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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.
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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
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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
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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
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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
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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-
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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 .
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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
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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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VOL. 70,
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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
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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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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
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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