itional Bure au of Standards

Library, K.W. Bldg

APR2 M962

NBS MONOGRAPH 38

Radiation Patterns in the Lower Ionosphere

And Fresnel Zones for Elevated Antennas

Over a Spherical Earth
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March 3, 1901,

as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the

national standards of measurement and the provision of means and methods for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development

of methods and instruments for testing materials, devices, and structures; advisory services to government

agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. Research projects are -also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau's unique

competence is required. The scope of activities is suggested by the listing of divisions and sections on the inside

of the back cover.

Publications

The results of the Bureau's research are published either in the Bureau's own series of publications or in the journals of professional and scientific societies. The Bureau itself publishes three periodicals available from

the Government Printing Office: The Journal of Research, published in four separate sections, presents complete

scientific and technical papers; the Technical News Bulletin presents summary and preliminary reports on work

in progress; and Basic Radio Propagation Predictions provides data for determining the best frequencies to use

for radio communications throughout the world. There are also five series of nonperiodical publications: Monographs, Applied Mathematics Series, Handboks, Miscellaneous Publications, and Technical Notes. A complete listing of the Bureau's publications can be found in National Bureau of Standards Circular 460, Publications of the National Bureau of Standards, 1901 to June 1947 ($1.25), and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957 ($1.50), and Miscellaneous Publication 240, July

1957 to June 1960 (includes titles of papers published in outside journals 1950 to 1959) ($2.25); available from

the Superintendent of Documents, Government Printing OiBce, Washington 25, D.C.
UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Radiation Patterns in the Lower Ionosphere and Fresnel Zones

for Elevated Antennas Over a Spherical Earth

R. G. Merrill and W. V. Mansfield

National Bureau of Standards Monograph 38

Issued April 2, 1962

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 70 cents

Contents

Page

1. Introduction 1 2. General procedures 1 3. Details of analysis 2 Refraction 2 Geometry 3

Interpolation and derivative correction 3

Divergence and defocusing 4

Voltage pattern in the ionosphere 4

Diffraction correction 6

4. General results 6

5. Lobe alinement antenna heights and angles of elevation and reflec- tion 7

6. The first Fresnel zone and obstacle criteria 16

Fresnel zone distances 16

Obstacle criteria 19

The zero phase surface 19

Applications 26

7. Summary 29

8. References and notes 29

9. Appendixes:

I. Antenna patterns for ^i: figures 1 to 32 30

II. Antenna patterns for figures 33 to 64 62

III. Comparison of patterns for ki and k^. figures 65 to 78 94

IV. Angles of elevation and reflection in the first Fresnel zone 108

V. Tables 125

VI. Mathematical discussion of the lobe shift 126

VII. Symbols and equations 127

iii
fIi
Radiation Patterns in the Lower Ionosphere and Fresnel Zones for

Elevated Antennas Over a Spherical Earth

R. G. Merrill and W. V. Mansfield

Ground reflection interference patterns in the lower ionosphere have been computed

for elevated antennas over a spherical earth. The computations incorporated parallax,

tropospheric refraction and defocusing, spherical divergence, and near-horizon diffraction. The following antenna siting parameters for VHF scatter propagation were computed from

the patterns:

1. Antenna height and elevation angle for placing the maximum of the first lobe at the path midpoint.

2. Distances from the antenna to the edges and to the quarter-wave contour of the

first Fresnel zone on the earth's surface.

3. Information for determining the effects of obstacles located in the first Fresnel zone.

1. Introduction

The purpose of this work is to give the results

of a detailed computation of antenna patterns in the lower ionosphere at VHF over a spherical

earth. The following results are presented which

have been computed from these patterns:

1. Antenna height and elevation angle for placing the maximum of the first lobe at the path

midpoint; this height will henceforth be called

the lobe alinement height.

2. Distances from the antenna to the edges

and to the quarter-wave contour of the first Fresnel zone on the earth's surface.

3. Information for determining the effects of

obstacles located in the first Fresnel zone. The

analysis of these effects incorporates a new concept and certain refinements within the limitations

of geometrical optics.

The antenna height-gain function has also been

computed from these patterns. [1] ' This function

shows that there is a range of heights lower than

the lobe alinement height which provides gain

greater than or equal to that of the lobe alinement

height.

The model used in the computations is defined

by : tropospheric refraction determined by surface

refractivity and a single gradient of refractivity

;

spherical divergence; defocusing due to refraction; a finite distance to the scattering stratmn in the

lower ionosphere; a near-horizon diffraction cor- rection; and horizontally polarized antennas.

Results are presented for an ionospheric scat- tering layer height of 85 km at frequencies of 30

to 55 Mc; these have been found to be the most

useful values [2] and they correspond to computed

path lengths of 1000 to 2400 km. Parameters

corresponding to "standard refraction" and a temperate over-water or tropical "wet" refraction were used.

Prior computations by Bailey, Bateman, and

Kirby [2] of lobe alinement antenna heights and

angles of elevation did not incorporate divergence

and defocusing.

2. General Procedures

Figures 1 and 2 show the geometry used in the

calculations. It is assumed that the great circle path is symmetrical about a line in the plane of

propagation drawn from the center of the earth

to the ionospheric scattering volume; this symmetry limits consideration to half the total path.

Tropospheric refraction is incorporated using a

bilinear model defined by surface refractivity and

a linear gradient of refractivity. Divergence due

to reflection from the curved surface of the earth

is computed by means of a simplification of the van der Pol-Bremmer geometrical derivation,

namely the assumption that there is no energy

outside the plane of propagation (cylindrical

earth). Divergence due to refractive defocusing

in the atmosphere is computed analogously to divergence for the direct ray and is incorporated

in the divergence of the reflected ray. In the refractivity model, a single linear gradient

is applied only up to that elevation at which the

refractivity vanishes (i.e., the index of refraction

' Figures in brackets indicate the literature references on page 29.

becomes unity). Above this height the refractivity is defined as zero. Model atmospheres

defined by Bean and Thayer [3] closely fit observed refractivity profiles, but limitations in significant figures with respect to wavelength in the computer program for ray tracing through

these atmospheres do not allow these models to be used to compute the interference patterns.

However, as will be shown below (sec. 4), comparison of the bUinear model used here with the

exponential reference atmosphere (the simplest

of the Bean and Thayer models) for the same

parameters shows that the former is entirely

satisfactory.

The spherical coefficient of reflection is taken to be —1.0 in accordance with Bremmer's anah'sis

[4]. For the maximum value of the grazing angle

(^) encountered in the present computations, the

least favorable conditions give a coefficient of

—0.9 at 40 Mc; for this case the gradient of the

reflection coefficient is negligible through the range

of ^ encountered so that the lobe position is virtually unaffected.

1
FOR USE OF 8 SEE FIG. 3

COMPLEMENT^

Figure 1. The geometry in the ionosphere.

E

r

1 V

/ ka

k

TO I

Figure 2. The geometry in the radio troposphere.

3. Details of Analysis

The parameters are the surface refractivity,

Ns ; the gradient of refractivity, AA^'; the height of

the antenna, h; the height of the ionospheric

scattering layer, /; and the frequency of transmission, /. The earth's radius a=6368 km (mean

value to the nearest km) and the velocity of light

in vacuo 2 c=2.99790 -lO'" cm/sec.

All equations numbered below also appear in a

single list as appendix VII.

Refraction. The radius of curvature of rays

passing through a linear atmosphere may be

made infinite by increasing the radius of the

earth [5]. The linear atmosphere is defined and

analyzed as follows. Let the variation with

height, z, of the index of refraction, n, be n=

ns-\-z(dn/dz), where is the index at the surface

and dn/dz is a constant. The modified index of

refraction which takes into account the earth's

curvature is defined by m=n(l + 3/a). Substitution gives m=[7ij+ 2 ((2 + Let this

be written as m=7is(l + 2/^a), where k is the

effective earth's radius factor. Equating the two

expressions for m we have k=l/[{l-\-{a-\-z) /ng}

(dn/dz)]. In terms of A^-units (defined by iV=

(n - 1

) • 1 0«) we have dn/dz

1 0-HNIdz= -^NlO-\

2 von Arderme, Tabellen 1956. In 1957 TJRSI and lUGO adopted (2.997925 ± 0.4-10-«)10'» cm/s.

2
so that k = l/[l —AN-lO-\a+z)/n,]. It is seen

that ^ is a function of z, but since a>>2 and 3

will be no greater than the height at which A^" vanishes (see eq (2) below), the effect of z is negligible. Further, for defining k, is negligibly

different from unity, whence we have

1 l-AiV-a-10-« (1)

For a given surface refractivity, the refractivity

vanishes at a height given by

E=Ns/m. (2)

The gradient, AN, is the change in A'^ per kilo- meter as measured 1 km above the earth's surface.

The height E will be called the radio tropopause

and the linear atmosphere from the surface to E

will be called the radio troposphere. (These must

be distinguished from the physical tropopause,

occuring at about 10 km, and the physical troposphere.)

Geometry. The geometry of the entire model is shown in figure 1 ; the geometry within the radio

troposphere over the ka earth is shown in figure 2. If ^, the angle of reflection (grazing angle), is given, the following quantities are obtained from

plane geometry and trigonometry

:

9ft=cos

ka

\ka+h cos ^ )— ^;

N:

sin dh n I 7 X {ka-\-h); cos 4i

=cos"' ^

ka

ka+E cos ^ M=^^{ka+E);

cos ^

ej=cos-' l^^^cos (^+Mj-i/'-^i

sin di ^ cos (lA+M^''^^^'

e=kid,+eE)+9r=e{^).

(3)

(4)

(5)

(6)

(7)

(8)

(9)

If A, the observed angle of elevation, is given, we

have:

a^A-d/ik-l);

6=^^^ (ka+E);

cos A

cos ..far

La

+E

cos (A+dE') ^^sin ie;-key) ^^^^^

(10)

(11)

(12)

cos {A+dE')'^—a=diA);

(13)

(14)

For the horizon case, 4'=^, we have:

(15)

:C0S .rf_ka_\

\ka+hj -^0 (horizon^); (16) _ _t /a+E ka \ ka \

Ka+I'ka+Ej Kka+Ej -cos

(17)

/a+E \ =cos '

( ^_|_^ COS 6eJ—6b

The formal relationship between A and rf/ is given

by

cos' La

+E ka+h

+1 ka+E • cos A

]

+(/:-l)cos-[|^.cos^]-^

=(^-l)cos-^(^cos^)

+A: j^cos"^ ^

-f-cos (;

ka

ka+h

a+E

a+I 'ka+E

cos^

ka

C0S\p^!

and it can be shown that in the general case

neither can be stated as a closed algebraic function of the other. All of these equations follow

directly from the geometry and trigonometry of

the problem.

Interpolation and derivative correction. In view

of (18), some approximation method for relating

A and is necessary. Since the angle 6 is the

angle in common between A and \p, and fewer

steps are required to compute 6 (A) than to compute 6 (if/), successive differences of 6 (A) were used

to compute the value A corresponding to a given

xf/. The families of angles and distances associated

with \l/ and A were computed independently using an arithmetic progression of difference hr, for the

successive values of the two angles, hi, being so chosen that the successive differences of 6 (A)

allow an inverse quadratic interpolation to be

used to compute the A corresponding to a given ^. The inverse interpolation as defined by Stirling's

central difference formula takes the following

forms

:

-^re(c) —hoUa+An] (19)

-4ra(c) —hz)Ui+An-

(20)

(21)
- A'g;-2 - A'Ci - V( - A'6>;-2 - A'Ci)^ - 8AK-2ieU - dn)

(22)

where 6' n<idn<Cd'n+u and 0„ is nearer d'n in (19)

and (20) and nearer d'n+i in (21) and (22). The

d{A) found by this interpolation agreed with d{yp)

in eight significant figures when the process was

carried out using 10 figures throughout (this being the normal capacity of the digital computer)

;

it was found that if agreement in 10 figures could

be approached, erratic behavior in the computed

voltage resultant near the horizon would be eliminated. A derivative correction was applied to the interpolated 6 {A) which is given by

^e=e{A,,^,,)-^{^|^n),

d{An(c)) ^ 2Ao . dSn AX-i+A%+2uAX-i

d(An(c)) 2hD

ddn aX_,+aX_,+2uA%-

(23)

(24)

(25)

(26)

(27)

where Uu and Ui are defined by (20) and (22),

respectively. The family Of angles and path

length segments defined by equations 10 to 17

is recomputed after each corrected interpolation.

This procedure achieved the desired result: it was

found that the recomputed path segments corre- sponded to the path segments defined by xp with

an error of ±3 mm (estimated from successive

differences); this error was sufficiently small with

respect to wavelength so that values of the voltage

resultant lay on a smooth curve throughout.

Divergence and defocusing. The spherical divergence derived from electromagnetic theory can

also be derived geometrically, as shown by van

der Pol and Bremmer [4,6]. Geometrically, the

divergence is the ratio of the cross section of a cone of rays at a given point after reflection from

a plane surface to the cross section at the same

point after reflection from a spherical surface. In the present computation the energy outside

the plane of propagation is defined as zero so that

linear dimensions rather than areas are used to compute the divergence. The "defocusing" of

both direct and reflected rays is defined analogously: the total bending by the radio troposphere

of rays at different elevation angles is different,

thus, in the plane of propagation, the ratio of the

distance between the ends of two "adjacent" rays

of given length without bending to the corresponding distance after the rays pass through the radio

troposphere defines the defocusing.

The geometry of this spherical divergence computation is shown in figure 3 ; refraction is included

in the computation although the bending is not

depicted in this figure as it was in figure 1. Figure

3A shows the shift in the rays with changing ^

if there were no reflecting surface; the plane

cross section of the cone is closely approximated

by the arc qi, which is defined as the linear

segment required. The rays are shown in

figure 1, and /3 is shown in figure 2. Letting

L be tlxe length of a ray, we have ii+H=

{U+U)I2 and 2i=(/32-/3i)L,+H. In figure_ 3B,

the ray lengths are the same but spherical divergence and refractive defocusing give the linear measure shown as q2. Angle 5 is shown in figure

1 ; since the difference (/(,£,) between successive

values of \p is so small that is negligible,

5i is taken equal to h+y^, and since 61—62 is small

enough so that the arc §2 may be taken as the

chord, we have 22= (^1— ^2)(<^+-^) and 32=

g2 cos 5i =22 sin 5i. Thus, Dii=qi/q2-

The geometry of the refractive defocusing of

the direct ray is shown in figure 4. Without

refraction the linear measure is as shown in figure

4A (see fig. 1) and is given by qi = (A2—Ai)Li+y^.

After bending through the atmosphere we have

the situation shown in figure 4B. Again, the

spacing is such that ai+i^+^i+H is taken to be

equal to aj+^i and the chord is taken to be equal

to the arC; We then have q2={6i—62){a-^ I) and

22=cos(q;j +^i) • sin(a]+^i) • 22- Thus, by definition, DD=qi/q2-

From these considerations we have the following

equations for the product of the spherical divergence and the tropospheric defocusing of the

reflected ray:

2[sin{rPn+6E,+6j^)]

(a+I) {6^-6„+{) ' ^

T)n=h{D,^^,+Dn^_,),B^,^^; (29)

and for the defocusing of the direct ray we have:

{Ar.+,^^^-An^^^J{Gn^Gn+,)ns^Rn-^Rn+l\;

n+i- 2 [sin ian+6^)]{a+I){6n-6n+i)

(30)

Dn=hiDn„^,+D,.^^),Dn=D,,^,. (31)

Voltage pattern in the ionosphere. The electrical path-length difference which determines the interference pattern is given by

L,= {Nn+Mn)ns+ Tn- {Gn ' U.+Bn), (32)

where 1 +Ns • 10~^ The tropospheric segments

have been multiplied by n^, which is a geometrical

correction giving the lengths of the curved rays over the earth with its normal radius {a).

4

The path-length difference in wavelengths is given by

Lef/c—Lto. (33)

The difference in electrical radians is

<{>= -^i„+0.5-[Z„+0.5]-(j-^^ j'27r, (34)

where —ir<<t>< +7r, and the lobe number is +

The square brackets denote the largest integer less than the improper fraction they enclose. Since

the D's represent the fraction of unit incident

power remaining after divergence and defocusing,

the square root of D/r^ will represent the voltage

field at a distance r from a unit power source; thus

the voltage contributions made by the direct and

reflected waves are given by

dD„

and

'Gn-ns +Rn

dR_ =

(35)

(36)

respectively. Figure 5 shows the phase relationship between d^^ and in the ionosphere or at the antenna, whence we have

and

, rsind) • dit~\

(37)

(38)

as the voltage resultant and its phase with respect

to the direct ray.

Figure 5. The voltage resultant.

Diffraction correction. Er=dD at the radio horizon because the divergence there is zero. Thus

there is a spurious rise in the interference lobe

pattern at the horizon and a spurious minimum

between the horizon and the first maximimi. In

order to eliminate this spurious portion of the

pattern, an exponential decay was imposed at the

point of inflection on the near side of the spurious

minimum and extended to the horizon. It was shown by Domb and Pryce [7] that the field just

beyond the horizon computed by diffraction theory can be merged with the free space field at the

inflection point on the horizon side of the first lobe

by a straight line when the patterns are plotted

logarithmically; this graphical procedure was used

to obtain approximate values of Er near the

horizon.

4. General Results

The analysis described in section 3 was programed for a digital computer so that, for a given

set of parameters, the program could be executed

for enough values of \p to obtain a smooth curve

of Er] it could also be executed over a wider range

of the parameters than was used for the present

work. The diffraction correction was done

grapliically.

\ alues of Er (37) for a given set of parameters

and an arithmetic progression of increasing \f/ (with

difference ho) are points on the interference lobe

pattern. Patterns for the following parameters

have been calculated: h=20, 50, 70, 100, 225, 500,

and 1000 m for each of /=30, 35, 40 45, 50, and

55 Mc at 7=85 km and E=S km with 1.342

(AiV=40.00, iV,=320.00) and A:2= 1.467 (AA^=

50.00, iV,=400.00). The patterns so computed

are given in appendixes I and II, and are compared

for kr and k2 in appendix III. We shall now compare results obtained from the

bilinear refractivity model described in section 3

with results obtained from the exponential refer- ence atmosphere defined by Bean and Thayer [3]. Surface distances and slant ranges have been computed using the exponential model for various angles of elevation from a surface terminal [8] ; these

distances correspond to the angular distance kds

+6/ and to the slant range M-ns+T in figure 1. A direct comparison [9] of the two models was made by computing surface distances and slant

ranges for the parameters defining ki and ^2 with

both models; the results are compared in table 1. The differences in distance are much less than

those given by the unbounded ka model: in the

latter case, with ^=4/3, the difference in surface dis- tance is some 85 km for the lower and 75 km for the high surface refractivities at a height of 73 km

where atmospheric bending becomes negligible [10]. It is now necessary to examine the difference in the positions of the antenna pattern in the ionosphere predicted by the two models. Since the ka

model is virtually exact for a given AA^ and Ns in

6
the first kilometer above the surface, and the

greatest antenna height considered is 1000 m, the

angular distance kd^ will be assumed to be correct. Figures 77 and 78 of appendix III give the differ- ence in lobe position for a 1000 m antenna at 35

and 50 Mc for ki and ^2. The difference in parameters represents a difference of 31 km in the

position of the horizon cutoff distance and a differ- ence of 23 km in the position of the first lobe maximum. From table 1 we may assume that the

exponential atmosphere will shift the first lobe maximum less than 10 km along the scattering layer

toward the antenna with respect to ki and less than

13 km with respect to ^2; noting that the lobes for the two refractivity profiles and a given frequency

are the same size and shape (though of different

amplitudes as well as at different distances), we

see that the effect would be negligible in view of

the fact that the vertical beamwidth of an antenna

at the given distances and associated angles would

prevent a horizontal shift of this amount from

being detected. The differences in lobe position

predicted hy the exponential and the bilinear

models for lower antennas may be deduced from

the remainder of appendix III, keeping in mind

that these differences decrease with decreasing

antenna height.

Nevertheless, it may be asked why an exponential model was not used. Apart from the

fact that this work was completed before the

reference atmospheres were defined, the error in correspondence between the direct and reflected

ray lengths must be within ±3 mm in order

to obtain a smooth pattern, regardless of the

model used, and sufficient significant figures are not obtainable from the present program for raj"

tracing through the exponential atmosphere.

It must be understood that no model atmosphere can represent anything but an average

Table 1. Bilinear versus exponential reference atmospheres

N. =320.00 AT, =400.00 AN=40.00 AN=50.00

ki = 1.342 ki = 1.467 E=8.00 km E =8.00 km

Distance, km Distance, km

Exp Bilin Diff Exp Bllin Dlff

0. 000 1107. 25 1116. 38 9.13 1129. 45 1142. 14 12. 69

. 029 1102. 90 1112. 11 9.21 1124.65 1137. 48 12. 83

. 057 1098. 57 1107.88 9. 31 1119.90 1132.85 12. 95

. 115 1090. 04 1099. 47 9. 43 1110. 54 1123.68 13. 14

. 229 1073. 39 1082. 91 9. 52 1092. 38 1105. 66 13. 28

!458 1041.56 1050. 87 9.31 1058. 01 1070. 94 12! 93

.859 989. 93 99H. 12 8. 19 1003. 01 1014. 27 11.26

1.719 892. 71 898. 00 5. 29 901.28 908. 40 7. 12 3.724 714. 98 716. 69 1.71 718. 89 721. 14 2.25 5.730 584. 88 585. 53 0. 65 536. 95 537. 81 0.86

11.459 365. 04 365. 14 .10 365. 56 365. 69 .13 Slant range, km Slant range, km

0. 000 1116. 56 1125. 67 9.11 1138. 79 1151. 46 12. 67

. 029 1112.20 1121.40 9. 20 1133. 99 1146.80 12. 81

. 057 1107. 88 1117. 16 9. 28 1129. 23 1142. 17 12. 94

. 115 1099. 34 1108. 75 9. 41 1119. 87 1132. 99 13. 12

. 229 1082. 68 1092. 19 9.51 1101.70 1114. 97 13. 27

!458 1050. 85 1060. 14 9. 29 1067. 32 1080. 23 12. 91 .859 999. 20 1007. 38 8. 18 1012. 30 1023. 54 11.24

1.719 901.98 907. 25 5.27 910. 56 917. 66 7. 10

3. 724 724. 39 726. 09 1.70 728. 30 730. 54 2. 24

5.730 594. 69 595. 33 0. 64 596. 75 597. 60 0.85

11.459 377. 13 377. 23 .10 377. 64 377. 77 .13 *Both models were computed in milllradians to facilitate comparison of the

results.

refractivity profile or the profile at some specific

time as defined by the parameters used. An

estimate of the effect of changing refractivity on the position of the interference lobe may be

obtained from the comparisons in appendix III and table 1 of the text, but local average refraction conditions must be known and taken into

account in any particular application.

A given antenna is at the lobe alinement height

for that path which is twice as long as the surface

distance from the antenna to the first lobe max- imum of its pattern. Thus, points on curves of

path length versus lobe alinement height can be read

directly from the patterns in the appendixes.

Detailed results from the pattern computations were used to compute Fresnel zone information.

5. Lobe Alinement Antenna Heights and Angles of Elevation and Reflection

Curves of antenna height versus the path

length, which is twice the surface distance to the first lobe maximum, are given in figures 6

A

and 6B using the parameters defining and k2 and 7=85 km. The plots result from smoothing

the values read from the interference patterns;

this smoothing was necessary because of the error

in matching the direct and reflected raj^s inherent

in Le (32) noted above, and the fact that the

lobe maximum may fall between two computed

points on the pattern (see the discussion of the

inverse interpolation, sec. 3). The smoothing was

done by successive difference techniques, the max- imum correction being 2 km; to aid in determining curves for values other than those given, the

smoothed points are listed in appendix V, tables

lA and IB, the third differences with respect to frequency (i.e., reading horizontally) being effec- tively constant, while differences along a curve

(i.e., reading vertically) show a nearlj^ exponential

behavior.

The angle of elevation (arrival or departure),

A, corresponding to the first lobe maximum is given in figures 7A to 7D. It must be remembered that the path length in figures 7A and

7C is the lobe alinement path length and that

the curves refer only to the lobe almement an- tenna height, whereas figures 7B and 7D may

be used with any path length for which a given

antenna height provides sufficient gain (see refer- ence 1). The lobe alinement heights given in figure 6

differ from those given in previous calculations.

The reason for this is that it is usually assumed

that the interference pattern is a function of the

path difference only, which means that the lobe

maximums occur when the path difference in

electrical radians (34) is zero; in the particular

7
8

o.

Q o o o-o

OOOO OOl ^ CO ;0 iri ^' rO' C\i — -p

10
ANTENNA HEIGHT, METERS

Figure 7B. Elevation and grazing angles versus antenna height (ki).

11
5338930 ' 3n9NV

12
case of reference 2, parallax and refraction cor- rections were applied to the least elevation angle

corresponding to zero path difference. However,

the present computations utilize (37), so that the

resultant is directly proportional to the product

of the divergence and defocusing of the reflected ray and to the defocusing of the direct ray, and

inversely proportional to the ray lengths. The

divergence and defocusing are shown in figure 8

;

their rates of change for the values of i/' obtaining

throughout the first lobe cause the maximum to be seen at other than <t>=0, and thus shift the lobe

maximum along the scattering layer toward the

antenna [11].

608365 0—62 2 13
9NISn00d3a QNV 30N39y3Aia

14
ooooooooooo — oodoooodo

9NISn00d3a QNV 30N39M3Aia

15
6. The First Fresnel Zone and Obstacle Criteria

Fresnel zone distances. The ground reflection

point for the value of xf/ corresponding to the

first lobe maximum will be called the geometric

ground reflection point to distinguish it from other

ground reflection points to be considered in

studying the first Fresnel zone. The surface

distance over the earth with radius a from the

antenna to the geometric ground reflection point

is found by multiplying (3) by ka for the proper

value of rp. These distances were subjected to

finite difference smoothing: for h=20, 50, and

70 m, the relationship between distance and

frequency is linear, while for h=100, 225, 500,

and 1000 m, the relationship is quadratic. For

antenna heights up to about 150 m, a spherical

earth appears to be plane, while for higher

antennas the curvature of the surface causes divergence to become significant in the computations: thus the functional relationship between

frequency and the distance to the geometric

ground reflection point changes from linear to quadratic with increasing height; this change

will also be seen in the distances discussed below.

Distances from the antenna to the geometric

ground reflection point are shown in figure 9 and the values from which the curves were drawn

are listed in appendix V, table 2. The radio

horizon line for ki satisfies the equation Hr—

4.13V^, where h is in meters and the radio hori- zon is in kilometers; for ^2, HR=4:.'i2^[h. For

comparison, the equation HR=-^2h expressed in

miles and feet is Z^b=4.12V^ expressed in kilometers and meters.

In the following discussions a reflected ray is always assumed to have one end at the antenna

and the other in the ionospheric scattering

volume.

The geometric ground reflection point is the

center of a family of Fresnel zones on the surface

of the earth. The phase at the antenna, relative

to the direct ray, of a ray reflected from any

other point along the great circle path can be

computed using various details from the antenna

pattern computations. With respect to the first Fresnel zone only, all reflection points for which

the reflected rays have equal phase with respect

to the direct ray define a continuous "contour"

line which is an ellipse on a plane and is "eggshaped" on a sphere; every such contour inter- sects the great circle path twice: once between

the antenna and the ground reflection point and

once between the ground refiection point and

the point on the earth's surface beneath the

scattering volume; the former point of intersection and its distance from the antenna will be referred to as near, and the latter and its distance SiS far. Distances from the antenna to the near and far points for which reflected rays

have phase differences relative to the direct ray

corresponding to a half-wave length (<^ =7r) and

to a quarter-wave length (<^ =7r/2) have been

computed in order to locate the first Fresnel zone on a spherical earth. The need for locating

the first Fresnel zone follows from the fact that

it contributes the major part of the reflected

energy seen at the antenna.

The Fresnel zone distances were computed as

follows. Referring to figure 1, let 0/ be the value

of 6 for the lobe maximum at a given frequency

and let SR/ be the total length of the corresponding reflected ray. The family of angles associated

with each value of x}/ used in computing the an- tenna pattern is examined for the following

conditions: kdh=df— {kds^"^ +61^"-^) (cf. (9)), where

these angles correspond to rf^n- The N'-^'' is then

computed (see fig. 1) and <S'i2^"' = (iV'"'+M^"')Ws +

T"^' (cf. (32)), where M'"' and T^"' are members

of the family of path segments associated with

\pn. SR'-"'' is then compared with *S'i?/-|-X/2 and

with SR/+X/4- When a required Fresnel distance

has been bracketed by two values of SR'-^\ a second order Lagrangian interpolation is used to

find the corresponding dn'-^'' ; ka times this angle

gives the required distance over the earth with

radius a. This procedure was satisfactory for

computing the far distances, but the near distances were undefined or inconsistent with one another

because of the extremely small slope of the function SR'"' in the corresponding range.

In order to find the near distances, a nonparallactic model was used which is described in

figure 10 (cf. figs. 1 and 2). The independent

variable r=0^^' —0^, from which 6'^ and N' can immediately be computed. W=2ka{svn. f/2) and

P=TFsin(i/'+f/2). The quantity compared with

X/4 or _X/2 is SN'-Nr, where SN'=N'+P.^ A

linear interpolation is made after the required

value is bracketed.

The use of (Si?/ in the computation implies that

the divergence at the various Fresnel zone points

is the same as the divergence at the geometric

ground reflection point; any resulting error in the

location of the computed Fresnel zone point is thus

directly proportional to its distance from the

groimd reflection point. Rough estimates show

that such an error will be detectable only in the

far distances for antenna heights of 200 m and

greater; however, far distances for such antenna

heights are either close to or beyond the radio

horizon.

The analyses for the far and near Fresnel zone

distances were combined in a single digital computer program which also computed the angles

relevant to the obstacle criteria which are discussed

below.

Distances to the near and far edges of the

Fresnel zone are presented in figure 11, while the near and far quarter-wave distances are presented

in figure 12. These distances are the results of

linear or quadratic (depending on antenna height)

16

I
Figure 10. Nonparallactic

geometry.

least squares fits to the computed values; these

regressions were done on a desk-sized digital computer. The values are listed in appendix V, tables

3 through 6. An overall picture of the position of

the first Fresnel zone is given in figure 13. It is emphasized that purely geometrical definitions

have been used to compute these distances.

The first Fresnel zone has been located along

the great circle from the antenna to the point

beneath the scattering voluine. Under the limitations of the model, points on the Fresnel zone boundary that are off the great circle path cannot

be determined; however, upper bounds of maximum width can be found by the plane earth

formula_ [2] for a given antenna height, viz,

w=4^/2h.

Obstacle criteria. The criteria for smoothness in the first Fresnel zone that have hitherto been suggested assume a plane earth, since the first zone

has been completely defined for that case. Deviations from smoothness are based on the geometrical analysis in figure 14A. The path dif- ference is Lr=2R sin \f/ : in terms of wavelength this

is 2Rsm\p/\, and in radians it is 47ri?sim/'/X. Rayleigh's criterion for distinguishing between rough

and smooth is to make this path difference less than Tr/4. Thus 4xi?sim/'/X<7r/4, or R<C\/{msmxP).

For the plane earth, A=\l/ and /i=X/(4sin^) ; hence

R<Ch/4:. Kerr [12] discusses other criteria based on the path difference analysis and emphasizes its crudity.

Assuming a receiving antenna over a spherical

earth, bounds for the angles of illimaination (elevation of the ionospheric scattering volume) and

reflection (elevation of the antenna) at any point

in the first Fresnel zone along the great circle path

have been obtained from a computation of these

angles at the edges and at the quarter-wave points

;

angles of illumination are designated by ^/p, and

angles of reflection by yp'p. Angles of illumination were computed using the nonparallactic model

(fig. 10) for both near and far distances. The

angle \1^f='^+^f, where ^F=OF—dn, dp being the

radian measure of the distance from the antenna

to the point in question; ^/'p is computed from the

sine law. The computed angles are plotted in appendix IV; figure 9 of appendix IV shows how

the variation of these angles throughout the length

of the zone may be estimated graphically.

A criterion based merely on height considerations ignores the shape of an obstacle: an object

may satisfy the height criterion mentioned above

and still eliminate the major portion of the re- flected energy, e.g., a building with a sloping roof, or a low hill, or just about any kind of depression.

Using geometrical optics, a simple vertical wall of

height R casts a shadow of length S=R/teLn\{/F in the direction of the antenna with respect to in- cident rays, and of length S'=R/yl/'F in the opposite direction with respect to reflected rays, where

the angles may be measured at R if the shadow is short enough to be considered as lying in a plane.

Thus, ignoring diffraction, an area with length

S-'rS' is eliminated from the zone as shown in figure 15. The zero phase surface. It will be noted that

the height criterion discussed above is stated in a completely negative manner, i.e., as a maximum

tolerated deviation, even though the path from the

top of the obstacle is shorter (by Lr) than that

from the smooth surface at the same distance.

Consider reflection from a surface that is slightly concave upwards: the divergence factor will be- come a convergence factor and focusing will occur. Thus, with respect to the present problem, surfaces may be defined which, when substituted for a spherical earth, render all reflected energy ex- actly in phase at the antenna and in the ionospheric scattering volume. Such surfaces form a family of ellipsoids having foci at the center of

the scattering volume and at the antenna: we

will find points on that member of the family that

is also tangent to the surface of the spherical earth

at the geometric ground reflection point.

Consider a point Pp in the first Fresnel zone

over a spherical earth (as previously defined

geometricaUy), and let the phase difference between the ray reflected at Pp and that reflected at the geometric ground reflection point be <f>: at a height Ro=\4>/2ir{sm\l/p-\-sinyJ/'p) on the

normal through Pp is a point P' such that the

length of the ray reflected at P' is the same as that of the ray reflected at the geometric ground

reflection point. That ellipsoidal surface which

passes through all such points wiU be called the

zero pJiase surface because any ray reflected from

it will be the same length as the ray reflected at the geometric ground reflection point.

It must be remembered that any point P' on

the zero phase surface is determined geometrically

19

21
SdSiaW ' XH9I3H VNN3iNV

22

24
25
Figure 14B. Generalized obstacle geometry.

in exactly the same manner as the Fresnel zone

distances. In particular, the effect of spherical

divergence on the length of the ray reflected at the geometric groimd reflection point has not been

removed even though the zero phase surface

implies convergence: this fact means that values

of Ro at far distances in the Fresnel zone will be

slightly high; parallax at the antenna, which can be detected at the near distances, has not been

taken into accoimt: this means that values of

Ro for the near distances will also be slightly high

(the necessary lack of significant figures in the

method of calculating the angles involved prohibits the inclusion of parallax).

The elevation of the zero phase surface over the

spherical earth at the previously computed Fresnel zone points is plotted in figure 16; these curves

are upper boimds for deviations from the smooth

spherical surface for "aiding obstacles" (see below)

because parallax was not incorporated and divergence was not removed.

Applications. An obstacle of height less than

Ro may significantly add to the energy at the

antenna if its upper surface has a slope between

that of the smooth surface of the earth and the

zero phase surface and has considerable longitudinal aspect, and its shadow doesn't extend

very much into the area bounded by the quarter- wave contour. The height and lateral aspect

Figure 15. Obstacle shadows.

of an obstacle determine the area eliminated from

the zone by shadowing (diffraction ignored).

A perfect zero phase surface above the first Fresnel zone will have a resultant field with a magnitude t/2 times that of the resultant from a

Fresnel zone on a plane. An approximate in- tegration shows that 71 percent of the energy

reflected from a Fresnel zone comes from within

the X/4 contour and 50 percent from within the

X/6 contour. Upper bounds for effects of the loss of portions of the first zone may be set by as- suming that the zone is of zero width ; a boimd for

tolerated loss may be taken as a sinusoidal function of position in the zone to get a measure of

the amount of surface which may be shaded or otherwise eliminated.

For holes with respect to a smooth surface and

for objects higher than Rq, some such criterion as the 7r/4 phase difference (Rayleigh) is reasonable

if combined with a knowledge of shading effects.

If the area immediately around the ground reflection point is satisfactory, a variation in an- tenna height falling within the 7r/4 criterion will

allow the use of a shorter antenna height apart

from the considerations in reference 1. Anything

which adds to the signal will also narrow the

beam width, and holes, etc., will widen it; the

narrowing represents a concentration of energy

and should not_be considered a disadvantage.

With w=4 y/2h as an upper bound for the maximum width, quantitative examination of the first Fresnel zone over a spherical earth has been

carried as far as the present model allows. It must be remembered that geometrical optics

only was used; furthermore, energy contributions

from the higher order zones have not been ex- amined: indeed, these are unknown for the plane

earth case.

26
27
J
7. Summary

Antenna heights for lobe ahnement at the

path midpoint in an 85 km scattering stratum

have been calculated at 30 through 55 Mc in steps of 5 Mc for antenna heights of 20, 50, 70,

100, 225, 500, and 1000 m and two refractivities

representing "standard refraction" and temperate

over-water or "wet" tropical conditions. The

calculations incorporated tropospheric refraction,

parallax, spherical divergence, refractive defocusing, and near-horizon diffraction. The calculations give somewhat higher antenna heights than

those published in reference 2; in the latter case, divergence and defocusing were not considered and

simpler allowances were made for refraction and

parallax. A comparison of the refractivity model

used in the present work with the exponential

reference atmosphere [3, 8] shows that the two are

effectively indistinguishable. The radiation patterns so computed are given in the first three

appendixes.

Distances from the antenna which locate the

first Fresnel zone on the spherical earth, and

angles of illumination and reflection within thezone,

have been given. Roughness criteria have been

given with respect to the zero phase surface

rather than the spherical surface of the earth.

8. References

[1] R. G. Merrill, Optimum antenna height for ionospheric scatter communication, IRE Transactions on Communications Systems CS-8, 14-19 (March

1960).

[2] 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 (October 1955).

[3] B. R. Bean and G. D. Thayer, Models of the atmospheric radio refractive index, Proc. IRE 47, 740-

755 (May 1959).

[4] H. Bremmer, Terrestrial radio waves: theory of propagation, pp. 90-92 (Elsevier Publishing Co., Amsterdam, 1949).

[5] J. C. Schelleng, C. R. Burrows, and E. B. Ferrell, Ultra-short wave propagation, Proc. IRE 21, 427-463 (March 1933).

[6] B. van der Pol and H. Bremmer, Further note on the propagation of radio waves over a finitely con- ducting spherical earth, Phil. Mag. [7] 27, 261-275

(March 1939).

The results of computing the antenna heightgain function have been published [1]: these show

that the optimum height is lower than the lobe

alinement height and that a broad range of lower

heights is essentially equivalent in gain to the lobe

alinement height.

The authors acknowledge the following contributions to this work: discussions with Richard

C. Kirby, Kenneth L. Bowles, Ernest K.

Smith, Bradford R. Bean, and Gordon D.

Thayer; contributions to the computer program by Mrs. Marie L. West; computations and

graphing by Loren P. Sims; computations by

Mrs. Charlotte I. Enfield and Alvin M. Gray;

extensive computer production work by Patricia

R. Lollar and Bonnie M. Laubach; computer

ray tracing for the refractivity model by Mrs.

Betty J. Weddle; valuable critical reading of the

text by Robert S. Cohen.

This work was carried out on behalf of the

U.S. Air Force under support extended by the

Ground Electronics Engineering and Installation

Agency, Rome, N.Y.

and Notes

[7] C. Domb and M. H. L. Pryce, The calculation of

field strength over a spherical earth, J. lEE 94, 325-339 (September 1947).

[8] B. R. Bean and G. D. Thayer, CRPL exponential

reference atmosphere, NBS Monograph 4 (October

29, 1959).

[9] Unpublished computation courtesy G. D. Thayer

and Mrs. B. J. Weddle.

[10] Bean and Thayer, Proc. IRE, loc. cit., fig. 8, p. 748.

[11] For a more detailed discussion of this point see appendix VI.

[12] Donald E. Kerr, editor, Propagation of short radio waves, M.I.T. Radiation Laboratory Series, vol. 13 pp. 411-418 (McGraw-Hill Book Co. New

York, N.Y., 1951).

608365 O— 3 29
snoAmiN 'aiaij

30
31
snoAmiH 'an3ii

32
SiiOAinm ' an3ij

33



snoAimiw ' an 31

J

37

39
o

o

tvl

40
Q lo

O

m

CM

—

—

O

snoAmiN 'a"i3ij

41
42

44
608365 0—62 4 45
46
47


SnOAimiH 'Q-\3\d

50
51
52
53

o

55

57
S Q

58




L
64
65

67
68
69

i

SiiOAniiw 'ai3ij

73

«

o

75
t

76
si"iOAn-|itN 'aiaij

608365 0—62 6 77


80


i



87
88
89

91


94
95



SnOAmiW 'Q13\A

99

101
102

Si-IOAimiW 'Q13U

104
105

snoAmiw '0131^

107
SW3i3W 'J.H9I3H VNN3iNV

108
Sa3X3W 'iH9l3H VNN3XNV

109

SH313N 'IHOIBH VNNBINV

111


Sd313W 'iH9l3H VNN3iNV

114
o o o o o o o o o o o in cvj — o « cj —

Sy3i3IN ' iH9l3H VNN3i.NV

115
SUBXaiN 'XHSOH VNN3XNV

116
117
SM3i3W 'iH9l3H VNN3iNV

118

SM3i3IN 'iHOI3H VNN3iNV

120

Sy3i3IN ' iH9l3H VNN3INV

122
Sa3J.3W 'iH913H VNN3iNV

123
0.0 05 1.0 1.5 2.0 2.5 ANGLE OF ILLUMINATION, DEGREES

3.0 3.5

Figure 9. Angles of illumination at 35 Mc. 124
Appendix V. Tables

Table 1. Lohe alinement antenna height, m

Path length Frequency (Mc)

30 35 40 45 50 55

Part A: fci refraction

km

1000 23.1 19.9 17.4 15.5 14.0 12.8

1100..- 26.6 22.8 19.9 17.7 16.0 14.7

1200 30.8 26.4 23.1 20.6 18.6 16.8

1300 35.9 30.8 26.9 23.9 21.5 19.4

1400 41.8 35.8 31.3 27.9 25.2 22.7

1500 49.5 42.5 37.2 33.1 29.7 26.6

1600 59.2 50.7 44.3 39.4 35.4 31.8

1700 72.9 62.2 54.3 48.4 43.7 39.3

1800 92.0 78.5 68.4 60.9 55.1 50.1

1900 122.0 103.5 90.3 80.8 73.5 66.8 2000-., 174.0 150.0 131.0 116.3 105.0 96.3

2100 285 246 216 193 175 161 2200 495 441 401 370 345 322

2300- - 883 819 763 715 675 643

Part B: fc 2 refraction

900 20.2 17.8 15.5 13.5 11.9 10.8 1000-.- 22.9 20.2 17.7 15.5 13.7 12.4

1100 26.4 23.2 20.3 17.8 15.8 14.3

1200 30.7 26.7 23.3 20.6 18.3 16.6

1300 35.5 30.9 27.1 24.0 21.5 19.6

1400 41.6 36.1 31.6 28.0 25.2 23.0

1500 .- 49.0 42.2 37.0 33.0 29.8 27.1 1600 58.0 49.9 43.9 39.4 35.7 32.2

1700 69.9 60.2 53.1 47.7 43.2 38.7 1800--- 87.0 74.7 65.6 58.8 53.2 47.8

1900 113.9 97.5 84.9 75.3 68.0 62.3

2000 157 135 117.3 103.4 93.0 85.6

2100 233 203 179 159 143 130

2200 ... 376 334 299 271 248 228 2300 658 593 540 496 459 427

2350 890 806 736 679 633 595

Table 2. Distance from antenna to ground reflection point, km

jVntenna

height

Frequency (Mc)

30 35 40 45 50 55

Part A : /;i refraction m

2050

70100

225

500 1000

0. 130 .898

1.74 3.60

16.2

47.7 90.9

0.154

1.036

2. 06

4. 23 17.9 50.4

93.4

0. 179 1.193

2. 38 4.84

19.6

52.7 95.6

0.203

1.351 2.69

5. 42

21.1 54.7 97.6

0. 228 1.509

3. 01

5. 98

22.5 56.4

99.3

0.252

1.666

3. 32

6. 51 23.8

57.8

100.7

Part B: k". refraction

Table 3. Distance from antenna to near edge of first Fresnel zone, km

Antenna

Frequency (Mc)

height

30 35 40 45 50 55

Part A: k\ refraction m

20 - - . - 0.022 0.027 0. 032 0. 036 0. 041 0.046

50 .161 .190 .218 . 247 .276 .304

70 .321 .378 .434 .490 .545 .599 100 .662 .777 .890 1.001 1.111 1.219 225-- 3.28 3. 79 4.29 4. 77 5.24 5.69

500 13.7 15.3 16.8 18.2 19.4 20.5

1000 37.0 39.8 42.3 44.6 46.6 48.4

Part B: refraction

20 0. 022 0. 027 0. 032 0. 036 0. 041 0. 046

50; .160 .189 .217 .246 . 274 .303

70 .321 .376 .432 .487 .543 .598

100 .661 .773 .885 .996 1.108 1.220

225 3.29 3.80 4. 31 4.80 5.29 5. 76 500 13.9 15.6 17.1 18.5 19.8 21.0

1000 38.0 40.8 43.4 45.7 47.8 49.7

Table 4. Distance from antenna to near quarter wave

point, km

Antenna

Frequency (Mc)

height

30 35 40 j 45 50 55

Part A: ti refraction m

20 0. 037 0.044 0. 061 0. 058 0. 065 0.072

50 .250 .295 .339 .383 .427 .470

70 .496 .585 . 672 .758 .842 .924

100 1.02 1.20 1.37 1.54 1.71 1.88

225 5. 05 5. 73 6. 42 7.10 7. 79 8. 47

500 19.6 21.5 23.3 24.9 26.4 27.8

1000 48.1 51.1 53.8 56.1 58.

1

59.8

PartB: /.-j refraction

20 0. 036 0.043 0.050 0. 058 0. 065 0. 072

50 .250 .294 .337 .380 .424 .468

70 .497 .582 .668 .753 .839 . 925 100 1.02 1.19 1.36 1.53 1.71 1.88

225 5. 04 5. 75 6. 47 7. 18 7. 90 8. 62

500 19. 7 21.8 23.7 25.5 27.1 28.5

1000 49.3 62.4 55.2 67.7 59.9 61.7

20...

50--.

70...

100-

.

225.. 500--

1000-

0.128

.866

1.74

3. 52

16.3

48.7 93.8

0. 162 1.023

2. 04 4.13

18.3

51.9

0. 176 1.179

2. 35 4.74

20.2

54.6

99.1

0.200

1.336

2.66 5.36

21.9 56.8

101.3

0. 224 1.493

2. 96

6. 97 23.5

58.7

103.2

0.247

1.650 3.26

6. 58

25.0 60.2

104.8

125
Table 5. Distance from antenna to far quarter

point, km

Table 6. Distance from antenna to far edge of first Fresnel

zone, km.

Antenna

Frequency (Mc) Antenna

Frequency (Mc)

height

30 35 40 45 50 55 height30 35 40 45 60 55

Part A: k\ refraction Part A: ti refraction m

20 0. 37 0.44 0. 52 0.60 0.68 0. 76

50 2. 78 3. 34 3. 89 4.44 5. 00 5. 55

70 5. 93 7. 00 8. 07 9.14 10.21 11.27

100 12.2 14. 2 16.

1

17.8 19.3 20.7 225 45.2 47.6 49.8 52.

1

54.4 56.7

500 96.0 97.4 98.8 100.

1

101.5 102.9

1000 149 150 161 151 152 153

Part B : ki refraction

20 0. 36 0. 43 0.50 0. 57 0.64 0. 71 50 2. 71 3.25 3. 80 4. 35 4. 89 5. 44

70 5. 84 6. 90 7. 95 9. 00 10. 05 11.10

100 12.0 14.0 15.9 17.7 19.4 20.9 225 45.5 48.8 51.9 54.6 56.9 58.8

500 98.

1

100.1 102.1 104.1 106.1 108.1 1000 154 155 156 158 159 160

TO 20 0.63 0. 75 0.86 0. 97 1.08 1.19

50 4. 51 6. 43 6. 32 7. 17 7. 99 8. 77

70 9. 02 10. 81 12. 46 13. 97 16. 34 16. 57

100 18.2 21.1 23.7 26.0 28.0 29.7

225 60.0 63.4 66.2 68.5 70.3 71. 6 500 117 118 119 120 121 122

1000 173 173 173 173 174 174

Part B : kt refraction

20 0. 58 0.70 0. 81 0. 93 1.04 1.16

50 4. 45 5. .30 6. 14 6. 99 7. 84 8. 69 70^_ - 9.17 10.65 12.13 13.60 15. 08 16. 66

100 17.9 20.7 23.3 25.7 27.9 29.9

225 61.3 64.7 67.9 70.8 73.4 75.7

500. _ 120 121 123 125 126 128

1000 179 179 180 180 181 181

Appendix VI. Mathematical Discussion of the Lobe Shift

The difference between the lobe alinement

heights given in reference 2 and those given in figure 6 is due primarily to the incorporation of

divergence and defocusing in the latter results. These quantities cause the first few lobe maximums

to occur at phase differences other than 4>~2-Kk

{k=Q, I, 2, . . .) between the direct and reflected

raja's. This is shown mathematically as follows:

Differentiating £'r (37) with respect to yp we obtain

dida) , dido) , , 7 / , ,\dxt) —r,—t-cfi5tt«(— sm 0) ^

+COS 4> (dr,

didn) , J dido)

0] (Al)

When 4> =2irk, Er —dn-^dr, and

didu) . dido)

d-^ J^ = 2irk d\p d\p

,k=0,l,2, . . . (A2)

Maximums occur at 4) =2irk only when this derivative vanishes. Similarly, minimums occur

onlv when

0 = ir(2t+l)

didp) didg)

d-p d^P (A3)

vanishes.

Using eq (35) and (36) and letting r^, and Tr be

the lengths of the direct and reflected rays, we

obtain

d{dn)_d /yz),

d^ d\l/\ Tr )

and

d{dr>)_d

dip drp

1- ' d{Dn) dirR)

dip dip

(A4)

1 d{Do) dirp)

2rn^'D^ dip dip

(A5)

knowing that the derivatives of Tr and are negative.

The behavior of d{DR)ldip and of d{PD)ldip is independent of <j> and can be quicldy deduced

from figure 8. Successive differences show that

Vr and Tjy (also independent of <p) are exponentially

decreasing functions of ip which asymptotically

approach 85 +0.002h km and 85 km, respectivel}', as ip approaches 7r/2. Thus the two derivatives

vanish as ip increases. Since the derivatives of Dd and Dr vanish almost immediately, the differ- ence in lobe alinement heights with increasing

ip (i.e., with decreasing antenna height), decreases

exponentially as shown in reference 1, figures 7 and 8. Similarly, since eq (A2) and (A3) are non-zero, they show that the minimums differ from zero voltage between the first few lobes, but hy less than the maximums differ from the in-phase sums

of the two components. This effect on the lobe

shape is greatest for small \p and is j^lainly evident

in appendix I, figures 27-32 and in appendix II, figures 59-64.

126
Appendix VII. Symbols and Equations

List of Symbols

A: Observed angle of elevation of direct ray at antenna (fig. 1). An(c) ' Interpolated value of A.

Anidc) ' Interpolated and corrected value

of A.

a: Mean earth's radius (6368 km).

c: Velocity of light in vacuo (2.99790-10'°

cm/sec).

Dj) : Refractive defocusing of direct ray.

Dr-. Product of spherical divergence and refractive defocusing of the direct ray.

di,: Voltage contribution of direct ray.

(Ir: Voltage contribution of reflected ray.

Er'. Voltage resultant (eq. 37).

E: Elevation of "radio tropopause" (eq. 2). /: Frequency (Mc).

G: Direct ray segment (fig. 2). Hr: Distance to radio horizon.

h: Antenna height.

ho- Difference of arithmetic progression of i// (eqs 19 and 21).

/: Ionospheric scattering layer height.

k: Efi^ective earth's radius factor for linear

atmosphere.

Lg-. Electrical path length difference.

Lr : Path difference between Fresnel zone surface

and obstacle surface.

Z/„: Path length difference in wavelengths.

M: Reflected ray segment (fig. 2). N: Reflected ray segment (fig. 2).

Ns'. Surface refractivity.

R : Direct ray segment (fig. 1).

Ro'- Elevation of zero phase surface.

SRf -. Total reflected r^y length at frequency / in Fresnel zone calculation. T: Reflected ray segment (fig. 1).

Uu'. "Upper" inverse interpolation factor (eqs

19-20).

Ui: "Lower" inverse interpolation factor (eqs

21-22). w. Width of first Fresnel zone over plane earth.

a: True angle of elevation (fig. 1).

jS: Elevation of reflected ray from antenna (figs. 2 and 3). AN : Gradient of refractivity.

A]A^: First and second order central differences.

8 : Angle used in spherical divergence computation (figs. 1 and 3).

f: Angle used in nonparallactic geometrv

(fig. 10).

6: Angular surface distance from antenna to point in the ionosphere.

^Efih,6i'- Angular surface distances (fig. 2). X: Wavelength.

4): Phase angle between direct and reflected rays.

(t>j!D'- Phase of voltage resultant with respect to direct ray.

yj/: Angle of ground reflection (grazing angle).

ij/p: Angle of illumination in first Fresnel zone

(elevation of- ionospheric scattering

volume).

i/'V: Angle of reflection in first Fresnel zone

(elevation of antenna).

Equations

Given and AN:

1 k=

l—LN-a-lO9ft =cos'

Given i/:

cos \p

6e=cos

M=^-^(ka+E)

cos l/'

9j=cos-M -La

±E

+/ cos (\p+dE.

cos {4^+0e)

e=k{en+eE)+ej=e{^p)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Given A:

0/ =cos-'(^|^±^cosA^-^ (10)

cc=A-eE'{k-l) (11)

^_s_m^

^^2) cos A

e' =cos-'[^t^cos {A^dE')^-ci=e{A)

^^sin ie'-keE') J

cos {A+dE )

(13)

(14)

127
Given 6=0:

i9ft =cos - ^ (^^^-p^J= —/lo (Horizon A) (16)

\a+I ka+Ej ^ \ka+Ej

cos 6/^-9/ (17)

_^/a+E ka

9/= cos ^ '

cos

= cos"

Given \{/ and A:

.J I a,-^E ka+h

r-La

cos A

+/ ka+E ]

= (^-1) cos-(^cos^)

. _,/a+E ka A

Inverse Interpolation:

(19) _ - A'i9;-2 - A 'g; - V( - A'e;-! - A' g;) 2 + 8A'g;-i (6>„ - g;)

(20)

An(c) —hj)Ui-{-A„^] (21)

- A'e:-2 - A'dn-i - V( - A'Ca -A'Ci)^ - 8A2e:-2(d -e„) 2A20^- (22)

where 0,^<0«<0^+i, for 6„ nearer 0^ in and nearer

^^-1 in Ui

Derivative Correction:

Ad=e{A„,,,)-e{rp„)

(23)

(24)

(25)

<^(-4n((;)) 2/ii, (26)

(27)

d(A„(c)) 2A/fl

where Uu and Ui are defined above.

Divergences of Direct and Reflected Rays: ^

['A«+i-'A.-(g;,„-g;,,+i)]

2[sin(;A.+^.„+^0]

[(7v„+A^„+i+M„+M„+0 7i,+r„+r.+i]

(a+/) (^„-^„+i)

Z>ie„=K^«TM+i+^«n-i), Dn=0 (29)

2 [sin (Q:„+0,)](a+/)(^„-0„+i)

(30)

(28)

• D,=i(D,^^.+D,^_^),D,=D,^^,^ (31)

Fai/i- Length Difference:

Le={Nn+Mn)ns+Tn-{Gn U.^R^)

where 7i, =l + |7V,|-10-« (32)

Frequency Dependent Quantities:

-=L.

0= |z.+0.5-[L„+0.5]-(J-^)|

where —7r<0<+7r.

Lobe number=[L„]-|-l

27r,

(33)

(34)

N.B. In (34) [ ] is the integral part of the improper

fraction.

d,

Gn ns+Rn

« (A^„^MJn,+r„

Er=-4d^^dD^^2dDdR cos <A

. _Ss\w<^-d^

4>ED=sm M —^

J

(35)

(36)

(37)

(38)

128

U. S. GOVERNMENT PRINTING OFFICE: 1962 O -608365
U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D.C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements.

Dielectrics. Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering

Metrology. Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological

Equipment. Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard

Reference Materials. Applied Analytical Research. Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer

Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental

Research. Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal PhysicsElectrolysis and Metal Deposition.

Mineral Products. Engineering Ceramics. Glass. Refractories. Enameled Metals. Crystal Growth.

Physical Properties. Constitution and Microstructure.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics.

Operations Research. Data Processing Systems. Components and Techniques. Digital Circuitry. Digital Systems. Analog

Systems. Applications Engineering.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic

Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical

Instruments. Basic Instrumentation.

Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy.

Molecular Kinetics. Mass Spectrometry.

Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic

Technical Services. Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere

Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements. Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High Frequency Electrical Standards. Radio Broadcast Service. Radio and Microwave

Materials. Atomic Frequency and Time Interval Standards. Electronic Cahbration Center. Millimeter- Wave Research. Microwave Circuit Standards. Radio Systems. High Frequency and Very High Frequency Research. Modulation Research. Antenna

Research. Navigation Systems.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere

Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

Radio Physics.

Circuit Standards.