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CALCULATED DIFFRACTION EFFECTS

AT VLF FROM A LOCALIZED

IONOSPHERIC DEPRESSION

JAMES R. WAIT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

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NATIONAL BUREAU OF STANDARDS

'Cecltnical ^Ylete 20S

Issued January 16, 1964

CALCULATED DIFFRACTION EFFECTS

AT VLFJFROM A LOCALIZED

IONOSPHERIC DEPRESSION

James R. Wait

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

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.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price 15<t

CONTENTS Page

Abstract 1

1. Introduction 1

2. The Uniform Guide ,. 1

3. The Non-Uniform Guide 2

4. Some Concrete Results 5

5. Some Final Remarks 6

6. References 7

7. Acknowledgements 7 Figures 8

CALCULATED DIFFRACTION EFFECTS AT VLF FROM A

LOCALIZED IONOSPHERIC DEPRESSION 1"

James R. Wait

Propagation of VLF radio waves in the earth-ionosphere

waveguide of non-uniform width is considered. The disturbed

region is permitted to be of finite extent. It is assumed that the height variations may be locally represented in terms of a propagation function S(x, y) which is a function of both x and y. Using

first-order scattering theory, calculations are presented for a disturbed region which is approximately rectangular in the horizontal

plane.

1. Introduction

It is now becoming recognized that the propagation of VLF radio waves is influenced by ionospheric

disturbances which occur along all or part of the great circle path between transmitter and receiver.

However, it is not always appreciated that ionospheric perturbation lying off the great circle path may

also have a significant effect. A quantitative estimate of these off -great -circle disturbances is of particular importance in connection with the detection of nuclear detonations. It is known that the x-ray radiation resulting from the explosion will produce additional ionization at D- and E-region

heights of the ionosphere. In this note, the results of some calculations are presented which bear on

this important problem. The detailed derivation of the required equations were given earlier f Wait,

1963]; however, for purposes of completeness, a brief discussion of the plausibility of the working

formulas is given here.

2. The Uniform Guide

In order to achieve tractability, without becoming encumbered in details, some simplifying

assumptions are made at this juncture. It is assumed that the space between the earth, of radius a, and the ionosphere is equivalent to a waveguide which supports only several modes of low attenuation.

In fact, for distances of the order of 4000 km or greater, it is necessary to consider only one mode for

frequencies of the order of 20 kc/s. Thus, at a distance d from the transmitting antenna, the vertical

electric field E is given approximately by [Wait, 1962]

„ . const. . . e

. -r— exp (-ikdSj) , (1) [sin(d/a)] 2

t The research reported here was supported by the Advanced Research Projects Agency under

ARPA Order No. 183-62.
if the height h and properties of the ionosphere did not vary. Here, k = 2 n /wavelength, and thus

k S is the (complex) propagation constant of the dominant mode. For example, the phase velocity

is (c/Re S,) where c is the velocity of light and the attenuation rate is (-k Im S,) nepers per unit

length. In what follows, the subscript- 1 on S is dropped.

3. The Non-Uniform Guide

In a first approach in treating the influence of variable ionospheric heights, one generalizes

equation (1) to allow for the dependence of S on the distance x, along the great circle, from the

transmitting antenna. Thus, [Wait, 1961]

E m

const. ^ expC-ik f S(x) dx) . (2) [sin(d/a)]2 J

q y

As indicated previously, this is a valid representation for an earth-ionosphere waveguide whose width

varies slowly along the great circle path. Obviously, such a simple formula ignores variations in the transverse or y direction. In particular, if the perturbed region lies off the great circle path, the above formula for E would give no information concerning the corresponding modification to the

field.A somewhat heuristic approach to the general problem is based on the concept that the integral in the exponent of equation (2) is the resultant of the individual contributions from the whole x, y plane.

S(x, y), which is regarded as a function of both x and y, will contribute to the modification of the

field whenever it departs significantly from the undisturbed constant value S . This suggests, with

a certain amount of hindsight, that the field under disturbed conditions should have the form

E = E°expf-ik \ 0(x)dxj ,

x,

(3)

where E is the field under undisturbed conditions and £2 (x) is a function of x, yet to be determined.

Here, the coordinates of the transmitter and receiver are (-x ,0) and (x1 ,0), respectively. In view

of the remarks made above, Q, (x) must involve an integral over the transverse direction y. Also,

the integrand must be weighted by S (x, y) - S which is the "local contrast;" and furthermore,

account must be taken of the different electrical lengths of the paths connecting transmitter and re- ceiver. After some consideration, it is found that [Wait, 1963]

A y? (x) 0(x) = Q-) a \ [S(x.y) - S°] exp(-ia2 y')dy , (4)

Yi( x)

where the y integration extends over the interval yx to y3 which encompasses the ionospheric

perturbations. In this equation

k S° r (X

> Xj

L(x - x){xl +x)J (5)

-2-
As shown before, [Wait, 1963] in a full analytical treatment of this problem, the factor -i <x s y

5

is the correction for the phase which retains only second-order terms in y. Also, it should be

mentioned that the form for a , given above, is strictly valid only on a flat earth. However, earth

curvature changes this factor by only a small amount, and for present purposes, this correction is ignored.

It is immediately evident, from the form of equation (4), that if S(x, y) is independent of y and

if (-Yi) an<i (* y?) are sufficiently large,

i +

2 n (x) s (~7 a

)

[ S ( x '°) - S°] exp (-i a

? yp

) d-

- oo

m S(x, 0) - S° . (6)

Inserting this value into equation (3) gives the required form of the modified field E when the ionospheric disturbance is effectively infinite in the transverse direction.

In order to present numerical data in a convenient form, it is desirable to normalize the results

to the ionospheric depression which has an infinite transverse direction. For example, the E field,

under disturbed conditions, may be written — = exp[-ik \ fi(x)dx] , (7) E -'6/2

where E is the corresponding field for an undisturbed ionosphere and 6 is the range of x which

encompasses the perturbation. Now, if the function S(x, y) - S is independent of y, it follows

that

6/2

Eq

= -^ = exp [-ik f [S(x) - S°] dx] (8) E

-6/2

The "normalized field anomaly" NFA may then be defined by the ratio

log(E/E°) —S-5— „_.

' = NFA . (9) log (E^/E )

It is obvious that NFA approaches one when the perturbation has an infinite transverse dimension.

Thus, a quantitative evaluation of NFA gives a good index of the effectiveness of an ionospheric dis- turbance of finite width.

For purposes of illustration, S(x, y) and S may be regarded as real. (In actual fact, they

have small imaginary parts.) Furthermore, it is assumed that S(x, y) differs from S only over

a rectangular region bounded by y: < y < ya and - bfl < x < b/Z as indicated in figure 1.

.3-
FIGURE 1 - PLAN VIEW OF SIMPLE IONOSPHERIC

PERTURBATION SHOWING LOCATIONS OF

TRANSMITTING AND RECEIVING ANTENNAS

(SCALE EXAGGERATED IN y DIRECTION)

Also, in order to achieve more simplicity, it is assumed that the difference or contrast function

S(x, y) - S is constant over this rectangular region. It is then found that

where

and

with

log(E/E°)

logtE^/E )

0^0* z

i = a

' • C-r) (~hr)

- -A « &M..-lM, *«]

2 ,XT + X 2

(10)
In obtaining equation (10), it has also been assumed that x and xx > > 6 , and S is replaced byunity.

4. Some Concrete Results

The normalized field anomaly NFA, as defined above, is a complex quantity. Its real part may

be defined as the normalized phase anomaly, NPA, while the imaginary part is defined as the normalized amplitude anomaly, NAA. Specifically,

NPA

and

where

= -|-[C(Z 2

) - CfZj) + S(Z

2

) - S^)] , (11)

= -y [S(Z

2

) - S^) - C(Z2 ) + CfZ^] , (12)

z z C(Z) = \ cos(^-tp ^)dt, (13) and S(Z) = \ sin f-y t ? ") d t ,

NAA

(14)

are Fresnel integrals [Born and Wolf, 1959].

The quantity NPA is a measure of the modification of the phase of signal ratio E/E . One

should note that

log E/E° = log |E/E°| + iarg(E/E°) , (15)

while

lo 8 E /*£ = iargfE^/E ) , (16)

since | E /E | is essentially unity when the imaginary part of S(x) is neglected. Thus

arg(EM

/E°)

.„,, arg (E/E°) NPA s —» S ' '—

, (17)

and

log |e/e° | i-Ie/e°| NAA s - —2—I—

1- a '—>— . (18) arg (E^/E ) argfE^/E )

To show numerical calculations, the following special case is considered; namely, x = xa = d/2 , which states that the perturbed region is equidistant from the two ends of the path. Also, the frequency

is taken to be 20 kc/s, corresponding to a wavelength, \ = 15 km.

In figure 2, NPA and NAA are shown plotted as a function of yT for the case where

v? - Yi = A — oo and d = x + Xj = 4000 km. It is noted that NPA asymptotically approaches

unity for yj sufficiently negative. This limit corresponds to an ionospheric perturbation which is effectly infinite in the transverse direction. Then, as yT is increased, in the positive direction, NPA

oscillates and at y: = it becomes exactly 1/2. Further increases of y1 reduce NPA until it asymptotically approaches zero. The latter limit corresponds to removing the ionospheric perturbation an " infinite "distance off the path.
It is interesting to note that the normalized amplitude anomaly, NAA, oscillates about zero

with maximum excursions when yj is in the vicinity of 0. Within the approximation that S(x, y)

is real, conservation of energy demands that the average value of NAA, with respect to y1

, is zero.The curves of NPA and NAA shown in figure 3 are for the same conditions as those in figure

2 except that d = 10, 000 km. Qualitatively, the results are very similar. The main difference is

that the period of the oscillations and the transition region is stretched out in the transverse direction.

Results of somewhat greater practical interest are shown in figure 4 where NPA is plotted as a

function of y = yt + (A/2) for various values of A the width of the perturbed region. (See figure

1 for relevant geometry. ) The wavelength is 15 km and the total range d is 4000 km. These curves

are even about y - 0, so only positive values are shown in the abscissa. Also, in order to con- m

serve space, the ordinates are shifted by 0. 5 for each curve. As expected, the curves in figure 4 show that NPA asymptotically approaches zero for values

of ym while it becomes approximately unity when y is near zero. Because of the interference from m

various diffracted waves , the structure of the field variations are rather complex although there is some semblance of a periodicity at larger values of ym. Curves in figure 5 are for the same conditions as for figure 4 except that now d = 10, 000 km.

The expected stretching of the abscissa is evident along with other changes in detail. A set of curves showing NAA for d = 4000 km and 10, 000 km are given in figures 6 and 7, respectively. These are for the same conditions as the corresponding NPA curves in figures 4 and

5. The NAA curves in figures 6 and 7 are even about y =0 and at the same time, the average m

value of NAA is zero with respect to ym .

5. Some Final Remarks

While the results in figures. 2 to 7 inclusive are for a wavelength of 15 km, the results may be

simply scaled to other wavelengths if corresponding changes are made in the other parameters. For

example, in a microwave model of this situation, \ might be 3. cm .leading to a scale factor

p = 15 km/3 cm = 5 x 10 . All other dimensions must then be divided by p if the diffraction patterns are to be invariant. For example, in figure 5, the total distance d = 20 meters and the 4 values of

A, reading from top to bottom, are 20 cm, 30 cm, 40 cm, and 60 cm. The correction for earth curvature has been implicitly neglected in all the above discussion. The

principal modification is to change the horizontal scale in figures 2 to 7 inclusive. It is not difficult

to show that the numerical values of y, or y should be multiplied by the factor X where m

and a is the radius of the earth. For example, this amounts to an increase of about 2 percent and

12 percent, respectively, for d = 4000 km and d = 10,000 km, respectively. Actually, X is equal

6-
to the ratio of the width of a Fresnel zone on a spherical earth to that on a flat earth. It is important to note that in a convenient microwave model such as a parallel plate waveguide, X would

be unity and yet it may be used to determine diffraction effects when X is not unity.

The results given in this note, although based on relatively simple theoretical concepts, should

give some insight into the behavior of the earth-ionosphere waveguide of non-uniform width. Some

of the methods used in this work bear a certain resemblance to an analysis carried out recently by

Crombie [1963] who argues from the viewpoint of physical optics. His final results appear to agree

with the formulas derived by the author [Wait, 1963] using analytical methods, and his conclusions are compatible with the numerical data given here.

6. References

Born, M. , and E. Wolf (1959).. Principles of optics (Pergamon Press, Oxford).

Crombie, D. D. (1963), On the effects of a small local ionospheric height depression on the propagation of a VLF radio signal (to be submitted to J. Res. NBS, Sec. D).

Wait, J. R. (Oct. 1961), Expected influence of a localized change of ionospheric height on VLF propagation, J. Geophys. Res. 66_, No. 10, 3119-3123. Wait, J. R. (1962), Electromagnetic Waves in Stratified Media (Pergamon Press, Oxford).

Wait, J. R. (Oct. 1963), On phase changes in VLF propagation induced by an ionospheric depression

of finite extent, ARPA Project Report (to be published).

7. Acknowledgements

I would like to thank Mrs. Carolen M. Jackson and Mrs. Lillie C. Walters for programming

the Fresnel integrals, Tom Griffiths for drafting the illustrations, and Mrs. Eileen A. Brackett

for assistance in preparing the manuscript. I would also like to thank D. D. Crombie and A. G. Jean

for some useful discussions of this subject.
d = 4000 km.

X = 15 km

A = oo

0.3

0.2

0.1

-0.1

-0.2

-0.3

200 yj(km)

FIGURE 2 - THE NORMALIZED PHASE ANOMALY, NPA, AND THE NORMALIZED AMPLITUDE

ANOMALY, NAA, AS A FUNCTION OF y, FOR A SEMI-INFINITE TYPE OF PERTURBATION

(i.e., A = oo), AND d = 4000 KM

200 y^km)

FIGURE 3 - THE NORMALIZED PHASE ANOMALY, NPA, AND THE NORMALIZED AMPLITUDE

ANOMALY, NAA, AS A FUNCTION OF yi FOR A SEMI-INFINITE TYPE OF PERTURBATION

(i.e., A = co), AND d = 10, 000 KM
4 5 6 y_ IN 100 km.

9 10

FIGURE 4 - NPA AS A FUNCTION OF y MEASURED TO THE CENTER OF A RECTANGULAR

m

SHAPE DEPRESSION FOR d = 4000 KM
4 5 6

ym

IN 100 km

7 8 10

FIGURE 5 NPA AS A FUNCTION OF y MEASURED TO THE CENTER OF A RECTANGULAR

m

SHAPE DEPRESSION FOR d = 10,000 KM

-10-
d = 4000 km., A = 15 km.

-0.4 J i I i I i I l_J.

1 3 4 5 6 1 8 9 10

ym IN 100 km. Tn

FIGURE 6 - NAA AS A FUNCTION OF y^ MEASURED TO THE CENTER OF A RECTANGULAR

SHAPE DEPRESSION FOR d = 4000 KM

-11-
y_m

IN 100 km

FIGURE 7 - NAA AS A FUNCTION OF y MEASURED TO THE CENTER OF A RECTANGULAR m

SHAPE DEPRESSION FOR d = 10,000 KM

-12-
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