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1/37

324 PROCEEDINGS OF THEEEE,OL. 70, NO. 4 , APRIL 1982

Radio Wave Scintillations in the Ionosphere

KUNG CHIE YEH,

FELLOW, EEE, AND CHAO-HAN

LIU,

FELLOW, EEE

Invited Paper

Absfruct-The phenomenon of scintillation of radio waves propagat-

ing through the ionosphere

is

reviewed in this paper. The emphasis is

on propagational

aspects,

including both theoretical and experhnental

results. The review opens with a discussion of the motivation

for

st*

chastic ormulation of he problem.

Based

on measurements rom

in-siru, radar, and propagationexperhnents, onospheric irregularities

are found to be characterized, m general, by a power-law spectrum.

While

earlier

measurements indicated a

spectral

ndex of about4, there

is

recent evidence showingthat the index may vary with the strength of

the irregularity and possibly a two-component spectrum may exist with

different

spectral

indices

for

large and small structures Several scintil-

lation theories including the

Phase

Screen, Rytov,

and

Parabolic

Equa-

tion

Method

(PEM)

are

discussed

next.Statistical parameters

of

the

signal

such

as

the average

signa,

scintillation index,

r m s

phase fluctua-

tions, orrelationunctions,powerpectra,distriiutions, tc.,

are

investipted. Effects of multiple scattering

arediscussed.

Expedmental

results

concerning

irregularity

structures and signal

s t a t i c s are

presented.

These

results

are

compared with theoreticalpredictions.The

agree-

ments are

&own

to

be

satisfactory in a large measure. Next, the tem-

poral behavior of a transionospheric radio signal

is

studied in terms of

a two-frequency mutual coherence functionand the temporal moments.

Results ncluding numerical simulations are discussedFinally,some

future efforts

in

ionospheric scintillation studies in the

reasof

transion-

aspheric communication and space and geophysics are recommended.

I. NTRODUCTION

I

A . History

of

onosphere Scintillation Studies

N

1946, Hey, Parsons, and Phillips

[

11 observed marked

short-period irregular fluctuations in the intensity of radio-

frequency (64MHz) radiation from the radio star Cygnus.

At first it was thought that the fluctuations were inherent in

the source itself. Subsequent observations indicated that there

was no correlationbetween fluctuations recorded at wo

stations 210 km apart, while fairly good correlation was found

for a separation of 4

km

[21,

[

31 .

This led to the suggestion

that the phenomenon was locally produced, probably in the

earths atmosphere. ndeed,as later observations confirmed

[4]

-[

l o ] , ths marked the f i i t observation of the ionosphere

scintillation phenomenon.

After he f i t artificial satellite was launched in1957, t

became possible to observe ionosphere cintillations using

radio transmissions from the satellite

[

1 1

-[

151

.

The interest

in the study of this phenomenon has continued in the ast two

decades. In general, the interests are twofold.

On

theone

hand, the study of the scintillation problem

is

directly related

to the transionosphericommunication roblemsuch

as

statistics of signal fading, channel modeling, ranging resolu-

tion , etc. On the other hand, scintillation data contain infor-

This

work was

supported by the Atmospheric Research Section of the

Manuscriptreceived September 18, 1981; revisedJanuary 18, 1982 .

National Science Foundation under Grant ATM 80-07039.

The authors are with the Department of Electrical Engineering,

Uni-

versity

of Illinois

at Urbana-Champaign, Urbana, IL

61801.

mation about he geophysical parameters of the ionosphere

and proper interpreation of the data is essential for a better

understanding of the physics and dynamics of the upper at-

mosphere.

As

observational data accumulated, it became

possible to discuss the global morphology of ionospheric

scintillation [ 161. n the early seventies, the discovery of

scintillation at gigahertz frequencies

[

171,

[

181 presented

an additional challenge to he field. Two satellite beacon

experiments specially designed for scintillation tudies, the

ATS-6 and the Wideband Satellite

[

191,

[

201, have provided

us with new observational data tha t helped

to

enhance

o m

knowledge of the scintillation phenomenon. These include

coherent multiple frequencydata orbothamplitude and

phase scintillations. Fig. 1 shows an example of such observa-

tions.Simultaneous multiteehnique observational compaigns

were carried out [21] which yielded valuable information

about the structures f the irregularities.

On the theoretical side, ionosphericscintillation was first

studied in terms of the

th n

phase screen theory

[

221, [231.

Advances in the study of wave propagation in random media

have helped in theeffort to develop aunifiedscintillation

theory [241. For weak scintillation, the single scatter theory

is

quite well established and experimental verifications of the

theoretical predictions have been demonstrated

in

many

in-

stances. The multiple scatter heory for strong scintillation

has also mademuch progress in recent years but here still

remains quite a few unresolved problems.

In

ths

review, the current status of the ionosphere scintilla-

tion of radio waves will be reviewed, both from the observa-

tional and theoretical points of view. The emphasis will be on

transionospheric radio wave propagation and signal statistics.

The morphology of ionospheric cintillation willbe the

subject of another review paper

[

251 and will not be discussed

here.

B. Motivat ion

for

Stochastic Formulation of he Problem

Wave propagation

is

concerned with the study of the space-

time fields that are transferred from one part of the medium

to another with

an

identifiable velocity of propagation. To

identify the velocity of propagation, one may choose to follow

a particular feature of the field such

as

the peak, the steep

rising edge, or the centroid.

As

it propagates the field may

change itsmagnitude, change itsshape, and even change its

velocity provided

ths

particular feature of the field can still

be identified and followed. Mathematically, wave propagation

problems are generally posed by

an

equation of the form

Lu

= q

(1.1)

where L is usually a linear differential operator and less fre-

quently an integro-differential operator or a tensor operator

0018-92 19/82/0400-0324$00.75

0

1982 IEEE


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A N D LIU:

RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE

325

1

0 10

5

eo

IS

3b

t S

35

4 0

io T

-10

4

0

4

40

o

5

I S

t0

e s

30

1 5

T IH E [8ECONOSI

I

3. -

i

2 .

- .

0.

0 10 tO

5

I S

30

LS

4 0

3s

,

i

-10

0

4

4 0

0

15

20

t S

30

3s

Fig. 1. Multifrequency amplitude and phase scintillation data from the

Time: 18:37:10 to 18:37:50

UT.

Data were detrended at

0.1 Hz.

DNA Wideband Satellite received at Poker Flat, AL, March

8,

1978.


3/37

326 PROCEEDINGS

OF THE

IEEE,

VOL.

IO,NO.

4,

APRIL 1982

when dealing with vector fields; u

is

the field or wave function,

scalar or vector, and

q is

the real source function. In posing

propagation problems in (1.1) we need t o specify:

i) Real source function q : Usually localized in space

and time.

ii)

Virhial source function

u o :

The ncident field uo satis-

fies the equation

uo =

0.

iii)

Shape and position of Boundaryonditions need

boundary surface

S :

be considered.

iv) Properties of propagating The operator L depends

on

medium:

these properties.

In many situations any one or a mixer of these four quanti-

ties may become very complex. When this is

so

the wave func-

tion is also expected t o be highly complex. In these cases one

may wish to adopt a stochastic approach as an alternate to the

usual

deterministic approach in solving (1.1).Generally,

stochastic approach

is

preferred if the information about the

above four quantities

is

incomplete and imprecise; or, even

when the four quantities are or can be specified exactly, the

mathematical demand in solving (1 l ) is too formidable a task;

or, even when (1.1) can be solved deterministically, the

ob-

tained results are not physically intuitive, nstructive, and

useful. In these cases, one adopts a statistical characterization

of any one or a mixer of these four quantities. If such a char-

acterization yields a stable and physically meaningful statisti-

cal characterization of

u ,

the stochasticapproach

is

then a

useful approach.

In the stochasticapproach one may classify the problem

according to which one of the four quantities

is

stochastic.

Therefore, instudies of excitation of fields by random sources,

the real source q is random; in studies of diffraction by partially

coherent fields, the incident field

u o is

ranqom; in studies of

scattering by bodies having random shapes and positions, the

boundary surface S is random; and in studies of diffraction

and propagation through random media, the operator

L

itself

is

random. In this way a large number of practical examples

have been discussed and classified in

[ 26] .

All these examples

are classified

as

belonging to one of these four classes for their

mixtures. According to this scheme of classification the study

of ionospheric cintillations would normally belong to he

classof problems dealing withdiffraction and propagation

througha andom medium. However, under certaincondi-

tions and sometimes in an effort to simplify the mathematical

task, the phase screen idea

is

advanced. In this case the prob-

lem can be classified as diffraction of partially coherent fields.

In adopting a statistical approach, one has in mind, at least

implicitly, two probability spaces: oneproability space for

the specification

of

the problem and one proability space for

the wave field. A point in the probability space corresponds

to a particular probability distribution that is used to charac-

terize the problem or the field.

Our

nterest in solving (1.1)

is

then to find the prescription that maps a point in the proba-

bility space of the problem onto a point n he probability

space of the field. Symbolically, the situation

is

represented

by Fig.

2 .

It should be realized that each point in the proba-

bility .space characterizes only the statistical properties. It is

entirely possible that two or more samples, known

as

realiza-

tions, may possess the same statistical properties,

as

usually

is

the case. An example of one such realization obtainedby

computer simulation

is

shown in Fig. 3 . Many such two-di-

mensional random surfaces

can

be generated

[ 2 7 ] ,

all having

the same statistical properties. If, for example, one

s

interested

in he behavior of radio rays, propagating in a luctuating

dielectric medium with certain statistical properties, one can

first use the specified statisticalproperties to realize many

o p p l n g

Probabl l i ty Space

of

the Prob lem Probab l l i t y Space

of

the Wave Function

Fig.2. A point

in

theprobabilityspace of theproblemspecifies he

probabilitydistribution of thedielectricpermittivity

or

electron

specifiesheprobabilitydistribution of the wave function. Our

density and apoint in theprobabilityspace of the wave function

interest

is

to find the mapping between these two probability spaces

as

depicted symbolically by this illustration.

H CORRELATIONENGTH

Fig. 3. Arealization of a wo-dimensionalrandomsurfacewith he

prescribed statistical properties. (After Youakim

e ta l .

[27].)

6

I

4 1 / I

-4

-6

I

I

I I I I I I I I

I 1

5 10 I5

Fig.

4.

Ray trajectories hroughrealizeddielectricmedia. All media

a value of 1.5 percent in

r m

fluctuations of refractive index. Statisti-

have identical

power

spectrum for the fluctuating refractive indexand

achieving the mappingdepictedin Fig.

.

(After Youakim

et al . [

281

.)

cal properties of the rays can be compiled from these traced rays, thus

media and then trace rays, all with identical initial conditions

in these realized media. The results foronesuchstudy are

shown in

Fig. 4 [ 2 8 ] .

The statistical behavior of the ray can

be obtained if a sufficiently large number of such rays have

been traced,

as

done

in

[ 2 8 ]

and

1291.

In

this

way, a method

known

as

the Monte Carlo method

is

thus constructed

so

that

the mapping between the two probability spaces

is

achieved.

Unfortunately, the Monte Carlo method is very cumbersome


4/37

YEH AND LIU: RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE

3 2 1

to apply, and one would rather use an analytical method

if

it

is

available. At the present ime, analytical methods are not

available in such a general framework. If one is willing to relax

his requirements by seeking a more modest answer, such as a

few finite numbers

of

moments instead of probability distribu-

tions, the problem usually becomes more mathematically

manageable. Even in uch cases, approximations reoften

needed and introduced to facilitateasolution.Theproblem

of ionospheric scintillations

is

no exception.

11.

CHARACTERIZATION F

I O N O S P H E R I C

IRREGULARITIES

A . Observational Evidence

The existence

of

ionospheric rregularities is required

to

explainmanyexperimentalobservations. The earliest

is

the

vertical soundingexperiment

[ 3 0 ]

in which a adarecho

is

received as the carrier frequency

is

swept from about

0.5

MHz

to

15

MHz. The received data are ypicallydisplayed n the

time delay (or virtualheight) versus frequency format. Nor-

mally the echo traces in such a display are very clean, showing

distinct onospheric ayers. On occasion, the echo races are

broadened and diffused for heights corresponding

to

the ion-

ospheric

F

region. When this happens the echoes are known

as spread

F

echoes and the irregularities that cause the spread

F

echoes are commonly called the spread

F

irregularities.

Many experimental techniques have been used to study these

spread

F

irregularities. A historical account of the experimen-

tal

effort can be found in [

3

1 . The experimental techniques

can be broadly grouped into two: remote sensing techniques

and

in-situ

measurements.

Most

remote sensing techniques

utilize adio waves and they canbe classified according

to

whether the radio waves arereflectedfrom,scatteredfrom,

or penetrating through the ionosphere. In a low-power opera-

tion the radio waves are normally reflected fromhe ionpsphere

in experiments such as vertical ionosonde, backscatter on@

sonde, and forward scatter onosonde. Such experiments are

useful in detecting the existence

of

spread F irregularities and

their results have been used in morphologicalstudies as re-

viewed by Herman

[ 3 2 ] .

As the radio frequency

is

increased

beyondsome value, theradio wavebegins to penetrate the

ionosphere and almost all of its electromagnetic energy escapes

into the outer space. Nevertheless there

is

a very small amount

of its energy that

is

scatteredback.Underquiescentcondi-

tions the backscattering

is

caused by ionospheric plasma fluc-

tuationsunder hermalagitations.Forsufficientlypowerful

radars the scattered signal may be strong enough to provide

us

withuseful nformation.Radarsoperatingon this principle

are nown as incoherentcatteradars

[ 3 3 ] - [ 3 5 ] .

In

monostatic mode the backscattered power is proportional to

the spectral content of electron density fluctuations at ne-half

of the radio wavelength. It mustbeunderstood, herefore,

that such radars can sense the irregularities only in a very nar-

row pectralwindow. On occasion,during the presence

of

spread

F

irregularities, the radar returns have been observed

to increase in power by80 dB in a matter of few minutes

[ 3 6 ] .

Thismeans that n a ewminutes he rregularity pectral

intensity can ncrease by

as

much

as

10' fold.This suggests

the highly dynamic nature of the phenomenon under study.

Recent experiments at the magnetic equator show that a cer-

tain type of spread

F

irregularities take the form

of

plumelike

structures and may be caused by Raleigh-Taylor instabilities

[ 3 7 ] .

Another remote sensing technique deals with scintilla-

tionmeasurementsand is thesubject of this review. Early

reviews on this subject have been made by Booker

[ 381

using

HORIZONTAL

SCALE I

SCALE ( k m )

MAGNETIC FIELD

(m)

00

100

IO

1 100 IO I

0.1

0.01

1

I

I

I

I

I

I

1

I

to I onosphere

Wander lng o f N orm a l

Mul t ip leormals

.-

v

e

of T lDs Phose

e

(Gravi tat ional ly

Sc ln t i l l a t lon

$

Anisotropic

1

H

t

I

Strongac

S c o t t i r i n g

and Trans-

equator la l

(M agne t i c o i l y

WAVE NUMBER

n i l )

Fig. 5. Acomposite spectrum summarizing intensityof ionospheric

irregularities

as

a function of wavenumber over a spatial scale from

the electron gyro-radius to the radius

of

earth.

(After

Booker

[ 461

.)

radiostars as sourcesand by YehandSwenson

[ 3 9 ]

using

radio satellites

as

sources. Because of the simplicity of experi-

ments, the scintillation observations canbe carried out atmany

stations. Globally

it

hasbeen ound tha t scintillationsare

most ntense n

two

auroral zones and the magnetic equator

[ 161 .

Both the spectra of scintillating phase

[ 4 0 ]

and scintil-

latingamplitude

[ 4 1 ]

have anasymptotic power-law depen-

dence,This suggests that he onospheric irregularitymust

have a power-law spectrum

as

well

[ 4 2 ]

.

More recent progress

on scintillationheories nd xperimentalobservations re

reviewed in later sections.

The other experimental technique has to do with measuring

ionospheric parameters

in situ.

This generally implies carrying

out measurements on boarda ocketora s'atellite. Probes

have been made to measure the density, temperature, electric

field, and ionic drifts. As far

as

scintillation

is

concerned, the

quantity of direct concern

is

the electron density fluctuation

A N . Characteristics of various types of A N are described in

[ 4 3 ] .

The power spectrum of A N

is

found t o follow a power

law

[ 4 4 ] , 4 5 ] ,

confirming theexpectations based on he

scintillation measurements

[401 ,

[

4

1 .

Therefore, the totality of

all

experimental evidence indicates

the existence of ionospheric irregularities over a wide spectral

range. This situation was best summarized by Booker [ 4 6 ] in

acompositespectrum eproduced n Fig.

5.

This composite


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328

PROCEEDINGS O F THE IEEE, VOL.

70,

NO.

4,

APRIL

1982

-

0045

0030

I

0015

I

0000

I

2345

Fig.

6.

Sample data of

136-MHz

ignals transmitted by the geostation-

ary satellite

SMSl

parked at

90'W

and eceivedatNatal,

Brazil

(35.23OW, .8S'S,

dip

-9.6') on

November

15-16, 1978.

The bot-

tom amplitude channel

is

approximately linear in decibels with a full

scale corresponding to

18

dB. The top and middle polarimeter out-

full-scale change corresponds to a rotation

of

180' or a

change of

puts

vary

linearly with the rotation of the

plane of

polarization.

A

1.89

X

10

el/m2 in electron content. The times given are in ocal

mean time with UT

=

LMT

+ 03

:

00. Two

successive depletions n

electron ontentwith ccompanied rapid scintillations are sepa-

rated by about 30

min

in time.

spectrum spans an eight-decade range, corresponding to scales

from the electron gyroradius t o the

e a r th

radius. In

ths

seven-

decade range, irregularities responsible for i:nospheric scintil-

lations vary from meters t o tens of kilometers.

At the present time, there

is

a great deal of interest in one

kind of equatorialscintillations associated with onospheric

bubbles. One example is depicted in Fig. 6, where the op

trace shows the amplitude of 136-MHz signals and the bottom

trace shows the Faraday rotation indicative of change in total

electron content (TEC) [47 ]. Notice the simultaneous increase

in scintillation intensity and rate,

as

indicated by the top chan-

nel, and the depletion in TEC by 5.7

X

10l6 el/m2

as

indi-

cated by thebot tom channel. While such bubble-associated

scintillations are of great interest, we must emember that

most observed irregularities at other geographic locations and

even at the magnetic equator are not associated with ioniza-

tion depletions. It

is

likely that there may exist many causa-

tive mechanisms. Readers interested in this ubject hould

consult a recent review [48].

B. Correlat ion Funct ions and Spectra

As

discussed in Section 11-A, there exists a large body of

experimental results which indicate that the electron density

in

the ionosphere can become highly complex and irregular.

When this

is

the case, it may be more convenient to describe

the propagation problem stochastically

as

discussed in Section

I-B. For

ths

purpose we must first deyr ibe the medium, by

its statistical properties. Thus et

A N ( r )

be the fluctuations

of electron number density from the background No. Depend-

ing on the problem, we may let

g = A N ( ; )

or let

= AN ;)/

N o z ) ;

n either case is assumed to be ahomogeneous ran-

dom field with a zero mean and a standard deviation

u t .

Its

autocorrelation function

s,

by definition,

B E ;I - ;2)

=

(E(;1)(;2

1)

(2.1)

where the angular brackets are used to denote the process of

ensemble averaging. By the Wiener-Khinchin theorem, he

correlation and the spectrum form a Fourier transform pair

m

(2.2b)

Since .

s

real, there must exist symmetry conditions

B E -;)

=

B E and

Qpg(-2) D E

( I?) . (2.3)

If the irregularities are $otrzpic, the correlation function in

(2.1) depends only on

( r , - r2 I.

In this case, the three-dimen-

sional Fourier transform given in (2.2) simplifies to

m

BE(')= I @,(K)K sinKrdK.2.4b)

r o

In someapplications, the one-dimensionaland two-dimen-

sional spectra are needed and they are defined, respectively, by

OD


6/37

YEH AND LIU: RADIOAVECINTILLATIONSN 329

For the special case of isotropic irregularities, the three-dimen-

sional spectrum is related to the one-dimensional spectrum by

The relation ( 2 .7 ) is useful for it prwides a means of deducing

the three-dimensional spectrum from a one-dimensional mea-

surement such

as

those carried out

in situ

by probes on a rocket

orasatellite. However, the sotropic property is paramount

in deriving the relation ( 2 .7 ) . In general when irregularities are

anisotropic, it is impossible to deduce @ E ( 2 ) from

V t

K ~ ) .

In he onosphere,probe measurements on board several

earlier satellites have a l yielded a power-law one-dimensional

spectrum of the form Vt a ; ' with m close to 2 [ 4 4 ] , [ 4 9 ] ,

irrespective of geographic locations and other conditions, for

spatial scales in a two-decade range from 7 0 m to 7 km.

As-

suming isotropic irregularities, these probe data would imply a

three-dimensional spectrum of the form

95 K ) 0: K - ~ (2 .8)

where the spectral index

p

must be close to

4

for

rn

close

to

2 ,

as is required by (2 .7) . This conclusion agrees closely with the

spectral index derived from the scintillation spectra of phase

[ 4 0] , [ SO] and of amplitude [ 4 1 1 ,

I511

by using,the phase

screen scintillation theory [ 4 2 ] or the Roytov solution [ 5 2 ] .

Thereare ndications,however, from ecentmultitechnique

measurements, that he pectral ndex

p

mayvary as the

strength of the irregularities changes [ 1621. The power spec-

trum maintains its power-law form to K >2 m-' (o r spatial

scale = 3 m) when the in-situ data are supplemented by the

radar data at 50 MHz [531 , [ 54] . There

s

indication, at least

sometimes, that such a spectrum can be extended to irregular-

ities as small as 11 cm 1551, [561. Nevertheless, onmathe-

matical and physical grounds, the power-law spectrum

( 2 .8 )

is

expected to be valid only within some inner scale and outer

scale. This is so because, mathematically, hemoments of

(2 .8)

may not all exist; some of the integrals will diverge unless

propercutoffsare ntroduced. Physically, adeparture rom

(2 .8)

is

expected near an inner scale where dissipation becomes

important and also near an outer scale at which the energy

feeding the instabilityoccurs.Recent ocket-bornebeacon

experiments

[

211 and in-situ measurements [2 1 I ] covering

more han five decades

of

scale

sizes

have shown a possible

two-component power-law spectrum for the equatorial rregu-

larities with a higher spectral index or the small structures.

To characterize the general power-law irregularity spectrum

withspectral index

p ,

Shkarofsky [ 5 7 ] introduceda fairly

general correlation-spectrum pair

where

ro

is

the nner scale and

I o

2 7 7 / ~ ~

s the outer scale,

and as such we must have KOrO


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330

PROCEEDINGS OF THE IEEE, VOL.

70,

NO. 4, APRIL

1982

(2.16)

where e is the electronic charge,

m

is its mass, eo is the free

space permittivity, w is the circular radio frequency, and

re

is

the classical electron radius. The quanti ty

A N ,

is the devia-

tion in the total lectron content defined by

AN,( p ) = A N (

p',

z) d z .

(2.17)

The correlation of the optical path separated by a distance p is

I

BA )

=(A@($)A@(; +;I = C 2 B ~ ~ , < ~ >2.18)

where

C =

eZ/2meow2. Since the electron content deviation

is given by (2.17), its correlation BAN,

can

be related to BAN

and

@AN

by

+ +

=

2lrZ fl@AN(;l,

0)

d 2 K l (2.19)

-00

-b

where K~ = ( K ~ , ,,). As

is

usually the case, the background

path

z

is much larger than the correlation length, the limits of

integration in the middle expression of (2.19) are extended to

- -DO and

00

asshown.Inserting (2.19) nto (2.18) relates di-

rectly the correlation of the optical path to the correlation of

ionospheric irregularities.

In the literature of wave propagation in random media, the

integrated correlation function occurs frequently and is usually

denoted by the symbol A, viz.,

00

AANG) =J BANG,)z.2.20)

Consequently, the electron content correlation is merely the

product of the propagation path z and the integrated correla-

tion unction (2.20). For he three-dimensionalcorrelation

function given by (2.9), A is found to be

-00

(2.21)

The corresponding one-dimensional spectrum

is

then

*

K ~ - ~ ) / ~

r o e ) . 2.22)

Equation (2.22) shows that for a three-dimensional spectrum

of the form K - ~

s

given by (2.1 l) , the one-dimensional speo

trum of the electron content is the form K ; ( ~ - ' ) . Notice the

change in the exponent.

D. Optical Path Structure Function

At times the electron density fluctuat ions and hence the opti-

cal path (2.15) contain a background trend so that they arenot

strictly homogeneous but only locally homogeneous [58 ]. In

these cases it

is

moreconvenient t o dealwith the structure

function D defined by

The structure function for the optical path DA$

p ' )

is just the

mean square value

of,

the optical path difference between two

points separated by

p

on the z = constant plane. Carrying out

several steps, this optical path structure function

can

be shown

to be

(2.24)

forpath lengths z greater than he correlation ength as

is

usually the case. The optical path structure function is there-

fore directly proportional to theelectroncontentstructye

function. If

A N i s

apm oge neo us random field, henDAN

r )

=

2 [BAN(O)

-

BAN(^)] which reduces (2.24) to

where the optical patn structure function is simply related to

the Correlation function of the electron content.

E.

Frozen Fields and Their Generalizations

In practice the fluctu:tion in electron density is a space-time

field and hence 5 = [ ( r , t). As such its space-time correlation

is

The space-time spectrum is given by the four-dimensional

Fourier transform

with its Fourier inversion. In experiments where radio energy

is scattered by ionospheric irregularities, the received wave

shows both a Doppler frequency shift and a slight broadening

of the spectrum. These effects,aspostulated in 59] and

[601, are caused by 1) the convection of scattering irregulari-

ties which is responsible for the Doppler shift, and 2) the time

variation of the irregularities which is responsible for he

Doppler broadening. For hemoment if we take only the

convection into account, the random field then satisfies

E( ; , t + t ' ) =

((;-

ZOt',) (2.28)

for which the space-time correlation has the form

B E ( ; , ) = E t ( ; - &t). (2.29)

In(2.28) and (2.29), z0 is the convection velocity. A field

that satisfies (2.28) is lfnown as the frozen field, since such a

field is convected with

uo as

if it were frozen. For frozen ields,

the correlation funct ion satisfies (2.29) and their space-time

spectrum satisfies

If this frozen field

is

also isotropic, we can show that


8/37

YEH AND LIU: RADIO WAVE SCINTILLATIONS INHE IONOSPHERE

3 3 1

where

W E

a)s

the frequency spectrum on time series

$(;,

t )

obtained by a fixed observer. The prime on W indicates dif-

ferentiation.Equation

(2 .31)

relates the spatial spectrum to

the frequency spectrumof an isotropic frozen random field.

When the pectrum is generalized to includenonfrozen

flows, we must take into account the possibility that irregular-

ities maychange with time

as

they move. Indoing

so

it

is

desirable to strikebalancebetween easonably imple

analyticexpression that can bemanipulatedmathematically

and he physical notion hat large irregularities are nearly

frozen, at least for a short time, and small irregularities are

in the dissipation range and hence can vary with time. After

considering these factors, Shkarofsky [

6

1 proposes to decom-

pose the spectrum

S

n the following way:

S E G , ) $ G ) ( 2 .32)

with the normalization

I

( w )

dw

= 1. ( 2 .33)

In the interest of not flooding t s review paper with too many

symbols, et he+argument of denote he Fourier domain.

For example (

K , t )

s obtained from

I?,

w ) y a one-dimen-

sional Fourier inversion with respect to w. With such a nota-

tion, the spectral decomposition scheme ( 2 .32) plus the nor-

malization

( 2 .33)

implies that

$ ( Z , t = O ) = l

BE(;, t = 0) =BE( ; ) . (2 .34)

Comparing

( 2 .32)

with

( 2 .30)

shows that

(2, w )= 6

w

+ ?

Go>

or

+ +

( t )

=

e - i K . v o t

( 2 .35)

for frozen flows. When flows are generalized

to

include dissi-

pations it

is

possible

to

propose many forms for [

61

1. If

the decay

is

caused entirely by velocityfluctuationswitha

standard deviation u u , 2 .35) can be generalized to

(2 .36)

The frozen field result of (2 .35) i s obtained from ( 2 .36) for

large irregularities

(viz.,

small K ) and short time as

is

desired

based on physicaleasoning discussed earlier. By Fourier

transforming ( 2 .36) with espect to t andsubstituting he

result in

( 2 .32) ,

the space-time spectrum becomes

and the corresponding correlation function becomes

00

Because of the presence of B E ( ? ) in the integrand, ; in the

exponent in

( 2 .38)

makes contribution to the ntegral only for

I

;

I

less than several correlation lengths. Therefore, as

t -+ m,

the triple integral

is

no longer a function

of

time which implies

B E t )

must have the asymptotic behavior

t- for

large times,

The velocity

Go

in

( 2 .38)

does no t necessarily have

to

be the

convective velocity of the fluid. In measurements made

in

situ

by probe carrying satellites and rockets,

So

becomes the veloc-

ity of the p:obe. The co2elation function of such in-situ data

i s ,

hen

B E

r t ) , ) where r

( t )= Got

describes the probe trajec-

tory

as

a function of time. A question that arises

is

whether

such an experimentally determinable correlation function can

yield the desirable information about the irregularity spectrum.

This problem has been investigated [

621

in what i s termed the

ambiguities

of

deducing the rest frame rregularity spectrum

from the moving frame spectrum. Let

PE(a)

e the spectrum

deduced in the moving frame, viz.,

00

PE a) ( 2 7 r ) - 1 I m B E ;(r), t ) - jut d t . ( 2 .39)

For a rectilinear motion of the probe we may :et

; ( t ) =z^uot

where

z^ is

a unit vector along the z-axis. Since a satellite ravels

with large velocities, the andom field as observed by+the

probe may be approximated

as

frozen. Consequently,

B E r t ) ,

t ) = Bg(z^uot)which when inserted into (2 .39) yields

(2 .40)

where V E s the one-dimensionalspectrumdefined in (2.5) .

Therefore, the moving frame spectrum

P E ~ )s

related to the

one-dimensional rest frame spectrum

V E C ,0, , )

by

( 2 .40)

with K , = a / u o under he rozen fieldassumption. If the

frozen field

is

isotropic, hededucedone-dimensional spec-

trum can in turn determine he three-dimensionalspectrum

by using

( 2 .7 ) .

If he frozen field

is

anisotropic and of the

kind discussed at he very end

of

Section 11-B, the hree-

dimensionalspectrum canbe ecovered onlywhen we also

know

g,, a,,,

andhe rientation

of

theprobemotion

relative to thecorrelation ellipse.

If the probe

is

moving slowly such as a rocket near the top

of

its

flight,

the frozen field assumption

is

no longer valid.

In this case the correlation function measured on the moving

frame becomes

As an example, let

B E = e- I r

2 2

( 2 .42)

then the ntegral in

(2 .41)

can be integrated to

give

( 2 .43)

Hencewhen

t I/ ,

the correla-

tion

( 2 .43)

approaches asymptotically

to

zero

as r - 3 ,

as d e

duced earlier.

In general, instead of a Gaussian correlation function

( 2 .42) ,

the integral in

(2 .41) i s

difficult

to

evaluate analytically. The

moving frame spectrum

Pc(w)

in this general case

is

related

2 2 2


9/37

3 3 2

PROCEEDINGS OF

THE

IEEE, VOL.

70,

NO. 4, APRIL 1982

TO

TRANSMITTER AT-CD

lO.O.2 I

-

ECEIVER

Fig. 7. Geometry of he ionospheric scintillation problem.

to the est frame spectrum@E

2)

by

-00

(2 .44)

for probes moving along the

z-axis

with a constant velocity

&.

The relation

(2 .44)

is complicated. By knowing

P o )

only,

it

does not seem possible to invert (2 .44) to get @E2) with-

out making additional assumptions.

111. S C I N T I L L A T I O NHEORIES

A . Statement of the Problem

With the statisticalcharacterization of the irregularities

as

discussed in Section 11, we can model the ionospheric scintilla-

tion phenomenon. Let us consider the situation hown jn Fig.

7.

A region

of

random irregular electron density structures

is

located from z = 0 o z = L . A time-harmonic electromagnetic

wave

is

incident2n the irregular slab at

z = 0

and received on

the ground at

( p ,

z ) . It will be assumed that the irregularity

slab can be characterized by a dielectric permittivity

e = (E) [

1

+

el i , . t )~

(3 .1)

where ( E )

is

the background average dielectricpermittivity

which for the onosphere is given by

(e) = (1 - f l O f

2 ) o

(3 .2)

and el(;, t ) is the fluctuating part characterizing the random

variations caused by the irregularities and is given by

Here,

f p o is

the plasma frequency corresponding to th e back-

ground electron density N o and f

s

the frequency of the inci-

dent wave. In the percentage fluctuation

A N / N o

= 5 the tem-

poral variations, caused by either the motion of irregularities

as in a frozen flow or the turbulence evolution as in a non-

frozen low, or both, are assumed to be much slower than

the period of the incidentwave.

As the wave propagates through he irregularity slab, to the

first order, only the phase is affected by the random fluctua-

tions in refractive index.

This

phase deviation

is

equal to

k o ( A 4 ) , where ko

is

the free space wavenumber and A @

s

the

optical path luctuation defined in (2.16). Therefore, after

the wave has emerged from the random slab,

its

phase front

is randomly modulated as shown in Fig. 7 . As this wave pr op

agates to t he ground, the distorted wave front will set up an

interference pat tern resulting in ampli tude fluctuations. This

diffraction process depends on the random deviations of the

curvature of the phase front which in turn is determined by

the size and strength distributions of the irregularities. Simple

geometric computation indicates that the major contribution

to the amplitude fluctuations on the ground comes from the

phase front deviations caused by irregularities of the sizes of

the order of dF =

d-,

which is the size of the first

Fresnel zone [ 6 3 ] . Basically, this simple picture describes

qualitatively the amplitude scintillation phenomenon when the

phase deviations are small. The wave front remains basically

coherent across each irregularity which acts to focus or de-

focus the rays. However, when the irregularities are strong

such that

e l is

relatively large, the phase deviations may be-

come

so

intense that the phase front is no longer coherent

across the irregularities larger than certain size. These irregu-

larities then lose their ability to focus or defocus the rays. The

interference scenario for the ampli tude fluctuation described

above therefore

is

no longer valid. Qualitatively, one would

expect he saturation of the amplitude fluctuation. Another

refinement of this qualitative picture

is

that when the irregu-

larity slab is thick one would expect to see ampli tude fluctua-

tions developing inside the slab such that as the wave emerges

from the slab it has suffered both phase and amplitude pertur-

bations. Hence, the development of the diffraction pattern on

the ground is affected by both factors.

In scintillation theories, one attempts tonvestigate quantita-

tively the various aspects of the phenomenon. Thestarting

point is the wave equation n electrodynamics.Under the

assumptions [

5

8]

i) the emporal variations

of

the irregularities aremuch

ii)

the characteristic size of the irregularities is much greater

the vector wave equation for theelectric field vector inside the

irregularity slab can be replaced by a scalar wave equation

slower than the wave period,

than the wavelength,

where

E

is a component of the electric field in phasor notation

and k 2 =

kg

( E ) .

Equation (3 .4) is a partial differential equation with random

coefficient, the solution of which, if available, will form the

basis for the scintillation theories. Unfor tunately, the general

solution of ( 3 .4 ) does not seem to be possible. One has t o

settle for various approximate solutions for different applica-

tions. To discuss these solutions, we f i t pecialize in the case

of normal incidence. The generalization of the results t o the

oblique incidence case will be discussed later in the develop

ment. For the normal incidence case, it

is

conven$nt to intro-

duce the complex amplitude for thewave field

u

( r )

Equation (3 .4) then yields an equation for the complex ampli-

tude

Based on this equation, an approach, known

as

the Parabolic

Equation Method (PEM), has been developed to treat prob-

lems of wave propagation in random media

[

241. The follow-

ing assumptions are made n this approach:

iii)

The Fresnel approximation in computing he ph ge of

the scattered field

is

valid, corresponding to

z

>>

I

>>

h

iv) Forward scattering: The wave is scattered mainly into a

small angular cone centered around hedirection of

propagation.

This

corresponds to

( e t ) z / l <

(SI) 0.

(3.17)

The correlation functions

(3.18)

where

@@(zl) s

the power spectrum for thephase

q5p)

given

by

@ ~ ( $ ~ ) = h 2 r ~ @ A N T ( $ ~ ) = 2 ~ L h 2 r ~ @ A N ( ~ ~ ,

).

(3.19)

From (3.18) and (3.19), we obtain the mean-square fluctua-

tions forx and S 1

/ r

+-

(3.20)

and the powerspectra for he log-amplitude and the phase

departure

a x ( 2 ~ )

Sin2 (K:Z/2k)@~($l)

= 2nLh2r,? Sh2 (Kf Z/2 k)@ ~~( 21,)

@s(;l)

= COS2

(K:Z/2k)@,#,($l)

=

2nLhr:

COS

(K:Z/~~)@AN($~,

).

3.21)

As mentioned above, the phase screen theory has been used

quite extensively in ionospheric scintillation work as well as

interplanetary and interstellar cintillations [4], 75] -[77] .

Although the derivation was specialized for an incident plane

wave, the results can be readily generalized to cases of spheri-

cal wave, beam wave

[

781, extended source

[

791, etc.

The expressions derived above are no longer valid

if

one

considers a deep screen where

is

no longer small. One

has to go back to (3.10) to derive general expressions for the

various parameters. Mercier [69] considered this problem in

somedetailand derived integral expressions for he higher

moments of the field. Recently, several authors have derived

analytic asymptotic expressions for he ntensity correlation

function and the spectrum [801-[ 861. Some of these results

will be discussed in latersections.

C. Theory

f o r

Weak Scint i l la t ion-Rytov Solut ion

When the effects of scattering on the amplitude of the wave

inside the irregularity slab are to be included

in

the treatment

of the scintillation phenomenon, one has to go back to (3.6)

and (3.7). With the substi tution of (3.15), (3.6) becomes

Under the assumption of weak scintillationsuch that he

higher order term (V$) can be neglected in (3.22), we ob-

tain

the equation for the Rytov solution241

, 581

(3.23)

The range of validity of

t is

solution has been discussed by

many authors 87]-[89].There is some evidence that he

Rytov solution may be applied t o ionospheric cintillation

data even for moderately strong scintillations [go].

The general solution of (3.23) can be obtained

as

exp [-jkl; - pI2/2(z

-

f)1

d p

(3.24)

where

o (p )

=

I n

u(p ,

0)

corresponds to the incident wave.

The field emerging from the bottom of the slab

is

given by

exp

[ JI (p;

L)] , which contains modifications for both ampli-

-tude and phase. The amplitude variations comeaboutfrom

the diffractional effects inside the slab,

as is

evident from the

second term in (3.24). The field on theground can be obtained

from (3.7) with

u

p,

L ) =

exp

[ (

L)]

as

its initial condi-

tion. The Rytov solution for (3.7)

is

exp -jk

l p

-

p

1/2(z

- L)1 d p

(3.25)

where

(pf, L ) s

obtained from (3.24).

Equation (3.25) gives the formal solution for theonospheric

scintillation problem under the Rytov approximation. It can

be used to derive the various statisticalparameters for he

wave field.

Again, let

us

specialize to a plane incident wave with unity

amplitude. Then the mean values x>

(Sl>= 0.

The power

spectra for

x, 1 ,

and the cross spectrum between

x

and SI or

the field on the ground are given, respectively, by [9 1 ]

?rk3fL K ?

@,s(K;) = in in- z - L/2)@&, 0).

K l

2k k

(3.26)

The correlation unctions can be obtained rom (3.26). We

note that by letting

L + 0

in the expressions for

@,,@s,

we

obtain he phase screen results (3.21)

if

thesubstitution

=

( r : h 4 / n z )@ A N

is

made.

Several aspects of

this

result are specially useful in the anal-


12/37

YEH

AND LIU: RADIO WAVE SCINTILLATIONS IN THE IONOSPHERE

335

0

2 4 6

0

10 I? 14 16 18 20

Fig. 8. Filterunctionormplitudecintillationlotted against

normalized wavenumber

Kfzlk.

Dashed line, marked

L = 0

km,

corresponds to the phase screen model.

ysis and interpretation of scintillation data. We shall consider

these points in the following.

I

ScintiZZation Index

S4: One of the most important

parameters in ionospheric scintillation study is the cintillation

index defined as the normalized variance of intensity of the

signal [ 9 1 ]

( I Z )

s f =

.

( 3 . 2 7 )

Other definitions for thecintillation index have been proposed

[ 9 2 ] , [ 9 3 ] , 4 4 ] . However, the

S4

indexhas beenadopted

by most investigators for digitally processed scintillation data.

For weak scintillations,

it

is easy to show

[9

1

s f

=4(X*).

From ( 3 . 2 6 ) and the definition of correlation unction, we

have

This quantity measures the severity of intensity scintillation

underhe weak scintillationssumption. The integraln

( 3 . 2 8 ) indicates that the contribution to the intensity scintilla-

tion from the irregularities is weighted by a spatial filter func-

tion, i.e., the expression in the square brackets of ( 3 . 2 8 ) . Fig.

8 shows the filter function versus K ' ( Z - L ) / k for three values

of the slab thickness

L .

The height of the slab

i s

3 5 0

km. The

oscillatorycharacter of the filter unction is known as the

Fresnel oscillation, which

is

more pronounced for a smallerL .

The irregularity spectrum is, in general, of a power-law type,

which decays as

K

increases. Therefore, heproduct of the

filter function and the spectrum has a maximum around

K fi

K F

= 2 n / d ~ ,orresponding to the

first

maximum

of

the filter

function. This is consistent with the intuitive pictureresented

in Section 111-A that when multiple scattering effects are no t

important, irregularities of sizes of the orderof the first Fresnel

zone are most effective in causing amplitude scintillation.

For a power-law spectrum QAN -

I P

with an outer scale

much greater than the Fresnel zone size, it is possible

to

show

from

( 3 . 2 8 )

that

[ 5 2 ]

s4

a ( 2

~ ) 1 4 - ( 2 + ~ ) / 4 . ( 3 . 2 9 )

This frequency dependence

of

the scintillation index has been

observed in many experiments, some f which will be discussed

in later sections.

From

( 3 . 2 8 )

we also note the dependenceof the scintillation

index on the hickness of the slab

L

' I 2 , and on rms A N .

2 ) Mean-Square Phase FZuctuations:

From ( 3 . 2 8 )we have

..

The phase filteMg function given in the square brackets of

( 3 . 3 0 ) is very different from the amplitude filtering function,

which as discussed in the lastsection shows Fresneleffects.

In fact, the major contribution to (St) omes from the large

irregularities. It

is

easy to show rom

( 3 . 3 0 )

that

(St)

s

proportional to

l / f z .

3)

FrequencyPowerSpectra: Inpractical ituations, the

irregularities in the ionosphere are in motion mostof the time.

This motion will cause the diffraction pattern on the ground

to

drift. This process

is

responsible for producing a temporal

variation of the signal received by a single receiver. In most

cases, for adio signals transmitted rom he geostationary

satellite, his is what one observes as the scintillation signal.

If the frozen-in'' assumption discussed in Section 11-E for the

irregularities is valid, then the temporal behavior of the signal

can be transformed into he spatial behavior. n other cases

where the radio signals are transmitted roma ransit satel-

lite, the speed

of

the satellite usually

is

much faster than the

drift speed of the irregularities

so

that the temporal variations

of

the signal received by a single receiver can be considered as

the result of the radio beam scanning over the spatial varia-

tions of frozen irregularities. In both cases, therelation be-

tween temporal and spatial variationss a simple translation by

themotion.The requency power pectrum of the signal

received at a single station denoted by

@ a)s

related to the

spatial power spectrum by 5 1

]

. ,-+-

where thecoordinate system

is

chosen uch that hedrift

velocity is in the x -z planewith the tranverse

( x

direction)

speed uo .


13/37

336 PROCEEDINGS

O F

THE EEE, VOL. 70, NO. 4, APRIL 1982

Substi tuting the expressions for spatial power spectra from

(3.26) into (3.31), we obtain the frequency power spectra for

the log-amplitude and phase, respectively,

where

For a power-law irregularity spectrum of the form K i p , the

general b'ehavior of

ax

and

9s an

be estimated. At the

high-

frequency end such that

5 >>5 ~

V ~ K F , oth

9,

and

vary asymptotically

as

a('-P)

And for

5


14/37

YEH A N D LIU: RADIO WAVECINTILLATIONS INHE IONOSPHERE

337

moment of the field

is

defined by

r m , n ( Z , S l , S z , . . . , S m ; S 1 , . . . r S n )

+ + + +

+ I

=(u1u2

* .1( u *

-

up)

(3.38)

+

where

ui = u ( q ,

z),

ui =

u(q ,

2).

the following equation [ 106

1

:

+ I

This

general moment for the field can be shown to satisfy

a r m * n

(z,

a

.

.

.

s

.

+

a 2

Sm,

Sn)

(3.39)

v; =

a2/ax;

+ a21ay,?

and

vj2 = a2/axjZ+ a2/ayj2.

For

z

> L ,

i.e., outside he slab, (3.39) is still valid

if

one sets

A A N

= 0

in the last term. Therefore, we have now a general

set of equations describing the behavior of the higher statisti-

cal moments of the scintillation signal.

This

set of equations

was first used t o develop a multiple-scatter scintillation theory

for the ionmphere ase in

[

531,

[

1071.

From the definition, we note that

rl,o

( u ) . The equations

for the averaged field thus become

z

>

L. (3.40)

For plane wave incident such that

02. = 0,

(3.40) yields the

solution, for >

L

( U ) = A ~

xp [ - ~ ~ A ~ L A ~ ( o ) / ~ I

A ~

xp [-&/21.

(3.41)

This agrees with the plane wave solution from he general

phase screen approach (3.12). We note that the measurement

of

( u )

will enable one to obtain the important parameter

4

for the ionospheric irregularity slab. In the following, we

shal l

presentsome esults obtained rom he scintillation theory

based on (3.39) and other equivalent versions of it. Emphasis

will

be onquanti ties hat are observed in he scintillation

experiments.

1) Mutual Coherence

Function:

Consider

1 1 ,1 = ( u ( z ,

p,

k)u * (z,

z

k)). The equation for r l , becomes

(3.42)

rl.,l s

known

as

the two-frequencywo-position mutual

&=1.6 9

C = I 5 5

c =2.97

-. -

AUSSIAN

POWER

LAW

X

i

i

01 0 2

0 3

5 4 0 5 0 6 0 7

08

09 I O

P / f O

Fig.

9. Contours

of constant correlation coefficient

C,,,

or frequency-

space eparations.

Both

power-law andGaussianrregularities are

included.

coherence function

[

105 .

The general analytical solution of

(3.42) is difficult and has not been obtained. Certain special

aspects of the equation are of interest. If one sets k

=

k in

(3.42) , one obtains the equation for the coherence function

r2

=(u(z, p)u*(z,

7 ) )

hich, for plane wave incidence, has

the analytic solution

r2(Z,p,Z)=A: exp

{ - ~ ~ A L [ A A N ( O ) - A A N ( Z -

?)I}

= A :

exp

[-

3 D+($

311

(3.43)

where De is the structure function+for the phase fluctuation

defined in (2.25). We note that forp= ;such that r2

=

( u 2 )

=

A :

from (3.43) which is consistent with the energy conserva-

tion requirements for forward scattering.

A

Rytov ype of solution or

rl ,

can be obtained from

(3.42) by writing

1 1 ,1 =

exp

( )

and neglecting the nonlinear

terms in the resulting equation for

.

Under this approxima-

tion, we have

[

1081,

[

1091

rl,

(z,

6,

?,

k,)

= ~ X P

$I (3.44)

{exp [jAkKf(z - L)/2kk] - exp [ j# r ~fz /2 kk ]}

eXp [ jzl (p

-

?)I d 2 K l / K f (3.45)

where Ak = k - k.

This

Rytov solution has been used to study pulse propaga-

tion in the ionosphere [ 1101 and t o characterize the transion-

ospheric communication channel

[

1 1 1

,

[

1 121.

Although certain asymptotic solutions of (3.42) have been

studied

[

1091, the general solution can only be obtained by

numerical ntegration. Fig. 9 shows some esults from such

computations

[

1 31.nransionospheric ommunication

applications, it

is

useful t o define a correlation coefficient for

the complex amplitude

I u

- ( u ) ) (U*

- ( U * ) ) ) I

I Iu - ( u ) 2

(Iu

-

(u)2)l

12

[( u 2 >-

(u>2)

((Ut? -

(u>2)]

I 2 *

c =

-

l r 1 , 1 ( s , ~ , k , k ) - u ) ( u * ) ~

(3.46)

In Fig. 9, C,

is

plotted for a set of values of normalized

parameters defined by


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338

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VOL. 70,

NO.

4,

APRIL 1982

C = 8nl:rzX

( ( A N ) )

to

=

L / k , l $

(=

z / k o l

A k

X=-

k + k

(3.47)

where lo

is

some characteristic size for the rregularities.

The contours in Fig. 9 indicate the level of correlation for

sig+& at different frequencies received at stat ions separated

by p . They can be used to study frequency and/or space diver-

sity schemes for transionospheric communication. The results

are given for both Gaussian and power-law irregularity spectra.

Recently, using data rom Wideband Satellite, the wo-fre

quency coherence function has been measured experimentally

[2121.

2)

Scintillation Index: r2,

computed or he same fre-

quency corresponds t o the coherence function known in the

literature

=r4 =r2 ,2u z , p l ) u z , p z > u * z , ~ ~ ) u * z , ~ ~ ) )

where the frequency dependence is omitted. From (3.39), we

have

Fig. 10. Scintillation ndex

S, as

a function of

r m s

A N computed

for

frequencies 125 MHz, 250 MHz, and 500 MHz. The irregularity slab

has

a

thickness of 50 km. The distance

between

the bottom

of

the

slab and the observer is 237.5 km. The background electron density

1012/m3

is

assumed with

p = 4

and an outerscale of 500 m.

(3.39) can

be

put into a dimensionless form

(3.48) with an initial condition

r4

= A :

at

z = (= 0.

Here

in

(3.54)

where

D@ 3 ) s

thetructureunctionorhe phase.

(=

Z / L T

41 = /IT & = IT q = KllT

+

Introducing new variables

-+

(3.55)

R = +

J l

+ & + &

+ & )

; =

;l - pz

+p;

-

3;)

and

P=j + p z - p - & i1

3

(51 -

pl

IT =

(87~rzC&X)-(/~)

(3.49) L T

= [8n2(2n)-P/

~ C & k ~ ~ Z ) ] - ( ~ p ) .3.56)

(3.48)

can

be transformed to

For z

>

L

,

he dimensionless equation

is

(3.50)

where

F is

the expression in the curly brackets

in

(3.48)

in-

volving the combination of the phase structure functions ex-

%ressed in the new variables. Note that F does not depend

on

R ;

this

is

due to the fact that he random field involved

is

homogeneous. If we specialize in plane wave incidence,

V R = 0

and we can set p = 0 in

F

without loss of generality. Equat ion

(3.50) then becomes

In erms of the power spectrum, the

F

function can be ex-

pressed

as

F ( ~ 1 , ; 2 ) = 4 ~ [ ( J @ ( z l ) ( 1 C O S ~ ~ - ; ~ )

+-

-OD

*

(1 - cos 21

;2)

dK1 (3.52)

where

(J@

is given in (3.19).

For a power-law irregularity spectrum of the form

a A N ( 2 1 )

=

C&

I Z ~-P

(3.53)

(3.5 7)

with

r4

at z =L computed from (3.56) as its initial condi-

tion.

Since (3.54) and (3.57) are dimensionless, their solutions

must be independent of the irregularity strength and the geom-

etry. Indeed, it will be possible to obtain a universal solution

for he problem [114], [1151. Equations (3.54) and (3.57)

consti tute the basis for mult iplescat ter ionospheric scintilla-

tion theory for intensity scintillations. Once

r4 s

known, one

can compute the scintillation index

S4.

s t =- 4((, , 1

-

1.

(3.58)

This

indicates that the scintillation index

is

a function of

(=

z / L T . From (3.28), i t can be shown tha t he scintillation

index

S40

computedunderRytov approximation (the sub-

script 0 indicates theRytovsolution) is proportional to (.

This

implies that the scintillation index

in

the general case

is

a

function of

S40

[ 1 151.

General analyticsolutions to (3.54) and (3.57) have not

been found although certain asymptotic solutions have been

obtained [85],

[

1161. Numerical solutions have been at-

tempted or some cases

[

1161,

[

1171.

Fig.

10 shows the

results from one of such computat ions [53]. The scintillation

1

A40


16/37

YEH AND LIU:ADIO IN THEONOSPHERE 339

ow

0.0

0.0 0.1 0.2

0.3 a4 0.5

S, AT 5 0 0 MHz

Fig. 1 1 . Spectral indices for t w o frequencies against scintillation index

S, . The ionosphere conditions are the same as those in Fig.

10.

2

L=SOkrn

f P=7 07MHz

3. L = 5 0 k r n

f,=IOMHz

Frequency

f

(MHz)

Fig. 12. Spectral ndex as a function of frequency for different ono-

spheres. The irregularity model is the same as in Fig.

10.

index S4 is plotted against the rms electron density fluctuation

for three different frequencies in the VHF and UHF bands.

We note that for small values of rms A N , S4 for

all

three fre-

quencies increases linearly with A N r m s , as predicted by the

weak scintillation theory (3.28). As ANrms increases, md-

tiplescatteringeffects become important , and saturation of

the scintillation index becomes apparent, first at he lower

frequency. This saturationeffect causes the frequency de-

pendence of S4 to depart from hat predicted by he weak

scintillation theory, viz., S4

af-,

= -(2 +p)/4 as given by

(3.29). For strong scintillations, the spectral index n

s

not a

constant any more; it depends on S 4 . Fig. 1 1 shows

wo

urves

of spectral index

as

function of S4 obtained from the same

numerical computations as in Fig. 10. We note hat for he

same ionosphericconditions, the spectral index curves are

different for different frequencies. This is due to the fact that

at different frequencies the degree of S4 saturation is different.

Fig. 12 shows the spectral index

n

as a funct ion of frequency

for different ionospheric conditions.

Althoughanalyticsolutions for (3.54) and (3.57)are not

available, over the years researchers have attempted to derive

asymptotic formulas for the scintillation index under strong

scintillationonditions, using the phase screenpproach

[801-[86]. The starting point is (3.8). With the assumption

of Gaussian statistics,

r4

t the bottom of the irregularity slab

can be computed. This can then be used as the initial condi-

tion for (3.57) t o yield an analytical expression for the power

spectrum function for the intensity on theround [81 [821

(3.59)

where

Again the phase structure function

D6

appears

in

the expres-

sion. The scintillation index S: can be obtained from (3.59)

-00

For weak scintillation, (3.59) can be approximated by ex-

panding exp

(g),

which

will

then lead to results

similar

to

those shown n (3.21) and (3.26). For power4aw ionospheric

irregularities of the form of (3.53) (valid for

K~

< K I I < ~ i ,

the scintillation index can be found explicitly [861

where J is a numerical factor dependent on the degree of an-

isotropy of the irregularities [861,

r

is the gamma function,

and { is the normalized propagation distance defined in (3.5 5).

As discussed above, the general solution for S4 will be a func-

tion of

5

only (3.58). From (3.62), it-follows that thegeneral

scintillation index

will

depend on S40 in a universal manner,

independent of the ionospheric condition and the propagation

geometry [ 1151. The parameter tp/ nd hence S40 can be

considered as the strengthparameter that characterizes the

level of scintillation for the onospheric applications.

Based on (3.591, asymptotic expressions for S4 and the

power spectrum for large values of

5

(or S40) have been de-

rived for different ranges of values for

p

[8 l l , 182 , [85 ,

[

861

.

For the case p

2

, the scintillation index is found to ex-

ceed unity for certain intermediate values of {. This is known

as focusing.

As

5 increases further, S4 approachesunity.

This behavior is also found in results from numerical compu-

tations [53],

[

1721.

3)

CorrelationFunction and CoherenceInterval: The cor-

relation fynction or he ntensity scintillation s given by

r4(5, l, r2 = 0). For weak scintillation this function can be

approximated by the results from the Rytovsolution (Fourier

transform of (3.26)). For strong scintillation, numerical solu-

tions of (3.54) and (3.57) give us

ths

correlation function.

Fig. 13 shows an example from such computation. Two inten-

sity correlation functions are shown for certain onospheric

conditions. It

is

interesting t o note the faster dropoff of the

correlation athe lower frequency, corresponding t o the

decorrelation for stronger scintillations. This decorrelation is

caused by multiple scattering of the wave from irregularities.

As

discussed in Section 111-A, at higher frequencies

so

that the

scintillation

is

in the single-scatter regime, the most dominant


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340

PROCEEDINGS OF THE IEEE, VOL. 70, NO. 4, APRIL 1982

D @ (Tc u o )=

1

(3 .64)

for the multiple-scatter regime. For the power-law irregularity

spectrum of the type

(3.531, it can

be shown that

[ 8 6 ]

f =5MHr

LI00krn

r ,=Wrn

Transverse C o a h t e

a (m)

Fig. 13. Transverseorrelation

functions

for the intensity f the

scintillating signal. Power-law irregularity

spectrum

with p

=

4.

O 5 0

00

200

5 o o I x x ) x ) ( x ,

Frequency

f (MHz)

Fig. 14. Correlation distance

as

a

function

of frequency.Ionosphere

condition

is

the same

as

in Fig.

13.

contributions to intensity scintillation come from irregularities

of the sizes approximately equal to the dimension of the first

Fresnel zone

4-j

Therefore, the correlation distance

from the intensity fluctuations should be approximately equal

to the Fresnel zone dimension which s proportional to l / g .

As

the requencydecreases, hemultiple catteringeffects

enter the picture and eventually dominate. These effects cause

decorrelation

so

that the correlation distance will decrease

as

the requency decreases. These two ompeting ontrolling

mechanisms for the correlation of the intensity at

high-

and

low-frequency imits will result n

a

maximumcorrelation

distanceoccurringat some intermediate requency. Fig.

14

shows two examples of such behaviy where the correlation

distance

is

defined

as

the distance

Ip

I at which the intensity

correlation

is

one-half of its maximum value

[ 5 3 .

Although

the results are for correlation distance, they cane transformed

to those for the coherence interval corresponding to the tem-

poralbehavior of the scintillating signal. If the frozen in

idea is valid, then the relation between correlation distance

I ,

and coherence ime

7, is

simply

T, = , / U O ,

where

uo

is

the

transverse drift speed.

In the phase screen approach, a more quantitative estimate

of T,

is

possible. It

can

be shown that for

p < 4 ,

the asymp

totic intensity correlation function for strong scintillations

is

given by

1861

r 4 ( z , & , o ) = 1 +exp [ - D @ ( ; ~ ) I .3 .63 )

Therefore, the coherence interval

,

can be defmed by

(3 .65)

where C

is

a parameter depending

on

the strength of the ir-

regularity and the propagation geometry.

The power spectrum for the intensity can be obtained from

the solution of (3.54) and (3.57). One approach is to Fourier

transform he woequations n andcarryoutcertain tera-

tive solutions for the resulting differential-integral equations

[581 , 1 181,

[

1 191. Some asymptotic results have been ob-

tained from the phase screen approach

[811-[83] .

The spec-

trum has the same high-frequency asymptote as for the weak

scintillation case, but the rolloff frequency

is

increased, indi-

cating a broadening of the spectrum which corresponds to the

decorrelation of the signal. There

is

also an ncrease in the

low-frequencycontent of thespectrum,corresponding to a

long tail of the correlation function.

In this section, we have presented the results of a multiple-

scatter theory for ionospheric scintillations based on the PEM.

Some related analytic results from phase screen theory are also

discussed.Recently,here have been somepromising new

developments using the path-integral method [

1201

-[

1233.

The method

is

especially suitable for strong scinti llations in

the saturation region.

The discussion of any scintillation theory

will

not be com-

plete

if

one does notmention heprobabilitydistributions

of the scintillating signals. Indeed,

t h i s is

an area that

is

least

developed in ionosphericscintillation heory.In hefollow-

ing section; a rief discussion on

this

subject will be given.

E. Probability Dismbutions of he Scintillating Signuls

Tostudy heprobabilitydistributions of thescintillation

signal theoretically, several approaches have been adopted in

the iterature. One

is

to use heuristicarguments to analyze

the scattering process and then apply the central-limit theorem

inrobabilitytudy to determinehe istribution. This

approach has led to the prediction of joint Gaussian distribu-

tions for the real and imaginary parts of the complex signal.

Application of the arguments to the Rytovsolutionresults

in the log-normal distribution for the intensity

[ 1241.

In the

weak scintillation regime, the theoretical predictions seem to

agree with theexperimentaldata

[ 1251.

There have been

many statistical theorems developed governing when the cen-

tral limit theorem can be used. However, these theorems are

difficult t o apply to propagation problems

[

1261.

The second

approach is to theoretically calculate first the moments of the

distribution and then omputehe istribution.n many

cases,

if

the moments of, say, the intensity of the signal are

known, the characteristic function z ( a ) can be determined

z(w)

=

(exp

(-jwI))

= I - j a m , + - m2 +

3

+ . .

- iw l2 ( - 1 0 ) ~

2 3

(3.66)

where

m,

= (r). The probabiliy distribution function for the

intensity p

( I )

can then be calculated

c 00


18/37

YEH AND LIU: RADIO WAVE SCINTILLATIONS IN THEONOSPHERE 34 1

For this approach t o work, the moments must satisfy certain

convergenceonditions [ 1271, [ 1281.Furthermore,he

moments themselves usually are not easy to obtain. In optical

scintillation problems several authors have attempted to use

thi s approach to determine the distribution [ 1291 ; he results

have not been very promising. Using the phase screen theory

to compute the moments for the intens ity, Mercier [691 has

shown that for a deep phase screen, the intensity of the signal

satisfies the Rayleigh distribution.

The third approach to the problem is to use the character-

istic unctions

[

1301.Some esults have been obtainedfor

the optical propagation case [ 13 1 .

To ths date, these approaches and efforts notwithstanding,

asatisfactory heory or heproabilitydistributions of the

variousparameters of the cintillating signal has not been

developed.ecently,orpticalropagationroblems,

several authors

[

1321 have adopted a practical procedure to

study hisproblem. This amounts toa trial anderror a p

proach in which a distribution based on plausible reasoning

is

taken

as

the basis for carrying out certain calculations for the

signal statistics. The computed results are then checked against

experimentaldata to see

if

thedistribution yields correct

predictions. Some insights can be gained from his ype of

investigation.

As

the bservational ataromonospheric

scintillationexperimentsaccumulate, t wibe desirable to

apply this technique o study the problem of probability distri-

butions of the signal.

F.

Polarization Scintillation

In previous discussions of this chapter, the background me-

dium is assumed to be isotropic. This

is

of course not exactly

true in the onosphere.Thepresence of the earth magnetic

field makes the ionosphere a magnetc-ionic medium and hence

anisotropic. Fortunately, most radio frequencies used in he

ionospheric cintillationexperimentsor in transionospheric

communications re all much higher thanheonospheric

electron gyrofrequency, which is roughly 1.4 MHz. Under the

high radio-frequency h i t , the chief magneto-ionic effect on

wave propagation is the Faraday effect

[

1331. The Faraday

effect is caused by continuouschange n elativephase be-

tween the two characteristic waves which are counter rotating

and ircularlypolarized.Each haracteristic wave will ex-

perience scattering if there are present electron density irregu-

larities. Under the high-frequency apcroximation a stochastic

wave equation for he electric field

E

can be derived and it

shows that hecharacteristic waves arenotcoupled by the

scattering process [ 1341. Making the weak and forward scat-

ter approximation, ths wave equation can be solved using the

Rytov method by assuming

~ ( i )1(i) e-pc(i)z

2

, i = O or x (3.68)

where 8') s the normalized ith characteristic vector (circular

in the present case),

k(')

is the propagation constantof the ith

mode, and @(j)s given by

[

1341

exp [ - jk( ' ) I ;

-

p'I2/2(z -

I)]

d ' p ' . (3.69)

Here

k

is

the propagation constant in the isotropic ionosphere

and

el

is givenby (3.3). Let the ncident wave be linearly

polarized with a unit amplitude, which when received in the

absence of irregularities is polarized along the x-axis. In the

presence of irregularities, the resultant wave can be obtained

by summing up the characteristic waves given by (3.68)

This

expression suggests that the resultant wave has a fluctuat-

ing phase given by Re (@( I +&))/2 and a fluctuating ampli-

tude given by Im (@('I@(x))/2. On the receiving plane, the

resultant is linearly polarized but its plane of polarization fluc-

tuates about the mean (in our case the mean

is

polarized along

the x-axis because of the choice of coordinate axes) with an

angle 52 =(@ ('I@(x))/2.Analytical xpressions orhe

variance of these fluctuations have been obtained for irregu-

larities with Gaussian spectrum [ 1341 and power-law spectrum

[

1351.Theyshow mportantdepolarizationeffectsup to

136 MHz in the ionosphere.

IV.

EXPERIMENTAL ESULTS

A .

Irregularity Structures

We have seen from the earlier discussions that the scintilla-

tion of radio signals

is

intimately related to the structure of

ionospheric irregularities, i.e., the space-time behavior of A N .

Even when restricted to the part of the structure or the spec-

trum that affects transionospheric radio aves only, the spatial

scaleswill range fromsubmeters to tens of kilometers. At

present there is no single experimental technique that is capable

of producing information over a volume of tens of kilometers

on each side with fine details down to submeter range instant

by instant. What one can hope for is to design an experiment

so

that a particular piece of information can be extracted. If

one desires more nformation,amultitechniqueexperiment

has to

be

designed,

as

has been done recently in many cam-

paigns [ 1361-[ 1381.

As

far as scintillation is concerned, one

is interested in knowing the horizontal size of the irregularity

patch, its height, its thickness, the background electron den-

sity, he variance of fractional electron density fluctuations,

and the irregularity pectrum. Only after possessing such

information on a global basis can one attempt to construct a

global scintillation model [ 1391. We review briefly such infor-

mation in the following.

At equator the irregularity patch size has been measured to

be up to 1000 km in the east-west direction with a preference

in the 150-300-km range

[

1401-[ 1421 and t o be 1000 km in

thenorth-southdirection [ 1431. This north-south size is

comparable to the airglow meaurements made recently, which

arendicative of regions of depleted lectronsorbubbles

[

1441,

[

1451. The east-west patch size

is

somewhat larger

than the average buble size of 70 km measured by the Faraday

station and drift methods [471; this is reasonable since it is

known that scintillations may exist even when the radio ray

path is outside of an equatorial bubble.

In

temperate latitudes,

the east-west patch size may exceed 1000 km and the north-

south size is generally of the order of several hundred kilo-

meters

[

1461

-[

1481. In all geographic regions, the nighttime

irregularities that produce scintillations are found o be mostly

embedded in the F region ionosphere from about 200 km to

1000 km [13 8], [14 5], [14 9]- [15 1], but daytime scintilla-

tionsare caused mainly by E region irregularities

[

1491,

[ 1521,

[

1531. The thickness of the patch

is

found to vary

from ens of kilometers to hundreds of kilometers

[

1381,


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1982

1141, [371,[1541,[1551. There

is

some evidence, at least

at temperate latitudes, that the fractional fluctuation of elec-

tron density

is

roughly uniform even though the background

plasma density may vary with height [ 1561. This means that

the electron density fluctuations near the

F

peak are generally

larger than that at other heights. The percent fluctuations in

electron density are usually very small, but can be

as

large

as

nearly 100 percent at the equator 1391.

In early days of scintillation study the irregularity spectrum

was assumed to be Gaussian mainly for mathematical con-

venience [231,

[

101 . The first suggestion that the spectrum

might follow a power-law form came from satellite scintilla-

tion data [40], [41

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