from 100 Mc to 10, 000 Mc,

They are appropriate for the climate of Washington, D. C. in August,

and take account of the decreasing rate of absorption as a ray is traced

from an antenna to the high elevations corresponding to a "scattering

volume. " (See Fig. 14).

Table III A in db

a

mc

100

Distance in Miles

200 500 700 1000

100 0.03 0. 1 0. 1 0. 1 0. 1 200 0. 1 0. 3 0.4 0.5 0.6

500 0.4 0.6 1.0 1. 1 1.3

1000 0.6 1.2 1.6 1.8 2.0

2000 0.8 1.6 2. 3 2.6 2.8

5000 1.2 2.4 3.3 3.7 4.1

0000 2. 3.7 5.0 5.4 6.0

62
63-

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11. Path Antenna Gain, G

P

The path antenna gain, G , may be determined from Fig. 43,

p which shows the loss in antenna gain, G, = G, + G^. - G . as a function to Li t r p*

of r\ and the ratio of angular distance, 0, to the half -power beamwidth,

Q, of each antenna. Note that 9 and Q are to be expressed in the same

units.

The results shown in Fig. 43 correspond to the case G. = G ,

£1, = Q = £2, and s = 1; however, G, is not very sensitive to changes

in s. For more general results, the reader is referred to Reference 9.

A good engineering approximation relating half -power beamwidth,

Jl, in milliradians to the free space gain Gt in decibels above an isotropic

radiator is given by

G

t =69.12 - 20 1og

1() S?t

, (11.1)

G - 69.12 - 20 log1n n , (11.2)

r 11) r

from -which it follows that

9/Q =[350 9xlO(Gt + Gr )/40]xlO-6

(11.3)

where and Q are in milliradians. The parameter r\ is defined by (8.4),

These formulas are derived for a parabolic dish of diameter D.

At a frequency f megacycles, mc

Gt e 20 log

1() D + 20 log1() £ - 52. 43 (1 1. 4)

in decibels above an isotropic radiator, where the actual diameter of

the dish, D , is expressed in feet, and it has been assumed that the

effective area of the dish is 56% of its actual area. The gain is that

expected with a 10 db tapered illumination, where the feed illuminates

the edge of the dish with 10 db less signal than its center.

64
-65-

LOSS IN ANTENNA GAIN, GL

assuming equal free space gains fy and Gr at the terminals of a symmetrical path

O.I 0.2 0.5 I 2 5 10 20 50 100

Ratio 8/£l of angular distance 9 to half-power antenna beamwidth, £1

Figure 43
-66-

1333 Nl N0I1VA313

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z' o

h- co

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L±J CD H -O

£Z (0 § <

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^_ ^

x:

u_ o 1 i- _l CO X— CO^

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O

O CD

UJ

O

CL

Li_ CO O _CD or

Q_ E

en z LO <

hcr II

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CDO

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CO Q

1333 Nl N0I1VA313
12. Calculations for Figure 13

The great circle path profile for the 176 -mile KBCL-FM path is

illustrated in Fig. 44 for the reference atmosphere corresponding to

surface refractivity Ng =301. For the first kilometer above the surface, this is the same as a "four -thirds earth" atmosphere, as may be

seen from Table II. Only the parameters shown in Fig. 44 are needed

from the terrain profile for each Ns value.

In this section are tabulated calculations of the theoretical curves

of Fig. 13 for each time block and for all -day, all-year conditions.

The reduced-to-sea level surface refractivity, N , for Time Block

No. 2 is read from Fig. 34 at the location of each four -thirds earth radio

horizon. These values, N = 313.0 and N = 316.3, correspond to heights

above sea level hj^t = 720' and hL,r = 900', and to surface refractivity

values Ng = 305.8 and Ns = 307.2, using (6. 1) from Section 6. The

average, Ng = 306. 5, is taken to represent winter afternoon conditions

along this path, and basic transmission less, Lb (p) is calculated for

p = 1, 10, 50, 90, and 99 per cent for Ns

= 301 and for Ns = 350; the

predicted values of L,|-

) (p) in Fig. 13 for Ns = 306. 5 are then obtained by

linear interpolation.

For KIXL-FM, d = 175.9 miles, fmc = 104.5 megacycles, hte = 553 feet, and hre = 32 feet. See Section 7. Table IV shows calculations

°* Lbms for a = 5280 miles (N s = 301) and a = 5896.66 miles (N s = 350),

using (3. 1). An appropriate section, equation, or figure is referenced

on each line of the table.

Table IV

Symbol Value for Ns = 301 Value for Ns = 350

( ht;

dLt 24.60 mi.

dLr 5.48 mi.

hLt 720'

hLr 900 ! 9et -3.869 mr.

9er 2.937 mr.

hrg )/5.28d 0. 129 mr.

d/2a 16. 657 mr.

a oo 12.917 mr.

£00 19.465 mr.

Qot 0. 789 mr.

24. 60 mi.

5. 48 mi.

720'

900'

-3. 626 mr.

2. 991 mr.

0. 129 mr.

14. 916 mr.

11. 419 mr.

17. 778 mr.

0. 546 mr.

Refe:rence

Fig. 44

Fig. 44

Fig. 44

Fig. 44

Eq. (5.5)

Eq. (5.6)

Eq. (5.7)

Eq. (5.7)

Eq. (5.7)

Eq. (5.8)

Eq. (5.9)

67
Table IV (Continued)

Symbol Value for Ns = 301 Value for Ns = 350 Reference

e or 3.975 mr. 3.921 mr. Eq. (5.10)

Ds 145.82 mi. 145.82 mi. Eq. (5.11)

1000

D

s /a 27. 617 mr. 24. 730 mr. Eq. (5.12)

e oo 32. 382 mr. 29. 197 mr. Eq. (5.12)

dst 81.1 mi. 82. 5 mi. Eq. (5.17)

d sr 64. 7 mi . 63. 3 mi. Eq. (5.18)

Aa 0.060 mr. 0.001 mr. Figs. 16&17

AP 0.027 mr. 0.000 mr. Figs.l6&17

ao 12.977 mr. 11.420 mr. Eq. (5.15)

Po 19.492 mr. 17.777 mr. Eq. (5.16)

9 32.469 mr. 29. 197 mr. Section 5c

s 0.6658 0.6424 Eq. (8.1)

s/(l+s)2 0.2399 0.2381 Eq. (8.2)

dmi 6 rad 5. 71 13 mi. 5. 1358 mi. Eq. (8.2)

K 1.3701 mi. 1. 2228 mi. Eq. (8.2)

hi 1. 1684 mi. 1.0456 mi. Eq. (8.3)

hcg 8, 720 feet 7, 840 feet Eq. (4.4)

y

. 0. 1907/mi. 0.2102/mi. Table II

*i 1.0891 1.0796 Eq. (8.5)

^S 1.0451 1.0282 Eq. (8.6)

G (ti) 164.2 db 164.0 db Fig. 36

AG 5.8 db 2.8 db Table II G(n) 170.0 db 166.8 db Eq. (9.1)

ho" hl 0.2017 mi. 0. 1772 mi. Eq. (9.2)

Fo 0.17 db 0. 16 db Eq. (9.2)

C(s) 2.08 2.35 Fig. 37

M(n) 0.01 0.01 Fig. 38

Fs 0.02 db 0.02 db Section 9c

vt 9.58 8.43 Eq. (8.7)

vr 0.833 0.760 Eq. (8.8)

q 0.0870 0.09015 Eq. (8.9)

AH 0.6 db 0.7 db Eq. (9.4)

**o 6.5 db 6.9 db Eq. (9.3)

30 log10 fmc 20 log10 d = 15.7 db

F(hcg, Ns ) 6.0 db 9.2 db Fig. 42

Lbms 186.0 db 180.0 db Eq. (3.1)

68
At this point, L^d is calculated by the methods of Reference 5.

For such a long path, L,^ > > Lbms> so t^iat Lbm = Lbms« [See (3.2)]

Calculations of Lfrd are listed in Table VII, however.

Because r| g is greater than 1, in Table IV it was not necessary to

calculate Ho (0). For Ns = 301, rj = 23.97 qs = 0.0579, and r2

= 1.388

using (8.9). Then, h(r 1

) = 1, h(r2

) = 0.715 from Fig. 40, and Ho

(0) = 5.6 db, from (9.5). For Ns = 350, Ho(0) =6.2 db.

Table V lists V(p, 0) as read from each time block (Table I) and for

all hours, using Figs. 2 - 12.

Table V

V (p, 0) in decibels

Ng = 301, = 32.47 milliradians:

Time Blocks

p 1 2 34 567 8 All hours

1% 15.5 12.0 14.9 21.8 12.3 18.3 22.1 18.1 19.1

10% 8.3 6.0 8.0 14.1 7.5 11.8 15.7 9.9 11.8

50% 1.0 0.0 1.1 6.7 2.7 5.1 9.5 1.5 3.5

90% -4.5 -4.3 -4.3 0.5 -1.7 -0.6 1.8 -4.8 -3.0

99% -8.8 -7.8 -8.6 -4.4 -5.4 -5.4 -4.5 -9.9 -7.7

Ns = 350, = 29.20 milliradians:

1% 17.0 13.0 16.0 22.5 12.6 19.6 24.3 19.4 20.3

10% 9.2 6.5 8.5 14.4 7.6 12.5 16.8 10.7 12.2

50% 1. 1 0.0 1.3 6.5 2.5 5.3 9.6 1.9 3.5

90% -4.9 -4.9 -5.2 -0.1 -2.1 -1.0 1.5 -5.3 -3.4

99% -9.5 -8.7 -10.4 -4.9 -5.9 -6.0 -5.1 -11.1 -8.8

Table VI lists Lb (p) for each time block and all hours for Ng = 30 1> 350, and as obtained by linear db interpolation for Ns = 306. 5. These

values are plotted in Fig. 13.

69
N = 301

s

Table VI

Li (p) in db

b

Time Blocks

p 1 2 3 4 5 6 7 8 All hours

\°Io 170. 5 174. 171. 1 164. 2 173. 7 167. 7 163.9 167.9 166.9

10% 177. 7 180. 178. 171. 9 178. 5 174.2 170. 3 176. 1 174.2

50% 185.0 186. 184.9 179. 3 183. 3 180.9 176. 5 184. 5 182.5

90% 190. 5 190. 3 190. 3 185. 5 187. 7 186. 6 184. 2 190. 8 189.0

99% 194. 8 193. 8 194.6 190.4 191. 4 191.4 190. 5 195.9 193.7

N =

s 350

1% 163.0 167. 164.0 157. 5 167.4 160. 4 155. 7 160.6 159.7

10% 170. 8 173. 5 171. 5 165.6 172. 4 167. 5 163. 2 169. 3 167. 8

50% 178.9 180. 178. 7 173. 5 177. 5 174. 7 170.4 178. 1 176.5

90% 184.9 184. 9 185.2 180. 1 182. 1 181.0 178. 5 185. 3 183.4

99% 189. 5 188. 7 190.4 184.9 185.9 186. 185. 1 191. 1 188. 8

N =

s 306. 5

1% 169. 7 173. 2 170. 3 163.4 173. 166.9 163. 167. 1 166. 1 10% 176.9 179. 3 177. 3 171. 2 177. 8 173. 4 169. 5 175. 3 173.5

50% 184. 3 185. 3 184. 2 178.6 182. 6 180.2 175. 8 183. 8 181. 8

90% 189. 8 189. 7 189. 7 184.9 187. 1 186. 183.6 190.2 188.4

99% 194. 2 193. 2 194. 1 189. 8 190. 8 190. 8 189.9 195.4 193. 1

In Table VII, calculations of L^^ from Section III and Appendix I of

Reference 5 are summarized. There is an error at the top of page 1513

of this reference, where the formula b° = 2b M - b' refers only to vertical

polarization. The parameters in Table VII are for horizontal polarization,

where b° = 180° - b'. References in this table are from Reference 5.

Parameters also used in calculating ^-j\)Tns are not repeated, L^d is

calculated from (43) of Appendix I, where a' = 3960 miles.

70
Tcible VE

Symbol Value for ISf = 301

s Value for N = 350s Reference

k=a/3960 4/3 1.48905 App. I, Ref. 5

al 0.014433 rad. 0.012679 rad. Sect. IH, (18) * Pi 0.018035 rad. 0.016518 rad. Sect. HI, (19)*

kt 1.4172 1.6433 Sect, in, (16) * kr 0.9076 0.9681 Sect. HI, (17)*

kte 0.7295 0.7295 Sect. Ill, (20) * ^re 0.6256 0.6256 Sect. Ill, (21) * C (kt) 1.0205 1.0721 App. I, (37) * C G

(kr) 0.8797 0.8988 App. I, (37) * Co(kte) 0.8179 0.8179 App. I, (37) * C (kre) 0.7771 0.7771 App. I, (37) * K(4/3) 0.0010 0.0010 App. I, Fig. 30*

The figure heading for Fig. 30, Appendix I, on page 1512 of

Reference 5 is misleading: The factor [473k1 ] 1 / 3 should read [4/(3^)]*' 3 or 1/Co^').

K(kt) 0.00098 0.00093 Fig. 30 * K(kr) 0.00114 0.00111 Fig. 30 * K(kte) 0.00122 0.00122 Fig. 30 * K(kre) 0.00129 0.00129 Fig. 30 * C[K(kte), b°] 61.63 db 61.63 db Fig. 33 * C[K(kre),b°] 61.63 db 61.63 db Fig. 33 * 16.6671og10fmc 33.65 db 33.65 db (43) *

10 log10 dmi 22.45 db 22.45 db (43) * Po(kt ) 1.6061 1.6061 Fig. 32(a)*

Po(M 1.6060 1.6060 Fig. 32(a)*

Po(kte) 1.6060 1.6060 Fig. 32(a)*

Po( kre) 1.6059 1.6059 Fig. 32(a)*

Denote the last term in (43) by the symbol D:

D = 488.69 f

mc

1/3 ^ (kt)C (kt) a

x

H po(kr) C (kr) j3 x

D 113. 10 db 105. 14 db (43) *

* See Reference 5.

71
Table VII (Continued)

Symbol Value for Ns = 301 Value for Ng = 350 Reference

Po (kte)/Co(kte)

P?(kre)/Co(kre)-

3. 153 3.153 (40) * 3.319 3.319 (40) * £te 1.290 1.290 (40) *

•1_ hre 0.0786 0.0786 (40) * G(hte) -8.2 db -8.2 db Fig. 35(c)*

G(hre) -16.7 db -16.7 db Fig. 35(c)*

101og10 C o(kt) 0.09 db 0.30 db (43) * 101og10 C o

(kr) -0.56 db -0.46 db (43) * Lbd 255.3 db 247.6 db (43) *

For N

g = 306. 5, Lbd = 254.4 db, as compared with Lfc(50) = 185. 2 db

in Time Block No. 2. (See Table VI).

ACKNOWLEDGMENT

Initial phases of the theoretical development and organization of data

analysis for this paper were under the supervision of Mrs. Frances T.

Daniel. Most of the detailed theoretical work was done by R. L. Abbott,

A. P. Bar sis, H. K. Hansen, W. J. Hartman, L. E. Vogler, and R. E.

Wilkerson, under the direction of P. L. Rice. Terrain profile data were

obtained at various times under the direction of W. V. Mansfield, P. H.

Elder, A. M. Ozanich, and J, H. Longbotham. Most of the detailed

calculation was supervised by Mrs. M. A. Schafer under the direction

of Mrs. Longley. R. P. Baptist, V. D. Brand, S. A. Buckel, M. W.

Clark, M. M. Coyle, C. S. Hays, R. V. Lisle, and V. H. Weitzel

helped with calculations and graph plotting.

* See Reference 5

72
REFERENCES

1. B.R. Bean and G.D. Thayer, " On models of the atmospheric

radio refractive index, " Proc. IRE, 47 (5) „ 740 - 755 (1959) .

2. K.A. Norton, " Transmission loss in radio propagation , " Proc. IRE , 41_, 146-152, ( 1953) . See also: K.A. Norton , "System loss in radio wave propagation, " J. Research NBS, 63-D

(Radio Propagation), 53, (1959) .

3. R.C. Kirby, "VHF propagation by ionospheric scattering - a survey of experimental results, " IRE Trans. Comm. Syst.

CS-4 (1) , 17-27, (1956).

4. K.A. Norton , "Point-to-point radio relaying via the scatter

mode of tropospheric propagation, " IRE Trans. Comm. Syst.

CS-4 (1), 39-49 , (1956) .

5. K.A. Norton, P. L. Rice, and L. E. Vogler, " The use of

angular distance in estimating transmission loss and fading range

for propagation through a turbulent atmosphere over irregular

terrain, " Proc. IRE , 43, 1488-1526, (1955).

6. H.G. Booker and W.E. Gordon, "A theory of radio scattering

in the troposphere, " Proc. IRE, 38 , 401-412 , (1950).

7. H. T. Friis, A. B. Crawford, D. C. Hogg, "A reflection theory

for propagation behind the horizon, " Bell Telephone System

Technical Journal, _36, 627-644 , (1957) (Monograph No. 2823) .

8. B. Josephson and F. Eklund, "Some microwave propagation

experiences from a just-below-horizon path, " IRE Trans.

Antenn. Prop. AP-6 , 176-178, (1958) .

9. W. J. Hartman and R. E. Wilkerson, "Path antenna gain in an

exponential atmosphere, " to be published in J. Research NBS,

63-D, November, 1959.

10. 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, (1955) .

73
11. H. G. Booker and W. E. Gordon, "The role of stratospheric

scattering in radio communication, " Cornell University Research

Report EE 308; September, 1956.

12. B. R. Bean and J. D. Horn, "On the climatology of the surface

values of radio refractivity of the earth's atmosphere, " to be

published in J. Research NBS, 63-D, (Radio Propagation),

September, 1959.

13. CCIR, "Atlas of ground-wave propagation curves for frequencies

between 30 Mc/s and 300 Mc/s, " published by 1 'Union international des Telecommunications in accordance with CCIR

Resolution No. 11, Geneva, 1955.

14. Note added July 28, 1956, to Reference 4,

15. B. R. Bean and R. L. Abbott, "Oxygen and water vapor absorption of radio waves in the atmosphere, " Geofis. Pura e Appl.

,

37 (2), 127-144, (1957).

16. Bradford R. Bean, "Some meteorological effects on scattered

radio waves," IRE Trans. , PGCS-4, 32-38, (1956).

74
Appendix

The Calculation of the Power Required for a Specified Grade of a Given Type of Service

The body of this report presents methods for predicting the basic

transmission loss, L)-) (p), exceeded by p per cent of the hourly medians

in a given time. Within each hour, signal variations may exhibit many

different types of fading, depending on equipment and antenna characteristics and upon what propagation mechanisms are dominant during

that hour. Various kinds of communication links (telegraph, teletype,

voice, facsimile, data links), and broadcasting and navigation facilities

are affected differently by various types of signal fading. For instance,

an hourly median signal-to-noise ratio of 38 db is listed in a recent report by Norton.!/ as desirable for good quality U.S. standard television

reception of a severely fading Rayleigh distributed signal; service fields

near a television broadcast station would not require such a large hourly

median signal-to-noise ratio, since there would be little fading within an

hour. Other types of service, such as FM voice, can get by with much

smaller hourly median predetection signal-to-noise ratios, even where

the same post -detection signal-to-noise ratio is desired,,

Concepts represented by the following symbols are explained in

other papers 2, 3, 4 / m Capital letters refer to quantities expressed in

decibels:

P = transmitter power in decibels above one watt, or dbw.

L = transmission line and matching network losses between transmitter and antenna. When the antennas are very near the

surface, then radiation resistance changes and this results in

an additional loss with vertical dipoles or gain with horizontal

dipoles which may be included in Lrt; a discussion of these

additional transmission loss terms is given in Appendix III of

Reference 3.

G = transmitting antenna gain in decibels above a theoretical

isotropic radiator. (G or G for half-wave dipole is 2. 15 db)

G = receiving antenna gain in db above an isotropic radiator.

L - transmission line and matching network losses between receiving antenna and receiver input terminals. (See also

Appendix III of Reference 3.)

v = volts available across receiver input.

75
r = receiver input impedance in ohms,

P = power available at the terminals of a loss -free receiving

antenna, dbw. Pa

- 20 log^Q v - 10 log^Q r + Lr . P = minimum acceptable value of Pa . For example: assuming

a minimum useful signal of 5 microvolts, r = 51 ohms and

L. r

= 1 db, the minimum acceptable power at the terminals

of the receiving antenna will be Pm = -106.02 -17.08+1.0 = -122. 1 dbw.

P = Pa - Lt = total radiated power in dbw.

E. R. P. = effective radiated power = P$ - Lj. + Gj- - 2. 15 in dbw.

L = Pr

- Pa

= transmission loss.

A = attenuation relative to free space.

Lf = Li - A = transmission loss in free space.

G <G+G = path antenna gain.

L. = L+G = transmission loss with isotropic radiators, called "basic

b p transmission loss". *

L^f = basic transmission loss in free space

= 36. 58 + 20 logiQ fmc + 20 log|Q dmi, where fmc is the radio

wave frequency in megacycles per second and dmi is the

distance in statute miles between transmitting and receiving

antenna terminals.

E = field strength in decibels above one microvolt per meter

(dbu) for one kilowatt E. R. P.

E, = free space field strength between half -wave dipoles.

R = predetection hourly median signal-to-noise ratio required to

provide a specified grade of service within the hour for which

the median signal level is given. The factor R is best determined experimentally, but it may also be calculated with

reasonable accuracy for some types of service by assuming

that the received signal is Rayleigh distributed about its

hourly median level. As an example, the calculated value

of R is 16.9 db to provide a 60 -word -per -minute frequency

shift telegraph service with not more than one-per-cent

binary error rate for a period of time of the order of one

hour during which the transmission conditions may be

* This definition differs from that used in Reference 2,

76
considered relatively constant, except for the rapid,

Rayleigh type, of fading J|/ . The factor R may be reduced

substantially by the use of diversity reception, this reduction

being larger, the larger the reliability required. For discussions of the expected diversity gain, reference may be

made to papers by Montgomery "> JJ , Mack_', Brennani',

and Staras 1Q /.

b = effective bandwidth of the receiver in cycles per second,

including allowance for differences between transmitter and

receiver oscillator frequencies and including the component

of the baseband required to accommodate the received modulation.

B =10 logiQ b (An absolute lower limit for the noise level is the

"Johnson noise" or "ktb noise". At 60° Fahrenheit, the ktb

noise level is equal to B-204 db).

F = effective noise figure, referred to the receiving antenna

terminals. This noise figure is the difference, or decibel

ratio, between the actual noise level (Pm-R) decibels and the

ktb noise level. Where external noise picked up by the receiving system is negligible, F is simply the decibel sum of

the receiver noise figure Fr and the transmission line and

matching network losses L. r . For carefully designed receivers

at 100 Mc or at 1000 Mc, Fr equals about 5 db or 10 db, respectively. F - (10 logio fmc - 5) decibels, approximately.

Noises originating within and external to the receiving

system add directly when expressed in watts, so in this case

it is not permissible to add decibels. F includes the effects

of the antenna noise (cosmic and man-made noise, for

example) as well as the receiver noise, together with the

receiving antenna circuit and transmission line loss. The

proper method for estimating the expected value of F is

discussed in detail in a recent National Bureau of Standards

circular 1 w. It is assumed that the receiver incorporates

gain adequate to ensure that the first circuit noise is detectable. As a practical matter, F is within 3 db equal to whichever is greater, Fr + Lr or Pn

- (B-204), where Pn is the

external noise picked up by the system, expressed in dbw.

Having once determined a value for Pm either from information

about a minimum useful signal or from information about R, F, and B,

it is a simple matter to compute the maximum allowable transmission

77
loss, L, the maximum allowable attenuation relative to free space, A,

or the minimum allowable field strength, E. The following relationships are useful:

P =P = R + F + B - 204 dbw. (1-1)

a m

P

r

= P

t - L

f

dbw. (1-2)

L = P -P , (1-3) r a

where this is interpreted as a maximum allowable transmission loss if

^adm= FVn» Definitions of Pamand P_ assume a matched load,

The difference (decibel ratio) between transmission loss L and

basic transmission loss L^ is the path antenna gain Gp, which is equal

to or less than the sum of free space gains Gt and Gr

. In free space,

the difference between Lf and L^f is Gj- + Gr

: L = L

b

- G

p

(1-4)

L

f = Lbf " (G

t + G

r

} {1 ' 5)

Attenuation relative to free space is defined in terms of the measured

transmission loss L and the loss Lf expected in free space. Thus, this

quantity, A, includes any effects of path antenna gain degradation (antennato-medium coupling loss), and is in general greater than the theoretical

attenuation relative to free space for isotropic antennas. Subtracting

(1-5) from (1-4):

where

A = L - L. = L. - L. , + GT

, (I -6)

I D DI J_j

G. = G. + G • - G > 0. (1-7) L t r p

Basic transmission loss in free space is known when frequency and

distance are given:

L,= 36.58 + 20 log. nf + 20 login d . (1-8) bf & 10 mc &10 mi *

78
Similarly, the free space field strength E^f dbu for one kw E. R. P.

using half-wave dipoles at both ends of a path is known:

E

bf= 102.79- 20 log

10

dmi . f 1 "')

Assuming Gl =0 for transmission between dipoles, the correlatives

of (1-4) through (1-7) may be written:

E=E

b

-GL(1-10)

Here is seen a major advantage of the concept of transmission loss,

which is a measurable quantity, while field strength is not, except

where a coherent wave front exists at the antenna and Gl = 0. Both E

and Efc> are plane wave field strengths that would have to exist at the

receiving antenna to produce the voltages measured at its terminals,

assuming antenna gains realized . E corresponds to the case where

antenna gains are not realized and E^ to the case where gains are

realized; other things being equal, antenna terminal voltages are less

in the one case than in the other, so E is less than Ejj by Gl decibels.

The correlative to (1-5) is

which says simply that in free space, free space gains are realized.

Attenuation relative to free space is obtained by subtracting (I- 10) from

(1-11):

A = E, - E = E. , - Eu + GT

. (1-12)

f bf b L

Compare (1-12) with (1-6). Equating these expressions and making use

of (1-8), (1-9), and (I- 10), a relationship emerges between L^ and E:

L. = 139.37 + 20 log f - E, b 10 mc b

= 139.37 + 20 log

1() fmc

- (E+GL ) , (1-13)

The meaning of (1-13) is the same as that of (5) in Reference 4, where

Gl is set equal to zero and in effect the statement is made that E underestimates El by the amount Gl»

79
APPENDIX

REFERENCES

1. K. A. Norton, "Point-to-point radio relaying via the scatter mode

of tropospheric propagation, " IRE Trans. Comm. Syst. , CS-4 (1),

39-49, (1956).

2. K. A. Norton, "Transmission loss in radio wave propagation,"

Proc. IRE, _41_, 146-152, (1953).

3. K. A. Norton, "Transmission loss in radio wave propagation: II, " NBS Technical Note No. 12, June, 1959. See also K. A. Norton,

"System loss in radio wave propagation, " J. Research NBS, 63-D

(Radio Propagation), 53, (1959).

4. K. A. Norton, P. L. Rice, and L. E. Vogler, "The use of angular

distance in estimating transmission loss and fading range for

propagation through a turbulent atmosphere over irregular terrain, " Proc. IRE, 43, (10) , 1488-1526, (1955).

5. G. Franklin Montgomery, "A comparison of amplitude and angle

modulation for narrow-band communication of binary- coded

messages in fluctuation noise," Proc. IRE, 42, 447-454, (1954).

6. G. Franklin Montgomery, "Message error in diversity frequencyshift reception," Proc. IRE, 42, 1184-1187, (1954).

7. G. Franklin Montgomery, "Ultrasonic switch aids diversity reception, " Electronics, 28, 169, (1955).

8. C. L. Mack, "Diversity reception in UHF long-range communications, " Proc. IRE, _431 1281-1289, (1955).

9. D. G. Brennan, "On the maximum signal-to-noise ratio realizable

from several noise signals, " Proc. IRE, 43, 1530, (1955).

10. H. Staras, "The statistics of combiner diversity," Proc. IRE,

44, (8), 1057-1058, (1956).

80
11. W. Q. Crichlow, D. F. Smith, R, N. Morton and W. R. Corliss,

"World-wide radio noise levels expected in the frequency band

from 10 kilocycles to 100 megacycles, " NBS Circular 557,

August 25, 1955; available from the Superintendent of Documents,

U. S. Government Printing Office, Washington 25, D. C, at

30 cents.

81


NBS