AFCRL-994

PB 161633

zBoulder laboratories

^o. 132

EVALUATION OF CONVOLUTION INTEGRALS

OCCURING IN THE THEORY

OF

MIXED PATH PROPAGATION

BY

J.R.JOHLER AND CM. LILLEY

SCIENTIFIC REPORT NO. 14

Contract Title: Antenna Patterns in the Presence of Finite Ground Surfaces,

CSO and A 58-40, dated March 31, 1958; Pro-61 -568, dated February 2, 1961

Prepared for Electronics Research Directorate

Air Force Cambridge Research Laboratories

Office of Aerospace Research

United States Air Force

Bedford, Massachusetts

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NBS Project Date of Report

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AFCRL-994

EVALUATION OF CONVOLUTION INTEGRALS OCCURRING IN

THE THEORY OF MIXED PATH PROPAGATION

by

J. R. Johler and C. M. Lilley

Scientific Report No. 14

Contract title: Antenna Patterns in the Presence of Finite

Ground Surfaces, CSO and A 58-40, dated March 31,

1958; Pro-61-568 dated February 2, 1961.

Prepared for

Electronics Research Directorate

Air Force Cambridge Research Laboratories

Office of Aerospace Research

United States Air Force

Bedford, Massachusetts

The research in this document has been sponsored by the Electronic Research Directorate of Air Force, Cambridge Research

Laboratories, Office of Aerospace Research (USAF) Bedford,

Massachusetts. The publication of this report does not necessarily constitute approval by the Air Force on the findings or

conclusions contained herein.

NATIONAL BUREAU OF STANDARDS

technical <V£ete

No. 132

EVALUATION OF CONVOLUTION INTEGRALS OCCURRING IN

THE THEORY OF MIXED PATH PROPAGATION

by

J. R. Johler and C. M. Lilley

November 8, 1961

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

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

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CONTENTS

Page

ABSTRACT " ~~f~

1. Introduction i

2. Theory 2

3. Computations 13

4. Conclusions 13

REFERENCES 14

EVALUATION OF CONVOLUTION INTEGRALS OCCURRING IN

THE THEORY OF MIXED PATH PROPAGATION

by

J. R. Johler and C. M. Lilley

ABSTRACT

The theory of propagation of electromagnetic waves around a sphere treats the smooth homogeneous case, i. e. , the case in which

the surface impedance of the sphere is uninterrupted by an abrupt

change in conductivity such as a land/sea boundary. It is known,

however, that such a theory can be extended to treat inhomogeneous,

irregular terrain by formulating certain convolution integrals which

utilize the smooth homogeneous formulas. The evaluation of these

integrals can be accomplished with dispatch on a large-scale

electronic computer with the aid of numerical analysis techniques.

The particular case of a land/sea boundary in a smooth, spherical

surface is illustrated for a variety of cases by evaluating the convolution integrals on a large-scale computer.

1. Introduction.

The theory of propagation of electromagnetic waves around a sphere usually assurr.es the earth to be smooth and homogeneous.

However, some extensions of the theory have been developed by

Monteath [1951] and Huffor d [1952]. More recently, Bremmer [1954]

developed the plane earth theory for abrupt conductivity changes and

the procedure was established in detail for plane and spherical

earth theory by Wait [1956, 1957, 1961]. The formulation of Wait [1961]

also developed procedures for the cases of transmitting and receiving

terminals at very great heights. Furutsu [1957] has also developed

methods for evaluating the field over sections of inhomogeneous

ground and in particular has developed a method especially convenient

for electronic computer computations in which the length of the

sections of inhomogeneous ground are large.
-2-

The formulation of this problem by the above mentioned authors,

especially Wait [1961 ], comprised convolution integrals involving an

integrand which can be calculated with the Watson [1918] type series

of residues. This paper demonstrates a method of evaluating these

integrals by an application of numerical analysis technique.

Detailed numerical calculations for the particular case of the land/

sea boundary are presented.

2. Theory. The analysis of the propagation over a finite surface can be more

conveniently formulated from the smooth, homogeneous earth theory

by utilizing the attenuation factor, F, directly instead of the field

(vertical electric field, E , for example), where

r E = 2 E F (!) r pr

and the attenuation factor, F, is defined as the ratio of the total Hertz

vector, n to the primary Hertz vector n , pr

F = F (co,d) = n/2n (2)

V

where the radiation field,

E = I I 10~ 7u) exp (iwt-ik , d)/d (3)

pr o 1 '

II = exp (iwt-ik d)/(-ik d) . (4)

The factor, F, can be evaluated with the aid of the series of

residues, [Bremmer, 1949], and in particular can be written,

[Johler, Kellar and Walters, 1956],

F = Ztrot

,, * 3 . I 2 \ s

x r

(k a) d/a yL [Zt - 1

' [2r

s 1/6 ] s = o

Xexp-j-i

s = 0, 1, 2, 3

,. .3 3 da adoca tt i ._.

( k

! a» Ts a I

+ JK +

4 i

' (5)
-3-

where t = time, seconds, a is the radius of the spherical earth along

which the vertical electric field, E ,is directed, kn is the wave number

r 1 of air at the surface of the earth, k = co r|,/c and n, is the index of

refraction of air, tu ~ 1 (c = 2. 997925( 10^ m/s, r\ ~ 1. 000338 at the

surface), or is a factor which accounts for the vertical lapse of the

index of refraction of air a ~ 0. 75 to 0. 85. h and h are the heights

of the transmitter and receiver respectively and fg(h), h= h , h , are

J. u the height gain factor (identical for either receiver or transmitter as a result of reciprocity). The conductivity dielectric - constant parameter, 6 = 6 emor 6 = 6 for vertical electric or vertical magnetic 5

polarization of the Hertz dipole, respectively, can be written

[Bremmer, 1949],

6 „ = o 1, (6)

2 .k

2

(k a) 3 |~ 2

t

- 1

6=^6. m (7)

, 2 e k

2

The wave number of the ground can be written,

(8)
-4-

The residues of the series (5) are defined by the special roots, t ,

of the differential equation of Riccati which has been formulated

[Johler, Walters and Lilley, 1959]

?-25Z t+1= 0. (9) dT

The height gain factors, f (h) = f (h.) or f (h ) are determined as

the ratio of two Hankel functions, H (z), [Bremmer, 19491,

v

(10) f s <h

' 'ft

\

+ * <y>

(2)

\

where

r = a +h

and

l T

(11)

v ~k,a +(k,a) t , (12)

s 1 1 s

which can be approximately written [Johler, Kellar and Walters,

1956] in terms of the tabulated Furry [1945] modified Hankel functions,

i

2 l ( Z

) I (k

i a) ""

S J

f (h) = (13)

/ r „,f < 2

> ' T

s

provided h <<C a »
-5-

The special case of a boundary (such as a land/sea boundary)

located a distance, d , from the transmitter located on land with

conductivity, cr and dielectric constant e

, or wave number of the

1 2,1

ground, k , is considered. A second finite ground surface located

around the receiver from the distance cL to d or the same boundary

located a distance, d , from the receiver such that d = d.. + d with a

conductivity cr , a dielectric constant e

. The field of the

ground wave at the receiver' is then, for a plane earth [Bremmer,

1949], and for a spherical earth [Wait, 1958, 1961 ],

E = 2 E F

r pr

(1,2)

(14)

where

F^^F^IM),

F<

1 ' 2

> = F

(1 >(d) T

ik

i d

M 2tt

z'

2

» -z

(1)

Z J

h. „(i) , , . „(2) F (d-p) F (|

'dp

NTp(d- P r (15)

where,

(d-d

2

>o)

Z = ^o = cm ~ 376. 7

o

(16)

ohms, the impedance of space,
-6-

k. =

CO

1

' (17)

Z*

1

* = Z

o k

2,1

1 -

'2,1

(18)

z

(2) = z

v-i

° k

2,2

1 -

'2,2

CO 1 o

:,l ~c [

e 2,1 "

1 w

I aj c

2

-, 2

(19)

CO '2, 2 c2 o

2, 2 CO

'2, P c[e

2,

- i

2 1

o- H c n 2

P o

CO

(20)

where the integer p, identifies the particular finite surface under

consideration. Thus, the smooth, homogeneous, spherical earthattenuation functions, F (d) = F (co, d), F (d-p) = F (co, d-p),

(2) (2) F (p) = F (co, p) are employed in the calculation of the attenuation

(1 2) functions F (co,d), for mixed propagation i. e. , propagation over

finite surfaces. The integral of the combination of the two attenuation

functions F (d-p)/\/d-p and F (p)/\fp appearing inside the

integral (15) is by convention called the convolution of these functions,

* ot

i.e., F

][ F

2 =

J F^t-r) F

2

(t) dr.

o'
-7-

An alternate form (of equation (15)), can be written,

d.

ik,d r

„(l) „(2)

(d) -

1

2tt

- Z

(1), v „(2)

r f^; ( P)f xw/ (d-p)

o V P (d-p)

dp

(21)

(d > d

x

)

In either case, equations (15, 21), the problem is clearly resolved as

the summation of the particular series of residues associated -with the

Watson [1918] transformation for the F, with various arguments

. ... (1) -U),- ^(2) (2) r

(l)

/ ,, _(1)., . function, F (p), F (d), F (d-p); or F (p), F (d), F (d-p),

whichever is appropriate; and the evaluation of a finite integral

between the limits and d or and d , whichever is appropriate.

This is accomplished quite readily with a Gaussian quadrature,

[Kopal, 1955]

1,2

f

J

f (p) d (p) = M

X

L

m=l

W f (p ) + e (M), m m

(22)

where e (M) is an error term which can be made arbitrarily small

by increasing M, and,

W = - dT _ H , m = 1, 2, 3 . . . M, m 2 1, 2 m

p = — d x + — d . m 2 1, 2 m 2 1, 2

(23)

(24)
-8-

The x 's are the Gaussian abscissas, and M determines the m

number of values of the integrand to be used in the quadrature. The

integral can be split into any number of finite integrals to get

convergence,

d a a d

1,2 1 2 1,2

j

f (p) dp = |f (p) dp + j* f (p) dp +... + J

f(p) dp. (25)

o o a. a

1 n

The quadrature problem is therefore the evaluation of a finite

integral,

b M

\ F (x) dx = ") a m=l

V

L.

where,

X = \ [ (b-a) x +(b +a) ]. (27) m 2 m

The Gaussian weights and abscissas can be determined from

the following:

1 M

\ f (x) dx = J

-1 m= 1 L

W =\ (b-a) H . (29) m 2 rn
The x 's are the roots of the Legendre polynomials defined by, m

±— (x

2

-l)

m ,2 mmi P (x), (30) m m

d

X

P (x) = 1

P

x (x) = x

^-hl -\

... . and,

(mv +1) P ,(x)+mP ,(x) = (2 m +1) x P (x). (31)

' m + 1 ' m-r '

s ' rri '

Upon determination of the roots, the weight coefficients, H , of m

the corresponding quadrature formulas are evaluated as follows:

Hm

(1-x

2

) [P (x )]

2

(32) m m m

Forty-eight Gaussian weights and abscissas were used in the

functions graphed in this paper, convergence being obtained by increasing the number of intervals of integration. These weights and

abscissas have been published by Kopal [1955], Davis and Rabinowitz

[1956].

The singularity, equation (21) for example,

1 l [p(d-p)]" 2 = [d(d-d

1 )]' 2,(d = d

1 ),

would apparently present some difficulties in numerical integration.
-10-

However, it should be noted that the Gaussian quadrature, equations

(24, 25, 26) does not include the end points of the interval of

integration. A 1 os s of precision, however, could possibly occur,

but this could be sidestepped by the use of multiple precision. In

any case, this did not become a problem for the examples presented

in this paper.

The Gaussian quadrature techniques do not utilize the values of

the integrand (15, 21) at the end points of integration. However, close

to zero the series of residues, equation (5), converges slowly and,

hence, the calculation of the attenuation function, F, becomes more

difficult with a considerable reduction in computation efficiency.

Since the Gaussian points, x , tend to cluster, equation (24) tends m

to cluster in the vicinity of d = 0, it is desirable to utilize the

plane earth approximation in this area. Indeed, quite high precision

can be obtained if the computation is preferred both by the series of

residues and the plane -earth theory such that a comparison is made

in the transition region to ascertain the computation precision. The

attenuation for the plane earth theory [Norton, 193 7] is,

2 , k.

2

k

4

F =

|

.' sin T ( 1 +R ) +( 1-R ) 'd +h

2

.2 . , 2 I e e

! -L;+_L sinT f(p)

kk k J

l k

l

cos T

f sin T \ , (33)
-11-

where,

f (p) = 1-i

J up exp (-p) erfc (i \Tp)

if,

ik

l D kl p

l = T — S1nrJ

(34)

(35)

where

-j: 2 2

d + h

(3 6)

then,

p =

p

l {

sin T

(3 7)

where h is the height of either the transmitter, h = h or the receiver, h = h and k and k are the wave numbers of the medium previously defined, equation (18, 19, 20), which can be specified as some

finite surface, k„ . The factor R : 2, p e coefficient for vertical polarization,finite surface, k„ . The factor R is the Fresnel reflection

2, p e

R

e

= R

e

(T) = {

k

2

2

C ° S T/k

l

2

' [k

2

2

" sin T ^

J

/ {

k COS T K

+ [k

2

/k

x - sin t]

Z j

(38)
-12-

where t = tan (d/h). Thus, the height gain previously discussed

for the residue series, equation (10, 12), is built into the plane -earth

thepry for short distances. It is now possible, with the aid of the

convolution integrals (15, 21), to elevate transmitting and/or receiving

dipoles and evaluate the field over mixed ground transmissions.

Indeed, in the limit whereupon the receiver or transmitter is

elevated to several earth radii; equation (10), the antenna pattern

over mixed ground, is described. Furthermore, the convolution

integrals can be developed for three or more finite ground surfaces

[Wait, 1961]. Thus, discontinuities located at distances, d , d +d

J. X .' u from the transmitter or d = d +d and d-d respectively from the

•eceiver, where d = d. +d +d can be written,

d

rft.2.3) r

(D f-T [

z'

2) -z(3) 1 C F

(1) (p)F(3) d-p)

dp

d1^2

d

l +d

2

ikd r y-' ..w I r F'"( p) r F'^(d.p)

F

(Z>

(n- P)F(3) (d-„)

"v/Tn - P) (d-ri)

d

i dp. (39)
-13-

3. Computations , The results of computations to illustrate technique are presented

in figures 1,2,3,4. The amplitude, figures 1,2 illustrate the

recovery effect of a land/sea boundary noted by Millington [1949].

There is, of course, a corresponding phase recovery, figures 3,4.

These computations were carried out on the CDC-1604 computer.

4. Conclusions . The results of this investigation indicate that the problem of

wave propagation over inhomogeneous or finite ground surfaces can

be treated with comparative ease on a large scale computer with the

aid of numerical analysis techniques applied to the convolution integrals and an adequate technique for summing the series of residues.

5. Acknowledgments.

The series of residues and the ground wave computer program

concept was developed by John D. Harper, Jr. and J. R. Johler of

Division 85.10 (Applied Electromagnetic Theory Section) and Harper

programed same for IBM-704 and CDC-1604 computers. The convolution integrals were programed for CDC-1604 by C. M. Lilley of

the same group.
-14-

REFERENCES

Bremmer, H. , Terrestrial radio waves; theory of propagation, Elsevier Publ. Co. , New York, New York,

(1949).

Bremmer, H. , The extension of Sommerfeld's formula for

the propagation of radio waves over a flat earth to

different conductivities of the soil, Physical XX, 441-

460, (1954).

Davis, P., P. Rabinowitz, Abscissas and weights for

Gaussian quadratures of high order, NBS, J. of Research

paper 2645, 35-37, (1956).

Furutsu, K. , J. of Radio Research Lab. (Japan) Vol. 2,

No. 10, eq. (7-1) (1955). See also Vol. 4, No. 16, No. 18,

(1957).

Hufford, G. A. , An integral equation approach to the problem

of wave propagation over an irregular surface, Qunt. Appl.

Math. , 9, 391, (1952).

Johler, J.R. , W. J. Kellar and L. C. Walters, Phase of the

low radio frequency ground wave, NBS Circular 573,

U.S. Govt. Printing Office, Washington, D. C.

(June 27, 1956).

Johler, J. R. , L. C. Walters and CM. Lilley, Low- and

very low- radio frequency tables of ground wave parameters for the spherical earth theory. The roots of

Riccati's differential equation, NBS Tech. Note, No. 7,

PB151366, U.S. Dept. of Commerce, Office of Tech.

Services, Washington 25, D, C. , (Feb. 1, 1959).

Johler, J. R. , L. C. Walters, CM. Lilley, Amplitude and

phase of the low- and very low- radio frequency ground

wave, NBS Tech. Note, No. 60, PB161561, U. S. Dept.

of Commerce, Office of Tech. Services, Washington 25,

D. C. , (June 1, I960).
-15-

Kopal, Z. , Numerical analysis, John Wiley and Sons, Inc. , New York, New York, p. 367, (1955),

Millington, G. , Ground-wave propagation across a land/

sea boundary, Nature, 164 , 114-45 (July, 1949), 163 , 128, (June 1949).

Millington, G. , Ground -wave propagation over an inhomogeneous smooth earth, Inst, of Elec. Eng. , Proc. Ill 96, 53-64, (1949).

Monteath, G. D. , Application of the compensation theorem

to certain radiation and propagation problems, Inst, of

Radio Eng. , London, Proc, 98, Part IV, 1-8, (1951).

Norton, K. A. , The propagation of radio waves over the

surface of the earth and in the upper atmosphere, Part II,

Proc. Inst, of Radio Eng. , No. 9, 25, (Sept. 1937).

Wait, J. R. , Mixed paths ground-wave propagation, short

distances, J. or Research of NBS, 57, No. 1, Research

paper 1687, 1-15, (July 1956).

Wait, J. R. and J. Householder, Mixed-path waves propagation, 2 layer distances, J. or Research of NBS, 59, No. 1, 19-26, (July 1957).

Wait, J. R. , On the theory of mixed-path, ground-wave

propagation on a spherical earth, J. of Research of

NBS, 65D, No. 4, (July-Aug. 1961).

Wait, J. R. , On the theory of propagation of electromagnetic

waves along a curved inhomogeneous surface, Can. J. Phys. 3_6, pp. 9-17, (Jan. 1958).

Watson, G. N. , The diffraction of electric waves by the earth,

Proc. Roy. Soc. , London, 95, p. 83, (1918-19).
-16-

LIST OF FIGURES

PAGE

Fig. 1 Amplitude of the ground wave as a function of

distance for a variety of land/sea boundary

locations, d , illustrating the recovery effect,

f = 1000 kc/s. 15

Fig. 2 Amplitude of the ground wave as a function of

distance for a variety of land/sea boundary

locations, d , illustrating the recovery effect,

f = 3000 kc/s, noted by Millington. 16

Fig. 3 Phase of the ground wave as a function of distance

for a variety of land/sea boundary locations, d , illustrating the recovery effect, f = 1000 kc/s. 17

Fig. 4 Phase of the ground wave as a function of distance

for a variety of land/sea boundary locations, d , illustrating the recovery effect, f = 3000 kc/s. 18
-17-

IU ' §^

1 1 1 | II 1

1 I: 1 | 1 1 1 1 1 1 1 |1 1 1 1:

IO" 5 r S. a =5 N. «2

= 80

J

IO" 6 r a-

-- 0005\

€ Z =I5 \

I

io-'

d, = IOO Mi 4

10"' d,= 200Mi-V"\\ \ 4

rr

IO" 9

IO"

10

IO"

11

IO"

12

IO"

13

UJ

h-UJ

in

d, = 300Mi<\\\

d,=400Mi-"\\\\\ \ - O

> 1 d,=500Mi'T\\\\

WW\\ -

UJ

rr

d, = 600Mi -T\\\

3 l\

Q- r

d,= 700Mi'-T\^

ft

: Hi IO"

14 \\W\u 1

drlOOO Mi - \\11

IO"

15 TRAM^MITTFR d BFrFIVFR \ WW? 1

V^^777^TOWwS5^-V \I 1 ^

IO"

16 V777' LAND SEA

it = 0.005 <r « tg = '5 <2 =

\\I 1

30 \| 1

IO"

17 Mo = 4tt(

f = 1000 k a =085

(V 7) \\\ \

:/s

\ 1 \ m-ia i .1 1 , , ill i ,,!,., ,i\\, \ i , ,,,"

10 100 1000 DISTANCE, d, STATUTE MILES

10000

Figure 1. Amplitude of the ground wave as a function of distance for

a variety of land/sea boundary locations, d , illustrating

the recovery effect, f = lOOO kc/s

.
•18-

50 100 150 200 250 500 DISTANCE, d, STATUTE MILES

Figure 2. Amplitude of the ground wave as a function of distance for a variety of land/sea boundary locations, d, illustrating

the recovery effect, f = 3000 kc/s, noted by Millington.
19-

&

o

<

uj"

</> <

100 iCOO DISTANCE, d, STATUTE MILES

10000

Figure 3. Phase of the ground wave as a function of distance for a variety of land/sea boundary locations, d\, illustrating the

recovery effect, f = 1000 kc/s.
20-

DISTANCE, d, STATUTE MILES

Figure 4. Phase of the ground wave as a function of distance for a variety of land/ sea boundary locations, dj, illustrating

the recovery effect, f = 3000 kc/s.
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Properties. Constitution and Microstructure. Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Digital Circuitry. Digital Systems. Analog Systems.

Applications Engineering.

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

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Molecular Kinetics. Mass Spectrometry.

Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research.

Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High Frequency Electrical Standards. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency ana Time Interval Standards. Electronic Calibration Center. Millimeter-Wave Research.

Microwave Circuit Standards. Radio Systems. High Frequency and Very High Frequency Research. Modulation Research. Antenna Research.

Navigation Systems.

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

Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.
NBS