NTIA Report 82~114

HF Ground Wave Propagation over Forested

and Built-up Terrain

D.A. Hill

u.s. DEPARTMENT OF COMMERCE

Malcolm Baldrige, Secretary

Bernard J. Wunder, Jr., Assistant Secretary

for Communications and Information

December 1982
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TABLE OF CONTENTS

LIST OF FIGURES.

LIST OF TABLES

ABSTRACT ...•

iv

vi

1

1.

2.

3 ..

INTRODUCTION. .

UNIFORM SLAB MODEL.

2.1 Integral Representation.

2.2 The Lateral Wave ..... 2.3 Normalized Surface Impedance

2.4 Height-Gain Functions..

2.5 Limit of Vanishing Slab.

INTEGRAL EQUATION APPROACH. . .

3.1 Formulation .

3.2 Integral Equation Solution

1

2

3

6

6

8

9

10

10

13

4. TWO-SECTION PATH....

4.1 Kirchhoff Theory

4.2 Comparison with Integral Equation.

15

15

20

5.

6.

7.

8.

9.

EQUIVALENT SLAB PARAMETERS. .

5.1 Forest Cover.

5.2 Built-up Areas

5.3 Snow Cover ..

SPECIFIC PATH CALCULATIONS ..

6.1 Smooth Paths in the Netherlands..

6.2 Irregular, Forested Terrain in West Germany.

CONCLUSIONS AND RECOMMENDATIONS

ACKNOWLEDGMENTS .

REFERENCES ....

22

24

24

26

30

30

35

50

53

53

APPENDIX A: ANTENNA WITHIN THE SLAB .

REFERENCES. . . . . . . . . .

APPENDIX B: KIRCHHOFF INTEGRATION.

REFERENCE . . .. ....

APPENDIX C: USER'S GUIDE, LISTING, AND SAMPLE OUTPUT FOR PROGRAM WAGSLAB.

iii

57

62

64

66

67
Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

LIST OF FIGURES

Vertical electric dipole over a uniaxial anisotropic

slab. . . . . . .

Equivalent geometry for the limit of a free-space slab

(Ehc = E

VC = 1) • • • . . .

Geometry for integral equation. The slab and ground parameters can vary as a function of x, but are constant in y..

Geometry for a two-section path with an elevation change D.

The section, x < 0, has a normalized surface impedance ba ,

and the section, x > 0, has a normalized surface impedance

8b · . . . . . . . . . . . . . . . . . . . . . . .

Propagation from a forest to a clearing for a frequency of

10 MHz. . . . . . . . . . . . . . . . . . . . .

Propagation from a clearing to a forest for a frequency of

10 MHz. . . . .. . . . . .

4

11

12

16

21

23

Figure 7.

Figure 8.

Relative permittivity E

V

versus building density B.

Magnitude of the spherical earth attenuation function

function of frequency and snowdep!~ D. Parameters:

E = E

h = 1.55, a = a

h v. v· = 2.5 x 10 Slm, E = 10, a g g

Sim . . .. .

as a

27

29

Figure 9.

Figure 10.

Figure 11.

Figure 12.

figure 13.

Smooth path in the Netherlands with receJ.vJ.ng sites at 60 km

and 80 km. The first 70 kID has good ground (E = 15 and

-2 g

a = 10 S/m), and the last 10 km has poor ground (E = 3

g -4 g and a

g = 10 S/m).... . . . . .. . .

Comparison of measurement with three theories at 2 MHz for a

smooth path in the Netherlands. Only the integral equation

solution takes into account the poor ground section shown in

Figure 9. . . . . . . . .

Comparison of measurement with three theories at 10 MHz for

a smooth path in the Netherlands. Only the integral equation solution takes into account the poor ground section

shown in Figure 9 • • • • • • • • • • • • • • • • • • •

Comparison of measurement with three theories at 30 MHz for a

smooth path in the Netherlands. Only the integral equation

solution takes into account the poor ground section shown in

f:Lgure 9. . . . . . . . . . . . . . . . . . . . .

Terrain profile for the path from Inneringen to Boblingen

iv

31

32

33

34

37
Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Figure 24.

Figure 25.

LIST OF FIGURES (continued)

Propagation from Inneringen to Boblingen at 2 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs. . . . • . . . .

Propagation from Inneringen to Boblingen at 5 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs. The circles are obtained from

multiple knife-edge diffraction theory .

Propagation from Inneringen to Boblingen at 10 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs..•..........

Propagation from Inneringen to Boblingen at 20 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs .

Propagation from Inneringen to Boblingen at 30 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs. The circles are obtained from

multiple knife-edge diffraction theory........•

Magnitude of height-gain function in forest slab an~ in air.

Parameters: D = 20 ~2 t.v = t.h = 1.1, crv = crh = 10 Slm,

E g = 10, and crg = lOS1m. . . . . . . . . . . . . . .

Spherical earth attenuation function for, bare ground and with

;forest cover. Parameters: D = 20 m, t. = t.h = 1.1, cr

v = -4 . -2 v

crh = 10 Slm, t.g = 10, and crg = 10 S/m ...••.•

Propagation from Inneringen to Boblingen (solid curve) and in

the reverse direction (dotted curve). Both cases are for

bare ground, and the frequency is 2 MHz . . . • . . . . . . .

Propagation from Boblingen to Inneringen (solid curve) and

in the reverse direction (dotted curve). Both cases are for

bare ground, and the frequency is 5 MHz . . . . . . . . . • .

Propagation from Inneringen to Boblingen (solid curve) and

in the reverse direction (dotted curve). Both cases are for

bare ground, and the frequency is 20 MHz .........•

Terrain profile for the path from Inneringen to Lechfeld. •

Propagation from Inneringen to Lechfeld (solid) and in the

reverse direction (dotted). Both curves are for bare ground,

and the frequency is 2 MHz •.......... 0 ••••••

v

38

39

40

41

42

43

44

46

47

48

49

51
Figure 26.

Figure A.I.

Table 1.

Table 2.

LIST OF FIGURES (continued)

Propagation from Inneringen to Lechfe1d (solid) and in the

reverse direction (dotted). Both curves are for bare ground,

and the frequency is 5 MHz •••...

Integration contours in the complex A plane

LIST OF TABLES

Characteristic Parameters of Typical Forests (Dence

and Tamir, 1969). . . . • . . • . • . . •

Magnitude and Phase of Surface Impedance 6 as a Function

-5 of Snow Depth D. Parameters: E

v

=Eh=1.55, 0v=oh=2.5 x 10

-2 Slm, Eg=10, 0g=10 S/m. . . . . . . . . . . . . . . •

vi

52

60

25

28
HF GROUND WAVE PROPAGATION OVER FORESTED AND BUILT-UP TERRAIN

D. A. Hill*

An integral equation method is present~~d for computing the vertically polarized field strength over irregular terrain which is covered

with forest, buildings, or snow. The terrain cover is modeled as an

equivalent slab, and a general computer code, WAGSLAB, is developed.

Numerous special cases are treated analytically, and comparisons are

made with numerical results from WAGSLAB.

Key words: flat earth; HF ground wave propagation; irregular terrain;

mixed path; lateral wave, slab model; spherical earth;

vertical polarization

1. INTRODUCTION

Integral equations (Hufford, 1952; Ott and Berry, 1970) have been found useful

for calculating the ground wave field over irregular, inhomogeneous terrain. More

recently Causebrook (1978a) has modified Hufford's integral equation to account for

the effect of built-up sections along the path and has been successful in modeling

MF propagation in the London vicinity. He assumed that the building heights were

small compared to a wavelength, and he did not consider the effect of antenna

height.

In this report, we extend Causebrook's general approach upward i~ frequency to

the HF band where the building and forest heights are not necessarily small and the

effect of the antenna height is important. The extension utilizes an anisotropic

slab to model forest, snow, or buildings along the path. We modify Ott's (197la)

computer program (WAGNER) to include the possibility of a lossy, anisotropic,

dielectric slab of arbitrary thickness over the ground. The slab parameters are

allowed to vary along the path just as the terrain height and earth conductivity

and dielectric constant are allowed to vary in program WAGNER. The source and

receiving antennas can be located either within or above the slab, and only vertical

pOlarization is considered. The same technique could be used for horizontal polarization, but the rapid attenuation makes horizontal polarization not useful for

ground wave propagation.

*The author is with the U.S. Department of Commerce, National Bureau of Standards,

Boulder, Colorado 80303.
The integral equation approach that is described in this report is a point-topoint prediction method and should be most useful for frequencies below 30 MHz.

For higher frequencies, the variability of the ground wave in time and space

becomes large, and an accurate point-to-point prediction becomes difficult. A

recent report (Hufford et al., 1982) describes an area prediction model for VHF,

UHF, and SHF frequencies.

The organization of this report is as follows: In Section 2, the fields of a

vertical electric dipole are analyzed for the dipole either within or above a uniform slab. The effective surface impedance and the height-gain functions are

derived. In Section 3, Ott's previous integral equation (197la) is extended to

include a slab above the earth by using surface impedance and height-gain approximations from Section 2. In Section 4, a two-section path is analyzed both by

Kirchhoff Theory and by the integral equation method. The analytical result from

the Kirchhoff Theory is useful in its own right and also provides a good check for

the revised integral equation. In Section 5, the values for the equivalent slab

parameters for forests, built-up areas, and snow are discussed. In Section 6,

specific path calculations are performed using the integral equation approach.

Section 7 includes a summary and recommendations for further work. In Appendix A,

both the integral representation and the asymptotic result are derived for the case

where the dipole is located within the anisotropic slab. Appendix B includes the

rather involved mathematical details of the Kirchhoff integration which is required

for the two-section path analysis in Section 4. Finally, Appendix C provides a

user's guide, a listing, and a sample output for program WAGSLAB which is essentially

an extension of Ott's program WAGNER.

Throughout the report, we present propagation results in terms of a normalized

attenuation function f which is the ratio of the actual electric field to twice the

free-space field. This is done in order to emphasize propagation effects and to

eliminate antenna effects. Thus f is unity for a flat, perfectly conducting ground.

If some form of transmission loss is desired, it can be easily computed from the

free space transmission loss and the magnitude of f.

2. UNIFORM SLAB MODEL

In this section, we analyze a uniform, anisotropic slab model for a vertical

electric dipole source. The asymptotic results for the uniform slab model (Wait,

1967a) can be cast in a very convenient form where the total field can be factored

into a product of twice the free space field times a ground-wave attenuation function times the height-gain functions for the source and observer heights. The

attenuation function depends on the surface impedance for a layered medium in the

2
same manner that the flat-earth attenuation function for a homogeneous earth (Wait,

1962) depends on the surface impedance of a homogeneous half-space. We utilize the

specific results for the surface impedance and height-gain functions of the uniform

slab in Section 3 where the slab parameters are allowed to change along the path.

Although we are primarily interested in the uniform slab model as a starting

point for the more general irregular terrain model of Section 3, the uniform slab

model is of interest in its own right for modeling uniformly forested paths (Tamir,

1967; Dence and Tamir, 1969; and Gordon and Hoyt, 1982). In this report, we also

use a slab to model built-up urban areas or snow accumulation. The specific slab

parameters appropriate for forests, urban areas, and snow are discussed in Section,

5.

2.1 Integral Representation

We consider first a vertical electric dipole located at a height h over a uniaxial anisotropic slab of thickness D as shown in Figure 1. The dipole has a current moment I ds, and the exp(iwt) time dependence is suppressed. The slab has

horizontal and vertical relative permittivities, E

h and E

V' respectively, and horizontal and vertical conductivities, a

h and a

v ' respectively. The region, z > 0, is

assumed to be free space with permittivity EO and zero conductivity. The region,

z < -D, is ground with relative permittivity E

g and conductivity agO We assume

free space permeability Po everywhere. In the following treatment it is convenient

to use the following expressions for the various relative complex dielectric constants

E = E + a ! (iw E ) gc g g 0

E

hc = Eh + ah!(iw EO) (1)

and

E = E + a !(iw E ) vc V v 0

The field in air (z>O) can be derived from the z component, IToz ' of an electric

Hertz vector which has the following integral form (Wait, 1967a):

where

IT

oz = ds fOOte-uolz-hl+ R (A) e-Uo(Z+h)]

4nw E II

o 0

3

A

- J CAp) dA

u 0

o

(2)
z

---r1t

h to 1 ...

. P

t ht Oh

D Eyt

Figure 1. Vertical electric dipole over a uniaxial anisotropic

slab.

4
K - 21 u

RI

1

(/I)

0 K 0 = K + 2 = 1 0 iw E: 0 0

(;\2

k

U = _ k 2 ) 2 k = W/c 0

K2 + Kl tanh(vO)

21 = Kl K

l + K2 tanh(vO)

K2 = ul(j~ E: E: ) '. 0 gc

and

Cylindrical coordinates (p,<j>,z) are used. The electric and magnetic fields in air

(z>O) are given by

E

op = E

oz

and

= -iw E:

o

arr

oz

ap (3)

E = H = H = 0 o</> op oz

An integral expression for the fields in the slab (0 > z > -0) can be derived

and is similar in form to (2). The flelds in the ground (z < -0) are not normally

of interest. Expressions for the fields both in air and within the slab when the

dipole source is located within the slab are derived in Appendix A.

5
2.2 The Lateral Wave

In general an evaluation of the fields involves a numerical integration as

indicated by (2). However, when kp is large and Izi is not large, integrals of

the type in (2) can be evaluated asymptotically. The result for the dipole in air

(h>O) was given by Wait (1967a), and the result for the dipole in the slab

(0 > h > -D) is derived in Appendix A. In either case, the vertical electric field

E either in air or in the slab can be cast into the following convenient form

z

E

z

Eo f(p) G(h) G(z) (4)

The result in (4) has the following interpretation. E is twice the field of

o

a dipole in free space or by image theory is the field when both the dipole and the

observer are located on a flat, perfect conductor:

E =

o

-iWll I ds

o (5)

The Sommerfeld attenuation function f(p) accounts for the imperfect conductivity of

the ground and the presence of the slab. For a perfectly conducting ground, p=O

and f(O)=l. In our case, the magnitude of the numerical distance p is large, and

f(p) is given by (Wait, 1962)

where

f(p) -1!(2p)

p = -ikpfl2!2

(6)

The normalized surface impedance fl will be defined and discussed in the following

section. Note that E f(p) has the expected p-2 variation which is associated with

o

the "lateral wave" (Tamir, 1967). The point we wish to make here is that the

"ground wave" has the same p-2 variation when Ipl is large. The "lateral wave" and

the "ground wave" are simply different names for the dominant part of the total

field near the ground when the horizontal distance is large. The height gain product, G(h) G(z), in (4) is independent of p and will be discussed in Section 2.4.

2.3 Normalized Surface Impedance

We define the surface impedance as the ratio of the tangential electric and

magnetic fields at the upper surface of the slab (z=O). From (2) and (3), we

derive the following

6
-E

op

Ho </>

z=O

= Z

1 (7)

where Zl is given by (2). The surface impedance Zl has units of ohms, and from (2)

it can be seen that it depends on the horizontal wavenumber A.

As indicated in Appendix A, the main contribution to the integral comes from

the region A k when p is large and the source and observer heights are small.

This is equivalent to saying that we can use the surface impedance value for a plane

wave at grazing incidence (A k) in ground wave propagation. In addition it is

convenient to normalize the surface impedance by dividing by the impedance of free

space no (=/~o/€o)· Thus, we define the normalized surface impedance as

= Zl/n I

o A=k

From (2) we can write in the following form

where

IE - 1 IE gc gc

and

v = ik/c: - K

o hc

(8)

(9)

Although (9) is derived for the case of a uniform slab, we use the same expression

for in Section 3 where the slab parameters are allowed to vary along the path.

There are a number of limiting cases which provide some insight into the

dependence of on the various parameters. If th~~ parameters of the slab approach

those of the ground (Eh = E = E ), then

c vc gc

= A = ~-=-T IE 2 ~gc gc (10)

The same limit (10) is obtained by letting D approach zero. In either case we are

back to a homogeneous half-space, and (10) is the known result for a homogeneous

half-space.

7
Another interesting limit is to let the slab thickness D approach infinity.

Then if V

o

is complex, tanh (voD) approaches unity and simplifies:

This is simply the surface impedance for an anisotropic half-space.

In many cases, the slab parameters are close to those of free space. So the

limit of E

hc and E

VC approaching unity is also of interest. In this case, the

following result is obtained

~IE =E =1 = ~2/(1+ikD~2) (12)

hc vc

This limit will be found useful in Section 2.5.

2.4 Height-Gain Functions

The height-gain function G(z) which appears in (4) takes on different forms

depending on whether z is positive or negative (Wait, 1967a):

where

{

G (z)

G(z) = 0

G (z)

s

z L 0

o > z > -D

(13)

Go(z) = 1 + ikz~

1 eVoz + R e-vo (2D+z)

E 1 -2v D vc + Reo

R =

and v

o' ~l' and ~2 are given by (9). As shown in Appendix A, the same height-gain

function applies to both the source and observer locations. This is simply a consequence of reciprocity since even the anisotropic slab is a reciprocal medium

(Monteath, 1973).

The limit of Go(z) as z approaches zero is unity

G (0) = 1

o

8

(14)
However, the limit of Gs(z) as z approaches zero is

G (0) = l/f:.

S vc (15)

This is a consequence of the continuity of normal current density at z=O.

2.5 Limit of Vanishing Slab

In the case of thin forest (Dence and Tamir, 1969), the slab parameters differ

only slightly from free space. Thus the limiting case when the slab actually

vanishes (Eh = f:. = 1) is of some practical interest as well as being a good c vc

check on the asymptotic solution given by (4). By physical reasoning we should

recover the result for a dipole at a height D+h and an observer at a height D+z

over homogeneous ground when f:.hc and f:.

vc

are both equal to unity.

When both hand z are positive, (4) becomes

E E (2-1

) G (h) G (z) z 0 p 0 0 (16)

E is

o

From (6) and (12), the second factor in (16) reduces

For the case where Eh and f:. are unity, reduces to the value in (12). c vc

not a function of E

h and f:. . C vc

to

-(1+ik])~2)2

= --=--- E =f:. =1 2 Pz hc vc

(17)

where P2=-ikp~22/2.

From (12) and (13), Go(h) reduces to

(18)

By substituting (17) and (18) into (16), the following result is obtained

= Eo (z-~J [1+ik6ZCD+hl] •

• ~+ik~2 (D+Z)] (19)

The second factor in (19) is ~he attenuation function for a homogeneous earth. The

third and fourth factors are the height-gain factors for the source and observer

9
respectively over a homogeneous earth. Thus the slab result reduces to the simpler

half-space result as illustrated in Figure 2.

3. INTEGRAL EQUATION APPROACH

For realistic propagation paths, the parameters will vary as a function of

position along the path. Analytical methods are not general enough to handle such

variations, but the integral equation approach (Hufford, 1952; Ott and Berry, 1970)

has been found useful for ground wave propagation over irregular, inhomogeneous

terrain. Ott's program WAGNER (Ott, 1971a and 1971b; Ott et al., 1979) has been

thoroughly tested and used on a wide variety of propagation paths. In this section

we extend Ott's integral equation method to include the presence of a lossy slab

over homogeneous earth. As in Ott's method, the properties of the earth (and also

the slab) are allowed to change along the path.

3.1 Formulation

A good description of the latest version of program WAGNER and the integral

equation on which it is based is given in Ott et ale (1979). The integral equation

utilizes the impedance boundary condition which we can generalize from the halfspace form to the slab form. The geometry is shown in Figure 3, and the slab and

ground parameters are allowed to vary with x but are independent of y.

We consider first the case where both the source and receiver heights are zero.

We define the attenuation function f(x) as the ratio of the vertical electric field

to Eo' twice the free-space field. The integral equation for f(x) is (Ott et al.,

1979) :

f(x) = W(x,O) - ~~ fcq e-ikHx,<l{y.«) W(x,<)

y(x)-y(s) + (L'I(s)-L'I ) W(x s) l [ x ]~ ds ,(20)

x-s a' ) s(x-s)

where

y2(X)

2x

W(x,s) = 1 - ilITPw(-lu)

10
SOURCE OBSERVER

---.----t----p-------.I

D+h D+z J Eo_---.,jL

Figure 2. Equivalent geometry for the limit of a free-space

slab (£h = £ = 1). c vc

11
y

/ORIGIN

-~e.::~-·X

INTEGRATION y(~)

RECEIVER

1-----------X--------------.1

Figure 3. Geometry for integral equation. The slab and ground parameters

can vary as a function of x, but are constant in y.

12
u = P[1 - y(x)-y(~) A(x-~)

r

00 _t 2

w(-Iu) -u erfc(ilU) 1 f e dt = e = i1T Iii'+ t "'"00

The normalized surface impedance A(~) is given by (9).

tion of distance along the path because it depends on ground constants, £ and

g

and ° , which are a function of distance

v

impedance A is evaluated at the source, a

and the properties of ware discussed by Abramowitz and Stegun (1964). The quantities x, ~, y(x), and y(~) are defined in Figure 3, and y'(~) is the slope dy/d~.

As indicated it is a func0g' and slab parameters, 0, £h' °h' £v'

along the path. The normalized surface

~=O.

To account for nonzero source and receiver heights, h

a

and h

r , we utilize the

height-gain functions discussed in Section 2.4. Thus we define an attenuation

function fh(x) which is the ratio of the vertical electric field to twice the freespace field in terms of the normalized attenuation function f(x) and the heightgain functions:

(21)

G is defined in (13), and the heights h

a and hr must be greater than .-0. That is,

the source and observer can be either in the slab or in air. G(ha ) is a function

of the ground and slab parameters at the source (~=O), and G(hr ) is a function of

the parameters at the observer (~=x). For the source and observer in air at the . + surface of the slab (hi=hr=O ), both height-gain functions are unity, and the two

attenuation functions are equal

fh(x) I + = h =h =0

a r

f(x) (22)

3.2 Integral Equation Solution

Before discussing the general solution of (20), it is useful to examine the

case of a uniform path. In this case, y(~)=y(x)=Y'(~)=O and A(~)=A. Thus the

a

integrand is zero, and f(x) is simply

f(x) = W(x,O)

= 1 - i I7fi) w( -IP) a a (23)

where

13
p = -ik~ 2x/2

a a

This result is the expected flat-earth attenuation function (Wait, 1962), and p is

a

the numerical distance. When Ip I is very small, then f(x) 1. For example, p a a

is zero for a perfectly conducting upper boundary (~=O). For large Ip I, the

a a

asymptotic form in (6) holds

-1 f(x) ---- 2 Pa

(24)

When either y or varies along the path, then the integral equation (20) must

be solved numerically. We use the same method of solution as Ott (197la) which is

a forward stepping solution in x (Wagner, 1953). Thus values of f (or f

h) are

obtained at discrete values of x along the profile. Since (20) is a Volterra

integral equation of the second kind, the value of f(x) depends only on the previously computed values of f(~) for < x. Physically this means that backscatter

is neglected.

For a given path calculation, the computer run time t r is roughly proportional

to the square of the number of x points, N. The number of points required to

x

maintain precision along a given path is proportional to length of the path Land

the radio frequency. Thus the run time can be written

t N 2 L2 • freq2

r x (25)

The constant of proportionality depends on how rapidly y and vary along the path

and is difficult to quantify.

Appendix C contains a user's guide, a listing, and a sample output for program

WAGSLAB. The program is an extension of program WAGNER (Ott et a1., 1979). The

primary differences are that WAGSLAB requires additional input data for the slab

parameters along the path and that the normalized surface impedance is the form

for a slab over ground as given by (9) rather than the form for a homogeneous earth

given by (10). The slab parameters are allowed to change discontinuously along the

path, and as a result will also change discontinuously. However, a linear height

interpolation scheme (Ott et al., 1979) is used to determine the height y(~) and

slope y'(~) so that y and y' will be well behaved.

14
The primary outputs are the magnitude and phase of f(x) and fh(x). We prefer

to use these normalized attenuation functions because the attenuation relative to

free space is an easy quantity to interpret and can be separated from the antenna

effects. If some form of path loss is preferred, the additional path loss L to

a

be added to the free-space path loss is

4. TWO-SECTION PATH

The uniform slab model which was discussed in Section 2 has the advantage of

yielding a rigorous analytical solution, but has the disadvantage of not providing

any information regarding the effects of changes in slab parameters along the path.

The integral equation approach described in Section 3 can handle arbitrary changes

in slab parameters along the path, but yields only numerical results which are

difficult to check. In this section, we treat the two-section path shown in Figure

4. Although this geometry has no exact solution, it is amenable to an approximate

analysis which yields a fairly simple analytical result as given in Section 4.1.

The analytical result in Section 4.1 is usenll in providing insight into the

effect of a forest clearing on propagation. In addition, the analytical result is

useful in providing a check- for the integral equation method. As seen in Section

4.2, the agreement between the two methods is remarkably good.

4.1 Kirchhoff Theory

The geometry is shown in Figure 4. The section, x < 0, has normalized surface

impedance ~a' and the section, x > 0, has normalized surface impedance ~b' A jump

in surface height of magnitude D occurs at x=O .. Since we employ the surface impedance boundary condition, there is no need to specify the details of the slab or

ground at this point. The region, x < 0, could represent a slab of thickness D as

in Figure 1 or it could just as well represent elevated ground as in a cliff

geometry.

The geometry in Figure 4 has been treated by Monteath (1973) for the case of

D=O, and here we extend his approach to allow for nonzero D. We take the source

and receiving vertical electric dipoles to be located at the surface. The mutual

impedance Zab between the two dipoles can be written as an integral over the vertical aperture S defined by z > D. If we make the Kirchhoff approximation that the

aperture fields of each dipole are those of a uniform structure (~a=~b and D=O),

then the mutual impedance is given by (Monteath, 1973)

15
I-'

0\

z

R

a ~4 R J

Figure 4. Geometry for a two-section path with an elevation change D. The section,

x < 0, has a normalized surface impedance ~a' and the section, x > 0, has

a normalized surface impedance ~b'

x
-ikr

e a

where

00 00 (l+Pa

)

= K

o

f dz f ----- D _00 r a

R y2 + (z - D)2 r "" +

a a 2 R

a

r

b "" ~+ y2 + z2

2 Rb

dy (26)

and P and P

b are reflection coefficients for vertical polarization. K is a cona 0

stant which depends on the dipole lengths, but the final result for the attenuation

function is independent of K. We make the Fresn~~l approximation where quadratic o

terms are kept in the phase but not in the amplitude

{ [

y2 (1 1) z2 eXf -ik - - + - + -- + . 2 R

a

R

b 2Rb

(Z-D)2]}d 2 R Y

a

(27)

The y integration in (27) can be done by stationary phase by using the following

result (Monteath, 1973)

_L exp {-ik [~2 (~a + ~)J }dy =

-i1T/4

e (28)

Substituting (28) into (27) and introducing a new constant K1, we have

00

f (1+P. ) ( 1+p b) • D a

{ [

Z 2 (z2-R:) 2 Jl }dz exp -ik 2 R

b

+

The reflection coefficients in (29) are given by

(29)

sin Ct - I:!.

a a

Pa = sin Ct + I:!. and Pb =

a a

17

sin Ctb - I:!.b

sin Ctb + I:!.b (30)
A convenient method of normalization is to consider the case of flat perfectly

conducting ground (L1a=L1b=D=O). In this case, the mutual impedance Zabo is given by

(Monteath, 1973)

Z

abo

2& K

l = -~k;----"'---

(ik) 2 (Ra+ R

b

)

(31)

In a manner consistent with the rest of this report, we define the attenuation

function f as the ratio of the mutual impedance Zab to that for perfect ground Zabo'

Thus f is given by

Zab

f = -Z- =

abo

k (ik) 2 IR +Rb 00

-----a J ( (l+Pa) l-i-Pb ) •

2& IRaRb D

From (30), the reflection coefficient terms in (32) are given by

(32)

and

1 + P = a

2 sin a.

a ----:-----;-- '" sin a. + L1

a a

2 sin a.

a

L1

a

(33)

where we have assumed that the angles a.a and a.b are small. Thus we can approximate

the sine terms

sin z - D and sin z

a. '" a.b '" - a R

a

Then (33) becomes

1 + P '" 2(z - D) and 1 + Pb '" 2z

a /). R /).b R a a b

Substituting (35) into (32) , we have

(34)

(35)

18
00

J z(z - D) •

o

(36)

The evaluation of the integral in (36) is rather tedious and is given in terms

of complementary error functions in Appendix B. However, for small values of 0,

f is given by

where

S '" 1 + 2H ( 1 5) + order H2

/;/A

0/ikj2 R

H and A a = = IR +\

R

b a

(37)

The leading term in f varies as (Ra+ Rb)-l and is independent of o. This is

actually the same result which Monteath (1973) obtained for 0=0:

For our purposes of modeling a discontinuous forest region, 0 is normally on the

order of 10 ITl and R

a

+ R

b is on the order of kilometers. Thus for the HF band, S

in (37) is well' approximated by unity, and f is well approximated by (38). This is

equivalent to saying that the change in surface impedance has a 'larger effect on the

propagation than does the change in elevation. This is consistent with some recent

mode matching calculations for a step change in height over a spherical earth (Hill

and Wait, 1982). In the case of ~a=~b=~' (38) reduces to the result for a uniform

path as given by (6).

Although the derivation has been carried out for each dipole antenna at the

surface as in Figure 4, we can generalize the results to nonzero heights. If the

antennas are located at heights h

a

and h

b , then a new attenuation function f

h is

given by

(39)

19
(40)

where G(ha) is the height-gain function for the left section of the path and G(hb)

is the height-gain function for the right section of the path. In either case, the

form of G is given by (13).

4.2 Comparison with Integral Equation

In this section, we compare the approximate solution of (39) with the integral

equation solution of (21). The same height-gain functions are used in each solution so that the comparison really involves the attenuation function f as calculated

by (37) for the aperture solution and by (20) for the integral equation solution.

Actually both solutions incorporate a number of approximations, but the approximations employed in the integral equation approach are less severe.

The specific case which we consider involves a forest-to-clearing path. For

the ground constants over the entire path, we take the conductivity 0g = 10-2 Sim

and the relative permittivity E = 10. For the forest slab parameters, we take

-4 g -4

(Ott and Wait, 1973): 0h=lO Slm, 0v=2.5 x 10 Slm, Eh=l.l, E

v=1.25, and D=lO m.

In Figure 5, results are shown for propagation from a forest to a clearing.

Both transmitting and receiving antennas are located at the surface of the earth.

Results are shown for three theories: the vertical aperture theory of Section 4.1,

the integral equation of Section 3, and an analytic continuation method of Tamir

(1977). For antenna separations less than 2 km, both antennas are in the forest

and the results are those of the uniform slab because all three theories neglect

backscatter. In this region, the three theories agree. For the receiving antenna

just beyond the forest, the vertical aperture and analytic continuation theories

predict a jump in the field strength and are not valid. However, for larger distances, the aperture theory and the integral equation are in good agreement. The

Tamir (1977) analytic continuation theory gives a value which is too high. It is

possible to show that Tamir's theory is roughly equivalent to replacing f in (37)

by

1 f '" -------

ik(Ra+Rb) l'lb 2

In other words the product of the two surface impedances l'la l'lb is replaced by l'lb 2 .

Since in this case the forest surface impedance l'l has a greater magnitude than the

a

ground surface impedance l'lb' Tamir's result yields a larger field. Only in the

limit of a very thin forest does the Tamir result agree with the integral equation

and the aperture theory. The increase in field strength in going from forest to

clearing was predicted earlier (Office of Telecommunications Technical Memorandum

20
10-1.-------------------

FOREST

Transmitter

Receiver

GROUND

1',

, ....... ....... ....... ....... ' .......

TAMIR, 1977"J ........

Aperture

Theory

o 0 0 0 Integral Equation

10-30~---~1 ----2~---.L3------J4

DISTANCE (km)

Figure 5. Propagation from a forest to a clearing for a frequency

of 10 MHz.

21
73-154, by Ott and Wait, limited distribution) and is analogous to the "recovery

effect" which has been calculated (Wait and Walters, 1963; Hill and Wait, 1981a)

and measured (Millington, 1949) for propagation from land to sea.

The problem of propagating from the clearing to the forest is equally interesting, and the three theories are shown in Figure 6. In this case, both antennas

are located 10 m above the ground even with the top of the forest. For short distances, the agreement with the integral equation is not perfect because the numerical distance p as defined by (6) is not large. For distances well beyond the forest

transition, the aperture and integral equation theories are in good agreement, but

the Tamir (1977) analytic continuation result again is high. Actually for the case

of the antennas even with the top of the forest, Tamir's analytic continuation

result is independent of the forest parameters.

Other cases for various slab parameters and frequencies have been studied, and

the trends are qualitatively similar. In order to. use the integral equation

approach, we actually replace the abrupt height change from the ground to the forest

with a linear transition. This is done automatically by program WAGSLAB so that

the terrain slope remains finite. A similar linear height interpolation method has

been used in program WAGNER (Ott et al., 1979).

5. EQUIVALENT SLAB PARAMETERS

The program WAGSLAB requires the user to supply the slab parameters along the

path. The five slab parameters are as pictured in Figure 1: height (D), horizontal

conductivity (oh)' horizontal relative permittivity (£h)' vertical conductivity

(0 ), and vertical relative permittivity (£). In the HF band, the wavelength v v

ranges from 10 m to 100 m. Consequently, objects such as trees and buildings are

not necessarily electrically small. However, the equivalent slab representation

seems to be the only tractable model for forests and urban areas at this time.

In this section, we review the literature and make some recommendations regarding the equivalent parameters for forests, urban areas, and snow cover. It is

hoped that future comparisons of theory and measurements for propagation through

forests and built-up areas will aid in the determination of the appropriate equivalent parameters. It might turn out that a multilayer slab (Ott and Wait, 1973a;

Cavalcante et al., 1982) is more appropriate for modeling forests and built-up

areas, but it is felt that our knowledge of the equivalent slab parameters is not

precise enough to justify such additional complexity at this time. However, such

an extension is possible simply by modifying the surface impedance and the heightgain function G.

22
10° -------------------...

Transmitter Receiver

GROUND

FOREST

° o......... /TAMIR.1977

..... .......... ............

--- o

o 0 0 0 Integra I Equation

4

10-3

023

L..- ......L- ---I ..L... -J

DISTANCE (km)

Figure 6. Propagation from a clearing to a forest for a frequency of

10 MHz.

23
5.1 Forest Cover

During the 1960's there was a great deal of activity in detemining the

electrical properties of forests in Southeast Asia (Jansky and Bailey, 1966; Parker

and Makarabhiromya, 1967). A later workshop (Wait et al., 1974) dealt with many

aspects of propagation in forested areas. A general conclusion (Dence and Tamir,

1969) was reached that the equivalent slab model for forests is generally valid for

frequencies below about 200 MHz.

Dence and Tamir (1969) refer to three. types of forest: "thin," "average," and

"dense." The parameters for the slab and the ground for these forest types are

given in Table 1. Note that the forest slab is assumed to be isotropic (Eh=Ev and

0h=ov)' The "average" forest is probably most characteristic of Southeast Asia.

A report by the Defense Intelligence Agency (1971) states that the forests of

West Germany fall between the "thin" and "average" forests of Dence and Tamir

(1969) in Table 1. However, the average tree height is 20 m which corresponds to

the "dense" forest of Table 1. Actually the forests of West Germany include both

coniferous and .deciduous trees, and it seems unlikely that one set of parameters

can characterize both types of forests well.

In order to obtain better slab parameters for German forests (or any other

forests), two approaches are possible. One is to make ground-wave transmission

measurements and to adjust the slab parameters in the theoretical model until a

good fit to the measured data is obtained. This is essentially the approach which

Causebrook (1978a) used to obtain parameters for the surface impedance of built-up

areas in the vicinity of London. A second approach is to attempt to measure the

equivalent slab parameters directly (Parker and Hagn, 1966). The first approach

has the disadvantage of not yielding any direct information on the slab parameters,

but has the advantage of yielding useful transmission results as well as data for

fitting slab parameters.

5.2 Built-up Areas

The primary difficulty in characterizing the effect of buildings on HF groundwave propagation is that the individual buildings are not necessarily small compared to a wavelength. For electrically small hemispherical losses on a perfectly

conducting ground, Wait (1959) has shown that an inductive surface impedance results

from induced electric and magnetic dipoles in the hemispheres. At MF, Causebrook

(1978a) has obtained good agreement between theory and propagation measurements by

representing built-up areas by an equivalent surface impedance.

24
Table 1. Characteristic Parameters of Typical Forests

(Dence and Tamir, 1969)

Forest Type

Parameter "Thin" "Average" "Dense"

Tree height D (m) 5 10 20

{

E = E

h 1.03 1.1 1.3

Forest v

cr = crh (S/m) 3 x 10-5 10-4 ;) x 10-4

v

{

E

g

5 20 50

Ground

10-3 10-2 10-1

cr (S/m) g

After attempting a number of approaches which proved to be unsatisfactory at

HF, we settled on a fairly simple approach which utilized Causebrook's (1978a) low

density theory. We start by taking the low frequency limit of as given by (9).

In this case. v D is small and we have

o

tanh (v D) '" v D

o 0 (41)

(42)

Substituting (41) into (9) and keeping only the leading term in v D, we have

o

+ v Dt> (1 _~2~) 01 2 t>l

For most cases, 1~21 will be much less than It>ll at low frequencies, and the last

term in (42) can be neglected to give the final result

= 2+ ikD (1 -f-) . vc

Causebrook's (1978a) low density theory gives the following result:

'" ~(O) + ikD [1 _ In(l + 10 B)] , 10 B

25

(43)

(44)
where B is the fractional building coverage of the ground and ~(O) is the impedance

at the surface of the ground. Thus we can equate ~2 to ~(O). The second terms in

(43) and (44) can then be equated to yield the following expression for €

vc

10 B

€ = vc In(l + 10 B) (45)

Since B is real and positive and less than one, (45) can be satisfied by the

following:

10 B

€v = In(l + 10 B) and crv = 0 (46)

B approaches zero, € approaches unity as it should for free

v

a function of B is given in Figure 7. Strictly speaking,

As the building density

space. A plot of € as

v

B should be limited to values much less than unity since (46) was derived on the

basis of low density. Although the previous result provides no information on the

horizontal parameters of the slab, €h and crh, it seems reasonable to equate them to

the vertical parameters on the assumption of roughly cubic buildings. Thus the

final result for the slab parameters is

10 B

€ = € = v h In(l + 10 B)

~d

(47)

For cases where built-up areas are mixed with forest, it seems reasonable to

increase €v' €h' cr

v ' and crh by the appropriate forestvrlues. Also, even though

the result in (47) is derived by matching the slab impedance to Causebrook's (1978a)

result in the low frequency (small kD) limit, we assume that the slab parameters are

frequency independent. This will mean that when kD is no longer small will be

different from Causebrook's (1978a) result in (44). However, since we have converted Causebrook's result into equivalent slab paramete~s, we no longer have the

restriction of small kD. Thus we can use (47) at HF. However, improvements in

(47) can probably be made once we have propagation measurements through built-up

areas at HF.

5.3 Snow Cover

The electrical properties of snow and ice have been reviewed by Evans (1965).

For dry snow, the dielectric constant of snow varies almost linearly with density

26
- --------------- ----------- ---------- -- - -- - -- --- --

5~-----------------

4

E 3 V

1

0.2 0.4

B

0.6 0.8 1.0

Figure 7. Relative permittivity Sv versus building density B.
from a value of unity for free space to approximately 3 at the density of ice. The

conductivity of dry snow is very small.

The presence of liquid water in snow can raise both the dielectric constant

and the conductivity dramatically (Evans, 1965). At a frequency of 1 MHz, Von Hippel

-6 (1954) has measured a dielectric constant of 1.2 and a conductivity of 1.85 x 10

Sim for freshly fallen snow and a dielectric constant of 1.55 and a conductivity of

-5 2.5 x 10 Sim for hard-packed snow followed by light rain. No anisotropy is noted

for snow by Evans (1965) or Von Hippel (1954).

To illustrate the effect of snow accumulation on propagation, we show the

change in surface impedance as a function of frequency in Table 2. We take the

larger value of permittivity for hard-packed, wet snow in order to show the largest

effect which is likely for a given snow depth. Although the assumption of a frequency independent conductivity is not correct, the loss tangent for snow is so

small that the error in conductivity is unimportant. As seen in Table 1, the effect

of increasing snow depth is to increase both the magnitude and phase of 6, but not

greatly. As seen by the flat-earth result in (6), an increase in 161 will result

in a decrease in field strength. In Figure 8 we illustrate the effect for a

spherical earth. The curves in Figure 8 were computed by a numerically efficient

£v = £h = 1.55, crv =

£ = 10, cr

g g

Table 2. Magnitude and Phase of Surface Impedance 6 as

of Snow Depth D. Parameters:

-5

crh = 2.5 x 10 Slm,

-2 = 10 Sim

a Function

0 0.5 m l.Om

3 MHz 0.128 L 39.8° 0.138L 42.6° 0.147L45.lo

10 MHz 0.218 L 29.2° 0.244 L 34.9° O.274L39.3°

30 MHz O.282L14.lo 0.338L25.5° 0.427 L 30.8°

28
10-1 _-__-----------------_

f =3 MHz

, ...........

........... '" .......... ..........

-~

f =10 MHz

Ifl

O=Om

------ 0= 0,5m

--- O=1,0m

40 60 80 100

x{km)

20

10-5 I-- ...L.... --'- ---L ..L..-. ....

o

Figure 8. Magnitude of the spherical earth attenuation function as a function of

frequency and snow depth D. Parameters: E

V = E

h = 1.55, r:J

v = erh = 2.5 x 10-5 81m, E

g = 10, erg = 10-2 81m.

29
method (Hill and Wait, 19S0) which has been used for calculations of ground-wave

propagation over sea ice (Hill and Wait, 1981b).

In the case of HF ground-wave propagation ove~ sea ice, a trapped surface wave

(Hill and Wait, 1981b) was supported because of the highly inductive surface

impedance. As seen in Table 2, a layer of snow over ground does not produce a

phase angle of much larger than 45°. Consequently, the main effect of the snow

layer is to increase the magnitude of and to decrease the ground-wave field

strength. For small depths the effect is small, but for depths of packed snow on

the order of a meter the effect becomes significant. As seen in Figure 8, the

effect is more prominent at the higher frequencies.

6. SPECIFIC PATH CALCULATIONS

In this section we apply program WAGSLAB to several paths. The smooth paths

in the Netherlands are a good check on WAGSLAB because the exact residue series

solution for a spherical earth (Hill and Wait, 1980) is available as a check. Both

earth curvature and atmospheric refraction are included in WAGSLAB by inputting the

appropriate earth radius. In all cases, we used the 4/3 earth radius value

(a::<S500 km) to account for normal atmospheric refraction (Bremmer, 1949; Wait,

1962), but other values could be used for different atmospheric conditions.

The irregular, forested terrain in southern West Germany presents a greater

challenge to the integral equation approach. We have no experimental results as

checks for these paths, but reciprocity provides a good theoretical check. Also,

we have found that the multiple knife-edge model of Vogler (1981) is in reasonably

good agreement with the integral equation solution for long, rough paths.

6.1 Smooth Paths in the Netherlands

Measurements of field strength and bit error rate have recently been carried

out on paths of 60 km and SO km in the Netherlands (Van der Vis, 1979). The frequencies range from 2 MHz to 30 MHz, and both vertical and horizontal polarization

were transmitted and received. Both paths had the same transmitter, and the two

paths were nearly in line. An examination of topographic maps of the area revealed

that both paths were over very smooth (less than a few meters in elevation change),

unforested terrain with no cities. The main feature of interest was a section of

low ground conductivity (dry, sandy soil) at the end of the 80 km path. Consequently, we modeled the path with two smooth sections as shown in Figure 9.

Unfortunately, no ground conductivity measurements were made, but we assume the

values suggested by Van der Vis (1979): E =15 and a =10- 2 Sim for the first

g g

30
\

\

'r--

,

60km

80km

.,

I

Receiver

/.-

I IOkrn

I

I

I

I

Receiver

Figure 9. Smooth path in the Netherlands with recelVlng sites at 60 km and

80 km. The first 70 km has good ground (8 = 15 and a = 10-2 S/m),

and the last 10 km has poor ground (E:g = 3

g

and a

g = glO-4 S/m).

70 km of good ground and 8 =3 and a =10-4 S/m for the remaining section of poor g g

~round.

In Figures 10-12, we show the magnitude of the attenuation function f along

the path for three different frequencies. As usual, the attenuation function is

defined as the ratio of the vertical electric field to that over flat, perfectly

conducting ground. Bot,h the transmitting and receiving antennas are assumed to be

located at the surface, and the measured data are adjusted to account for the antenna

heights by assuming the height-gain function in (13). In each case the integral

equation result shows a rapid drop in field strength beyond 70 km because of the

decrease in ground constants. The measured data a.re consistent with this drop, but

a detailed comparison is not possible because measurements are available only at

60 km and 80 km. It appears that the ground conductivity and dielectric constant

which were suggested by Van der Vis (1979) for the first 70 km were a little too

high. Lower values would have provided a better fit with the measurements.

For comparison in Figures 10-12, we also show the spherical earth and flat

earth results for a uniform path (Hill and Wait, 1980). Because the integral equation solution assumes no backscatter, it should agree with the spherical earth theory

over the first 70 km. The accuracy of the integral equation solution becomes

31
f =2 MHz

If I

x

o

- - - - Flat Earth

Spherica I Earth

000 0 0 Integral Equation

x x Measurement

o

o

o

40 60 80

10-3 L...._..1.-_...L-_--L.-_--'-_---L_--J__1..-----I

o 20

x(km)

Figure 10. Comparison of measurement with three theories at 2 ~ffiz for a

smooth path in the Netherlands. Only the integral equation

solution takes into account the poor ground section shown in

Figure 9.

32
10-1 ~-----------------

f=10MHz

10- 3

X

0

0

0

--- Fiat Earth

Spherical Earth

000 0 Integra I Equation

X X Measurement

10- 4

0 20 40 60 80

x(km)

Figure 11. Comparison of measurement with three theories at 10 MHz for a

smooth path in the Netherlands. Only the integral equation

solution takes into account the poor ground section shown in

Figure 9.

33
10-2 r""'T'------------------.....,

f=30MHz

............

- x

o o 0

- - - Flat Earth

-- Spherica I Earth

o 0 0 0 Integral Equation

X X Measurement

20 60 80

10-5 ~_........_....L..._...L.-_....I-_-L._.....L.._.....L._......I

o

Figure 12. Comparison of measurement with three theories at 30 MHz for a

smooth path in the Netherlands. Only the integral equation

solution takes into account the poor ground section shown in

Figure 9.

34
harder to maintain at higher frequencies and longer paths. At 30 MHz, it was

necessary to sample approximately every 40 m (equals 4 wavelengths) along the path

to obtain the agreement (which is not perfect) in Figure 12. This resulted in 2000

sample points along the path and an execution time of approximately 15 min on a

large mainframe computer. This rather long run time decreases rapidly as the

frequency is decreased as indicated by (25). Thus program WAGSLAB can be considered very efficient for the lower portion of the HF band but only marginally

efficient at 30 MHz. It should probably not be used above 30 MHz except for very

short paths.

Also shown for comparison in Figures 10-12 is the flat earth result for a uniform path. It is easy to see that earth curvature is not particularly important at

2 MHz but is quite important at 30 MHz. An approximate upper bound in path length

and frequency for the validity of flat earth theory can be obtained from examining

the spherical earth theory. The earth curvature can normally be neglected when the

following inequality is satisfied

(~a )1/3 (~) « 1 (48)

From examination of numerical results (Hill and Wait, 1980), we can set the

right-hand side of (48) equal to 0.2.

radius), then the maximum path length

spherical earth theory is given by

If we assume a = 8500 km (= 4/3 earth

x for good agreement between flat and

max

x

max '" 38.1 (km)

f 1/3

MHz

(49)

where f

MH is the frequency in megahertz. From (49), x = 30.2 km at 2 MHz, and

z max

x = 12.3 km at 30 MHz. For x greater than x , the flat earth theory predicts max max

a field strength which is too high as seen in Figures 10-12. For example, in

Figure 12 at a frequency of 30 MHz and a distance of 80 km, the flat earth field

strength is a. factor of 2.43 greater (= 7.7 dB) than the spherical earth result.

6.2 Irregular, Forested Terrain in West Germany

In order to provide a more demanding test for program WAGSLAB, we examined two

paths over irregular, forested terrain in southern West Germany. We have no experimental data for these paths at this time, but there has been some planning for

future HF ground wave measurements in the area (NTIA Technical Memorandum 82-80

by Kissick and Adams, limited distribution).

35
The path which we considered in most detail is a 56.6 km path from Inneringen

to Bob1ingen. The terrain profile shown in Figure 13 was generated from the files

of the Defense Mapping Agency. The terrain profile checks very well with the profile which we obtained by hand from 1:50,000 scale maps of the area. The path contains numerous forested sections and three short sections with buildings. For the

forested sections, we chose a height D of 20 m and the "average" electrical properties of Table 1. For the urban areas we did not have available information on

building height and density. We chose a building height D of 10 m and a building

density B of 0.2 (Causebrook, 1978a). From (47), this yields dielectric constants

of E

v

=Eh=1.82. For conductivity we took the value for "thin" forest, cr

v

=crh=3 x 10-5

Slm, rather than crv=crh=O as given by (47) because it seems likely that some trees

are present in the urban areas. The actual locations of the forest and urban slabs

are given by the input data in the sample computer run in Appendix C. We also had

to guess at the ground parameters and chose the following values for fairly good

-2 ground over the entire path: E =10 and cr =10 S/m.

g g

In Figures 14-18, we show propagation from Inneringen to Boblingen both with

and without the forest and building slabs at five frequencies from 2 MHz to 30 MHz.

In all cases, both the transmitting and receiving antennas are located at the

surface of the ground (ha=hr = -D). As a result, the dotted curves which include

the slabs are more irregular because the height-gain function G(hr ) is discontinuous

as the receiving antenna moves from bare ground into a slab. This results in a

discontinuous value of f

h which is proportional to G(hr ) as shown by (21). The

dependence of G on height both in and above the forest slab is shown in Figure 19.

In most cases, IG(hr ) I is less than unity in the slab and increases from unity

above the slab.

The height-gain effect can be considered a local effect on the field, but the

normalized surface impedance ~(s) influences the field at all points for x >s as

seen by (20). In most cases, I~I is greater in the forest region than over bare

ground. Consequently, the dotted curves in Figures 14-18 which include the forest

and urban slabs generally lie below the solid curves for bare ground. At the

higher frequencies where the terrain roughness is more important, the effect of the

slabs is less dominant. The analytical solution for a spherical earth (Hill and

Wait, 1981b) is shown in Figure 20 for both bare ground and forest slab cover. The

forest cover has a large effect at 5 MHz, but a much smaller effect at 20 MHz and

30 MHz where diffraction loss becomes important.

Two difficulties with the integral equation approach are that computation time

becomes large for long, rough paths at high frequencies as indicated by (25) and

36
1200

1000

· . . . . . . . . . . . . . . . . · . · . .

. . . . . . . . . . . . . . . . . .. . . . . . . . . . .. · . . . · . . . .. . . . . . . .... . . . . . . . . . . . . . . · . . . . . . . . . . . · . . . . . . . . . . . . . . . . . . . .. .. · . . . . . . . . . . . . . . . . .

600

400

10 200

o

INNERI~N

20 40 50 56.& B08Ll~N

Figure 13. Terrain profile for the path from Inneringen to Bob1ingen.

37
\0. -.

..

..

..

..

..

..

.. ..

..... ,-

. -.- .. ..... ..

.. .. . .. ... . . -.-

..

: . ..

. .. . . . . ..

.:

...

..

.. .. ....

.-., . .:-,.: -. . -..

......

..

..

... ...

-....

..

i

- ...

.....

.- ...- .. ..::- ..

.. ...

. I

•01

30 40 50

x

10 20

•001 ....._ ......_ ........--l.._......._ ......__"'-_......_ ......_.....l.._......l._......i...J

o

Figure 14. Propagation from Inneringen to Bob1ingen at 2 MHz. The

solid curve is for bare ground, and the dotted curve

includes the forest and urban slabs.

38
·-.. .' . .....

,,~ .. .. "'.. ...... ..\ .. .. "'.. " : .' ..'.. .' .. ':

," '.

'. •001 .. "

'.

. . :... -. . . ,"

."

10 20 30 40

•000 I L------:----:---- ----lL.-..J

o

x

Figure 15. -Propagation from Inneringen to Boblingen at 5 MHz. The solid curve

is for bare ground, and the dotted curve includes the forest and

urban slabs. The circles are obtained from multiple knife-edge

diffraction theory.

39
• 1 •

•001

.. .

·

·

· .•.

~.

..

.. ,

..

., .. . . . . ..

.. .. -. ..

.... ...

•

•

30 40

x

10 20

•0001 ....._ ....._ ......_"'--_...l-_..I.-_.J..._~_"""-_ ......_....;..J

o

Figure 16. Propagation from Inneringen to Boblin~en at 10 MHz. The

solid curve is for bare ground, and the dotted curve includes

the forest and urban slabs.

40
· I

.01

.....

.c:

.....

.....

0.

!

.001

, 00 0

0

00 It 00

00 It 00 " : 0

t, o'

.0001 " o' 00 :

o 10 20 30

x

40 50

Figure 17" Propagation from Inneringen to Boblingen at 20 MHz. The solid

curve is for bare ground, and the dotted curve includes the

forest and urban slabs 0

41
• 1

.01

, .. " .. -. , . \ .

.0001

o 10 20

':

':

':

..

:

30

x

40

"..

•

• ,

•

•

•

Figure 18, Propagation from Inneringen to Boblingen at 30 MHz. The solid

curve is for bare ground, and the dotted curve includes the

forest and urban slabs, The circles are obtained from multiple

knife-edge diffraction theory,

42
2.5,....--------------------

2.0

f=30MHz--__ J

I

I

1.5

1.0

f=20 MHz

f=5 MHz

-

r------~ ,'/,.-

O.5~-_ .~

...~~~ FOREST+

AIR

-5 o 5 10

h(m)

-15 -10

O'---__...L-__--L .L...-__--L..__---J__----J

-20

Figure 19. Magnitude of height-gain function in forest slab and in air.

Parameters: 0 = 20 m, E

V = E

h = 1.1, cr

v = crh = 10-4 S/m,

E

g = 10, and cr

g = 10-2 S/m.

43
f=5MHz--~

BARE GROUND

- --- FOREST COVER

~-f=5MHz

f=20 MHz

f=30MHz .......:::::~-~

10 20 30

x(km)

40 50 60

Figure 20. Spherical earth attenuation function for bare ground and with forest

cover. Parameters: D = 20 m, E

V = = 1.1, a

v = ah = 10-4 Slm,

E = 10 and a = 10-2 S/m. g , g

44
also precision loss occurs for long rough paths. It is generally difficult to predict the precise limitations of the integral equation approach (Ott, 1971b), but

generally the results start to appear noisy with large oscillations when the integral

equation approach starts to fail. This behavior is difficult to identify because

the actual field variation for rough paths at high frequencies contains oscillations

due to multipath. A good example is the change in character of the Inneringen to

Bollingen results as the frequency is increased in Figures 14-18. The 30 MHz curve

is much noisier than the 2 MHz curve. Also the average signal level does not continue to drop for large distances. To check the behavior, we compared the integral

equation result with Vogler's (1981) multiple knife edge calculation for the same

path. Vogler's results are shown at 5 MHz in Figure 15 and at 30 MHz in Figure 18,

and each point was obtained by fitting the terrain profile with ten equivalent

knife edges. The agreement is slightly better at 30 MHz, but is fairly good even

at 5 MHz. Vogler's results at 30 MHz actually predict a higher field strength at

56.6 km than at 30 km because of the shadowing which occurs from the steep terrain

slope between 25 km and 30 km.

A second check for the integral equation solution is reciprocity. If we

reverse the transmitting and receiving points (Monteath, 1973), the results should

be unchanged. In Figures 21-23 we show the integral equation result for propagation

in both directions over the Inningen to Boblingen path. If reciprocity is satisfied,

then the two results at the end of the path (x=56,.6 km) should be identical.

Because of the way the terrain profile is handled" we would actually expect some

small differences. It is clear that reciprocity is very well satisfied at 2 MHz

and 5 MHz, but that some difference occurs at 20 MHz. We can conclude that some

accuracy is lost at 20 MHz. Perhaps agreement could be improved by increasing the

number of sample points, but we did not attempt this. The good agreement with the

multiple knife edge results at 30 MHz in Figure 18 indicates that the integral

equation solution can produce useful results at the upper end of the HF band even

if some accuracy is lost.

A second terrain profile which runs from Inneringen to Lechfeld is shown in

Figure 24. This path is longer (116.5 km), but much smoother than the Inneringen

to Boblingen path in Figure 13. Also this path is less heavily forested than the

Inneringen to Boblingen path and runs through a great deal of farm and pasture land.

Consequently, we did not consider any forest or building effects and used E =10 and

g

cr =10- 2 S/M for the ground parameters over the entire path. Again we set both

g

antenna heights hand h equal to zero.

a r

45
...

... .1

t

..

...

..... e•

. . -. - e.- .... ,.. ... . .' ., ... ,. ..- -.:..'.

• 30 40 50

X

10 20

•01 '--_~--I_-.L_.....L_....J._-J._-L._-.L..._..J.._....L..._..u

o

Figure 21. Propagation from Inneringen to BoblingenCsolid curve) and in

the reV~rse direction (dotted curve). Both cases are for bare

ground, and the frequency is 2 MHz.

46
• J

.01

..

....":. . :.-.,: ... -..-..,...... '..' ,

.

• 30 40 50

x

10 20

•001 ~_..J..--....l-._"""'_--L.._--L._--J,_--1__.L...-_.L...._-'--_....L.-I

o

Figure 22. Propagation from Boblingen to Inneringen (solid curve) and

in the reverse direction (dotted curve). Both cases are for

bare ground, and the frequency is 5 MHz.

47
• 1

,

..

.. .

.- ..

..'

..' . " .,... '\ ..'~ .. '\ ...." '\ :.:....:. .. ..".. . ."'.....

:'..

I". ••

....

, I

..

30 40 50

X

10 20

•0001 ..... "'--_......_-'-_......_.....l.._--'-_....;._.....:.__~_'""-J

o

Figure 23. Propagation from Inneringen to Boblingen

in the reverse direction (dotted curve).

for bare ground, and the frequency is 20

(solid curve) and

Both cases are

MHz.

48
1200

1000

200

o

ItoIIIERltaN

116.5

LECHFELO

Figure 24. Terrain profile for the path from Inneringen to Lechfeld.

49
In Figures 25 and 26, we show propagation in both directions at frequencies of

2 MHz and 5 MHz. Since the results are in close agreement at the ends of the paths,

reciprocity is well satisfied and the accuracy is probably quite good. We did not

run WAGS LAB for higher frequencies over such a long path because of the large computer time which would have been required.

7. CONCLUSIONS AND RECOMMENDATIONS

The main purpose of this report is to describe a theoretical method for predicting HF ground wave propagation over irregular terrain with forest, building, or

snow cover. The only method that appears to be general enough for detailed predictions over arbitrary paths is the integral equation method (Ott, 1971a). The

approach which was adopted and described in Section 3 was to generalize Ott's program WAGNER (Ott et al., 1979) to allow for the effect of a lossy, anisotropic slab

over the earth. The slab parameters can be chosen to match those of forest, snow,

or buildings as discussed in Section S. Although the approach was applied to the

HF band (3 MHz - 30 MHz), it actually becomes more efficient as the frequency is

decreased. For frequencies above HF, it is probably only useful for short, relatively smooth paths. The new computer code is called WAGSLAB and a user's guide,

listing, and sample output are given in Appendix C.

Numerous special cases were considered in this report because they are good

checks for WAGSLAB and because they are amenable to analytical solution. For

uniform paths which are short enough to neglect earth curvature, the uniform slab

model (Tamir, 1967; Wait, 1967a) is adequate. This model is described in Section

2, and the results are cast into a form convenient for ground wave propagation.

When the path has two sections as in a forest-to-clearing case, an approximate

analytical solution can be derived. This solution is derived and compared with the

integral equation solution in Section 4. When the path is uniform, but is long

enough for curvature to be important, then spherical earth theory (Hill and Wait,

1981b) can be used. It is compared with the integral equation solution in Section

6.1.

Two long, rough paths were analyzed by program WAGSLAB in Section 6.2. No

experimental results were available, but some comparisons were made with multiple

knife edge diffraction (Vogler, 1981). Also, reciprocity was checked by running

the terrain profiles in each direction. This appears to provide an excellent check

for asymmetric paths.

This report represents only a first attempt at ground wave prediction over

irregular, forested, and built-up terrain. Numerous improvements and extensions

are probably possible, but we recommend the following: (1) detailed comparisons

50
·.... ..

22 44 66 88 110

•01 '---..L..--..I--....l..--..J-_-1..._.....J.._......I.._--l._--I. L-J

o

x

Figure 25. Propagation from Inneringen to Lechfe1d (solid) and in the

reverse direction (dotted). Both curves are for bare ground,

and the frequency is 2 MHz.

51
. I

.01 . .....-.. . ..

...... .. .. ,

:

22 44 88 110

.001 ~_..I...-_..L....._..l-_...L..-_...L..-_.J...._.J...._..L...._...J...._..I.....J

o

x

Figure 26. Propagation from Inneringen to Lechfeld (solid) and

in the reverse direction (dotted). Both curves are

for bare ground, and the frequency is 5 MHz.

52
with measurements over a variety of radial paths containing forest, building, and

snow cover (NTIA Technical Memorandum 82~0 by Kissick and Adams, limited distribution); (2) a complete study of the limits of WAGSLAB for long paths and high frequencies; (3) consideration of a switch from the integral equation approach to

multiple knife-edge diffraction for long, rough paths; and (4) the addition of an

additional height-gain function for the magnetic field which would be applicable to

reception with loop antennas. The height-gain functions for the vertical electric

field and the horizontal magnetic field are equal in free space, but are different

within the slab medium. This difference has been observed experimentally in cities

and forests at MF by Causebrook (1978b). In addition it might be possible after a

comparison with measurements to improve the theory for the equivalent slab parameters

for cities or to infer the equivalent parameters directly from measurements.

8. ACKNOWLEDGMENTS

The Defense Nuclear Agency (DNA) supported the research described in this

report. Major Bernard Romanik and Dr. Patrick Crowley of DNA initiated the idea to

develop a prediction program for HF ground wave propagation over irregular terrain

with forest, building, or snow cover. Mrs. Rita Reasoner of ITS provided invaluable

computer programming assistance. Mr. Lewis Vogler of ITS provided the multiple

knife-edge diffraction calculations. Discussions with Dr. William Kissick of ITS

and Dr. Randolph Ott of COMSAT were very helpful.

9. REFERENCES

Abramowitz, M., and I. A. Stegun (1964), Handbook of Mathematical Functions,

National Bureau of Standards, AMS 55.

Bremmer, H. (1949), Terrestrial Radio Waves (Elsevier Publishing Company, Amsterdam).

Causebrook, J. H. (1978a), Medium-wave propagation in built-up areas, Proc. lEE

125, pp. 804-808.

Causebrook, J. H. (1978b), Electric/magnetic field ratios of ground waves in a

realistic terrain, Electronics Letters !!, pp. 614-615.

Cavalcante, G. P. D. S., D. A. Rogers, and A. J. Giarola (1982), Analysis of electromagnetic wave propagation in multilayered media using dyadic Green's functions, Radio Sci. ~, pp. 503-508.

Defense Intelligence Agency (1971), Basic phenomena involved in MF and HF radio

communications, ST-CS-06-13-7l, Appendix III.

Dence, D., and T. Tamir (1969), Radio loss of lateral waves in forest environments,

Radio Sci. !, pp. 307-318.

53
Evans, S. (1965), Electrical properties of ice and snow--a review, J. Glaciology

42, pp. 773-792.

Gordon, G. A., and E. Hoyt (1982), An estimate of HF/VHF surface-wave communication

link reaches in the West German forest environment, RDA-TR-119930-00l, R&D

Associates, Marina del Rey, California.

Hill, D. A., and J. R. Wait (1980), Ground wave attenuation function for a spherical

earth with arbitrary surface impedance, Radio Sci. ~, pp. 637-643.

Hill, D. A., and J. R. Wait (198la), HF ground wave propagation over mixed land,

sea, and sea-ice paths, IEEE Trans. Geosci. Remote Sensing GE-19, pp. 210-216.

Hill, D. A., and J. R. Wait (198lb), HF radio wave transmission over sea ice and

remote sensing possibilities, IEEE Trans. Geosci. Remote Sensing GE-19, pp.

204-209.

Hill, D. A., and J. R. Wait (1982), Ground wave propagation over a mixed path with

an elevation change, IEEE Trans. Ant. Prop. AP-30, pp. 139-141.

Hufford, G. A. (1952), An integral equation approach to the problem of wave propagation over an irregular terrain, Quart. J. App1. Math. ~, pp. 391-404.

Hufford, G. A., A. G. Longley, and W. A. Kissick (1982), A guide to the use of the

ITS irregular terrain model in the area prediction mode, NTIA Report 82-100.

Jansky and Bailey Research and Engineering Department (1966), Tropical propagation

research, final report, Vol. 1, Atlantic Research Corp., Alexandria, Virginia.

Millington, G. (1949), Ground wave propagation over an inhomogeneous smooth earth,

Pt. 1, Proc. lEE 96, pp. 53-64.

Monteath, G. D. (1973), Applications of the Electromagnetic Reciprocity Principle

(Pergamon Press, Oxford).

Ott, R. H. (197la), An alternative integral equation for propagation over irregular

terrain, 2, Radio Sci. ~, pp. 429-435.

Ott, R. H. (197lb), A new method for predicting HF ground wave attenuation over

inhomogeneous, irregular terrain, U.S. Dept. of Commerce, Res. Rept. No.

OT/TRER 7, January (NTIS Accession No. AB721179).

Ott, R. H., and L. A. Berry, (1970), An alternative integral equation for propagation over irregular terrain, Radio Sci. ~, pp. 767-771.

Ott, R. H., L. E. Vogler, and G. A. Hufford (1979), Ground wave propagation over

irregular, inhomogeneous terrain: comparison of calculations and measurements,

NTIA Report 79-20 (NTIS Accession No. PB 298668/AS).

Ott, R. H., and J. R. Wait (1973), Excitation mechanisms for transmission through

forest-covered and vegetated media, U.S. Department of Commerce, Office of

Telecommunications, Boulder, Colorado, Technical Report No. ACC-ACO-8-73 prepared for U.S. Army Communications Command, Ft. Huachuca, Arizona (NTIS

Accession No. AD 771-915).

54
Parker, H. W., and G. H. Hagn (1966), Feasibility study of the use of open-wire

transmission lines capacitors and cavities to measure the electrical properties

of vegetation, Stanford Res. Inst., Special Tech. Rept. 29, Menlo Park, CA.

Parker, H. W., and W. Makarabhiromya (1967), Electric constants measured in vegetation and in earth at five sites in Thailand, Stanford Res. Inst., Special

Technical Report 43, Menlo Park, California.

Tamir, T. (1967), On radio-wave propagation in forest environments, IEEE Trans.

Ant. Prop. AP-15, pp. 806-817.

Tamir, T. (1977), Radio wave propagation along mixed paths in forest environments,

IEEE Trans. Ant. Prop. AP-25, pp. 471-477.

Van der Vis (1979), Measurements on short distance HF data links, National Defense

Research Organization TNO, Report IR 1979-72.

Vogler, L. E. (1981), The attenuation of e1ectrorrlagnetic waves by multiple knifeedge diffraction, NTIA Report 81-86 (NTIS Accession No. PB 82-139239).

Von Hippel, A. (1954), Dielectric Materials and A~p1ications (Technology Press of

MIT, Cambridge).

Wagner, C. (1953), On the numerical solution of volterra integral equations, J.

Math. Phys. ~, pp. 289-401.

Wait, J. R. (1959), Guiding of electromagnetic waves by uniformly rough surfaces,

Pts. 1 and 2, IEEE Trans. Ant. Prop. AP-7, pp. 5154-5168.

Wait, J. R. (1962), Electromagnetic Waves in Stratified Media (Pergamon Press, New

York).

Wait, J. R. (1967a), Radiation from dipoles in an idealized jungle environment,

Radio Sci. ~, pp. 747-750.

Wait, J. R. (1967b), Asymptotic theory for dipole radiation in the presence of a

lossy slab lying on a conducting half-space, IEEE Trans. Ant. Prop. AP-15, pp.

645-648.

Wait, J. R., R. H. Ott, and T. Telfer (1974), Workshop on radio systems in forested

and or vegetated environments, Technical Report No. ACC-ACO-1-74, U.S. Army

Communications Command, Fort Huachuca, Arizona.

Wait, J. R., and L. C. Walters (1963), Curves for ground wave propagation over

mixed land and sea paths, IEEE Trans. Ant. Prop. AP-11, pp. 38-45.

55
I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I
APPENDIX A: ANTENNA WITHIN THE SLAB

In this appendix, we consider the case where the vertical electric dipole source

is located within the slab. The geometry is shown in Figure 1, but the antenna

height h is now negative: 0 > h > -D.

Following the notation in Section 2.1, the fields in air (z > 0) can be derived

from the z component, IT

oz ' of an electric Hertz vector of the following form:

IT

oz =

00

I ds f To(A) e-uoz u

A ~JO(Ap) dA

4TIiw£0 0 0 (A-I)

where To(A) is an unknown transmission coefficient and all other symbols are

defined in Section 2.1. The three nonzero field components, Eop ' Eoz ' and Ho~' are

given by (3).

In the ground (z < -D), the z component, ITgz ' of the Hertz vector can be

written

IT

gz

00

= I ds f Tg(A) eU(Z+D) ~l JO(Ap) dA 4TIiw£ £

o gc 0

(A-2)

where T (A) is an unknown transmission coefficient and all other symbols are defined

g

in Section 2.1. The three nonzero field components, Egp ' EgZ ' and Hg~' are given

by

a2IT , ,2 ) E = gz E = (-Y~ + ITgZ (A-3) gP ap crz gz

and

a IT

H = -iw£ £ gz

g~ o gc ap

Within the anisotropic slab (-0 < z < 0), it is most convenient to work directly

with the azimuthal magnetic field, H~. Clemmow (1966) has given the expression for

the field in an infinite medium which we can generalize to the appropriate form

within a slab:

H~ = I ~~ KIJ

1 (Ap) [e-vi z-hl + A(A) e

VZ

+ B(A) e-vzJ (~2) dA (A-4)

where A(A) and B(A) are unknown coefficients and all other symbols are defined in

Section 2.1. From Maxwell's curl equation, the two nonzero components of the

electric field, E and E , can be written

p z

57
-1 a H

E = -~ P iWE: E:h az

o c

and

E K (3

= (p H~) (A-5) z iw£ E:h P ()p o c

The four unknowns, To(A), Tg(A), A(A), and B(A), can be determined from the

continuity of tangential electric and magnetic fields at the interfaces:

(E - E ) I

°P P z=o

= 0

(Ho~ - H,J I = 0 'I' z=o

and

(E - E ) I

p gp z=-D = 0

(H - H ) I ;:: 0 g~ z=-D

(A-6)

By substituting the field expressions into (A6) and performing some algebra, we

obtain the following expressions for the unknowns:

A(A) = Ro

evh [1 + R

g

e-V(2D+2h)] /5

-v(2D+h) r

e

2Vh B(A) = R ] /5 g

e L1 + R .. 0

K u (l+R ) vh

e

[1 e-V(2D+2h)] T (A) o 0 = + R

0 v 5 g (A-7)

where 5 = 1 - R R

o g

-2vD

e ,

58
Kl - K K

l - K

R 0 R 2 = = 0 Kl + K g Kl + K

0 2

are defined in Section 2.1. This completes the formal inte- and all other symbols

gral representation of the fields in all three regions.

The integral forms for the fields can be evaluated asymptotically when the

horizontal distance p is large. We are primarily interested in the vertical electric field, and from (A-I) and (3), the vertical electric field in air (z > 0) can

be written

E

oz = I ds oof T (A) e -uoz (~) J (Ap) dA

4TIiw€

000

0 u 0 (A-8)

The Bessel function in (A-8) can be replaced by Hankel functions (Abramowitz and

Stegun, 1964), and the integration range in (A-8) can be extended to _00 to yield

the following form (Wait, 1967b):

(A-9)

We follow the method which Wait (1967b) used for a dipole above an isotropic

dielectric slab. The integral in (A-9) can be deformed around the branch cuts which

result from branch points at k and kg(=-i Yg) as shown in Figure A.l. In addition,

there may be one or more surface wave poles at A=ks which result from poles of

To(A) (= zeros of S) as given by (A-6). The contributions from the contour Cg and

any surface wave poles are exponentially attenuated for large p, and we keep only

the branch cut integral C :

o

E

oz

I ds f

4TIjw€o C

o

H (2) (Ap) dA

o (A-lO)

In order to evaluate (A-lO) , we first rewrite To(A) in a form which is equivalent to that in (A-7):

T (A) = o

€ [l+R e- 2VD] vc g

(A-H)

Near A=k, (A-II) can be approximated

59
Im(A)

ORIGINAL

CONTOUR

k Re(A)

O DEFORMED

X -----~ CONTOUR

kS

Figure A.I. Integration contours in the complex A plane.

60
1

E

vc 1 R -2vD

+ e

g

A=k (A-12)

The first factor in (A-12) is identically equal to Gs(h) as given by (13). Thus,

(A-12) can be written

Substituting (A-13) into (A-lO) and approximating A2 by k 2 , we have

(A-l3)

E '" oz

I ds k2 G (h)

4'1Tiw E s f

o c

o

(A-14)

Wait (1967b) has evaluated the integral over C :

o

f

'"1 + ik~z -ikp lk ~2p2 e C 0

o

Substituting (A-IS) into (A-14), we can write E in the final form

oz

(A-IS)

(A-16)

where Eo is given by (S), f(p) is given by (6), and Gs and Go are given by (13).

This form is consistent with the result in (4)" but disagrees with the result for

the dipole in the slab given by Wait (1967a). He has an extra factor of K

2 multiplying Gs(h) which leads to a height-gain function which is different for the source

and observer in the slab. More recently, -Wait (private communication) has agreed

that the factor K 2 is spurious and that (A-16) is the correct result. Thus the

height-gain function in (13) applies to both the source and observer, and reciprocity

is satisfied.

For the case of both the dipole and the observer located within the slab, Ez

is obtained from (A-4) and (A-S):

61
(A-17)

=

For large p, we again include only the C contour in Figure 27. Also, the first

o

factor in (A-17) does not contribute, and E

z

is approximately given by

(A-18)

By expanding A(A) and B(A) about A=k, (A-18) can be put into a form similar to

(A-14) :

In this case, the integral over Co is given by

-ikp J _e__

C i k t. 2 p2

o 0

From (A-19) and (A-20), E can be written

z

(A-19)

(A-20)

As expected, the same height-gain function G appears for both the dipole height h

s

and observer height z since both are located in the slab.

REFERENCES

Abramowitz, M., and I. A. Stegun (1964), Handbook of Mathematical Functions, National

Bureau of Standards, AMS 55.

Clemmow, P. C. (1966), The Pland Wave Spectrum of Electromagnetic Fields (Pergamon

Press, Oxford), Section 8.1.5.

Wait, J. R. (1967a), Radiation from dipoles in an idealized jungle environment, Radio

Sci. !, pp. 747-750.

62
Wait, J. R. (l967b), Asymptotic theory for dipole radiation in the presence of a

lossy slab lying on a conducting half-space, IEEE Trans. Ant. Prop., AP-15,

pp. 645-648.

63
APPENDIX B. KIRCHHOFF INTEGRATION

In order to evaluate (36), we first rewrite f in the following manner:

/2 'ik(Ra + 1\)

f =I TI (R )3/2 I

a b a--b

where

00 1 2 2 l I = bz(z - D) exp -ik [ZZRb

+ (~ ~aD) J~ dz

The problem now is to evaluate I.

We first make the substitution, z' = z - D:

(B-l)

OOf '. [(z' + D)2 I = (z' + D) z' expJ-1k 2 R

o l b

z ,2 ] l

+ 2 R

a

) dz' (B-2)

By completing the square in the exponent, (B-2) can be rewritten:

. 2 00

I = [-lk D 1 f (" D)' . exp 2 (R + R ) J z + Z •

abo

We now make a second $ubstitution:

(B-3)

u = I~k (B-4)

After some algebra, this allows (B-3) to be written

. k D2

exp -1 2(R

a

+ R

B

I = [ik (.!- +.L ] 3/2 Iu

2 Rb R

a

where

64

(B-5)
00

-u2

I = J (u2 + al u + a

o) e du

u

u.Q,

u.Q, = D I ik 2(~a + ~)

and

The integral in (8-5) can be evaluated in terms of the complementary error function

(Abramowitz and Stegun, 1964):

erfc(u.Q,) du (B-6)

Using (8-6) and doing one integration by parts, we can write (B-5) as

(B-7)

For the special case of D=O, we have Un = a = a =0, and I is given by 0 1 u

;; ;; = erfc(O) = 4 (B-8)

This result leads us to rewrite I :

u

I

u (B-9)

65
Using (B-1), (B-5), and (B-9), we can now obtain the following final result

for f:

(B-lO)

where

2 r.- 2 H -A H2

S = (1 + 2 H ) erfc(vA H) + -- -- e

ITI/A

H -

and

The exponential factor, exp(-H2), has magnitude unity and only results in a phase

delay. All of the dependence of the magnitude of f on D is contained in S. For

most cases, IHI is small. In this case we can expand S in a power series in H by

using the small argument expansion of the complementary error functioij (Abramowitz

and Stegun, 1964)

(B-11)

The magnitude of S is then approximated by

(B-12)

REFERENCE

Abramowitz, M., and I. A. Stegun (1964), Handbook of Mathematical Functions, National

Bureau of Standards, AMS 55.

66
APPENDIX C. USER'S GUIDE, LISTING, AND SAMPLE OUTPUT FOR

PROGRAM WAGSLAB

The input data are described as follows. Most quantities are pictured in

Figure 3 (or 1). Note on Card 3 that the antenna heights are referred to the

ground, not the top of the slab.

Card 1: KIND, TO (110, FlO.O)

KIND = beginning type of distance at which F(x) will

be evaluated.

1 is specified on next cards

o is equidistant

TO = total distance in kilometers

Card 2A: XCI)

XCI) =

(8FlO.5)

specific distances in kilometers at which F(x)

will be evaluated.

IF KIND = 1 use these cards to begin givin~

specific distances until you are done or want

to change to equidistant points.

Terminate this set of distances with a o.

Card 2Bl: NED (12)

NED =number (limited to 50) of consecutive sections

with F(x) evaluated at equidistant points.

(This should agree with the number of pairs of

DEP and FINT).

Card 2B2: (DEP(I), FINT(I), I = 1,NED) (8FlO.0)

DEP(I) =distance in kilometers at which this set of

equidistant points ends.

FINT(I) = interval of these equidistant points in

kilometers.

If KIND = 0, use these cards to begin.

Note: There may be a series of 2A and 2B cards to reach the total

distance given on card 1.

Card 3: HA, FREQ, HAR, AKM (4FlO.5)

HA = transmitter antenna height in kilometers =ha-D

FREQ = frequency in MHz

HAR = receiver- antenna height in kilometers = hr-D

AKM = effective earth radius in kilometers

Card 4: ID (8AlO)

10 = path identification.

67
SIGX(I)

EPSX(I)

ISLAB

Tel)

EH(I)

EV(I)

SH(I)

SV(I)

Card 5:

Card 6 toM:

Card M+1:

Card M+2 to

end:

N, IXUNITS, IZUNITS, REFEL (3!10,F10.1)

N c number of points on terrain profile

IXUNITS = 0, distance input in kilometers = 1, distance input in statute miles

IZUNITS = 0, height input in meters = 1, height input in feet

REPEL = reference elevation in IZUNITS(=height at d=O)

(X(I), I = 1,N) (4(F10.2,F10.0))

XCI) = terrain distances in IXUNITS

Z(I) = terrain heights in IZUNITS

NGC (110)

NGC = number of sets of ground constants

(limited to 50).

(DX(I)~ SIGX(I), EPSX(I), ISLAB, T(I), EH(I), EV(I), SH(I),

SV(I), I = 1,NGC) (F8.2,F8.6,F8.2,I2,3F8.2,2F8.6)

DX(I) =maximum distance in kilometers for given set of

ground constants

=conductivity of ground = C1 (S/m) = relative permittivity of gfound = E:

g = 0 is no slab in this section

= 1 slab ground constants to follow

=thickness of slab in meters =0

=hori7onta1 rel~tive pe:mi~t~vity for slab = E:h = vert1ca1 relat1ve permlttlv1ty for slab = E:

= horizontal conductivity for slab = C1h (S/m)v = vertical conductivity for slab = C1

V {S/m)

68
A listing of program WAGSLAB and all subroutines is given below. Program

WAGSLAB is essentially a modification of program WAGNER (Ott et al., 1979) to allow

for the effect of a slab over the ground.

PROGRAM ~AGSLA8CI~PUT,OUTpUT,TAPE60-INPUT,TAPEZt

c

C MlJDIFICATION OF IIAGNER FOR DAVE HILL; 518Z, To ACCOMMODATE SLA8S OF

C SNOW ,BUILDINGS, AND VEGETATION. M4IN CHANGES IN SUBROUTINE IN

C USI~G GROUNO CONSTANTS TO COMPUTE OELTA CIKPEDANCEt. GHAClt

C ANO GHRCNGC) (DEPENDENT ON PARTICULAR SETS OF GROUNO CONSTANTS)

CARE COMPUTEO IN IN ANO USED IN FH(X) WHICH IS COMPUTEO IN WAGSLA8.

CHAND HP ARE MOOIFIED IN TERRANE WHEN A SLAO IS PRESENT.

C VEPTICAL POLARIZATION IS SET CONSTANT. PREVIOUS REFERENCES 10

C HA ARE ELIMINATEO.

C

OIME~SION IPOLIZI

OlMENSION ADABI3),AOGHI3) 0IME~SION FIZOuO),R13CZOOOt,R14C2000),R15(ZOOO),R16(ZOOO), CR17CZOoOI,~leCZOOo),R19C2000t,R20C~OOOt,RZICZOOO)

CUMMON 111 HA,HAR,AKM CO~MON 12/ O,H,HP

COMMON 131 OELTAR,WAVf CU~MON /41 FRF.Q,POL COM~ON 151 NG,A8C48t,GH(48)

COMMON 161 N,X(2001),INDEX

COMMJN IINPUTI DUM~CI03Z0),IDC8t,REFEL

CQMMON IGCXI NGC,DXCSO),ETAXC501,DELTAX(50),SIGX(50t,EPSXC50t, CTI501,SHC50"SV(50),~H(50),EVC501'GHA,GHRC50)

CO~MaN/PMF/FLOS(ZOOO),O~MCZOOot

DUUBlE PRECISION D48,OGH C~MPLEX FE_H,f,ALA~Z,SUM,DELTAR,ETAR

COMPLEx KER~L,PO,Pl,P2,P3,P4,CTMP

COMPLEX ETA,OELTA

COMPLEX GHA,GHR,ETAX,DELTAX,FH

DATA CNG-5)

DATA CAOAB-.9C61198459,.5384693101,O.)

DATA CADGH-.2369268851,.47B6286104,.568888a8888)

CANGCZ'-AT4N2CIIMAGClt,REAL(Zt,

IPOLll)-8H YERTICSIPOLCZ)-8HHORIZONT

C

C READ GAUSSIA~ QUADRATURE ABCISSAS AND WEIGHTS

C

NRsCNG+l)/2

DO 1 L-IINR

OABsAOABClI

UGH-AOGHI U

J=NG-Ltl

ABCU-OAfl

Ai'CJI--AflCLt

GHIU-OGH

1 GHCJI-GHCL'

C

C C~ll SUBROUTINE DI5T1 TO SET UP OISTA~CE ARRAY X IN ~ETERS.

C SlA~r WITH xczt. xClt-O. HAS ALREADY BEEN SET.

THE DISTANces 00 NQT HA~E TO BE EQUALLY SPACED.

C SU~~JUTINE DI5TX SHOULD TeST N TO ~E LESS THANO~ EQUAL TO 2000.

C

4 qU-O.

"'(U m l1.,Ool

CtLl lHSTl"

c

C MAKE SURE THEQE ARE AT LEAST 4 DISTANCES.

C

IF IH.GE.4' GO TO 2

PRINT 18

CAll EXIT

C

2 SOPTXZ-SORTIXCZII

SQPIX3=SCRTIXC3'1

SQPTX4-SQRT(XI411

69
01s5vRTlX(2)+Xl3).Xl41).lXlZ)+lSQRTX~-SORTX31+Xl3).lSORTXZ-SQRTX"

ItXl4'*lSQRTX3-SQ~TXZ))

R1=~(3).Xl4)+lSORTX4-SQRTX31101

R2=XlZ,*X(4)+t$QRJX2-$QRTX41101

R3s~(2)+Xl3)+lSORTX3-SQPTX211Dl

R4=(tZ)+(SQRTX4++3-SQRTX3.+3'+Xl31+CSQRTXZ++3-SQRTX4 ••3)+Xl4).CSO lRr~3.t3-SQRTX2++3)I/Ol

P5-S0RTlX(3)+Xl4))+lXl3)-Xl4)"Ol

P6-SQRTlX(Z)+Xl41).lxt4)-XCZ))IOl

R7=SQRTlXlZ)+XC31)+lXlZ)-Xl3))/Ol

R8=lSORTXZ+CSQ~TX3+.3-S0RTX4•• 3)+SQRTX3.lSQRTX4•• 3-SQRTXZ ••3)+SQRT

1<4.lSQRIXZt+3-S~RTX3**3)"OI

R~-SORTlXl31+X(4»)*lSQRTX4-SQRTX3)/Ol

RIC-SQRllXlZ)+Xl4')+(SQRTX2-S0RTX4)/DI

R11-SQRTlXll)+Xtl))+lSORTX3-SQRTX2)/Ol

R1Zs(SQRTXZ+tXl~)-Xf3)+SQRTX3+CXC21-XC41)+SQRTX4.(Xl3)-Xt2)))/01

00 3 M-5,N

MI-M-!

M?-M-Z

OELT~l-X(M)-Xt~l)

OElTA2·X(M)-Xl~Z)

OZ-lOELTAI-DELJAZ)+OELTA1+0ELTAZ

~13(~'-XtM11·XlKZI·lXCMZ'-XCM1')/DZ

R14l~'=XtM)+XlMZ)+CXtM)-XCMZ')IOZ

R15t~)=XlM'tXlM1).tXCM1)-XCM)'/DZ

P16lM)-lXlMl'++Z-XCMZ)•• Z,/DZ Rl1l~I-CXCM2).+2-Xl~)+.Z)/D2

R1BtM)-txl~)+*Z-XtHl)+.Z"02

R19(1,=tXtM2)-XtMl)I/DZ

RZOCM'-lXlM)-XCM2')/OZ

3 R21l~)-lXlMl)-XCM.'/DZ

C

C REAO I~ANSM{TTER HEIGHT, FPEQUF.NCY, RECEIVER HEIGHT AND EARTH RADIUS.

C COL OESC~tPJION

C 1-10 TRANSMITTER HEIGHT. KM

C 11-2J fREOUENCV. MHZ

C 21-30 P[CEI~ER HEIGHT, KM

C 31-40 EARTH RADIUS. KM

C

REAO 19, HA.FREQ.H~q,AKM

POl-l.

; NOIE THAT ~ERT[CAl POLAR[ZATI~N IS NO~ CONSTANT.

c 4aQ=~A~*1.F+3

HA=HA*1.E3

KPCL'POl

AlA~·l.9979~5E7/FREQ

WAVE=b.Z831853071AlAM

AlAMZ-llO.10710b7812,O~1011067812'/SQRT (ALAM)

TO=SEEONDCOVM) .

c

C lOCP ON DISTANCE

C

00 14 I-1,N

INOEX-I

IF l1 .NE. 11 GO TO 51

CAll TEPRAN2CXCI),H.HP,ETA,OELTA,ETAR,DElTAR,CONO,EPS'

PRINT 20, FREQ.IPOllKPOl),AKM,HA,HAR

GO T~ 52

51 CAll TE~PANE (XCI),H,HP,ETA.OElTA,ETAR,OELTAR,CONO,EPS)

52 CONTINUE

IF ll.EO.1) GO TO 33

n-XlI)+lH.*2"C2.+XCI))

FCI)-FEWHCH.XCI))

IF lI.lE.b) GO TU 13

C

C J - 2 THROUGH 4

C

SUMc(O.,O.)

00 9 J-Z,4

PO-PlsPZ-P3-CO.,O.,

K-J-1

XPZ-O.S.lXCJ)+XCK))

XM2=O.5*CXlJ)-XC~)'

70
00 7 M-l,NG ~0-XJZ+A8CHI·XI1Z

C1MP-KERNLCXO'.GHCMI

PI-Pl+CTMP.SQRTCXOI

p2-P2+C1Mp·.<0

P3-P3+C1MP+SQRTCXO'·+3

IF CK.NE.11 GO TO 6

'O=O.z5+~ci'*Cl.+A9CH" •• Z PO-pO+SCRTC'(OI·KERNlCXO'·GHC~'

GO 1'") 7

6 po-po+eT,..p

7 corn {NUE

Pl=PltXM2

P2-P2*XI'I2

1'3-1'1*=<112

IF IK.NEo!J GO TO B

PO-PO*SC~TCXIJII

GO T!1 9

6 PO-POUf'l2

9 SUM·SU1+PO+~4*Pl+Ra.PZ+PIZ*p3+FCZ'.CR1*Pl+R5*pZ+R9*p3'+FC3J.CRZ.pl

L+P6*P2+Rl~.P3'+FC41*CR3*pl+R7·PZ+Rl1+p3'

C

C J = 5 1H~OUGH 1-1

C

11-1-1

()lJ 1L J-5,Il

t'O=PZ=P4-C,).,O.1

XP2=J.5tIXCJI+XIJ-111

~112·0.J*IXCJI-~(J-l"

')0 1') '~"'l.NG

XO"'XP2+AI3CI1I*XI1~

CTHp",KERNLC,(O'*GHCI1I

PO"'1'0+C1I'1p

I' Z- pZ +C TI" P. XI)

10 p4-P4+C1MP*XO.+Z

PO-PO*XI1Z

PZ-pZUI12

p4=P4*XI12

11 SUM-SUM+FCJ-aJ*CR15CJJ*PO+RI8CJJ.p2+RZICJ,*p4J+FeJ-IJ*eR14fJJ.PO+R

1l7CJI*PZ+RZOCJI*p4,+FIJ'+CR13eJ,+PO+R16CJJ.PZ+R19CJJ.P4J

C

C J-l

C

THETA=4SIN CSC~TCXCIIJ/XCII"

CTHETA-COS eTHETA'

PO-PZ-P4'" co., O. J

00 12 H-l,NG

TEMP"'1.-0.Z5*CTHETA.+Z*C1.+A9(I1JI**2

XO-XCII*TEI1P

CT~P·SQRTCXC[J-XO'.KERNLCXOJ.GHC~'

po-pO+Cl/'lp

PZ-PZ+C1HptTEMP

1Z p4-p4+C1I1P+TEI1P**Z

PO-pO*CTHETA*SQRTCXCII'

PZ-pZ+CTHET4*SQPTCXCIJJ··3

P4=P4*CTHETA*SQRT(XCIJ,··5

F(I)sCFCIJ-AL~~Z.CSU~+F(I-Z).CR15(It.pO+R18eIJ+pZ+RZleIJ+p4J+FCll)

I*CR14CI'*PO+R11CI'.P2+RZOCI,.P4"'/(l.+ALA"Z.CR13eIJ.pO+R16CIJ+PZ+.

2R19IU*p4J)

13 AHPsCABSCFCIJJ

pHA·C~NGCFCI"

IGCC-l

00 30 I GC -11 NGC

IF C~CI' .LE. QXCIGC" GO TO 3Z

IGCDs IGC+ 1

30 CONTINUE

32 CONTINUE

FH-FII'.GHA·GHRCIGCOJ

A1"1 PH" CA 8S (FH I

PHAH-CANGCFfi'

RR2"'20.·ALOGIOC~AVE*XCII'AMPHJ

!lTl-~R2

FlOSCl''''13Q.37+20.*AlOGlOCFREQJ-RQZ

OK''\( II-XC 11*.001 TIME=5ECONCCOU~)-TO

GO TO 34

71
33 PI! INl 24 P~INT 26, IOXIIGCI,SIGXIIGCI,EPSXIIGC),TIIGC',SHIIGC),SVIIGC),

CfHIIGC',EVIIGC),QElTAXIIGt),GHRIIGC), IGC-l,HGC)

prJ ItH 2~, GHA P~ I In 21

GU TO 14

34 CflN r P;UE

PRINT 22, Xll),H,A~P,PHA,AMPH,PHAH,TIME,FlOSlI',BTl

~RIIE lZ,*) XtI"l00r.,AMPH

14 O'~ 11 Ni.!E

GO n 4

C

18 FOR~AT ItONU~8ER OF OISTANCES < 4+)

19 FORMAT 14FlO.5'

20 FORMAT I*OFREQUfNCY .t,FLO.2,lOX,AB,tAl POlARIZATION*,8X,*EARTH RA

CDIUS ·*FIO.O,. KM*I

CIX,*TRANSMITlER ANTENNA HEIGHT .*F7.3,* METERSt,lOX,*RECEIVER ANTE

CNtH HEIGHT .tF7.3,t I'ETERS*)

21 FORMAT 1119X,*X*,lOX,*Z*,16X,*FIX,*,

C24X,tFHIX).,13X,*TIMINGt,2X,*FIElD STR*.3X,*aTl*1

C8X,*IM,t,8X,tCM)t,6X,tKAG*,

CIOX,tARG*,12X,*MAG*,lOX,tARGt,12X,*ISECt*,4X,tC08U,*t

22 FORMAT I*Ot,F12.2,FIO.l,E15.5,E13.5,

CE15.5,EI3.5,FIO.3,FIO.Z,F8.2t

23 FORMAT l/lX,tGHA~*2F8.41

24 FORMAT IlllX,tOISTANCE SIGMA EPSILON SLA8 SH *,

Ct SV EH EV IMPEDANCE *,

C* GHU/J

26 FORMAT CIX,F7.0,F9.6,F7.Z,F5.0,2X,2F9.6,2X,2F7.2,2X,ZF8.4,

C2X,2F8.4)

END

COMPLEX FUNCTION KERNLlXO)

COMMON III HA,HAR,AKM

COMMQN 121 O,H,HP

COMMQN 131 DElTAR,WAVE

COMMON 141 FREO,POl

COMMON 151 NG,lBI4B),GH(481

CUMMON 161 ~X,XC20011,I

COMPLEX FEWH,OElTA,OELTAR,ETA,ETA~

CALL TE~RA~E CXO,HO,HPO,ETA,DElTA,ETAR,OElTAR,CQND,EPSJ

XMS·XII)-XO

HO·H-HO

Rl-SORTlXO**Z)

RW-WAVetlXO+[(HOt.21/(2.*XOJI+XHS+IIHO**21/CZ.*XHSI)-D)

KERNl-CMPlXlCOS fRWI,-SIN (~WII*SORT(X(I)'(Rl.XHSI)*CIHPO+OELTA-OE

llTAR)tFEWHlHO,XMS)-IHD/XMS))

RfTURN

END

72
CO~PLEX FUNCTIONWERF(lll, CO~PLEX 1,ZZZ,ZV,V,ZZ,C,W,S

DIMENSION C(lZ', W(5,4)

EQUIVALENCE (S,C(lZ))

LOGICAL LZZ

OA1. (C(l,-(.O,-.56418958J5')

OATA «((W(I,J',I-l,5),J-l,4'-(1.,.O',

(3.678794411714423f-Ol,6.071577058413937E-Ol),

X (1.831563688873418E-OZ,3.400Z6Z17066066ZE-Ol),

(1.Z340ge040866768E-04JZ.Ol1573170376004~-Ol),

X (1.lZS35174719Z646E-07,1.4595358990015Zee-Ol),

X (4.Z75835761558070E-Ol,O.OOOOOOOOOOOOOOOE+OO),

X (3.041442052569126E-Ol,Z.08218938Z0Z8316E-OlJ,

(1.40Z395813662779E-Ol,2.ZZZ134401796991E-OlJ,

X (6.5317777Z8904697E-OZ,1.739183154163490E-OIJ,

( (3.628145649998664E-02,1.356389510006551E-Ol',

X (Z.553956763105058E-Ol,O.OOOOOOOOOOOOOOOE+OOJ,

X (2.1849Zt15Z748907E-Ol,9.299780939Z60166E-OZJ,

X (1.4795275951Z0158E-Ol,1.311797170842178E-Ol',

X (9.27107664Zb4433ZF-OZ,1.Z8316962228Z615E-Ol',

X 15.968692961044590E-02,1.132100561Z44882E-Ol',

X 11.7900l1511813930E-Ol,O.OOOOOOOOOOOOOOOE+OOJ,

X 11.6426l13639Z9661E-Ol,5.019713513524966E-02),

X 11.307514696b96522E-Ol,6.11126504774547ZE-OZ.'

X 19.640250558304439E-OZ,9.123632600421Z58E-02.,

X (6.979096164964750E-OZ,8.9340000Z4036461E-02J'

XX-REAUizz.

YY-AIMAG(ZZZJ

X-ABS (XX,

Y-ABS (VY) l-C~PL X()f,Y J

LZZ- .FAlSE.

IF (X.GE.It.5.0~.Y.GE.3.5J GO TO 6

I-X+.5

J-Y+.5

V-CMPLXIFl~ATII"FlOAT(J')

Z'I-Z-V

C(Z )-WI I+lIJ+U

AI-0.

O!J 11-3,12

AI-Al-.5

CII)-IV*C(I-l)+C(I-Z),'AI

I CONTINUE

J-IZ

011 2 l-Z111

JeJ-l

Z S-S*ZV+C(JJ

3 IF (YY.GE.O.) GO TO 4

IF (.NOT.LZZI ZZ-l*Z

S-Z.tCEXP(-ZZ)-S

IF IXx.GT.O., i-CONJGIS)

GO 'T:J 5

4 1~ (XX.LI.O.I ;=CONJG(S)

5 loIERF-S

QETI.l~N

6 Ll2".TOIU~.

l2=l+Z

S-Z*I(O.,O.4613135Z79J/(Z2-0.l90l635092J+IO.,O.09999Z16168JI

1(ZZ-1.7B449Z7465J+IO.,O.0026638938748)'IZZ-5.52534374379.J

GO TI) 3

ENO

73
CQMPLEX FUNCTIONfE~HlHD,(O'

CUMMnN 131 DELTAR,WAVE

COMPLEX TEMP,Q,l,lZ,ZZ,HWERF,WERFI,WERF,IWERF,DElTAR

TEMP-lO.707106781Z,-0.707106781Z'+SQRTC.5tWAVE'

Xl)2=::;QR100'

Q--rEMPtHO/XD2

Z-rEMptOElTAR*XOZ+Q

Zl=-Z

Zl" AI!'IAGC lZ'

IF lll.lT.O •• OR.IABSCREAlCZZ".lT.6 •• ANO.II.lT.6." GO TO 1

12-Z1*t2

HWE~F·IIZ-2."CZltlZZ-3.5"

GO TJ 2

1 WERfZ"WERFIZZ'

~WERF-ZZ-O.5t~eRFZ/(ZZtWERfZ+lO.,-O.5b418958)'

2 ZWERF-Z+HWERf

FEWH-lQtZWERF-O.5"lZtZWERF-O.5'

RETU'lN

END

SU8ROUTINE TERaANE lX,H,HP,ETA,OElTA,ETAR,OELTAR,COND,EPS,

COMMON/INPUTI TD(5000"THTC5000,

COMMUN IGCXI NGC,UXI50"ETAX(50),DElTAXC50),SIGXCSO),EPSX(SO"

:T(50"SHCSO"SVI50"EHC50',EVI50',GHA,GHRI50,

COMMON III HA,HAR,AK~

COMPLEX ET4X,OElTAX,ETAR,OELTAR,ETA,DElTA

C

C COMPUTE HEIGHT, SLOPE, CONDUCTIVITY AND DIELECTRIC CONSTANT AT X.

C

7 CONTINUE

If (X .GT. TOlN" X-TOIN'

A-I000.*41<"

HP--XIA

H-.5tXtHP

00 6 I-2,N

IF (1( .Gr. TDlllI GO TO 6

H-H+THT(I-l'+I(X-TOlI-ll"ITOlII-TDlI-l,,)tlTHTlI'-THTll-I)'

HP=HP+C1HTlI'-THTCI-l)"CTOII,-TOll-ll)

GO TO 9

6 CONTINU.E

9 CUNTOiUE

00 2 1-1,50

IF 1)( .lE. (OX(I,+.000111 GO TO 4

2 caNTl~UE

4 ETA-ETAX(I)

OElTA-CElTAXII)

CONrJ"SlGXU,

EPS=EPSlll II

C

C MODIFICATION Uf HAND HP fOR PRESENCE OF SLAB.

C

IF (I .EQ. U GO TO 10

H-H+TII-l)+I(X-OXII-III/IDXIII-DXII-II')+lTIII-Tll-lll

HP=HP+IT(1)-Tll-lI1/10XII'-DXII-111

GO TO 12

10 CUNTINUE

HsH+Tlll

12 CONTINUE

I1ETUP.N

EN'''''' TEIIPlNl

CALL lNIH,

ETAR-ETAXlll

DElrARaDElTAXILI

GO HI7

Pin

74
SV3RfJUTINE OIsrX

C

C SUBilJUTINE TO FIll DISTANCE 4RRAY. USE EITHER SPECIFIC

C OISIA~!CESIKINO- l)ORCOMPUlEOI~HtKES AlEOUIDISTANT

CPOINrS IKIND - 0), A.CD,/1!3INATION OF THE TIIOKINOSCAN BE

C USEO. All VAPIAi3lI:S APE READ' IN IN KIL0I1E.TERS4ND. THE

C OISTANCE ARRAY IS FILLED IN METERS.

c CO~M~N /6/ N,XIZ001),INDEX

DIMENSION DEP(50),FINTI50)

N-2

READ 100, KINO~Tn

IF IEOF(60)) B,Z

2 CONTINUE

TO/l=1'0*I.E+3

IF IKINo) 4,H

4 00 6 L-N,2001,8

K=l +7

~EAD 101, IXII), I-l,K)

01) 6 J=l,S

N"l+J-l

IF 1~IN) .LE. 0.) GO TO 10

XIN)"XIN)*1.E+3

6 CONTINUE

7 P~INT 102

8 CAll EXIT

1\.1 IF IXIN-ll .l T. TOM) GO TO 12

Ii=!Ij-l

RETljRN

12 Q.EAO 103, NEO,IDEPII),FINHI), I-1,NED)

00 20 I-1,NEO

OEPM=OEPII)·I.E+3

FINTM=FINTII)*I.E+3

S"-.1 tF INTM . 00 14 J-thZQOl

JS=J

XIJ)=FINT~+XIJ-l)

If IXIJ) .GE. ITOM-SV)) GO TO 16

IF IXIJ) .GE. IOEP~l-SV)) GO TO 18 lit CONTPlI.H

GJ TO 7

16 XIJS)-TDM

N=J5

RET OR N

18 X(JSI-OEPM

{I .. J S+1

2lJ CONTINUE

GO TO 4

100 FORMAT IIIO,FIO.D)

101 FOP "AT 18FIO.5) 1~2 FOMMAI I*ONUM9ER UF PIST4NCES EXCEEDS OIHENSION*)

lC3 FUQMAT IIZ/(8F1G.O)1

EN [)

75
SU~KOUT[NE [NINI

CUM":,)"l IINPiJTI'(15000), Z(5000), HIZ0), 81120h UlItO), AMltO), to

1I81,REFH

COMJoiON Hexl ~GC,OX(50),ETAXI50),DELTAXI50)'SIGXI50),EPSX(50),

CH 50) , SHI 50), SVI 50) , EHI 50 ) , EV( 50) , GH A, GHR ( 50)

COMMaN III H4,H~R,AK~

COMMON Iltl FREQ,POL

COMPLEX ETAX,EHC,EVC,~K,~,Dl,D2,rVT,OELTAX,CTAHH

COMPLEX GHA,GHR,R,FH,FHT,EI

JATA El/IO.I1." FHIANTI-1.+EI*~N·OEL1AXII)*(ANT-TIII.

FHTI4NT)aICExPI-V+ITIJ)-ANT»+R+CEXPI-V+IT(II+ANTI»1

C(EVC+l1.+R*CEXP(-Z.*V*TIII'»

c

C INPUT

C

C 10 - 1PEN1[F[CATION . C t~ = "lUMBER OF OATA POIN'lS

C IXUNITS -0, DISTANCES INPUT IN KILOHETER~

C (XUNITS -I, DISTANCES INPUT IN MILES

C IZUNITS - O~ HEIGHTS INPUT IN METERS

C IZUNITS a 1, HEIGHTS INPUT IN FEET

C REFEL - REFERENCE ELEVATION IN METERS

C

1502 FORMAT 18A10)

1503 FORMAT (2X,8AlO'

1504 FO~~Ar (4(FI0.5,FI0.1I'

1,05 FQPMATIIt(FlO.2,F10.0»

1506 FQRMAT 110X,110,2F15.5)

1510 FOQMAT 13IlO,FI0.1)

1511 FURMAT IZX,+NUMBER UF P~OFILE DATA POINTS IS*llOI

CZX,+REFERENCE ELEVATION IS+F10.Z,* METERS*I

CZX,*PATH PROFILE AS PUT IN FOlLO#S*/)

1512 FORMAT III. THE ~UM3ER OF SCilLED D4TA POINTS HAS EXCEEDED 5000 OR

ITHE NUMBER OF GROUND CONSTANT PAIRS HAS EXCEEDED 50*'

1513 FU~~4r C*OAOJUSTED PATH PROFILE FCllO~S.'Z3X,. 0 IN HETERS*,3Xt

I+HT IN "EYERS*I

1514 FORMAT I1Hl)

1515 FORMAT 1+0*12,* OISTANCE A~O GROUND CONSTANT PAIRS FOlLOW*'

l11X,*0 IN KH*,3X,*SIGMA+,ZX,*EPSIlON*1 CC10X,F~.3,F8.4,F9.0)1

1516 FO~~~T CfB.2,FB.b,FB.2,IZ'3F8.Z,ZFB.6)

FTOM '" .3048

i;>EIIIJ 1502, 10

PQ[NT 1514

PkINT 1503, 10 ~EAQ 1510, N.]XUNITS,IlUNITS,REFEl

If CIlUNITS .E~. 11 PEFEL-PEFEl*FTOM

IF l'l .GT. 500'» GO TO 110

PRINT 1511, N,~F;EL

"QEAD 1~C5, IJCII,ZIII, l-l,NI

IlRINT 1504, IXII" Z(I" I- 1, Nl

[F ('l .LT. 1) GO TO 1£0

PRINT 1513

XCljt'5 T"lCCO.

IF C!XUNlTS .EO. U XCO~ST-1609.3

lCLllljT=l.

[F CllUNITS .EQ. 11 lCONSr-fTOM

IJO 105 I • 11 N

Z[II"llll*lCONST-R~FFL

HII=)((lltXCONST

PRINT 1506, I, X(II, lCI)

105 COf.lTIt.lUE

lZ0 CONTINUE

~EAf) 1510, NGC

IF Ir-.GC .GT. 50) GO TO 110

4N"Z.t3.14l59?654+FREO/299.1925

00 130 I-I,NGe ~EAO 1516, O)(I),SIGX(I),EP5XllltISLAB,T(I),EHII),EVIII,SH(I),

CS,-"(1 ) E1AXIII"C~PlXIEPSXII),-11q15 ••S1GXII)/FREQi

It I1SlAB .HE. 0) GO TO 125

DELTAX(II-CSQWTIETAX(I)-l.)

OElTAX(I)-OElTAXIII/FTAXIII

IF II .Ee. 1) GHA"'fHIHA)

GHPCII"FHIHARI

76
GO T'l 130

125 CONfINUE

EHCzCMPlXlEHl(t,-17975 •• SHllt/F~EOt

EVC=CMPlXlEVllt,-17975.+SVCI)/FREO)

AK-EHClEVC

V-CMPlX(O.,~M)·CSORTCEHC-AKt

U1-C$ORTlEHC-AK)'~HC

02 zCSORTlETAXCll-1.)/ETAX11)

TVT-CTANHlVtTlIJ)

OELT~X(I)-Dl·llD2+01.TVT)/CD1+D2.TVT)'

IF (1 .NE. 1) GO TO lZ7

IF (flIt .Gf. HA) GO TO 126

GHA"fHlHA,

GU TO 127

126 CONTINUE

R-CDI-OZ'/lOl+OZ'

GHAzFIH lHAl

127 COI'ITINI.IE

IF nIl) .GT. HHt GO TO 126

GHRllt-FHlH~Il'

GO TO 130

128 COt>lTINUE

pz I01-02t/I01+0Z)

GHj;' II )-FHTCHARt

130 DXlI)-OXII)t1COO.

RFTl'!HI

110 em'll INUE

P!llHT 1512

END

COMPLEX FUNcrIO~ CTANHlzt

CO'1PlEX Z,t)

U·CEllPIZ.tzt

CTANH-lU-l.'/lU+1.t

END

77
A sample output for program WAGSLAB is given below:

JNNERINGI:N ra 80HINGE~

NU!'IBER OF P~I)F-ll E DATA f'OINT) IS 115

REFERENCI: I:ll:~ATION IS 810.00 METERS

P4fH PROfIlE t,S PUT IN FOLLOWS

0.00000 810.0 .53000 820.0 .80000 827.0 1.00000 620.0

1.40000 800.0 2.55000 180.0 3.35000 71)0.0 3.60000 735.0

3.8000() 760.0 4.95000 760.0 5.80000 780.0 6~20000 780.0

8.80000 780.0 9.00000 768.0 10.05000 780.0 U.30900 180.0

1l.6500() 740.0 11.75000 100.0 11.95000 700.0 13.10000 120.0

13.600CO 730.0 14010000 720.0 15.40000 140.0 1&.20000 1&0.0

11.2,000 180.0 11.8(001) 780.0 18.0COOO 790.0 18.15000 800.0

B.OtlOCO 800.0 19.~0000 740.0 19.80000 770.0 20.20000 7,.0.0

20.40000 740.0 20.80000 160.0 22.00000 760.0 22.10000 780.0

23.05\l()0 800 •.0 23.20000 820.0 23.60000 820.0 24.08000 7ao.o

24.30000 B20.0 24.~5000 820. o· 25.15000 800.0 25.90000 &"0.0

26.05000 640.0 26.2'0000 720.0 26.60000 670.0 2&.80000 705.0

Z7.100CO 620.0 21.30000 700.0 27.10000 &00.0 28.30000 520.0

28.70000 500.0 29.00000 480.0 29.25000 "80.0 29.35000 500.0

30.00000 500.0 30.20000 4" (). 0 30.60000 440.0 30.85000 lt20.0

31.35000 400.0 32.30000 400.0 32.45000 380.0 33.75000 380.0

33.(11)000 360.0 35.20000 3"0.0 36.8COOO 340.0 36.90000 3&0.0

37.60000 360.0 37.65000 380.0 38.10000 380.0 38.50000 360.0

36.65000 34C.O 38.75000 320.0 39.30000 320.0 39.50000 380.0

39.80000 420.0 'tl.OOOOO 420.0 41.93000 440.0 43.75000 HO.O

44.10000 460.0 44.40000 "80.0 1t4.75000 480.0 45.00000 500.0

45.35000 500.0 45.05000 4&0.0 "5.<15000 400.0 It&.10000 370.0

"6.60000 440.0 47.05000 420.0 47.25000 "40.0 41.55000 ,.60.0

4S.tGOOD 460.0 48.65000 440.0 "9.30000 380.0 lt9.50000 400.0

4<1.95000 (ti5.0 5').30000 400.0 50.55000 380.0 50.75000 380.0

50.95000 400.0 51.65000 400.0 51.75000 380.0 5Z.Z0000 380.0

52.30000 400.0 52.55000 420.0 53.65000 420.0 5,..00000 400.0

54.35000 400.0 54.8-5000 "20.0 55.00000 440.0 55.55000 4'25.0

55.<10000 440.0 56.15000 460.0 56.63000 "65.0

ADJUSTEO PATH PROFILE FOllOWS

0 IN !'lE TE RS HT IN METERS

1 0.00000 0.00000

2 530.00000 IO.OCOoo

3 800.00000 17.0000C

4 1000.00000 10.00000

5 !'tOO .OOOCO -10.0COOO

6 2550.00000 -30.00000

1 3350.00000 -50.00000

8 3000.00000 -15.COOOO

q JBoe.ooooo -50.ceooo

10 ~q50.0JOOO -50.00000

11 580J.00000 -30.0COOO

12 6200.00COO -30.00000

13 \.lsoo.QOOOO -30.00000

14 '1000.00000 -42.00COO

15 1)050.COOOO -30.0COOO

16 11300.00000 -30.('COOO

17 1165C .00000 -1:>.00000

18 11 7%. OOOCO -HO.OCOCO

19 U95C .OOCOO -110.COOOO

20 1110!J.OUO()(j -90.00000

21 13()OU. :)0000 -80.00000

22 1~10C.JOCOO -JiJ.orcoc

23 Ij4'JO.JOOCC -7] .OCO()(i

24 1'>200.00000 -50.0000e

(5 172';)1.) .')OCOO -3C.CC00C

26 t 7 .3 'J {) • (' (' (' 01.1 -30.0CC'OL

27 l':lOOO.OOOOO -20.00000

26 16750.00000 -10.00000

29 1'1000.00000 -10.COOOO

30 1'1400.00000 -70.00000

78
31 1'1800.00000 -40.00000

32 20200.00000 -70.00000

33 20400.00000 -70.00000

34 20800.00000 -50.0('000

35 22000.00000 -50.00000

36 22700.00000 -30.00000

37 ~3050.00000 -10.00000

38 23200.00000 10.00000

39 23600.00000 10.00000

40 24060.00000 -30.00000

41 24300.00000 10.0eooo

42 24850.00000 10.00000

43 25150.0000C -10.0COOO

44 25900.00000 -170.0COOo

45 26050.00000 -170.00000

46 26200.00000 -90.00000

47 26~OO.00OOO -140.00000

48 26600.000CO -105.00000

49 27100.00000 -190.00000

50 21300.00000 -110.00000

51 27700.00000 -210.00000

52 28300.00000 -290.00000

53 28700.00000 -310.00000

54 2'iOOO.00OOO -330.00000

55 2'1250.00000 -330.00000

56 2'} 350.00000 -310.00000

57 30000.00000 -310. 00000

58 30200.00000 -370.00000

59 30600.00000 -370.00000

60 30650.00000 -390.00000

61 31350.00000 -410.00000

62 32300.00000 -410.00000

&3 32450.00000 -430.00000

64 33750.00000 -430.00000

65 33900.0000C -450.00000

66 35200.00000 -470.00000

67 3"'800.000CO -470.00000

68 36900.00000 -450.00000

69 37600.00000 -450.00000

70 31850.00000 -430.00000

71 H100.00COO -430.00000

72 33500.00000 -450.00000

73 36650.00000 -470.00000

74 38750.\.10000 -4'l0. cceoe15 3HOO.OOOOC -490.00000

76 39500.00000 -430.0COOO

77 3/600.0000e -390.00000

18 HOuC.00000 -31i).OOOOO

79 41930.00000 -370.00000

80 43750.00000 -371J.00000

81 44100. (lOO 00' -350.00000

82 44400.00000 -:330.00000

63 44750.00CCC -330.COCOO

t14 45:>00.')0000 -310.00000

85 4;]50.00000 -3l).or-ooc

IH) 4,650.'JCCOO -350.00000

e7 4:i95().OOCOO -410.0CGDl:

88 46100.00000 -440.00000

89 4'l600."'('C(lO -370.0(,000

90 't 7')50 .O(''.IOC -]90.00COO

91 47250.00000 -370.00000

92 47550.00000 -350.00000

93 4·'3600.00000 -350.00000

94 4t1650.00')OC -370.0('000

95 49300.00000 -430.00000

96 49500.00000 -410.00000

97 49950.00000 -395.00000

98 5HOO.ooooo -410.0COOO

'l9 505!10.00000 -430.00000

100 5J750.0000e -430.00000

1,n 5(H50.0000C -410.00000

102 51650.00000 -41.o.ceooo

103 51750.00000 -430.00000

104 52200.00000 -430.00000

105 52300.00000 -410.0000C

79
106

107

108

109

110

111

112

113

114

115

52550.00000

53650.00000

54000.00000

54350.00000

H1:l50.0000C

nooo.OOOOO

55550.00000

j5900.00000

5H50.0000f)

50630.00000

-3QO.CCOOO

-HO.OOOOO

-410.00000

-410.00000

-390.00000

-370.00000

-365.00000

-.370.00000

-no.oocoo

-345.00000

80
FREOUENCY • 2.00 ~ERrIC~L POLARIZATION EARTH RADIUS· 6500. KM

TRAN5"ITT~R 'NTENNA HEIGHT - 0.000 ~ETERS RECEIVER ANTENNA HEIGHT - 0.000 METERS

DISTANCf SIGil EPSILON SLA~ SH SV EH EV HIPEDANCE GHR

00

f-'

500.

1080.

3510.

5080.

6750.

7320.

8400.

9110.

10770.

11560.

11960.

12920.

19460.

20280.

25030.

28100.

29~40.

30 It 60.

32630.

33680.

345!10.

35750.

~6~20.

37700.

38750.

39360.

39120.

"l'HO.

41; 00.

488C:).

5lF,7r.J.

520 u",.

56630.:

.010QOO

.0lJOOO

.onooa

.0IOOOtl

.01;)000

.01JOOO

• () 10CCO

.010000

.01:>000

.01)000

.01000U

.OLOOOO

.('UOOO

.010000

.01)')00

• OUOO\)

.CIOUOO

• ul00JO

.01'JOOO

.01'Jl.lOO

.010COO

• QUOOO

.OlJOJO

.010000

.OlJOOv

• ~J1:j J00

• Cl~utHJ

• ('1:)(00

• () 100,)')

.Olooe:>

.vIJCOJ

• 'J})'J\JO

.01000J

10.00

10.00

10.00

10.00

10.00

10.00

10.00

10.00

10.00

10.00

10.0C

10.00

10.00

10.00

lO.OC

10.0C

10.00

lo.eo

10.00

10.00

10.00

lO.oa

IO.C\)

10.C)O

1(). ::Jl)

1) .CO

IO.v\)

I). \1

!J.t:r;

!Q.OC

10. :)0

Iv. '..<J

10.00

O.

20.

O.

20.

O.

20.

O.

20.

O.

20.

O.

10.

O.

20.

O•

20.

O.

20.

O.

20.

O•

20.

10.

O.

10.

O.

')(1

L'" •

I).

20.

o• 7.0.

::'.

20.

0.000000

.00011)0

0.000000

.OCOI00

0.000000

.000100

0.000000

.000100

0.000000

.000100

0.000000

.000030

o.oeoooo

.0eOlOO

O.OOOOJO

.0e0100

o.OCOOOO

.OC0100

0.000000

.0l'0100

o.ooocoo

.000100

.000030

0.000000

.000030

o.cceQue

.OCOIOO

.;) .I.:CO.))O

.OCQue

o.lj(COOO

.0001)(,

Ci.e'\JO\)')(.

.000100

0.000000

.000100

0.000000

.000100

o.oocooo

.000100

0.000000

.000100

0.000000

.000100

0.000000

.000030

0.000000

• OOClOO

0.000000

.000100

0.000000

.OJ0100

0.000000

.000100

0.000000

.000100

.0-)0030

0.000000

.000030

0.000000

.GOOIOO

v.Geooua

.0)00100

o.OOCOOQ

.oaGIOC

o.O)COOO

.000100

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.82

0.00

1.10

0.00

1010

0.00

1010

O.co

1.10

0.00

1010

1.82

0.00

1.82

0.00

1.10

0.00

1010

0.00

lole

c.oo

1.10

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.10

0.00

1.82

0.00

1.10

0.00

1.10

0.00

1010

O.OC

1.10

0.00

1.10

1.82

0.00

1.82

0.00

1010

0.00

1010

0.00

1.10

O.O'l'

1.10

.0787

.5442

.0787

• 54ft 2

.0787

• 54~2 .0787

.5442

.0787

• 541t 2

.0787

.1435

.0787

.5442"

.0787

.5442

.0787

.5H2

.0787

.5442

.0787

.5442

.1435

.0787

.1435

.0787

.544 " .0787

.5442

.0737

.5442

.0797

• 51,42

.0697

.2888

.0697

.2888

.0697

.2888

.0697

.2888

.0697

.2888

.0697

.2748

.0697

.2888

.0697

.2688,

.0697

.2a88

.0697

.2888

.0697

.2888

.2748

.0697

.2748

.0697

.2888

.0697

.2886

.0697

.2868

.0697

.2888

1.0000

.6644

1.0000

.6644

1.0000

.6641,

1.0000

.664ft

1.0000

.6641,

1.0000

.6104

1.0000

.6641,

1.0.000

.6641,

1.0000

.6641,

1.0000

.(;64ft

1.0000

.6644

.6104

1.0000

.6104

1.0000

• f,644

1.0000

.66""

1.0000

.6644

1.0000

.6644

0.0000

.1684

0.0000

.1684

0.0000

.1684

0.0000

.168ft

0.0000

.1684

0.0000

.0286

0.0000

.1684

0.0000

.lba4

0.0000

.1684

0.0000

.1684

0.0000

.1684

.0286

0.0000

.0286

0.0000

.1684

0.0000

.1664

0.0000

.1684

0.0000

.1684

GHA- 1.0000 0.0000

X

(I'll

z

(M I MAG

FlXl

ARG MAG

FHlXl

ARG

TIMING fIELD STR

(SECI (08UJ

8TL

200.00

400.00

600.00

800.00

1000.0C

3.8

7.5

15.2

27.3

27.2

.96930E+OO -.38983E+00

.95300E+00 -.55580E+OO

.95266E+00 -.69678E+00

.Q7126E+00 -.83019E+00

.94560E+00 -.91~92E+OO

.96930E+00 -.38983E+00

.95300E+00 -.55580E+00

.65300E+00 -.I,4850E+OO

.6&515E+00 -.581Q1E+00

.64816E+00 -.66661,E+00

.180

.181,

0185

.181

.189

126.65

120."8

113.68

111.35

109.18

18.74

24. en

31.11

34.04

36.21
1200.0C 18.9 .96455E+00 -.25096E+01 .96455E+00 -.Z5896E+01 .200 111.05 34.34

1400.Clj 7.3 .84870E+00 -.26652E+01 .84870E+00 -.Z6652E+01 .211 108.60 36.80

11)00.00 2.1 .7b305E+OO -.26492EtOl .76305E+00 -.2649Z£+01 .224 106.51 38.88

1800.00 -3.1 .69663E+00 -.26693E+01 .69663£+00 -.26693£+01 .237 10tt.70 ttO.69

2000.00 -8.2 .M87lEtOO -.26960E+01 .64871E+00 -.26960H01 .253 103.16 1t2.23

2200.0( -13.4 .61157E+OO -.27255E+Ol .61157£+00 -.27255£+01 .269 101.82 43.57

2400.00 -18.6 • 58130'E+O\l -.27562£+01 .58130E+00 -.27562£+01 .285 100.63 44.76

2600.00 -24.2 • 55178f: +00 -.27955E+Ol .55176£+00 -.27955E+Ol .304 99.tt8 45.91

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50000.00

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53000.00 -551.2 .66578E-OZ -.31169E+01 .45636e-02 -.Z8686E+01 61.298 31.64 113.75

5320').00 -551.6 .64807E-02 -.31169E+01 • 4ft422 e-o 2 -.28687E+01 67.916 ,,1.38 114.01

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53600.00 -552.3 .627B51:-02 -.31079£+01 .43036E-02 -.28596E+01 69.152 31.04 114.35

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FORM NTIA-29 U.S. DEPARTMENT OF COMMERCE

(4-80) NAT'L. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION NO. 2. Gov't Accession No. 3. Recipient's Accession No.

NTIA Report 82-114

4. TITLE AND SUBTITLE 5. Publication Date

HF Ground ~~a ve Propagation over Forested and November 1982

Built-up Terrain 6. Performing Organization Code

7. AUTH~R(S) 9. Project/Task/Work Unit No.

D. . Hill

8. PERFORMING ORGANIZATION NAME AND ADDRESS

U. S. Department of Commerce 9104493

National Telecommunications and 10. Contract/Grant No.

I'nformat~on ~dTinistration. !~nst'tuta or e Taommup'oca8,sn Sciences 2:> Broa [way. BOil er 8 3 3

11. Sponsoring Qrganization Name and Address 12. Type of .Report and Period Covered

Llefense Nuclear Agency NTIA Report ~Jashington, DC 20305 13.

14. SUPPLEMENTARY NOTES

15. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature

survey. mention it here.)

An integral equation method i s presented for computing the vertically

polarized field strength over irregular terrain which is covered with

forest, buildings, or snow. The terrain cover i s modeled as an

equivalent slab, and a general computer code, WAGSLAB, is developed.

Numerous special cases are treated analytically, and comparisons

are made with numerical results from WAGSLAB.

16. Key Words (Alphabetical order, separated by semicolons)

flat earth; HF ground wave propagation; irregular terrain; mixed

path; lateral wave, slab model; spherical earth; vertical

polarization

17. AVAILABILITY STATEMENT 18. Security Class. (This report) 20. Number of pages

Kl UNLIMITED. Unclassified 100

19. Security Class. (This page) 21. Price:

0 FOR OFFICIAL DISTRIBUTION.

Unclassified

-tru.s. GOVERNMENT PRINTING OFFICE:1983-677-389 I 93