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Reprinted from AMERICAN JoURNAL oF PHYSICS, Vol. 35, No.9, 869-870, September 1967

Printed in U. s. A.

Nature of Radio

Waves Reflected from an Undulating Surface

G. L. AusTIN

Department of Physics, University of Canterbury, Christchurch, New Zealand

(Received 5 December 1966)

The character of signals reflected from a sinusoidal-shaped surface is discussed with par-ticular reference to the ionosphere and an experimental verification of the essential features obtained using a model and microwaves is presented.

T

HE problem of calculating the total signal received due to reflections from a sinusoidal layer in the ionosphere is important because of current discussions about the existence of inter-nal gravity waves.1 _{The problem has been solved}

exactly by Rayleigh2 _{and for a shallow sinusoidal}

layer by Hewish. 3 _{The first of these solutions}

has been found to be too complex for normal use and the second is not applicable to gravity waves several radio wavelengths high. It is evident from a consideration of the method due to Rayleigh that a large contribution to the reflected radio wave is only made from points on the gravity wave near a specular reflection. White-head* and Landmark5 _{have considered curved}

surfaces of the type exhibiting only one specular reflection and Landmark, in the above paper, also considers the specular reflections from two plane surfaces inclined to the horizontal. A

Gaussian surface has been considered by White-head4, and Bramley6 _{discussed a geometry}

some-what. similar to that which is described below but did not point out the effects of tl1e gravity-wave parameters on the type of fading observed.

The equation of the reflecting surface is as-sumed to be

z

= 1/J where 1/J is a periodic

func-tion of 'X. and the time t with a mean height h

and an amplitude a (see Fig. 1). We consider specifically

1f

a sin ( h wt)

+

h,

1 C. 0. Hines, Can.

J.

Phys. 38, 1441 ( 1960).

2 Lord Rayleigh, Theory of Sound (MacMillan and

Co. Ltd., London, 1937), Vol. II, p. 89.

3 A. Hewish, Proc. Roy. Soc. (London) A209, 81

(1961).

4

J.

D. Whitehead,

J.

Atmosph. Terr. Phys. 9, 269

( 1956).

J.

Atmosph. Terr. Phys. 24, 715 ( 1962).

5 B. Landmark,

J.

Atmosph. Ten, Phys. 10, 288

(1957).

6 E. N. Bramley, Proe. Roy. Soc. (London) A220,

39 (1953).

FlG. 1. Parameters of the layer.

where k is the wavenumber of the gravity wave

and w is its frequency. For a specular reflection,

we have

d1f/dx - x/tfl.

Neglecting a in comparison with h, we obtain cos (kx-wt) x/ah k.

This equation may be solved either graphically or numerically to give the values of x for points where specular reflections occur.

Let

kx- wt-u.,

then

cos tt

= -

u/a k2 _{h wt/a k}2 _h.

The graphical method of solution is indicated in Fig. 2. It may be seen directly from this figure that if lc

<

(a h)--% then only one specular re-flection can ever occur at any one time. The condition that three specular reflections should always occur is that the straight line should just

Fm. 2. Graphical solution.

870 G. L. AUSTIN

0 50 d·crn

h= 75cm

0 50 d·cm

100

100

FIG. 3. Microwave measurement 1.

FIG. 4. Microwave measurement 2.

FIG. 5. Microwave measurement 3.

touch the cos u curve in two places. This condi-tion is given by u

=

tan u which immediately gives that three or more specular reflections will always occur when k

> (

4.49/ a h) '12. If k lies

between these limits then the records of ampli-tude against time should show deep fading with several reflections and shallow fading with only one reflection.

In order to check this theory, a model of an ionospheric gravity wave was constructed. It

consisted of a piece of hardboard approximately 250 X 100 em bent so as to form an approximate sinusoidal surface with a wavelength of 100 em and an amplitude of 5 em. This was supported on a wooden frame and covered with aluminium foil. A 3-cm microwave generator and detector

h=125cm

~ .\~j..

..

-

~

_...

, ..

\·~.-· ·~

.. · ... · ·-. )

_~

~

I

_{... + •} II • v'\,,w•

0 50 d·cm 100

h = 150crn

0 50 d ... cm 100

t

·~ ~ ~ ~·

•

~..!

· · II / . ·

J\

c: '

··V \ 'I' \

I \

~

I \ ···'·

. <1> ... ,, \ ~ f ....

c:

... ... ... ,.._

+ ... ... t I

- '•

\

...

\

..

~· '

h=200cm

0 50 d-cm 100

FIG. 6. Microwave measurement 4.

FIG. 7. Microwave measurement 5.

FIG. 8. Microwave

measurement 6 .

were then set up on an optical bench to simu-late the transmitter and receiver, respectively. The limits obtained above give that for one specular reflection h

<

50 em while for three specular reflections h

>

225 em. Figures 3-8 show the result of plotting the intensity received against distance along the optical bench for dif-ferent values of h in the region defined above. The heavy line represents the region of deep fading predicted by this theory. It may be seen that the results cross check as required.

[image:173.595.61.526.89.441.2]

Amplitude

Dist:l.'ibutiom;

Resul tir,.:_ ::.·orr. Cur·-.rature

Distributions

of a

Reflector*

G,.L .. AUSTIN and

J.r.tt.

TEORPE

Physics Depurtmont

University of

Ganterbur'J

·christchurch, New Zealand.

A s:Lul)le theory of focusing is discussed and ampli tuO.e prot~.0i2.::.. -;.J

dis J;:ributions are related to distributions of curvature at the re:;:-.lect,:.:..~.

'''he results are e.pplicable to amplitude distributions of radio sign-:.ls

reflected fror;:. the

ionosphere.

1

1. Intrcduction

In a :recent paper (Khastgi:r et

al,

1968) the observed a.'":lplitudes rz.:turned from sporadic E layers frequently shoy; double peaked amplj.tuC.e

p::.·ob~bility &istributions. The authors su,::e;ested that th.e t1o 11e:::lcs

cotlld be duo to r,:flections from t'.N sirr:ul taneous and indeLJendent

superimposed processes. In this note we shall extend the analysis of

Yihitehead (

'\962)

who interpreted the observed fading il.1 terms of

focusint;; to show tc.at a. satisfactory explanation can be obtained frcm.

2

2. Observed Distributionb

If the radio wave returned from the ionosphere is scattered from

irregularities in randmn motion then the amplitude distribution will be

the fumiliar Reyleigh (

1899)

distribution and if there is a constant specular component as well then the distribution will be that of Rice

(1945).

However, man;y people have observed reflections from the D, E

ami F regions and obtained a.mplituC.e distributions which are not very

similar to those above. The major features w.i"ich require some

e;.."?lana.-tion are the occasional occurl'ence of distribue;.."?lana.-tions having two peaks

r.hich ue will call '1; shaped 1 _{and. the lo·.-1 freq~Jency of occurrence of}

amplitudes of very siiall value civinc rise to an 'offset'.

~he 1_{11 scaped' distributions have been noted by Dasgupta and}

Vij (

1960)

and Kushnerevsky and Zayarnaya (

1960)

for the ~, region and Khastgir et al (

1968)

for sporadic· E., We have noted. 11 shaped

dis-tributions from the E region. The offset, which is the dis?lacement f:.";_.::.

zero of the lower bound of the amplitude distr~bution, hasl:een previous:.y

3.

Analysis

Consider a reflecting surface having a cylindrical undulation

~~th

a

radius of curvature R and a range h from a radar

(fig.

1).

Following Whitehead

(1956, 1962)

simple geometry gives the width of

tho area illuminated in the region of the reooivGr as

3

( 1)

where L is the width of the illulf.inated region of the reflector

assumed constant and much larger than a Fresnel zone. The distance

from the reflector to the virtual image of the transmitter is v. If we

now eliminate v from

(1)

by using the usual mirror equation we obtain 1

=

2L

t

1 - hjR

I

The amplitude A

1 of the signal received is determined by the area

illwLinated in the :region of the receiver and is given from (2) by

If ·we define tne norn:alized curvature h/R as f we may then write

A 2

=

1

1

I

1 -

fl

(2)

(4)

The probability distribution of the received amplitudes is the

same as the distribution off, except that the amJ?litude probabilities

are weighted in proportion to the area over which that amplitude occurs.

This area is inversely proportional to A 1

2

and consequently

Now from

(4)

we obtain

d.A1

df

=

2

-A3

1

'

and consequently

4

(6)

(7)

5

4•

Comments on the liumerical Results

It may be noted that all the derived amplitude distribution::~ show

a distinct offr.et and~omparing the distributions of amplitude for

normal oistributions of f with two different standard deviations (curves

a and b'in figure 2), we see that the offset

can

be interpreted to indicate the relative spread of P'(f). If a suitable spectrum of sinusoidal

atmospheric waves is considered the distribution of f is J,! shaped (fig. 2,

curVe c) but as may be seen this does not neces:w.rily produce an M

shaped amplitude distribution. Curve d in fiGUre 2, however, does lead

5 •

.Amplitude Distribution for the Second Reflection

If the amplitude, E

2

, measured at the ground of the radio wave

twice reflected from

th~

ionosphere is re-defined by

6

2E

=

_.6

p

(8)

where

p

is the apparent reflection coefficient, then A

2

is unity for

a plane reflector. Geomet1•ical optics similar to that of section

3

then gives

A

2

2

-1

i (

1-f) ( 1-2f)l

so that combining

(5)

and

(9)

we obtain

(9)

( 10)

The distribution of the second reflection amplitude for the normal

curvature distribution (curve a in figure 2) is shown in figure

4~

7

6.

Lirni tB.tions of iha Analysis

It may appear tha.

t

the analysis would ~l~eak down if the curved scree were 1_rough1_{• This has been} invostiga.t~d _{by Whitehead ( 1960) and he s::owecl}

that the screen should have a polar diagram which has a reflecting. cone

"

ancle of not more than a fev1 degrees. 1'he E and F regions have been

in-vestigated byBrigga anc1 Phillips (19.50) who came to the conclusion that

this conditjon vms satisfied for radio mwelenc.;ths t,;reater than about 200m.

It is evident that the very lnrge values of arnvlitude predicted for perfect

focusinc events at f

=

1 for example will not occu1~ for the roueh screen but these have a very low probabilit _ _. due to the 1/A 2 term ir. the analysis.

There is, !·.ow;;ver, a theoretical difficulty with the method in that

for cases ;;Len f is r;reater than + 1 the condition. for three or more

specular reflections to occur at the s1liJle time is satisfied (Brarr.la.ly,

1953

and Austin, 1967). It is impossible to take t}:is into account wi thou-;,

making some asstm;ptions about the shape of the screen. However,

invJ5-tigation indicates tho. t the result of neglecting the second<3.l:'Y reflections

p,...ints is to alter the 11I1lplitude distribution by a few percent.

It micht be thought that this analysis could be used in reverse to

obtain P'(f) from theroadily mea.sur€lable P(A

1). However (4) does not

uniquely define f if A

1 is known since there are two values of f related

by f₁

+f

₂= 2 which both. satisfy the equation. The two solutions co:crespor::6. to the cases where the focus occurs above or below grounci level and there

is no wa:y in which the ~elative probabilities maybe assigned unless P'(f)

8

7.

Conclusions

'i/e have tried to show that simple geometrical o;;tics provides

"'

an explanation oft he observed characteristics of reflected rad.io 7;ave

nm~:litudes. The theory has b·JOn primarily aimed at the totally reflected signal from the ::..:; and F regions but as far as we are awn.re no

8.

References

Austin,

G.~.

(1967),

Nature

of

radio waves reflected from an undulating

sur:f'o.oe,

Am.

J •.

Phys.

,l2,

No. 9,

869-870.

Bramley, E.N.

(1953),

Direction-finding studies of large-scale ionospheric

h•••oe;ularities, Proo. Roy. Soc. A,

ggg,

39-61.

Brisgs, B.H. and G.J. Phillips

(1950),

A Study of horizontal irregularities

of the ionosphere, Proc. Phys. Soc .. Lond. B,

.§2.,

907-923.

Dasgupta, P. and K.K. Vij

(1960),

Statistical

ana~sis

of fading of a

single downoorning wave, J. Atmosph. Terr. Phys. 18, No.4, 265-275.

Khastgir, S.R., S. Ganguly, and S. Samanta

(1968),

The double-peak

dist-ribution of the radio wave returned from the -sporadic E layer,

Radio Sci. _2, (New Series),

I•:o.

6,

535-537.

Kushnerev sl<y, J. V. eLnd E. S. Zayarnaya (

1960),

The dri:ft of small scale

irregularities in the F

2

-layer, Some Ionospheric results obtained

during the I.G.Y., Ed. \,'.J.G. Beynon, 313-321 (Elsevier Publ. Co.,

London).

Piggott,

'li.l~., 'II.J.G.

Be,:,mon, G.i.l.

Broim,

and C.G. Little

(1957),

?2-"''

meastu•ement of ionospheric absorption, Annals of the

I.

G. Y.,

,.2,

171-226.

Pit;gott, W

.R. (

1960),

Discrepancies in the ionospheric absorption de(1uced.

from tho first order and multiple order reflections, Some

Iono-spheric Results obtained during the International Geophysical Year,

Rayleigh, Lord ( 1899), Collected works

j,,

495 (Camb. Univ •. Press).

Rice,

s.o. (

194-5), I·,!athematical analysis of randorr, noise, Bell. Syst.

Teoh. J.

1J±,

No;' 1, 46-108.

10

von Biel, H.A. (1965), 1965 Solar eclipse partial reflection experiment,

CAL Re1lort No. VS-2025-P-1, Cornell Aeronautical La bora tory, Inc.

Whitehead, J.D. (1956), The focusing of short radio waves reflected from

the ionosphe1·e, J. Atrr.osph. 'L'err. Phys.

,2,

269-275 •

. Whitehead, J.D., (1960), Focusing of radio

w

ves reflected from a rough curved ionosphere, Aust.

J.

Pi~s.

j2,

621-624.

i'fl.iteh0ad, J.D. (1962), Distribution of echo amplitudes from an undulating

FIGtJ:R:B CAPTIOTS

Figure

1

Geometr,y of the model considered.

Figure

2

Distributions of f:

a) normal, standard deviation one

b) normal, standard deviation two

c) for a spectrum of sine waves;

P'

(f)

<411!:

(x

2

'l:f/4

2 2/

6)-i .

2

- 7t f 1

sl.n x dx

d) which gives rise to an

l1l

shaped distribution of A

1

•

Figure

3

Distributions of

a.mplitu(;e,

A

₁

corresponding to the

distributions of f in figure 2.

Figure

4

Distribution of the amplitude, A

2

of

iile

second

:..~ eflev;ion

[image:185.595.39.515.71.503.2]

h

v

[image:186.595.44.469.196.757.2]

I

3~

1

0

I

FIG. B. 4

(\

/ I

[image:189.608.32.524.97.742.2]

Radio Science, Volume 4, Number 1, pages 35-40, January 1969

On the nature of the irregularities that produce

partial reflections of radio waves from the lower io11osphere (70-100 km)

G. L. Austin

Physics Department, University of Canterbury, Christchurch, New Zealand

A. H. Manson

Institute of Space and Atmospheric Studies University of Saskatchewan, Saskatoon, Canada.

(Received Aprill, 1968; revised August 22, 1968.)

The irregularities that produce the partial reflection of radio waves ( 1.5-6.0 MHz) from

the D region are investigated. Experimental evidence indicates that reflection usually occurs

from localized irregularities rather than from a region of smaller scale turbulence.

1. INTRODUCTION

In recent years there has been a growing number of experiments in which medium and high frequency radio waves (1.5-6.0 MHz) are partially reflected from the lower ionosphere. One of the first quantita-tive experiments was the differential absorption ex-periment (DAE) for measuring electron number densities [Gardner and Pawsey, 1953]. These experi-ments assumed that the refractive index irregularities were due to changes in electron density and that the reflection coefficient of these irregularities could be calculated by using the Fresnel equation

R

=

(n2 - nt)/(n2 n1) (1) where n1, n2 are the refractive indices for an

electro-magnetic wave in regions of the ionosphere where the refractive index changes rapidly within one wave-length. It was then shown that the amplitude ratio of the ordinary and extraordinary components of the partial reflections as a function of height could be used to produce an electron number density profile. Gardner and Pawsey considered two main types of ionization irregularity. The first is a sharply bounded, horizontally stratified irregularity of extent com-parable with a Fresnel zone, to which the Fresnel equation clearly applies. The other type is a series of random irregularities or blobs. Gardner and

Pawsey [1953] concluded that the details of the reflection mechanism made no difference to the elec-tron number density profiles obtained by the DAE. It is very important to know more about the nature and cause of these D-region irregularities, both for

35

their possible relationship to the dynamics of the upper atmosphere and for their effect ou the DAE and other radio wave experiments. Several experi-mental and theoretical methods for investigating the nature of the irregularities are discussed in this paper. In section 3, reflection coefficients for two extreme cases are developed. In section 4, the percentage changes in electron number density required to give the observed partial reflections are found by using these equations in conjunction with measurements of reflection coefficients and electron number density profiles. A possible origin of these irregularities is discussed briefly in section 5. Arising from the devel-opment in section 3, experimental tests are suggested whereby a definitive statement can be made about the actual reflection mechanism (section 6). The consequences of these ionospheric models upon radio wave experiments are discussed in the con-cluding section.

(B, B, c, c., E, E'

'

H, N(h),

P,

R,

2. NOTATION

total number of blobs in the scattering region. total number o( layers in the scattering region. velocity of light in vacuo.

velocity of sound.

energy density observed on the ground accord-ing to saccord-ingle layer theory.

energy density observed on the ground accord-ing to saccord-ingle blob theory.

height of reflecting level.

electron number density as a function of height. peak power transmitted in the pulse.